The Engineering Analysis and Design of the Aircraft Dynamics Model For the FAA Target Generation Facility

Transcription

1 The Engineering Anlysis nd Design of the Aircrft Dynmics Model For the FAA Trget Genertion Fcility Mrk Peters Michel A. Konyk Prepred for: Scott Doucett ANG-E161 Simultion Brnch, Lbortory Services Division Federl Avition Administrtion Willim J. Hughes Technicl Center Atlntic City, NJ Under: Air Trffic Engineering Co., LLC 107 Route 9 South Plermo, NJ 083 FAA Prime Contrct No. DTFAWA-10A-0000 October 01

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3 Acknowledgments This project ws supported under contrct No. DTFA03-94-C-004 (CADSS) with the Federl Avition Administrtion Willim J. Hughes Technicl Center, Atlntic City, NJ, under subcontrct to System Resources Corportion (SRC). The uthor cknowledges the insightful project direction provided by Mr. Dn Wrburton of the FAA nd Mr. George Chchis of SRC, during the course of this work. The uthor lso wishes to cknowledge the vrious technicl contributions nd reviews from the members of the Segull tem. Specificlly, the uthor cknowledges Dr. John Sorensen nd Tysen Mueller for their contributions to the error modeling sections of the Nvigtor development, nd George Hunter nd JC Gillmore for their ledership nd guidnce throughout the project. Mrk Peters September, 1999 i

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5 Tble of Contents 1. INTRODUCTION BACKGROUND SCOPE ORGANIZATION.... THE AIRCRAFT EQUATIONS OF MOTION THE DEFINITION OF THE BODY FRAME AND THE INERTIAL FRAME...5. DEFINITION OF FLIGHT MECHANICS NOMENCLATURE THE WIND AND STABILITY REFERENCE FRAMES THE DERIVATION OF THE SIX DEGREE OF FREEDOM EQUATIONS OF MOTION THE MODAL PROPERTIES OF THE SIX DEGREE OF FREEDOM MODEL The Short Period Mode The Phugoid Mode The Dutch Roll Mode The Roll Mode The Spirl Mode SIMPLIFYING THE EQUATIONS OF MOTION TO FOUR DEGREES OF FREEDOM THE ADDITION OF WINDS TRAJECTORY PROPAGATION OVER AN EARTH MODEL Elliptic Erth Reference Frmes The Erth Model nd its Kinemtic Prmeters Kinemtics Relting the Coordinte Frmes Trnsforming DIS to Lt/Lon THE DERIVED STATE VARIABLES THE AIRFRAME MODEL THE ENGINE MODEL THE STANDARD ATMOSPHERE MODEL INTEGRATION TECHNIQUES The Second Order Runge-Kutt Method...54 iii

6 .13. The First Order Euler Method THE INTEGRATION OF THE ROLL EQUATION The open loop roll rte nd roll ngle equtions The closed loop system An Anlyticl Solution in the Discrete Time Intervl DESIGN OF THE LATERAL CONTROL SYSTEM THE EXAMINATION OF THE LONGITUDINAL DYNAMICS THE LINEAR MODEL OF THE LONGITUDINAL DYNAMICS THE ANALYSIS OF LONGITUDINAL AIRCRAFT MODAL PROPERTIES The Chrcteristic Polynomil of n LTD System Modl Properties of the Longitudinl Dynmics TRANSFER FUNCTION ANALYSIS OF THE LONGITUDINAL DYNAMICS The ḣ C L Trnsfer Function The M C L Trnsfer function The ḣ T Trnsfer function THE FEEDBACK CONTROL SYSTEM FOR LONGITUDINAL CONTROL THE GENERAL CONTROL LAW MANIPULATING AN LTD STATE-SPACE WITH INTEGRAL CONTROL AN ANALYSIS OF THE EFFECTS OF FEEDBACK CONTROL ON THE MODAL PROPERTIES FEEDBACK CONTROLLER DESIGN Lift Coefficient Control of Altitude Rte Lift Coefficient Control of Speed Controlling Speed nd Altitude Rte Simultneously THE SUPPORTING FUNCTIONAL LOGIC OF THE LONGITUDINAL CONTROL SYSTEM CONTROL STRATEGIES Altitude Rte Controller - Feeding Bck Altitude Rte Only Speed Controller - Feeding Bck Speed Only Dul Controller - Feeding Bck Speed nd Altitude Rte iv

7 5.1.4 Altitude Cpture Speed Cpture DIVIDING THE FLIGHT ENVELOPE INTO REGIONS The Speed-Altitude Plne Tking off Lnding Region Mngement THROTTLING IN REGIONS 1 THROUGH THE SELECTION OF GAINS THE AIRCRAFT S FLIGHT ENVELOPE DETERMINING ACCEPTABLE MODAL PROPERTIES CHOOSING A SINGLE REFERENCE CONDITION FOR GAIN CALCULATION EVALUATING SYSTEM PERFORMANCE WITH SCHEDULED GAINS THE LATERAL DIRECTIONAL CONTROL LAWS THE BANK ANGLE CAPTURE ALGORITHM THE HEADING CAPTURE ALGORITHM USING THE BANK ANGLE CAPTURE AND HEADING CAPTURE ALGORITHMS TO EXECUTE A TURN LATERAL GUIDANCE AND NAVIGATION ROUTE LEGS AND WAYPOINTS Pth nd Termintor Concept Leg Termintors Geometric Nodes of Route Direct-To-Fix Leg Trck-To-Fix Guidnce Course-To-Fix Guidnce Rdius-To-Fix Guidnce LATERAL GUIDANCE Trcking Geogrphic Pth Following Course Lw Trnsition LATERAL GUIDANCE MANAGER v

8 8.3.1 Route Following Route Cpture Hold Heding Cpture BASIC ALGORITHMS REQUIRED FOR COMPLETE FUNCTIONALITY Clculting the Aircrft Turn Rdius Determining Distnce to go Along Segment Rhumb Line Bering Rhumb Line Distnce Creting Vectors Representing Segments NAVIGATION ERROR MODELING VOR/DME NAVIGATION Determining Aircrft Position from VOR/DME Sttion Determining the Proper VOR/DME Sttion to use for Nvigtion GPS NAVIGATION ILS LOCALIZER ERROR MODEL THE LONGITUDINAL GUIDANCE SYSTEM AIRCRAFT DEVICE DEPLOYMENT Flps on Approch Flps on Tkeoff Speed Brkes Spoiler Lnding Ger Ground Brking PREPARING FOR APPROACH AND LANDING Energy Grdient Distnce Remining to Runwy CROSSING RESTRICTIONS IN DESCENT FLIGHT TECHNICAL ERROR OPERATIONAL DETAILS...04 vi

9 Piloted Flight Technicl Error FMS Flight Technicl Error ILS Flight Technicl Error MODEL VERIFICATION AND VALIDATION CONSTANT AIRSPEED CLIMBS AND DESCENTS MACH/CAS DESCENTS AND CAS/MACH CLIMBS SPEED CHANGES SPEED CHANGES DURING CLIMBS AND DESCENTS AUTOMATIC ROUTE CAPTURE VECTORED ROUTE CAPTURE INITIAL FIX ROUTE CAPTURE SEGMENT TRANSITION FLIGHT TECHNICAL ERROR NAVIGATION ERRORS GPS Nvigtion Error VOR/DME Error TERMINAL FLIGHT PHASES Tke-Off Lnding CONCLUSIONS...34 APPENDIX A...37 ANALYSIS OF THE TRANSFER FUNCTIONS OF THE LONGITUDINAL DYNAMICS...37 APPENDIX B ND ORDER GAUSS-MARKOV MODEL FOR GPS CLOCK ERROR DUE TO SELECTIVE AVAILABILITY DISCRETIZING THE CONTINUOUS ND ORDER GAUSS MARKOV PROCESS CALIBRATION OF PARAMETERS GLOSSARY...54 ACRONYMS...56 REFERENCES vii

10 List of Figures Figure.1. The body fixed reference frme ligned with n ircrft...5 Figure.. The reltionship between body nd inertil reference frmes...6 Figure.3. The 3--1 Euler sequence of rottions used to quntify the ircrft's orienttion...6 Figure.4. The forces, moments, velocity components nd ngulr rtes of n ircrft..8 Figure.5. Illustrtion of the stbility nd wind coordinte systems...10 Figure.6. Illustrtion of the Short Period mode cusing oscilltions bout the ircrft s center of grvity...18 Figure.7. Illustrtion of the Phugoid mode...18 Figure.8. Illustrtion of the Dutch Roll mode...19 Figure.9. Illustrtion of the Roll mode...19 Figure.10. Illustrtion of n unstble Spirl mode...0 Figure.11. The DIS frme nd the surfce frme...33 Figure.1. The longitude nd ltitude rottions...33 Figure.13. The Spheroidl Erth model...36 Figure.14. Tringle Representing Slope of Meridionl Ellipse nd Incrementl Ltitude Chnge...36 Figure.15. Alternte Tringle of Reltionship Between Geodetic nd Prmetric Ltitude on the Meridionl Ellipse...37 Figure.16 Position Vector on the Spheroid...39 Figure.17. Mximum thrust vs ltitude for DC-9/MD Figure.18. Fuel Consumption t mximum thrust (both engines) for DC-9/MD Figure.19. Temperture vs ltitude for the stndrd dy tmosphere...51 Figure.0. The speed of sound vrition with ltitude for the stndrd dy tmosphere...5 Figure.1. Pressure vrition with ltitude for the stndrd tmosphere...53 Figure.. Density vrition with ltitude for the stndrd tmosphere...54 viii

11 Figure.3. Roll mode response to 30 degree desired bnk ngle...64 Figure 3.1. Comprison of liner nd nonliner models with 0.01 perturbtion from the reference lift coefficient Figure 3.. Comprison of liner nd nonliner models with 0. perturbtion from the reference lift coefficient Figure 3.3. An illustrtion of drg vs irspeed t constnt ltitude, highlighting the nonminimum phse behvior of the trnsfer function of the linerized system Figure 3.4. The root locus of proportionl control pplied to the M C L trnsfer function Figure 4.1. Block digrm for the longitudinl control lw...80 Figure 4.. Effects of proportionl feedbck to the lift coefficient...86 Figure 4.3. Effects of integrl feedbck to the lift coefficient...87 Figure 4.4 Effects of Proportionl Feedbck to the Thrust...87 Figure 4.5 Effects of Integrl Feedbck to the Thrust...88 Figure 4.6. System response to 0.1 step in lift coefficient...91 Figure 4.7. Root Loci of k p (left) nd k 14 i 14 (right) successive loop closures...91 Figure 4.8. The effects of n incresed proportionl gin k p Figure 4.9. System response to 1000 ft/min commnded rte of climb...93 Figure k b root locus for the Mch Cpture controller Figure k p root locus for the Mch Cpture controller Figure 4.1. Simulted Mch Cpture...99 Figure k i (left) nd k 1 b (right) root loci for Mch cpture Figure Simultion of the Completed Mch Cpture Figure Simultion of Mch cpture with slower dynmics Figure The initil k p (left plot) nd k 14 i (right plot) loop closures for region Figure The finl increse in k p to chieve dequte dmping Figure Simultion of commnded 1000ft/min rte of climb without ny feedbck to thrust Figure Root loci for k p (left plot) nd k i (right plot) ix

12 Figure 4.0. Simultion of commnded 1000ft/min climb rte using the finl controller for region Figure 5.1: Illustrtion of n ircrft cpturing n ltitude Figure 5.: Altitude Rte During Region Trnsition Figure 5.3. Flow digrm for clculting K h Figure 5.4. The Speed-Altitude Plne Figure 5.5. The constnt energy line on the speed-ltitude plne Figure 5.6. Illustrtion of the pproximtion for the constnt energy line using the digonl cut cross the stedy, level flight region (region 7) of the speed-ltitude plne Figure 5.7. The speed-ltitude plne in terms of indicted irspeed Figure 6.1. An exmple flight envelope Figure 6.. The flight envelope for DC-9 in terms of CL nd True Airspeed Figure 6.3. The locus of Phugoid poles for the entire flight envelope of DC-9 in the clen configurtion Figure 6.4. The locus of closed loop phugoid poles in for ltitude-rte-only feedbck for the entire flight envelope of DC-9 in the clen configurtion Figure 6.5. The locus of closed loop Phugoid poles for speed-only feedbck for the entire flight envelope of DC-9 in the clen configurtion Figure 6.6. The locus of closed loop Phugoid poles in the stedy, level flight region for the entire flight envelope of DC-9 in the clen configurtion Figure 6.7. The locus of closed loop Phugoid poles in the stedy, level flight region for the entire flight envelope of DC-9 with full flps deployed Figure 7.1. Block digrm for heding feedbck Figure 7.. Simultion of Aircrft Executing Turn Figure 8.1: Fly-by nd fly-over wypoints followed by trck-to-fix legs (source: FAA AIM) Figure 8.: Fly-over wypoint followed by direct-to-fix leg (source: FAA AIM)...16 Figure 8.3: Course-to-fix leg showing the rhumb line nd the ircrft's nvigtion to it (source: FAA AIM) Figure 8.4: Rdius-to-fix leg (source: FAA AIM) Figure 8.5. Illustrtion of the difference between ground trck nd heding x

13 Figure 8.6. Illustrtion of the ircrft in route following mode Figure 8.7. Logic for insuring desired ground trck is within proper boundries Figure 8.8. Illustrtion of distnce clcultion geometry Figure 8.9: Prmeters of Right Turn Rdius-To-Fix Leg Figure An illustrtion of the dilemm of whether to mke right of left turn to heding Figure Algorithm for cpturing heding...17 Figure 8.1. An ircrft turning to fix Figure Guidnce lgorithm for Direct-to-fix leg with fly-over termintion point173 Figure 8.14 Aircrft Cnnot Enter the Cpture Hlo Figure Illustrtion of segment trnsition distnce Figure The geometry of segments tht intersect t n obtuse ngle Figure Geometric representtion of segments djoined t n cute ngle Figure Illustrtion of geometry ssocited with n ircrft merging onto segment when ircrft is heding in the direction of the segment Figure An ircrft merging onto segment which is pointed in direction opposite of the ircrft's current velocity Figure 8.0. Logic for Automtic Route Cpture Guidnce Figure 8.1. Scenrio of n ircrft determining which segment to cpture...18 Figure 8.. A scenrio demonstrting the filure of criterion # Figure 8.3. Regions where both criterion #1 nd criterion # fil Figure 8.4. Flow chrt detiling segment determintion logic Figure 8.5. Illustrtion of ircrft using 45 degree intercept Figure 8.6. Illustrtion of two ircrft cpturing segment using dynmic fix Figure 8.7. Determining n offset fix (dynmic fix) loction Figure 8.8. Route cpture with fixed heding guidnce Figure 8.9. Logic for determining whether or not heding will intercept segment189 Figure Cylindricl mpping of sphericl Erth model. Shown re two fixes nd the constnt heding route between the fixes Figure 8.31 Geometry relting ltitude, longitude, nd bering on sphericl erth..19 Figure 9.1. An illustrtion of the rnge nd bering from the sttion xi

14 Figure 9.. Illustrtion of the qudrnts of the compss rose with respect to VOR/DME sttion Figure 9.3. Logic for determining if the current VOR/DME used for nvigtion should be chnged...19 Figure 9.4. Logic for determining the pproprite VOR/DME for the next segment Figure 9.5. An illustrtion of the geometry used to determine which VOR/DME should be used for segments without VOR/DME Figure 9.6. Logic For determining which VOR/DME to use when no VOR/DME lies long route Figure 9.7. Mesured ILS Loclizer Bering Devition Angle Errors Figure 9.8. Simulted ILS Loclizer Bering Devition Angle Errors Figure 10.1: Flp Deployment Algorithm...03 Figure 10.: Compring needed nd ctul energy grdients to determine urgency...07 Figure 10.3: Procedure for incresing urgency (i.e., deploying drg devices)...07 Figure 10.4: Illustrtion of Assumed Vectored Approch...10 Figure 10.5: Algorithm for Determintion of Distnce Remining to Runwy...1 Figure Simultion Window for TGF-test...07 Figure 1.. Comprison of Pseudocontrol (blck) nd TGF-test (gry) in constnt indicted irspeed climb nd 80kt...09 Figure 1.3. A comprison of Pseudocontrol (blck) nd TGF-test (gry) in descent t constnt indicted irspeed of 300 kts...10 Figure 1.4. Comprison of Pseudocontrol (blck) nd TGF-test (gry) performing Mch/CAS descent from 30,000 ft to 10,000ft using n MD80 t 130,000lb...11 Figure 1.5. A decelertion of n MD80 from 350kts to 50kts while t 10,000ft using Pseudocontrol (blck) nd TGF-test (gry) simultion tools...1 Figure 1.6. An MD80 ccelerting from 50kts to 350kts while mintining 10,000ft using Pseudocontrol (blck) nd TGF-test (gry) simultion tools...13 Figure 1.7. An MD80 decelerting from Mch 0.8 to Mch 0.6 while mintining 30,000ft using Pseudocontrol (blck) nd TGF-test (gry) simultion tools...14 Figure 1.8. An MD80 ccelerting from Mch 0.6 to Mch 0.8 while mintining 5,000ft using Pseudocontrol (blck) nd TGF-test (gry) simultion tools...15 xii

15 Figure 1.9. An MD80 in climb with vrious speed chnges using the TGF-test simultion...16 Figure Automtic route cpture with the ircrft close to the route but heded in the wrong direction...17 Figure Automtic route cpture with n ircrft fr from the cpture segment.18 Figure 1.1. Automtic route cpture with the ircrft heded perpendiculr to route19 Figure Automtic route cpture with ircrft in n mbiguous region between segments...0 Figure Vectored route cpture from n mbiguous position...1 Figure Vectored route cpture when the vectored heding tends to be in the opposite direction of the route... Figure The initil fix cpture lgorithm being used to vector n ircrft bck to the beginning of the route...3 Figure An ircrft cpturing route from behind using the initil fix cpture lgorithm...4 Figure Piloted flight technicl error of n MD80 trveling t 50kts nd 5000ft6 Figure Piloted flight technicl error of n MD80 trveling t 300kts nd 30,000ft...6 Figure 1.0. FMS flight technicl error of n MD80 trveling t 50kts nd 5000ft7 Figure 1.1. FMS flight technicl error of n MD80 trveling t 300kts nd 30,000ft7 Figure 1.. An ircrft trjectory using GPS nvigtion...8 Figure 1.3. An ircrft flying segment using VOR/DME nvigtion where both endpoints re VOR/DME sttions...9 Figure 1.4. An ircrft flying route comprised of VOR/DME sttions with intersections between the VOR/DME sttions...30 Figure 1.5. An MD80 t 130,000lbs tking off with rottion speed of 150KIAS..3 Figure 1.6. Longitudinl view of n MD80 on finl pproch nd lnding...33 Figure 1.7. A top view of n MD80 on finl pproch to lnding...34 Figure A.1: The root loci of proportionl nd integrl lift coefficient control of ltitude rte considering the thrust control of speed...44 Figure B.1 GPS Receiver Mesurement Geometry xiii

16 Figure B.. Monte Crlo Simulted SA Position Errors Figure B.3. Monte Crlo Simulted SA Velocity Errors xiv

17 List of Tbles Tble.1. Definition of flight mechnics nomenclture...8 Tble.. Summry of kinemtic nd dynmic equtions of motion...17 Tble.3. The equtions of motion for the 4-DOF model...6 Tble.4. The equtions of motion including wind grdients...30 Tble.5. Common Lplce Trnsform Pirs...6 Tble 3.1. Stbility nd control derivtives of f V Tble 3.. Stbility nd control derivtives of fγ Tble 3.3. Stbility nd control derivtives of f h Tble 4.1. The gin scheduling equtions for lift coefficient control of speed Tble 5.1: Speed - Altitude Plne Boolens Tble 5. Desired Descent Speed Schedule for Jet nd Turboprop Aircrft Prepring for Approch Tble 5.3 Desired Descent Speed Schedule for Piston Prepring for Approch Tble 6.1 Mrker key to Figures Tble 10.1 Flp Deployment Schedule for Jet nd Turboprop Aircrft on Approch...0 Tble 10. Flp Deployment Schedule for Piston on Approch...0 Tble 1.1 Error Sttistics Summry...5 Tble B.1. Observed Locl Coordinte Position Root-Men-Squre (rms) Errors Tble B. Simplified vs. Observed SA Position nd Velocity Model Prmeters xv

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19 List of Symbols * IAS M A A cl b B B δb TA δb TG Semi-minor xis of the Erth Temperture lpse rte Angle between intersecting segments Speed of sound in the tmosphere The nominl decelertion fctor for n ircrft in terms of IAS The nominl decelertion fctor for n ircrft in terms of Mch Stte mtrix of stte spce representtion Closed loop stte mtrix Semi-mjor xis of the Erth Input mtrix of stte spce representtion The bering from sttion The irborne receiver bering bis The ground bsed bering bis δb ILS, FTE The bering devition ngle for ILS flight technicl error B B I The totl bering error The totl bering devition error for loclizer modeling B I, A The irborne bering devition error for loclizer modeling B I, G The ground bsed bering devition error for loclizer modeling B cl Closed loop input mtrix c c j C C L C Lmx C Lnom C Li Scle fctor for Guss Mrkov process Thrust specific fuel consumption Output mtrix of stte spce representtion Lift coefficient Mximum lift coefficient Nominl lift coefficient of the ircrft Integrted portion of lift coefficient signl xvii

20 C Lp C Lo C Lα C Lδe C Do C D C f1 C f C f3 C f4 C M16 C Tc,1 C Tc, C Tc,3 C θ d d c d r d offset D D D brke D fix0 Proportionl portion of lift coefficient signl Zero lift lift coefficient Lift curve slope Vrition of lift coefficient with elevtor deflection Zero lift drg coefficient Drg coefficient BADA coefficient for determining fuel burn BADA coefficient for determining fuel burn BADA coefficient for determining fuel burn BADA coefficient for determining fuel burn Compressibility coefficient BADA coefficient for determining mximum thrust BADA coefficient for determining mximum thrust BADA coefficient for determining mximum thrust Cosine of θ The distnce to go long segment Distnce to constrint Rhumb line distnce of segment The distnce the dynmic fix is plced from the leding fix Drg Crry through term mtrix of stte spce representtion Brking force Distnce the ircrft is from fix e 1 Error between desired indicted irspeed & ctul indicted irspeed e 1 d Error between bsolute desired IAS & rmped desired IAS e 1 r Error between rmped desired indicted irspeed & indicted irspeed e Error between desired Mch & ctul Mch e d Error between bsolute desired Mch & rmped desired Mch xviii

21 e r Error between rmped desired Mch & ctul Mch e 3 Error between desired ltitude nd ctul ltitude e 3 d Error between bsolute desired ltitude & rmped desired ltitude e 3 r Error between rmped desired ltitude nd ctul ltitude e3 ref Reference vlue which determines speed ltitude plne loction e s Generic speed error used in speed ltitude plne e 4 Error between desired ltitude rte nd ctul ltitude rte e 4 d Error bsolute desired ḣ & rmped desired ḣ e 4 r Error between rmped desired ltitude rte nd ctul ltitude rte e 5 Error between desired heding nd ctul heding e 3 r Distnce between the constnt energy line nd zero ltitude error e left _ turn Left turn error between desired heding nd ctul heding e right _ turn Right turn error between desired heding nd ctul heding e e At E t Error vector Stte trnsition mtrix Totl energy E td f f min f Desired totl energy Fuel burn rte Minimum fuel burn rte Flttening prmeter for the Erth f V Function for clculting V fγ Function for clculting γ f h g h h irport h con h d Function for clculting ḣ Grvittionl ccelertion Altitude Airport elevtion Constrint ltitude Desired ltitude xix

22 h di h dr ḣ dr ḣ est h error h GS The desired ltitude of n ircrft entering Region 1 or 4 from climb Rmped desired ltitude Rmped desired ltitude rte Nominl verge descent rte for n ircrft The ltitude error used to define the speed ltitude plne Commnded ltitude from the glideslope h trg et Trget ltitude H H x Angulr momentum vector Angulr momentum bout the x-body xis H y H z I CL I T I x I y I z I yz I xz I xy IAS error J k f k lg k ψ Angulr momentum bout the y-body xis Angulr momentum bout the z-body xis Stte vrible representing the integrted portion of the lift coefficient Stte vrible representing the integrted portion of the throttle Moment of inerti bout the x-body xis Moment of inerti bout the y-body xis Moment of inerti bout the z-body xis Product of inerti Product of inerti Product of inerti The indicted irspeed error used to define the speed ltitude plne Jcobin for solving systems of equtions A term which determines the rte of chnge of rmped speed vlues A term used to chrcterize engine spooling lgs Feedbck gin for heding cpture K Induced drg fctor 1 πear K ḣ A term used to smooth trnsitions into Region 7 from climbs nd descents K b K i Proportionl gin mtrix in the feedbck pth Integrl gin mtrix xx

23 K p k p k φ Proportionl gin mtrix in the feedforwrd pth Gin feeding roll rte bck to ilerons Gin feeding roll ngle bck to ilerons K re K ε l l dyn l e l offset l turn L L L p L δ M M con M nomc M nomd M M d M dr M error M o M α M m m δ e m IAS Constnt for clculting ṙ e Constnt for clculting ε Longitude Longitude of the dynmic fix Longitude of the ircrft s estimted position Segment trnsition distnce Distnce from segment t which n ircrft should turn Lift Rolling moment Rolling moment vrition with roll rte Rolling moment vrition with elevtor deflection Mch number Constrint Mch number The nominl Mch used for CAS/Mch climbs The nominl Mch used for Mch/CAS descents Pitching moment Desired Mch number Desired Mch number The Mch error used to define the Mch speed ltitude plne Zero ngle of ttck pitching moment Pitching moment vrition with ngle of ttck Pitching moment vrition with elevtor deflection Aircrft mss Slope of the constnt energy line Slope of the constnt energy line for IAS bsed flight xxi

24 m Mch N n n B n s p p p mb p sl q q r r CT r IT r e r el r l r lf r t δr FTE R R s R e R op S w SP error S θ t c Slope of the constnt energy line for Mch bsed flight Ywing moment Lod fctor Scled Gussin white noise A unit vector norml to segment Roll rte Pressure Ambient pressure Se level pressure Pitch rte Dynmic pressure Yw rte Aircrft s lterl position from the loclizer Aircrft s in trck rnge to runwy Rdius of the Erth Rdius of the Erth t given ltitude The distnce point on the Erth s surfce is from the polr xis A vector from the ircrft s loction to the leding fix of segment Aircrft rdius of turn The lterl position error Idel gs constnt A vector describing segment long route A vector from the center of the Erth to point on the surfce A vector from the center of the Erth to n ircrft Aircrft reference or wing re Generic speed error boundry used to define speed ltitude plne Sine of θ Time to constrint xxii

25 t m T T p T i T T mb T mx T min T sl u u PCL u PT u icl u it u FTE δv FTE V V r V V V p V stll V c V d V IAS V IASnom c V IASnomd V IASd Time required for mneuver Thrust Proportionl portion of thrust signl Integrted portion of thrust signl Temperture Ambient temperture Mximum thrust Minimum thrust Se level temperture Velocity of ircrft in the x-body direction Proportionl lift coefficient input signl Proportionl thrust input signl Integrted lift coefficient input signl Integrted thrust input signl Zero men unity vrince Gussin white noise The lterl position error rte Body frme velocity vector Rottion speed Climb out speed True irspeed Approch speed Stll speed Aicrft speed with respect to the Erth s surfce Desired true irspeed Indicted irspeed The nominl CAS used for CAS/Mch climbs The nominl CAS used for Mch/CAS descents Desired indicted irspeed xxiii

26 V IASdr V IAScon V GS V w V x V y V wx V wy v w w W W igf W cgf X X DIS x x x c x r x c x DIS x e x in x st x s x w x b Desired rmped indicted irspeed Constrint indicted irspeed Ground speed Wind velocity Aircrft speed with respect to the Erth s surfce in Northerly direction Aircrft speed with respect to the Erth s surfce in Esterly direction Wind velocity with respect to the Erth s surfce in Northerly direction Wind velocity with respect to the Erth s surfce in Esterly direction Body frme side velocity in the y-body direction Body frme velocity in the z-body direction Gussin white noise Aircrft weight Wind in-trck grdient fctor Wind cross-trck grdient fctor X-body force X DIS position Distnce from point on the Erth s surfce to the Equtoril plne Merctor projection x-coordinte X loction of the ircrft in flt Erth coordintes Northerly distnce of VOR in miles from reference point X-xis in the c-frme X-xis in the DIS frme X-xis in the ECEF (Erth Centered Erth Fixed) frme X-xis in n intermedite frme X-xis in the stbility xis system X-xis in the surfce or NED (North-Est-Down) frme X-xis in the wind xis X-xis in the body frme xxiv

27 x i y y y d Y Y DIS y r y c ŷ c ŷ e ŷ DIS ŷ in ŷ st ŷ s ŷ w ŷ b ŷ i Z Z DIS z 1 ẑ c ẑ e ẑ DIS ẑ in ẑ st ẑ s ẑ w ẑ b X-xis in the flt Erth inertil frme Output vector Merctor projection y-coordinte Desired output vector Y force in the y-body direction Aircrft position long the y-dis xis Esterly distnce of VOR in miles from reference point Y loction of the ircrft in flt Erth coordintes Y-xis in the geocentric reference frme Y-xis in the ECEF (Erth Centered Erth Fixed) frme Y-xis in the DIS frme Y-xis in n intermedite reference frme Y-xis in the stbility reference frme Y-xis in the NED (North-Est-Down) frme Y-xis in the wind reference frme Y-xis in the body reference frme Y-xis in the flt Erth inertil reference frme Z force in the z-body direction The position of the ircrft long the z-dis xis The zero of the ḣ C L trnsfer function Z-xis of the geocentric reference frme Z-xis of the ECEF(Erth Centered Erth Fixed) reference frme Z-xis of the DIS reference frme Z-xis of the intermedite reference frme Z-xis of the stbility reference frme Z-xis of the surfce or NED(North Est Down) reference frme Z-xis of the wind xis reference frme Z-xis of the body reference frme xxv

28 ẑ i α β β β B Z-xis of the flt Erth inertil reference frme Angle of ttck Side-slip ngle Dmping fctor for Guss Mrkov process ( s) Sptil dmping fctor for loclizer error modeling δ Distnce from rdil δ Aileron deflection δ e δ r δ ψ t ε γ γ γ p Elevtor deflection Lterl offset from flight technicl error Difference between ground trck nd heding Time step Difference between geodetic nd geocentric ltitude Rtio of specific hets Flight pth ngle reltive to the ir mss Approch ngle λ Geocentric ltitude µ Geodetic ltitude µ e Ltitude of the ircrft s estimted position µ dyn Geodetic ltitude of the dynmic fix ρ ρ δρ VDA δρ VDG ρ ρ sl σ p σ v σ pn Rnge to nv-id (used in nvigtion error modeling) The totl rnge error (used in nvigtion error modeling) Airborne receiver rnge bis Ground bsed rnge bis Air density Se level ir density Stndrd devition of the position in nd order Guss Mrkov process Stndrd devition of the rte term in nd order Guss Mrkov process North position stndrd devition for GPS modeling xxvi

29 σ pe σ vn σ ve ψ ψ GT ψ GTd ψ d ψ r ψ t ψ δψ fte θ φ φ des ω n ω p Est position stndrd devition for GPS modeling North velocity stndrd devition for GPS modeling Est velocity stndrd devition for GPS modeling Heding ngle Ground trck ngle Desired ground trck ngle Desired heding ngle Rdil bering or bering of segment Intercept ngle tht the ircrft mkes with the segment The difference between the ircrft s ground trck nd heding The heding bis from flight technicl error Pitch ngle Roll ngle Desired roll ngle Nturl frequency Nturl frequency of the Phugoid mode b ω i Angulr velocity between the i frme nd the b frme δω ILS, FTE The bering devition rte for ILS flight technicl error ζ ζ p v ntf v nlf v ptf v plf τ Dmping rtio Dmping rtio of the Phugoid mode A unit vector from the triling fix to the next VOR/DME A unit vector from the leding fix to the next VOR/DME A unit vector from the triling fix to the previous VOR/DME A unit vector from the leding fix to the previous VOR/DME Time constnt ssocited with lnding ltitude trcking xxvii

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31 1. Introduction This document presents the nlyticl development of the ircrft dynmics model for the enhncement of the functionl cpbilities of the FAA Trget Genertion Fcility (TGF). The work ws originlly conducted under Segull Technology s contrct No. DTFA03-94-C-0004 (CADSS) with the FAA. Specificlly, this document discusses the detiled engineering design nd softwre implementtion of n Aircrft Dynmics Model (ADM) suitble for incorportion into the FAA TGF simultions t the FAA Willim J. Hughes Technicl Center, Atlntic City, NJ. The model is designed to be implemented on computers locted within the fcility, nd to work in conjunction with softwre models of rdr, dt links, nd other Air Trffic Mngement (ATM) equipment to provide reltime simultion of ircrft operting within the Ntionl Airspce System (NAS). This introductory section provides brief bckground into the project s well s discusses the scope nd orgniztion of the document. 1.1 Bckground The FAA Willim J. Hughes Technicl Center conducts reserch nd development to investigte emerging Air Trffic Control (ATC) & ATM technologies, ssocited pplictions, nd ATC processes nd procedures. Inherent in these efforts is the requirement to emulte rel-world opertionl conditions in lbortory environments. This requirement extends cross the opertionl domins (e.g., Terminl, En Route, nd Ocenic). Much of this work requires the estblishment of opertionl test beds encompssing current opertionl s well s emerging prototype ATC systems. These test beds re frequently used to conduct studies tht simulte the opertionl conditions found or desired in the ssocited domin. In the mjority of cses, it is necessry to provide relistic representtion of ir trffic scenrios to evlute the system, process, or procedure being evluted. The Technicl Center s TGF provides this cpbility by producing simulted primry nd secondry rdr trgets to the system under test. To the gretest extent possible, TGF-produced trgets must ccurtely reflect the flight dynmics of the ircrft tht they represent. Currently, the ircrft dynmics incorported in the TGF re bsed on the first principles of physics nd eronutics. The models provide the performnce chrcteristics needed to support high fidelity simultions. The TGF incorportes fuel burn models environmentl (wether) effects. Additionlly, the modeled ircrft re representtive of commercil ir trffic in the US Ntionl Air Spce (NAS). As future simultions re developed or brought to the Technicl Center, higher fidelity will be required to identify NAS opertionl sfety nd performnce issues. The TGF is prepred to increse its fidelity nd opertionl connectivity required to meet the -1-

32 demnds by the other FAA progrms nd simultors. The gol of this project hs been to develop nd mintin high-fidelity simultion cpbility to meet the needs of the FAA in operting, testing, nd evluting its NAS. 1. Scope This document provides defendble, theoreticl foundtion for the engineering theory, principles nd lgorithmic design of the Aircrft Dynmics Model. The engineering nlysis strts with the first principles of ircrft flight mechnics nd derives 6-degreeof-freedom model. Simplifying ssumptions re presented nd the model is reduced to 4 degrees of freedom. The propgtion of the ircrft on the surfce of the Erth is discussed long with ll the necessry reference frmes to support ll current conventions nd interfces. The modeling the effects of wind re included. The flight control system necessry to fly the ircrft through the fundmentl mneuvers of climb, descent, turning to heding, nd speed chnges is discussed. The control theory necessry to implement the flight control system is discussed in detil. The guidnce system used to pln the ircrft s pth nd to cpture nd follow routes is presented. Different types of route cpture methods re discussed such s the utomtic nd vectored route cpture methods. Nvigtion systems nd their error models re presented, including the modeling of VOR/DME, ILS, nd GPS nvigtion. The guidnce nd nvigtion models re used by the lgorithms tht meet speed nd ltitude constrints, nd so these re presented long with them. Pilot modeling nd pilot flight technicl error re lso included. The document concludes with section on verifiction nd vlidtion, the process by which the vrious fetures of the simultion re tested nd verified. 1.3 Orgniztion Since much of the nlysis is of highly technicl nture, n effort hs been mde to orgnize the document so tht specific topics re esy to ccess. This is done to void the need to red the entire document to find specific point. The document is orgnized into 13 sections, ech of which is summrized here. Section provides detiled nlysis of the ircrft equtions of motion. The 4 degree-of-freedom ircrft model is derived from first principles. All trjectory propgtion mteril is lso covered. The numericl integrtion techniques re lso discussed. Section 3 develops liner model of the longitudinl dynmics nd nlyzes the longitudinl modl properties of the system. Trnsfer function nlysis of the longitudinl dynmics is lso performed Section 4 provides detiled nlysis of the feedbck control spects of the longitudinl control system. The section provides insight into control strtegies for cpturing desired stte. There re different feedbck control strtegies for different --

33 flight phses; ech of these control systems is discussed, nd strtegy for clculting the required gins is developed. Section 5 dels with the design of the feedbck control lgorithms tht stbilize the ircrft nd drive it to the desired stte. It lso provides the lgorithms for mnging the control strtegies nd trnsitioning between them. Section 6 documents the decision process tht led to the finl conclusion tht gin scheduling would not be necessry. By crefully choosing the reference flight condition, it is possible to choose one set of gins tht will work for the ircrft s entire flight envelope. Section 7 discusses the lterl directionl control system Section 8 discusses the lterl guidnce system. The purpose of the lterl guidnce system is to steer the ircrft to follow routes or other simultion pilot commnds. Section 9 discusses the nvigtion systems nd nvigtion error modeling The purpose of nvigtion error modeling is to model the vrinces tht occur in ircrft flight pths s result of imperfect nvigtion informtion. Section 10 discusses the logic used to mke ircrft meet speed nd ltitude constrints t fixes nd to meet the restrictions of terminl flight phses (tke-off nd lnding). Section 11 documents the flight technicl error lgorithms tht model the inbility of the pilot or utopilot to steer the ircrft perfectly long the desired course. Section 1 describes verifiction nd vlidtion of the ircrft simultion. The section strts with testing of climbs, descents, nd speed chnges nd continues with testing of the guidnce lgorithms. The nvigtion error nd flight technicl error re lso tested. Finlly, the terminl flight phses re tested. -3-

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35 . The Aircrft Equtions of Motion The purpose of this section is to provide theoreticl foundtion for the ircrft equtions of motion tht re used in the TGF ircrft simultion. The foundtion will strt with the definition of reference frmes. Once the reference frmes hve been defined, we will derive the equtions of motion for the full six-degree-of-freedom (DOF) model. Then, we will pply severl ssumptions to the equtions of motion to reduce the 6-DOF model to 4-DOF model..1 The Definition of the Body Frme nd the Inertil Frme As discussed in Nelson (1989), the two mjor reference frmes used in the derivtion of the ircrft equtions of motion re the ircrft body-fixed reference frme (denoted with b subscript) nd the inertil reference frme (denoted with n i subscript). The ircrft body frme s origin is fixed t the ircrft s center of grvity. The body frme hs its ˆx b - xis ligned with the nose of the ircrft so tht the ircrft s nose points in the positive ˆx b direction. The positive ˆy b direction points out long the ircrft s strbord wing. The ˆz b xis points down to complete right hnded coordinte frme. Figure.1 shows the body fixed reference frme. y^b x^b z^b Figure.1. The body fixed reference frme ligned with n ircrft The inertil reference frme is fixed on point on the Erth s surfce nd is ligned so tht the positive ˆx i xis points to true North nd the positive ˆy i xis points to true Est. The ˆz i xis points down nd is norml to the surfce of the Erth. This frme is commonly referred to s the North-Est-Down, or NED frme. The body reference frme cn ssume ny orienttion with respect to the inertil frme. Figure. illustrtes the reltionship between the body nd inertil reference frmes. The orienttion of the body frme with respect to the inertil frme is usully described by n Euler sequence of rottions. The ordering of the rottions is criticl to the orienttion of -5-

36 the body frme. It is difficult to visulize the ctul sequence of rottions in single drwing, so the sequence is illustrted with three seprte drwings. Figure.3 shows the Euler sequence of rottions tht is used to quntify the ircrft s orienttion. x^b y^b z^b y^i x^i z^i Figure.. The reltionship between body nd inertil reference frmes x^i ψ x^1 x^ z^i,z^1 y^1, y^ θ x^, x^b y^i x^1 y^ φ y^1 y^b z^ z^b z^1 Figure.3. The 3--1 Euler sequence of rottions used to quntify the ircrft's orienttion z^ The first rottion is through the ngle ψ bout the ˆz i xis to n intermedite reference frme, which is rbitrrily denoted with 1 subscript. The second rottion is through the ngle θ bout the ŷ 1 xis to nother intermedite reference frme, which is denoted with subscript. The finl rottion is through the ngle φ bout the xˆ xis to the body frme. The ngles ψ, θ, nd φ re referred to s the heding, pitch, nd roll ngles, respectively. -6-

37 The conversion between the inertil frme nd the body frme of the ircrft is ccomplished using direction cosine mtrices. The first cosine mtrix estblishes the reltionship between the inertil frme nd the first intermedite reference frme. Equtions (.1), (.), nd (.3) quntify the reltionship between the individul rottions. The nomenclture C θ nd S θ re simplified nottion for cosθ nd sinθ. This is done for ll trigonometric mnipultions to simplify the ultimte expression. From the rottion sequence shown in Figure.3, one cn write the following direction cosine mtrices shown in Equtions (.1) through (.3). ˆx 1 C 0 ˆ ψ Sψ xi ˆ y S C 0 ˆ = ψ ψ yi ˆz i 1 ˆz 1 (.1) ˆx C 0 ˆ θ Sθ x1 ˆ y ˆ = y1 ˆz S 0 C ˆ θ θ z 1 (.) ˆx ˆ b x ˆ y 0 ˆ b = Cφ S y ˆz b 0 Sφ C z φ ˆ φ (.3) The product of these three direction cosine mtrices results in the complete conversion between the inertil frme nd the body frme s shown in Eqution (.5). ˆx ˆ b Cθ Sθ Cψ Sψ xi ˆ y ˆ b = Cφ Sφ Sψ Cψ yi ˆz ˆ b Sφ C φ Sθ C θ z i ˆx ˆ b Cθ Cψ Cθ Sψ Sθ xi ˆ y ˆ b = Cφ Sψ + Sφ Sθ Cψ CφCψ + Sφ Sθ Sψ SφCθ yi ˆz ˆ b Sφ Sψ Cφ Sθ Cψ SφCψ Cφ Sθ Sψ CφC + + θ z i (.4) (.5) The inverse of Eqution (.5) is shown in Eqution (.6). -7-

38 ˆx ˆ i Cθ Cψ Cφ Sψ + Sφ Sθ Cψ Sφ Sψ + Cφ Sθ Cψ xb ˆ y ˆ i = Cθ Sψ CφCψ + Sφ Sθ Sψ SφCψ + Cφ Sθ Sψ yb ˆz ˆ i Sθ SφCθ CφC θ z b (.6). Definition of Flight Mechnics Nomenclture Next, certin flight mechnics nomenclture must be defined. This nomenclture consists of the vrious liner nd ngulr velocities ssocited with the motion of the ircrft s well s the forces nd moments tht re pplied. Figure.4 provides n illustrtion of the nomenclture s it pplies to the ircrft. Y,v M,q y^b L,p x^b X,u N,r z^b Z,w Figure.4. The forces, moments, velocity components nd ngulr rtes of n ircrft Tble.1 summrizes the nomenclture definition so tht the mthemticl symbols cn be ssocited with the proper terminology. The reder will note tht the term L is used to notte the rolling moment. Further long in the text, L will lso be used for lift. This is n unfortunte consequence of the merging of two engineering disciplines, dynmics nd control nd erodynmics. To void confusion, this document will notte the rolling moment using L insted of L, which is reserved for lift. Tble.1. Definition of flight mechnics nomenclture Roll Axis ˆx b Pitch Axis ˆy b Yw Axis ˆz b Angulr rtes p q r Velocity components u v w Aerodynmic Force Components X Y Z Aerodynmic Moment Components L M N Moments of Inerti Products of Inerti I x I y I yz I xz I z I xy -8-

39 It is importnt to note tht the forces defined in Tble.1 re ligned with the body frme. These forces do not directly coincide with the more commonly known erodynmic forces of lift nd drg. The forces of lift nd drg re defined with respect to nother reference frme, the wind frme, discussed in the following section..3 The Wind nd Stbility Reference Frmes We need to derive two dditionl reference frmes to resolve the reltionship between the commonly known erodynmic forces of lift nd drg nd the body forces of the 6-DOF model. These reference frmes re the stbility frme nd the wind frme s defined by Stevens & Lewis (199). The wind frme is used to describe the motion of the ir mss reltive to the ircrft body frme nd llows us to describe the erodynmic forces on the ircrft. These stbility nd wind frmes chrcterize the ngle of ttck, α, nd the side-slip ngle, β. These erodynmic ngles re defined by mens of coordinte rottions from the body frme. The first rottion, bout the ˆy b xis, defines the stbility frme nd the ngle is the ngle of ttck, α. With no sideslip, α is the ngle between the ircrft ˆx b xis nd the ircrft velocity vector reltive to the surrounding ir mss. The ngle of ttck is positive if the rottion bout the ˆy b xis ws negtive. This bckwrds definition is the unfortunte result of merging the disciplines of erodynmics nd clssicl kinemtics. The second rottion leds to the wind frme, nd the side-slip ngle is the ngle between the stbility frme nd the wind frme. An ircrft hs sideslip if its velocity vector reltive to the ir mss is not in the plne defined by ˆx b -ˆz b. The rottion is bout the z- xis of the stbility frme, ˆz st, nd β is defined s positive if the rottion bout the ˆz st xis is positive. The wind frme s x-xis, ˆx w, is ligned with the ircrft s velocity vector, which is the vector sum of the body frme velocities, V = uxˆ + vyˆ + wzˆ. The b b b other xes, ˆy w nd ˆz w, re orthogonl to ˆx w nd to ech other. Figure.5 illustrtes the orienttion of the x-xes of the stbility nd wind frmes with respect to the body frme. Equtions (.7) - (.9) show the direction cosine mtrices tht define the trnsformtions between the coordinte frmes. ˆx 0 ˆ st Cα Sα xb ˆ y ˆ st = yb ˆz 0 ˆ st Sα C α z b (.7) -9-

40 x^b β α x^st x^w V y^b z^b Figure.5. Illustrtion of the stbility nd wind coordinte systems ˆx 0 ˆ w Cβ Sβ xst ˆ y 0 ˆ w = Sβ Cβ yst ˆz ˆ w z st ˆx ˆ w CβCα Sβ Cβ Sα xb ˆ y ˆ w = SβCα Cβ Sβ Sα yb ˆz 0 ˆ w Sα C α z b (.8) (.9) Using the direction cosine mtrices, we cn derive expressions for the ngles, α nd β. We strt with the definition of true irspeed. The true irspeed of n ircrft, V, is defined s the mgnitude of the ircrft s velocity reltive to the ir mss surrounding the ircrft. By definition, the only component of this velocity is long the ˆx w xis of the wind frme. Tht is to sy the totl ircrft velocity is ligned with the ˆx w xis. Written in eqution form, V = V xˆ w. Using the inverse of the direction cosine mtrix in eqution (.9), we cn define the body frme velocities in terms of the true irspeed nd the ngles α nd β. ux ˆ ˆ b CβCα SβCα Sα V xw vy ˆ b = Sβ Cβ 0 0 wz ˆ b Cβ Sα Sβ Sα C α 0 (.10) The three resulting sclr equtions re shown below in Equtions (.11)-(.13). -10-

41 u = V C C (.11) β α v = (.1) VS β w = V C S (.13) β α Rerrnging Eqution (.1) gives us n expression for side-slip. β v V 1 = sin (.14) Tking the quotient of w/u, we cn derive n expression for ngle of ttck s shown in Equtions (.15)-(.16). w VCβ Sα Sα = = = tnα (.15) u V C C C β α α α w u w u 1 = tn (.16) Assuming tht the ngle of ttck is smll, it cn be pproximted s just the rtio w/u. Often, this expression is used to substitute α for w. Using the wind reference frme, we cn resolve the reltionship between the commonly known erodynmic forces of lift, drg nd thrust (L, D, nd T, respectively), nd the body forces of the 6 DOF model. We cn see from the direction cosine mtrix (.9) tht if we model the erodynmic forces on n ircrft in terms of lift, drg, nd thrust, equtions (.17) through (.1) re expressions for X, Y, nd Z forces in the body frme. The ircrft weight is not included becuse it is not n erodynmic force. F ˆ ˆ ˆ = Txb Lzw Dxw (.17) Xx ˆ ˆ ˆ b CβCα SβCα Sα Dxw Txb Yy ˆ 0 0ˆ 0ˆ b = Sβ Cβ yw + yb Zz ˆ ˆ 0ˆ b Cβ Sα Sβ Sα C α Lz w z b (.18) X = T DCβCα + LSα (.19) -11-

42 Y = DS β (.0) Z = DCβSα LCα (.1) Note tht the ctul forces nd moments of the full 6 DOF model include much more thn simply lift, drg, nd thrust. Unstedy erodynmics ply lrge role in the determintion of the complete force nd moment model. There is one more term tht must be formlly defined: the flight pth ngle, γ. The flight pth ngle is the ngle tht velocity vector mkes with the horizontl. If the velocity vector is expressed in surfce coordintes, the verticl component is simply the vector mgnitude multiplied by the sine of the flight pth ngle. We cn develop n expression for the flight pth ngle reltive to the ir, γ, by trnsforming the ir speed vector to the surfce frme. This is ccomplished vi sequence of rottions between the wind frme nd the surfce frme. V Cθ 0 Sθ Cψ Sψ 0 CβCα SβCα Sα V ˆ xw x V = 0 Cφ Sφ Sψ Cψ 0 Sβ Cβ 0 0 y V 0 S C S 0 C C S S S C 0 γ sin φ φ θ θ β α β α α Working with the third row eqution, the ir speed flls out nd we cn solve for γ. 1 ( α β θ ( φ β φ α β ) θ ) γ = sin C C S S S + C S C C (.) The subscript on the flight pth ngle denotes tht it is n erodynmic flight pth ngle. This is to sy tht it is the ircrft s flight pth ngle reltive to the ir mss. The ircrft s flight pth ngle reltive to the ground is generlly different becuse of the influence of winds. We cn see from Eqution (.) tht if both α nd β re zero, the Euler ngle θ reduces to γ..4 The Derivtion of the Six Degree of Freedom Equtions of Motion Once the reference frmes nd nomenclture re defined, the derivtion of the equtions of motion is strightforwrd. The liner equtions of motion re derived by summing the forces to the time rte of chnge of liner momentum ( mss ccelertion ). The ccelertion of the ircrft s velocity is determined using the bsic kinemtic eqution, which sttes tht the totl ccelertion of the ircrft with respect to the inertil frme is -1-

43 equl to the derivtive of the velocity vector with respect to the body frme plus the cross product of the ngulr velocity between the inertil nd body frmes nd the velocity vector. The bsic kinemtic eqution is shown in Eqution (.3) where dv V i = = + ω V (.3) dt t b i is the ccelertion of the ircrft with respect to the inertil frme dv is the totl time derivtive of the velocity vector dt V is the derivtive of the velocity vector s seen in the body frme t b ω is the ngulr velocity of the body frme reltive to the inertil frme: ω = pxˆ + qyˆ + rzˆ b b b V is the velocity vector in the body frme: V = uxˆ + vyˆ + wzˆ. b b b The expression for the ircrft s ccelertion is shown in Eqution (.4) nd simplified in Eqution (.5). dv = ux ˆ ˆ ˆ ( ˆ ˆ ˆ ) ( ˆ ˆ b + vy b + wz b + pxb + qyb + rzb uxb + vyb + wzˆ b ) (.4) dt dv dt ( ) ( ) ( ) = u + qw rv xˆ + v + ru pw yˆ + w + pv qu zˆ (.5) b b b To complete the equtions of motion, we must equte the ccelertion terms to the pplied forces ccording to Newton s second lw (F=m). Tble.1 summrizes the erodynmic forces pplied to the ircrft; however, the ircrft weight must lso be considered. The ircrft s weight lwys cts downwrd in the ˆz i direction. Using the direction cosine mtrix, the ircrft s weight (mg) cn be represented in body frme coordintes. mgz ˆ = mgs ˆx + mgc S ˆy + mgc Cˆz (.6) i θ b θ φ b θ φ b where m is the ircrft s mss g is the grvittionl ccelertion. -13-

44 Summing the forces nd equting the force terms yields the finl expression. ( θ ) + ( + θ φ ) + ( + θ φ ) X mgs ˆx Y mgc S ˆy Z mgc C ˆz b b b ( ) ( ) ( ) = m u + qw rv ˆx + m v + ru pw ˆy + m w + pv qu ˆz b b b (.7) Eqution (.7) cn be broken down into its individul components to yield the three force equtions of motion. ( ) X mgs = m u + qw rv (.8) θ ( ) Y + mgc S = m v + ru pw (.9) θ φ ( ) Z + mgc C = m w + pv qu (.30) θ φ The moment equtions re equl to the time rte of chnge of ngulr momentum. The ngulr momentum of the ircrft is equl to the inerti mtrix multiplied by the ngulr velocities. The expression for ngulr momentum is shown in Eqution (.31) where the symbol H is used to denote the ngulr momentum. H 0 ˆ x I x I xz pxb H 0 0 ˆ y = I y qyb H 0 ˆ z I zx I z rz b (.31) Becuse ircrft re symmetric, two products of inerti, I yz nd I xy, re zero nd therefore re eliminted from the ngulr momentum expression. Eqution (.31) cn be expnded to three sclr equtions. H = I px ˆ I rz ˆ (.3) H x x b xz b = I qy ˆ (.33) y y b H = I px ˆ + I rz ˆ (.34) z zx b z b The time rte of chnge of ech of these expressions is clculted using the Bsic Kinemtic Eqution of the form shown in Eqution (.35). -14-

45 dh dt H = + ω H (.35) t b When the kinemtic expressions re summed to their respective moments, the three moment equtions re derived. The three moment equtions re shown in Equtions (.36)-(.38). For convenience -15-

46 Tble. summrizes the fundmentl kinemtic nd dynmic equtions of motion. ( ) ( ) L = I p I r + pq + qr I I (.36) x xz z y ( ) ( ) M = I q + rp I I + I p r (.37) y x z xz ( ) N = I p + I r + pq I I + I qr (.38) xz z y x xz So fr we hve developed the full 6-DOF equtions of motion, which re quite involved. The next step in developing 6-DOF model, well beyond the scope of this discussion, would be to derive expressions for the forces nd moments tht ct on the ircrft. For complete discussion, refer to Nelson (1989). The forces nd moments tht re used in the 6 DOF model re quite different from the simplified subset presented in Equtions (.19) through (.1). The true forces nd moments re complicted expressions tht require estimtes of unstedy erodynmic dt to hndle properly. Our next tsk is to simplify the 6-DOF equtions of motion to 4-DOF equtions using some simplifying ssumptions. However, first we will briefly discuss the modl chrcteristics of the 6-DOF model so we better understnd which chrcteristics re most likely going to influence trjectory propgtion..5 The modl Properties of the Six Degree of Freedom Model Before simplifying the equtions of motion to 4-DOF, it is useful to discuss the five modes of motion ssocited with the 6-DOF model. Three of the modes re second order nd two of the modes re first order mking up n 8 th order system. These modes re: -16-

47 Grouping Tble.. Summry of kinemtic nd dynmic equtions of motion Equtions Force Equtions Moment Equtions Body Angulr Velocities in terms of Euler ngles nd Euler rtes ( ) ( ) X mgs = m u + qw rv (.8) θ Y + mgc S = m v + ru pw (.9) θ φ ( ) Z + mgc C = m w + pv qu (.30) θ φ ( ) ( ) ( ) ( ) ( ) L = I p I r + pq + qr I I (.36) x xz z y M = I q + rp I I + I p r (.37) y x z xz N = I p + I r + pq I I + I qr (.38) xz z y x xz p = φ ψ (.39) S θ q = θ C + ψ C S (.40) φ θ φ r = ψ C C θ S (.41) θ φ φ Euler rtes in terms of Euler ngles nd body ngulr velocities θ = qc rs (.4) φ φ φ = p + qs T + rc T (.43) φ θ φ θ ( qsφ rcφ ) ψ = + sec θ (.44) Short Period (Longitudinl plne) Phugoid (Longitudinl plne) Dutch Roll (Lterl-Directionl plne) Roll (Lterl-Directionl plne) Spirl (Lterl-Directionl plne). The longitudinl dynmics control the forwrd speed nd ltitude of the ircrft. There re two second-order oscilltory modes comprising the longitudinl dynmics. These modes re referred to s the short period nd the Phugoid mode. The lterl-directionl dynmics consist of one second-order mode nd two first-order modes. These modes control the turning dynmics of the ircrft within the lterl plne. -17-

48 .5.1 The Short Period Mode The short period mode is nmed becuse it is the fster of the two modes. It is the mode tht defines the ircrft s pitching bout its center of grvity. The short period mode controls the dynmics between elevtor deflection nd the ircrft s resulting lift coefficient. Generlly, the short period mode is over ten times fster thn the Phugoid mode..5. The Phugoid Mode The Phugoid mode is the slower of the two longitudinl modes. We cn think of the Phugoid mode s grdul interchnge between potentil nd kinetic energy bout some equilibrium ltitude nd irspeed. The Phugoid mode is chrcterized by chnges in pitch ttitude, ltitude, nd velocity t nerly constnt lift coefficient. Usully, the Phugoid is over ten times slower thn the short period mode nd therefore the Phugoid will hve the dominnt influence over the ircrft s trjectory. This is illustrted in Figure.7. Figure.6. Illustrtion of the Short Period mode cusing oscilltions bout the ircrft s center of grvity Figure.7. Illustrtion of the Phugoid mode.5.3 The Dutch Roll Mode The Dutch Roll mode is the only oscilltory mode of the lterl directionl dynmics nd is combintion of ywing nd rolling oscilltions. The Dutch Roll gets its nme from its resemblnce to the weving motion of n ice skter. The Dutch Roll mode is mostly n nnoynce to the pilot nd pssengers. The pilot cn esily dmp out the motion of the Dutch roll. The Dutch roll is illustrted in Figure

49 Figure.8. Illustrtion of the Dutch Roll mode.5.4 The Roll Mode The roll mode chrcterizes how fst n ircrft cn chieve stedy stte roll rte fter n ileron deflection. It is first-order mode nd therefore does not oscillte. The roll mode cn influence the trjectory of n ircrft by cusing dely between the time turn is commnded nd when stedy stte turn rte is chieved. The Roll mode is illustrted in Figure.9. Figure.9. Illustrtion of the Roll mode.5.5 The Spirl Mode The spirl mode chrcterizes n ircrft s spirl stbility bout the verticl xis. This mode controls whether or not n ircrft returns to level flight fter smll perturbtion in roll ngle. When this mode is unstble, the ircrft will hve tendency to deprt from level flight nd enter spirl dive. -19-

50 Figure.10. Illustrtion of n unstble Spirl mode If the mode is stble, the ircrft remins in level flight. Usully the mode is stble. Even if the mode is not stble, the pilot will compenste to mintin stright nd level flight..6 Simplifying the Equtions of Motion to Four Degrees of Freedom The first step to simplifying the 6-DOF equtions of motion is to mke two ssumptions bout the ircrft in flight. These ssumptions re tht: 1. The ircrft s pitch dynmics, chrcterized by the short period mode, re fst enough to be ssumed instntneous.. The pilot mintins coordinted flight. We first concentrte on the implictions of coordinted flight. Constrining the ircrft to coordinted flight implies tht the side-slip ngle is lwys zero. This in turn implies tht there is never ny side velocity, v, ny side-force Y, or ny ywing moment N. This reduces Eqution (.9), the side force eqution, to Eqution (.45). The reduced Eqution (.45) is no longer differentil eqution. Furthermore, the yw rte derivtive, ṙ, is neglected, reducing the ywing moment differentil eqution to n lgebric expression s shown in Eqution (.46). The rolling moment derivtive is removed by the substitution of the rolling moment eqution in for ṗ. gcθ Sφ = ( ru pw) (.45) ( ) ( ) L + I xz pq qr I z I y 0 = I xz + pq( I y I x ) + I xzqr (.46) I x -0-

51 Similrly, the side velocity, v, drops out of the other force equtions s well. ( ) X mgs = m u + qw (.47) θ ( ) Z + mgc C = m w qu (.48) θ φ Now, concentrte on the first ssumption tht the ircrft s pitch dynmics re fst enough to be neglected. This implies tht the ircrft is ble to commnd n ngle of ttck instntneously. Therefore, we neglect the derivtives, ẇ nd q. The terms ( p r ) nd rp in the pitching moment eqution re second order effects nd cn be neglected. The pitching moments re, therefore, ssumed to be in equilibrium, consistent with instntneous ngle of ttck dynmics. M = 0 (.49) The net effect is tht the Z force eqution is reduced to n lgebric expression. ( ) Z + mgc C = m qu (.50) θ φ This leds to n explicit expression for the pitch rte. Z mgcθ Cφ q = (.51) mu To solve for ngle of ttck, the pitching moment, M, must be expnded into its individul terms s shown in Eqution (.5). This is merely formlity becuse we will shortly show how we cn remove ngle of ttck entirely. M = M + M α + M δ = 0 (.5) o α δ e e The terms in (.5) re: M : The zero ngle of ttck pitching moment o M α : A derivtive relting pitching moment chnges to chnges in ngle of ttck M δ : A derivtive relting the effect of elevtor deflection on ngle of ttck. e Solving for ngle of ttck yields Eqution (.53). -1-

52 M o α = M M α δ e δ e (.53) The stedy-stte ngle of ttck is n lgebric function of the elevtor deflection, exclusively. This implies tht the lift coefficient, C, lso is very nerly n exclusive function of the elevtor deflection s shown in Equtions (.54) nd (.55). L C = C + C α + C δ (.54) L Lo Lα Lδ e e M M δ C C C C Mα o δe e L = L + o Lα + L δ δe e (.55) where C L is the zero ngle of ttck lift coefficient o CL α is the lift curve slope with respect to ngle of ttck CL δ is the effect of elevtor deflection on lift coefficient. e Although the ngle of ttck hs been removed from our equtions, we cn still clculte it for nimtion purposes. Furthermore, the elevtor deflection cn be completely bypssed in fvor of the ircrft s lift coefficient s the primry longitudinl control input. This is convenient becuse the terms relting lift coefficient, ngle of ttck nd elevtor deflection re not provided in commonly vilble ircrft models. Using equtions (.5) nd (.54), we cn derive n expression for the ngle of ttck tht depends only on the coefficient of lift. This cn be used for disply purposes, nd the reder is reminded tht the expression ssumes instntneous ngle of ttck dynmics. α = M C o L C L + C o Lδ e M δe M C C α Lα Lδ e M δe (.56) This eqution cn be re-written in simpler form, requiring only two irfrme constnts. C α = C L L o C L α (.57) --

53 The primed constnt in the numertor of eqution (.57) is n effective trim lift coefficient, while the primed constnt in the denomintor is n effective pitch sensitivity. For disply purposes, suitble nominl vlues re 0. nd 10, respectively, to yield the ngle of ttck in rdins. Returning to the model development, we ssume tht the ngle of ttck is smll. This ssumption combined with the previous ssumption of coordinted flight implies tht the wind nd body frmes re very nerly ligned with ech other. The lignment of the wind nd body frmes implies the following: 1. True irspeed, V, nd u re ligned. Therefore V cn be substituted for u in the differentil equtions.. The lift force, defined s pointing in the ˆz w direction, is now ligned with the Z force in Eqution (.30). (Z = -L) 3. The drg force, defined s pointing in the ˆx w, is ligned with the Thrust, defined s being ligned with the ˆx b xis. (X = T D) 4. The Euler ngle θ reduces to γ, the flight pth ngle. These simplifictions gretly reduce the equtions of motion. Equtions (.58) through (.60) show wht remins of our differentil equtions. T D mgsθ = mv (.58) L mgcθ Cφ q = (.59) mv ( ) ( ) L = I p I pq + qr I I (.60) x xz z y These equtions need to be rerrnged into useful form. We strt with Eqution (.58), which cn esily be rerrnged s n expression for true irspeed. T D V = gsθ (.61) m Rerrnging Eqution (.59) into n expression for flight pth ngle tkes more steps. We strt by relting the pitch rte, q, to the Euler ngle θ by using the reltions from -3-

54 Tble.. L mgcθ Cφ Cθ Sφ θ = ψ (.6) mv C C φ φ Using -4-

55 Tble. gin, we cn substitute for yw rte, r, in terms of Euler ngles in Eqution (.45). gcθ Sφ Sφ ψ = + θ (.63) V C C C C θ φ θ φ Combining equtions (.6) nd (.63) result in the finl expressions for θ nd ψ. LCφ mgcθ θ = (.64) mv LSφ ψ = (.65) mv C θ Using the fct tht the flight pth ngle nd the pitch ngle re identicl for this model, we substitute γ for θ. This protects us from confusion tht might rise s result of our simplified Euler ngle expressions. LC mgc φ γ = (.66) mv γ LSφ ψ = (.67) mv C γ The finl eqution of motion to mnipulte is the rolling moment eqution. This eqution governs the rte t which n ircrft cn estblish bnk ngle. The rolling moment eqution s written is function of roll nd pitch rtes nd the moments of inerti. ( ) ( ) L = I p I pq + qr I I (.68) x xz z y We choose to neglect the higher order terms nd reduce the rolling moment eqution to its liner form of L = I p. We will use stndrd stbility nd control derivtives to define the rolling moment. x where L p + L = I p (.69) δ δ p x -5-

56 L p is the rolling moment derivtive with respect to roll rte L δ is the rolling moment derivtive with respect to ileron deflection δ is the ileron deflection. The prmeter, δ, is more ccurtely linerized ileron deflection prmeter. At smll deflection ngles it is mthemticlly equivlent to the ileron deflection. However, this liner form of the lterl dynmics llows us lter to solve the system nlyticlly while removing the ileron deflection prmeter from the system of equtions. We will tke one further step to redefine our derivtives to include 1/ I x so tht we cn write our differentil eqution in first order form. p = L p + L δ δ (.70) p Nme Tble.3. The equtions of motion for the 4-DOF model Eqution True Airspeed Eqution T D V = gsγ (.61) m The Flight Pth Angle Eqution The Heding Eqution The Roll Rte Eqution LC mgc φ γ = (.66) mv γ LSφ ψ = (.67) mv C p γ p = L p + L δ δ (.70) Since we do not hve much dt for L p nd L δ, we will hve to use engineering judgment s to the best vlues for specific ircrft. These numbers will be terms tht must be twekble so tht the user cn tune them to suit. The finl form of the equtions of motion, without wind effects, is shown in Tble.3. An lterntive derivtion of the equtions of motion my be found in Muki (199). -6-

57 .7 The Addition of Winds The erodynmic forces on n ircrft re creted by its motion reltive to the surrounding ir. The ir itself is in motion reltive to the inertil frme. If the wind is constnt, then the ir mss is not ccelerting nd we need only consider the vector sum of the ir mss nd the ircrft s ir speed to propgte over the erth model. If the wind is not constnt, then we must consider the contributions of n ccelerting frme to the equtions of motion. We mke two importnt simplifying ssumptions before implementing winds. The first is tht the wind contins no verticl component. V = V xˆ + V yˆ (.71) w wx i wy i where V w is the ir mss velocity with respect to the inertil frme V wx is the x component of the ir mss velocity ligned with true North V wy is the y component of the ir mss velocity ligned with true Est. The second ssumption is tht the wind vries only with ltitude; tht is, the lterl nd temporl vritions re zero, dv w /dx = 0, dv w /dy = 0, dv w /dt = 0. The ddition of winds into the system is done by creting yet nother reference frme. This frme, the ir mss frme, is inserted in between the inertil frme nd the wind nd body frmes. Recll tht the wind nd body frmes re equivlent for our simplified model. The ir mss frme s orienttion is ligned with the inertil frme, but moves t constnt velocity with respect to the inertil frme. The ircrft s motion s described in the previous sections is now considered to be with respect to the ir mss frme nd not the inertil frme. Knowing tht the ircrft s velocity is ligned with the ˆx w xis we cn determine the ircrft s speed reltive to the ir mss in inertil coordintes. V ˆ ˆ C C x γ ψ i Cγ C ψ Cφ Sψ + Sφ Sγ C ψ Sφ Sψ + C S C φ γ ψ V xw V ˆ 0ˆ Cγ S y ψ i = Cγ S C C S S S S C C S S y ψ φ ψ + φ γ ψ φ ψ + φ γ ψ w V ˆ 0ˆ Sγ z i Sγ S C C C z φ γ φ γ w (.7) The velocity of the ir mss reltive to the inertil frme is presented below. The reder will note tht we ssume there is no verticl component to the wind. -7-

58 The totl ircrft velocity with respect to the inertil frme is then the sum of the ir mss velocity with respect to the inertil frme nd the ircrft s velocity with respect to the ir mss. The term V i is the ircrft s velocity with respect to the inertil frme or the sum of the ircrft s true irspeed nd the wind velocity. ( ) ( ) V = V + V C C xˆ + V + V C S yˆ V S zˆ (.73) i wx γ ψ i wy γ ψ i γ i The terms in Eqution (.73) cn be rewritten to introduce the terms V x, V y, nd ḣ. V = V xˆ + V yˆ hz ˆ (.74) i x i y i i These terms re defined s follows: V x is the velocity of the ircrft with respect to the inertil frme in the true North direction V is the velocity of the ircrft with respect to the inertil frme in the true Est y direction ḣ is the ltitude rte or verticl speed of the ircrft. Lter, the terms V x nd V y re used when defining the ltitude rte nd longitude rte of the ircrft. Assuming tht the winds re constnt, Eqution (.73) is sufficient for modeling the dynmics. However, if the winds re not constnt, then wind grdient terms must be dded to the differentil equtions. Often, winds vry with ltitude. If this is the cse, the equtions of motion must be re-derived ccounting for the vritions in winds with respect to ltitude. As the body frme is ccelerting reltive to the ir mss, we introduce n inertil force due to the vrition in wind speed, F w. In bove eqution, V in nd V cr re the chnge in ir mss velocity in-trck nd crosstrck, respectively. Cθ 0 S V θ in Fw = m Sφ Sθ Cφ SφCθ V cr (.75) Cφ Sθ Sφ CφCθ 0 Now the Force Equtions re updted to include, F w. -8-

59 ( + qw rv) X mgs mv θ incθ = m u (.76) ( + ru pw) Y + mgc S mv ins S mv crc = m v θ φ θ φ φ (.77) ( + pv qu) Z + mgc C mv ins C + mv cr S = m w θ φ θ φ φ (.78) The sme simplifictions used to simplify Equtions (.8)-(.30) in re pplied Equtions (.76)-(.78). T D m gs V C = γ (.79) in γ V gc rv (.80) γ S Vin S S VcrC = φ γ φ φ L + gc m γ C φ V S C + V S = qv (.81) in γ φ cr φ Eqution (.79) is the new True Airspeed Eqution. To determine the effects on the Flight Pth Angle Eqution nd the Heding Eqution, use -9-

60 Tble. nd Equtions (.80) & (.81). θ L mv C θ in θ cr φ θ φ = + (.8) φ gc V V V S V V S C φ C S ψ C φ gsφ V Sθ Sφ V in S cr φ ψ = + θ (.83) V C V C C V C C C φ θ φ θ θ φ Solving the system of equtions given by Equtions (.8) & (.83) gives the updted Flight Pth Angle Eqution nd the Heding Eqution in terms of V in nd V cr. LC gc V S θ + mv V V φ θ in θ = (.84) LSφ V cr ψ = (.85) mv C V C θ To find V in nd V cr, use W igf nd W cgf,the in-trck grdient fctor nd the cross-trck grdient fctor, respectively. They re defined s follows: θ W igf dv dv wx wy = Cψ + S ψ (.86) dh dh W cgf dv dv wx wy = Sψ + C ψ (.87) dh dh W igf nd Wcgf re relted to V in nd V cr by the simple reltions below: V V in in = h = Wigf h = Wigf VSγ h (.88) V V cr cr = h = Wcgf h = WcgfVSγ h (.89) The resulting equtions of motion including wind terms re shown in Tble.4 Tble.4. The equtions of motion including wind grdients -30-

61 Eqution True Airspeed Eqution Nme T D V = gsγ W igfvsγ C γ (.90) m Flight Pth Angle Eqution LCφ mgcγ γ = + Wigf S γ (.91) mv Heding Eqution LS φ γ = W (.9) ψ mv C γ cgf S C γ Roll Rte Eqution p = L p + L δ δ (.93) p.8 Trjectory Propgtion over n Erth Model The kinemtics of n ircrft hve been determined in coordinte system fixed to the surfce of the erth. We now turn our ttention to propgting the ircrft over the surfce of the Erth. The equtions tht chrcterize this motion re developed in three steps. These steps re: 1. Develop set of reference frmes. Introduce n Erth model (the propgtion equtions re dependent on the selected model). 3. Develop the kinemtic expressions relting the ircrft s velocity in the surfce frme to chnges in ltitude nd longitude..8.1 Elliptic Erth Reference Frmes There re three mjor reference frmes used for the nlysis. These reference frmes re DIS Coordintes - Since it is necessry tht the TGF simultor conform to the DIS stndrd (Institute of Electricl nd Electronics Engineers [IEEE], 1993) for representing ircrft trjectory propgtion, nd the DIS frme is n Erth-centered, Erth-fixed frme with rectngulr coordintes, we choose to use it for our development. We will use 'DIS' subscript to describe the coordintes of this frme. Geodetic Coordintes - This frme (lso known s Ltitude-Longitude-Altitude) is commonly used in nvigtion. It uses ngulr coordintes (geodetic ltitude nd -31-

62 longitude) nd position prmeter (ltitude). All re referenced to the locl surfce of the erth model. No tg is used to identify this frme, s context lone is sufficient. Surfce Coordintes - A reference frme on the surfce of the erth ligned such tht the x-y plne is tngent to the surfce with the x-xis pointing to the north nd the y- xis pointing to the est. This frme is often referred to s the North-Est-Down (NED) frme. We will use n s subscript for brevity of nottion when writing equtions. Surfce Coordintes nd Geodetic Coordintes re orthogonlly ligned. The observnt reder will notice tht the ircrft equtions of motion were clculted ssuming flt Erth nd tht we here ssume the development frme ws the North- Est-Down frme. This implies necessrily tht erth rottion nd the vrition of the grvity vector with position over the erth were ignored in developing the ircrft equtions of motion. This simplifiction limits our mthemticl model to the flight of ircrft only. The model will not properly hndle the flight of sub-orbitl crft nd spcecrft such s intercontinentl bllistic missiles, stellites, or the spce shuttle. The model is dequte for ll vehicles trveling under Mch 3. For trjectory propgtion, since we cnnot ssume flt Erth, the originl inertil reference frme, denoted with n i subscript, is modified for the elliptic Erth. Thus, we lign our newly defined surfce frme, with the inertil frme i. The surfce frme then moves with the ircrft so s to provide frme tht is tngent to the Erth s surfce for interfcing with trjectory propgtion equtions, nd prllel to the ircrft s horizontl plne of flight for interfcing with the ircrft's flt-erth dynmics. All velocities tht were originlly defined with respect to the i frme, re now tken to be with respect to the s frme. Figure.11 shows the reltionship between the DIS nd surfce frmes. The DIS-frme is fixed in the center of the erth with the ẑdis -xis out the North pole. The Plne described by the xˆ DIS - nd the ŷdis -xes is in the plne of the equtor. The xˆ DIS xis is through zero degrees longitude, or the prime meridin. The surfce (NED) frme is tngent to the surfce of the erth nd is centered in n object propgting long the Erth's surfce. It is denoted with n 's' subscript for surfce. The ˆz s xis points downwrd nd is norml to the surfce of the erth. The ˆ s x - nd ˆ s y -xes define plne tngent to the surfce of the erth. The longitude nd ltitude ngles describe the rottion between the two frmes. -3-

63 Figure.11. The DIS frme nd the surfce frme Figure.1. The longitude nd ltitude rottions The orienttion of the reference frmes is such tht t zero degrees longitude nd ltitude, the two reference frmes re orthogonlly ligned. Furthermore, the surfce frme conforms to the orienttion normlly used with ircrft s shown in Figure.. Figure. shows the reltionship between the surfce frme nd the ircrft's body fixed frme. The ngulr rottion from the DIS-frme to the s-frme cn be thought of s n Euler sequence of rottions. The ordered rottions re illustrted in Figure.1. The first rottion is longitude. Longitude (l ) is rotted bout the positive ẑdis -xis in right hnded sense from the 'DIS' frme to n intermedite frme noted with n 'in'. The ẑdis -xis coincides with the xˆ in -xis in the intermedite frme. The rottion yields direction cosine mtrix in Eqution (.94). -33-

64 ˆx in ˆx DIS ˆ yin = sinl cosl 0 ˆ ydis ˆz in cosl sinl 0 ˆz DIS (.94) The geodetic ltitude ( µ ) is rotted bout the positive ŷin -xis in left-hnded sense from the 'in' frme to the surfce frme denoted with 's'. The trnsformtion to the surfce frme defined in this wy, s shown in Eqution (.95), keeps geodetic ltitude positive in the Northern hemisphere. ˆx s cos µ 0 sin µ ˆx in ˆ ys = ˆ yin ˆz s sin µ 0 cos µ ˆz in (.95) The product of these two mtrices is the complete direction cosine mtrix between the two reference frmes. ˆx s cos µ 0 sin µ ˆx DIS ˆ ys = sinl cosl 0 ˆ ydis ˆz s sin µ 0 cos µ cosl sinl 0 ˆz DIS ˆx s sin µ cosl sin µ sinl cos µ ˆx DIS ˆ y = sinl cosl 0 ˆy s ˆz s cos µ cosl cos µ sinl sin µ DIS ˆz DIS (.96) (.97) The inverse of rottion mtrix is simply its trnspose, nd so the trnsformtion of vector from the NED-frme to the DIS-frme is esily obtined. As n exmple, the velocity in DIS coordintes is given by V DIS sin µ cosl sinl cos µ cosl = sin µ sinl cosl cos µ sinl V cos µ 0 sin µ s (.98) The rottion mtrix of eqution (.97) cn trnsform vector described in the DIS coordinte system to the North-Est-Down coordinte system. The unfortunte consequence of this formt is tht the geodetic ltitude nd longitude of the origin of the vector re unknown to the vector described in DIS coordintes, nd so eqution (.97) is not very useful. This problem is resolved in section.8.4 below. -34-

65 .8. The Erth Model nd its Kinemtic Prmeters The TGF Aircrft Dynmics Model uses the WGS-84 erth model, s defined by Ntionl Imgery nd Mpping Agency (NIMA, 1997). The WGS-84 erth model is spheroidl erth model s shown in Figure.13 where: is the equtoril rdius, or the semi-mjor xis of the spheroid, which is meters, or ft b is the semi-minor xis given by b = (1-f) where f is the Erth Flttening Prmeter. (1/f = ) A polr cross section of the spheroidl erth is depicted in Figure.13. It illustrtes the flttening of the spheroid t the poles s well s grphicl representtion of three commonly used ltitude definitions: geocentric ltitude, λ, prmetric ltitude, θ, nd geodetic ltitude, µ. The side view illustrtes how the cross-section of the flttened erth cn be represented s rottion from circle. The ngle subtended in this rottion is the rcsine of the eccentricity, e, of the meridionl ellipse. The point O is the center of the erth. B is the North Pole. The semi-mjor (equtoril) xis, OA, of the meridionl ellipse hs length, the semi-minor (polr) xis, OB hs length b. From the digrm we cn see tht b = cos(rcsin(e)). The eccentricity of the ellipse is defined s e = 1 b -35-

66 Figure.13. The Spheroidl Erth model sin θ dθ R µ dµ µ b cosθ dθ Figure.14. Tringle Representing Slope of Meridionl Ellipse nd Incrementl Ltitude Chnge Mthemticlly, it is most convenient to describe the geometry of the ellipse vi the prmetric ltitude, θ. A point P on the ellipse hs coordintes ( cos θ, b sin θ). The point P' is the point on the circumscribing circle (of rdius ) the sme distnce from the polr xis s P. The ltitude used in nvigtion is the geodetic ltitude, µ, which is defined s the ngle between the northerly horizon t P nd the polr xis. This definition yields reltionship between the prmetric nd geodetic ltitudes: the tngent of the geodetic ltitude is the negtive inverse of the slope of the spheroid t P (i.e., the chnge -36-

67 in equtoril displcement divided by the chnge in the polewrd displcement). This is illustrted in Figure.14. R µ is the rdius of curvture of the meridionl rc t P. ( cosθ ) ( sinθ ) d sinθdθ tn µ = = d b b cosθ dθ tn µ = tnθ (.99) b b sin µ θ cosµ Figure.15. Alternte Tringle of Reltionship Between Geodetic nd Prmetric Ltitude on the Meridionl Ellipse It should be noted tht eqution (.99) pplies only to points on the spheroid. From inspection of eqution (.99) we cn crete nother tringle (Figure.15). From inspection of Figure.15 we write sinθ = b sin bsin µ µ + cos µ sinθ = bsin µ ( 1 e ) sin µ + ( 1 sin µ ) sinθ 1 e = sin 1 sin µ e µ (.100) Similrly, cosθ = b sin cos µ µ + cos µ -37-

68 cosθ = ( e ) cos µ 1 sin µ + cos µ cosθ 1 = cos 1 e sin µ µ (.101) From Figure.14 the displcement long the meridin is given by, R d = sin + b cos d µ µ θ θ θ = 1 e cos θ dθ = 1 e ( 1 e sin µ ) 3 dµ So tht the rdius of curvture of the meridionl ellipse t P is given by, R µ = 1 e ( 1 e sin µ ) 3 (.10) We here define one more prmeter tht becomes useful in our kinemtic nlysis. This prmeter is lbeled s D µ in Figure.13 nd we shll cll this prmeter the Geodetic Distnce to the Polr Axis. It should not be confused with the rdius of curvture t P, which is somewht lrger. By inspection, Dµ cos µ = cosθ From eqution (.101) D µ = 1 1 e sin µ (.103) A similrly defined prmeter, lthough not illustrted, is the Geodetic Distnce to the Equtor, D µ. eq From eqution (.100) Dµ sin µ = bsinθ eq -38-

69 D µ eq = b 1 e 1 e sin µ eq ( 1 ) D = D e (.104) µ µ.8.3 Kinemtics Relting the Coordinte Frmes The velocity vector eqution (.74), hs been developed from the ircrft erodynmics in terms of the surfce, or NED, coordintes. We cn equte this velocity vector to the time rte of chnge of the position vector in the surfce coordinte system. The position vector, R op, is esily expressed in terms of DIS coordintes. We trnsform R op to surfce coordintes vi eqution (.97). Once this trnsformtion is done, R op is expressed in terms of ltitude nd longitude. Consequently, the time rte of chnge of R op will yield velocity vector in terms of ltitude nd longitude rtes. With the velocity vector expressed in terms of ircrft dynmics nd in terms of ltitude nd longitude rtes, we cn solve for the ltitude nd longitude rtes for propgtion over our erth model. Figure.16 Position Vector on the Spheroid -39-

70 By inspection of Figure.16, the component of xis to the point P is, R op prllel to the equtor from the polr Req = ( Dµ + h)cos µ (.105) This is resolved long the xˆ DIS nd ŷ DIS xes by multiplying by the cosine nd sine of the longitude. Similrly, the component of the point P is R op prllel to the polr xis from the equtor to ( µ ) ( ( ) ) eq µ z = D + h sin µ = D 1 e + h sin µ (.106) DIS And so the position vector in DIS coordintes is obtined immeditely in two forms by inspection. R DIS op ( θ + h µ ) ( θ h µ ) cos cos cosl = cos + cos sinl (.107) bsinθ + hsin µ R DIS op ( Dµ + h) ( Dµ h) µ ( ) cos µ cosl = + cos µ sinl (.108) ( D 1 e + h) sin µ The position vector in surfce coordintes is obtined by pre-multiplying by the rottion mtrix in eqution (.97) R s op sin µ cosl sin µ sinl cos µ DIS = sin cos 0 l l Rop cos µ cosl cos µ sinl sin µ e Dµ sin µ cos µ s R op = 0 (.109) Dµ ( 1 e sin µ ) h We need to tke the derivtive of the position vector to get n expression for the ircrft s velocity. The derivtive is defined in (.110). -40-

71 dr dt R = + ( ω R op ) (.110) t s s op op s From eqution (.103) e D sin cos s µ µ µ Rop = 0 t t Dµ ( 1 e sin µ ) h ( ) e D sin cos cos sin s µ µ µ e µ Dµ µ µ R op = 0 t D µ ( 1 e sin µ ) + e µ Dµ sin µ cos µ h e sin cos e sin cos D = µ µ µ = µ D µ µ (.111) 3 ( 1 e ) ( ) sin µ 1 e sin µ µ µ 4 e sin µ cos µ µ Dµ e µ D µ ( cos µ sin µ ) ( 1 e sin ) s µ Rop = 0 t µ e Dµ sin µ cos µ h The ngulr rottion of the surfce frme reltive to the Erth-fixed DIS frme is esily defined in mixed coordintes, keeping in mind tht the ltitude rottion is negtive rottion bout the yˆs -xis. ω = l z ˆ ˆDIS µ y s Trnsforming the longitude rottion to surfce coordintes vi eqution (.97), we get l cos µ s ω = µ (.11) sin µ l Expnding the cross product term, -41-

72 l cos µ e Dµ sin µ cos µ s = µ 0 sin µ l Dµ ( 1 e sin µ ) h ( ω Rop ) ( ω Rop ) ( 1 sin ) ( Dµ h) cos µ Dµ e µ + µ h s = l + µ µ e Dµ sin µ cos µ we end up with the finl expression for velocity in Eqution (.113). dr dt op s 4 e sin µ cos µ µ Dµ e µ D µ ( cos µ sin µ ) ( 1 e sin µ ) µ Dµ ( 1 e sin µ ) + µ h = 0 + l ( Dµ + h) cos µ µ e D sin cos h µ µ µ µ e Dµ sin µ cos µ dr dt op s ( Rµ + h) µ = l ( Dµ + h)cos µ h (.113) From eqution (.74), the velocity of the ircrft in the surfce frme is V s V x = V y h (.74) We cn set the x nd y components of eqution (.74) equl to those of eqution (.113) nd solve for the ltitude nd longitude rtes. The z components produce n identity. ( Rµ + h) V µ x V y = l ( Dµ + h)cos µ h h -4-

73 These expressions cn be rerrnged in terms of l nd µ s shown in equtions (.114). Vx µ = R + h µ Vy l = ( Dµ + h)cos µ (.114).8.4 Trnsforming DIS to Lt/Lon There my be future need to trnsform DIS coordintes to the stndrd position coordintes of ltitude, longitude, nd ltitude, e.g., when TGF is integrted with externl ircrft motion pplictions tht output DIS coordintes. The equtions trnsforming DIS to ltitude-longitude position re the inverse of eqution (.108). Unfortuntely, n ccurte representtion of the inverse of eqution (.108) is unginly. An equivlent itertive lgorithm (Borkowski, 1989) used to obtin the needed ccurcy is more estheticlly ppeling. The equtoril position vector, R eq, s defined in eqution (.105), is the vector sum of the x- nd y- components of the DIS coordintes of R op. R = x + y eq DIS DIS The longitude is obtined from DIS coordintes immeditely. y DIS l = rctn (.115) xdis From eqution (.107), we hve R = cosθ + hcos µ eq z = bsinθ + hsin µ DIS (.116) Eliminting h nd µ, this system of equtions cn be shown to reduce to the form shown in Borkowski (1989) in which we need to solve for the prmetric ltitude, θ. Borkowski showed tht this form is superior in simplicity nd ccurcy over the rnge of ltitudes. f ( ) ( ) The prmeters Ω nd c re constnt for ech position. θ = sin θ Ω c sin θ = 0 (.117) -43-

74 1 bz DIS Ω = tn R eq c = b ( Req ) + ( bzdis ) (.118) As with the inverse of eqution (.108), n nlyticl solution of eqution (.117) is unginly. An itertive pproch using the Newton-Rphson method provides suitble ccurcy nd speedy convergence. The Newton-Rphson method requires the derivtive of f(θ). ( ) ( ) f θ = cos θ Ω c cos θ (.119) A suitble first pproximtion to the prmetric ltitude is provided by ssuming zero ltitude. This is obtined from equtions (.116). 1 zdis θ0 = tn br eq (.10) Subsequent vlues of the prmetric ltitude in the Newton-Rphson itertive method re defined by the following eqution. θ θ f = n+ 1 n f ( θn ) ( θ ) n (.11) To obtin one-foot ccurcy requires θ settling within bout10-9. Once θ is determined, the geodetic ltitude, µ, is obtined from eqution (.99). This llows the use of equtions (.116) to find the ltitude. By inserting the trigonometric identity, µ µ sin + cos = 1 eqution (.116) cn be mnipulted to eqution (.1). ( eq ) ( DIS ) h = R cosθ cos µ + z bsinθ sin µ (.1) There re simpler forms for obtining the ltitude, but Borkowski prefers eqution (.1) for its uniform ccurcy over the rnge of ltitudes. -44-

75 .9 The Derived Stte Vribles The equtions of motion re referred to s the stte equtions becuse they re the fundmentl equtions tht govern the ircrft s motion. Ech stte eqution is nmed for the stte vrible for which it clcultes derivtive. In our cse, we hve four stte vribles V γ ψ p tht re governed by the equtions in Tble.3. There re lso other importnt vlues tht re not stte vribles but rther functions of the stte vribles. We cll these vlues derived stte vribles. There re five importnt derived stte vribles: V IAS : The indicted Airspeed M : The Mch number ψ GT : The ground trck heding V GS : The ground speed ψ : The turn rte f : The fuel flow/burn rte W : The ircrft weight h : The ircrft ltitude The indicted irspeed is the speed mesurement indicted on n Aneroid type irspeed indictor tht is hooked to n ircrft s pitot sttic system. The irspeed indictor mesures the difference between the sttic nd rm-ir pressures nd pproximtes n irspeed from the pressure difference. The indicted irspeed is not good estimte of the true irspeed. At higher ltitudes the difference between indicted nd true irspeed my be in error by s much s 100 kts. To simulte the reding on n irspeed indictor, Eqution (.13) is used to convert from Mch number to indicted irspeed. ( γ 1) γ γ ( 1) * p γ 1 γ VIAS = 0 1+ M γ 1 p o (.13) The terms in Eqution (.13) re defined s follows: *: γ: p: p 0 : The speed of sound. The rtio of specific hets for ir (not to be confused with the flight pth ngle). γ = 1. 4 under norml conditions. The mbient pressure. The se-level pressure. -45-

76 To convert indicted irspeed to Mch number, requires rerrnging Eqution (.13). Since this lgebric mnipultion is not trivil, only the result is provided here. M γ 1 γ γ γ 1 p o γ 1 V IAS = * + γ 1 p 0 (.14) The Mch number is the rtio of the true irspeed nd the speed of sound s shown in Eqution (.15). M V = (.15) * The ground speed nd the ground trck heding re derived from the velocity terms first presented in Eqution (.74). V = V + V (.16) G x y ψ GT V 1 y = tn Vx (.17) The turn rte of the ircrft is clculted using the heding eqution. Eqution (.65) is reprinted here for convenience. LSφ ψ = (.65) mv C γ There re two derived stte vribles tht re not merely functions of the integrted sttes. These vlues must be integrted; however, we seprte them from the forml integrtion of the differentil equtions becuse they do not require the rigorous integrtion procedure used to numericlly integrte the stte equtions. These two derived sttes re ltitude, h, nd ircrft weight, W. The ltitude is simply the integrtion of the ltitude rte nd the ircrft weight is the integrtion of the fuel burn rte. The method of integrtion is discussed in the numeric integrtion section. -46-

77 .10 The Airfrme Model The irfrme model is dpted from Segull Technology s AMT (Aircrft Modeling Tool), which is dpted from EUROCONTROL s Bse of Aircrft Dt (BADA) (Nuic, 009). The primry purpose of the irfrme model is to clculte the erodynmic forces pplied to the ircrft. These forces re lift nd drg s defined in Anderson, The lift of the ircrft is clculted using Eqution (.18). L = qs w C L (.18) The drg of the ircrft is clculted using Eqution (.19). The terms for these equtions re s follows: D = qs w C D (.19) L D S w q C D C L Lift Drg Wing Reference Are Dynmic pressure Drg coefficient Lift coefficient To clculte the dynmic pressure Eqution (.130). is used. 1 q ρv = (.130) where the terms in the eqution re defined s follows: ρ V ir density true irspeed The ir density is obtined from the tmosphere model, which is discussed in Section.1. The lift coefficient is n input tht is generted by the control lws. The drg coefficient is clculted using Eqution (.131). where ( ) min drg C = C + K C C (.131) D Do L L -47-

78 C D The zero lift drg coefficient o C L min drg The lift coefficient t minimum drg K The induced drg coefficient The drg polr eqution comes from clssicl incompressible erodynmics. The compressibility effects of high speed flight re currently neglected. Ech ircrft in the simultion hs five flp settings nd spoiler tht cn be deployed when needed. The flp settings re nmed for their respective flight phses. Ech flp setting is described by its own drg polr. Clen configurtion Initil climb configurtion Tke off configurtion Approch configurtion Lnding configurtion 5 x Altitude ft Mximum Thrust (Both engines) lbs x 10 4 Figure.17. Mximum thrust vs ltitude for DC-9/MD80.11 The Engine Model The engine model is responsible for providing two importnt prmeters to the rest of the model. These prmeters re the mximum thrust vilble nd the fuel burn rte. We use the BADA prmeters (Nuic, 009) to perform these clcultions. The form of the mthemticl models for thrust nd fuel burn depends on the ircrft s instlled configurtion for thrust production. The ADM uses three seprte mthemticl thrust/fuel burn models for three engine types: turbofn, turboprop (i.e., gs-turbine engine ttched to propeller), nd piston-prop (i.e., internl combustion engine ttched to propeller). As n exmple of the implementtion, we present here the thrust nd fuel -48-

79 burn models for the turbofn engine instlltion. While the models for other instlltions re of similr formt, the reder is referred to Nuic, 009 for further detil. The mximum thrust vilble to the turbofn-powered ircrft t ny given time is function of the locl ir density. Figure.17 shows the mximum thrust vilble for DC-9 ircrft s the ltitude is incresed. The thrust nd fuel burn models represent the ir density in terms of pressure ltitude. The mximum vilble thrust is computed using Eqution (.13). h Tmx = CT 1 + C c, 1 T h c, 3 C Tc, (.13) where: h is ltitude T mx is the mximum climb thrust C T c, i, re the BADA coefficients fitting ctul thrust performnce to ltitude The fuel burn rte is clculted using the following equtions. V η = C f + 1 C 1 f (.133) f = ηt (.134) f min h = C f 1 3 C f4 (.135) where: 1. η is the thrust specific fuel consumption. V is the true irspeed 3. T is the Thrust 4. f the fuel flow rte 5. f min is the minimum fuel rte 6. h is the ltitude -49-

80 7. C f, re the BADA coefficients fitting ctul fuel burn performnce to ltitude nd i speed The fuel flow rte is normlly clculted using Equtions (.133) nd (.134); however, there is lower bound on the fuel burn, which is clculted using Eqution (.135). If the fuel burn clculted using Equtions (.133) nd (.134) is lower thn the minimum fuel burn rte, the minimum fuel burn rte is returned s the fuel burn rte. Figure.18 shows the fuel burn rte for DC-9 t mximum thrust for vrious irspeeds. 5 x Altitude ft KTAS 00 KTAS KTAS Fuel Consumption lb/hr x 10 4 Figure.18. Fuel Consumption t mximum thrust (both engines) for DC-9/MD80.1 The Stndrd Atmosphere Model Since ircrft operte in the Erth s tmosphere nd their lift nd drg chrcteristics depend on the properties of tht tmosphere, it is essentil to be ble to define these properties. To do this, the Interntionl Stndrd Atmosphere (ISA) model (Interntionl Civil Avition Orgniztion, 000) is implemented. The derivtion of the governing equtions is omitted since they re commonly vilble (e.g., Anderson, 1989). There re two seprte regions to the Erth s tmosphere tht we re concerned with. The first region is the grdient region where temperture drops off linerly with ltitude. The grdient region spns from the Erth s surfce to 11 km (~36100 ft). The second is n isotherml region where the temperture is constnt. The isotherml region extends to 0 km (~65600 ft). Our model is not concerned with ltitudes bove 0 km. Figure.19 illustrtes the temperture vrition of the stndrd tmosphere. -50-

81 8 x Altitude ft Temperture R Figure.19. Temperture vs ltitude for the stndrd dy tmosphere In the grdient lyer (less thn 11 km), the temperture of the mbient ir surrounding the ircrft is clculted using Eqution (.136). T = T h (.136) mb sl where T sl : The se level temperture (15.0 C) T mb : The mbient temperture h : The ltitude : The temperture lpse rte in the grdient lyer (6.5 C/km) In the isotherml lyer, the temperture stys constnt t C. ( ) T = T 11km = C (.137) mb sl The speed of sound is strictly function of the mbient ir temperture. It is clculted using the thermodynmic reltion in Eqution (.138) -51-

82 8 x Altitude ft Speed of Sound fps Figure.0. The speed of sound vrition with ltitude for the stndrd dy tmosphere * = γ RT (.138) where * is the speed of sound, nd γ is the rtio of specific hets for ir. c p γ = (.139) c v where c p is the constnt pressure specific het nd c v is the constnt volume specific het. The term R is the idel gs constnt nd T is the bsolute mbient ir temperture. Figure.0 shows the reltionship between the speed of sound nd ltitude for stndrd dy. Figure.1 illustrtes the pressure vrition with ltitude for the stndrd dy. If the ircrft is in the grdient lyer, the pressure rtio of the ircrft is clculted using Eqution (.140). p mb T mb = p T sl sl g R (.140) -5-

83 8 x Altitude ft Sttic Pressure lb/ft^ Figure.1. Pressure vrition with ltitude for the stndrd tmosphere where p mb : The mbient pressure. p sl : The se level pressure. g : Grvittionl ccelertion. R: Idel gs constnt. If the ircrft is in the isotherml lyer, the pressure rtio of the ircrft is clculted using Eqution (.141). = psl p g p RT h 11km mb p ( )( ) iso e iso sl (.141) where the subscript iso refers to conditions t the bottom of the isotherml lyer (11 km). The finl eqution, which clcultes mbient density, is vlid regrdless of the tmospheric region. It is the idel gs eqution of stte. p mb ρ mb = (.14) RTmb The reltionship between density nd ltitude is illustrted in Figure

84 8 x Altitude ft Density slug/ft^3 x 10 3 Figure.. Density vrition with ltitude for the stndrd tmosphere.13 Integrtion Techniques The TGF simultion s designed requires the rel time integrtion of series of nonliner differentil equtions nd one liner differentil eqution, Eqution (.93). Since nonliner differentil equtions cnnot be solved nlyticlly, some type of numericl integrtion method must be employed. There re mny techniques vilble, so it is importnt to find technique well-suited to the needs of prticulr problem. There re severl items to consider when choosing numericl lgorithm. These re: 1. Accurcy required.. Frequency of the dynmics to be simulted. 3. The computtionl efficiency required. 4. The stbility of the lgorithm. The most demnding integrtion requirements for the TGF project stem from the Phugoid mode of the longitudinl dynmics. This mode generlly hs period of 30 sec, which is not very fst. Therefore, sophisticted numericl lgorithm need not be pplied. Furthermore, the integrtions tht re not influenced by the Phugoid mode require even less computtionl precision. For the Phugoid influenced equtions, good second order method should suffice. For the non-phugoid influenced equtions, first order method is quite dequte The Second Order Runge-Kutt Method A second order Runge-Kutt method is chosen for those equtions tht re influenced by the Phugoid mode. This method, is simple nd stble. It is self strting nd does not require informtion from previous time steps. It is slightly more computtionlly expensive thn other methods such s n Adms-Bshforth method, but the use of the Adms-Bshforth method did not prove s stble s the Runge-Kutt method nd -54-

85 required more complex lgorithm becuse it is not self strting. These methods re discussed in detil in Hoffmn (199), or ny other numericl method text. The second order Runge-Kutt lgorithm is summrized below. Consider stte vector, X(k), t time step, k, nd, t(k), the time t step, k. These re the inputs to the numericl integrtor. X( k) x1 ( k) x ( k) x ( ) n k = (.143) It is our objective to updte the stte vector to the next time step t (k+1). To do this we must first clculte K 0, the initil term of the Runge-Kutt integrtion sequence. The numericl integrtion routine does not ctully do this. Insted, it uses series of functions of the sttes nd the independent vrible, time, s shown in Eqution(.144). For our problem, the functions f 1 - f n re the stte equtions (.61), (.66), nd (.67). K ko 1 f1( x1 ( k), x( k), xn ( k), t( k),...) k f ( x ( k), x ( k), x ( k), t( k),...) k fn( x1 ( k), x( k), xn ( k), t( k),...) o n o 1 n 0 = = t (.144) The numericl integrtion routine will tke K 0 nd dd it to the originl stte vector t time step (k) nd then send the results bck to the derivtive functions. This results in K 1. K 1 k1 f 1( x1 ( k) + ko, x 1 ( k) ko,, x ( ), ( ),...) n k + ko + t k + t n k1 f( x1 ( k) + ko, x 1 ( k) ko,, x ( ), ( ),...) n k + ko + t k + t n = = t k1 f ( 1( ), 1 ( ),, ( ), ( ),...) n n x k + ko x k ko x n k + ko + t k + t n (.145) The finl step is to determine, X(k+1). -55-

86 X( k 1) x1 ( k) ko k 1 1 x ( k) 1 k k xn( k) ko k n 1 n o 1 + = + + (.146) The equtions tht re integrted using the second order Runge-Kutt technique re Equtions (.61), (.66), nd (.67) or the true irspeed, flight pth ngle, nd the heding ngle stte vrible equtions..13. The First Order Euler Method The first order Euler method is rgubly the simplest numericl integrtion routine vilble. Under most conditions, it is not considered dequte for ctul simultion, but rther is used only s n instructionl exmple. However it is very inexpensive computtionlly, nd is more thn dequte for the very slow chnges in ltitude, position nd weight chnges occurring in the TGF model. Using the sme X(k) vector defined in (.143), the next time step, X(k+1), is esily clculted using Eqution (.147). x1 ( k) f1( x1 ( k), x( k), xn ( k), t( k),...) x ( k) f( x1 ( k), x ( k), xn ( k), t( k),...) ( k 1) X + = + t x ( k) f ( x ( k), x ( k), x ( k), t( k),...) n n 1 n (.147) The quntities tht re integrted by this method re s follows: Ltitude Longitude Weight Altitude.14 The Integrtion of the Roll Eqution The roll eqution is unique in our simultion becuse it is the only differentil eqution tht is liner. Becuse the eqution is liner, no numericl integrtion technique need be pplied. Furthermore, we cn perform the loop closures of our lterl directionl control logic within the nlytic solution itself, eliminting the ileron deflection prmeter, δ, vi liner control lw. -56-

87 .14.1 The open loop roll rte nd roll ngle equtions The roll mode is governed by Eqution (.93), which is reprinted below. It is convenient to ssign φ, the derivtive of the roll ngle, to p, the roll rte. This yields Eqution (.148). These two equtions yield the second order dynmics tht chrcterize n ircrft s roll ngle in response to deflection of the ilerons. p = L p + L δ δ (.93) p φ = p (.148) It is convenient to rrnge the equtions into stte spce representtion. p Lp 0 p L δ = + δ 1 0 φ 0 φ (.149) The system of eqution (.149) yields the lterl dynmic response to commnded ilerons. The desired system is one in which desired roll ngle is commnded nd the ircrft responds ccordingly, nd so we move on to closing this loop..14. The closed loop system We here continue to describe the lterl dynmic system, equtions (.149), s liner, time-dependent (LTD) system of equtions. The system follows the common LTD vector-mtrix form, ( t) ( t) ( t) x = Ax + Bu, (.150) in which the stte vector, x, contins the roll rte, p, nd the roll ngle, φ, nd the control vector, u, contins only the ileron deflection prmeter, δ. We ccordingly define liner feedbck controller tht is bsed on the difference between the desired nd ctul stte. The LTD system then becomes, u = K x ( - x ) des { ( des - x) } ( ) ( ) x = Ax + B K x x = A BK x + BK x des, (.151) -57-

88 In this cse, the gin mtrix is (x1). k p K = k φ Expnding into the nomenclture of our lterl dynmics, the closed-loop stte spce system is p Lp Lδ k p Lδ k p φ Lδ k p Lδ k p φ des = + φ 1 0 φ 0 0 φ des (.15) where p des is the desired roll rte nd φ des is the desired roll ngle. We do not ctully llow commnd p des so we cn eliminte it from Eqution (.15). p Lp Lδ k 0 0 p Lδ k p L k φ δ φ = + φ 1 0 φ 0 0 φ des (.153) The system we hve developed is second-order, liner system. It is completely defined by the modl properties of the stndrd second-order system, which we cn determine by compring the chrcteristic eqution of eqution (.153) to tht of the stndrd secondorder system. ( I - A ) ( ) det s = s L Lδ k s Lδ kφ = s + ζω s + ω cl p p n n The modl properties re determined by inspection. n ( L Lδ k ) ζωn = p p ω = L k δ φ (.154) We cn completely define the lterl dynmic response of different ircrft models by specifying the nturl frequency, ω n, nd dmping rtio, ζ. We hve essentilly replced four prmeters with two. Additionlly, the ileron deflection prmeter hs been eliminted. The primry control prmeter in its plce is the desired roll ngle. We rewrite the LTD system s function of the two modl properties nd the desired roll ngle. -58-

89 p ζω 0 0 n ω p n ωn = + φ 1 0 φ 0 0 φ des (.155).14.3 An Anlyticl Solution in the Discrete Time Intervl Our system is still in the form of eqution (.150), with the control vector, u, defined s the desired stte vector. As stted erlier, becuse the system is liner, we cn develop n nlyticl solution in the time intervl of our computer simultion, thereby eliminting the need for numericl integrtion. The stte vector is known t the beginning of the intervl, the control vector is known nd constnt in the intervl, nd so we cn solve eqution (.150) nlyticlly for the stte vector t the end of the intervl. If the control vector were time-dependent in the intervl, the well-known (e.g., Ogt, 1990, eq ) nlyticl solution to eqution (.150) is given by eqution (.156). t 0 ( t τ ) At A x( t) e x(0) e Bu ( τ ) dτ (.156) = + The stte trnsition mtrix, e At, is defined s the inverse Lplce trnsform of the chrcteristic mtrix. e At L ( si A) 1 1 However, since the control vector of eqution (.155) is constnt in the discrete-time simultion intervl from 0 t t, we cn derive simpler solution. The control vector is one-dimensionl, contining only the desired bnk ngle nd is independent of time in the intervl. Let us rewrite eqution (.155) in shortened nottion, replcing the constnt closed-loop mtrices with A cl nd B cl, respectively. ( ) ( ) x t = A x t + B φ cl cl des We trnsform this eqution to the Lplce domin nd solve for x(s) x ( s) = ( si Acl ) x 0 + ( si Acl ) B clφdes (.157) s Eqution (.157) is now convenient nd trnsformble function of the Lplce vrible. We rrive t solution in the time domin by tking the inverse Lplce trnsform. This solution is identicl to the solution obtined from eqution (.156) with constnt control. -59-

90 x ( t) = L ( si Acl ) x 0 + L ( si Acl ) Bcl φ des s (.158) To llow for the complete rnge of lterl dynmic design, we choose to hndle the solution differently for different vlues of the dmping rtio, ζ. For the cse of the underdmped (ζ < 1) system, in which the system poles re complex conjugte pirs nd the solution is oscilltory, it is convenient to keep eqution (.157) in the nottion of the modl properties. s ω ω n n ( 1) s + ζω ns + ωn s + ζωns + ω s + ζω n ns + ωn x ( s) = x 0 + φ des (.159) 1 ωn s + ζω ω n ω s( s n s + ζωns + ωn s + ζωns + ω + ζω n ns + ωn ) For the criticlly dmped system (ζ = 1), the chrcteristic eqution contins double pole t s=-ω n nd eqution (.157) cn be rerrnged into more convenient form. s 1 1 ( ω n ) ( ωn ) ( ) ( ) ( ) s + ωn s + ωn s + ωn x ( s) = x 0 + φdes (.160) 1 s + ω n 1 ( ω ) ( s + ωn ) ( s + ωn ) s( s + ωn ) For the over-dmped system, the system poles re rel. We solve the chrcteristic eqution for the system poles t s=- nd s=-b, where, 1, b = ζω n 1± 1 ζ We cn now rerrnge eqution (.157) in terms of nd b. s 1 1 ( b) ( s + )( s + b) ( s + )( s + b) ( s + )( s + b) x ( s) = x 0 + bφ des (.161) 1 s + ( + b) 1 ( s + )( s + b) ( s + )( s + b) s( s + )( s + b) Tble.5 presents Lplce trnsforms pplicble to eqution (.157), s found in commonly vilble Lplce trnsform sources. From these Lplce trnsforms we cn -60-

91 write the solutions to equtions (.158) - (.161) in the time domin. First we write the solution for the under-dmped cse. ωn ζωt ( ωn ζ t θ ) e ( ωn ζ t) 1 sin 1 sin 1 p( t) 1 ζ 1 ζ p( 0 ζω ) nt e = φ ( t) 1 1 φ( 0) sin ( ωn 1 ζ t) sin ( ωn 1 ζ t + θ ) ωn 1 ζ 1 ζ (.16) ωn ζωt e sin ( ωn 1 ζ t) 1 ζ + φ des 1 ζωt 1 e sin ( ωn 1 ζ t + θ ) 1 ζ Next we write the solution for the criticlly-dmped cse. ( ) ( ) ( 0) ( ) ωnt ωnt ωnt ωnt p t e ωnte ω p n te ωn te = φ ω des nt ωnt ωnt + ωnt ωnt (.163) φ t te e + ω 0 nte φ 1 e ωnte And, finlly, we write the solution for the over-dmped cse. 1 bt t b t bt b t bt ( ) ( be e ) ( e e ) ( 0) ( e e p t ) b b p b = + φdes (.164) φ ( t) 1 t bt 1 t bt φ ( 0) 1 bt t ( e e ) ( be e ) 1 + ( be e ) b b b Equtions (.16) - (.164) (ll of which re in the form of eqution (.158)) provide n nlyticl solution for the stte vector t ny time, t, when the initil vlue of the stte vector (t t=0) is known. We cn pply this solution cross discrete time intervl, t, by setting the initil stte vector, [p(0);φ(0)] to tht t the strt of the intervl nd solving for [p( t);φ( t)] using the relevnt eqution. The solution t the end of the intervl is then constnt mtrix function of the initil stte nd the desired bnk ngle. It is importnt to note tht the constnt mtrices need only be clculted once for given ircrft nd time step size. (For vrying time step, the mtrices would need to be clculted ech time.) Once the initil clcultions re mde, the reltions used to updte the stte from one time step to nother re simple. For instnce, consider n ircrft with criticlly dmped lterl dynmics defined s below. -61-

92 Tble.5. Common Lplce Trnsform Pirs L L L Lplce Domin Time Domin ( ω ζ t) ω ω s + ζω s + ω n n = e sin 1 n 1 ζ 1 n n ζωt ( t ) s 1 1 ζ = e sin ω 1 ζ θ, θ = tn n s + ζω s + ω n n 1 ζ ζ ζωt 1 1 ( ω ζ t θ ) s + ζω 1 = e sin 1 + n s + ζω s + ω n n 1 ζ 1 n ζωt 1 L ( ) L L L L 1 L L L L ω 1 n ζωt = 1 e sin ω 1 ζ t + θ n s ( s + ζω s + ω ) n n 1 ζ ( )( ) 1 1 = s + s + b b s ( )( ) t bt ( e e ) 1 1 = s + s + b b bt t ( be e ) bt t 1 ( be e ) = + s ( s + )( s + b) b b 1 s + ( + b) ( )( ) 1 = s + s + b b t bt ( be e ) 1 1 ( s + ) s ( s + ) = te 1 t ( + ) t d t = ( te ) = e te dt t t [ e te ] 1 1 = s s 1 1 s + ( s + ) = e + te 1 t t t ωn = rd sec ζ = 1 Discretizing the system for 0.5 sec time step, using Eqution (.163), the system becomes. p( k + 1) p( k) = + φ φ k φ k ( ).. ( ). des ( k) (.165) -6-

93 If the stte vector (roll rte nd ngle) is zero t the beginning of the intervl nd the control (desired roll ngle) is 30 (0.53 rd), eqution (.165) would yield roll rte of 1. deg/sec nd roll ngle of t the end of the intervl. Eqution (.165) is the only clcultion tht must be mde to updte between time steps..15 Design of the Lterl Control System We need to determine pproprite vlues for the modl properties of the lterl dynmics. While the lterl dynmic system ws modeled using the stbility derivtives, L δ nd L p, nd the control gins, k p nd k φ, it ws reveled bove (refer to equtions (.154)) tht we need only determine pproprite vlues for the frequency nd dmping rtio of the lterl dynmics. Let s nlyze the step response to the criticlly system with zero initil condition. Tble 16. of Rymer (1999) provides the specifictions for the rolling performnce of militry ircrft per MIL-F-8785 B. From this tble, we cn conclude tht similr commercil ircrft should be cpble of chieving 30 bnk in seconds. We shll ssume tht this time guideline refers to settling time, T s, of bout four time constnts. T s 4 = ζω n From eqution (.163), the roll eqution with zero initil condition is s follows. n n ( ) ( 1 t t t e ω ω = nte ) φ ω φ Figure.3 shows the response of this eqution using 30-degree desired bnk ngle, nturl frequency of rd/sec. The reder will note tht the modl properties used to crete Figure.3 correspond to time constnt of two seconds. At settling time of two seconds, the roll ngle is within bout 10% of the desired stte. We cn crete ircrft models with fster lterl dynmics by incresing the time constnt, ζω n. des -63-

94 Figure.3. Roll mode response to 30 degree desired bnk ngle -64-

95 THIS PAGE INTENTIONALLY LEFT BLANK -65-

96 3. The Exmintion of the Longitudinl Dynmics It cn be rgued tht the mjority of the effort put forth to build successful feedbck control system is spent trying to understnd the plnt tht is to be controlled. This certinly is the cse with the nonliner longitudinl ircrft dynmics. The insight developed is fundmentl tool used to mke intelligent decisions regrding feedbck control strtegy. This chpter dels with the development of solid insight into the plnt dynmics, in this cse the longitudinl dynmics of the ircrft. To develop insight, severl tsks re performed. These tsks re: Develop liner model of the longitudinl dynmics Anlysis of longitudinl modl properties Trnsfer function nlysis of the longitudinl dynmics. The liner model of the ircrft dynmics is fundmentl tool tht llows for the modl nlysis nd the exmintion of the response to inputs vi the trnsfer functions. Modl nlysis of the liner model enbles identifiction of the physicl properties tht ffect the modl properties of the system. Finlly, certin trnsfer functions re creted from the liner model tht give insight into the different feedbck control strtegies tht cn be used. It should be noted tht the liner model of the longitudinl dynmics is not replcing its nonliner prents in the dynmic simultion; it is merely being used s lerning tool. 3.1 The Liner Model of the Longitudinl Dynmics The modeling equtions for the ircrft dynmics, s presented in Tble.3, re nonliner with the exception of Eqution (.70). This nonlinerity limits our bility to perform n in depth study into the behvior of the system of equtions nd lso precludes the design of feedbck control system. To overcome this limittion, common pproch in feedbck control is to develop linerized version of the system of equtions. In our liner modeling of the system of equtions, we choose to seprte the longitudinl dynmics from the lterl-directionl dynmics. We cn do this becuse the longitudinl modes nd the lterl-directionl modes re only lightly coupled (Nelson, 1989). For the longitudinl cse, we constrin the ircrft to not turn. The generl form of non-liner dynmic system is representtion of the stte vector s non-liner differentil eqution, nd the output vector s seprte non-liner function. Both functions re generlly dependent on the stte, control, nd time

97 (, t) (, t) x = f x,u y = g x,u (3.166) In this system, x is the system s (in our cse, the ircrft s) stte vector, u is the system s control vector, y is the system s output vector. The control lw is user-defined nd is typiclly bsed on driving the observble output to desired condition. While it is not typicl to express generlized control lw, we cn present one here to illustrte tht, once the stte nd desired output re known, the system hs unique solution. u = u( y, y d ) In clssicl control theory, liner, time-dependent (LTD) stte-spce is represented by the following system. ( t) ( t) ( t) ( t) ( t) ( t) x = Ax + Bu y = Cx + Du (3.167) In this system, A, B, C, nd D re constnt mtrices. The stte vector is collection of vribles tht completely describes the system s stte t ny given time. Our longitudinl stte vector includes true irspeed, V, flight pth ngle, γ, nd ltitude, h. The control vector is the system s control inputs. Our longitudinl control vector includes lift coefficient, C L nd thrust, T. In discrete time, if the stte nd the control inputs t given time-step re known, the stte eqution yields the stte t the next time-step. The output vector contins those prmeters tht re redily mesurble by the controller. The ircrft (nd, consequently, the ircrft s controller) my not know its true irspeed or its flight pth ngle, but it is ble to mesure its indicted irspeed, V IAS, Mch number, M, ltitude, h, nd ltitude rte, ḣ, nd these prmeters mke up the output vector. Equtions (.61) nd (.66) re the focus of the longitudinl dynmics. These two equtions chrcterize the Phugoid mode of the ircrft. To get the pproprite ltitude informtion, we include the verticl component of the vector eqution (.73) to our system of equtions, s shown in eqution (3.168). Altitude rte, ḣ, nd ltitude, h, re both needed for the feedbck control of the longitudinl dynmics. The equtions for the longitudinl dynmics re repeted here. V T D mgsinγ = m (.61)

98 γ = L cosφ mg cosγ mv (.66) h = V sinγ (3.168) Within these equtions re the terms L nd D, which re functions of the stte vribles. However, the terms L nd D re not explicitly defined in terms of the stte vribles. To do this we need to know the erodynmic chrcteristics. Using the irfrme equtions of section.9, we cn express lift nd drg nd their relted coefficients. As discussed in Section.6, we know tht the lift coefficient will be treted s control input to the system, nd will not be function of the sttes. L = 1 ρ V S C (3.169) w L The ircrft drg cn be similrly expressed. D = 1 ρ V S C (3.170) w D where the drg coefficient, C D, of eqution (.131) is restted here. ( ) min drg C = C + K C C (.131) D Do L L We need to express the totl drg in terms of the drg coefficient. ( ) D = ρv S C + K C C 1 w Do L Lmin drg (3.171) These reltions for lift nd drg need to be substituted into the stte equtions. Equtions (.61) nd (3.171) re combined to get the explicit stte eqution for true irspeed. ( ) min drg 1 T ρv S w CD + K C sin o L C L mg γ V = (3.17) m Equtions (.66) nd (3.169) re combined to get the explicit flight pth ngle eqution. γ = 1 ρv SwCL cosφ mg cosγ mv (3.173)

99 We cn represent Equtions (3.17), (3.173), nd (3.168) simply s functions of the stte nd control vribles. This is our non-liner system. V = f ( V, γ, h, C, T) V L γ = f ( V, γ, h, C, T ) γ L h = f ( V, γ, h, C, T ) h L We wish to express the longitudinl dynmics s n LTD stte-spce, s in eqution (3.167). This will fcilitte n nlysis of the modl properties. It will lso fcilitte the computtion of feedbck gins in Section 4. To crete liner system from non-liner system, we perform Tylor series expnsion bout reference condition (identified with subscript 0 ) nd, ssuming perturbtions from the reference condition re smll, keep only first-order terms. f V V V fv fv fv f V o + = fv + V h T C o + γ V γ h T C o o fγ fγ fγ fγ f γ γ + γ γ γ o = f + V o + + h + T + C V γ h T C o o o o L o o o L o h h fh fh fh fh fh + = f V h T C o h + o + γ V γ h T C o Similrly, the prmeters of the output vector re repeted here. o o o L o L L L ( γ 1) γ γ ( 1) * p γ 1 γ VIAS = 0 1 M γ 1 p o (.119) V M = (.11) * h = h h = V sinγ (3.168)

100 Once gin, we cn represent these equtions s functions of the stte nd control vribles. V = g ( V, γ, h, C, T ) IAS VIAS L M = g ( V, γ, h, C, T ) M L h = g ( V, γ, h, C, T ) h L h = g ( V, γ, h, C, T ) = f ( V, γ, h, C, T ) h L h L The linerized perturbtion equtions of the output vector re, g g g g g V V g V h T C VIAS VIAS VIAS VIAS VIAS IAS + IAS = V + o IAS o + γ L V γ o h T C o o o L o g g g g g M + M = g + V + γ + h + T + C o M M M M M M o L V γ h o o o T o CL o g g g g g h + h = g + V + γ + h + T + C o h h h h h h o L V γ h o o o T o CL o f f f f f h + h = f + V + + h + T + C o h h h h h h o γ L V γ o h T C o o o L o A prtil derivtive of system eqution is clled stbility derivtive if it is with respect to stte vrible nd control derivtive if it is with respect to control vrible. The stbility nd control derivtives for the longitudinl model re orgnized in tbulr form nd presented in Tble 3.6, Tble 3.7, nd Tble 3.8. It is importnt to note tht these derivtives re derived specificlly for the 4 DOF model tht we hve constructed. These derivtives re similr, but not interchngeble with the clssic stbility nd control derivtives of the full 6 DOF equtions of motion such s those found in Nelson (1989) or Stevens nd Lewis (199). By inspection, we see tht the prtil derivtives of gv for V IAS g V, M nd g V IAS gh. From equtions (.119) nd (.11), h, g M, nd g h re ll zero except

101 - 6 - * * 1 M V g V V = = (3.174) ( ) 1 ( 1) * V IAS o g p M V V p γ γ γ γ γ γ = + + ( ) 1 1 * ( 1) ( 1) 0 * 1 ( 1) IAS o o V o p p M M M p p g V p M p γ γ γ γ γ γ γ γ γ γ γ γ = + + (3.175) 1 g h h = The LTD stte-spce representtion of the linerized longitudinl dynmics (per eqution (3.167)) is shown in equtions (3.176) nd (3.177). V h f V f f h f V f f h f V f f h V h f C f T f C f T f C f T C T V V V h h h V L V L h L h L γ γ γ γ γ γ γ γ γ γ L N M M M O Q P P P = L N M M M M M M M O Q P P P P P P P L N M M M O Q P P P + L N M M M M M M M O Q P P P P P P P L NM O QP (3.176) IAS IAS L h h L h h h V V V V M M C V h T h f f h C T f f f V h γ γ = + (3.177)

102 All of the derivtives in these constnt mtrices re evluted t the reference condition; the nottion indicting tht explicitly hs been removed. Numerous derivtives re zero for ll cses. These derivtives re: f h fγ fγ f f fh = = = = = = 0 T h h C T V h h L Tble 3.6. Stbility nd control derivtives of f V Stte/Input Derivtive of f V V ( ) f ρv Sw CD + K C o L C Lmin drg γ h V V = (3.178) m fv = g cos γ (3.179) γ fv = 0 (3.180) h f ρv SwK ( C V L CL ) min drg C L C L = (3.181) m fv = 1 (3.18) T T m Tble 3.7. Stbility nd control derivtives of fγ Stte/Input V γ h C L T f γ V f γ γ f γ ρsref CL g = + m V Derivtive of fγ cos γ (3.183) g = sin γ (3.184) V = 0 (3.185) h f γ ρv S ref = (3.186) C m f γ L = 0 (3.187) T

103 Tble 3.8. Stbility nd control derivtives of f h Stte/Input Derivtive of f h V γ fh V fh γ = sin γ (3.188) = V cos γ (3.189) h f h = 0 (3.190) h C L f C = 0 (3.191) h L T f T = 0 (3.19) h Furthermore, if we ssume tht the ircrft s reference condition is level flight (γ = 0 ), we cn set other derivtives to zero. f γ γ fh = = 0 V Modifying our stte equtions results in Equtions (3.193) nd (3.194). These equtions represent the finl form of the linerized model. L NM O P Q V γ h P = L NM fv f V 0 V γ fγ 0 0 V fh 0 0 γ O L NM QP O f f V CL T f γ γ CL hp + 0 M 0 0 Q L M NM V V O L NM QP C L T O QP (3.193) VIAS 0 0 V VIAS 0 0 M V 0 0 M 0 0 C V L = γ + h 0 0 T h h 0 0 f h 0 0 γ (3.194) It is useful to compre the simultion results of the liner model to the results from the nonliner model. The expected result is tht the liner model will gree with the nonliner model for very smll perturbtions from the reference condition. As the

104 perturbtions from the equilibrium condition become lrger, the liner model will not follow the nonliner dynmics. This behvior is seen in Figure 3.4 nd Figure 3.5. Both figures show the time histories of the three longitudinl sttes long with the ltitude rte s clculted by the liner nd nonliner models. Figure 3.4 shows the models response to smll perturbtion or chnge in the nominl or reference lift coefficient. As cn be seen from the time histories of Figure 3.4, the mtch between the two models is good. However, s the perturbtion or chnge in the nominl or reference lift coefficient becomes lrger, the liner model fils to reflect ccurtely the behvior of the nonliner dynmics. This is the limittion of using liner models. To ccount for this limittion, mny liner models, ll referenced bout different reference conditions, re used to ccurtely model the ircrft s performnce throughout the entire flight envelope. Figure 3.4. Comprison of liner nd nonliner models with 0.01 perturbtion from the reference lift coefficient

105 Figure 3.5. Comprison of liner nd nonliner models with 0. perturbtion from the reference lift coefficient 3. The Anlysis of Longitudinl Aircrft Modl Properties 3..1 The Chrcteristic Polynomil of n LTD System To solve the system of equtions (3.167), we use Lplce trnsformtion, remembering tht the mtrices re constnt. There is no loss in generlity in ssuming the initil conditions re zero. ( ) ( 0 ) = ( ) ( ) ( ) ( s) ( s) ( s) sx s x t sx s = Ax s + Bu s y = Cx + Du (3.195) The stte nd output vectors cn be expressed in terms of the control vector. 1 ( s) ( s ) ( s) x = I - A Bu 1 ( s) ( s ) ( s) y = C I - A B + D u (3.196) Equtions (3.196) express the open loop dynmics in terms of known control input. The polynomil given by det ( si - A ) is known s the chrcteristic polynomil of the LTD

106 system. It is of the sme order s the stte vector. Anlysis of the chrcteristic polynomil revels the modl properties of the system. 3.. Modl Properties of the Longitudinl Dynmics The min limittion of liner models is tht they re vlid only for limited rnge round the reference conditions tht were used to crete them. To model n entire flight envelope of n ircrft, mny liner models, ech hving its own set of reference conditions, must be developed. Immeditely, one cn then see the dvntge to hving liner model tht is function of s few reference vlues s possible. For instnce, if the ircrft s liner model vried only with true irspeed, it would mke for simple one dimensionl set of liner models, ech with different true irspeed reference. However, we cn see through observtion tht the liner model of the longitudinl dynmics for given ircrft is fundmentlly function of three vrying prmeters. These re: V : The ircrft s true irspeed ρ: The ir density m : The mss of the ircrft The fct tht there re three terms immeditely presents n inconvenience. Any set of liner models must be three-dimensionl. For instnce, even modest number of vritions, sy 10 true irspeeds, 10 msses, nd 10 different ir densities, would yield 1000 reference conditions nd hence 1000 liner models. It is, therefore, very desirble to eliminte vrying prmeter if possible. Elimintion of vrying prmeter is the ttempt of this section. Consider the stte spce representtion of the system s shown in Eqution (3.193). This system of equtions contins three stte equtions, the first two of which chrcterize the Phugoid longitudinl mode (the V nd γ equtions). The third stte eqution, the h eqution, contributes only to the clcultion of ltitude nd does not ffect the Phugoid mode. L NM O P Q V γ h P = L NM fv f V 0 V γ fγ 0 0 V fh 0 0 γ O L NM QP O f f V CL T f γ γ CL hp + 0 M 0 0 Q L M NM V V O L NM QP C L T O QP (3.193)

107 Becuse the h eqution does not contribute to the Phugoid dynmics, we choose to ignore it in the following nlysis. Ignoring the h eqution reduces the stte equtions to Eqution (3.197). L N M V P = γ O Q L NM f V V f γ V O P fv fv f V γ V L O CL T γ f γ P NM 0 M 0 C Q L M QP + N L O L QP NM C L T The chrcteristic polynomil cn be written immeditely by inspection. O QP (3.197) V s fv s fv f γ γ V (3.198) When we substitute for the ctul derivtives we see tht the chrcteristic polynomil expnds to the following. ( ) ρv S C K C C + ρsref CL g + + cosγ + cosγ m m V w D0 L L ref s s g (3.199) We know from clssicl control theory tht the nturl frequency nd dmping rtio re represented in the chrcteristic polynomil s s + ζω s + ω. Therefore, we cn ssign the lst term to equl the squre of the Phugoid frequency, ω p. n n ω p F HG ρsref CL g = g cosγ + cosγ (3.00) m V I KJ We cn gin insight from this eqution. First, if the flight pth ngle is smll, the reltion cn be reduced to Eqution (3.01). ω p F HG ρgsref CL g = + (3.01) m V I KJ

108 The rtio g V is likely to dominte this term t low speeds nd be smll t high speeds. It is function of true irspeed squred, which is proportionl to dynmic pressure. Consider the other term. It is inversely proportionl to the mss. This suggests tht s the mss goes down, the frequency goes up. This is true providing tht the lift coefficient does not chnge. However, it is likely tht the lift coefficient will chnge with mss becuse the pilot will lwys tend to trim the ircrft for given flight condition. If we ssume tht lift equls weight or is close to equling weight for the vst number of flight conditions we cn write the reltion for mss s seen in Eqution (3.0). qs C = mg (3.0) ref L Using (3.0) we cn substitute for C L in Eqution (3.01) resulting the Phugoid expression in Eqution (3.03). ρgs ref mg g ω p = + m qs ref V (3.03) Cnceling terms leves (3.04). ω p F HG ρg g = + q V I KJ (3.04) Finlly, noting tht the dynmic pressure is function of density nd true irspeed, ω p F HG I KJ ρg g = + (3.05) ρv V we cn see tht the Phugoid frequency for the trimmed ircrft is entirely function of true irspeed s shown in Eqution (3.06). g g (3.06) = F H G I K J = ω p V V This is n interesting result becuse it suggests tht the frequency of the Phugoid is not function of the weight of the ircrft or the ltitude t which the ircrft is flying

109 Moving to the dmping rtio of the Phugoid mode, we cn express the dmping rtio using the middle term of the chrcteristic polynomil if we divide by ω p s shown in Eqution (3.07). ζ p ( ) ρv S C + K C C = g m V w D0 L L ref (3.07) Unfortuntely, there is no simplifiction tht reduces the dmping rtio to single function of ny prmeter tht we hve so fr defined. This expression for dmping implies tht the only wy to schedule gins to control the Phugoid is to hve threedimensionl tbles consisting of ircrft weight, true irspeed nd ltitude (ir density). This cretes lrge computtionl burden nd requires the storge of mny scheduled feedbck gins. It is desirble to somehow reduce the schedule to two-dimensionl tble. One solution is to substitute dynmic pressure for the density nd true irspeed. This substitution effectively ssumes tht chnges in speed nd ltitude (density) cn be interchngeble. However, we cn see tht density nd speed work independently of ech other. While it my be cceptble to schedule vs. dynmic pressure, it is only n pproximtion. It is better if nother quntity cn be found. Working towrds simplified expression for dmping, we revisit Eqution (3.07). Assuming trimmed ircrft, we cn substitute Eqution (3.0) into Eqution (3.08) for weight, mg. After some lgebric mnipultion, the finl result is shown in Eqution (3.09). ζ p = V g ( ) ref ρv S C + K C C m w D0 L L ( ) ref S C K C C ρv + ζ p = mg w D0 L L ( ) ref qs C + K C C ζ p = qs C w D0 L L w L ( ) ref C + K C C ζ p = C D0 L L L

110 ζ p = 1 L f D (3.09) We see tht the dmping is function of the lift to drg rtio of the ircrft. An lterntive derivtion is contined in Nelson (1989), which comes to the sme bsic conclusion. Unfortuntely, it is impossible for the control logic or ny sophisticted instrument to ctully mesure the lift to drg rtio of the ircrft. However, the lift to drg rtio is lwys function of the lift coefficient t given time. So, while the lift to drg rtio cn not be known, we cn pproximtely mesure the trim lift coefficient t ny given time in the flight nd know tht the lift coefficient corresponds to prticulr loction on the drg polr nd hence prticulr L/D. Therefore, the conclusion is tht the modl properties for the ircrft re uniquely described by the trim lift coefficient nd the true irspeed. However this does not consider the control derivtives. There re three control derivtives to be considered s shown in Equtions (3.10) - (3.1). f γ C L f C ρv Sref = (3.10) m V w L L ρv S KC = (3.11) m fv = 1 (3.1) T m Consider the expression for the lift coefficient in Eqution (3.13). If the lift coefficient nd the true irspeed re known, it is possible to solve for the rtio between the ir density nd the ircrft s mss. This cn be seen in Eqution (3.14). C L mg = ρs V w (3.13) ρ m g = C S V (3.14) L w While it is not possible to solve for the mss nd density directly, we cn solve for the rtio, which is then ll tht is needed to define two of the control derivtives s shown in Equtions (3.15) nd (3.16)

111 f γ C L f C ρ V S =F I H m K ref ρ V = V SwKCL L m (3.15) (3.16) The finl control derivtive, the fv T derivtive, is obviously function of mss only. Therefore, it is not completely possible to define the whole system using two prmeters. However, hving only one term tht is function of mss is fr more convenient thn hving the mss term throughout the model. All gin clcultions cn be done using nominl mss nd vrying true irspeed nd trim lift coefficient. To ccommodte different ircrft msses only one term need be chnged. 3.3 Trnsfer Function Anlysis of the Longitudinl Dynmics In Section 3. we investigted the modl property vrition of the longitudinl dynmics. The modl property nlysis, by defining the dmping nd frequency of the mode, defined the chrcteristic polynomil of the system. The purpose of this section is to outline the trnsfer functions tht re most likely to dominte the longitudinl dynmics. A trnsfer function is n isoltion of the ffects of n input (or driving) function on n output (or response) function. Becuse the trnsfer functions include the zeros of the system s well s the poles (the roots of the chrcteristic polynomil), the trnsfer functions provide more thorough understnding of the dynmic system. The limittion of trnsfer function nlysis (in ddition to the nlysis limittions lredy imposed by linerizing the system) is tht it focuses ttention on the isolted effects of single input on single output. For this reson, we use trnsfer function nlysis only to revel trends in liner feedbck control. The output eqution of the system (3.196) cn be expressed in trnsfer function form. where y ( s) = G ( s) u ( s) (3.17) ( s) ( s ) 1 G = C I - A B + D (3.18) We cn plug in the system mtrices to obtin the trnsfer function mtrix for our system. First we modify the output eqution, eqution (3.194). Notice the ddition of fctor of - 7 -

112 60 in the f h γ term; this fctor is to convert the ltitude rte from ft/sec to ft/min so tht the output mtches the verticl speed indictor in cockpit. VIAS 0 0 V VIAS 0 0 M V 0 0 M 0 0 C V L = γ + h 0 0 T h h 0 0 f h γ (3.19) Our trnsfer function mtrix is obtined by plugging the system mtrices of equtions (3.193) nd (3.19) into eqution (3.18). G ( s) V f IAS V V f V f IAS γ V fv IAS s + s s V CL V γ CL V T M fv M f V f γ M f V s + s s 1 V CL V γ CL V T = f f f V f V f γ f h V f f h V f f γ f γ γ f s s s h V + s V V γ V γ CL γ V CL V γ T f f f h γ V f V f f f f γ h γ V 60s + s 60s γ V CL V CL γ V T (3.0) The chrcteristic polynomil discussed in the previous section is lso the denomintor of our trnsfer functions. This is true for ny stte spce system. Since the chrcteristic polynomil hs lredy been investigted, there is no need to repet the nlysis here. The nlysis here will focus on the numertors of the trnsfer functions. Severl trnsfer functions were selected for study to determine the cceptbility of using the two control inputs (thrust nd lift coefficient) for vrious tsks. The trnsfer functions tht were singled out for study nd discussed here re the ḣ C L (g 41 ), M C L (g 1 ), nd ḣ T (g 4 ) trnsfer functions The ḣ C L Trnsfer Function The ḣ C L trnsfer function reltes n ircrft s response in ltitude rte to chnges in the lift coefficient. The term of eqution (3.0) corresponding to the ḣ C L trnsfer function is expressed below

113 f f f f f f h γ γ V V γ 60 s + h γ CL V CL V CL = g41 = C f L V f V f γ s s V γ V (3.1) We cn rrnge the numertor of the trnsfer function to show clerly the zeros. We cn see tht we hve one zero, which is highlighted in Eqution (3.3). We represent the zero with the symbol z 1. ḣ C L = 60 F HG fh γ F G fγ fv fv f I γ V CL V s C f LKJ G + γ HG HG CL V s fv s fv f γ γ V F f γ C L II J KJKJ (3.) F fγ fv fv V CL V z = 1 G f HG C f γ L γ C L I KJ (3.3) We cn not gin much insight from the symbolic representtions of the derivtives lone, so it is necessry substitute in for the derivtives from the tbles in Section 3.1. Expnding the inner terms of the zero we hve: z 1 = F HG Simplifying, ρv S KC m F HG w L ref L ρs C g ρv S C + KC ρv S 0 + cosγ + m V KJ m m ρv S z 1 m I ref V Sw =F I H KCL C m K D0 d i w D L ref I KJ (3.4) ρ d i (3.5) Bsiclly, wht the zero shows us is tht if the prsite drg (zero lift drg) is smller thn the induced drg, the zero of this trnsfer function will be positive. The cross over

114 point is when the ircrft is operting t its mximum lift to drg rtio. Figure 3.6 illustrtes the flight regime where the zero of the ḣ C L switches sign. The condition when zero of trnsfer function is positive is referred to s nonminimum phse system becuse of such system s tendency to immeditely move in the opposite direction of the finl stedy stte vlue. The ḣ C L is even more insidious becuse it hs negtive DC gin. DC gin (the nme is tken from electric circuits) is the vlue of the trnsfer function t s = 0. It revels the gin of the stedy-stte response to the input. A negtive DC gin insures tht the finl output will be negtive of wht ws commnded. A simple discussion suffices to provide n intuitive understnding of wht is hppening. Consider wht hppens when n ircrft is flying so slow tht the drg is higher thn it would be if it were to fly fster (see Figure 3.6). To fly slower the ircrft ctully needs more thrust. However, the ddition of thrust tends to ccelerte the ircrft nd hence lower the drg, which mkes the ircrft fly fster. Likewise, if n ircrft in this condition tries to climb, the result of incresing the lift coefficient will tend to slow the ircrft. However, insted of slowing down nd reducing drg, which would normlly llow climb, the ircrft sees n increse in drg. A pilot must be creful to dd throttle nd chnge lift coefficient simultneously to control flight in this regime. Bck Side of Thrust Curve (Positive zero) Totl drg Point of zero crossover (Negtive zero) Prsite drg Induced drg Figure 3.6. An illustrtion of drg vs irspeed t constnt ltitude, highlighting the non-minimum phse behvior of the trnsfer function of the linerized system

115 Our trnsfer function nlysis of the linerized system implies tht lift coefficient control of ltitude rte is stble on the front side of the thrust curve, nd unstble on the bckside. Bsed on this nlysis, one might ssume tht control is difficult or impossible on the bckside of the thrust curve. Fortuntely, the linerized system nd trnsfer function nlysis fil to ccurtely predict the stbility of our system dynmics on the bckside of the thrust curve. In generl, the system is not operting with constnt thrust while using lift coefficient to control ltitude rte. An increse in thrust is typiclly vilble to correct the non-minimum phse behvior. And, while it is not typicl mneuver to descend t idle thrust while mintining commnded ltitude rte, such mneuver is still possible, prtly becuse idle thrust increses in descent nd prtly becuse the speed is llowed to vry in such mneuver. And lift coefficient control of ltitude rte is vible control strtegy throughout the flight envelope The M C L Trnsfer function The M C L trnsfer function chrcterizes the response in Mch number with chnges in lift coefficient. The term of eqution (3.0) corresponding to the M C L trnsfer function is shown below. fv f γ M f γ C s + V C f V L L V M CL = g 1 = C f L V f V f γ s s V γ V (3.6) Next, the zero of the trnsfer function cn be nlyzed s shown in Equtions (3.7) nd (3.8). f fv f γ ρv Sref g γ CL = m (3.7) fv ρv SwKC L C m L fv f γ γ CL g = (3.8) fv V KC L C L

116 We cn see from Eqution (3.8) tht the zero is lwys negtive. This is fvorble result becuse it mens tht Mch number cn lwys be controlled effectively by the lift coefficient. However, the control design will not necessrily be trivil. Consider the following exmple of trnsfer function. M C L f (3.9) s = s s The root locus of this trnsfer function shows us tht feedbck control of Mch number using the lift coefficient would be difficult. Proportionl feedbck tends to drive the poles unstble s seen in Figure 3.7. When the poles do converge on the rel xis, one pole heds towrds positive infinity nd the other heds towrds the zero. We cn see from Figure 3.7 tht it will be necessry to feedbck something in ddition to Mch for stbility Img Axis Rel Axis Figure 3.7. The root locus of proportionl control pplied to the M C L trnsfer function The ḣ T Trnsfer function Consider the throttle to ltitude rte trnsfer function. The term of eqution (3.0) corresponding to the ḣ T trnsfer function is shown below

117 h T = g4 = s h γ V 60 f f f γ V T f f f s V γ V V V γ (3.30) This trnsfer function hs no zeros, which mkes it very well behved. Using throttle to control ltitude rte should not present design chllenge nd is not ffected by the bck side of the thrust curve. Applying throttle while the lift coefficient is held constnt will lwys mke the ircrft climb regrdless of the flight condition

118 4. The Feedbck Control System for Longitudinl Control In the previous Chpters, the bulk of the nlysis effort ws spent on the derivtion of the physicl model of the ircrft. The physicl model of the ircrft consisted of the dynmic equtions, which model the ircrft s performnce, nd the kinemtic equtions, which chrcterize the ircrft s propgtion over the surfce of the Erth. The purpose of building such n intricte model is to insure the fidelity of modeling ctul ircrft in flight. The min dvntge of high fidelity ircrft model is tht it ccurtely models the performnce nd hndling chrcteristics of n ircrft. Accordingly, the disdvntge of high fidelity ircrft model is tht it ccurtely models the performnce nd hndling chrcteristics of the ircrft. To mke the ircrft model follow desired trjectory, the ircrft model must be flown by pilot in the sme sense tht the ctul ircrft must be flown. Argubly, the longitudinl control system is the most complicted nd sensitive prt of the entire simultion. This Chpter is the first of three Chpters tht cover the longitudinl control system. The purpose of the longitudinl control system is to provide mens of utomting two fundmentl ircrft mneuvers. These mneuvers re: Altitude chnge nd ltitude cpture Speed chnge nd speed cpture Generlly, the functionlity of the longitudinl control system cn be divided into two distinct clsses of lgorithms: feedbck control nd supporting functionl logic. This Chpter dels with the design of the feedbck control lgorithms to stbilize the ircrft nd drive it to the desired stte. There re different feedbck control lgorithms for different flight phses, nd ech of these is discussed long with strtegy for clculting the required gins. 4.1 The Generl Control Lw Our generl control lw for the longitudinl dynmics is the sme regrdless of where in the flight envelope the ircrft is operting. The only rel difference in controlling flight within different regions of the flight envelope is the gins tht re used. The generl control lw frmework llows for ny output vrible to be fed bck to ny input vrible. The generl frmework for proportionl-integrl (PI) control lw is shown in block digrm form in Figure 4.1. The terms in the block digrm re defined s follows: y d is the desired output vector y is the ctul output vector -79-

119 e is the error vector equivlent to y d - y K p is the proportionl gin mtrix in the feed-forwrd pth. K i is the integrl gin mtrix (lso in the feed-forwrd pth) K b is the proportionl gin mtrix in the feedbck pth x = Ax + Bu is the linerized stte eqution for the longitudinl dynmics y = Cx + Du is the linerized output eqution for the longitudinl dynmics From experience in deling with the longitudinl dynmics, we hve seen tht proportionl control using the pproprite output error, e, is sufficient to chieve the dynmic response desired. Integrl control is then dded to eliminte stedy-stte error. The finl feedbck loop, the one using K b, is designed to llow proportionl feedbck control of certin output vribles without ffecting the zeros of the trnsfer functions. Gins in the feedbck pth ffect only the modl properties of system; gins in the feedforwrd pth ffect the dynmics of the system while t the sme time driving the stte error to zero. In certin instnces, it is necessry to mke use of the stbiliztion offered by feeding bck prticulr output while not driving tht output to ny prticulr vlue. Figure 4.1. Block digrm for the longitudinl control lw The reder is reminded tht the linerized system is simply convenient pproximtion to the ctul, non-liner system tht is used for simultion. The convenience of the liner pproximtion is in nlyzing the effects of our controller on the modl properties of the system to help us choose the controller. Once the controller is designed, we need to verify tht it produces stble nd relistic behvior in the ctul, non-liner simultion model. The generl form of LTD stte-spce is restted here. ( t ) ( t ) ( t ) ( t) ( t) ( t ) x = Ax + Bu y = Cx + Du (4.1) Our generl control lw is defined in the time-domin s, -80-

120 ( t) ( t) ( t) ( t) t p 0 i b u = K e + K e dt - K y (4.) Becuse the generl control lw llows for ny output vrible to be fed bck to ny input vrible, ech gin mtrix is of dimension (n x l), nd is of the form, K p k p k 11 p k 1 p 1, l k p 1 = k p k n,1 pn, l (4.3) where, n is the number of control vribles, l is the number of output vribles, nd the subscript, p, refers to the proportionl gin mtrix. The form the integrl nd feedbck gin mtrices is the sme except tht subscripts i nd b re substituted for the subscript p. 4. Mnipulting n LTD Stte-Spce with Integrl Control Integrl control dds system poles to our stte-spce. In effect, it chnges the order of our LTD system. This cn be seen explicitly if we modify our LTD system to include the differentil equtions dded by our PI controller. Let s define proportionl nd integrl control vectors s, ( t) ( t) ( t) ( t) = K e( t) u = K e - K y u p p b i i (4.4) nd let s define the integrted control s, u = t ( t) ( ) I u 0 i t dt Then the control vector, eqution (4.), cn be written in terms of its seprted proportionl nd integrl terms. ( t) = ( t) + ( t) u u I Clerly, the derivtive of the integrted control vector, I u, is the integrl control vector, u i. u p u ( t) = u ( t) I (4.5) i -81-

121 The stte eqution cn be rewritten in terms of the seprted control vector. ( t ) ( t) ( t ) + ( t ) x = Ax + B up I ( t) ( t) ( t) + ( t) p u x = Ax + Bu BI (4.6) Combining equtions (4.5) nd (4.6) nd the output eqution into one LTD system gives, u ( ) ( ) ( t) ( t) ( ) ( ) ( t) ( t) ( ) ( ) x t A B x t B 0m, n up t = + I t t 0 0 I 0 I u t u n, m n u n n i x up y ( t) = [ C D ] + D 0l, n Iu ui (4.7) From eqution (4.4), the control lw cn be written s, ( t) ( t) up K K p = e - y ui K i 0n, m b ( t) ( t) (4.8) m is the number of stte vribles in the stte vector (3 in our current exmple), n is the number of control vribles in the control vector (), l is the number of output vribles in the output vector (4), 0 n is squre, n x n mtrix of zeros, nd 0 n,m is n x m mtrix of zeros. Equtions (4.7) nd (4.8) represent our new LTD system with liner, PI controller. For the purposes of nlyzing the modl properties of our system, consider tht the stte, control, nd output vectors re simply perturbtions from reference condition. Without ffecting the system s chrcteristic polynomil, it cn be ssumed tht our desired result is to return to tht reference condition. Then the output vector, y d, is vector of zeros nd the error vector, e, becomes the negtive of the output vector, y. d ( ) e = y - y = 0 - y = -y Then the control lw, defined by eqution (4.4), cn be written s, ( t) ( t) up Kp + K b = y ui K i l ( t) -8-

122 Since, in this liner pproximtion, there is no wy to differentite between the effects of proportionl gin in the feedbck pth, K b, nd proportionl gin in the feed-forwrd pth, K p, there is no use in considering both nd so K b is neglected. u u p i ( t) ( t) K p = y K i ( t) (4.9) Equtions (4.7) still follow the formt of n LTD system: the new A-mtrix is squre, the new stte nd control re still time-dependent row vectors, nd the new A-, B-, C-, nd D-mtrices re constnt. Once combined with the stndrd output-bsed feedbck control lw of eqution (4.9), we hve complete LTD feedbck control system conducive to modl nlysis. It is convenient form tht dds the integrl terms (nd their resultnt poles) to the stte vector to fcilitte the nlysis of modl properties nd clcultion of gins. 4.3 An Anlysis of the Effects of Feedbck Control on the Modl Properties Until this point, the nlysis of this chpter is generlized to ny LTD system with PI controller defined by eqution (4.), but we now begin to tilor the nlysis to our model. When we combine our LTD system of equtions (4.7) with our originl stte, control, nd output vectors nd our A-, B-, C- nd D-mtrices s defined in Chpter 3, we get, fv f V f V f V f V f V 0 V γ CL T V C L T V fγ f 0 0 γ f u p γ C L γ γ V C L h C u p T = L f h 0 0 h I 0 0 ui C I L C C L L γ I u it T I T VIAS 0 0 V V V 0 0 u p IAS M γ C L M V h 0 u pt = + h I uic C L L h f 0 0 h ui I T T γ -83-

123 And the generl form of our LTD system is, fv f V f V f V 0 fv f V V 0 0 γ CL T V V CL T fγ f u pc γ γ L γ f γ V C L h u p = h + CL f T h I C IC L ui C L L γ I I u i T T T VIAS V V VIAS M γ M V (4.10) * = h h ICL h f h I T γ Following the form of the generl gin mtrix of eqution (4.3), our proportionl, integrl, nd feedbck gin mtrices re, k p k 11 p k 1 p k 13 p 14 K p = (4.11) k p k 1 p k p k 3 p4 ki k 11 i k 1 i k 13 i 14 K i = (4.1) ki k 1 i k i k 3 i4 K b kb k 11 b k 1 b k 13 b 14 = kb k 1 b k b k 3 b4 (4.13) Substituting into eqution (4.9), we hve our control lw. * Note tht the ḣ eqution hs dded fctor of 60 to its f h γ term. This is to convert the output from ft sec to ft min s would be red on rel verticl speed indictor. -84-

124 up C L k p k 11 p k 1 p k 13 p 14 VIAS up k T p k 1 p k p k 3 p M 4 = u i k C L i k 11 i k 1 i k 13 i h 14 u k h i k 1 i k i k 3 i 4 it (4.14) There re 16 gins in our gin mtrix, ech representing the effect of feedbck of prticulr output prmeter to prticulr control prmeter. To gin insight into the effect ech feedbck gin is likely to hve, we exmine the root locus of ech gin. Consider Boeing t reference weight of 198,000 lbs in stedy, level flight t 300 knots indicted irspeed (KIAS) nd 30,000 feet. For this exmple, the LTD system of equtions (4.10) becomes, V V u pc L γ γ u pt h = h ui I C I C C L L L I I u T T it V VIAS γ M = h h ICL h I T Figure 4. - Figure 4.5 show how the phugoid mode poles re moved by vrious types of feedbck. Figure 4. illustrtes the system s behvior with different proportionl feedbck to the lift coefficient. The first subplot shows the positive feedbck of indicted irspeed to the lift coefficient, which we see is unstble. A moment s reflection on the nture of the system provides intuitive verifiction. An increse in the lift coefficient results in higher drg, which serves to slow the ircrft. This implies tht lowering the lift coefficient would serve to increse speed. The second subplot verifies this nd shows negtive speed feedbck to the lift coefficient provides for stble control. The third subplot shows the effect of ltitude feedbck to lift coefficient, which we see tends to shoot the phugoid poles up long the imginry xis. This will increse the phugoid nturl frequency while reducing the dmping of the system. -85-

125 The fourth subplot shows the effect of ltitude rte feedbck to lift coefficient. Here we see well behved loop. Feeding bck the ltitude rte tends to dmpen the system. This effect cn be intuitively verified by remembering tht rte terms usully do increse the dmping of system. Figure 4.3 shows the effects of integrl feedbck to the lift coefficient. The second subplot shows problem with instbilities of integrl speed feedbck to the lift coefficient. If we choose to use negtive integrl speed feedbck to the lift coefficient, we will need to feed some output in the feedbck pth to keep it stble. The fourth subplot shows tht we need to be creful with low dmping of integrl ltitude rte feedbck to the lift coefficient. Figure 4.4 shows the effect of proportionl feedbck to the thrust. The first subplot shows positive feedbck of indicted irspeed to the thrust. The locus is well behved nd tends to increse the dmping of the system. Positive feedbck of indicted irspeed or Mch to the thrust is good choice for the control of speed. The fourth subplot shows ltitude rte feedbck to the thrust. The ltitude rte here tends to hve the sme effect on the Phugoid poles tht ltitude feedbck hd to the lift coefficient. Altitude rte feedbck to the thrust is not good choice for controlling ltitude rte or ltitude. Similrly, the third subplot shows tht ltitude feedbck to the thrust drives the system unstble Figure 4.. Effects of proportionl feedbck to the lift coefficient -86-

126 Figure 4.3. Effects of integrl feedbck to the lift coefficient Figure 4.4 Effects of Proportionl Feedbck to the Thrust -87-

127 Figure 4.5 Effects of Integrl Feedbck to the Thrust Figure 4.5 shows the effects of integrl feedbck to the thrust. The third nd fourth subplots confirm tht it is not wise to control ltitude or ltitude rte with thrust becuse the system becomes unstble s soon s we pply control. The first subplot shows tht integrl feedbck of speed to the thrust tends to shoot the phugoid poles up the imginry xis, incresing the nturl frequency nd lowering the dmping of the response. This implies tht we need to be mindful of low dmping when controlling speed with the thrust. 4.4 Feedbck Controller Design In the previous section, we nlyzed the effects of feedbck control on the modl properties of the LTD system in order to gin insight into how feedbck control would ffect our nonliner, four-degree-of-freedom model. In this section, we continue to use the LTD system to design feedbck controllers for severl different regions of n ircrft s flight regime. The LTD system provides suitble pproximtion to the nonliner system so tht feedbck controllers designed using the LTD system will produce similr results in our nonliner model. Ech controller is customized to ttin the desired flight configurtion using smooth trnsitions tht re typicl of commercil ircrft flight. -88-

128 Let us summrize our conclusions from the nlysis of the root loci of the LTD system. The primry control input in n ircrft is the control stick (i.e., the lift coefficient), so it should be the primry control in our controller s well. Figure 4. nd Figure 4.3 show tht we cn successfully control speed with the lift coefficient, s long s we use negtive feedbck, nd s long s we feed output (e.g., ltitude rte) in the feedbck pth to keep the controller stble. Figure 4. nd Figure 4.3 lso show tht we cn successfully control ltitude rte with the lift coefficient. Figure 4.4 nd Figure 4.5 show tht we cn successfully control speed with the throttle (i.e., thrust). Our bsic strtegy will be to use lift coefficient to control ltitude rte during speed chnges; use lift coefficient to control speed during ltitude chnges, while feeding bck ltitude rte to keep the system stble; nd use thrust to control speed nd lift coefficient to control ltitude rte when we need to control both. Our control lw llows us to feedbck ll outputs to both inputs; lthough only frction of the gins re used in given feedbck system. To design our controllers, we will use our full control lw, s defined by eqution (4.8) nd expnded here using the gin mtrix definitions of equtions (4.11), (4.1), nd (4.13). up CL k p k 11 p k 1 p k 13 p e 14 1 kb k 11 b k 1 b k 13 b VIAS 14 u P k T p k 1 p k p k 3 p e 4 kb k 1 b k b k M 3 b 4 u = i h C L ki ki ki ki e u ki k 1 i k i k 3 i e i h T (4.15) The control inputs to the system split into their proportionl nd integrted prts. The control inputs u nd u P re the proportionl portions of the lift coefficient nd thrust T PCL respectively, nd ui CL nd u i re the integrted portions of lift coefficient nd thrust. T Lift Coefficient Control of Altitude Rte There re severl flight regimes in which pilot uses the control stick to cpture or mintin ltitude rte while llowing the speed to chnge. During level flight ccelertions nd decelertions, pilot will preset the thrust (mximum thrust for ccelertion, idle thrust for decelertion) nd llow the speed to chnge ccordingly while using the control stick to mintin level flight (i.e., zero ltitude rte). Alterntively, the pilot my wish to cpture nd mintin desired, non-zero ltitude rte nd let the speed chnge s it my (i.e., during ccelertions in climb or decelertions in descent). -89-

129 For these regions, we need controller tht uses lift coefficient to control ltitude rte but does not modulte thrust. In the truest sense, the controller is not controlling speed; however, in ctulity the speed is controlled becuse we preset the thrust ccording to the desired direction of the speed chnge. When the speed ners the desired stedy, level flight condition, we switch to controller tht simultneously controls speed nd ltitude rte. To ccomplish the gols of this controller, we need simply to commnd n ltitude rte for ll time. For this reson we cn simplify the output eqution of the LTD system of eqution (4.10) to, V f h h = γ γ h In doing so, we cn see tht the dynmics of the system re governed exclusively by the trnsfer function s defined in Chpter 3. Consider DC-9 trveling t ft nd h CL 578 ft/sec nd weighing 140,000lbs. The LTD system is shown below with its reduced control nd output vectors. The phugoid eigenvlues for the open loop system re locted t ± i, which corresponds to nturl frequency of rd/sec nd dmping rtio of A plot of the step response of the system to 0.1 chnge in lift coefficient is shown in Figure 4.6. V V γ γ upc L = + h h 0 0 ui C L I I C 0 1 C L L (4.16) V h = γ h -90-

130 h_dot ft/min Time (secs) Figure 4.6. System response to 0.1 step in lift coefficient Figure 4.7. Root Loci of k p 14 (left) nd i 14 k (right) successive loop closures It is desired to increse the frequency nd dmping of the system while hving zero stedy stte error. We strt with proportionl ltitude rte feedbck to the lift coefficient, k p 14, shown in the left side of Figure 4.7. A vlue of k p = 6 10 moves the poles to loction of ± 0.056i with dmping rtio of nd nturl frequency of rd/sec. We dd integrl control to gurntee zero stedy stte error, not frequency chnges; but we must be creful bout the ssocited decrese in dmping we noticed in our preliminry nlysis. As seen on the right hlf of Figure 4.7, we move the poles with -5 k i = to loction of i 14 ±. In doing so, we increse the frequency of the mode to rd/sec but we reduce the dmping to To correct -91-

131 for the low dmping we cn increse the gin k p once gin nd move the Phugoid 14 poles to the left s shown in Figure 4.8. The finl vlues chosen for k p nd k 14 i re nd The finl step response is shown in Figure 4.9. Notice tht the system chieves the 1000 ft/min climb rte with zero stedy stte error. Furthermore, the time history for the lift coefficient is well within cceptble bounds. Note however tht the lift coefficient initilly is rther ggressive. In the next chpter, we define limits to the mount control input cn chnge in time-step so tht such ggressive control ction is mde more relistic. The method of successive loop closures is good for initil work nd illustrtion of the system dynmics, but it is tedious if mny gins must be chosen or specific dynmic properties re desired quickly. Furthermore, for the purpose of scheduling gins, the method of choosing gins must be utomted. Automting gin scheduling is resonbly strightforwrd once control scheme hs been estblished. Since the control logic is simple, the method of pole plcement cn be completely nlytic. This explicit method of pole plcement is outlined in Brogn (1991). Its limittion is tht it is cumbersome nd useful only for low-order systems, however its simplicity gives dded flexibility in gin selection Figure 4.8. The effects of n incresed proportionl gin k p 14-9-

132 h_dot ft/min CL sec Figure 4.9. System response to 1000 ft/min commnded rte of climb The process strts with replcing the individul terms in the system of equtions with plceholders to simplify the finl expressions. Furthermore, since we re not feeding bck ltitude nd it does not contribute to the dynmics, we cn remove it from the stte eqution. V b V b γ CL u pc 1 0 b1 1 0 L γ b ui CL I C 0 1 L = + I 0 V h = 0 c4 0 γ I CL We wnt to be ble to control the eigenvlues of the closed loop A-mtrix, which is defined s follows. A cl = A - BKC 11 1 b11 b k p14 A 1 0 b 1 b1 0 cl = 0 c4 0 0 ki [ ] -93-

133 Simplifying, 11 1 b11 b11 0 k p14 A 1 0 b 1 b1 0 cl = 0 c4 0 ki b b 0 0 c k 0 A = cl [ ] p b 1 b1 0 0 c4ki b11 0 b11c 4k p b A cl = 1 0 b1c4k p c4ki 0 14 we hve the finl closed loop A-mtrix in Eqution (4.17) b11c 4k p b A cl = 1 b1c4 k p b 14 1 (4.17) 0 c4ki 0 14 Next, we need to clculte the chrcteristic polynomil for the A-mtrix. s + b c k b s cl = s + b c k b 0 c4ki s 14 [ I - A ] p p14 1 Finlly, we hve n expression for the chrcteristic polynomil shown in (4.18). [ I - A cl ] ( )( ) det s = s s + c b k p s + b c ki ( 1 c4b11k p )( 1s) + + ( s 11 )( b1 )( c4ki ) = s + c b k s s c b k s + b c k p p i14 s + c b k s + b c k s b c k p i i14-94-

134 det 3 [ si - A cl ] = s + ( c4b1k p ) s + ( 4 11 p ) i p ( 1b11 c4ki 11b 1c4ki ) c b k b c k c b k s (4.18) From this we cn determine wht gins re necessry to chieve the desired chrcteristic polynomil. We hve the following form of the chrcteristic polynomil: 3 [ I - A ] det s = c s + c s + c s + c cl Therefore, we cn set ech coefficient equl to its corresponding term in Eqution (4.18). c 1 = 1 ( p ) 14 ( i ) 14 p14 p14 ( i i ) c = c b k c = b c k c b k + c b k c = c b k b c k Being ble to define the coefficients in terms of the gins is helpful; however, we relly wnt to determine the gins from the coefficients to be sure of getting the correct response. This is problem becuse we hve three linerly independent equtions nd two unknown gins. In order to solve this system, we will hve to leve one of the modl properties of the third order system s n unknown. The chrcteristic polynomil in terms of modl properties is, [ si - A cl ] = ( s p1 )( s + ζω s + ω ) det n n where ζ nd ω n re the dmping rtio nd nturl frequency of the oscilltory motion nd p 1 is the system poll. We cn expnd this form of the chrcteristic polynomil nd set the coefficients equl to those in the previous form. ( p ) 14 ( ) ( ) ζω p = c b k n ω ζω p = b c k c b k + c b k n n i p p 1 ω p = c b k b c k n i i14-95-

135 ( p ) ζω p = b c k n ( ) ( ) ω ζω p = b b c k + b c k n n p i ω p = b b c k n i14 This system of equtions is liner with constnt coefficients nd cn be written in mtrix form. b1c4 0 1 k p ζωn ( 1b11 11b 1) c4 0 ζωn k i ω 14 n 1 = ( 1b11 11b 1) c4 ω n p 0 1 This mtrix eqution is esily solved for the gins. I dmping rtio of 0.9 nd nturl frequency of.0 re suggested Lift Coefficient Control of Speed During climbs nd descents, pilot typiclly mintins constnt irspeed or Mch number during the ltitude chnge. The pilot will preset the thrust (climb thrust for climbs, idle thrust for descents) nd llow the ltitude rte to vry ccordingly while using the control stick to mintin speed. For these regions, we need controller tht uses lift coefficient to control speed but does not modulte thrust. Since the ircrft hs two mesurements for speed, Mch nd indicted irspeed, both speeds hve to be considered in seprte nlyses. However, since the solutions re identicl with the exception of few chnges in feedbck gins, only the Mch cse is discussed. The gol of the feedbck controller is to cpture given speed by djusting the lift coefficient. This mens tht the system is minly governed by the M C trnsfer function discussed in Section However, feedbck of speed to the lift coefficient hs problem with low dmping s illustrted in Figure 4.. The feedbck control strtegy to fix the low dmping problem hs lredy been touched upon in Section Our bsic strtegy is to build proportionl -plus- integrl controller for cpturing Mch, nd then feedbck ltitude rte in the feedbck pth to increse the dmping of the system. We will need two of the four outputs to complete the design, so the output eqution of the LTD system of eqution (4.10) cn be simplified to, L -96-

136 M 0 0 V M V = γ h f h h γ Consider the sme DC-9 trveling t ft nd 578 ft/sec nd weighing 140,000lbs. The open loop dynmics for the system re the sme s shown in eqution (4.16), nd our new output eqution is 5 V M = γ 4 h h Initilly, the phugoid eigenvlues re locted t ± i, which correspond to nturl frequency of rd/sec nd dmping rtio of (which is very low dmping). We strt the loop closures by closing the feedbck pth k b s shown in Figure Setting k b = moves the poles to ± i, which correspond to 14 nturl frequency of rd/sec nd dmping rtio of ( more suitble dmping rtio). Applying proportionl Mch feedbck to the lift coefficient, we increse the frequency of the poles to 0.0 rd/sec, which we initilly think is good vlue s shown in Figure The resulting feedbck gin is k = Note the negtive vlue 1 of k p. This mkes intuitive sense becuse n increse in speed should be the result of 1 lower lift coefficient. It is lso consistent with our conclusions from Figure 4.. To test the prtilly built controller, we ttempt commnd to Mch 0.7. From the simultion shown in Figure 4.1, we find out tht, while we like the response, the required control effort is excessive. The lift coefficient drops nerly to -1. Relizing tht too much control effort is required, we drop the frequency down to 0.1 rd/sec, which corresponds to gin of k = With this reduction in gin, the poles sit t ± i with 1 p frequency of rd/sec nd dmping rtio of p -97-

137 Figure k b root locus for the Mch Cpture controller 14 The next loop closure, the integrl control, is tricky becuse the integrl control tends to send the poles towrds the right hlf plne. Integrl control lso pulls the integrtor pole wy from zero long the negtive rel xis. A gin of k = 0. 1 is pplied, which tends 1 to line up the rel prt of the conjugte pir with the integrtor pole s shown in Figure The poles move to ± i where the frequency is rd/sec nd the dmping is Since the dmping is low, more ltitude rte is pplied. The ltitude rte is pplied until the locus strts to curve bck inwrd nd hed towrds the 5 imginry xis. The gin is set to k b = where the frequency of the system is rd/sec nd the dmping is i Figure k p1 root locus for the Mch Cpture controller -98-

138 We lso note tht the integrtor pole is moved to The complete system is simulted s shown in Figure From Figure 4.14 we see tht the controller cptures Mch of 0.7 nd tht the lift coefficient is not unresonble. However, the speed of the response is low MACH CL sec Figure 4.1. Simulted Mch Cpture Figure k i (left) nd 1 k b 14 (right) root loci for Mch cpture In dding the extr gin to k b we diminished our frequency so we would probbly wnt 14 to redo the design if this set of gins were ctully going to be used. However, the ctul -99-

139 gins used re clculted utomticlly by the gin scheduling lgorithm where we cn choose exctly the modl properties tht we wnt MACH CL sec Figure Simultion of the Completed Mch Cpture We hve seen from the control lw design tht successive loop closures cn be difficult. In this exmple, one cn see tht just bout ny set of modl properties could be chieved; however, the tril nd error pproch is certin to tke considerble mount of time. It is still vluble to mnully close the loops t lest once becuse it helps to build our understnding of the system dynmics. For instnce, we now know tht high system frequencies require n excessive control force. For this system, it is best to schedule nturl frequencies on the order of 0.1 rd/sec nd no higher. For the purpose of scheduling the ctul gins, the utomted method is presented next. Tking the LTD system of eqution (4.10), we remove indicted irspeed nd ltitude from the output eqution nd we remove ltitude nd integrted thrust from the stte eqution. V b V b γ CL u pc 1 0 b1 1 0 L γ b ui CL I C 0 1 L = + I

140 y M c V 0 0 γ IC L 1 = h = 0 c4 0 The bsic control lw s developed erlier is, ki 1 CL = k p + e 1 + kb h 14 s where ( ) e = M M d Note tht for the purpose of determining eigenvlues, the gins k p nd k 14 b hve 14 identicl effects. Therefore we substitute k b into the mtrix loction reserved for k 14 p14 for the purposes of gin scheduling only. The gin mtrix becomes, k p k 1 b 14 K = ki 0 1 The closed loop form of the equtions (A-BKC) is, A - BKC b b 0 k k c p1 b14 1 = 1 0 b 1 b1 0 ki 0 0 c b11 b11 0 k p c 1 1 kb c = 1 0 b1 b1 0 ki c b b k c b k c p b14 4 = 1 0 b1 b1k p c 1 1 b1kb c k c 0 0 i

141 11 b11 k p c b11k b c 14 4 b 11 = 1 b1k p c 1 1 b1kb c 14 4 b1 ki c The chrcteristic polynomil is clculted next. det s + b k c + b k c b s cl = det + b k c s + b k c b + ki c s [ I - A ] p b p1 1 1 b [ si - A cl ] = s ( s 11 + b11k c )( ) ( )( )( ) 1 1 s + b1k c b11k c 14 4 b1 + k c 1 1 ( b11 )( s + b1kb c4 )( + ki c1 ) s ( 1 + b11k b c4 )( 1 + b1k p c1 ) det p b b i Summing like terms yields the chrcteristic polynomil. det det 3 [ I - Acl ] = + ( ) ( ) ( ) + ( 1 ( 1 + b1k p c )) ( ) 1 1 s + 1b1k i c 1 1 ( c4b11 kb ( )) ( ) b1k p c 1 1 s c4b11 kb b 14 1ki c ( ki b11c 1) s + ( c4b11b 1kb ki c1 ) [ ] ( ) s s c b k s b k c s b k c c b k s 4 1 b p p b I - A ( ) ( ) ( ) k b k c s 3 s cl = s + c4b1 b p1 1 ( 1 1 b1k p c ) (( ) ) b11k p c 1 1 c4b1kb 14 ( c4b11 kb ( )) ( ) b1k p c ki b 1 11c 1 s + + ( 1b1k i c1 ) ( c4b11k b b1ki c1 ) ( c4b11b 1kb ki c1 ) The coefficients of the terms re difficult to mnge so some effort is pplied to simplifying them. We wish to express the chrcteristic polynomil in the form, 3 [ I - A ] det s = C s + C s + C s + C Setting the two forms equl, we cn solve for the coefficients. -10-

142 3 s C = : 1 1 ( ) ( ) s C c b k b k c : = 4 1 b p1 1 C = c b k + b k c 4 1 b14 11 p ( ( )) ( ) + ( + ) + p ( ) 1 p1 b14 ( 3b11 kb ) ( ) 14 1 b1k p c 1 1 ki b 1 11c1 1 s : C3 = [ b1k c b11k c1 c4b1k ] C = [ + b k c c b k + c b b k k c p b p1 b c b k c b b k k c + k b c b p1 b14 1 i ] C 3 = + b k c c b k + c b k + k b c p b b14 i ( ) ( ) ( ) s C = b k c b b k k c + b b k k c 0 : i b14 i b14 i1 1 C = b k c i1 1 Assuming tht the gin k b is known, we cn solve for the other gins. 14 k k p1 i1 C c4b1kb = (4.19) b c C b c = (4.0) Plugging these gins bck into the C 3 eqution yields, Simplifying, C b C c b k + c 4 1 b = b11c 1 C c b k + c b k + b c b b b1c

143 C C b c b k + c c b k c b k C b c 4 1 b = b b b11c 11 1b1c 1 b C b c b b C b C = + k + ( c b ) k + ( c b ) k b b b14 b11 b11 b11 1b1 1b1c 4b 1 1b1C 111 b1 C4b 11 C3 = 1c4b11 11c 4b1 kb b11 b11 b11 1b1 nd we cn solve for the gin k b. 14 k b14 b C b C b C + + = 1b1c 4b 1 1c4b11 11c4b 1 b b11 b11 1b1 (4.1) Becuse we hve three gins, we see tht we were ble to specify ll the coefficients of the chrcteristic polynomil. This enbles us to plce the poles rbitrrily. We cn choose the frequency nd dmping of the system poles nd the loction of the integrtor pole. From our mnul loop closing, we lso hve some ide of wht vlues mke good gins. We cn specify the coefficients in terms of the desired modl properties nd the loction of the integrtor pole, p, by expressing the chrcteristic polynomil in terms of its roots. 3 [ si - A ] = C1s + Cs + C3s + C4 = ( s p)( s + ζω s + ω ) det p p C = 1 C C C = p + ζω = pζω + ω = pω n n n n (4.) Error! Not vlid bookmrk self-reference. summrizes the gin scheduling equtions used for this controller. Looking t Figure 4.14, we see tht the lift coefficient is still commnded rther violently. Therefore we decide to specify slower dynmics. Using equtions in Error! Not vlid bookmrk self-reference. we cn try mny different modl properties quickly. After some experimenttion, we choose the following modl properties: -104-

144 Tble 4.9. The gin scheduling equtions for lift coefficient control of speed C = 1 C C C = p + ζω = pζω + ω = pω n n n n (4.) k b14 b C b C b C + + = 1b1c 4b 1 1c4b11 11c4b 1 b b11 b11 1b1 (4.1) k i1 C b c 4 = (4.0) k p1 C c4b1kb = (4.19) b c MACH CL sec Figure Simultion of Mch cpture with slower dynmics -105-

145 ω = 0. 05rd/sec n ζ = 0. 7 p = The gins for this set of modl properties re k = , k = , nd 1 1 k b = Figure 4.15 shows the simultion results. With the slower dynmics, the lift coefficient no longer hs its initil shrp dip; however the system tkes 40 seconds longer to cpture the desired Mch number. This performnce penlty is cceptble becuse n extr 40 seconds is not much time in the course of n entire flight Controlling Speed nd Altitude Rte Simultneously Mintining specified speed nd specified ltitude rte simultneously (s in stedy, level flight) requires feedbck control of both the thrust nd lift coefficient. Becuse of this, the controller for this flight regime is the most complicted of our controllers. It is lso the most used controller, becuse the ircrft spends most of its time controlling both speed nd ltitude (s in stedy, level flight). The design gols of this controller re lso the most mbitious. Here, we desire to drive both the speed of the ircrft nd ltitude rte to some commnded vlues. Consider the sme DC-9 trveling t ft nd 578 ft/sec nd weighing 140,000lbs. The open loop dynmics for system re the sme s shown in eqution (4.16); however, becuse of the ddition of the throttle feedbck control, we need to use the full stte eqution of the LTD system in eqution (4.10). With the given flight condition, the stte eqution becomes, p V V γ γ h = h I C I L CL I I T T (4.3) up C L up T ui CL u it i -106-

146 Since we wnt to be ble to drive the ircrft to prticulr speed nd ltitude rte independently of ech other, we must be very creful how we rrnge the feedbck in the feed-forwrd pth. Feedbck to ny one input cnnot be used to drive two independent errors to zero simultneously. By observing effects of feedbck in Figure 4. nd Figure 4.3, we see tht the throttle is much more dept t controlling speed thn ltitude rte. This is becuse the feedbck of speed to the throttle tends to increse the dmping of the longitudinl mode. Similrly, the lift coefficient is much better control of ltitude rte thn speed. Therefore we only llow speed feedbck to the throttle nd ltitude rte feedbck to the lift coefficient. For purposes of this demonstrtion, we use Mch in the feedbck. The control lws tht use indicted irspeed re identicl in form. Initilly, the Phugoid eigenvlues re locted t ± i, which corresponds to nturl frequency of rd/sec nd dmping rtio of Initilly, we close the lift coefficient feedbck pths in mnner very similr to wht ws done in Section We close the proportionl loop first using k p. We increse the proportionl gin 14 to k p = s shown in Figure This results in longitudinl mode poles t ± 0.465i. Our modl properties re ω p = rd nd ζ p = Our dmping rtio is cceptble; however, our frequency is low. We djust the frequency when we dd integrl control. As shown in the second plot of Figure 4.16, we increse the integrl 5 gin to k i = This moves the Phugoid poles to ±0.3954i where the 14 modl properties re ω p = 0.4 rd nd ζ p = While this results in n cceptble frequency, the dmping rtio is too low. Therefore, we increse the proportionl gin to k p = s shown in Figure This results in longitudinl mode poles of ± 0.850i with the corresponding modl properties of ω p = 0.4 rd nd ζ p = The position of the integrtor pole is img 0 img rel Figure The initil k p 14 (left plot) nd rel k i 14 (right plot) loop closures for region

147 img rel Figure The finl increse in k p 14 to chieve dequte dmping h_dot (ft/min) MACH CL 0.6 Thrust (lbs) Time (sec) Figure Simultion of commnded 1000ft/min rte of climb without ny feedbck to thrust At this point we cn illustrte the system dynmics by commnding 1000 ft/min climb. The simultion results (Figure 4.18) show response of ltitude rte nd Mch number s well s the lift coefficient nd thrust inputs. We cn see tht we pproximtely chieve the 1000 ft/min climb rte; however, we hve smll error of pproximtely 5 ft/min

148 This error is due to the slow integrtor pole. We ignore this smll error for now. We lso see tht the Mch number tpers off during the climb. This is expected t the moment becuse there is no feedbck to the throttle to mintin Mch (the thrust is constnt) img 0 img rel rel Figure Root loci for k p (left plot) nd k i (right plot) We lso see tht the controller slowly increses the lift coefficient to mintin the rte of climb s the ircrft slows down. If the controller is left to continue, it will stll the ircrft. Adding feedbck to thrust corrects the problem. Feedbck to thrust is initited 6 using proportionl control. We increse the gin k p to s shown on the left side of Figure The longitudinl mode is virtully unffected by the proportionl feedbck to throttle. The biggest influence of the proportionl feedbck to the thrust is to move the integrtor pole for the lift coefficient feedbck frther negtive. The lift coefficient s integrtor pole is moved to This is ctully desirble becuse it reduces the stedy stte error in ltitude rte. The finl loop closure, integrted thrust is closed next. As shown on the right hnd side of Figure 4.19, the integrted feedbck k i to thrust hs virtully no effect on the longitudinl dynmics; rther, it tends to drw the two integrtor poles together. If the gin is incresed further, the integrtor poles re drwn together nd become complex conjugtes. This in turn cretes nother longitudinl oscilltion, which is slower thn the dominnt mode. This extr mode is undesirble, so 5 we stop the integrted feedbck t k i = The finl system is described s: complex conjugte poles: ± 0.836i Thrust integrtor pole : Lift coefficient integrtor pole: Modl properties: ζ = ω = rd/sec n -109-

149 h_dot (ft/min) MACH Thrust (lbs) CL x Time (sec) Figure 4.0. Simultion of commnded 1000ft/min climb rte using the finl controller for region 7 The finl system is shown simulting commnded ltitude rte of 1000 ft/min in Figure 4.0. Note tht with the ddition of thrust feedbck, the throttle is djusted simultneously to mintin the desired speed. As with the other regions, the exercise of successive loop closures is useful but not prcticl for the tsk of scheduling mny different conditions; therefore, n utomted pproch is developed. With the ddition of throttle feedbck, the problem becomes much more complex. Agin, we will substitute plce holders into the LTD system to mke the lgebr more mngeble. V u pc b b V b b L γ 1 0 b1 0 γ b u pt = + I C I C u L L ic L I I T T u it (4.4) -110-

150 M c y = = h V γ IT 1 0 c4 0 0 ICL We need to close the loops round the throttle nd the lift coefficient. The control lw is summrized s Lp p14 d Li i14 d ( ) ( ) C = k h h C = k h h p p d i i d ( ) ( ) T = k M M T = k M M The closed loop A-mtrix, A cl = A - BKC is, A cl b b b b p b1 0 b k p c k i 0 c = k 0 i1 0 k A = cl b b b b c k p b1 0 b c11k p c4ki c11k i b b1 b c k p b c k p b1 0 0 b1c4 k p A cl = c4k 0 0 i c11k i b1 c11k p 1 1 b11c 4k p b b1 1 b1c4 k p b A cl = 0 c4ki c11k i

151 The chrcteristic polynomil of the closed-loop A-mtrix is, det s 11 + b1c 11k p b11c 4k p b b1 1 s + b1c4k p b I - A cl = det 0 c4ki s 0 (4.5) 14 c11k i 0 0 s 1 [ s ] Simplifying Eqution (4.5), + b c k b b det [ si - A cl ] = + ( c11k i ) det s + b 1 1c4k p b c4ki s p s + b c k + b c k b ( s) det s + b c k b 0 c4ki s p p p14 1 [ si - A cl ] = ( c11k ) ( ) ( ) 1 b1 s b1b 1c4k s b 14 1b1c4 k 14 ( s 11 + b1 c11k p )( s + b1c4 k p )( s) det i p i [ ] ( s) + ( 1)( c4ki )( b ) ( b1 )( c4ki )( s 11 b1 c11k p ) 14 1 ( 1 b11c 4k p )( 1 )( s) det si - A cl = b1 c11k i s b 1 1b1c4 k p c 14 11ki s c 1 11ki b 1 1b1c4 ki 14 ( s )( s 11 b1 c11k p )( s b ) 1 1c4k p14 1b11 c4ki s b ( ) 14 1c4ki s 14 11s b1 c11k p s 1 1 ( 1 b11c 4k p ) s

152 det det [ s ] I - A cl = b c k s b b c k c k s c k b b c k 1 11 i p14 11 i1 11 i i14 s + s b c k s b c k s + b c k s b c k b c k s p1 1 4 p p p1 1 4 p14 b c k s b c k s + b c k s b c k b c k i i i i s b c k s [ si - A cl ] 4 + ( 1) s p14 = 3 ( 11 b1 c11k p b ) 1 1c4k p s 14 ( p ) p1 1 4 p i p i1 ( b1b 1c4k p c ) 14 11ki 1 1b11c 4ki b 14 1c4ki b1c4k i b 14 1c11k p s 1 ( c11k i b1b 1c4ki ) b c k b c k b c k b c k + b c k b c k s The eqution is of the form, 4 3 [ si - A cl ] = C1s + Cs + C3s + C4s + C5 = ( s p1 )( s p )( s + ζω + ω ) det n n p1 s The coefficients of the chrcteristic polynomil, in terms of the modl properties tht we specify for the design (two integrtor poles, the dmping rtio nd nturl frequency), re shown below. C = 1 1 n ( ) ζωn ( ) ( ) C = ζω p + p 1 C = ω p + p + p p 3 n 1 1 C = ω p + p + ζω p p C 4 n 1 n 1 = ω p p 5 n 1 The coefficients of the chrcteristic polynomil, in terms of the system prmeters, re shown below

153 C = 1 1 ( p ) 1 p14 ( p ) 14 p1 p14 i14 p14 i1 ( p ) 14 i1 i14 i14 i14 p1 ( c ki b b c k ) C = b c k b c k C = b c k b c k b c k b c k + b c k b c k C = b b c k c k b c k + b c k b c k b c k C = i 14 Since we hve four gins nd four coefficients, we re ble to rbitrrily plce the poles. Becuse the lgebr is complex in the solution of these equtions, we simplify the equtions with following substitutions. C = C + 11 C = C C = C 4 4 C = C 5 5 x x x x x = + b c = + b c 1 4 = b c = b c b c = + b c x x x = + b c = + b c = + b b c c x = + b c b c x x = + b c b c = c b b c Finlly, we hve the following system of equtions. C = 1 1 ( p ) 1 p14 ( p ) 14 p1 p14 i14 p14 i1 ( p ) 14 i1 i14 i14 p1 ( x ki ki ) C = x k + x k 1 C = x k + x k k + x k + x k + x k C = x k k + x k + x k k C = To obtin solution, we use lgebr to extrct s much informtion from the system of equtions s possible nd then employ simple itertive routine. This method proved to be stisfctory nd is now outlined. First, we cn simplify the C 3 eqution. (( ) p p p i i ) C = x + x k + x k k + x k + x k dding C to C 3, -114-

154 C xk p14 C = ( x + x ) k + x k + x k + x k p14 4 p14 5 i14 7 i1 x1 x 4 C C = ( x + x ) k + ( C k x k ) + x + x k p14 p14 p i 1 x1 x11k i1 x 4 C C = ( x + x ) k + ( C k x k ) + x + x k p14 p14 p i 1 x1 x11k i1 We end up with qudrtic eqution in terms of k p where k 14 i1 coefficients s shown in Eqution (3.10). is term in the x 4 C 5 xk p ( x ) x6 + C k p + C 14 3 x5 + x7ki = 0 1 x 1 x11k i1 (4.6) Using C 5, C 4, nd C we cn get ki in terms of k 1 p s shown in Eqution (3.1). 14 C C 5 xk p 14 C 4 = x8k p + x 14 9ki + x 14 10ki 14 x11k i x 14 1 C x k x C k x k k C x k ( ) 5 8 p i = i + 14 i14 p14 x11 x1 C x k x C C k = + x k + k x k k 5 8 p i14 9 i14 i14 p14 i14 x11 x1 C x k x C + + ki x 14 k p k 14 i C 14 4ki = p14 10 x9ki 14 x11 x1 x10c C 5x8k p14 x9 + xk p14 ki C 14 4ki + = 0 14 x1 x11 k i14 x10c C 5x8k C 4 ± C 4 4 x9 + xk p14 x1 x11 = x10c x9 + xk p14 x1 p

155 k i1 x10c C 5x8k C 4 ± C 4 4 x9 + xk p14 C x 5 1 x11 = x11 x10c x9 + xk p14 x1 p14 (4.7) 4 Since we know tht k p is lwys positive nd on the order of 10, simple itertion 14 lgorithm ws developed tht begins with n initil k p, clcultes k 14 i using Eqution 1 (3.13), nd then tests the solution using Eqution (3.14). Generlly, the method yields two sets of workble gins. The solution chosen of the two workble sets is the solution with the lowest throttle gins. This wy the throttle is modulted the lest

156 5. The Supporting Functionl Logic of the Longitudinl Control System In the previous chpter, we developed the feedbck control lgorithms tht stbilize the ircrft nd drive it to the desired condition. The result ws our bsic, liner control model, eqution (4.), which cn be expressed in mtrix/opertor nottion s, u = Ke (5.8) where u is the control vector (in our cse, lift coefficient nd thrust), e is the output error vector (which contins the indicted irspeed, Mch number, ltitude, nd ltitude rte), nd K is the constnt mtrix opertor of proportionl, integrl, nd feedbck gins. Our gol is to squeeze stndrd piloting strtegies into tht bsic, liner model nd use it to control our non-liner dynmics. But bsic, liner control model is too simple to cpture the complexities of piloting n ircrft throughout its entire flight envelope. In generl, pilot will provide different control inputs for given error vector bsed on the flight regime. As n illustrtion of the different piloting strtegies used in different flight regimes, consider the following. During tke-off, the pilot sets tke-off thrust nd nominl stick loction nd holds them constnt until the ircrft reches its rottion speed. In climb, the pilot fixes the thrust t climb thrust nd uses the control stick to cpture either irspeed or ltitude rte. In descent, the pilot fixes the thrust t descent thrust nd uses the control stick to cpture either irspeed or ltitude rte. Our liner controller my be ble to use the sme lift coefficient controller s for climbs, but the fixed thrust vlue is different. In stedy, level flight, the pilot uses the throttle to hold the irspeed t the desired vlue nd the control stick to hold the ltitude. So there re cses in which there is no modultion of control inputs t ll, cses in which only one control prmeter is modulted, nd cses in which they re both modulted. Since different flight regimes require different control strtegies, we will need to develop bsic, liner controller for ech flight regime. The purpose of this chpter is to define these different flight regimes, or regions, nd the conditions tht bound them, nd to develop the lgorithms for determining the desired condition within those regions. In the previous chpter, we introduced the concept of the desired output. Becuse our system uses multiple regions nd multiple controllers, we need to differentite between the desired output tht ech mthemticl controller sees nd the commnded output tht

157 the system (in the rel world this would be the pilot) is trying to ttin. The commnded output, y c, typiclly comes from the user interfce. Ech region will use supporting functionl logic to select time-vrying desired output, y d (t), tht defines pth to the commnded output, y c. The desired output is then pssed to the region s controller nd the controller determines the control inputs needed to follow tht pth. 5.1 Control Strtegies In this section, we briefly revisit the discussions of the previous chpter on the different control strtegies used for different phses of flight nd explicitly stte the control lw used for ech Altitude Rte Controller - Feeding Bck Altitude Rte Only A pilot will ccomplish speed chnges in level flight typiclly by fixing the throttle ppropritely (dvnced for speed increse, reduced for speed decrese) nd using the control stick to mintin level flight. To insure tht the ircrft stys in level flight, the pilot will rely primrily on the ltimeter nd the ttitude indictor. In sense, he is using feedbck control on his ltitude rte (with desired rte of zero) nd llowing the ircrft to ccelerte (or decelerte). Alterntively, pilot cn use the control stick to mintin constnt, non-zero ltitude rte while keeping the throttle fixed. He cn lso follow vrying ltitude rte tht my be determined by specific geometric pth (i.e., following glide slope or flight pth ngle). The pilot cn lso use the control stick to gin blnce of irspeed ccelertion nd verticl speed. This mounts to dividing the chnging totl energy between chnges in potentil nd kinetic energy. It is lso possible to exchnge potentil nd kinetic energy without ffecting the ircrft s totl energy much t ll (e.g., descending nd ccelerting). By djusting the control stick, the pilot cn control how much energy goes to chnging irspeed nd how much goes to chnging ltitude. All of these regimes re exmples of ltitude rte control. The TGF simultor cn use ltitude-rte-only feedbck to follow constnt or time-vrying ltitude rte using this control strtegy. All tht remins is for the region (i.e., the supporting functionl logic) to tell the controller the desired ltitude rte, ḣ d. For ltitude-rte-only feedbck, eqution (5.8) becomes ( ) ( ( ) ( )) ( ) C t = k h t h t + k h t h t dt k h t (5.9) ( ) ( ) ( ) L p14 d i14 d b

158 5.1. Speed Controller - Feeding Bck Speed Only Climbs nd descents t constnt irspeed re lso typiclly ccomplished without modultion of the throttle. For climbs, the throttle is dvnced to the desired climb power, nd for descents, the power is reduced to descent power. The control stick is then djusted to mintin the proper irspeed while the ltitude is llowed to chnge. The pilot uses informtion from the irspeed (or Mch) indictor to djust the control stick to mintin speed. This type of control is fundmentlly different from the speed chnge in level flight becuse the control stick is now controlling speed insted of ltitude rte. The ltitude rte is left to vry ccording to the throttle setting. The TGF simultor cn use speed-only feedbck to ccomplish ltitude chnges using this control strtegy. All tht remins is for the region (i.e., the supporting functionl logic) to tell the controller the desired speed, V IAS, or Mch number, M. For indicted irspeed feedbck, eqution (5.8) becomes ( ) ( ( ) ( )) ( ) ( ) ( ) ( ) CL t = k p V 11 IAS t V d IAS t + ki V 11 IAS t V d IAS t dt kb V 11 IAS t (5.30) nd for Mch feedbck, eqution (5.8) becomes ( ) = ( ( ) ( )) + ( ( ) ( )) ( ) CL t k p M 1 d t M t ki M 1 d t M t dt kb M t (5.31) Dul Controller - Feeding Bck Speed nd Altitude Rte A pilot will mintin stedy, level flight typiclly by using the control stick to fly level nd djusting the throttle to hold the desired speed. To insure tht the ircrft stys in level flight, the pilot will monitor the ltimeter nd the ttitude indictor. To insure tht the ircrft holds speed, the pilot will monitor the irspeed (or Mch) indictor. The TGF simultor cn use this strtegy to mintin stedy, level flight, to fly constnt verticl speed t specified irspeed, or to follow specific speed-ltitude profile, s on pproch nd lnding. All tht remins is for the region (i.e., the supporting functionl logic) to tell the controller the desired speed nd ltitude rte. For IAS-bsed control, eqution (5.8) becomes ( ) ( ( ) ( )) ( ) ( ) ( ( ) ( )) ( ) d d ( ) ( ) ( ) C t = k h t h t + k h t h t dt k h t L p14 d i14 d b14 ( ) ( ) ( ) T t = k V t V t + k V t V t dt k V t p1 IAS IAS i1 IAS IAS b1 IAS (5.3) nd for Mch feedbck, eqution (5.8) becomes

159 ( ) ( ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) C t = k h t h t + k h t h t dt k h t L p14 d i14 d b14 ( ) ( ) ( ) ( ) ( ) T t = k M t M t + k M t M t dt k M t p d i d b (5.33) Altitude Cpture Cpturing ltitude is fundmentl function tht the longitudinl dynmics must perform; however, ltitude is never feedbck prmeter directly in the controllers designed for our ADM. Insted ltitude rte is commnded in mnner such tht ltitude cpture is obtined. Initilly it ws not cler tht this ws the best solution to the problem. In fct mny direct ltitude feedbck strtegies were tried, nd the stte spce model still llows for ltitude feedbck. However, insted of direct ltitude feedbck, the ltitude error is used to determine n pproprite vlue for ḣ d, the desired ltitude rte. The reson for this decision is bsed primrily on the need for smooth trnsition between control region tht does not control ltitude rte, nd one tht does. Altitude Error (ft) Descent Rte (fpm) Distnce Trveled (nm) Figure 5.1: Illustrtion of n ircrft cpturing n ltitude For regions in which the ltitude is cptured, the desired ltitude rte is clculted by pplying gin to the difference between the desired ltitude nd the ctul ltitude. d h ( ) h = K h h (5.34) d We set nominl vlue of K 7 1 h =. Figure 5.1 shows how the descent rte vries min over the finl 70 feet nd the smooth cpture of the ltitude when K = 7 h. There is still problem to contend with t region trnsition. Consider n ircrft in, descent t idle throttle t specified irspeed s shown in Figure 5.. The ircrft

160 descends t whtever rte is required to mintin the commnded irspeed with n idle throttle; consequently, there is no direct control over the rte of descent. There is likely to be n undesired trnsient when the ircrft mkes the trnsition to region with controlled ltitude rte. The cuse of the undesirble trnsient is the fct tht the ircrft s ltitude rte upon entering the new region nd the desired ltitude rte derived from Eqution (5.34) will not necessrily mtch. The mismtch cuses the control lw to drive the initil error to zero with excessive control inputs. Figure 5. shows the mismtch nd the sudden increse in the descent rte. To solve this problem, the initil desired ltitude rte is set equl to the ircrft s current ltitude rte. The mesured (current) ltitude nd ltitude rte re used to clculte the vlue of K by rerrnging Eqution (5.34). h Descent Rte K h initil = current ( h h) d h current Descent Profile (5.35) 600 Before region trnsition 600 Before region trnsition Boundry of control Altitude Error (ft) h current Kh = initil ( hd h) current Descent Rte (fpm) 350 h cu rren t K h = in itil ( hd h ) current Distnce Trveled (nm) Figure 5.: Altitude Rte During Region Trnsition When the ircrft is within 70 feet of the trget ltitude, K is returned to its nominl h vlue of seven regrdless of the initil clculted vlue. Seventy feet is chosen becuse it yields n ltitude rte ner 500 fpm, stndrd vlue for ltitude cpture estblished by Federl Air Regultions. This chnge in K does yield n unrelistic jump in the ltitude rte just before ltitude cpture but it is much smller thn t region trnsition nd it ensures tht the ltitude is cptured in timely mnner. It lso prevents the previous descent or climb from ffecting the continuing cruise performnce of the ircrft tht h

161 would come from preserving either n unusully low or high logic for clculting K h K. Figure 5.3 shows the h Speed Cpture The feedbck control lws were derived for smll errors. In the stedy, level flight control region, which uses speed control, the speed error is never more thn 10 knots. This is suitble speed to keep the control inputs from being excessive. But in the climb nd descent regions, which lso use speed control, it is possible for the speed error to be 100 knots or more. Our controller would produce excessive control inputs to correct this error. To void these excessive control inputs, we must devise vrying desired speed profile tht will produce cceptble control inputs nd speed rtes. A constnt desired ccelertion of one knot per second ws chosen for its simplicity nd suitbility in producing n cceptble trnsient response. The initil desired speed upon entering speed-controlled region is the ctul speed. K h In ltitude rte control region? NO K = 1 h YES K = 1 h YES K h initil = current ( h h) d h current NO e 3 <70 YES K = 7 h NO New K h

162 Figure 5.3. Flow digrm for clculting K h V IASd 0 = V IAS V = V + V t IASd IASd IASd (5.36) where ( 1 kt ) * sign ( ) c V = V V sec IASd IAS IAS The function, sign(x), returns +1 if x 0 nd -1 otherwise. Once the indicted irspeed is within five knots of the commnded vlue, the desired speed is simply the commnded speed. For Mch control, the equtions for desired Mch re similr. A constnt desired ccelertion of 0.0 Mch per second is selected. It is equivlent to n ccelertion of one knot per second t 5,000 feet ltitude nd 300 knots. M d 0 = M M = M + M t d d d (5.37) where ( ) * sign ( c ) M d = M M sec 5. Dividing the Flight Envelope into Regions Becuse of the need for different controllers nd different supporting logic in different regimes of the flight envelope, we need to divide the flight envelope into regions. Ech region s control lw, supporting logic, nd desired output reflect the pilot s decision logic in bringing the ircrft from its ctul stte to its desired stte. All of the supporting functionl logic for the longitudinl control system centers on the concept of the speed-ltitude plne. The speed-ltitude plne ws used by Muki (199) during the development of Pseudocontrol, the originl control system developed for the Pseudo Aircrft Simultion (PAS) system developed for NASA Ames. The speed-ltitude plne hs been revisited nd dpted for the TGF project nd hs undergone extensive modifiction since it ws used in PAS

163 Altitude error (ft) 0 0 IAS error (kts) Figure 5.4. The Speed-Altitude Plne 5..1 The Speed-Altitude Plne The speed-ltitude plne is used to represent the ircrft s ctul position reltive to its commnded position, with ltitude error plotted on the verticl xis nd speed error plotted on the horizontl xis. It llows us to split the ircrft s en route flight envelope into different control regions. A digrm of the speed-ltitude plne is shown in Figure 5.4. The bsic purpose of the speed-ltitude plne is to emulte the wy pilot mkes decisions bout flying n ircrft. We define n re in the immedite vicinity of the commnded condition in which the ircrft is in stedy, level flight. We set up speed error nd ltitude error bounds to define this stedy, level flight region. We then need to define wht to do outside of these bounds in order to bring the ircrft within them. The speed-ltitude plne provides mens of generlizing these reltionships nd lso provides insight into wht the throttle setting should be bsed on the ircrft s current energy level The Line of Constnt Energy The line of constnt energy illustrtes how n ircrft cn hve the sme energy level t different speeds nd ltitudes. It is mens of determining whether the ircrft is low on - 1 -

164 energy or high on energy with respect to its commnded condition; tht is, does the ircrft need to dd thrust or reduce thrust to ttin the commnded condition. Originlly, the line of constnt energy ws bsed on true energy clcultions for the ircrft but s it evolved it becme simply gross representtion of the ircrft s energy. This section discusses the evolution of the line of constnt energy. The totl energy of n ircrft is the sum of its potentil nd kinetic energies. The terms re defined s follows: E: The totl ircrft energy m: The ircrft mss g: The grvittionl ccelertion h: The ltitude V : The true irspeed E = K.E. + P.E. = mgh + ½ mv The eqution cn lso be written in terms of the energy per unit mss. e = gh + ½ V (5.38) The energy tht the ircrft would hve t some commnded stte cn be written similrly, e = gh + V (5.39) 1 c c c where e c is the totl energy t the commnded stte, h c is the commnded ltitude, nd V c is the commnded true irspeed. Notice tht it is quite possible for e c = e without the ircrft ctully being t the commnded stte. Tht is to sy, the ircrft could hve the right mount of energy but be either fst nd low or high nd slow. This is illustrted in Figure 5.5 which shows the line of constnt energy. The x- nd y-xes on Figure 5.5 re the speed error nd ltitude error from some commnded stte shown t the point (0,0). If the ircrft s current stte lies on the constnt energy line, the ircrft lredy hs enough energy to ttin the commnded stte. Therefore, the mount of throttle djustment needed is miniml. However, if the ircrft lies below the constnt energy line, the ircrft needs energy to ttin the commnded stte. Likewise, if the ircrft is bove the energy line, the ircrft hs excess energy nd must lose energy to chieve the commnded stte

165 The constnt energy curve is prbol, s illustrted by the qudrtic reltionship between ltitude nd speed in eqution (5.38). For prcticl control implementtion, liner pproximtion to the constnt energy curve in the speed-ltitude plne is used, s shown in Figure 5.5. However, even this pproximtion is simplified becuse of trnsition problems between regions. The finl form of the line of constnt energy pproximtes the ctul energy curve by forming digonl cross the stedy, level flight region, s shown in Figure 5.6. Altitude Error (ft) 0 Actul Constnt Energy Curve Approximted Constnt Energy Line 0 Speed Error (knots) Figure 5.5. The constnt energy line on the speed-ltitude plne

166 h error Region 7 line of constnt energy pproximtion h error SP error SP error Figure 5.6. Illustrtion of the pproximtion for the constnt energy line using the digonl cut cross the stedy, level flight region (region 7) of the speed-ltitude plne Bounding the Speed-Altitude Plne The speed error bound on the stedy, level flight region is 10 knots. The simultion engineers chose this vlue becuse it is bound typiclly used in rel flight; n ircrft within ten knots of controller-ssigned speed is considered to be in complince with tht speed. The simultion engineers chose the ltitude error bounds so tht the slope of the digonl of the stedy, level flight region is the sme s the slope of the constnt energy line in the vicinity of flight condition tht is representtive of en route flight: 300 knots 5,000 ft. At 5,000 feet ltitude in the stndrd tmosphere, 300 knots IAS corresponds to 78 fps true irspeed (TAS) nd ten knot IAS speed difference corresponds to bout fps speed difference in TAS. The slope of the constnt energy curve t this point is given by the derivtive of eqution (5.38). Assuming tht grvity is constnt, 1 d ( e) = d ( gh) + d V de = g dh + V dv (5.40) Using discrete nottion, nd substituting in the vlues corresponding to our speed chnge nd flight condition, we cn clculte the corresponding ltitude chnge tht would keep the totl energy constnt

167 e = g h + V V ( 0) = ( 3. ft ) h + ( 78 ft )( ft s s ) s h = 497 ft s We choose to round to 500 feet for the ltitude error. The prmeters needed to define the regions of the speed-ltitude plne re the speed error, ltitude error, nd slope of the constnt energy line. These prmeters nominlly re set s follows: h error : This is the ltitude error used to bound the stedy, level flight region nd, consequently, the slope of the constnt energy line of the speed-ltitude plne. The nominl vlue is 500 feet. SP error : This is the speed error used to bound the stedy, level flight region. It needs to be defined in terms of knots for IAS-bsed control nd in terms of Mch number for Mch-bsed control. IAS-bsed: SP error = IAS error = 10 knots. Aircrft re typiclly expected to hold their speeds within 10 knots. Mch-bsed: SP error = M error = 0.0. This is the Mch error used to bound the stedy, level flight region in the Mch speed-ltitude plne. The vlue corresponds to 10 knot IAS speed chnge t 300 knots IAS nd 5,000 feet in the stndrd tmosphere The Regions of the Speed-Altitude Plne The speed-ltitude plne is divided into nine different regions. Ech region hs different combintion of control lw nd supporting functionl logic. The speed error on the speed-ltitude plne is either represented in knots of indicted irspeed or in Mch number. Figure 5.7 shows the regions of the speed-ltitude plne in terms of indicted irspeed. The regions re enumerted below with their vrious functions. Stedy, Level Flight (Region 7): In Region 7, the ircrft is sufficiently close to converging on desired stte. The pilot modultes the control stick nd the throttle simultneously to cpture the desired stte. Climbing & Accelerting (Region 1): In Region 1, the ircrft is low nd slow nd is, therefore, low on energy. The throttle is set to full nd the pilot climbs nd ccelertes the ircrft, using the control stick to mintin blnce of irspeed ccelertion nd climb rte. Level Accelertion (Region ): In Region, the ircrft is slow enough to be low on energy; therefore, the throttle is set to full. The pilot uses the control stick to cpture nd mintin the desired ltitude while ccelerting into Region

168 h i g h Altitude error (ft) -h error 0 h error s l o w 1 7 l o 6 w f s t 5 55 IAS error 0 -IASerror V IAS error (kts) Figure 5.7. The speed-ltitude plne in terms of indicted irspeed Descending & Accelerting (Region 5): This is n energy trde region, mening tht the ircrft is trding potentil energy for kinetic energy. The throttle is set to idle nd lift coefficient is used to cpture desired ltitude rte tht is constnt multiple of the ltitude error. Stedy Descent (Region 3): In Region 3, the ircrft is high enough to hve excess energy; therefore, the throttle is set to idle. The pilot uses the control stick to cpture nd mintin the desired speed while descending into Region 7. Descending & Decelerting (Region 4): In Region 4, the ircrft is high nd fst nd is, therefore, high on energy. The throttle is set to idle nd the pilot descends nd decelertes the ircrft, using the control stick to mintin blnce of irspeed decelertion nd descent rte tht is weighted towrds descending. Level Decelertion (Region 5): In Region 5, the ircrft is fst enough to hve excess energy; therefore, the throttle is set to idle. The pilot uses the control stick to cpture nd mintin the desired ltitude while decelerting into Region 7. Climbing & Decelerting (Region 55): This is n energy trde region, mening tht the ircrft is trding kinetic energy for potentil energy. The throttle is set to idle nd

169 lift coefficient is used to cpture desired ltitude rte tht is constnt multiple of the ltitude error. Stedy Climb (Region 6): In Region 6, the ircrft is low enough to be low on energy; therefore, the throttle is set to full. The pilot uses the control stick to cpture nd mintin the desired speed while climbing into Region 7. The reder should notice similrity between the bsic strtegy in ech of the regions nd the discussion of Section 4.1.1, where the bsic piloting strtegies for different types of mneuvers re outlined. The speed-ltitude plne is mens of mechnizing the control strtegy tht pilot would use depending on the ircrft s stte reltive to the commnded stte Region Mngement in the Speed-Altitude Plne The error between the ircrft s commnded nd ctul sttes is compred to the error bounds of the speed-ltitude plne. This determines the ircrft s control region in the speed-ltitude plne. The bsolute errors between the ircrft s ctul nd commnded sttes, e 1 through e 4, re defined below. The terms re defined s follows: e1 = VIASc VIAS (5.41) e = M c M (5.4) e3 = hc h (5.43) e = h h (5.44) 4 c V IASc : The commnded indicted irspeed (kts) V IAS : The ctul indicted irspeed (kts) M c : The commnded Mch number M: The ctul Mch number h c : The commnded ltitude (ft) h: The ctul ltitude (ft) ḣ : The commnded ltitude rte (ft/min) c ḣ : The ctul ltitude rte (ft/min) We cn use Boolen expressions to define four true-flse prmeters tht coincide with the bounds nd definitions of the speed-ltitude plne. These re used to id the flow lgorithm (illustrted below) tht determines which region logic to use. For exmple, if the commnded speed is more thn ten knots below the ctul speed, the ircrft is defined to be fst. These speed-ltitude plne Boolens re defined in Tble

170 LOW 3 error e > h HIGH 3 error e < h SLOW s error e > SP FAST s error e < SP LOWENERGY e > ( m e ) 3 * s Tble 5.1: Speed - Altitude Plne Boolens This section describes the control strtegies nd supporting functionl logic used in ech of the regions of the speed-ltitude plne Stedy, Level Flight (Region 7) Region 7 is the most highly used region becuse it represents the ircrft t or ner stedy, level flight. The ircrft is close enough to desired stte tht our liner controller cn cpture tht stte without excessive control inputs. Both the lift coefficient nd the thrust re controlled. The control lw is given by equtions (5.3) nd (5.33) for IASbsed nd Mch-bsed control, respectively. The desired ltitude rte is given by eqution (5.34) nd the desired speed is given by equtions (5.36) nd (5.37) for IASbsed nd Mch-bsed control, respectively Accelerting (Region ) In Region, the ircrft is slow enough to be lowenergy, but it my be bove or below its commnded ltitude. Becuse the ircrft is lowenergy, the throttle is dvnced to full throttle. The system uses the lift coefficient to cpture nd mintin the desired ltitude rte while ccelerting into Region 7. The control lw is given by eqution (5.9) with the desired ltitude rte given by eqution (5.34) (the sme s for Region 7) Descending & Accelerting (Region 5) Region 5 is n energy trde region. Throttle is set to idle nd the desired ltitude rte is constnt multiple of the ltitude error., The control lw is given by eqution (5.9) with the desired ltitude rte given by eqution (5.34). K must be lrge enough so tht the desired descent rte is lrge enough to permit ccelertion with n idle throttle. If the descent rte is too smll, the energy loss of negtive (T-D) (becuse of the idle throttle) will force decelertion. A vlue of K = 1 is suggested. h An lterntive to setting constnt idle throttle is to set the throttle so tht the thrust produced is equl to the drg. With no energy being lost, the ircrft would be forced to ccelerte while cpturing the desired descent rte. From eqution (.59), with T=D, h

171 h V = g sin γ = g V The problem from softwre engineering stndpoint is tht the drg is not redily vilble to the throttle controller. However, the bove eqution cn be used to clculte desired ccelertion nd new throttle controller could be developed tht cptures tht desired ccelertion. This pproch hs shown promise in preliminry tests, but hs not been implemented. It requires dul controller, with lift coefficient controlling ltitude rte nd throttle controlling ccelertion Decelerting (Region 5) In Region 5, the ircrft is fst enough to be not lowenergy, but it my be bove or below its commnded ltitude. It hs more energy thn it needs so the throttle is reduced to idle. It is the sme s Region except for the throttle setting. The system uses the lift coefficient to cpture nd mintin the desired ltitude rte while decelerting into Region 7. The control lw is given by eqution (5.9) with the desired ltitude rte given by eqution (5.34) (the sme s for Region 7) Climbing & Decelerting (Region 55) Region 55 is subset of Region 5 tht uses constnt descent rte; tht is, the greter of its current ltitude rte nd 500 fpm. ( ) h = mx h,500 fpm d Region 5 is n energy trde region. Throttle is set to mx full nd the desired ltitude rte is constnt multiple of the ltitude error., The control lw is given by eqution (5.9) with the desired ltitude rte given by eqution (5.34). K must be lrge enough h so tht the desired climb rte is lrge enough to permit decelertion with full throttle. If the climb rte is too smll, the energy gin of positive (T-D) (becuse of the full throttle) will force n ccelertion. A vlue of K = 1 is suggested. An lterntive to setting constnt full throttle is to set the throttle so tht the thrust produced is equl to the drg. With no energy being lost, the ircrft would be forced to decelerte while cpturing the desired climb rte. From eqution (.59), with T=D, h V = g sin γ = g V The problem from softwre engineering stndpoint is tht the drg is not redily vilble to the throttle controller. However, the bove eqution cn be used to clculte h

172 desired ccelertion nd new throttle controller could be developed tht cptures tht desired ccelertion. This pproch hs shown promise in preliminry tests, but hs not been implemented. It requires dul controller, with lift coefficient controlling ltitude rte nd throttle controlling ccelertion Descending (Region 3) In Region 3, the ircrft is high enough to be not lowenergy. It my be bove or below its desired speed. Becuse the ircrft is not lowenergy, the thrust is set to idle. The ircrft uses lift coefficient control to cpture its desired speed while descending into Region 7. The control lw is given by eqution (5.30) nd eqution (5.31) for IAS-bsed nd Mch-bsed control, respectively. The desired speed is given by equtions (5.36) nd (5.37) for IAS-bsed nd Mch-bsed control, respectively Climbing (Region 6) In Region 6, the ircrft is low enough to be lowenergy. It my be bove or below its desired speed. Becuse the ircrft is lowenergy, the thrust is set to full. The ircrft uses lift coefficient control to cpture its desired speed while climbing into Region 7. The control lw is given by eqution (5.30) nd eqution (5.31) for IAS-bsed nd Mchbsed control, respectively. The desired speed is given by equtions (5.36) nd (5.37) for IAS-bsed nd Mch-bsed control, respectively Climbing & Accelerting (Region 1) In Region 1, the ircrft is low nd slow nd is, therefore, lowenergy. The throttle is set to full nd the system uses ltitude rte feedbck to mintin blnce of irspeed ccelertion nd climb rte tht is weighted towrds ccelertion. The control lw is given by eqution (5.9). Since the system uses ltitude rte feedbck, we need to determine the desired ltitude rte tht will yield the blnce of irspeed ccelertion nd climb rte mentioned bove. With the throttle set to full, the ircrft s thrust is typiclly greter thn its drg. This mens tht the ircrft s totl energy is incresing. We need to determine how much of this energy increse goes towrds ccelerting nd how much goes towrds climbing. An eqution relting ccelertion nd ltitude rte is given by eqution (.59), T D T D h V = g sinγ = g (.59) m m V which cn be written in the form of the chnging energy. ( T D) V = gh + VV m (5.45)

173 Note tht this eqution is presented in BADA s eqution (3.1-1). Note lso tht this eqution is the time derivtive of eqution (5.38). We cn rewrite this eqution to express n ircrft s desired ltitude rte in terms of n energy rtio, ER. h d ( T D) V ( T D) V V V mg ( 1+ ER) = = mg 1+ gh (5.46) The energy rtio, ER, is n expression of the rtio of chnging kinetic energy to chnging potentil energy. We cn use it to express how much of the chnging energy (i.e., the thrust energy being dded to the system) goes towrds chnging speed nd how much goes towrds chnging ltitude. BADA uses the sme pproch, but expresses eqution (5.46) in terms of n energy shre fctor s function of Mch number. For ccelertion in climb, BADA recommends n energy shre fctor tht corresponds to n energy rtio of ER =.3, nd tht is wht we use for our controller. With the desired ltitude rte determined by eqution (5.46) we cn use n ltitude rte controller to cpture tht ltitude rte Descending & Decelerting (Region 4) In Region 4, the ircrft is high nd fst nd is, therefore, not lowenergy. The throttle is set to idle nd the system uses ltitude rte feedbck to mintin blnce of irspeed decelertion nd descent rte tht is weighted towrds the descent. The control lw is given by eqution (5.9). Just s in Region 1, we need to determine the desired ltitude rte tht will yield the blnce of chnging irspeed nd ltitude. For decelertion in descent, BADA recommends n energy shre fctor tht corresponds to n energy rtio of ER =.3; however, our tests hve shown tht n energy rtio of ER = 1.0 is more suitble. To determine the desired ltitude rte for Region 4, we use eqution (5.46) with the throttle set to idle thrust nd the energy rtio set to ER = Constnt Verticl Speed (CVS) Mneuvers (Formerly Region 8) When verticl speed is commnded, the ircrft is performing constnt verticl speed (CVS) mneuver. The mneuver is completed when the ircrft becomes not fst nd not slow. There re three seprte CVS mneuvers: idle-thrust CVS, mx-thrust CVS, nd dul-control CVS. If the speed error is within SP error, the ircrft is in dul-control CVS. Otherwise, if the ltitude error is positive, the ircrft is in mx-thrust CVS mneuver nd if the ltitude error is negtive, the ircrft is in n idle-thrust CVS mneuver

174 Dul-Control CVS Mneuver The control lw is given by eqution (5.3) nd eqution (5.33) for IAS-bsed nd Mchbsed control, respectively the sme s for Region 7. The desired speed is given by equtions (5.36) nd (5.37) for IAS-bsed nd Mch-bsed control, respectively. The desired ltitude rte is s commnded. Once in dul-control CVS, if the ircrft becomes slow, tht mens it is losing speed. It is expected tht this cn hppen only in climbing mneuver in which the ircrft does not hve enough thrust or power to mintin irspeed nd verticl speed concurrently. If left in the CVS climb, the speed will only continue to fll. In this cse, the ircrft must brek out of the CVS mneuver nd into Mch or CAS climb. Once in dul-control CVS, if the ircrft becomes fst, tht mens it is gining speed. It is expected tht this cn hppen only in descent mneuver in which the ircrft does not hve enough drg to mintin irspeed nd verticl speed concurrently. In this cse, there is n lterntive to breking out of the CVS mneuver: deploy speed brkes. If the speed error moves bck within ½ the SP error (i.e., five knots), speed brkes cn be turned off. It is possible for the ircrft to get in cycle of turning speed brkes on nd off n unrelistic scenrio but this should yield cceptble performnce. With speed brkes on, if the ircrft continues to gin speed (i.e., if the speed error exceeds 1 ½ times the SP error - i.e., 15 knots), it shll brek out of the CVS descent nd into Mch or CAS descent Idle-Thrust CVS Mneuver In this mneuver, the ircrft is not lowenergy nd is descending. The ltitude error is necessrily negtive (n entry condition) nd the commnded verticl speed is negtive. The speed is not controlled; the mneuver is entirely independent of speed behvior. The throttle is set to idle nd the system uses ltitude rte feedbck to mintin the commnded verticl speed. The control lw is given by eqution (5.9). This mneuver lso needs brek-out condition. It is necessry for the success of this mneuver tht the ircrft s energy rte is negtive. This is the purpose of setting idle thrust. As n exmple, if the commnded descent rte is so high tht the ircrft must ccelerte to extreme speeds in order to cpture the commnded descent rte, the ircrft s energy my ctully be incresing, even though the throttle is t idle. Therefore, if the energy increment, s given by eqution (5.40), is positive, the ircrft cn first deploy speed brkes. If the energy increment is still positive, the ircrft shll brek out of the CVS descent nd into Mch or CAS descent Mx-Thrust CVS Mneuver

175 In this mneuver, the ircrft is lowenergy nd is climbing. The ltitude error is necessrily positive (n entry condition) nd the commnded verticl speed is positive. The speed is not controlled; the mneuver is entirely independent of speed behvior. The throttle is set to mx climb thrust nd the system uses ltitude rte feedbck to mintin the commnded verticl speed. The control lw is given by eqution (5.9). This mneuver lso needs brek-out condition. It is necessry for the success of this mneuver tht the ircrft s energy rte is positive; this is the purpose of setting mx climb thrust. As n exmple, if the commnded climb rte is so high tht the ircrft must fll to dngerously low speeds in order to cpture the commnded climb rte, the ircrft s energy my ctully be decresing, even though the throttle is t its mximum. Therefore, if the energy increment, s given by eqution (5.40), is negtive, the ircrft shll brek out of the CVS climb nd into Mch or CAS climb. 5.. Tking off If n ircrft is initited in the simultion s tke-off, the guidnce module genertes sequence of legs tht will tke the ircrft to its low-ltitude cruising speed nd 6000 feet ltitude. Tht sequence is the tke-off ground run, rottion, lift-off, initil climb, nd cruise t 6000 feet ltitude. The tke-off ground run, rottion, nd lift-off legs re the tke-off legs. An ircrft cnnot enter either of these legs unless it is initited s tkeoff nd it must successively stisfy the boundry conditions of ech leg to progress into the initil climb. Additionlly, these legs use their own controllers nd re, therefore, given their own regions. The initil climb nd level cruise legs use the region mngement of the speed-ltitude plne Tke-Off Ground Run (Region 1) The tke-off ground run region uses n open-loop controller to ccelerte the ircrft until it is within V R of its rottion speed while keeping on the ground. The region specifies constnt control inputs. A lift coefficient of zero is specified to keep the ircrft from lifting off during the ground run nd tke-off thrust is specified to give the ircrft its mximum ccelertion. C L = 0 T = C T TTO mx (5.47) Boundry Condition: V IAS < V R - V R The constnt V R is defined the sme for ll ircrft. It represents the speed t which the simulted ircrft enters the rottion region, region 13. Conceptully, it is intended to represent the increment below rottions speed t which the pilot begins to pull bck on

176 the control stick to lift off the runwy. Its vlue in NextGen is selected so s to provide mple time for the controller of region 13 to rmp up to the ircrft s rottion lift coefficient. V R = 0 knots 5... Rottion (Region 13) The sole purpose of the rottion region is to rmp the lift coefficient up from zero to the ircrft s rottion lift coefficient s function of speed. At first look, this requires n open-loop controller tht rmps the lift coefficient between these two points s function of time, but this would require unique controller form. TGF progrmmers nd engineers decided tht it would be esiest to implement controller of the sme form s ll the other controllers, i.e., of the form of eqution (5.8). This requires the development of gins tht cuse the feedbck controller to mimic our desired open-loop behvior. Eqution (5.8) is shortened form of eqution (4.), which is rewritten here. ( t) ( t) ( t) ( t) t p 0 i b u = K e + K e dt - K y (4.) We desire lift coefficient behvior tht increses stedily. Since indicted irspeed increses with time during the ground run, we simplify our controller by selecting tht output only. We lso specify tht the desired speed is the rottion speed throughout the region. The single input, single output (SISO) system is then, ( ( )) ( ) ( ) ( ( )) ( ) L IAS p11 R IAS t 0 i11 R IAS b11 IAS C V t = k V V t + k V V t dt - k V t (5.48) The lift coefficient needed to lift the ircrft off the runwy t rottion speed is dubbed the rottion lift coefficient. C L R = W 1 ρv R S We would like the lift coefficient to chnge from zero t the beginning of region 13 to the rottion lift coefficient when the ircrft reches its rottion speed. At rottion speed, we wnt the rte of chnge of the lift coefficient with speed to fltten out so tht we don t overshoot it nd impose undo drg on the ircrft. We, therefore, specify the following conditions for our controller. C L (V R - V R ) =

177 d dv C L (V R ) = IAS C L R ( CL ( VR )) = 0 The bound of region 13 is defined by ircrft speed only. Once the ircrft speed is greter thn the rottion speed, control is pssed to region 14. A lower bound is not specified; once control is pssed from region 1, the ircrft stys within region 13 until the boundry condition is stisfied. Boundry Condition: V IAS V R To simplify integrtion, we ssume the indicted irspeed in region 13 hs constnt ccelertion; i.e., it is of the following form. IAS IAS IAS ( ) V = V t V = constnt (5.49) We trnsform equtions (5.48) nd (5.49) into functions of the indicted irspeed. 1 dt = V IAS dv IAS VIAS 1 C ( V ) = k ( V V ) + k ( V V ) dv - k V ki VIAS 11 C ( V ) = k ( V V ) + ( V V ) dv - k V L IAS p11 R IAS V i11 R IAS IAS b11 IAS R VR VIAS L IAS p11 R IAS R IAS IAS b11 IAS V VR VR IAS ( ) = ( ) C V k V V L IAS p11 R IAS ( ) ( ) k V V V + - V i11 IAS R R VR (( VIAS ) ( VR VR )) kb V 11 IAS IAS We now hve n eqution relting the lift coefficient to the indicted irspeed. ( ) ki V 11 IAS VR VR L ( IAS ) = p ( ) 11 R IAS - b11 IAS V IAS C V k V V k V (5.50) The derivtive of eqution (5.50) is,

178 d dv IAS k C V = k V V k i11 ( ( )) ( ) L IAS p11 IAS R b11 VIAS (5.51) The initil conditions plugged into equtions (5.50) nd (5.51) yield the following equtions. i11 ( ( )) = [ ] ([ ] ) ([ ] ) ki V 11 R VR VR VR L ( R R ) = p ( [ ]) 11 R R R - b11 R R V IAS ki V 11 R VR VR L ( R ) = p ( [ ]) [ ] 11 R R - b11 R = LR V IAS L R p11 R R b11 VIAS [ ] C V V k V V V k V V = 0 C V k V V k V C d dv IAS k C V k ( V V ) k = 0 [ ] k V k V V = 0 p11 R b11 R R k V i11 IAS V R - k V b11 R LR k k = 0 p11 b11 = C This system yields the following solution for our region 13 controller. k k k p11 i11 b11 = 0 V = = 0 C IAS LR VR (5.5) The region 13 controller is given by eqution (5.48) Lift-Off (Region 14) When the ircrft reches its rottion speed, the pilot pulls bck on the control stick to lift off the runwy, but mintins tke-off thrust nd tke-off flps. He uses this configurtion until he reches the mneuver ltitude. NextGen opertes similrly: control is pssed to region 14 t rottion speed nd the ircrft remins in region 14 control until it reches its mneuver ltitude, set to 400 feet AGL in NextGen. Therefore, we need controller tht returns tke-off thrust nd uses the lift coefficient to control speed. In this sense, region 14 is similr to region 6 (constnt speed climbs) except tht it uses tke-off thrust

179 T = C T V d T TO = V R mx Boundry Condition: h 400 ft AGL The controller is given by eqution (5.30). Additionlly, s soon s we enter this region, we cn rise the lnding ger, which mounts to removing the ger drg coefficient from the drg eqution. C D ger 5..3 Lnding Region Mngement An ircrft tht hs received the commnd to begin the lnding sequence will remin in the speed-ltitude plne regions until it reches its glide slope. Once the ircrft reches the glide slope, it begins the lnding sequence nd progresses through regions 9, 10, nd 11. A new commnded ltitude will move control bck to the speed-ltitude plne mnger (i.e., lnding is borted). It is ssumed tht Mch-bsed control is not used for ny lnding mneuver, so Mch-bsed controllers re not considered Prepring for Approch In mny cses, n ircrft is instructed to begin preprtions for pproch. An unrestricted ircrft my begin bleeding off energy by decelerting on its own. In the rel world, pilots will prepre for the pproch in this wy, prticulrly if they re fmilir with the pproch profile for their rrivl irport. Tbles Tble 5. nd Tble 5.3 outline the desired speed profile s function of ltitude for n unrestricted ircrft tht is on pproch. The schedule comes from section 4 of the BADA User Mnul. This schedule is used for ircrft before nd on the glide slope. For intermedite ltitudes, liner interpoltion is used to determine the desired speed. In the tble, V des,1 is the ircrft s preferred descent speed below 10,000 feet ltitude, vilble from BADA. Altitude (ft) Desired Speed, V IAS (kts) d 0 1 3V. stllld V + 5. stllld V stllld V + 0. stllld V stllld 6000 MIN(V des,1,0) 10,000 MIN(V des,1,50) Tble 5. Desired Descent Speed Schedule for Jet nd Turboprop Aircrft Prepring for Approch

180 Altitude (ft) Desired Speed, V IAS (kts) d 0 1 3V. stllld V + 5. stllld V stllld V + 0. stllld V stllld 10,000 V des,1 Tble 5.3 Desired Descent Speed Schedule for Piston Prepring for Approch Approch (Region 9) Becuse the ircrft is following the glide slope ( liner ltitude profile) nd preferred speed profile, it needs controller tht feeds bck speed nd ltitude rte. The control lw is given by eqution (5.3). The only difference is in the development of the desired output. Region 9 specifies the ircrft s speed profile s function of distnce from the runwy threshold. Alterntively, the ircrft cn be speed restricted with commnded speed. Either wy, region 9 specifies desired ltitude rte tht will hve the ircrft descend long the glide slope. The guidnce module defines the ltitude profile bsed on the ircrft s distnce from the glide slope ntenn of its ssigned runwy. The guidnce module defines distnce, d GS, s the distnce prllel to the runwy from the glide slope ntenn to the ircrft. The ircrft s desired ltitude is function of tht distnce nd the ngle of the glide slope ntenn s signl, γ GS. h = h = d tnγ (5.53) d GS GS GS The ircrft s desired ltitude rte must consider the locl glide slope ltitude s well s the ircrft s ltitude reltive to the glide slope. As in Region 7, eqution (5.34) is used to determine the ltitude tht would correct the ircrft s ltitude error, but we must lso consider tht the desired ltitude is chnging per eqution (5.53). We dd the derivtive of eqution (5.53) to eqution (5.34) to get the desired ltitude rte for Region 9. d h = K h h + d dt ( ) ( ) tn d h d GS GS γ

181 The time derivtive of the ircrft s distnce from the runwy threshold is the ircrft s groundspeed projected into the plne of the loclizer. If we ssume tht the ircrft cptures the loclizer t the sme time s the glide slope, we needn t concern ourselves with the ngle between the ircrft s ground pth nd the loclizer. This is resonble ssumption tht mkes the eqution much simpler. Additionlly, since the ground speed nd the distnce from the loclizer re defined in opposite directions, negtive sign results. h = K h h V γ (5.54) ( ) tn d h d G GS Lnding Flre (Region 10) When the ircrft is within 100 ft bove the runwy, it begins the lnding flre. The lnding flre region (region 10) is region of heightened control sensitivity to mtch the desired flre profile. The desired profile is qudrtic reltionship between height bove the runwy nd desired ltitude rte. It ws designed to be tngent to 3 glide slope t 100 ft bove the runwy nd to touch down on the runwy t 1 ft/s. ( ) ( ) h = h h 0. h h 1 d rwy rwy In this eqution, height bove the runwy, (h-h rwy ), is in feet nd the desired ltitude rte, ḣ d, is in ft/s. The desired speed is the ircrft s lnding speed, V LD. The control lw is given by eqution (5.3) Lnding Ground Run (Region 11) Once the ircrft touches down, it enters the lnding ground run region (region 11). In the lnding ground run, the thrust is throttled bck to idle, the lift coefficient is set to zero, nd the drg coefficient is incresed by the spoiler coefficient (to simulte the deployment of the spoiler). There is no feedbck control in this region. When the speed is 0 knots below the lnding speed, the ircrft is terminted from the simultion. C L = 0 T = C T Tidle mx (5.55) 5.3 Throttling in Regions 1 Through 6 When the thrust controller returns thrust tht differs from the current thrust, we must consider tht the thrust difference my not be vilble in one time step; prticulrly when the difference is the full rnge from idle to mx thrust, nd prticulrly when the engine is turbine engine. To ccount for this, simple spooling lg hs been dded to the controller to limit the mximum mount tht the thrust cn chnge in time step

182 T mx Tmx = t (5.56) klg The term k lg is the lg fctor. Conceptully, it is the time it tkes the engine to spool from zero thrust to mximum thrust. While spooling is not pplicble to piston engines, k lg is still used, just t much smller vlue. The current vlues for the different engines re presented below. turbofns: turboprops: pistons: k lg = 0 sec k lg = 5 sec k lg = sec We hve chosen to model engine spooling in the longitudinl control logic rther thn in the engine. Certinly, the rel turbofn ircrft hs spooling in the engine; however, spooling in the engine would introduce troublesome nonlinerity into the open loop dynmics. Such nonlinerities would undoubtedly expnd the control logic nd increse the number of gins needed, nd they would require extensive engineering effort to develop sufficient control lw. Furthermore, the loction of the spooling hs no bering on the perceived motion of the ircrft

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184 6. The Selection of Gins In Section 5, we determined feedbck control strtegies for ech of the regions in the speed-ltitude plne. Furthermore, much effort ws expended to develop mens to clculte cceptble gins for the different regions. The purpose for these computtionl methods ws two-fold. First, there were mny ircrft models to develop. Mnully determining gins using root locus or bode techniques would be time consuming nd would require skilled controls engineer. Secondly, it ws expected tht ech ircrft would need to hve schedule of gins to provide sufficient performnce throughout the ircrft s flight envelope. Therefore, ech ircrft would require gins to be clculted t mny different reference conditions. However, by crefully choosing the reference flight condition, it is possible to choose one set of gins tht will work for the ircrft s entire flight envelope. This section documents the decision process tht led to the finl conclusion tht gin scheduling would not be necessry. 6.1 The Aircrft s Flight Envelope For given ircrft weight, there re generlly two prmeters tht define the ircrft s flight envelope: ltitude nd irspeed. The flight envelope, shown in Figure 6.1, illustrtes how fst nd slow the ircrft cn fly nd how high the ircrft cn fly. Altitude Airspeed Figure 6.1. An exmple flight envelope Of course, s the ircrft s weight chnges, so do portions of the flight envelope of the ircrft; therefore, there re three prmeters tht ffect the ircrft s dynmic condition. In n effort to remove one of these prmeters, Section 3 demonstrted tht the flight envelope could be represented with true irspeed nd lift coefficient rther thn true -139-

185 irspeed, ltitude nd weight. The phugoid dynmics could be represented exclusively in terms of lift coefficient nd true irspeed if trimmed ircrft is ssumed. The only term tht vried with weight ws the thrust control derivtive, which contributes to the forcing function of the system. Therefore, the flight envelope cn be modeled s shown in Figure 6.. Only resonble trim conditions re considered. Figure 6. shows the high-weight/high-ltitude nd lowweight/low-ltitude stll condition boundries, which encompss n re lbeled s the useble rnge. This useble rnge represents ll possible trim conditions for the ircrft. Outside of tht rnge, the ircrft either must fly too fst, (i.e. fster thn Mch 0.9) or must hve ir denser thn se level. Note tht while the Region 7 controller my use this entire envelope, in the other regions, the envelope is bounded on the left by the mx L/D lift coefficient. The interesting observtion here is tht the useble re is rther smll when compred to the totl rnge of vible lift coefficients nd true irspeeds. One must sk the question, how much modl property vrition cn there be within this rnge? To nswer this question, the ircrft s flight envelope ws represented in yet nother wy. The locus of ll possible phugoid poles for the entire flight envelope ws plotted on single grph. The ircrft s speed, ltitude, nd weight were vried encompssing the entire rnge of resonble trim conditions. The results re shown in Figure 6.3. From the locus we cn see three extremes. Roughly, the points re: ± 0.056i ωn = rd ξ 0 19 p sec p =. trimmble irspeed ± 0.04i ωn = rd ξ 0 17 p sec p =. trimmble irspeed ± 0.099i ωn = rd ξ p sec p =. trimmble irspeed. High lift coefficient t highest Low lift coefficient t highest High lift coefficient t lowest While there is considerble vrition in frequency nd dmping, it is plin to see tht the ircrft needs n increse in dmping to hve cceptble modl properties

186 Figure 6.. The flight envelope for DC-9 in terms of CL nd True Airspeed Figure 6.3. The locus of Phugoid poles for the entire flight envelope of DC-9 in the clen configurtion -141-

187 6. Determining Acceptble Modl Properties In previous sections of the document, considerble effort ws mde to determine gins tht would yield desirble trnsient responses for the ircrft s phugoid dynmics. Generlly, determining wht is desirble is esy. Most dynmic systems re considered to be well behved if they hve dmping rtio of 0.7 nd frequency sufficiently high enough to remove trnsients quickly. A hrder question to nswer however is wht dynmic properties re sufficient. Tht is to sy while it my be obvious wht is desirble, it my not t ll be obvious how to determine the rnge of cceptble vlues. Different types of opertions my be more or less sensitive to poor uncontrolled dynmics. This section ddresses the question of just how precisely the phugoid dynmics must be held to specified set of modl properties. There is no precise nswer to this question. However, from observtion of the ircrft flying with vried modl properties, one cn conclude tht very wide rnge of properties is cceptble. First, consider the most importnt stte vribles in the longitudinl dynmics from the pilot s point of view: speed nd ltitude. The control system must be ble to drive the ircrft to different speeds nd ltitudes. Consider chnges in ltitude. The feedbck control systems don t use ltitude explicitly, but rther feedbck its derivtive, ltitude rte. Becuse of the integrl reltionship between the two, ltitude tends to be insensitive to smll trnsients in ltitude rte. Furthermore, the creful design of the desired output vector minimizes lrge errors tht would produce undesirble trnsients. In ddition, the stedy stte error between the desired output vector nd the ctul output vector is of little concern until the desired output reches the commnded output.. Zero stedy stte error is importnt only when the commnded output vlues hve been reched. This is in contrst, of course, to mission such s precise terrin following where the error in following time-vrying output vector would be criticl. While it is difficult to put rnge on cceptble modl properties, we cn stte some generl guidelines tht re bsed purely on observtion. These re: The dmping rtio of the mode is more importnt thn the frequency The dmping rtio cn vry from roughly nd chieve stisfctory performnce The frequency cn vry from 0. 1 rd sec to 1. 0 rd sec nd still yield cceptble results. This is rther lrge rnge which suggests tht control system with even meger performnce is likely to be cceptble. Most importntly, however, such wide rnge suggests tht single set of gins, if chosen crefully, could ccommodte the entire flight envelope of the ircrft. -14-

188 6.3 Choosing Single Reference Condition for Gin Clcultion To choose single reference condition tht would serve s representtive of the whole flight envelope, severl conclusions from Section 3 must be revisited. These conclusions re: ζ p = 1 L f D ; The dmping of the mode is inversely proportionl to the L/D rtio ω p g ; The frequency of the phugoid is inversely proportionl to the true V = F H G I K J irspeed. The lift coefficient tht yields the highest L/D rtio, C L ( L, hs the lowest dmping. D ) mx This occurs t the bottom of the thrust curve, i.e., t the bucket speed. Any vrition in the lift coefficient on either side of the thrust curve will yield decrese in the L/D rtio nd, therefore, n increse in phugoid dmping.. Furthermore, the lowest true irspeed will hve the highest phugoid frequency. Using this informtion the following reference condition ws chosen. Choose the trim condition for the mximum L/D rtio Using the lift coefficient for mximum L/D, trim the ircrft with the lowest possible true irspeed. Generlly this is done by choosing low ltitude nd low weight. The rtionle for gin selection is s follows: Since it is nturl for the phugoid dmping to increse, select gins t the reference condition tht puts the dmping ner the lower bound of cceptble. As the lift coefficient vries, the dmping will increse nd fll within the cceptble rnge. Since the phugoid frequency decreses with incresing speed, select gins to put the frequency nd the reference condition ner the top of the cceptble frequency rnge. As the velocity increses, the frequency will come down nd sty within the cceptble rnge. Of course, in prctice there is no gurntee tht system ugmented with feedbck control will mintin ny of its open loop tendencies so the rtionle s stted is merely vgue guideline. In relity different properties work better, however the stted reference condition did turn out to be good choice

189 6.4 Evluting System Performnce with Scheduled Gins In prctice, the ircrft dynmics did not vry s predicted when feedbck control ws pplied; however, through some experimenttion, the following modl properties were found to yield fvorble results. For jet (turbofn) nd turboprop ircrft: Integrtor pole loctions: -0.0, -0.5 ω p = 0.5 rd/sec, ζ p = 0.9 For piston ircrft: Integrtor pole loctions: -0.0, -0.5 ω p = 0.40 rd/sec, ζ p = 0.9 Gins were clculted for the DC-9 ircrft in ll control regions. Using these desired modl properties the loci of poles over the entire flight envelope re plotted in Figure Figure 6.7. A key to the figures is presented in Tble 6.1. The poles re plotted only for the corners of the flight envelope s defined by the lift coefficient, ltitude, nd weight. Tble 6.1 Mrker key to Figure Figure 6.7 Outer mrker - Lift Coefficient box front-side circle bucket dimond bck-side Middle mrker - Altitude box se level circle mid-rnge ltitude dimond service ceiling Inner mrker - Weight + empty weight x mid-rnge weight dot mx tke-off weight Consider ltitude-rte-only feedbck. The locus of closed loop poles produced for the ircrft is in Figure 6.4. Notice tht the lowest phugoid dmping occurs on the bck-side of the thrust curve, insted of t the bucket speed s predicted erlier. The highest phugoid frequencies occur t se level, s predicted. One should note the loction of the integrl poles since they stry into the right-hlf-plne. This mens tht our nlysis of the liner system is predicting instbilities ner stll speed, prticulrly t low weights nd low ltitudes. We will hve to vlidte this flight condition by nlyzing the trnsient response of the non-liner system. It is useful to note tht we should expect significnt difference between the non-liner system nd our liner pproximtion outside of the -144-

190 stedy, level flight region becuse we stry more from the reference condition used for the lineriztion. Figure 6.5 shows similr locus for speed-only feedbck. The trend is similr to tht of Figure 6.4, except tht the integrl poles do not extend into the right-hlf-plne. While this region demonstrtes greter stbility, we should continue with n nlysis of the nonliner trnsient response. Figure 6.6 is root locus plot for the stedy, level flight region. We note similr trends s in Figure 6.4 nd Figure 6.5: the lowest dmping occurs on the bck-side of the power curve nd the highest phugoid frequencies occur t low weight nd low ltitude. However, inspection of the locus does show tht the modl properties for the entire flight envelope do fll within the generl guidelines set forth in Section Figure 6.4. The locus of closed loop phugoid poles in for ltitude-rte-only feedbck for the entire flight envelope of DC-9 in the clen configurtion -145-

191 Figure 6.5. The locus of closed loop Phugoid poles for speed-only feedbck for the entire flight envelope of DC-9 in the clen configurtion Figure 6.6. The locus of closed loop Phugoid poles in the stedy, level flight region for the entire flight envelope of DC-9 in the clen configurtion -146-

192 Unlike the other Regions, the stedy, level flight controller my use the bck-side of the thrust curve. There re two PI (proportionl/integrl) compenstors t work nd therefore there re two integrl poles. In some prts of the envelope the integrl poles become complex. This is seen in the locus of points grouped closer to the imginry xis. (See Figure 6.7 s well). This still is not concern becuse the integrtor poles remin within the cceptble guidelines for modl property selection. Finlly, high lift devices re considered. If the ircrft cn operte with single set of gins when high lift devices re employed s well s during clen configurtion, the number of required gins for the system cn be cut by fctor of 5 (voiding different set of gins for ech flp setting). To explore this possibility, the full flp cse of the DC- 9 is considered. Figure 6.7 shows the locus of points within the flight envelope under the full flp condition. As cn be seen, the mjority of flight conditions remin cceptble. However, in some cses, the integrl poles become complex nd re much slower thn the Phugoid poles. This cn cuse problem t low speed in tht the ircrft my not cpture the desired irspeeds s crisply nd clenly s it does t higher speeds. This condition presents bit of dilemm. During most of the simultion development, it ws ssumed tht low speed flight would require dditionl control lws nd gins. Now we cn see tht one set of gins cn fly the ircrft through its entire flight envelope; however, performnce could be diminished in the low speed rnge. Here is clssic trde-off between precision nd simplicity

193 Figure 6.7. The locus of closed loop Phugoid poles in the stedy, level flight region for the entire flight envelope of DC-9 with full flps deployed A more sophisticted control lw will fly the irplne more precisely; however, it would lso require extr code, nd more gins. For right now, since performnce is still resonbly good in the low speed rnge, the decision hs been mde to cpitlize on this unexpected result to simplify the control system. However, future requirements for other ircrft types my require more complex control system to ddress the needs of low speed flight

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195 7. The Lterl Directionl Control Lws The lterl directionl control lws re considerbly simpler thn the control lws required to fly the ircrft longitudinlly. The reson for the dded simplicity is tht one of the governing differentil equtions, the roll eqution, cn be modeled with liner pproximtion. This pproximtion is used for two resons. The first reson is tht n ccurte model of the full nonliner roll dynmics is not essentil to the modeling of ccurte trjectories. The second is tht the roll response is so hevily ugmented by the pilot or utopilot tht the dynmics of turn rte cpture is much more dependent on control inputs thn the ctul roll dynmics. The decision to pproximte the speedy roll mode permits us to ese computtionl effort by selecting hlf-second time step. Such long time step is not dequte to cpture the roll dynmics, though it is more thn dequte to cpture the slower phugoid dynmics. Becuse our roll mode is liner, we cn close feedbck loops nlyticlly without requiring the sme detil tht is done with the longitudinl dynmics. The min loop closures for the turning dynmics were lredy closed nlyticlly in Section nd imbedded directly into the open loop dynmics mking the desired bnk ngle, φ des, the primry input to the roll eqution. The four topics for discussion re: 1. The bnk ngle cpture lgorithm. The heding cpture lgorithm 3. Using the Bnk Angle Cpture nd Heding Cpture Algorithms to execute turn 4. Deciding which wy to turn 7.1 The Bnk Angle Cpture Algorithm The bnk ngle cpture lgorithm is the mjor kernel of the lterl directionl control lw. Consider the discretized bnk ngle system derived in Section.14. In generl, the lterl stte t the (k+1)th time step is function of the lterl stte t the kth time step nd the desired bnk ngle. p φ k + 1 p = A + φ φ B k des The constnt mtrices, A nd B, re functions of the modl properties (ζ,ω n ) nd the time intervl

196 Sometimes it is desirble to commnd specific turn rte. Since the turn rte eqution is very nerly liner function of the roll ngle, we simply choose to djust our commnded bnk ngle rther thn creting nother feedbck control loop to drive the system to commnded turn rte. Clculting the required bnk ngle is done by rerrnging eqution (.67). φ d 1F = sin H G m ψv C L γ I KJ (7.57) 7. The Heding Cpture Algorithm The heding cpture lgorithm is designed to cpture specified heding. To cpture given heding, we feed bck the desired heding to the bnk ngle using the control lw shown in eqution (7.58). φ = k ψ ψ d ψb d g (7.58) To predict the effect of this feedbck control lw, we must first dd the heding eqution to our stte spce model. Consider the linerized version of the turn rte eqution which finds its wy into our stte mtrix. dψ d LS φ LCφ = = φ dφ dφ mvc γ mv Cγ 0 (7.59) dψ = dp 0 (7.60) If we ssign our reference condition for the lineriztion to be φ = 0.0, then φ = φ. Furthermore, if we note tht for the bulk of the flight the lift equls the weight nd the flight pth ngle is ner zero, we cn simplify eqution (7.59) to eqution (7.61). dψ g φ dφ V (7.61) Arrnging the system of equtions in stte spce we hve eqution (7.6)

197 L NM O P L M p Lp L kp L k 0 φ g ψqp = M 0 0 V N O L Q PNM p L k δ δ φ δ φ φ ψ O L QP + NM 0 0 O QP φ des (7.6) When we close proportionl loop round the system with k ψ s our feedbck gin, s shown in Figure 7.1, we see tht the new closed loop system is eqution (7.63). There is n integrl reltionship between the heding nd the roll ngle so zero stedy stte error is chieved without the use of integrl control. ψ d k ψ ẋ = Ax + Bu ψ L NM O P L M Figure 7.1. Block digrm for heding feedbck p Lp L kp L k L k k φ g ψqp = M 0 0 V N O L Q PNM p φ ψ O L QP + NM L k k δ δ φ δ φ ψ δ φ ψ 0 0 O QP ψ d (7.63) Further verifying tht integrl control is unnecessry, we see tht the trnsfer function tht chrcterizes the reltionship between ψ nd ψ d, eqution (7.64), hs DC gin of 1. ψ ( p p ) g L k k V δ φ ψ ψ = d 3 s L Lδ k s + L k s L k k δ φ + δ φ ψ V g (7.64) 7.3 Using the Bnk Angle Cpture nd Heding Cpture Algorithms to Execute Turn When turning, the heding cpture lgorithm cnnot be used for lrge heding errors. The reson is tht the heding cpture lgorithm will commnd bnk ngle proportionl to the heding error. If the heding error is lrge, the control lw will commnd n unresonbly lrge bnk ngle such s 180 degrees. This bnk ngle would correspond to n inverted ircrft nd certinly does not mke the ircrft turn ny fster. Therefore, the heding cpture lgorithm is used only when the heding error is less thn 15 degrees

198 For errors greter thn 15 degrees, the bnk ngle control lw is used to commnd constnt turn rte in the direction of minimizing the heding error. Nominlly, bnk ngle of 30 degrees is used. Consider the following simultion exmple. The simultion prmeters re s follows: V = 300 ft/sec L p = L δ = k p =.836 k φ =.756 Although the ADM uses k ψ = 1, this simultion uses the following feedbck gin, which ccounts for vritions in irspeed. V kψ = (7.65) g We simulte turn to the right from heding of 0 degrees to heding of 100 degrees s shown in the simultion results in Figure 7.. Initilly, the ircrft rolls to the right to chieve bnk ngle of 30 degrees. The ircrft holds the bnk ngle nd stedily turns towrd heding of 100 degrees. When the ircrft is within 15 degrees of the desired heding, the heding cpture lgorithm tkes over nd drives the remining heding error to zero. -15-

199 Roll Rte (deg/sec) Bnk Angle (deg) Heding (deg) Time (sec) Figure 7.. Simultion of Aircrft Executing Turn -153-

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201 8. Lterl Guidnce nd Nvigtion The TGF Simultor s Lterl Guidnce Mnger breks lterl instructions into sequence of bsic lterl mneuvers. These lterl mneuvers re described in this chpter long with descriptions of how they re flown using the lterl control lws lredy described. It is the intent of the Lterl Guidnce nd Nvigtion System to follow the concepts nd terminology of FAA Are Nvigtion (RNAV) procedures. The source for most of the RNAV concepts nd procedures in this section is the web version of the FAA Aeronuticl Informtion Mnul (Federl Avition Administrtion, 010). 8.1 Route Legs nd Wypoints Routes re bsic lterl pths. For the TGF simultor, routes re defined in ccordnce with RNAV processes nd procedures. These definitions re provided here so tht we cn define route nvigtion procedures in the subsequent sections Pth nd Termintor Concept A leg type describes the desired pth preceding route node. Leg types re identified by two-letter code tht describes the pth (e.g., heding, course, trck, etc.) nd the termintion point (route node). The "pth nd termintor concept" sttes tht every leg of route hs termintion point nd pth type into tht termintion point. Route legs in the TGF simultor re defined in this sme wy. There re four leg types: direct-to-fix (DF), trck-to-fix (TF), course-to-fix (CF), nd rdius-to-fix (RF). Only the TF leg type requires preceding fix to define the pth. Figure 8.1: Fly-by nd fly-over wypoints followed by trck-to-fix legs (source: FAA AIM) -161-

202 8.1. Leg Termintors Geometric Nodes of Route Wypoints re geogrphic points on the surfce of the Erth, defined by ltitude/longitude coordintes. The TGF Simultor requires geogrphic representtion of the leg termintion points (route nodes), so wypoints re commonly used. In keeping with RNAV concepts, wypoints used s route nodes cn be defined s "fly-over" or "flyby." Fly-by wypoints re used when n ircrft should begin turn to the next course prior to reching the wypoint seprting the two route segments. Fly-over wypoints re used when the ircrft must fly over the point prior to strting turn. Fly-over nd fly-by wypoints re illustrted in Figure Direct-To-Fix Leg A Direct-to-fix (DF) leg type is the direct pth from the ircrft's current position to the leg termintion point (route node), regrdless of course or trck (see Figure 8.). It is not stright line or geogrphiclly fixed pth, but includes the pth of the ircrft's turn towrds the fix. The ctul pth flown will vry with ircrft type nd speed. The pth is defined by the ircrft s bering to the termintion fix. Figure 8.: Fly-over wypoint followed by direct-to-fix leg (source: FAA AIM) Trck-To-Fix Guidnce A Trck-to-fix (TF) leg type is defined s the constnt course pth (i.e., rhumb line) to the route node from preceding route node. For this reson, trck-to-fix legs re sometimes clled point-to-point legs. The preceding route node is typiclly the termintion fix of the previous leg. With the beginning nd ending nodes of the trck-tofix leg defined, it is esy to clculte the rhumb line trck (see Section 8.5), which is followed using Ground Trck Guidnce. Figure 8.1 illustrtes trck-to-fix leg Course-To-Fix Guidnce A Course-to-fix (CF) leg type is defined s the constnt course pth (i.e., rhumb line) to the route node long the specified course (or heding). A course is ssumed to be mgnetic course, unless otherwise specified. Course-to-fix nd trck-to-fix legs re similr in tht they both define rhumb lines. The difference is tht in course-to-fix leg, -16-

203 the rhumb line is defined by the termintion fix (route node) nd the specified course, nd in trck-to-fix leg, the rhumb line is defined its end points. In both cses, the rhumb line is followed using Ground Trck Guidnce. Figure 8.3: Course-to-fix leg showing the rhumb line nd the ircrft's nvigtion to it (source: FAA AIM) Rdius-To-Fix Guidnce A Rdius-to-fix (RF) leg is defined s constnt rdius circulr pth round defined turn center tht termintes t fix. In ddition to the termintion fix, the RF definition must include the turn center point nd the turn direction (i.e., right turn or left turn). In the TGF Simultor, the strt point of the RF leg is not defined so tht ny RF leg defines complete circle. The ircrft nvigtes to the circulr pth nd moves in the direction specified. The end condition for the RF leg is defined by the ending zimuth (regrdless of trcking error), which is determined from the termintion fix nd the turn center. The circulr pth is geometric pth over the erth nd is followed using Ground Trck Guidnce nd Bnk Angle Cpture guidnce. A rdius-to-fix leg is illustrted in Figure 8.4. The prmeters of rdius-to-fix leg re illustrted in Figure 8.9. Figure 8.4: Rdius-to-fix leg (source: FAA AIM) -163-

204 8. Lterl Guidnce The purpose of the lterl guidnce system is to steer the ircrft to follow routes or other lterl pths. There re three bsic lterl mneuvers, which re used in combintion to chieve lterl guidnce. These re: Trck Geogrphic pth Follow Course Lw Trnsition Every lterl mneuver in the TGF Simultor is mde up of these three bsic mneuvers. The intent of this section is to describe these three bsic mneuvers in terms of the two bsic lterl control lws (heding control nd bnk ngle control) described in Chpter 7. Once this description is completed, we will hve complete mpping of ny lterl mneuver to the bsic control lws of the simultor Trcking Geogrphic Pth In this type of mneuver, n ircrft is intending to fly specific geogrphic pth over the surfce of the erth. Currently, the TGF simultor is cpble of nvigting two types of geogrphic pths: circulr rcs, nd rhumb lines. The simultor must compre the ircrft s position on the surfce of the erth to the desired geogrphic pth nd pply lterl control inputs to correct ny lterl errors. This is ccomplished using ground trck guidnce. The end condition of Trck Geogrphic Pth mneuver is triggered by distnce from (or, cpture hlo round) the pth s termintion fix. For fly-by wypoints, this distnce is djusted to llow for the trnsition (described below). The ground trck guidnce lgorithm is the bsic guidnce lgorithm used for nvigting geogrphic pths (routes nd fixes). These pths re trnslted into ground trck zimuth s function of some independent vrible nd then trcked. The ground trck zimuth is the ngle between the ircrft s ground trck nd true North. Under zero wind condition, the ground trck zimuth is the sme s the ircrft s heding. In the presence of wind, the ground trck zimuth will differ from the ircrft heding s illustrted in Figure 8.5. To nvigte geogrphic pth, the ircrft must follow given ground trck zimuth rther thn specific heding; yet the lterl control system is designed only to turn to desired heding. The lterl guidnce must bis its heding commnds to the lterl control system with correction fctor tht ccounts for winds. To ccommodte this requirement, the lterl guidnce mesures the difference between the heding nd the ground trck zimuth, which we denote s ψ. Our trcking lgorithm mkes uses the following nomenclture: -164-

205 ψ: The ircrft s heding, in degrees. ψ GT : The ircrft s ground trck zimuth, in degrees. δψ w : The wind bis. ψ d : The desired heding. ψ : The desired ground trck zimuth. GTd N Aircrft heding ψ GT ψ V w δψ ψ w Aircrft ground trck V cos γ Wind V GT Figure 8.5. Illustrtion of the difference between ground trck nd heding We define the wind bis s the difference between the ircrft s ground trck zimuth nd its heding. δψ w ψ GT - ψ (1.1) The ircrft s ground trck zimuth nd heding re vilble from the ircrft dynmics. The wind bis, δψ w, is then used to djust the desired ground trck so tht the ircrft will trck properly. The result is the desired heding. ψ = ψ δψ (1.) d GTd w Eqution (3.1) is not intended s n ccurte representtion of the vector lgebr grphiclly depicted in Figure 8.5; it merely shows the use of the wind bis s correction fctor. Its simplicity does not compromise its intent, which is to cpture the desired ground trck zimuth. Anlysis of the wind bis requires comprison of the zimuth of the ircrft s velocity vector, ψ, with the ground trck zimuth, ψ GT; but in Chpter, we stted the ssumption tht the ircrft is lwys trimmed for coordinted flight; i.e., the sideslip is lwys zero. This mens tht the zimuth of the ircrft s velocity vector, ψ, is coincident with the ircrft s heding, ψ. Therefore, for our purposes, the nlysis is eqully ccurte in compring ψ nd ψ GT

206 Trcking Rhumb Line Trck-to-fix nd course-to-fix legs re composed of rhumb lines. When trcking rhumb line, the trcking lgorithm commnds the ground trck of the ircrft bsed on: the ircrft s lterl distnce from the rhumb line, the rhumb line's bering, nd the ircrft s rdius of turn. This is illustrted in Figure 8.6. The intercept ngle for the given segment is function of how fr the ircrft is lterlly from the segment. Segment r t 1 r t Figure 8.6. Illustrtion of the ircrft in route following mode The intercept reches mximum of 45 degrees when the ircrft is one-hlf turn rdius wy from the segment. The intercept ngle is bounded t 45 degrees. Equtions (1.3) nd (1.4) re used to determine the ircrft s desired ground trck. First, ψ is clculted using Eqution (1.3). If the result hs mgnitude greter thn 45 degrees, the nswer is bounded t 45 degrees using Eqution (1.4). The rtio δ δ is used to preserve the sign of the originl vlue. Note tht the lterl distnce term,δ, mintins sign convention of positive vlues on the right side of segment nd negtive vlue on the left side of the segment. This solution is dpted from the originl System Segment Specifiction (Federl Avition Administrtion, 1993). δ o ψ = 90 + δψ, ψ < 45 (1.3) r t FTE δ ψ = 45, ψ 45 o (1.4) δ -166-

207 ψ = ψ ψ (1.5) GTd r The terms re defined s follows: δ : The ircrft s lterl distnce from the cpture segment (nm) ψ r : The cpture segment s bering. (degrees) r t : The ircrft s turn rdius. (nm) ψ GT : The ircrft s desired ground trck (degrees) d δψ FTE : The heding bis from flight technicl error (degrees) As with ll other heding commnds, the term ψ GT needs to be djusted to keep vlues d within the degree rnge. The logic for this opertion is shown in Figure 8.7. The flight technicl error (Chpter 11) is sent to the heding-bsed course guidnce in the form of lterl offset, denoted s δ r. The lterl error offset is then relted to heding bis using Eqution (1.6). FTE δψ FTE 90 = δ rfte (1.6) r t ψ GTd > 360 No ψ GTd < 0 No Yes Yes ψ GTd = ψ 360 GTd ψ GTd = ψ GTd Figure 8.7. Logic for insuring desired ground trck is within proper boundries The terms in the eqution re defined s follows: δψ FTE : The heding bis creted from flight technicl error (degrees). r t : The turn rdius for the ircrft t the current speed (nm)

208 δ rfte : The lterl offset from flight technicl error (nm). The flight technicl error conversion from lterl distnce to heding bis mimics Eqution (1.3) in form nd cuses the course guidnce lgorithm to produce the lterl offset error of δ r in the flight pth. FTE r lf d R s n s δ Figure 8.8. Illustrtion of distnce clcultion geometry Determining Lterl Distnce from Rhumb Line The ircrft s lterl distnce from rhumb line is clculted using vector opertions. The dot product is tken of the position vector from the ircrft s loction to the leding fix of the segment nd unit vector norml to the vector describing the segment itself. The expression is best represented mthemticlly in Eqution (1.7) nd Figure 8.8. To insure tht the desired ground trck for segment cpture is correct, it is necessry for δ to be negtive when on the left side of the segment nd positive on the right side of the segment. δ = r nˆ (1.7) lf s The unit norml is represented in Eqution (1.8). Rs R y sx nˆ = xˆ yˆ R R s s s s s (1.8) The terms in the equtions re defined s follows: δ: The lterl distnce from the segment r lf : A vector from the ircrft s position to the leding fix -168-

209 nˆ s : A unit vector norml to the segment R s : A vector representing the segment Trcking Circulr Arc When flying rdius-to-fix leg, n ircrft is trcking circulr rc pth on the surfce of the Erth. Trcking to this circulr rc is performed similrly to the rhumb line trcking nd the sme pproch is used here. Figure 8.6 pplies eqully to this discussion. Becuse the ircrft is following constnt rdius turn, it is ssumed to be in constnt bnk ngle turn. While this ssumption is gross representtion of the turn, the lterl correction term will hndle ny discrepncies. The constnt bnk turn is followed using the Bnk Angle Cpture lgorithm described erlier. The nominl bnk ngle for the turn, φ t, is given by the prmeters of the rdiusto-fix leg. The heding djustment, ψ, is clculted s with rhumb line trcking, using equtions (1.3) nd (1.4). The heding djustment is then converted to bnk ngle djustment, using heding gin, k ψ, similr to tht defined in eqution (7.8). The nominl vlue for the heding gin is k ψ = 1. k ψ φ = ψ (1.9) Then, similr to eqution (1.5), the desired bnk ngle is given by φ = φ φ (1.10) d t Determining Lterl Distnce from Circulr Arc Route Leg For the purposes of correcting the cross-trck error when trcking geogrphic circulr pth (i.e., rdius-to-fix leg), n ircrft s lterl distnce, δ, from circulr rc is here defined s the difference between the rc's rdius nd the ircrft's distnce from the turn center, s illustrted in Figure 8.9. To mintin the sign convention tht lterl distnce is positive when to the right of the pth, step function is defined here tht is bsed on the turn direction specified in the RF definition. f TD + 1, right turns = 1, left turns (1.11) The lterl distnce is then, -169-

210 TD ( ) δ = f r d (1.1) where r t nd d re defined s lwys positive. As stted erlier, circulr pths re trcked using the Bnk Angle Cpture lgorithm, which requires desired bnk ngle. Using eqution (1.1), the heding correction, ψ, is clculted using equtions (1.3) nd (1.4). The bnk ngle correction is clculted using eqution (1.9), nd the desired bnk ngle is clculted using eqution (1.10). t Figure 8.9: Prmeters of Right Turn Rdius-To-Fix Leg 8.. Following Course Lw In this type of mneuver, n ircrft is intending to fly specific compss course (heding) lw. The course lw my be constnt (e.g., the ircrft is trying to mintin mgnetic heding of 190 ) or function of some independent vrible, like position (e.g., direct-to-fix leg is course lw tht mtches the ircrft s bering to the fix, which is function of the ircrft s position). The end condition of course lw my be time-bsed or position-bsed. A course lw mneuver mkes use of the Heding or Bnk Angle control lws. As described in Chpter 7, the lterl guidnce mnger must decide which direction to turn nd whether or not bnk ngle control is needed to bring the heding error into region in which liner heding cpture cn be used

211 The strtegy for deciding whether to use bnk ngle cpture in the heding guidnce is worth repeting. The heding cpture lgorithm is used only when the heding error is less thn 15 degrees. For errors greter thn 15 degrees, the bnk ngle control lw is used to commnd bnked turn in the direction of minimizing the heding error Deciding Which Wy to Turn The TGF simultor s user interfce llows for commnded left nd right turns. Often, however the ircrft is left to mke tht decision itself. This my be the cse for constnt heding instructions nd for direct-to-fix instructions. In this cse, the ircrft must choose which direction of turn is the shortest. Either left turn or right turn will work, but one turn is shorter. The dilemm is illustrted in Figure To the humn, it is obvious tht right turn is pproprite for the sitution; however, the logic required to mke the utopilot come to the sme conclusion is not trivil. The following logic determines the mgnitude nd the sign of the heding error, referred to s e 5 in the simultion code. The first tsk is to determine the mgnitude of the heding errors to the left nd right, symboliclly represented s e left_turn nd e right_turn. 360 Current heding 70 Left Turn Right Turn Desired heding Figure An illustrtion of the dilemm of whether to mke right of left turn to heding if ψ d > ψ e right_turn = (ψ d - ψ) e left_turn = (ψ d - ψ) if ψ d < ψ -171-

212 e right_turn = (ψ d - ψ) e left_turn = (ψ d - ψ) Next, the bsolute vlue of e left_turn nd e right_turn re compred to determine which is smller. The ctul heding error, e 5, is set equl to the smller of these two errors. It is convenient to use the convention tht turning errors to the left re lwys negtive nd turning errors to the right re lwys positive. This corresponds nicely to the bnk ngle convention where bnks to the right re considered positive nd bnks to the left re negtive. Therefore, there is no need to djust the previously developed control lws to mke sure tht the ircrft turns in the desired direction when commnded Cpturing Heding when the Direction of Turn is Specified The introduction of the left or right turn vribility dds complexity to the system. When the ircrft is constrined to turn in only one direction, it will likely overshoot slightly nd insted of turning bck to the heding, it will continue turning in the specified direction for nother 360 degrees. There needs to be distinction mde for the initil cpture of the heding. Figure 8.10 shows the lgorithm for determining whether or not heding hs been cptured. Essentilly, the ircrft turns in the specified direction until the heding error is within 5, t which point the simultor uses the logic of the previous section to determine the turn direction on its own. The term e 5 is the error in heding. Is Heding Cptured? No 0 e 5 < 5. 0 No Yes Yes Heding is now Cptured Determine Left or Right Turn Clculte e 5 Figure Algorithm for cpturing heding -17-

213 8...3 Direct-To-Fix Guidnce Direct-to-fix guidnce is used in severl different super-mneuvers, including Fix Cpture, Route Following, nd Hold Mneuvers. To fly direct-to-fix leg, it is necessry to know the ircrft's rnge nd bering to the fix. (Algorithms for clculting rnge nd bering to fix re discussed lter.) Once the bering to the fix is known, the Course Lw logic is used to turn the ircrft to tht bering. This control strtegy is effective s long s the ircrft is sufficiently fr wy from the fix. This geometry is illustrted in Figure 8.1. North Fix Point of ground trck nd heding convergence Initil Azimuth Finl Azimuth R Rnge Figure 8.1. An ircrft turning to fix Clculte bering nd distnce to fix Commnd ircrft to bering no is ircrft within cpture hlo? yes Terminte leg guidnce Figure Guidnce lgorithm for Direct-to-fix leg with fly-over termintion point -173-

214 The bering to fix chnges constntly s the ircrft moves, except when it is flying directly t the fix. By using the ircrft s bering to the fix s its desired heding, the ircrft is gurnteed to be flying to tht fix. When the ircrft is within preset cpture limit distnce (nominlly, 0.1 nm) of the fix, the fix is considered cptured. Figure 8.13 illustrtes the direct-to-fix lgorithm. When the cpture "hlo" round the fix is entirely within the turning circle of the ircrft, s illustrted in Figure 8.14, the ircrft cnnot cpture the fix. The simultion should perform check to see if the cpture hlo is within the turning circle of the ircrft nd, if so, to consider the fix cptured. Figure 8.14 Aircrft Cnnot Enter the Cpture Hlo 8..3 Trnsition A trnsition mneuver is used to blend lterl mneuvers. The intent of the trnsition is to blend lterl mneuver onto subsequent geogrphic pth. But the indictor used by the TGF Simultor for determining the need for blended trnsition is the fly-by wypoint, not the subsequent pth; nd there is no requirement tht the mneuver succeeding flyby wypoint is geogrphic pth. Nor is there requirement tht the fly-by wypoint lie on the geogrphic pth defined in the succeeding mneuver. For this reson, the trnsition following mneuver with fly-by wypoint is defined by tht sme mneuver, nd not by the subsequent geogrphic pth tht is to be blended. The unfortunte consequence of this pproch is tht it is possible to define trnsitions tht do not ctully blend the subsequent mneuver; however, this will not crete lterl guidnce nomlies, s the -174-

215 lterl guidnce system will simply correct to the current pth, regrdless of the size of the pth error. Therefore, it is left to the route designer to ensure lterl pth continuity. There is one exception in which trnsition is not indicted by fly-by wypoint, nd tht is the constnt course merge to route mneuver. This is constnt course lw with n end condition defined by proximity to route. In this cse, the trnsition is defined by subsequent geogrphic pth. The Route Segment Selection lgorithm is used to determine the route segment to be intercepted. As the ircrft pproches the route, the selected segment my chnge, so the selection lgorithm repets throughout the mneuver. Segment trnsition distnce Segment Figure Illustrtion of segment trnsition distnce Fly-By Wypoint Trnsitions A fly-by wypoint (s illustrted in Figure 8.1) indictes tht the ircrft is to blend its pth into subsequent lterl mneuver. To ccomplish this, the TGF guidnce mnger cretes "trnsition" phse during which the ircrft uses the bnk ngle cpture lgorithm to fly constnt bnk ngle (nominlly 30 ) until the ircrft is within n cceptble heding error (nominlly, 5 ) of the initil heding of the subsequent lterl mneuver. 3 This estblishes the end condition of the trnsition phse. To ccomplish this trnsition, the guidnce mnger needs to determine the point long the current segment t which the trnsition should begin. This point is defined by the distnce from the 3 If the initil heding of the subsequent mneuver is not defined, it is tken s the rhumb line bering between the consecutive termintion fixes. If rdius-to-fix leg is involved in the trnsition (preceding or triling), the course vector t the beginning nd terminting fixes should be used in determining the course difference,

216 termintion fix. This distnce is termed the segment trnsition distnce nd is illustrted by Figure The mgnitude of this distnce is ffected by the turning rdius of the ircrft nd hence the speed of the ircrft. The trnsition is typiclly constnt bnk ngle lw nd its end condition is typiclly heding-bsed (e.g., 30 bnk until ttining the initil heding of the subsequent geogrphic pth). A trnsition mneuver mkes use of the Bnk Angle control lw. We encounter dilemm in choosing bnk ngle for trnsition mneuvers. Stndrd procedures (Federl Avition Administrtion, 010) cll for 3 degrees/second turn, but experience nd trck dt show tht such turns re much shrper thn is common. Turning with 30 degree bnk ngle is not unrelistic but such turns re shrper thn most turns seen in rdr trck dt. The TGF hs seen better correltion with trck dt when using 17 degree turns. Perhps the choice of bnk ngle is better correlted to ircrft type or ltitude of the turn. Currently, the TGF uses constnt defult desired bnk ngle (nominlly, 30 ), but reserch my be necessry to determine more suitble lw. Figure The geometry of segments tht intersect t n obtuse ngle -176-

217 l offset r t r t h Figure Geometric representtion of segments djoined t n cute ngle To determine the segment trnsition distnce, two different drwings of the scenrio re presented in Figure 8.16 nd Figure Figure 8.16 shows the more common cse where segments intersect t obtuse ngles. Figure 8.17 shows the less common cse where segments intersect t cute ngles. It will be shown tht the eqution for the segment trnsition distnce is the sme in both cses. The course (or heding) difference between the segments, ψ, cn be clculted using the definition of the dot product of the course vectors nd the ngle, (shown in Figure 8.17 but omitted from Figure 8.16), between them. The ngles nd ψ re supplementry ngles (ie, they sum to 180 ) The segment trnsition distnce, l offset, is clculted by observing in Figure 8.16 tht two isosceles tringles re formed creting kite like pttern. We cn then bisect the ngle nd form two right tringles. Trigonometry cn then be used to clculte the segment trnsition distnce bsed on the difference of the hedings of the two segments nd the turn rdius, r t, of the ircrft. A lg fctor, k, (suggested vlue, 1.3) is dded to llow for mrgin of error since turn dynmics re not instntneous. The eqution is vlid for the cute ngle cse of Figure 8.17 s well. l offset ψ = kr ttn (1.13) -177-

218 8..3. Merge to Route Trnsitions When n ircrft is following constnt course lw to intercept route, it must guge when it should strt to turn to merge clenly onto the route. Generlly, the distnce tht is required is function of the ircrft s speed nd the intercept ngle tht the ircrft hs with the segment. It is very similr clcultion to tht used for fly-by wypoint trnsitions. Figure 8.18 illustrtes the geometry of n ircrft merging onto segment The lgorithm requires the ircrft s true irspeed nd heding nd vector describing the segment. V G : Aircrft s ground speed ψ: Aircrft s heding over the ground R s : A vector describing segment. First, vector, V, representing the ircrft s velocity is creted from the ircrft s speed nd heding. We cn see from the geometry in Figure 8.18, tht the problem is similr to the fly-by wypoint trnsition problem. We cn lso see tht the distnce t which the ircrft should turn, l turn, is the projection of l offset onto line norml to the segment. Modifying eqution (3.14), we get l turn ψ = krttn sin ( ψ ) (1.14) Using trigonometric identities, eqution (1.14) cn be simplified to eqution (1.15). Rs 180- r t l turn V c r t Figure Illustrtion of geometry ssocited with n ircrft merging onto segment when ircrft is heding in the direction of the segment -178-

219 l offset Rs l turn r t V c r t Figure An ircrft merging onto segment which is pointed in direction opposite of the ircrft's current velocity turn t ( 1 ) l = kr cos ψ (1.15) When the ircrft is tending to hed in the direction opposite the direction of the segment, more distnce is needed to turn becuse the ircrft must completely chnge the direction of flight to fly long the segment. This cse is illustrted in Figure However, Eqution (3.15) is still vlid s cn be verified from inspection of the geometry in Figure Lterl Guidnce Mnger The Lterl Guidnce Mnger cretes sequence of the bsic lterl mneuvers detiled bove bsed on wht re best described s super-mneuvers. The Lterl Guidnce Mnger ccepts the following super-mneuvers. Route Following Route Cpture Heding Cpture Hold Aircrft Following -179-

220 Aircrft Grouping The Aircrft Following nd Aircrft Grouping super-mneuvers re ppliction-specific mneuvers nd re not covered in this document. Hs segment been determined? No Determine Cpture Segment Yes Hs dynmic fix been estblished long the pth? Yes No Determine Dynmic Fix Steer towrds dynmic fix using Fix Cpture Guidnce Determine if it is time to turn onto the route Is it time to merge onto route? Yes No Determine if the segment should be dvnced. Terminte Automtic Route Cpture Guidnce nd initite Route Following Guidnce Should segment be dvnced? Yes Advnce Segment No Figure 8.0. Logic for Automtic Route Cpture Guidnce -180-

221 8.3.1 Route Following Bsed on the ircrft s specified route, the Lterl Guidnce Mnger cretes sequence of mneuvers to guide the ircrft long tht route. The most common route is sequence of trck-to-fix legs with fly-by wypoints. In ccordnce with the procedures of the preceding sections of this chpter, the Lterl Guidnce Mnger will crete the rhumb lines defined by the legs nd the trnsition mneuvers between them s well s the end conditions for ech mneuver Route Cpture The Lterl Guidnce Mnger will define pth to the ircrft s specified route bsed on one of three cpture types: Automtic, Direct, or Heding Intercept. It will lso crete the trnsition to the route, if necessry, nd the mneuvers of the route itself Automtic Route Cpture For utomtic route cpture, the Lterl Guidnce Mnger must select cpture segment from the ircrft s route, then crete point on tht segment to trget for the cpture. The route cpture then becomes identicl to Direct Route Cpture. The lgorithm is described in the flow digrm illustrted in Figure 8.0. The procedure used in utomtic route cpture guidnce involves determining cpture segment, determining dynmic fix, nd leg trnsition Selecting Cpture Segment There re three criteri tht re used to determine the pproprite segment to cpture. These criteri use three prmeters: 1. The Dot Products of the ircrft s loction reltive to the leding nd triling fixes nd the segment s vector R s.. The lterl distnce from the segment. 3. The closest triling fix. Criterion #1 is summrized s: If segment s position vector, when dotted with position vector from the ircrft s loction to the leding fix, yields positive vlue nd if segment s position vector, when dotted with position vector from the ircrft s loction to the triling fix of the sme segment, yields negtive vlue, then the ircrft should cpture the segment. r R < 0 r R < 0 r R > 0 1 s1 s 3 s3 r R < 0 r R > 0 r R > 0 s1 3 s 4 s3-181-

222 R s3 r 4 r 3 R s r r 1 R s1 Figure 8.1. Scenrio of n ircrft determining which segment to cpture Consider the following scenrio shown in Figure 8.1. There re three segments in the route nd the ircrft must determine which segment to cpture. To do this the position vectors r 1 through r 4 re determined. These vectors re then dotted with the position vectors of ech segment. If the dot product between position vector from the ircrft loction to the leding fix of segment nd the segment position vector is positive, then the ircrft tends to be behind the leding fix of segment. Likewise, if the dot product is negtive, the ircrft will be hed of the fix. For segment to be good choice for cpture, the ircrft should be behind the leding fix nd in front of the triling fix. In the scenrio in Figure 8.1, we see tht the dot products for the first segment re both negtive. Therefore the ircrft is in front of the first segment. For the second segment, the dot product to the triling fix is negtive while the dot product to the leding fix is positive. The second segment would therefore be n cceptble choice for cpture. Looking t the third segment, both dot products re positive so the segment is in front of the ircrft. Initilly, this test lone ws thought to be sufficient to determine which segment should be cptured; however, it is not. Consider the next cse shown in Figure 8.. When the required dot products re clculted, it is shown tht both segments meet the requirements for cpture. Therefore criterion # clcultes the lterl distnce from every segment deemed cceptble by -18-

223 criterion #1. The closest segment is cptured. In Figure 8., the first segment is chosen becuse it is the closest to the ircrft. R s r r 3 R s1 r 1 Figure 8.. A scenrio demonstrting the filure of criterion #1 C B A Figure 8.3. Regions where both criterion #1 nd criterion # fil These two criteri re not sufficient to hndle ll cses. Consider the cses where no segment is cceptble s defined by criterion #1. These cses re illustrted in Figure 8.3. Criterion #1 will fil to yield ny segment for cpture if its dot product -183-

224 requirements re not met. This often occurs when the ircrft is sufficiently behind or in front of the route s shown in regions A nd C of Figure 8.3. There is lso nother ded region where two segments meet s shown in region B. If n ircrft is in this region, criterion #1 will not find segment. In this cse, criterion #3 is used. Criterion #3 checks the ircrft s distnce from every segment s triling fix. It then chooses to cpture the segment tht is ssocited with the closest triling fix. Figure 8.4 illustrtes the segment determintion logic. Step # 1 Step # Evlute next Segment Does segment meet criteri #1? Yes Reject Segment No Did ny segment meet criteri #1 & # No Determine Segment using criteri #3 Yes Stop Is the distnce to this segment shorter thn the current selected segment? No Yes Mke Segment the Cpture Segment Figure 8.4. Flow chrt detiling segment determintion logic The Dynmic Fix The dynmic fix is n rbitrry fix tht is creted by the system t convenient loction long the selected cpture segment to be used in direct-to-fix mneuver. This is more robust lgorithm thn n rbitrry intercept pproch. For instnce, once the cpture segment is determined, the ircrft could be given n intercept heding of 45 degrees nd intercept the segment. However, this method hs some inherent limittions. First, the ircrft would lwys intercept using 45 degrees regrdless of how fr the ircrft ws wy from the segment. An ircrft fr wy from the cpture segment might pss the segment before ever cpturing it. This sitution is illustrted in Figure

225 Close ircrft with 45 intercept cptures segment Fr wy ircrft with 45 intercept misses segment Figure 8.5. Illustrtion of ircrft using 45 degree intercept To void the problem of ircrft overshooting cpture segments, dynmic fix is plced on the segment to be cptured, nd the ircrft is commnded to fly towrd the dynmic fix using direct-to-fix guidnce. This sitution is illustrted in Figure 8.6. In this cse the further ircrft nturlly uses lrger intercept ngle. This system insures tht the proper segment is cptured nd lso provides some pprent vriety in intercept ngles so tht ll ircrft do not pper to behve the sme. To clculte dynmic fix, first consider the drwing in Figure

226 Dynmic fix Both ircrft cpture segment Figure 8.6. Illustrtion of two ircrft cpturing segment using dynmic fix r 1 Dynmic Fix d 3r t Figure 8.7. Determining n offset fix (dynmic fix) loction The distnce, d, left to trvel on given segment is determined by dotting r 1, the position vector from the ircrft to the leding fix, with unit vector in the direction of the segment, rˆ s. The dynmic fix distnce from the leding fix, d offset, is somewht rbitrrily chosen to be three turn rdii less thn the distnce d. The turn rdius of the ircrft is notted, r t

227 The three-turn-rdius distnce ws chosen to insure tht the ircrft, regrdless of its initil orienttion or position, cn cpture the dynmic fix while still mintining the proper generl direction long the route. Generlly, two turn rdii would be sufficient, providing the ircrft does not speed up during the mneuver. However, three turn rdii re used s fctor of sfety just in cse the ircrft increses its speed nd hence its turn rdius during the mneuver. d = r ˆ r (1.16) 1 s d = d 3r (1.17) offset t We cn crete vector, R offset, describing the loction of the dynmic fix where xˆ s is unit vector pointing true North nd ŷ s is unit vector pointing true Est. R = d cosψ ˆx d sin ψ ˆy (1.18) offset offset r s offset r s However, to use the direct-to-fix guidnce, the fix must be represented in terms of ltitude nd longitude. We cn pproximte the ltitude of the fix by converting the xˆ s component of the R offset vector to degree vlue s done in Eqution (1.19). Similrly, the longitude cn be clculted in Eqution (1.0). o 360 µ dyn = µ leding doffsetcos ψ r (1.19) π r e o 360 ldyn = lleding doffsetsinψ r π rcos µ e dyn (1.0) Direct Route Cpture In Direct Route Cpture, the cpture point is known. The Lterl Guidnce Mnger builds direct-to-fix mneuver to the cpture point, trnsition to the route, nd the mneuvers of the route itself Heding intercept Route Cpture For Heding Intercept Route Cpture, the Lterl Guidnce Mnger builds heding mneuver with position-bsed end condition. This mneuver must constntly monitor the ircrft s position reltive to the route nd evlute the selection of cpture segment nd the coincident trnsition mneuver. The Mnger ttches the trnsition mneuver nd the mneuvers of the route itself. It should be noted tht the lgorithm hs no control over the initil heding, nd so there is no gurntee tht the course will intercept the -187-

228 route. The Mnger will, however, provide wrning if the ircrft is unlikely to intercept the route. Figure 8.8 shows the bsic lgorithm for the guidnce lw. It should be noted tht even though the ircrft is being vectored, it is necessry to determine the cpture segment so tht the ircrft knows when to merge onto the route. Determine cpture segment Is it time to turn onto the route? Yes No Continue to follow commnded heding Terminte route cpture with fixed heding guidnce nd ctivte Route Following Guidnce Figure 8.8. Route cpture with fixed heding guidnce Determining Whether the Heding will Intercept Route While the lgorithm hs no control of the ircrft s initil heding, the lgorithm will provide wrning if the heding chosen by the user is unlikely to intercept the given route. Bsiclly, the lgorithm mesures whether or not the intercept ngle crosses segment nd is relted to the route following lgorithm. The intercept ngle ψ t is clculted using Eqution (3.16). ψ t = ψ r - ψ d (1.1) The terms re s follows: ψ t : The intercept ngle tht the ircrft heding mkes with the segment ψ r : The bering of the segment ψ d : The desired heding The resulting number is used in the logic presented in Figure

229 8.3.3 Hold The Hold super-mneuver is used by the Lterl Guidnce Mnger when hold is commnded or when the fix of direct-to-fix commnd is not prt of the ircrft s specified route. Holds in the TGF simultor re constructed in ccordnce with holding procedures in the AIM. The Lterl Guidnce Mnger builds direct-to-fix mneuver followed by hold entry (direct entry, prllel entry, or terdrop entry) nd then sequence of course-to-fix nd rdil legs. The reder is referred to the AIM for further informtion ψ t = ψ r ψ d Yes ψ t < 180 ψ = ψ No ψ t > 180 Yes ψ = ψ 360 No t t t t ψ t > 0 ψ t < 0 & or & δ < 0 δ > 0 No Aircrft will likely cpture segment Yes Aircrft will likely NOT cpture segment Figure 8.9. Logic for determining whether or not heding will intercept segment Heding Cpture The Heding super-mneuver is simply constnt heding course lw. 8.4 Bsic Algorithms Required for Complete Functionlity To mke the guidnce system operte properly, some lower level functions re required. These functions re: Clculting the ircrft s turn rdius Determining the lterl distnce to segment Determining the distnce to go long segment Determining the Rhumb line bering nd distnce to fix -189-

230 8.4.1 Clculting the Aircrft Turn Rdius To clculte n ircrft s turn rdius, r t, stndrd eqution is used from Anderson (1989), Eqution (1.), where V is the ircrft s speed in the surfce coordinte system, g is the grvittionl ccelertion, nd n is the ircrft lod fctor. r t = g V n 1 (1.) The lod fctor for the ircrft is clculted by considering Eqution (1.3) where L is the lift of the ircrft, φ is the bnk ngle, nd W is the weight of the ircrft. In the simultion, we will ssume tht the ircrft lwys will provide enough lift to mintin level flight which is implied by the equlity of Eqution (1.3). L cosφ = W (1.3) The lod fctor of n ircrft is defined s the lift over the weight. Assuming enough lift is provided to mintin level flight, the lod fctor cn be determined s n exclusive function of bnk ngle. L L 1 n = = = W L cos φ cos φ (1.4) Plugging eqution (1.4) into eqution (1.), we get r t V = (1.5) g tnφ 8.4. Determining Distnce to go Along Segment Using the sme nomenclture nd geometry presented in Figure 8.8, the distnce left to trvel long the segment, d, cn be clculted using Eqution (1.6). d = r lf R R s s (1.6) Rhumb Line Bering The rhumb line is line of constnt course, or heding. The distnce between the two fixes using the rhumb line cn be much greter thn gret circle rc if the fixes re fr prt

231 For sphericl erth, rhumb line is stright line drwn between the two points on Merctor projection of the erth s surfce. The merctor projection seprtely mps ltitude nd longitude to plnr surfce. The rhumb line is then the hypotenuse of the tringle formed by the projection of the ltitude chnge nd longitude chnge onto tht plnr surfce. Our derived equtions should be consistent with the Merctor trnsformtion equtions of Clrke (1995). Fixes Constnt heding route Ltitude(deg) Longitude(deg) Figure Cylindricl mpping of sphericl Erth model. Shown re two fixes nd the constnt heding route between the fixes. North r cosλ dl Fi x ds r dλ Est -191-

232 Figure 8.31 Geometry relting ltitude, longitude, nd bering on sphericl erth Referring to Figure 8.31, the chnges in ltitude nd longitude long rhumb line on sphericl erth re given by, which cn be rewritten s, Upon integrting, we get, tnψ = r cos λ dl r dλ tnψ = cos λ dl dλ (1.7) dλ dl = tnψ cos λ l l l dl = tnψ l λ λ dλ cos λ l = tnψ ln secλ + tn λ π λ l = tnψ ln tn + 4 π λ π λ1 l l1 tnψ = ln tn + ln tn λ λ λ λ 1 1 (1.8) Solving for the true heding, we get, tnψ ( l l ) 1 = π λ π λ1 ln tn + ln tn This is consistent with the equtions for the equtoril Merctor projection s presented in Clrke (1995). The longitudes must be nlyzed so tht the shorter pth round the world is chosen. The following lgorithm will normlize the longitude chnge for our purposes. -19-

233 l = l l 1 { ( ) } While l > 180 l = l + sign l *360 (1.9) And the true heding eqution becomes, l tnψ = π λ π λ1 ln tn + ln tn (1.30) Becuse the rctngent hs rnge (-90,90 ) nd we wnt heding in the rnge (0,360 ), we need to be creful bout how we solve this eqution. We, therefore, develop the 360 rctngent function. π λ π λ1 ψ = rctn360 ln tn + ln tn +, l 4 4 (1.31) Arctngent Function In this section, we develop n lgorithm for the rctngent of rtio of Crtesin coordintes in the rnge (0,360 ). Becuse the rtio is of the ordinte to the bsciss, we cn dpt the rnge of the rctngent function per the qudrnt of the coordinte pir. Function θ = rctn 360 (δ bsciss,δ ordinte ) if δ bsciss 0 nd δ ordinte 0 θ δ rctn ordinte = δ bsciss if δ bsciss 0 nd δ ordinte 0 else end δ ordinte θ = rctn δ bsciss δ ordinte θ = rctn δ bsciss -193-

234 8.4.4 Rhumb Line Distnce Once gin referring to Figure 8.31, the incrementl distnce long rhumb line on sphericl erth is given by, which integrtes to, s re dλ cosψ = ds re dλ ds = cosψ ( λ λ ) r e 1 = (1.3) cosψ Absolute vlue is used becuse we wnt positive distnce. Eqution (1.3) does not pply for est-west rhumb lines. For these cses we use n lternte reltion, lso obtined from the geometry of Figure Since the ltitude is constnt, this integrtes to, ds = r cos λ dl (1.33) EW e s = r cos λ l (1.34) EW The question rises s to wht erth rdius to use in the clcultions. A good pproximtion to the sphericl erth rdius is to use the locl rdius of the first point in the WGS-84 erth model, which is given by using equtions (.107) nd (.108). e Creting Vectors Representing Segments There is need to represent segments s vectors. To crete vector, mgnitude nd bering re required. Generlly, the rhumb line informtion is used. A segment s bering is the rhumb line bering between the two endpoint fixes tht mke up the segment, nd the segment s length is the rhumb line distnce between the two fixes. The vector components re represented in the surfce frme s shown in Eqution (1.35) R = s cosψ ˆx + s sin ψ ˆy (1.35) s s s s s s s -194-

235 The nomenclture is defined s follows: R s : The vector representing the segment s s : The rhumb line distnce of the segment ψ s : The rhumb line bering between the triling nd leding fixes of the segment ˆx, ˆ y : Unit vectors representing the x, y components of the surfce frme. s s -195-

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237 9. Nvigtion Error Modeling The purpose of nvigtion error modeling is to model the vrinces which occur in ircrft flight pths s result of imperfect informtion. Three different nvigtion systems re modeled: These re: VOR/DME nvigtion GPS nvigtion ILS nvigtion The two nvigtion types generlly used for en route types of opertion re VOR/DME nd GPS nvigtion. The ILS model is used only for pproch to lnding. All of the nvigtion error models perform similrly in tht they crete perturbed estimte of the ircrft s loction for the guidnce system to use s n input. Therefore, the nvigtion error models ll return ltitude-longitude pir which represents the ircrft s position s determined by imperfect nvigtion. 9.1 VOR/DME Nvigtion Aircrft which use VOR/DME nvigtion re relying on network of ground bsed VOR trnsmitters for bering informtion nd DME for distnce informtion. The ircrft use rnge nd bering informtion from VOR/DME sttions of known position to estimte the position of the ircrft. However, pure VOR/DME nvigtion puts more constrints on the problem in tht ircrft usully lwys fly either to or from VOR/DME sttion long predetermined rdil rther thn using some re nvigtion (RNAV) technique. Therefore, not only does the ircrft s position need to be clculted, but lso technique to determine which VOR/DME is most pproprite for nvigtion must lso be determined. The process of VOR/DME nvigtion cn be broken into two prts. These re: Determining the ircrft position from given VOR/DME sttion Determining which VOR/DME sttion is best used for nvigtion Determining Aircrft Position from VOR/DME Sttion The VOR trnsmitters send line-of-sight RF signl tht provides bering of the irborne receiver with respect to mgnetic north. In the following discussion, it is ssumed tht the mgnetic bering ngle, hs been corrected with the mgnetic bering correction, to yield the geodetic bering ngle, B: The VOR/DME nvigtion error model tkes perfect informtion bout the ircrft s position nd corrupts it ccording to the rnge nd bering bises for given VOR/DME. This corrupted ircrft position informtion is sent to the guidnce system which guides the ircrft using the corrupted informtion

238 Consider the illustrtion in Figure 9.1. The estimted ircrft position is in error from the ctul ircrft position by certin rnge error, ρ, nd bering error, B. Generlly, the rnge nd bering from the sttion to the ircrft is clculted using the rhumb line bering nd distnce lgorithms in Section 8. ρ: The rnge to the sttion (nm) B: The bering from the sttion (deg) ρ: The totl rnge error (nm) B: The totl bering error B VOR/DME Estimted Aircrft Position B ρ+ ρ ρ Actul Aircrft Position Figure 9.1. An illustrtion of the rnge nd bering from the sttion The ctul ircrft position in the NED or surfce frme from the sttion is represented in (x,y) coordintes s defined in equtions (3.10) nd (3.1) where the terms x ct nd y ct re the ctul (x,y) loctions for the ircrft. xct = ρ cos B (1.1) yct = ρ sin B (1.) The estimted loction of the ircrft s determined from the rnge nd bering error is defined by equtions (3.13) nd (3.14). f f xest = ρ + ρ cos B + B (1.3) f f yest = ρ + ρ sin B + B (1.4) -188-

239 The position error cn be represented with x nd y. Considering the x eqution first, we cn write f f x = ρ + ρ cos B + B ρ cos B (1.5) f x = ρ + ρ cos Bcos B sin Bsin B ρ cos B (1.6) x = ρ cos Bcos B ρ sin Bsin B + ρ cos Bcos B ρ sin Bsin B ρ cos B (1.7) Linerizing with respect to the error bises, we hve eqution (3.15). f x B ref + x B ρ = ρ cos Bsin B B ρ sin Bcos B B ref ref ρ ref ρ cos Bsin B B ρ sin Bcos B B ref + cos Bcos B ρ sin Bsin B ρ ref ref ref (1.8) Assuming reference condition of ρ = B = 0, the linerized eqution reduces to eqution (1.9). x = ρ Bsin B + ρ cos B (1.9) The y eqution cn be mnipulted similrly. f f y = ρ + ρ sin B + B ρ sin B (1.10) f y = ρ + ρ sin Bcos B + cos Bsin B ρ sin B (1.11) y = ρ sin Bcos B + ρ cos Bsin B + ρ sin Bcos B + ρ cos Bsin B ρ sin B (1.1) Linerizing, we hve eqution (1.13). f y ρ ref y ρ + B = sin Bcos B ρ + cos Bsin B ρ ref ref B ref ρsin Bsin B B + ρcos Bcos B B ref ρsin Bsin B B + ρcos Bcos B B ref ref ref (1.13) -189-

240 Assuming reference condition of ρ = B = 0, the linerized eqution reduces to eqution (5.56). y = ρ sin B + ρ B cos B (1.14) Arrnging in (1.9) nd (5.56) in Mtrix/Vector form, we hve eqution (1.14) L NM x y O L QP = NM cos B ρ sin B B BQP L N M ρo sin ρ cos B Q P O (1.15) Generlly, both the ground sttion nd the irborne receiving equipment contribute to the rnge error nd bis error. The terms δρ VDA nd δb TA chrcterize the irborne receiver bises. These terms re rndomly generted when the ircrft is initilized. The VOR/DME sttion hs errors, δρ VDG nd δb TG,which need to be obtined from the VOR/DME sttion itself. Depending on wht qudrnt the ircrft is in with respect to the VOR/DME, the bis cn be different. A VOR/DME sttion needs some wy of returning the correct bis informtion when prompted with the bering from the sttion, B. Figure 9. illustrtes the reltionship between the VOR/DME nd the four qudrnts. For now, it my be esier to only hve one bis per VOR/DME. 00 IV Ι VOR/DME ΙΙΙ ΙΙ 180 Figure 9.. Illustrtion of the qudrnts of the compss rose with respect to VOR/DME sttion -190-

241 When the irborne nd ground sttion bises re summed, they cn be inserted into eqution (1.15) resulting in eqution (1.3). L NM x y O L QP = NM cos B ρ sin B OL QP d NM δρ + δρ VDG VDA d TG Ti A sin B ρ cos B δb + δb io QP (1.16) However, the position of the ircrft is represented in terms of longitude nd ltitude. Therefore conversion must be mde. Two conversion fctors re used. These re: nm deg µ : Nuticl miles per degree of ltitude nm deg : Nuticl miles per degree of longitude l Equtions (1.17) through (1.19) re used to clculte these conversion fctors. = 360 nm πre deg µ (1.17) r = r cos µ (1.18) l e c nm deg l π rl = (1.19) 360 The terms in the equtions re defined s follows: r e : The rdius of the Erth in nuticl miles r l : The rdius from the polr xis to the surfce of the Erth t given ltitude µ c : The ircrft s current ltitude Finlly, the ircrft s estimted (corrupted) position cn be clculted using equtions (3.16) nd (1.1). x µ e = µ c + nm deg l y = l + deg l e c nm µ (1.0) (1.1) -191-

242 The terms in the equtions re defined s follows: µ c : The ircrft s current ctul ltitude l c : The ircrft s current ctul longitude µ e : The ircrft s estimted ltitude l e : The ircrft s estimted longitude The estimted vlues re the finl return vlues. Hs Current Segment Been Chnged? No Is Current Segment VOR/DME segment? No Yes Yes Evlute the new segment to determine the current VOR/DME nd whether or not the current segment is VOR/DME segment Is the triling fix of the segment being used for nvigtion? Is the ircrft pssed the hlfwy point on the segment? Yes Yes No No Do Nothing Switch current VOR/DME to the leding fix of the current segment Figure 9.3. Logic for determining if the current VOR/DME used for nvigtion should be chnged -19-

243 The DME ground equipment ccurcy is 0.05nm (1σ) while the irborne equipment ccurcy is 0.5nm (1σ) or 1.5% (1σ) of rnge, whichever is greter (AC90-45A 4 ). The ccurcy of the VOR ground equipment is (1σ) while the irborne equipment is (1σ) (AC90-45A). By fr, the gretest contributor to the nvigtion error is the bering ccurcy. The DME error plys smll role. Slnt rnge error is not ccounted for explicitly in the model. This is becuse the ltitude of ech VOR/DME, which would need to be known to mke the clcultion, is not known. Furthermore, since the slnt rnge error of given sitution cn be estimted by the pilot, the pilot is likely to compenste for it when crossing fixes nd cpturing rdils. Therefore, slnt rnge error is unlikely to contribute gretly to the nvigtion error Determining the Proper VOR/DME Sttion to use for Nvigtion When n ircrft is nvigting using VOR/DME nvigtion, the pilot must tune in the VOR/DME which is ssocited with the prticulr segment which he/she is flying. The proper nv-id informtion would be retrieved from the chrt used for nvigtion. This level of relism does not exist explicitly in the TGF simultion becuse cpturing every detil nd nunce of the low ltitude victor routes nd high ltitude jet routes would be prohibitively expensive to implement. Therefore, the victor nd jet routes re not being explicitly followed. Rther, the ircrft only hs knowledge of the fixes on the route nd whether or not those fixes re VOR/DME sttions or intersections. Becuse of this simplifiction, the nvigtion system must look t the vilble nv-ids long the route nd determine which one would be most pproprite to use for nvigtion. Figure 9.3 contins the logic which is used to determine whether or not the current VOR/DME should be switched. The logic cn lso be expressed s set of the following rules: A fix is either VOR/DME or n intersection A segment is defined by two fixes which re locted t the endpoints of the segment If one of the fixes ssocited with the segment is VOR/DME, tht VOR/DME is used for nvigtion. If both fixes ssocited with the segment re VOR/DME s, then the VOR/DME closest to the ircrft is used for nvigtion. If neither fix is VOR/DME, then serch is done to find the best VOR/DME long the route to use. 4 AC90-45A: Advisory Circulr Approvl of Are Nvigtion Systems for Use in the U.S. Ntionl Airspce System -193-

244 It is lso worth noting tht the nvigtion lgorithms hve nothing to do with switching segments. However, VOR/DME nvigtion needs to be wre of switches when they occur. If the current VOR/DME needs to be switched, the logic in Figure 9.4 must be used. No Is the triling fix VOR/DME? No Is the leding fix VOR/DME? Yes Yes Neither Fix is VOR/DME Use logic for segments without VOR/DMEs Is the triling fix VOR/DME? Yes No Set the current segment s VOR/DME segment Set current VOR/DME to the leding fix Set current VOR/DME to the triling fix Figure 9.4. Logic for determining the pproprite VOR/DME for the next segment Generlly, there re either one or two VOR/DMEs on the segment. When there is only one VOR/DME, tht VOR/DME is used. If there re two VOR/DMEs on the segment, the ircrft must use the closest one. When the segment does not hve VOR/DME ssocited with it, one must be determined. The lgorithm must decide which of two VOR/DMEs is most pproprite. These two VOR/ DMEs re: The previous VOR/DME which ws used for nvigtion on the lst segment The next VOR/DME tht lies long the route but not on the current segment Such scenrio is illustrted in Figure 9.5. The ircrft lies on segment which does not hve VOR/DME but it is in between two segments tht do hve VOR/DME. From -194-

245 inspection of the drwing, we cn see tht the next VOR/DME long the route is much better choice becuse the current segment lies long rdil of the next VOR long the route. Previous VOR/DME Next VOR/DME v nlf v plf v ntf v ptf Intersection Intersection Figure 9.5. An illustrtion of the geometry used to determine which VOR/DME should be used for segments without VOR/DME To lgorithmiclly drw the sme conclusion, there re four unit vectors tht must be clculted. The clcultions cn be mde with bering clcultion lgorithm long with the vector tool of choice. It is impertive to the function of this lgorithm tht the vectors be unit vectors nd not vectors of unequl mgnitudes. These vectors re v ntf : A unit vector from the triling fix to the next VOR/DME v nlf : A unit vector from the leding fix to the next VOR/DME v ptf : A unit vector from the triling fix to the previous VOR/DME v plf : A unit vector from the leding fix to the previous VOR/DME The unit vectors ssocited with ech VOR/DME re then dotted with ech other. Idelly, the dot product is equl to unity for perfect mtch between VOR/DME nd segment. However, for the purposes of the lgorithm, we choose the higher vlue of equtions (1.)nd (1.35)s shown in Figure 9.6. vn v tf n (1.) lf v v (1.3) p tf p lf The higher vlue indictes tht the vectors re pointing nerly in the sme direction. This indictes tht the segment lies long rdil to the VOR/DME in question which mkes it good cndidte for nvigtion. There will be cses when neither VOR/DME is pproprite for nvigtion. In this cse, the lgorithm still chooses the highest dot product; however, it cn not relly be sid tht the ircrft is following rdil To or From VOR. The ircrft is re nvigting -195-

246 insted. This is not necessrily relistic procedure for n ircrft tht is flying using VOR/DME nvigtion; however, when such nomlies in the flight pln occur, it is best tht the ircrft continue to fly rther thn indicte n exception. Determine the next VOR/DME long the route Clculte v v v v p tf p lf n tf n lf If evn vnj > evp v tf lf tf pj lf Use Previous VOR/DME Use Next VOR/DME Figure 9.6. Logic For determining which VOR/DME to use when no VOR/DME lies long route 9. GPS Nvigtion There re number of error sources tht contribute to the ircrft GPS position nd velocity error. Up until 000, the dominnt error ws the GPS stellite clock error. GPS stellite clock error is n intentionl degrdtion of the GPS signl clled Selective Avilbility (SA). It ws implemented in 1990 to deny full position nd velocity ccurcy to unuthorized users fter initil testing of the GPS system reveled ccurcies much better thn nticipted. In My of 000, the United Sttes stopped degrding GPS performnce with SA. The clock error of SA in GPS nvigtion ws formerly modeled using nd Order Guss- Mrkov model. This is discussed in Appendix B. The nlysis in Appendix B develops closed-form nd order difference eqution for nd Order Guss-Mrkov process tht forms the bsis for implementing the process in code

247 Other sources of GPS nvigtion error re so much less significnt thn tht of GPS stellite clock error (from SA) tht they do not merit inclusion in this model of Aircrft Dynmics. 9.3 ILS Loclizer Error Model For n ILS loclizer, the mesured lterl devition is the ngle, B I. This cn be converted into lterl position error s follows. The slnt rnge, r IT to the runwy, is pproximted using the rhumb line distnce lgorithm. Then the lterl position, r CT is: rct = rit BI (1.4) The devition ngle is comprised of errors from ground bsed equipment nd irborne equipment s shown in eqution (1.5) where B I, G is the ground bsed component nd B I, A is the irborne component. BI BI, G + BI, A (1.5) A number of references hve determined tht the ground-bsed component of the loclizer error is not simple rndom bis. Insted, it vries with the distnce from the runwy. A convenient model for this error source is to tret it s sptil first-order Guss Mrkov s shown in eqution (1.6). By tht is ment tht the error does not vry with time but with the loction of the receiver from the ILS loclizer trnsmitter. where, nbf: s Scled Gussin white noise β B f: s Sptil dmping fctor f f f f BI, G s = β B s BI, G s + nb s (1.6) s ds ds =F I H dt vit t dt dt K = f (1.7) β Bf t = vitf t β Bf s (1.8) f f f f I, G B I, G B B t = β t B t + n t (1.9) -197-

248 The ctul ILS loclizer bem bending errors for five different irports re illustrted in Figure 9.7. A set of five simulted ILS loclizer bem bending errors using the bove sttisticl model is presented in Figure 9.8. Figure 9.7. Mesured ILS Loclizer Bering Devition Angle Errors -198-

249 0. 5 I L S L o c l i z e r E r r o r s Bering (deg) D i s o w P t h t n c e f r m R u n y ( n m ) 0. 5 I L S L o c l i z e r E r r o r s Bering (deg) P t h D i s t n c e f r o m R u n w y ( n m ) 0. 5 I L S L o c l i z e r E r r o r s Bering (deg) D i s o w P t h t n c e f r m R u n y ( n m ) 0. 5 I L S L o c l i z e r E r r o r s Bering (deg) D i s o w P t h t n c e f r m R u n y ( n m ) 0. 5 I L S L o c l i z e r E r r o r s Bering (deg) D i s o w P t h t n c e f r m R u n y ( n m ) Figure 9.8. Simulted ILS Loclizer Bering Devition Angle Errors -199-

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251 10. The Longitudinl Guidnce System When n ircrft flies long route it is often necessry to hve the ircrft utomticlly meet speed nd ltitude constrints tht re plced on fixes long the route. These re generlly termed crossing restrictions. One cse where crossing restrictions re often used is when modeling flight long Stndrd Arrivl Route (STAR). When the flight pln contins STAR for the pilot to follow, it is ssumed tht the pilot hs t lest textul description of the STAR nd will mke pproprite speed nd ltitude chnges s published. This chpter describes procedures by TGF-simulted ircrft in meeting or prepring to meet longitudinl route restrictions. The procedures will involve the deployment of highlift devices nd trgeting of desired speeds nd ltitudes Aircrft Device Deployment Aircrft devices include high-lift devices (ie, wing flps nd slts) nd lnding ger. This section describes their deployment Flps on Approch For ll ircrft, the flps (or, more ppropritely, high-lift devices) re modeled, s they re in BADA, with lift nd drg coefficient vlues corresponding to the flp nd slt deployment for the different wing configurtions for the ircrft. The five configurtions re cruise, initil climb, tke-off, pproch, nd lnding. We hve chnged the nmes to flps0, flps1, flps, flps3, nd flps4, respectively, to remove the impliction tht prticulr configurtion is used in prticulr phse of flight. Typiclly, piston ircrft hve only 4 configurtions, so flps3 nd flps4 re identicl. The stndrd flp schedule on pproch is presented in Tbles Tble 10.1 nd Tble 10.. This is the schedule tht is followed under stndrd pproch. Once the ircrft s speed drops below the given flp set speed, tht configurtion is set. Flps my be extended on more ggressive schedule if the ircrft needs to lose energy more quickly, but never when the ircrft is bove the flp limit speed. The flp limit speed is 30 knots bove the flp set speed for jets nd turboprops nd 0 knots bove the flp set speed for pistons. Flp Configurtion Set Speed (kts) flps0 flps1 1 3V stllld flps 1 3V stllld flps3 1 3V stllld flps4 1 3V stllld Tble 10.1 Flp Deployment Schedule for Jet nd Turboprop Aircrft on Approch -01-

252 Flp Configurtion Set Speed (kts) flps0 flps1 1 3V stllld flps 1 3V stllld flps3 1 3V stllld flps4 1 3V stllld Tble 10. Flp Deployment Schedule for Piston on Approch Flps on Tkeoff The lgorithm for the deployment of flps in tkeoff flight is presented in Figure Bsiclly, tkeoff flps re deployed for tkeoff nd retrcted s the ircrft climbs. Flps re not deployed in cruising flight Speed Brkes Speed brkes re modeled in the simultor s n increment, C D brkes, to the profile drg coefficient. Speed brkes re deployed if n ircrft is sked to expedite its descent. Algorithms within the simultor guidnce module will deploy speed brkes in the cse tht the ircrft is predicted to be unble to rech n ltitude-fix restriction. Speed brkes my lso be deployed in the lnding sequence if the ircrft is well bove the glide slope. Figure 10.1: Flp Deployment Algorithm -0-

253 Spoiler The spoiler is modeled in the simultor s n increment, C D spoiler, to the profile drg coefficient nd complete loss of lift. It is used in the Touchdown region (Region 10) only nd is deployed immeditely when the ircrft s ltitude is the sme s the runwy ltitude. Region 10 uses n open-loop controller with the thrust set to touchdown thrust nd the lift coefficient set to zero ( consequence of spoiler deployment) Lnding Ger Lnding ger re modeled in the simultor s n increment, C D ger, to the profile drg coefficient. They re extended for the tkeoff sequence nd retrcted once the ircrft reches 400 feet AGL. They re lso extended during the lnding sequence once the ircrft hs pssed the outer mrker of its ssigned runwy Ground Brking When the brking function ws initilly conceived, it ws thought tht some number from dt file would be red into the irfrme model, nd n increse in drg would result from some sttic brking force. However, the ircrft dt files did not hve ny informtion regrding brking force. Eqution (5.1) is proposed ground breking model tht ssumes tht the brking force is 30% of the ircrft s weight. This pproximtion is convenient becuse it does not rely on n independent ground brking prmeter in the ircrft dt files. However, ground breking is not currently implemented in the TGF simultor. D brke = 0.3W (5.1) 10. Prepring for Approch nd Lnding A pilot tht is pproching terminl irspce cn infer certin detils bout his descent nd decelertion. He my hve flown in the irspce before nd so he knows how controllers usully guide him in to the runwy. He my hve witnessed the pths of the ircrft in front of him. In either cse, he knows tht it would behoove him to begin either descending or decelerting or both within the implied constrints of the irspce or the constrints imposed on him by the controller. c These re detils tht re difficult to convey to computer progrm tht requires bsolutes in the form of trget speeds nd ltitudes. At the very lest, we need to inform the simultor tht the ircrft will soon begin its pproch to the irport nd tht it my begin descending nd decelerting within imposed constrints. This cuses the ircrft to -03-

254 mke decisions on how quickly it needs to lose energy nd whether or not to deploy drg incresing devices Energy Grdient In relity, pilots do not mke clcultions compring the energy grdient needed with the ircrft s bility to lose energy, but they do mke intuitive judgments of this comprison. Our technique of modeling this judgment is to mke those clcultions. The simultor will clculte its current mechnicl energy stte (bsed on speed nd ltitude) nd trget mechnicl energy (bsed on preferred speed nd ltitude upon crossing some fix, e.g., the outer mrker) nd divide the difference by the distnce to the fix nd compre tht to its bility to lose energy. If the ircrft needs to increse its bility to lose energy, it cn deploy drg incresing device, such s speed brkes or flps. This is referred to s dirtying the ircrft. An ircrft s specific mechnicl energy is the sum of its specific kinetic energy nd specific potentil energy, s defined by its speed nd ltitude. This ws first shown in eqution (5.11). V G e = + gh (5.11) We define n ircrft s energy grdient s the rte of chnge of specific mechnicl energy per unit chnge in ground distnce. de eg (10.1) dx The ircrft s ctul energy grdient is mesure of its bility to lose energy. Combining eqution (10.1) with eqution (5.11), we get, dv dh dv dt dh dt = + = + dx dx dx dt dx dt h eg = V G + g (10.) V G G eg VG g VG g G Compring this to eqution (5.19) gives n eqution for the ircrft s energy grdient in terms of thrust, drg, nd mss. T D eg = (10.3) m -04-

255 The energy grdient needed is defined s the difference between the trget nd current specific mechnicl energies divided by the ground distnce to the trget. This cn be represented in discrete nottion. eg e e x t 0 n = (10.4) An ircrft on pproch will be descending nd decelerting nd will, therefore, hve negtive energy grdient. An ircrft cn decrese its energy grdient (i.e., mke it more negtive) by deploying drg incresing devices. A comprison of the energy grdient, eq. (10.3), with the needed energy grdient, eq. (10.4), will determine if this is necessry. If the needed energy grdient is less thn the ctul energy grdient, deployment of drg incresing device my be necessry. Under norml pproch conditions, n ircrft s energy grdient will decrese (i.e., become more negtive) s it slows down nd deploys flps in preprtion for lnding. Becuse the ircrft is typiclly in clen configurtion bove bout 8000 feet ltitude, the needed energy grdient my be less thn the ctul energy grdient even though the ircrft is following typicl glide rtio nd there is no urgency for dirtying the ircrft. The lgorithm developed for drg device deployment must consider this. If the ircrft is on norml glide rtio bove 8000 feet, the ircrft will not increse its urgency. The glide rtio is something tht irline pilots normlly consider on pproch. A glide rtio of 3:1, distnce:height, with distnce to the runwy mesured in nuticl miles nd height mesure in thousnds of feet bove ground level (AGL), is considered typicl. For exmple, n ircrft tht is 30 nm from the runwy nd t 10,000 feet hs glide rtio of 3:1. The eqution for glide rtio is presented below. x( nm) glidertio = (10.5) h ( 1000 ft) If n ircrft is not on norml glide rtio bove 8000 feet nd its needed energy grdient is less thn its ctul grdient, it will hve trouble losing energy in its descent. The simultor will mrk this ircrft s needing to increse its urgency in getting down. This cn men deploying flps, speed brkes, or lnding ger s pproprite. The simultor will respond to n incresing urgency by first trying to set the next increment of flps (fter checking to see tht the ircrft is below the flp limit speed), then the speed brkes, then the lnding ger. The lgorithm for compring the needed nd ctul energy AGL -05-

256 grdients for incresing urgency is presented in Figure 10.. The lgorithm for the deployment of drg devices is presented in Figure h < 8000 nd glidertio.5? no no eg < eg n? yes speed brkes on nd eg no sb < eg n? no yes yes Increse urgency Turn off speed brkes Exit Figure 10.: Compring needed nd ctul energy grdients to determine urgency Increse urgency in V > (V limit ) next flp? yes Speed brke on? yes V > (V limit ) ger? yes no no no set next flp Deploy speed brkes Deploy lnding ger Return Figure 10.3: Procedure for incresing urgency (i.e., deploying drg devices) -06-

257 Speed brkes re convenient drg device used to increse n ircrft s bility to lose energy. When the speed brkes re deployed, they increse the ircrft s prsite drg coefficient by n increment given in the ircrft dt file. Speed brkes typiclly work so well tht fter period of use, they re no longer needed. If speed brkes re deployed nd the ctul energy grdient is less thn the needed energy grdient, we cn compre the needed energy grdient ginst the ircrft s energy grdient without speed brkes to see if they cn be retrcted. The ircrft s mss nd descent thrust re independent of speed brke deployment. From eq. (10.3), the ircrft s energy grdient without speed brkes is given by the following. eg no sb T D = m no sb The difference between the energy grdients with nd without speed brkes is then clculted by subtrcting eq. (10.3). eg no sb ( + ) ( + ) 1 D V CD KC 0 L CD KC 0 L no sb no sb D eg = = m ρ m The speed brkes ffect neither the induced drg coefficient nor the lift coefficient. eg no sb ( ) ( ) 1 ρv C C 1 V C eg = = m m D0 D 0 no sb ρ D speedbrkes So the difference in the energy grdient in the two configurtions is function of the prsite drg coefficient increment of the speed brkes. We cn simplify this expression further by using n pproximtion for the lift coefficient. In stedy, wings-level descent, the lift is equl to the verticl component of weight. L = Wcos γ C L Wcos γ = 1 ρv Neglecting the flight pth ngle, γ, we get, -07-

258 C L mg =, 1 ρv resonble pproximtion for the lift coefficient. The energy grdient without speed brkes is then, eg no sb gcd = eg + speedbrkes (10.6) C L 10.. Distnce Remining to Runwy In order to clculte the needed energy grdient of eqution (10.4) nd the glide rtio of eqution (10.5), we need to determine the distnce remining to the runwy. This cnnot be clculted simply s the distnce between the ircrft s current position nd the runwy threshold becuse the ircrft my not be pointed t the runwy. In the cse of simple pproch, n ircrft is following pre-determined route to the runwy nd so the distnce to the runwy is known. But in mny cses, the ircrft does not know its directed route to the runwy. If the ircrft is on downwind leg, it could be vectored to the pproch by the controller t ny time. In this cse, we hve to mke ssumptions bout the typicl pproch Distnce Remining to Runwy long Filed Route If the ircrft s lterl guidnce system is following filed route nd the lst fix on the route is irport t which the ircrft is lnding, then the distnce remining to the runwy is the distnce left long the route. If the irport is not the lst fix, then the irport fix is ppended to the filed route nd the distnce remining to the runwy is the distnce left long the route. For the purpose of clculting the energy grdient on pproch, the remining route distnce is typiclly limited to mximum of 0 nm. As discussed in Chpter 8, TGF routes re typiclly Trck-to-Fix (TF) segments with flyby wypoints. An estimte of the remining route distnce cn be obtined by dding the distnces, S, of the segments between the fixes, but this ignores tht the segments re shortened by flying by the termintion fixes when trnsitioning between segments. We cn get better estimte by predicting the ircrft s curved pth in flying by the termintion fixes. Let us ssume tht the pth flown by n ircrft in trnsitioning between TF segments with fly-by wypoints is constnt rdius turn with constnt bnk ngle (ie, instntneous bnk dynmics). Using this ssumption nd referring to Figure 8.16, the length of the trnsition rc is given by, r t ψ. (The ssumed constnt turn rdius, r t, is obtined from eqution (8.5) using the speed in the route nd stndrd bnk ngle.) -08-

259 For ech trnsition, the route distnce is shortened by twice the offset length nd incresed by the trnsition rc length. For n segments, our estimte of the route distnce is n n 1 i ( t offset ) RouteDistnce = S + r l i= 1 i= 1 Using eqution (8.13) for l offset nd ignoring the lg fctor, k, this becomes n n 1 ψ RouteDistnce = Si + rt ψ tn i= 1 i= 1 (10.7) Distnce Remining to Runwy for Vectored Aircrft If the ircrft s lterl guidnce system is following vector, we hve to mke ssumptions concerning the vectored pproch. These ssumptions re bsed on typicl vectored pproch tht meets Federl Air Regultions (FARs). They re s follows: Aircrft will merge onto the loclizer bering on 30 intercept, nd Aircrft will merge onto the loclizer bering t lest 3 nm before the outer mrker. We ssume tht the ircrft will fly its current heding until crossing the 30 intercept, then follow the 30 intercept to the loclizer bering, then follow the loclizer bering to the runwy. This pproch is illustrted in Figure In order to develop n lgorithm for the ssumed vectored pproch, the following vribles re defined. P vector from position 3 nm from the outer mrker (long the loclizer bering) to the ircrft d P-rwy the distnce between position 3 nm from the outer mrker to the runwy threshold ψ p the bering from position 3 nm from the outer mrker (long the loclizer bering) to the ircrft ψ h the ircrft heding (Actully, the ground trck heding is the ccurte ngle to use, but the ircrft heding will yield suitble pproximtion.) the loclizer bering ψ loc Note: ll ngles nd ngle differences re defined in the rnge [0,360]. -09-

260 ψ h P d 1 ψ loc OM 3nm ψ int 30 d Figure 10.4: Illustrtion of Assumed Vectored Approch By inspection of Figure 10.4, we write the following vector eqution. d ψ d ψ = P = P ψ int 1 h P Resolving into est - north components, we get, d sinψ d sinψ = Psinψ int 1 d cosψ dcosψ = Pcosψ int 1 h h P P In mtrix form, sinψ h sinψint d1 sinψ P = P cosψ cosψ d cosψ h int P Solving the mtrix eqution, 1 = d 1 d1 sinψ h sinψint sinψ P P d = cosψ cosψ cosψ h int P d1 P cosψint sinψint sinψ P d = sinψ cosψ cosψ sinψ cosψ sinψ cosψ d int h int h h h P P sinψ Pcosψint cosψ Psinψint sinψintcosψ h cosψintsinψ sinψcos cos sin h P ψ h ψ P ψ h From the trigonometric identity, -10-

261 The solution for d 1 nd d becomes, or d sin cos b cos sin b sin ( b) sin ( ψ P ψint ) ( ψ ψ ) 1 P d = sin ( ψ h ψint ) sin P h P, ψ p, ψ h, ψ loc ψ int = ψ loc + 30 yes 30 (ψ p - ψ loc ) 180 no ψ int = ψ loc 30 yes 180 (ψ p - ψ loc ) 330 no x = d c-rwy sin d1 = P sin d sin = P sin ( ψ P ψint ) ( ψint ψ h ) ( ψ P ψ h ) ( ψ ψ ) int h d1 < 0 or d < 0 no yes x = 0 nm x = d 1 + d + d P-rwy x Figure 10.5: Algorithm for Determintion of Distnce Remining to Runwy -11-

262 sin d1 = P sin d sin = P sin ( ψ P ψint ) ( ψint ψ h ) ( ψ P ψ h ) ( ψ ψ ) int h (10.8) And the distnce remining to the runwy is given by x = d 1 + d + d P-rwy (10.9) If either d 1 or d is negtive, then the heding will not intersect the loclizer intercept. In this cse, we ssume x = 0 nm. If the ircrft is between the left nd right loclizer intercepts, we will ssume tht the distnce remining to the runwy is the stright line distnce. (This ssumptions fils to consider tht the ircrft my be between the intercepts nd moving wy from the runwy, but tht is n unlikely cse.) Filing this cse, if the ircrft is left of the loclizer bering, it is ssumed tht the ircrft will cross the left intercept, if right, then the right intercept. In ny cse, if the clculted vlue for the distnce remining to the runwy is greter thn 0 nm, then we set x = 0 nm. Figure 10.5 illustrtes this lgorithm Crossing Restrictions in Descent When n ircrft is sked to descend nd cross fix t specific ltitude, norml procedure is for the ircrft to mintin the current ltitude for s long s possible. This requires clcultion of the top of descent point. The reson for the dely in beginning the descent is tht ircrft re more efficient t higher ltitudes nd so prefer to fly there. Conversely, there is no need to dely climb if the crossing restriction is bove the current ltitude becuse the ircrft prefers to be t the higher ltitude sooner rther thn lter. Experience shows tht ircrft in constnt CAS descent below the tropopuse hve very little vrition in flight pth ngle. Our pproch in the TGF model is to mke quick estimte of the top of descent point, pply smll mrgin of sfety (i.e., mke the glide slope shllower), nd to fly with dul control long the clculted flight pth ngle in the sme mnner we fly long n ILS. We mke considertion for lternte vritions in true irspeed by estimting the energy rtio. Since we will hve no prior knowledge of wind vrition with ltitude, we will neglect the effects of wind. The first step is to mke n estimte of the flight pth ngle. Eqution (.59) is rewritten here. -1-

263 T D V = g sin γ (.59) m Once gin, we re-write this in the form of chnging energy, removing the subscript,, referring to the irmss. The energy rtio, ER, is defined s nd so the flight pth ngle is given by T D g sin γ + V = m sin γ 1 V + V = T D g h W VV V dv T D sin γ 1+ sin γ 1 gh = + = g dh W V dv ER = g dh T D sin γ = W 1 ( 1+ ER) (10.10) To mke n estimte of the flight pth ngle, we first need n estimte of the energy rtio. We clculte the energy rtio verged over the entire descent, from point 1 to point, using n verge vlue for the velocity. Our pproximtion for the energy rtio over the entire descent is given by V + V1 1 ER = = g h h g h h ( V V1 ) ( ) ( V V ) ( ) 1 1 (10.11) Then we solve for the flight pth ngle using eqution (10.10) nd the top of descent vlues for thrust, drg, nd weight. To llow for some mrgin of sfety, the desired flight pth ngle 10% shllower. -13-

264 γ d T D 1 = 0.9 rcsin W ( 1+ ER) (10.1) The next step is to obtin the distnce to the crossing fix, d fix nd the top of descent distnce, d TOD. For ircrft flying route, the distnce to the crossing fix is the distnce long the route to the fix. For ircrft on vectors, this is the rhumb line distnce to the fix. The top of descent is the distnce from the fix t which the flight pth intersects the current ltitude. The ircrft will mintin stedy, level flight t the current ltitude nd distnce until one nuticl mile before the top of descent. d TOD h h tn γ 1 = (10.13) d Once in the descent, the desired ltitude is the height of the desired flight pth t distnce d fix from the crossing fix. hd = h d fix tn γ d (10.14) Recll tht the control system needs desired ltitude rte, not desired ltitude. As in Region 7 for stedy, level flight, eqution (5.7) is used to determine the ltitude rte tht would correct the ircrft s ltitude error, but we must lso consider tht the desired ltitude is chnging per eqution (10.14). We dd the derivtive of eqution (10.14) to eqution (5.7) to get the desired ltitude rte. d h = K h h d dt ( ) ( ) tn d h d fix d The time derivtive of the ircrft s distnce from the fix is simply the negtive of the ircrft s groundspeed, since the ground speed nd the distnce from the fix re defined positive in opposite directions. γ ( ) tn h = K h h + V γ (10.15) d h d G d. The desired speed during the descent is given either by the preferred speed profile for the prticulr ircrft type in descent or tht given per the speed restriction nd eqution (5.9) or (5.10) for IAS-bsed or Mch-bsed speed control. -14-

265 11. Flight Technicl Error The flight technicl error (FTE) is the inbility or inexctness of the pilot or utopilot to steer the ircrft perfectly long the desired course. If the ircrft is steered by n utopilot, it is the error in steering the ircrft perfectly long the intended course. The wypoint nd nvigtion id errors re independent of the FTE. Field dt indicte tht there is rndom lterl FTE component tht exists long the route segments. For the FMS-guided ircrft, the rndom en route wnder ws found to be 0.13 nm. (1σ) while for the non-fms-guided (piloted) ircrft, it ws found to be 0.7 nm (1σ), with period vrying from roughly 4 to 8 minutes during the en route flight segment (Hunter, 1996). A resonble model for this rndom lterl position wnder, δr FTE, is described by second order Guss Mrkov process: F HG δr FTE 0, 1, δv - ω, - β ω FTE I KJ =L N M 0 0 O δrfte c Q P F H G I δv K J +F H G 0 I u K J FTE FTE (1.1) The terms in the expression re defined s follows: δr FTE : The lterl position error (nm) δv FTE : The lterl position error velocity (nm/sec) c: The scle fctor of the forcing function ω 0 : The nturl frequency of the system β : The dmping of the system u FTE : The zero men unity vrince Gussin white noise For terminl flight segment during ILS loclizer guidnce, it ws found tht the lterl wnder tended to increse linerly with the distnce from the runwy (Timoteo, 1990). This suggests tht FTE bsed on bering devition ngle wnder is more pproprite during the terminl flight phse. Therefore, rndom wnder of 0.4 degrees (1σ) with n pproximte time constnt of 90 seconds is pproprite. In the cse of the ILS, the second order Guss Mrkov process is written in terms of bering devition ngle wnder s shown in eqution (1.13). -03-

266 F H G δb ILS, FTE,, δbils, FTE δ J L = c Ω N M ω, - β ω δω ILS, FTE ufte ILS, FTE I K 0 0 O Q P F H G I K J +F H G I K J (1.) The new terms in eqution (3.1) re s follows: δb ILS, FTE : The bering devition ngle (deg) δω ILS, FTE : The bering devition rte (deg/sec) 11.1 Opertionl Detils The flight technicl error is quite simple to implement using the Guss Mrkov processes with vlid error prmeters. The three types of flight technicl error only operte when the ircrft is operting under the route following guidnce system. This guidnce lgorithm must prompt the prticulr flight technicl error model being used for n updte to the lterl position devition, δr FTE Piloted Flight Technicl Error The piloted flight technicl error proceeds once the Guss Mrkov process hs been initilized. For ech time step tht the piloted flight technicl error is used, the Guss Mrkov process is dvnced one time step nd vlue for δr FTE is returned. The piloted flight technicl error uses Guss Mrkov process with the following prmeters: β = 0. 50: The dmping term. σ p = 0. 7 nm: The stndrd devition of the position σ v = nm sec: The stndrd devition of the velocity FMS Flight Technicl Error The FMS flight technicl error proceeds once the Guss Mrkov process hs been initilized. For ech time step tht the FMS flight technicl error is to be used, the Guss Mrkov process is dvnced one time step nd vlue for δr FTE is returned. The FMS flight technicl error uses Guss Mrkov process with the following prmeters: β = 0. 50: The dmping term. σ p = nm : The stndrd devition of the position σ v = nm sec: The stndrd devition of the velocity ILS Flight Technicl Error ILS flight technicl error is more complex becuse the Guss Mrkov process is set up to return n ngulr devition from the pth rther thn liner distnce. Therefore the liner distnce must be clculted from the ngulr devition δb ILS, FTE which is returned -04-

267 in degrees. To get the lterl offset, eqution (3.15) is used where d s is the distnce to go to the loclizer. Generlly, the ILS is modeled s two segment route, where the first cpture segment is from some rbitrry initil pproch fix to the finl pproch fix, nd the second segment is from the finl pproch fix (the beginning of the glide slope for ILS pproches) to the loclizer. Therefore, the distnce tht the ircrft is to the loclizer cn be clculted by using the rhumb line distnce to fix lgorithm. π δ r d sin δb 180 = F H I K ILS, FTE s ILS, FTE (1.3) The ILS flight technicl error uses Guss Mrkov process with the following prmeters: β = 0. 50: The dmping term. o σ p = 0. 3 : The stndrd devition of the position deg σ v = sec : The stndrd devition of the velocity -05-

268 THIS PAGE INTENTIONALLY LEFT BLANK 1. Model Verifiction nd Vlidtion Verifiction nd vlidtion of the lgorithms used to develop the TGF simultion ws ccomplished primrily by using smll JAVA tool tht served s testing pltform for the lgorithmic development. This tool, which is nmed TGF-test, llowed for rel time mnipultion of ircrft trjectories on the screen nd lso monitored mny of the ircrft s stte vribles on the screen in the form of stripchrts. All lgorithms tht re coded in the min TGF simultion were first tested nd evluted in the TGF-test simultion. The min screen of the TGF-test lgorithm is shown in Figure 1.1 where ircrft trjectories re superimposed over n electronic mp of fixes nd routes. The ircrft icon, which represents the flying ircrft, shows the heding orienttion so the difference between ground trck nd heding cn lso be viewed visully. -06-

269 Figure 1.1. Simultion Window for TGF-test The spects of the TGF simultion tht needed verifiction re s follows: Constnt irspeed climbs nd descents Mch/CAS descents nd CAS/Mch climbs Speed chnges during climbs nd descents Automtic route cpture Vectored route cpture Initil fix route cpture Segment trnsition Flight technicl errors Nvigtion errors Tke-off nd lnding 07

270 When pproprite, the TGF simultion model ws compred to Pseudocontrol, the ircrft dynmics kernel of PAS. PAS, the NASA tool for trjectory genertion, hs been considered s n cceptble bseline for ircrft performnce. Such cses include the verifiction of climb nd descent performnce s well s speed chnges. For other opertions, such s route cpture nd route following, visully inspecting the mneuvers is sufficient to insure proper opertion. 1.1 Constnt Airspeed Climbs nd Descents The PAS model in Pseudocontrol uses much higher fidelity ircrft nd engine models thn wht the TGF model uses, so it is expected tht there would be some vrition in performnce. Generlly, however, the difference in the ctul trjectories generted by the simultions is negligible. While the trjectories re nerly identicl, the TGF model does not produce fuel burn estimtes which re s ccurte s the PAS model becuse PAS uses mny more coefficients in the model. Two comprisons of Pseudocontrol nd TGF-test re presented in this section. The first comprison between Pseudocontrol nd TGF-test is shown in Figure 1.. Figure 1. illustrtes n MD-80 t 10,000 ft nd 80kts s it initites constnt indicted irspeed climb to 30,000ft. Four stripchrts re shown in the plot, ech representing different ircrft stte vrible. These re Mch, indicted irspeed, ltitude, nd the lift coefficient. The Pseudocontrol plots re represented with the drk line nd the TGF-test plots re shown in gry. The simultion shows good mtch between the two models. Initilly, there is smll fluctution in the indicted irspeed of both models while the climb is estblished. Once the climb is estblished, both models hold the pproprite 80kt irspeed. The ircrft climb nerly identiclly in terms of ltitude trcking. This is very importnt since the ir trffic controllers re sensitive to the chnging rte t which ltitude increses. The Mch plot shows tht both models trck 08

271 0.8 Mch IAS (kt) x 104 Altitude ft CL time (sec) Figure 1.. Comprison of Pseudocontrol (blck) nd TGF-test (gry) in constnt indicted irspeed climb nd 80kt Mch number identiclly s well. Considering the lower fidelity model represented in the TGFtest system, over Pseudocontrol, the dt mtch is quite good. A similr comprison is mde in descent. An MD80 weighing 130,000lbs is commnded to descend from 30,000ft to 10,000ft t n irspeed of 300kts. The descent is shown in Figure 1.3. When the ircrft initite the descent, there is some fluctution in the indicted irspeed. In this exmple, the TGF-test model hs lrger fluctution thn the Pseudocontrol simultion, but the fluctution is still only 1.5kts. This smll fluctution is cceptble. Once the descent is estblished, both ircrft hold the commnded irspeed well. The ltitude profile of the TGF-test ircrft mtches the Pseudocontrol ircrft well nd the Mch number vries properly lso. 1. Mch/CAS descents nd CAS/Mch Climbs The idle thrust Mch/CAS descent nd full thrust CAS/Mch climb re importnt fetures of the ircrft simultion becuse jet irliners re most likely to use these types of mneuvers for climbs nd descents. Becuse the mneuvers re similr, this section will considers s its only test cse, the idle thrust descent. Full thrust CAS/Mch climbs re simply reverses of the descents. 09

272 When n idle descent is initited, the throttle is pulled idle nd the pilot descends t rte so tht the ircrft mintins the desired Mch of the desired Mch/CAS pir. At low speed nd low ltitude, convention dicttes tht the speed of ircrft be mesured in terms of indicted irspeed. Therefore t some point during the descent, the pilot will cpture the desired CAS of the Mch/CAS pir. As the pilot descends t constnt Mch, the indicted irspeed meter will show n increse in speed. At some point during the descent, the indicted irspeed meter will red the desired indicted irspeed for the 0.8 Mch IAS (kt) x 104 Altitude ft CL time (sec) Figure 1.3. A comprison of Pseudocontrol (blck) nd TGF-test (gry) in descent t constnt indicted irspeed of 300 kts descent. At this point, the pilot trcks the desired CAS insted of the Mch. Typiclly there re four stges to Mch/CAS descent. These stges re: 1. Chnge speed from the cruising Mch, M 1, to the descent Mch, M.. Descend t M. The ircrft descends tm until reching predetermined CAS. 3. Descend t constnt CAS. The ircrft descends t its constnt descent CAS until it reches the metering fix crossing ltitude, where it levels off. 4. Decelerte to the metering fix crossing speed. Finlly, the ircrft decelertes to 50 kt, the metering fix crossing speed of typiclly 50kt. 10

273 For this section s comprison between Pseudocontrol nd TGF-test, we strt with n MD80 t 30,000ft in cruise t M0.76. The ircrft initites Mch/CAS descent with the following speeds: (M0.76/30kt). The ircrft then levels out t 10,000 ft mintining 30kts. Figure 1.4 shows the mneuver. The mtch between the Mch nd indicted irspeeds is good nd the trnsition between Mch nd indicted irspeed is smooth without ny undesirble trnsients. Similrly, the level off t 10,000 ft is smooth without ny overshoot. Most importntly, the ltitude profiles for both simultions mtch very well. 0.8 Mch IAS (kt) x 104 Altitude ft CL time (sec) Figure 1.4. Comprison of Pseudocontrol (blck) nd TGF-test (gry) performing Mch/CAS descent from 30,000 ft to 10,000ft using n MD80 t 130,000lb 1.3 Speed Chnges Speed chnges, while they do not tke much time in the course of flight, do give good indiction of the model s fidelity in terms of drg nd thrust. Consider n ccelertion. For the ircrft to hve the proper ccelertion, in speed up mneuver, the excess thrust must be correct. This excess thrust is function of the totl vilble thrust nd the totl drg. If either is off, the ccelertion will not be right. However, errors in either could cncel ech other out. For instnce, high drg number could be cnceled by high thrust vlue. Decelertions, becuse they re performed t idle thrust, tend to remove the thrust from the system so the primry decelertion fctor is the ircrft s drg. If ccelertions nd decelertions re both studied, 11

274 generlly conclusions bout both the drg nd thrust cn be mde. The decelertion gives insight into the fidelity of the drg 0.8 Mch IAS (kt) x 104 Altitude ft CL time (sec) Figure 1.5. A decelertion of n MD80 from 350kts to 50kts while t 10,000ft using Pseudocontrol (blck) nd TGF-test (gry) simultion tools model, nd the ccelertion gives insight into the thrust to drg rtio. Assuming drg informtion is known from the decelertion, thrust informtion cn be derived from the ccelertion. First, decelertion is considered. An MD80 t 130,000lbs cruising t 10,000 ft is slowed down from 350kts to 50kts. Figure 1.5 shows the decelertion mneuver. The two simultions show good greement during the slowdown with no undesirble trnsients in either simultion. Furthermore, ltitude is held constnt t 10,000ft. From this plot we cn ssume tht the drg informtion in the MD80 model is ccurte. Next, n ccelertion is considered. The sme MD80 is ccelerted from cruise condition of 50kts nd 10,000ft to 350kts while mintining ltitude. The mneuver is shown in Figure 1.6. The slope of the speed curves for both simultions mtch very well, suggesting tht the thrust model for the ircrft is working well. The ltitude is held resonbly well; however, the TGFtest model does show slight tendency to let the 1

275 0.8 Mch IAS (kt) x 104 Altitude ft CL time (sec) Figure 1.6. An MD80 ccelerting from 50kts to 350kts while mintining 10,000ft using Pseudocontrol (blck) nd TGF-test (gry) simultion tools ltitude drift slightly. The ltitude vrince shown on the plot is on the order of 10ft, so it is not mjor concern. From the results in Figure 1.5 nd Figure 1.6 we conclude tht the thrust nd drg models for this ircrft re good. Speed chnges re lso performed t higher speeds to test the compressibility drg model. First decelertion is considered s shown in Figure 1.7. An MD80 t 30,000ft nd Mch 0.8 is decelerted to Mch 0.6. There is slight discrepncy here between the Mch numbers of the two models. The TGF-test simultion tkes longer to enter the decelertion wheres the Pseudocontrol model enters the decelertion immeditely. One reson for this gentle initition of the mneuver is tht the TGF-test simultion models engine spooling wheres the Pseudocontrol model does not. Once the decelertion is estblished, notice tht the two Mch lines re nerly prllel. This suggests tht the rte of ccelertion is very close but the slow initition time on the prt of TGF-test offsets the mneuver. The result is tht the mneuver tkes 7-10 seconds longer with TGF-test thn it does with Pseudocontrol. Since the rte of ccelertion is nerly the sme, we know tht the drg models re very close. The difference in initition is explined by the differences in the spooling lgs of the engines nd some differences in control system design. 13

276 If the longer decelertion is problem, the effectiveness of the spooling lgs could be decresed. Presently this is not concern. 0.8 Mch IAS (kt) x 104 Altitude ft 3 CL time (sec) Figure 1.7. An MD80 decelerting from Mch 0.8 to Mch 0.6 while mintining 30,000ft using Pseudocontrol (blck) nd TGF-test (gry) simultion tools The Mch ccelertion is shown in Figure 1.8, where the MD80 ccelertes from Mch 0.6 to 0.8 while mintining 5000 ft of ltitude. Here we do not see the sme spooling lgs in the initition of the mneuver. Since the spooling lgs re the sme for both increses nd decreses in thrust level in the TGF-test model, there is no esy explntion for the discrepncy. At ny rte, the two models re virtully identicl in the Mch ccelertion. 1.4 Speed Chnges during Climbs nd Descents One of the more insidious problems encountered during the design of the longitudinl control system ws the problem of chnging speeds during climbs nd descents. Section 4. of the longitudinl control system discusses the problem in depth nd explins how the ultimte solution to the problem ws the rmping of inputs. Generlly, the problem centered round the fct tht lrge speed chnges while climbing or descending ws not nticipted, nd the feedbck control system ws only set up to hndle smll chnges. This ment tht the control system hd 14

277 rther high gins to keep the speed errors smll. When the high gins were pplied to the lrge errors, the mneuvers becme 1 Mch IAS (kt) x 104 Altitude ft CL time (sec) Figure 1.8. An MD80 ccelerting from Mch 0.6 to Mch 0.8 while mintining 5,000ft using Pseudocontrol (blck) nd TGF-test (gry) simultion tools violent. It ws esy to see wht ws hppening. During climb nd descent, the control stick rther thn the throttle is used to control speed. Therefore, when the feedbck control system sw lrge errors from user, the control stick ws moved violently to correct the sitution. This section uses TGF-test to demonstrte the speed chnges during climbs to illustrte the stbility of the mneuvers using the rmped inputs. Figure 1.9 illustrtes n MD80 which is initilly t 10,000ft nd 50kts. A climb is initited to 30,0000ft. During the climb, the speed of the ircrft is first incresed to 30kts nd then reduced to 80kts. Finlly the speed is incresed to 300kts where it is held until the ircrft completes the climb. The point of the plot is tht speed chnges re mde grcefully nd do not cuse ny unusul ptterns in the ltitude profile other thn the norml ffects ssocited with the chnge of speed during climb. Furthermore, ttention to the lift coefficient curve shows tht the control system is well behved nd is not commnding unresonble control inputs. There lso isn t ny chtter in the system. 15

278 0.8 Mch IAS (kt) x 104 Altitude ft CL time (sec) Figure 1.9. An MD80 in climb with vrious speed chnges using the TGF-test simultion This is direct result of the rmped inputs preventing lrge errors from occurring in Regions 3 nd Automtic Route Cpture The TGF-test lgorithm uses severl route cpture lgorithms to cpture routes. The first of these is the utomtic route cpture lgorithm which cptures route regrdless of the ircrft s loction with respect to the route or its orienttion. To verify this lwys works, mny test cses were run. Generlly, the key spects which determine good cpture re whether or not the ircrft chooses resonble segment, nd whether or not overshoot is excessive. Some of the more difficult test cses re presented in this section. The first cpture, shown in Figure 1.10, illustrtes the cse where the ircrft is very close to segment long route but heded in the wrong direction. When the route cpture lgorithm is initited, the utomtic route cpture lgorithm rightly chooses the segment which is closest to the ircrft. Then the lgorithm clcultes dynmic fix for intercept long the route. However, becuse the ircrft is lredy so close to the segment, the ircrft penetrtes the 1 turn rdii boundry defining the route following lgorithm 16

279 ltitude longitude Figure Automtic route cpture with the ircrft close to the route but heded in the wrong direction long before the ircrft is ble to converge on heding to the dynmic fix. Therefore, this prticulr scenrio is test of the route following lgorithm more thn it is test of the utomtic route cpture lgorithm. It demonstrtes tht the route following lgorithm cn hndle lrge discrepncies in ircrft heding reltive to the bering of the segment. The ircrft mkes right turn towrds the segment nd then slightly overshoots the segment. This overshoot is unvoidble becuse the ircrft is well within turn rdii of the segment before the mneuver is initited. The performnce shown in Figure 1.10 is exctly wht is desired. The next route cpture mneuver, shown in Figure 1.11, demonstrtes the cse where the ircrft is t considerble distnce from the cpture segment. This route cpture demonstrtes the use of the dynmic fix. As stted in Section , the dynmic fix is n imginry fix which is creted by the system t some loction long segment nd is used s point of reference for cpture. When the utomtic route cpture lgorithm ws first conceived, it seemed s though the most obvious method of cpturing the route ws to fly some intercept heding to the route. For instnce, once the cpture segment ws determined, the ircrft could be given n intercept heding of 45 degrees nd intercept the segment. However, this method seemed to hve some inherent limittions. First, the ircrft would lwys intercept using 45 17

280 ltitude longitude Figure Automtic route cpture with n ircrft fr from the cpture segment degrees regrdless of how fr the ircrft ws wy from the segment. An ircrft fr wy from the cpture segment might pss the segment before ever cpturing it. Figure 1.11 illustrtes n exmple of this type of cpture sitution. To void the problem of ircrft overshooting cpture segments, dynmic fix is plced on the segment to be cptured, nd the ircrft is commnded to fly towrd the dynmic fix. In this cse the further ircrft nturlly uses lrger intercept ngle s seen in Figure This system insures tht the proper segment is cptured nd lso provides some pprent vriety in intercept ngles so tht ll ircrft do not pper to behve the sme. The cpture lgorithm does exctly wht is expected. The utomtic route cpture guidnce converges on heding tht leds directly to the dynmic fix, nd then cptures the route when the ircrft is within 1 turn rdii of the segment. There is no overshoot becuse the ircrft is given enough spce to mneuver in this exmple. The trck in Figure 1.11 lso demonstrtes one other importnt performnce trit of the lgorithm. The lgorithm switches segments nd cptures the next segment long the route without overshoot in spite of the cute ngle which joins the segments. This exmple demonstrtes the effectiveness of the segment trnsition lgorithms t ssuring 18

281 smooth segment trnsition regrdless of segment geometry. The cse shown in Figure 1.11 is more difficult segment trnsition mneuver ltitude longitude Figure 1.1. Automtic route cpture with the ircrft heded perpendiculr to route Figure 1.1 is the lest senstionl of the cpture exmples in tht it demonstrtes rther esy nd likely scenrio. The ircrft is reltively close to the cpture segment nd heding in the generl direction of the segment. Therefore, the ircrft cptures the segment with ese. This exmple does show how the dynmic fix does mke the cpture ngles vry s function of the distnce tht the ircrft is from the segment. Being reltively close in this exmple, the ircrft tkes smller intercept ngle which is pprent in the difference between the originl heding nd the intercept heding. Note tht in this cse, the originl heding would hve intercepted the segment s well. When the ircrft is well within 1 turn rdii, the ircrft turns to intercept the segment without ny overshoot. This is exctly wht is desired. The finl utomtic route cpture exmple, shown in Figure 1.13, demonstrtes the cpture of route from n mbiguous re reltive to the route. As discussed in Section 8.4.1, there is ded region where two segments meet s shown in Figure If n ircrft is in this region, the norml segment determintion lgorithms will find tht the ircrft is in front of the triling 19

282 segment, behind the leding segment, nd will not return cpture segment. In this cse, nother criteri is used which checks the ircrft s ltitude longitude Figure Automtic route cpture with ircrft in n mbiguous region between segments distnce from every segment s triling fix. It then chooses to cpture the segment tht is ssocited with the closest triling fix. In Figure 1.13, the ltter logic is used to determine the pproprite cpture segment. Once the pproprite segment is determined, the ircrft is flown towrds dynmic fix nd ultimtely cptures the segment with no overshoot. 1.6 Vectored Route Cpture The vectored route cpture lgorithm steers the ircrft long user specified heding until the ircrft intercepts the route. Ech time step, the lgorithm determines which segment is best to cpture nd, ech time step, the lgorithm determines if it is time to merge onto the route. It should be noted tht the lgorithm hs no control over the initil heding. Therefore, if the user supplied heding steers the ircrft wy from the route, the guidnce lw is not ble to do nything bout it lthough it will provide wrning if the ircrft is unlikely to intercept the route. It is importnt tht this lgorithm work properly becuse it is the most hevily used cpture lgorithm. 0

283 ltitude longitude Figure Vectored route cpture from n mbiguous position Becuse the vectored route cpture lgorithm does not choose the intercept heding, it must be prepred to work with ll intercept hedings, including those which my be poor choices. To test the vectored route cpture lgorithm severl poor choices re provided. The first scenrio, shown in Figure 1.14, demonstrtes vectored heding where the choice of cpture segments is mbiguous. Becuse the lgorithm cn not control the heding of the ircrft, it is possible tht the best cpture segment could chnge depending on the ircrft s flight pth. In our scenrio, the ircrft is initilly pointed so tht the flight pth will intercept the front end of the triling segment; however, the best segment for cpture is ctully the next segment long the route. The ircrft relizes this nd turns onto tht segment without overshoot when the segment proximity permits. Figure 1.14 lso shows the ircrft trnsitioning to new segment while on the route. This mneuver provides nother exmple of the segment trnsition logic t work where the trnsition between segments occurs without ny overshoot. This is good performnce. Notice tht the ircrft trck is stopped just s the ircrft strts to mke the turn onto the finl segment t the route. 1

284 The finl exmple of vectored route cpture is shown in Figure In this exmple, the ircrft is commnded to cpture route while being vectored on heding which tends to be in the opposite direction of the route. This type of cpture is prticulrly chllenging becuse the ircrft must mke such lrge turn to cpture the segment. As cn be seen in Figure 1.15, the ircrft mintins the vectored heding until it is time ltitude longitude Figure Vectored route cpture when the vectored heding tends to be in the opposite direction of the route to merge onto the route. Once the pproprite distnce is reched, the ircrft merges with the route with no overshoot. 1.7 Initil Fix Route Cpture One requirement for the simultion ws tht the ircrft hd to be ble to strt flying long route by pssing through the initil fix. This type of route cpture lgorithm is ctully the lest complicted becuse there is no need to determine cpture segment nd there is no need to determine when to initite route following. The route following is initited s soon s the ircrft psses through the initil fix. Severl exmples of this mneuver re shown. The first exmple, shown in Figure 1.16, illustrtes the unlikely cse where the ircrft is commnded to fly bck to the beginning of the route from some loction in front of the route. This mneuver turned out to be very hndy for testing, but it is not likely to be used much in prctice. It is interesting to see how this lgorithm differs from the other cpture lgorithms. The

285 lgorithm directs the ircrft to fly to the initil fix. However, unlike the utomtic route cpture lgorithm, the initil fix cpture flies the ircrft through the initil fix before cpturing the route. This method gurntees overshoot when used in this ltitude longitude Figure The initil fix cpture lgorithm being used to vector n ircrft bck to the beginning of the route fshion. Nturlly, this lgorithm is relly designed to ssure tht n ircrft behind route cptures the route by flying through the first fix. This cse is illustrted in Figure Notice tht when the lgorithm is used s it is intended, the overshoot is minimized. However, there will usully be some overshoot becuse the lgorithm is constrined to fly through the initil fix. 1.8 Segment Trnsition The segment trnsition lgorithm controls when the ircrft initites turn from one segment to nother. The lgorithm must initite turn with enough spce to smoothly mke the trnsition. Becuse so mny exmples of segment trnsition re contined in the plots deling with route cpture, no dditionl plots re presented here. The reder should observe the trnsitions in Figure 1.10, Figure 1.11, Figure 1.14, nd Figure Figure 1.10 shows typicl segment trnsition where the difference in bering ngle between segments is smll. The ircrft mnges these without difficulty. The other exmples ll del with trnsitions between segments with lrge bering differences. These re more complex becuse the mount of spce llotted for the trnsition is 3

286 ltitude longitude Figure An ircrft cpturing route from behind using the initil fix cpture lgorithm more constrined. Excessive overshoot occurs if the trnsition logic isn t designed to hndle these cses. As cn be seen from the plots, the current segment trnsition logic hs no difficulty with these types of mneuvers. 1.9 Flight Technicl Error The flight technicl error (FTE) is the inbility or the inttention cusing the pilot to steer the ircrft perfectly long the desired course. If the ircrft is steered by n utopilot, it is the error in steering the ircrft perfectly long the intended course. In terms of the previous nvigtion error sources, the FTE is considered to be the guidnce nd control error, where only the guidnce error is included. The wypoint nd nvigtion id errors re not included. For the TGF simultion, there re two distinct flight technicl errors modeled. The first flight technicl error is the piloted flight technicl error, the error ssocited with humn pilot following route. The second error is the FMS error, the error ssocited with n FMS driven utopilot guiding the ircrft. Generlly, the only difference between these errors is the mgnitude of the stndrd devition nd the frequency of the mode. The sttistics summry is reprinted in Tble 1.1 4

287 Tble 1.1 Error Sttistics Summry Error Source Type Bis (1σ) Bet FMS Enroute FTE nd Order 0.13 nm 0.7 Guss Mrkov 5. kts Piloted Enroute FTE nd Order Guss Mrkov 0.7 nm 4 kts 0.5 Generlly, the only mens of testing the finl lgorithms is to mke sure tht the lterl vrition stys within the stndrd devition requirements. This is difficult to do using grphicl representtion of trjectories, so generlly tbulted dt were used to verify the flight technicl error. However, the trjectories provide richer understnding of wht the flight technicl dt does to the simultion. Four plots re presented in this section demonstrting the en route flight technicl error developed for the simultion. These plots were ll creted using closed route so tht n ircrft would fly round the sme route creting Monte-Crlo simultion. The route ws flown t different speeds using both piloted nd FMS flight technicl error. Figure 1.18 shows n ircrft flying t 50kts nd 5000ft. This is reltively slow configurtion so the trjectories show considerble mount of wobbling bck nd forth long the route. This is becuse the frequency of the lterl flight technicl error is not function of the ircrft s speed. Becuse the ircrft covers greter distnce per unit of time when moving fster, the flight technicl error lwys ppers to be more extreme in slower ircrft. This point is ptly demonstrted when Figure 1.18 is compred to Figure 1.19 where piloted ircrft is trveling t 300kts nd 30,000ft. Even though the sme flight technicl error is in effect, the fster ircrft ppers to hve less wobble in it trjectory, lthough the lterl offset of both plots is bout the sme. The finl two plots show the sme ircrft using FMS. Figure 1.0 shows the ircrft moving t 50kts nd 5000ft. Figure 1.9 shows the ircrft moving t 300kts nd 30,000ft. Since the FMS hs only bout 0% of the lterl offset tht the piloted flight technicl error hs, the ircrft trjectories re much tighter. This reduction in lterl offset over the piloted flight technicl error is redily pprent; however, it is not impossible to get sense for the wobble of the ircrft long the route in either plot, with this scle of distnce. 5

288 ltitude longitude Figure Piloted flight technicl error of n MD80 trveling t 50kts nd 5000ft ltitude longitude Figure Piloted flight technicl error of n MD80 trveling t 300kts nd 30,000ft 6

289 ltitude longitude Figure 1.0. FMS flight technicl error of n MD80 trveling t 50kts nd 5000ft ltitude longitude Figure 1.1. FMS flight technicl error of n MD80 trveling t 300kts nd 30,000ft 7

290 1.10 Nvigtion Errors The purpose of nvigtion error modeling is to model the vrinces which occur in ircrft flight pths s result of imperfect informtion. The two nvigtion types generlly used within the simultion t this point for en route types of opertion re VOR/DME nd GPS nvigtion. All of the nvigtion models perform similrly in tht they crete perturbed estimte of the ircrft s loction for the guidnce system to use s n input. Therefore, the nvigtion error models ll return ltitude longitude pir which represents the ircrft s position s determined by imperfect nvigtion GPS Nvigtion Error The GPS nvigtion error is so smll tht it is generlly undetectble, for the typicl scles used. For the purposes of completeness, plot of GPS error over route is generted, but for en route purposes, GPS error is smll enough to wrrnt ignoring it completely. Figure 1. shows n ircrft flying route. Unfortuntely, there is no wy of seeing the ircrft trck becuse it is so superbly hidden by the route line itself. Even exploded views of this plot fil to revel ny interesting vrition between GPS nd the route ltitude longitude Figure 1.. An ircrft trjectory using GPS nvigtion 8

291 1.10. VOR/DME Error The nvigtion errors ssocited with VOR/DME error is much more interesting thn those ssocited with GPS from modeling point of view. Generlly, VOR/DME nvigtion systems hve bises in their ngulr mesurements which tend to mke lterl offset errors grow s function of distnce from the nv-id. The errors lso produce interesting quirks when VOR receivers re switched during nvigtion. For instnce, n ircrft tht is following one VOR for portion of segment my develop lterl offset on one side of the segment nd upon switching to the next VOR, immeditely develop lterl offset on the other side of the segment. The first scenrio considered is shown in Figure 1.3 where n ircrft is trcking segment which hs VOR/DME t ech endpoint. The northerly most sttion hs slight esterly bis which tends to mke the ircrft trck to the est of center. The southerly VOR/DME tends to hve slight westerly bis which mkes the ircrft trck to the west of center ltitude longitude Figure 1.3. An ircrft flying segment using VOR/DME nvigtion where both endpoints re VOR/DME sttions The ircrft is heding southwest long the segment, nd it first cptures the route shortly fter pssing the northerly VOR/DME. The ircrft then trcks the course using signl from the northerly VOR/DME. As the ircrft progresses long the route, the ircrft drifts to the est slightly. When the ircrft psses the midpoint of the segment, the ircrft switches nv-ids nd 9

292 uses the southerly VOR/DME insted. Upon tuning in this nv-id, the ircrft crosses over the route nd trcks on the other side becuse of the slight westerly bis ssocited with the southern VOR/DME sttion. As the ircrft pproches the sttion, the error becomes smller s would be expected. The second scenrio, shown in Figure 1.4, illustrtes n ircrft following route which hs VOR/DME sttion t ech of its endpoints. However, there re two intersections in between the VOR/DME sttions mking three distinct segments. Becuse of this rrngement, the center segment does not hve VOR/DME sttion ssocited with it. In this sitution, the lgorithm must choose between the southerly VOR/DME nd the northerly VOR/DME to determine which is most pproprite to use for nvigtion. From visul inspection, the humn pilot would utomticlly choose the northerly most VOR/DME becuse the center segment is nerly perfectly ligned with rdil from tht sttion. Using the southerly VOR/DME would require some sort of re nvigtion technique x: Intersection O: VOR/DME ltitude longitude Figure 1.4. An ircrft flying route comprised of VOR/DME sttions with intersections between the VOR/DME sttions 30

293 The ircrft is initilly heded northest on the first segment using the southerly VOR/DME for nvigtion. Becuse this is the only VOR/DME ssocited with the first segment, the ircrft does not switch nv-ids. However, when the ircrft switches to the middle segment, it must mke decision bout which VOR/DME sttion to use. As discussed in the lst prgrph, the northerly VOR/DME is the best one to use nd in fct, we see tht the northerly VOR/DME is the one chosen by the lgorithm for nvigtion. We know this by observing the bis tht the ircrft tkes fter switching between the first segment nd the middle segment. The ircrft hs n esterly bis which is ssocited with the northerly VOR/DME. This bis is continully decresed s the ircrft flies further towrds the VOR/DME nd crosses onto the finl segment. Eventully, when the ircrft crosses the northern VOR/DME, the bis is reduced to zero. There is one other chrcteristic to note regrding the crossing of segment 1 onto segment. Notice tht the ircrft is using DME to determine the end of the route rther thn the intersection of VOR/DME rdils. If the intersection of VOR/DME rdils ws used to determine the fix loction, the ircrft would hve estimted the fix loction to be northest of the ctul fix loction. Rther, the ircrft estimtes the end of the route with ner perfection in spite of the VOR rdil bises. This is trit of n ircrft equipped with DME s opposed to one which only hs VOR nvigtion. This segment trnsition phenomen is one of the distinguishing chrcteristics of VOR/DME nvigtion s opposed to VOR/VOR nvigtion which is not currently modeled in the system Terminl Flight Phses The terminl flight phses for the ircrft consist of tke off nd lnding. These mneuvers re different thn ny which hve been considered so fr becuse they require tht the ircrft fly slowly nd interct with the ground Tke-Off During tke-off, the ircrft initilly ccelertes down the runwy with the lnding ger initilly supporting ll of the ircrft weight. This phse of tke-off is referred to s the ground roll. The lift coefficient is held t zero. Since we ssume coordinted flight, we do not concern ourselves with keeping the ircrft on the centerline. When the ircrft reches rottion speed, the lift coefficient of the ircrft is incresed until the ircrft leves the ground nd strts climbing. The lnding ger is retrcted s soon s the ircrft hs climbed severl hundred feet. Once the lnding ger is retrcted, the mximum vilble throttle is reduced to 90% of mximum possible throttle. Next, the ircrft ccelertes to new speed of 10 kt. While speeding up, the flps re retrcted s the proper speeds re obtined. Retrction of the flps is the lst commnds issued by the nvigtor for tke-off. Figure 1.5 illustrtes the tke-off of n MD80 ircrft weighing 130,000lbs. The ircrft initites the tkeoff ground roll t 0 seconds into the strip chrt recording. Roughly 40 seconds lter, the ircrft hs chieved sufficient speed for rottion. Looking t the lift coefficient plot, 31

294 we see the initil lift coefficient spike which is the control system executing the rottion. As the ircrft strts to lift off the ground we see the speed initilly stbilize right round the rottion speed. Then, the ircrft strts to ccelerte t slower rte towrds 10kts. Upon reching 10kts, the ircrft holds constnt speed nd 0.4 Mch IAS (kt) Altitude ft CL time (sec) Figure 1.5. An MD80 t 130,000lbs tking off with rottion speed of 150KIAS continues to climb. Notice tht the rte of climb increses when the ircrft stbilizes t 10kts. One of the importnt fetures of this tke-off exmple to notice is the simultneous ccelertion nd climb between 60sec nd 10sec. From Figure 1.1, we know tht on tkeoff the ircrft climbs nd ccelertes simultneously. To cuse this, specil rmping of the speed input is performed s discussed in Section Lnding The finl pproch nd lnding is quite possibly the most difficult of ll mneuvers to simulte. The ircrft must utomticlly follow n pproch to n irport, mintining the pproprite ltitudes ll long the pth, nd then cpture the ILS loclizer nd glide slope for the finl verticl descent. Finlly, the ircrft must touchdown. All through the mneuver, the control logic must monitor nd commnd the proper ircrft speed nd mke sure tht the pproprite 3

295 flp settings re deployed. Furthermore, the entire finl pproch is flown on the bck side of the thrust curve, the most difficult flight regime for the control logic. The terminl flight phses re lso the most difficult to verify. Pseudocontrol does not lnd ircrft so there is no cceptble bseline for insuring tht the longitudinl dynmics re proper. Therefore, verifiction of the longitudinl performnce on lnding nd tke-off consisted of mking sure tht the ircrft performnce conformed to the performnce dt which were used to crete the lgorithms. Such dt re contined in Figures 1.1 nd 1.. First, the longitudinl dynmics is considered. Figure 1.6 shows n MD80 on finl pproch nd lnding to n irport. By the time Figure 1.6 strts recording the pproch, the ircrft hs lredy slowed to 170kts nd hs cptured the loclizer. At roughly 55 seconds, the ircrft cptures the glide slope. The ircrft simultneously slows down to its finl pproch speed of 130kts. At roughly 180 seconds, the ircrft touches down nd the brkes re pplied. The speed reduces quickly nd the ircrft is brought to stndstill by 10 seconds. 0.5 Mch IAS (kt) Altitude ft CL time (sec) Figure 1.6. Longitudinl view of n MD80 on finl pproch nd lnding 33

296 Figure 1.7 illustrtes the top view of the sme MD80 from Figure 1.6. The ircrft hs lredy cptured the loclizer when the trck strts. The ircrft is heding southest long the ILS. The first circle long the route is n initil pproch fix locted 0 miles from the irport threshold, which is mrked by n X. The second circle, which is 5 miles from the threshold, is the finl pproch fix for the ILS. For this exmple, both the flight technicl error nd the ILS bem bending model re in effect. However, s cn be seen from the trck in Figure 1.7, little vrition is seen. This is expected considering the smll stndrd devitions which re mesured from ctul dt. The ircrft touches down ner the threshold nd stops moving bout 3000 ft lter ltitude longitude Figure 1.7. A top view of n MD80 on finl pproch to lnding 1.1 Conclusions The testing tht ws done to verify nd vlidte the TGF simultion gives us high degree of confidence tht the models contined herein hve sufficient fidelity for use s trget generting tool. Generlly, there were two mens of verifying nd vlidting the system. For the longitudinl dynmics, quntittive mesuring tool ws needed to insure tht the ircrft performnce ws 34

297 relistic. The tool used ws Pseudocontrol, the ircrft dynmics kernel of PAS. PAS, the NASA tool for trjectory genertion, hs been ccepted s n cceptble bseline for ircrft performnce. For the guidnce opertions, such s route cpture nd route following, visully inspecting the mneuvers is sufficient to insure proper opertion. Repeted testing of lgorithms ws done to insure tht the route cpture nd route following lgorithms would cpture the route from ll different initil conditions. Exmples of the most difficult initil conditions hve been discussed in this section. Flight technicl error nd nvigtion error ws vlidted by mking sure tht the modeled vrinces conformed to the rel flight dt sttistics which were used to construct the models. Terminl flight phses were the most difficult to verify becuse of the lck of informtion vilble for ctul ircrft descents nd lndings. Informtion from flight hndbooks nd performnce mnuls ws used to verify s best possible the ircrft performnce on lnding. 35

298 Appendix A - Anlysis of the Trnsfer Functions of the Longitudinl Dynmics In the originl version of this document (in section 3.3), n nlysis ws conducted on the trnsfer function between lift coefficient input nd n ltitude rte output for singleinput, single-output (SISO) system. This ws intended to provide insight to the controllbility of ltitude rte using lift coefficient. There ws concern bout the righthlf-plne zeros in the trnsfer function when the ircrft is flying on the bck-side of the thrust curve. The nlysis concluded tht control reversl occurred in this regime mking it impossible to mintin stble flight when using lift coefficient to control ltitude rte. The nlysis filed to consider tht in stedy, level flight on the bck-side of the thrust curve, the system is not SISO; thrust is used to control speed while lift coefficient is used to control ltitude rte. And in level ccelertion where the system is SISO, the linerized system of equtions (3.7) nd (3.8) does not pply. The reder is reminded tht the system is linerized bout stedy, level reference condition in which thrust equls drg. In level ccelertion, the throttle is dvnced to full nd is greter thn drg nd energy is being dded to the system. A similr rgument cn be mde for level decelertion. To get n indiction of the response of the ltitude rte to lift coefficient input in stedy, level flight, we cn modify the LTD system of equtions to bsorb the thrust control nd then nlyze the system s SISO, LTD system. The purpose of this ppendix is to blend the optimized thrust control gins of Chpter 4 into the LTD system nd then nlyze the h trnsfer function. The nlysis shows tht, in this dul input, dul output system, upcl thrust is djusted to counterct the dverse effect of the chnging lift coefficient on the speed so tht the commnded ltitude rte cn be cptured. Equtions (4.6) re restted here. V up C b b V b b L γ 1 0 b1 0 γ b u PT = + I C I C u L L ic L I I T T u it (4.6) -37-

299 y V M c γ I CL IT 1 = = h c 4 The control lw is reduced from eqution (4.14). up 0 C L k p 14 up 0 T k p M u = = u i 0 k C L i h 14 u ki 0 it In order to incorporte the thrust control into the LTD system, the thrust nd lift coefficient controls must first be seprted. We lso re-introduce the vectors s explicit functions of time. up C L up CL 0 0 k p u P 0 u T P M k T p M u = = u u + = 0 0 k h 0 0 h icl ic i L 14 u 0 ui 0 0 k 0 i T i T u u + u -K y - K y ( t) = ( t) ( t) = ( t) ( t) Then, ssuming K T is known. CL T CL T ( ) ( t) ( t) ( t) ( t) C ( t) ( t) L T = Ax ( t) + Bu C ( t) + Bu ( t) ( ) ( t) ( t) L T = A - BK TC x + BuCL ( t) ( t) C ( t) L ( t) ( t) x = Ax + Bu = Ax + B u + u y = Cx + Du u = -K y CL CL We rrive t the trnsfer function by converting to the lplce domin nd solving the mtrix lgebr. -38-

300 sx ( s) = ( A - BK TC) x ( s) + BuC ( s) L { si - ( A - BK TC) } x ( s) = BuC ( s) L -1 x ( s) = { si - ( A - BK TC) } BuC ( s) -1 ( ) C ( ) L ( ) { ( T )} We first solve for the closed-loop A mtrix. y s = C si - A - BK C B + D u s (A.1) L ( A - BK C) T 11 1 b b1 b11 b b 0 b k 0 c ki p 1 = c b11 b1 b11 b b1 0 b k p c = ki c b11 b1 b1 k p c b = ki c b1 k p c 1 1 b11 b = b ki c Then, -39-

301 { si ( A - BK TC) } s b1 k p c 1 1 b11 b1 0 s b s s k c = i 1 ( p ) s b k c b b s b = 0 0 s 0 k c i Computtion of the inverse of this mtrix is not trivil. We rrived t the solution using the lgebric mtrix inversion functions of MATLAB. s -40-

302 { si - ( A - BK C) } T 1 = b s + b s b 1 s 1 s + + b k c s b k c b s b b b k c b s b b k c 11 1 p i p i 1 b s s s s + + b k c s b k c 11 1 p i s k c i 1 k c s + + b k c s 1 i 1 b s + b 11 1 p k c s i 1 s s s + + b k c s + b k c 11 1 p 1 1 i

303 { } 1 Becuse of the form of the C-mtrix, which pre-multiplies the si - ( A - BK C) mtrix in the eqution (A.1), the { } 1 row of the si - ( A - BK C) pge spce.) T h u pcl trnsfer function is concerned only with second mtrix. (Note: trnsposed nottion is used to conserve T { si - ( A - BK C) } T 1 (,:) 1 s + ( 11 + b1 k p c ) 1 s + b1 ki c 1 s b s + ( b + b b k p c + b ) s + b b k c s 1b1 = s i 1 ( 11 1 p 1) ( 1 i 1 1 1) s + + b k c s + b k c T Post-multiplying the B-mtrix, we get, { si - ( A - BK C) } T 1 (,:) nd pre-multiplying the C-mtrix, we get, ( p ) s b1 k c 1 s + b1 ki c 1 1b11 + b1 s 1b1 b s + ( b + b b k p c + b ) s + b b k c s 1b1 B = s i 1 ( 11 1 p 1) ( 1 i 1 1 1) s b k c s b k c T -4-

304 (,:) ( p ) s b1 k c 1 s + b1 ki c 1 1b11 c4 + b1c4 s 1b1 c4 b1s + ( 11b 1 + b1b 1k p c ) 1 + 1b11 s + b1b 1ki c 1 c s 1b1 c4 1 C s { si - ( A - BK C) } B = T ( 11 1 p 1 ) ( 1 i 1 1 1) s + + b k c s + b k c 4 T The h u pcl { } 1 C I - A - BK C B T trnsfer function is element (,1) of the s ( ) ( p ) s + + b k c s + b k c i 1 1b11 c4 + b1c4 h s u pc L s + ( 11 + b1 k p c ) ( ) 1 s + b1 ki c 1 11 = h = c u pc L 4 ( p ) + ( p ) + ( 1 1 i 1 1) b s + b + b + b b c k s + b b c k i ( ) s s b c k s b c k mtrix. (A.) We verify tht by setting we get eqution (3.67). k = k = 0 nd substituting in the mtrix prtil derivtives, p1 i1 p CL ( ) h b s + b + b = c u s s f f f f f f b b c V V γ V γ h 11 =, 1 =, 1 =, 11 =, 1 =, 4 = 60 V γ V CL CL γ fγ f V f γ f γ f V s + + h f C h L V CL V CL = 60 u pc L γ f V fv fγ s s V γ V (3.67) But of course, we re concerned with nlyzing the trnsfer function with the optimized thrust gins left in. For B763 t stll speed nd 30,000 ft. eqution (A.) becomes, -43-

305 h 1640s + 631s u p CL s s s = ( ) which hs complex conjugte zeros t z 1, = ± In other words, thrust control of speed moves the h trnsfer function zeros into the left-hlf plne, thereby u pcl erdicting the non-minimum phse system. The integrl control trnsfer function, h, is the sme s eqution (A.) except with n dded integrl pole t s = 0. The uicl root loci of proportionl nd integrl h CL trnsfer functions with thrust control of speed re shown in Figure A.1. The figure shows tht lift coefficient control of ltitude rte is well-behved s long s speed is controlled by thrust simultneously. We do, however, hve to be mindful of low dmping in this re, s indicted by the locus moving up the imginry xis of the integrl control root locus plot. Figure A.1: The root loci of proportionl nd integrl lift coefficient control of ltitude rte considering the thrust control of speed -44-

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308 Appendix B - nd Order Guss-Mrkov model for GPS Clock Error due to Selective Avilbility When GPS position nd velocity error ws first modeled, the dominnt error ws the GPS stellite clock error. Figure B.1 illustrtes the bsic stellite geometry. GPS stellite clock error is n intentionl degrdtion of the GPS signl clled Selective Avilbility (SA). It ws implemented in 1990 to deny full position nd velocity ccurcy to unuthorized users fter initil testing of the GPS system reveled ccurcies much better thn nticipted. In My of 000, the United Sttes stopped degrding GPS performnce with SA. The nlysis supporting the modeling of GPS stellite SA illumintes the implementtion of the nd Order Guss-Mrkov utility routines used to model other nvigtion errors, such s Flight Technicl Error. Since the exct model for GPS stellite SA clock error is clssified, number of uthors hve pproximted it using second order Guss Mrkov model in the pseudornge domin (Prkinson & Spilker, 1996). These models cn be used to determine the GPS receiver position nd velocity errors s follows. Strting with the SA clock error model for ech visible stellite, the SA clock pseudornge nd rnge rte errors cn be obtined. These errors cn then be trnslted into locl coordinte frme, such s locl estnorth-up (ENU) frme using the methodology described erlier. An equivlent locl coordinte system SA position nd velocity error model cn be formulted, to void the need to model the GPS stellite orbits which re required to determine the line-of-sight direction cosines. It lso voids the need for Lest Squres filter

309 GPS SATELLITE Z r R SAT R AC AIRCRAFT Y X Figure B.1 GPS Receiver Mesurement Geometry The pproch tht is used is to strt with second-order Guss Mrkov SA pseudornge model. Then, by djusting the prmeters of this model to mtch observed locl coordinte SA position error sttistics, it is possible to obtin simplified SA position nd velocity error model. In generl, when the uncorrelted SA pseudornge nd rnge rte errors re trnslted into SA position nd velocity errors, the resulting position nd velocity errors re correlted. The fundmentl simplifying ssumption tht will be used is to ssume tht these correltions re negligible. If the GPS stellites were locted directly overhed nd exctly on the horizon t the four crdinl directions, the correltions would indeed be zero

310 Tble B.1. Observed Locl Coordinte Position Root-Men-Squre (rms) Errors Coordinte Position Error Mesured Averge Dily Vritions over 30 Dys Est 3 m 15% North 31 m 14% Horizontl 41.5* m & 37.5** m 10% Verticl 67 m 10% * Bsed on observed horizontl position errors ** Bsed on observed stedy stte horizontl position time difference errors The vilble field dt, obtined from Timoteo (199) consists of the position error sttistics of Tble B.1. In ddition to the dy-to-dy vribility, there is lso ltitude dependence for the verticl error, prticulrly for ltitudes greter thn 60 degrees. In ddition, field dt ws bstrcted from Timoteo (199) to describe the horizontl temporl decorreltion. A generl second-order Guss Mrkov model is described by the second-order differentil eqution (Gelb, 1974): x + βω x + ω x = cw (B.1) 0 0 Expressed s stte-spce eqution, (B.1) becomes F O Q P F H GI K J + F H GI K J x x HG I v K J L = N M v c w ω βω 0 0 (B.) where, x,v = error nd error derivtive (v = x) β = dmping fctor ω = nturl frequency 0 w = Gussin white noise c = scle fctor equtions (B.1) or (1.14) constitutes the simplified SA position nd velocity error model. There is one set of these equtions for ech est, north, nd verticl component

311 Discretizing the Continuous nd Order Guss Mrkov Process To implement nd order Guss Mrkov process in code, it must be discretized. The corresponding closed-form nd order difference eqution is shown in eqution (B.3) (Prkinson & Spilker, 1996): x φ φ x γ γ w = + v v w φ1 φ 1 0 γ k + k k (B.3) The discrete input mtrix hs three terms nd the stte trnsition mtrix hs four terms within them tht need to be clculted s shown in equtions (B.4)nd (B.5). L Φ = N M φ11 φ1o Q P φ φ 1 L Γ = N M γ 11 γ 1O Q P 0 γ (B.4) (B.5) To clculte the stte trnsition mtrix vribles, it is necessry to clculte some preliminry terms. These terms re defined in equtions (B.6) nd (B.7). ω σ v σ 0 = F H G I K J p (B.6) ω σ v σ F 1 = H G I K J 1 β p (B.7) The stte trnsition mtrix terms re then defined in equtions (1.3) through (1.4). b 1 g b 0 1g b 1 g βω 0 t φ11 = e cos ω t + β ω / ω sin ω t b g b g (B.8) 0 t φ / ω e sin ω t (B.9) = βω φ 1 = ω 0 φ1 (B.10) φ βω 0 t = e cos ω t β ω / ω sin ω t b 1 g b 0 1g b 1 g (B.11)

312 To clculte the discrete input mtrix, n dditionl term is needed which is shown in eqution (B.1). F HG 3 c = 4β σ v σ The terms for the discrete input mtrix re shown in equtions (B.13) through (B.15) p I KJ (B.1) Q Q / Q (B.13) γ 11 = 11 1 γ 1 = Q / Q (B.14) 1 γ = Q (B.15) where the terms Q 11, Q 1, nd Q, re terms of the white noise error covrince mtrix nd re defined in equtions (B.16) through (B.18). Q 11 c = 4βω 3 0 ω0 1 e ω1 βω t 0 βω0 t [ e [ 1 cos( ω t) ] c Q1 = Q1 = 4ω Q c = 4βω 0 1 ω0 1 e ω1 βω t 0 [ 1 β cos( ω t) + β( ω /ω ) sin( ω t) ] 1 1 [ 1 β cos( ω t) β( ω /ω ) sin( ω t) ] (B.16) (B.17) (B.18) The stte vribles of the process, x nd v, re initilized using equtions (B.19) nd (B.0) where the terms w 1 nd w re unit vrince discrete Gussin white noise. (Gussin rndom numbers). x v = σ p w 1 (B.19) = σ v w (B.0)

313 Clibrtion of Prmeters The next step is to select the three unknown prmeters, σ p, σ v, nd β, to mtch the observed dt sttistics. The results re summrized in Tble B.. A one-hour smple history for ll three locl position nd velocity SA error components is illustrted in Figure B. nd Figure B.3. Tble B. Simplified vs. Observed SA Position nd Velocity Model Prmeters Prmeter Symbol Observed Predicted North Position Sigm σ pn 31 m 31 m Est Position Sigm σ pe 3 m 3 m Verticl Position Sigm σ pv 67 m 67 m North Velocity Sigm σ vn 0.38 m/s Est Velocity Sigm σ ve 0.39 m/s Verticl Velocity Sigm σ vv 0.8 m/s Dmping Fctor β 0.55 Nturl Frequency ω ω c Horizontl Position Sigm σ ph 41.5* m & 37.5** m 37.3 m Horizontl Position Correltion ρ NE -0.13* & -0.9** -0.3 * Bsed on observed horizontl position errors ** Bsed on observed stedy stte horizontl position time difference errors in Figure B

314 0.05 G P S S A P OS ITION E RRO RS EAST (nm) NORTH (nm) VERTICAL (kft) TIM E (s ec ) Figure B.. Monte Crlo Simulted SA Position Errors G P S S A V E LO CITY E RRO RS EAST (kts) NORTH (kts) VERTICAL (fps) TIM E (s ec ) Figure B.3. Monte Crlo Simulted SA Velocity Errors

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316 Glossry Azimuth Bering Drg Dutch Roll Mode Empty Weight Flight Pth Angle Fuel Weight Geocentric Ltitude Geodetic Ltitude Ground Trck Heding Ground Trck Speed Heding Indicted Airspeed Lift Mch Number Mgnetic [Azimuth] An ngle mesured reltive to the ground-bsed coordinte system (i.e., true north). The zimuth direction of the position vector from one point to nother (e.g., from n ircrft to fix). The component of erodynmic force cting prllel to the ircrft's longitudinl xis nd in direction opposite the thrust. It is defined positive in the direction of the negtive x- xis of the body xis system. A coupled roll nd yw motion tht is often insufficiently dmped. The weight of fully opertionl ircrft without fuel or pylod. The ngle tht the true irspeed vector mkes with horizontl plne. The fuel cpcity of the ircrft. The ngle between line from center of the erth to the given point nd the equtoril plne. The ngle between line perpendiculr to the surfce of the ellipsoidl erth t the given point nd the equtoril plne. The ngle tht the ircrft's ground speed vector mkes with the ground-bsed coordinte system (i.e., true-north). This is the zimuth of the ircrft s velocity vector. The difference between ground trck heding nd true heding is due to the wind. The speed of the ircrft over ground. In other words, it is the mgnitude of the ircrft s true irspeed projected to horizontl plne. The zimuth of the ircrft's nose (i.e., longitudinl xis). This is the speed shown by n ircrft's irspeed indictor, s clculted from the mesured locl dynmic pressure. Its difference from true irspeed increses with ltitude. (Also known s Clibrted Airspeed.) The component of erodynmic force cting norml to the plne formed by the true irspeed vector nd the ircrft's lterl xis. The rtio of the true irspeed to the locl speed of sound. An zimuth ngle (e.g., heding, bering) mesured reltive to mgnetic north. -54-

317 Pylod Weight Phugoid Mode Pitch Angle Rhumb Line Roll Angle Short Period Mode Thrust True [Azimuth] True Airspeed The pylod cpcity of the ircrft. An oscilltory mode of ircrft dynmics in which kinetic nd potentil energy re exchnged. The ngle of ttck is minly unchnged. The ngle tht the ircrft's longitudinl xis mkes with the ground. (Also known s Elevtion Angle) A stright line on Merctor projection of the erth. It is convenient in nvigtion becuse it yields the constnt bering to be followed for nvigting between the two end points of the rhumb line. The ngle tht the ircrft's lterl xis mkes with the ground. (Also known s Bnk Angle) An oscilltory motion in the xis of rottion of pitch. The ngle of ttck is constntly chnging. This mode is typiclly much fster thn the phugoid. The thrust force creted by the ircrft's engines. Acts long the ircrft's longitudinl xis nd is defined positive in the direction of the x-xis of the body xis system. An zimuth ngle (e.g., heding, bering) mesured reltive to true north. The ctul speed of the ircrft reltive to the surrounding ir mss. -55-

318 Acronyms ADM AGL AMT ATC ATM BADA CAS DIS DME DOF ECEF FAA FTE GPS IAS LTD NAS NED RNAV SISO TGF VOR ircrft dynmics model bove ground level ircrft modeling tool ir trffic control ir trffic mngement Bse of Aircrft Dt clibrted irspeed Distributed Interctive Simultion (DIS provides militry stndrd erth coordinte system.) Distnce Mesure Equipment degree of freedom erth-centered, erth-fixed Federl Avition Administrtion flight technicl error Globl Positioning System indicted irspeed liner, time-dependent Ntionl Airspce System the North-Est-Down coordinte system Rdio Nvigtion single input, single output Trget Genertion Fcility VHF Omnidirectionl Rnge nvigtion system -56-

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320 References Anderson, J.D. (1989). Introduction To Flight. New York: McGrw-Hill. Federl Avition Administrtion (1993), System Segment Specifiction, TGF-SY01- Rev 1,, Willim J. Hughes Technicl Center, Atlntic City, NJ. Borkowski, K. M. (1989). Accurte Algorithms to Trnsform Geocentric to Geodetic Coordintes. Bulletin Géodesique. 63, Brogn, W.L. (1991). Modern Control Theory. Englewood Cliffs, NJ: Prentice Hll, Inc. Clrk, K. C. (1995). Anlyticl nd Computer Crtogrphy. Englewood Cliffs, NJ: Prentice Hll, Inc. Institute of Electricl nd Electronics Engineers. (1993). IEEE Stndrd for Informtion Technology -- Protocols for Distributed Interctive Simultion Applictions--Entity Informtion nd Interction, (IEEE ), Wshington, DC: IEEE. Federl Avition Administrtion. (010). Aeronuticl Informtion Mnul (AIM). Wshington, D.C.: U.S. Government Printing Office. Gelb, A. (1974). Applied Optiml Estimtion. Cmbridge, MA: The M.I.T. Press. Hoffmn, J.D. (199). Numericl Methods for Engineers nd Scientists. New York: McGrw-Hill. Hunter, G. (1996). CTAS Error Sensitivity, Fuel Efficiency, nd Throughput Benefits Anlysis, (Tech. Report ). Los Gtos, CA: Segull Technology. Interntionl Civil Avition Orgniztion. (000). Mnul of the ICAO Stndrd Atmosphere: extended to 80 kilometres(6 500 feet) (3rd ed., 1993, Doc 7488). Montrel: ICAO. Kyton, M., Fried, W.R. (1997). Avionics Nvigtion Systems. New York: John Wiley & Sons, Inc

321 Muki, C. (199). Design nd Anlysis of Aircrft Dynmics Models for the ATC Simultion t NASA Ames Reserch Center (Tech. Report ). Los Gtos, CA: Segull Technology. Nelson, R.C. (1989). Flight Stbility nd Automtic Control. New York: McGrw-Hill. Ntionl Imgery nd Mpping Agency (1997). Deprtment of Defense World Geodetic System 1984, Its Definition nd Reltionships With Locl Geodetic Systems (3 rd ed., Technicl Report TR8350.). Wshington, DC: NIMA. Nuic, A. (009). User Mnul for the Bse of Aircrft Dt (BADA) (Rev. 3.7, Tech. Rep. No ). Brétigny-sur-Orge, Frnce: EUROCONTROL Experimentl Centre. Ogt, K. (1990). Modern Control Engineering. Englewood Cliffs, NJ: Prentice Hll. Prkinson, B.W., Spilker, J.J. (1996). Globl Positioning System: Theory nd Applictions. Wshington, DC: Americn Institute of Aeronutics nd Astronutics. Rymer, D. P. (1999). Aircrft Design: A Conceptul Approch (3 rd ed.). Reston, VA: Americn Institute of Aeronutics nd Astronutics. Stevens, B.L., Lewis, F. L. (199). Aircrft Control nd Simultion. New York: John Wiley & sons, Inc. Thoms, J., Timoteo, D. (1990). Chicgo O Hre Simultneous ILS Approch Dt Collection nd Anlysis (DOT/FAA/CT-TN90/11). Atlntic City, NJ: Federl Avition Administrtion Willim J. Hughes Technicl Center