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-