Stochastic Six-Degree-of-Freedom Flight Simulator for Passively Controlled High-Power Rockets

Transcription

1 Stochastc Sx-Degree-of-Freedom Flght for Passvely Controlled Hgh-Power s Smon Box 1 ; Chrstopher M. Bshop 2 ; and Hugh Hunt 3 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Abstract: Ths paper presents a method for smulatng the flght of a passvely controlled rocket n sx degrees of freedom, and the descent under parachute n three degrees of freedom. Also presented s a method for modelng the uncertanty n both the rocket dynamcs and the atmospherc condtons usng stochastc parameters and the Monte Carlo method. Included wthn ths, we present a method for quantfyng the uncertanty n the atmospherc condtons usng hstorcal atmospherc data. The core smulaton algorthm s a numercal ntegraton of the rocket s equatons of moton usng the Runge-Kutta-Fehlberg method. The poston of the rocket s center of mass s descrbed usng three dmensonal Cartesan coordnates and the rocket s orentaton s descrbed usng quaternons. Input parameters to the smulator are made stochastc by addng Gaussan nose. In the case of atmospherc parameters, the varance of the nose s a functon of alttude and nose at adjacent alttudes s correlated. The core smulaton algorthm, wth stochastc parameters, s run wthn a Monte Carlo wrapper to evaluate the overall uncertanty n the rocket s flght path. The results of a demonstraton of the smulator, where t was used to predct the flght of real rocket, show the rocket landng wthn the 1 area predcted by the smulaton. Also lateral acceleraton durng weather cockng, whch was measured n the test, shows a strong correlaton wth smulated values. DOI: 1.161/ ASCE AS CE base subject headngs: Stochastc processes; Smulaton; Monte Carlo method; Degrees of freedom; Passve control; Spacecraft. Author keywords: ; Stochastc; Smulaton; Flght; Unguded; Machne learnng; Monte Carlo; 6DOF; Parachute; HPR. Introducton Background 1 Mcrosoft Research Cambrdge; presently, Research Fellow, Unv. of Southampton, SO19 7JG, Unted Kngdom correspondng author. E-mal: s.box@soton.ac.uk 2 Dstngushed Scentst, Mcrosoft Research Cambrdge, 7 JJ Thompston Avenue Cambrdge, CB3 FB, Unted Kngdom. 3 Senor Lecturer n Engneerng, Dept. of Engneerng, Unv. of Cambrdge, Trumpngton Street, Cambrdge, CB2 1PZ, Unted Kngdom. Note. Ths manuscrpt was submtted on March 17, 29; approved on March 1, 21; publshed onlne on March 3, 21. Dscusson perod open untl June 1, 211; separate dscussons must be submtted for ndvdual papers. Ths paper s part of the Journal of Aerospace Engneerng, Vol. 24, No. 1, January 1, 211. ASCE, ISSN /211/ /$25.. Hgh-power rocketry HPR s both a popular hobby for amateur enthusasts and an academc actvty wth a number of unverstes usng HPR as a teachng and research tool. A hgh-power rocket s defned as a rocket wth total mpulse of between 16 Ns and 4,96 Ns Natonal Fre Protecton Assocaton NFPA 22. Sold fuel HPR motors consstng of an ammonum perchlorate NH 4 ClO 4 and powdered alumnum Al mx are sold commercally. Some rockets are alternatvely powered by hybrd motors usng a lqud oxdzer, e.g., ntrous Oxde N 2 O and a sold fuel, e.g., hydroxyl-termnated polybutadene. Typcally, the rockets are passvely controlled and fly to a maxmum alttude of between 1 and 5 km, but rockets have been flown much hgher 13 km Weatherll 29. At apogee, t s usual for the rockets to deploy a parachute for safe recovery to earth. The rockets generally carry some avoncs and sensors payload to record flght data. They may also carry addtonal sensors, for example, to record atmospherc measurements. s are flown at scheduled meetngs organzed by rocketry clubs and regulated by natonal rocketry organzatons such as the U.K. ry Assocaton UKRA and, n the Unted s, the Natonal Assocaton of ry NAR. The Web stes of these organzatons are good resources for more nformaton on HPR. Motvaton There are two prncpal motvatons for a stochastc sx-degreeof-freedom DOF flght smulator for passvely controlled rockets. The frst s as a tool for predctng the landng locaton and the second s as a tool for desgn. As the rockets are passvely controlled, the fler cannot control the landng locaton of the rocket after t has been launched. Therefore, accurate predctons of the rocket s landng locaton are mportant for safe flyng. The stochastc element of the smulator s partcularly mportant for ths applcaton as t makes t possble to quantfy the uncertanty n the landng poston and the probablty that the rocket wll land n a gven area. As a desgn tool, the a 6DOF flght smulator allows the rocket desgner to fly prototype rocket desgns vrtually to assess performance and optmze aspects of desgn such as the margn of stablty defned later n the artcle. Contents Ths paper presents a methodology for smulatng HPR flghts usng a 6DOF smulator for the rocket ascent and a 3DOF smulator for the parachute descent. It s assumed that the rocket s an JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 / 31 J. Aerosp. Eng :31-45.

2 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. axsymmetrc rgd body and s passvely controlled. It s also assumed that the dynamc and aerodynamc propertes of the rocket and parachute are known and can be suppled as nputs to the smulator. These propertes can be found through expermentaton e.g., usng models n a wnd tunnel, or they can be estmated from the rocket/parachute geometry. A method for the latter s presented n Box et al. 29. We begn by descrbng the software archtecture used for the core smulaton algorthms of the rocket smulator and the parachute smulator. We then proceed to show how ths archtecture can be extended to smulate dfferent flght scenaros, such as multstage flghts. We then descrbe n detal the dynamc models that are used by the smulator to solve the rocket and parachute equatons of moton. Ths ncludes descrptons of all the dynamc and aerodynamc data that must be suppled to the smulator and step-by-step solutons to the equatons of moton. Up to that pont, the smulaton method descrbed s entrely determnstc. In followng sectons, we show how to extend the smulator to perform stochastc smulatons usng the Monte Carlo method. Some of the most mportant stochastc parameters n the smulaton are those descrbng atmospherc condtons, n partcular the wnd speed. The wnd speed and drecton have a sgnfcant effect on the flght path of an HPR rocket. Therefore we present a method for quantfyng the uncertanty n the forecast wnd speed data and modelng ths uncertanty n the stochastc smulatons. The smulaton method presented n ths paper has not been fully verfed expermentally. However, n the fnal secton we present a demonstraton of the smulator where smulaton output s compared wth data recorded durng an HPR rocket flght. The results of ths early test ndcate that the method could be useful. In the remander of ths ntroducton, we present a bref explanaton of how ths research fts wthn the context of prevous work, and a note on passve control aerodynamcs. Context Prevous work e.g., Duncan and Ensey 1964; Nassr et al. 24 has shown how to formulate the 6DOF equatons of moton for a passvely controlled rocket and demonstrated the use of the Monte Carlo method to probe the senstvty of the flght path to varaton n the rocket s dynamc parameters. The man contrbutons of ths paper are enumerated below. 1. An updated formulaton of the rocket s equatons of moton accountng for modern computatonal technques. In partcular, we use quaternons to descrbe the rocket s rotatonal orentaton; the computatonal advantages of quaternons for rgd body dynamcs smulaton are descrbed n Baraff A software archtecture for combnng rocket and parachute models to smulate rocket flghts wth parachute recovery, ncludng multstage flghts, flghts wth parachute falure and stochastc flght smulatons. 3. A quanttatve method for estmatng the uncertanty n the atmospherc condtons, where the varance n both wnd speed and wnd drecton s a functon of alttude and correlated at adjacent alttudes. 32 / JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 (a) (b) Centre of Mass Centre of Pressure F A F A F N F N Passve Control Aerodynamcs Drectonal Stablty A passvely controlled or unguded rocket derves ts stablty from fns lke a dart or an arrow. There s no actve control or steerng to correct or adjust the rocket s trajectory after launch. The addton of fns moves the center of pressure toward the rear of the rocket. The center of pressure s the pont on the rocket through whch all aerodynamc forces can be assumed to act. In order for the rocket flght to be stable, the center of pressure must be aft of the center of mass Fg. 1. The dstance between the centers of pressure and mass s the margn of stablty. The angle between the drecton of arflow over the rocket and the rocket s roll axs s known as the angle of attack. In the case of a stable rocket, the aerodynamc forces wll act to reduce the angle of attack to zero, but n the case of an unstable rocket, the opposte s true. Weather Cockng Durng the launchng stage of a passvely controlled rocket flght, the trajectory of the rocket s constraned by a launch tower. Ths allows the rocket to buld up some speed and hence aerodynamc stablty before the constrant s removed. As the rocket clears the launch tower, and f there s a crosswnd, the rocket wll be travelng wth an angle of attack. The effects of the drectonal stablty wll cause the rocket to rotate nto the crosswnd and reduce the angle of attack. The angular momentum of the rocket wll then cause an overrotaton leadng to a characterstc damped oscllaton n the rocket s angular poston. Ths phenomenon s known as weather cockng and, as wll be shown later n ths artcle, t s mportant when analyzng the effcacy of the smulator. Archtecture Dsplacng Moment Restorng Moment In ths secton, we present a graphcal overvew of the software archtecture usng block dagrams. We frst descrbe the core α UNSTABLE STABLE Fg. 1. a Dagrams of drectonally stable; b drectonally unstable rockets. The atmosphere relatve velocty vector s marked V and the aerodynamc force s shown n two orthogonal components axal and normal. The angle s the angle of attack. α V V J. Aerosp. Eng :31-45.

3 Output Output Parachute Yes Yes Intal t, X, Q, P L, Stop? No RKF45 Updated t, X, Q, P, L Intal t, X, P Stop? No RKF45 Updated t, X, P Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Parameter base Dynamc Model Dervatve X &, Q &, F, τ Fg. 2. Code block dagram for the core rocket smulaton routne smulaton algorthms that are used to ntegrate numercally the equatons of moton for the rocket and the parachute. Then we show how these can be used to construct dfferent flght scenaros. Core Smulaton Algorthms The core rocket smulator works by numercally ntegratng the rocket s equatons of moton over tme usng the 4th/5th order Runge-Kutta-Fehlberg algorthm. Ths algorthm s well descrbed n numercal computng lterature. A good example can be found n Press et al. 27. A block dagram summarzng the rocket smulator s shown n Fg. 2. The state of the rocket at any tme durng the smulaton s descrbed by four vectors: X s a vector descrbng the poston of the rocket s center of mass n global Cartesan coordnates, whch are algned wth a tangent plane to the Earth s surface at the launch ste. Q s a quaternon descrbng the rocket s orentaton, P and L are vectors descrbng the rocket s lnear and angular momentum, respectvely. The ntal tme t and state X,Q,P,L are passed to the Runge-Kutta-Fehlberg block, labeled RKF45 n Fg. 2. The RKF45 block passes values of tme and state to the rocket dynamc model, whch solves the equatons of moton to get the dervatves of the four state vectors wth respect to tme. To do ths, the model requres addtonal data. These are the parameters descrbng the dynamc and aerodynamc propertes of the rocket, the thrust of the rocket and the atmospherc condtons. The model gets these data from the parameters database. The state dervatves are: the lnear velocty Ẋ, the quaternon dervatve Q, the Force F and the Torque. These are passed back to RKF45. A detaled descrpton of the rocket dynamc model and parameters data are gven n next secton. When the RKF45 algorthm completes ts tme step, t returns a new tme t and new state X,Q,P,L. The new state s n turn passed back to RKF45 as the startng state for the next step. Ths loop contnues untl the stop condton s satsfed. Varous stop condtons can be used. For example: to make the smulaton termnate when the rocket reaches apogee, the vertcal momentum of the rocket can be used as a stop trgger. The parachute smulator Fg. 3 works n a very smlar way to the rocket smulator and the parachute dynamc model and parameters are descrbed n the next secton. One mportant dfference s that rotatons are not modeled n the parachute smulaton so there are only lnear state vectors X,P and state dervatve vectors Ẋ,F. Parameter base Parachute Dynamc Model Dervatve X,F & Fg. 3. Code block dagram for the core parachute smulaton routne Full Flght Smulaton The rocket and parachute smulators depcted n Fgs. 2 and 3 can be combned to smulate a complete flght. The block dagram for a smple flght s shown n Fg. 4. The rocket smulator s run frst and termnates when the rocket reaches apogee. Then, the fnal state of the rocket smulaton s passed to the parachute smulaton as the ntal state. The rocket s descent under parachute s then smulated untl t reaches the ground. An example flght path from ths type of smulaton s shown n Fg. 5. The launch pad coordnates are,, and the wnd drecton s predomnantly from the southwest. It can be seen that durng the ascent, the rocket heads upwnd a short dstance as t clmbs to 3,5 m. At apogee, a drogue parachute s deployed and the rocket s blown back downwnd as t descends to an alttude of 3 m. Here, a second man parachute s deployed and the ncrease n drag can be seen n the path of the rocket. Fnally, the rocket lands approxmately 85 m north and 65 m east of the launch pad. More Complex Flght Scenaros In addton to the smple scenaro shown n Fg. 4, more complex flght scenaros can be constructed usng the rocket and parachute smulators as buldng blocks. Two examples are the smulaton of a parachute falure and the smulaton of a two stage rocket. These are dscussed below. A parachute deployment falure durng an HPR rocket flght can pose a safety hazard; therefore, t s useful to be able to smulate ths scenaro. Fg. 6 shows the block dagram for a full flght smulaton whch ncludes the possblty of a parachute deployment falure. In ths case, the rocket smulaton termnates Parameter base Intal Flght Program (1-Stage) Parachute Intal Parachute Output Fg. 4. Code block dagram for a smple rocket flght program JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 / 33 J. Aerosp. Eng :31-45.

4 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Alttude (m) Fg. 5. Plot of a smulated rocket flght path generated usng the program shown n Fg. 4 when the rocket reaches apogee, then there s a decson over whether the parachute deployment wll fal. Ths choce can be made through user nput or t can be random wth an assgned probablty. If the decson s for a successful parachute deployment, then the fnal state of the rocket at apogee s passed to the Parameter base Intal Flght Program, wth parachute falure Next step Intal Output Yes Fal? No Parachute Fg. 6. Code block dagram for a rocket flght program wth parachute falure Alttude (m) Fg. 7. Plot of two smulated rocket flght paths, one for a successful flght sold and one wth parachute falure dotted parachute smulator as the ntal state. Ths s effectvely the same scenaro as shown n Fg. 4. However, f the parachute deployment fals, then the fnal state s passed to the rocket smulator agan and a ballstc descent s smulated. Fg. 7 shows example flght paths for the two scenaros. Note that ths approach can only smulate a complete parachute deploy- Parameter base Full Intal (Full ) Upper Stage Intal Booster Stage Intal Stage Separaton (Upper Stage) Output Flght Program (2-Stage) (Boost Stage) Parachute Parachute Fg. 8. Code block dagram for a two-stage rocket flght program 34 / JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 J. Aerosp. Eng :31-45.

5 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Alttude (m) ment falure,.e., the parachute stays completely wthn the rocket. In order to model a partal deployment, a new dynamc model would be requred. The block dagram for a two stage rocket flght s shown n Fg. 8. In ths case, the database contans dynamc, aerodynamc and thrust data for the rocket n three confguratons; frst, the complete rocket as t s on the launch pad and then the upper stage and the booster stage after separaton. The frst rocket smulaton covers the flght of the complete rocket from launch up to the tme of separaton. At stage separaton, new ntal states for the upper and booster stages are calculated from the state pror to separaton. The flghts of the upper and booster stages are then smulated to ther respectve apogees. Followng ths, the two parachute descents are smulated as usual. Fg. 9 shows an example of the smulated flght paths for a two stage rocket flght. Dynamc Models Fg. 9. Plot of the smulated flght path for a two-stage rocket. After separaton, the paths of both the upper and booster stages are shown. Dynamc Model Ths secton descrbes n detal the rocket dynamc model that was ntroduced n the prevous secton. The task of the model s to solve the rocket equatons of moton. That s to take the rocket s current tme t and state X,Q,P,L and calculate the state dervatves Ẋ,Q,F,. We begn by defnng the nputs to the dynamc model and the constant values used n the calculaton. Table 1. Constants Used n the Dynamc Models Symbol Value Descrpton 1.4 Rato of specfc heats for ar R 287 Gas constant for ar dfference n specfc heats K Reference temperature Pa s Reference dynamc vscosty C 12 K Sutherland s constant M E kg Mass of the Earth r E 6,378,1 m Radus of the Earth G m 3 /kg s Unversal gravtatonal constant Y A, 1,, Reference yaw axs P A,,1, Reference ptch axs R A,,,1 Reference roll axs Constants and Inputs For reference, Table 1 shows all the constants that are used n the rocket dynamc model. There are two types of nputs to the rocket dynamc model, as can be seen n Fg. 2: the current state and the parameters. The current state vectors are summarzed n Table 2. Table 3 summarzes the parameters that are stored n the database. Most of these data are a functon of some quantty lke tme t or alttude z, therefore they are stored n tables and the values can be nterpolated as requred by the model. The aerodynamc Table 2. Dynamc Model Perod Vectors Symbol Elements Descrpton t Tme P P x, P y, P z Lnear momentum vector L L x,l y,l z Angular momentum vector Q s,v x,v y,v z Quaternon X x,y,z Poston vector Table 3. Parameters That Are Stored n the base for the Model Symbol Functon of Descrpton T t Thrust M t Mass X cm t Dstance of the center of mass from the nose tp I xx t Moments of nerta about the rocket s yaw axs I yy t Moments of nerta about the rocket s ptch axs I zz t Moments of nerta about the rocket s roll axs C da t Thrust dampng coeffcent C A R,, M Coeffcent of axal aerodynamc force C N R,, M Coeffcent of normal force C R R, f, M Coeffcent of aerodynamc roll torque X cp Dstance of the center of pressure from the nose tp W x, y, z Wndspeed vector x,y,z Atmospherc densty A x,y,z Atmospherc temperature X RB body length A RB body cross-sectonal area maxmum X f Dstance of the plane of the fn s centers from the nose tp r f Roll moment arm Fn cant angle JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 / 35 J. Aerosp. Eng :31-45.

6 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. coeffcents C A, C N, and C R can be functons of up to three quanttes Reynolds number R, angle of attack and Mach number M. However, some methods for estmatng these coeffcents assume that they are ndependent of one or more of these quanttes. For example, a common method of estmatng C N descrbed n Box et al. 29 assumes that C N s ndependent of Reynolds number R. Calculatng the Dervatves To calculate the state dervatves, the rocket dynamc model uses the current tme and state to access the database and get the values of thrust, mass, aerodynamc coeffcents, wndspeed, etc., that apply to that tme and state. Ths secton presents the equatons for calculatng the state dervatves wth these data. The method assumes that the rocket s axsymmetrc. Poston and Orentaton. The poston of the rocket s center of mass n global Cartesan coordnates s gven by X. The quaternon vector Q descrbes the rocket s rotatonal orentaton. Specfcally, Q descrbes a rotatonal transformaton between a reference orentaton and the current orentaton. The transformaton can be descrbed as a rotaton of radans about a rotaton axs a passng through the center of mass of the rocket. The elements of the quaternon vector are Q= s,v = s,v x,v y,v z where s = cos 2 v x = sn 2 a x v y = sn 2 a y v z = sn 2 a z Q can be converted to a rotaton matrx R usng the followng transformaton: 1 = 1 2v 2 2 y 2v z 2v x v y 2sv z 2v x v z +2sv y 2 R 2v x v y +2s v z 1 2v 2 2 x 2v z 2v y v z 2sv x 2 2v x v z 2sv y 2v y v z +2sv x 1 2v 2 x 2v y The unt vectors descrbng the yaw, ptch and roll axes of the rocket n ts current orentaton can be calculated usng R and the reference orentatons Table 1 T Y A = RY A, T P A = RP A, T R A = RR A, Lnear and Angular Veloctes. The Earth-relatve lnear velocty vector of the rocket s center of mass s gven by Ẋ = P/M 4 The angular velocty vector for the rocket s calculated usng 36 / JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY = RI 1 R T L T where I =reference nerta tensor, defned usng values for the rocket s moments of nerta from the database = I xx I I yy 6 I zz The angular velocty vector and the quaternon Q are used to calculate the quaternon dervatve Q ṡ = 1 v 2 7 v = 1 s + v 2 8 Q = ṡ,v Angle of Attack, Reynolds Number, and Mach Number. In order to calculate the forces and torques on the rocket, we need to recover the aerodynamc coeffcents from the database. For ths, we must know the angle of attack, the Reynolds number and the Mach number of the rocket n ts current state. The angle of attack s defned as the angle between the unt vector descrbng the rocket s roll axs and the rocket s apparent velocty vector V, so t s gven by = cos 1 Vˆ R A 1 where the symbol n Vˆ ndcates that the vector V has been normalzed. The apparent velocty vector V s the velocty of the rocket s center of pressure relatve to the atmosphere. Unfortunately, the locaton of the center of pressure X cp must be recovered from the database and may tself be a functon of. If ths s the case, then the atmosphere relatve velocty of the center of mass V cm can be used as an approxmaton of V. Ths s gven by V cm = Ẋ + W 11 Ths gves an approxmate cm whch neglects the effects of the rocket s own rotaton on. Provded the rocket s angular velocty s small compared wth ts forward velocty, ths approxmaton s good enough to select X cp from the database. Then the apparent velocty V s gven by V = V cm + V 12 where V =lnear velocty vector of the center of pressure due to the angular velocty of the rocket and s gven by V = X sn cos 1 R A ˆ R A 13 Here the drecton of the vector s gven by the cross product of the unt vectors for the roll axs and the axs of rotaton; and the magntude s the angular velocty multpled by the perpendcular dstance between the center of pressure and the axs of rotaton, gven by X sn cos 1 R A ˆ where X = X cp X cm and ˆ = normalzed angular velocty vector. If there s a large dfference between and cm, then the estmated value of can be used to extract a new value of X cp from the database and then an updated can be calculated, thus 5 9 J. Aerosp. Eng :31-45.

7 can be found teratvely. The Reynolds number of the rocket s gven by N f = 1 N =1 f, 22 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. R = VX RB 14 where V= V, X RB =length of the rocket body and =knematc vscosty gven by + C = A + C 3/2 A The Mach number of the rocket s gven by V M = R A where R A =local speed of sound Fn Angle of Attack. In some rocket desgns, the fns are canted wth a small angle to nduce roll. If the fns are canted, then even f the angle of attack of the rocket s zero, the fns wll have an angle of attack f equal to the cant angle. Ths wll cause the rocket to ncrease roll velocty untl the fn angle of attack s zero. The centers of pressure for each of the rockets fns le on a crcle whch can be descrbed by the dstance of ts center from the nose tp X f and ts radus r f. The plane of the crcle s perpendcular to the rocket s axs. Because the rocket may have an angle of attack and angular velocty, the f of each of the fns may be dfferent. In order to mantan axsymmetry and because we are unconcerned wth the number of fns, we can estmate a mean fn angle of attack f by fndng the angle of attack at N evenly spaced ponts on the crcle and takng an average. For the results presented n ths paper N=4. If P b =a pont on the crcle n rocket reference coordnates, then the coordnates of the pont n space are P = RP b + X 17 The lnear velocty of ths pont due the rocket s angular velocty s gven by V p,, = S sn cos 1 Ŝ ˆ Ŝ ˆ 18 where S =X P. The total velocty vector for the pont P s then gven by V p, = V p,, + Ẋ + W 19 If l s a unt vector along the shortest path from the rocket s axs to P, then we can defne the quaternon that descrbes the cant angle of an magnary fn at pont P Q c = cos 2,sn 2 l R A 2 Whch usng Eq. 2 gves R c. Then the fn angle of attack at pont P s gven by f, = 2 cos 1 Vˆ p, R c l 21 Unlke rocket angle of attack fn angle of attack f, as defned n ths model, can be ether postve or negatve dependng on whch sde of the fn t s. The mean fn angle of attack and mean fn veloctes can then be calculated over the N ponts used V f = 1 V p, 23 N =1 f s used to recover the correct value of the coeffcent of roll C R from the parameters database and V f s used n the calculaton of roll torque n Eq. 35. Force and Torque. The force vector on the rocket can be expressed as the sum of four component vectors F = F T + F g + F A + F N 24 F T =thrust vector, whch acts n the opposte drecton of the roll axs, so s gven by F T = TR A F g =gravty vector, whch s assumed to be F g =,, Mg T where g s calculated usng N M E g = r E + z 2 27 F A and F N aerodynamc forces on the rocket broken down nto axal and normal components, respectvely. The magntude of the axal aerodynamc force s F A = 1 2 V2 A RB C A 28 where V= V. The axal force vector acts n the opposte drecton to the vector of the rocket s roll axs, so the force vector s gven by F A = F A R A The magntude of the normal aerodynamc force s F N = 1 2 V2 A RB C N 29 3 The normal aerodynamc force acts n a drecton that s orthogonal to the roll axs R A and n the plane formed by the roll axs and the apparent velocty vector V F N = F N R A R A Vˆ 31 The torque vector on the rocket can be expressed as the sum of three component vectors = N + da + R 32 N =torque on the rocket due to the normal force s calculated as N = F N X R A Vˆ 33 where X = X cp X cm =moment arm. When a rocket rotates about a transverse axs durng the thrustng phase of flght, hot gas wthn the motor tube wll be accelerated laterally. Ths produces a dampng moment called thrust dampng. The torque on the rocket due to thrust dampng da s modeled by da = C da RmR 1 34 JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 / 37 J. Aerosp. Eng :31-45.

8 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Table 4. Parachute Dynamc Model Vectors Symbol Elements Descrpton t Tme P P x, P y, P z Lnear momentum vector X x,y,z Poston vector where m=dagonal matrx wth elements 1 1 on the man dagonal. R s the roll torque on the rocket due to fn cant. The magntude of ths torque can be expressed as R = 1 2 V 2 fa RB C R r f 35 where r f =roll moment arm and C R can be postve or negatve descrbng clockwse or antclockwse roll, respectvely. The roll torque s about the rocket s roll axs, therefore the torque vector s defned by R = R R A 36 In some cases, roll torque can be generated by angled thrust from the rocket s nozzle ether ntentonally or otherwse. A method for modelng ths s not presented explctly here. To account for ths, an addtonal term would have to be added to Eq. 32. Parachute Dynamc Model Constants and Inputs The parachute dynamc model works n a very smlar way to the rocket model except that rotatons are gnored. For examples of more complex parachute models see, Dobrokhodov et al. 23; Km and Peskn 26. Tables 4 and 5 defne the state vectors and parameters for the model. The parachute model does not use any constants that are not already lsted n Table 1. Table 5 shows the parachute coeffcent of drag C D and parachute area A P as functons alttude z. Ths s so that the parachute model can smulate the descent of rockets that deploy two or more parachutes at dfferent alttudes durng the descent. The values of C D and A P are not nterpolated between alttudes as wth wnd speed or densty but rather dfferent values apply to dfferent alttude ranges. Calculatng the Dervatves In ths secton, we show the equatons for calculatng the state dervatves n the parachute model. Velocty. gven by The earth relatve velocty of the parachute Ẋ s Table 5. Parameters That Are Stored n the base for the Parachute Model Symbol Functon of Descrpton W x, y, z Wndspeed vector x,y,z Atmospherc densty A x,y,z Atmospherc temperature C D z Parachute coeffcent of drag A P z Parachute area M Mass of rocket and parachute 38 / JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 Parameter base Intal Collated Output Yes Stop? Ẋ = P/M 37 The atmosphere relatve velocty apparent velocty of the parachute s gven by V = Ẋ + W where W = wndspeed vector. 38 Force The force vector on the parachute can be expressed as the sum of two components F = F D + F g 39 F D =drag force vector, the magntude of the drag force on the parachute s gven by F D = 1 2 V2 C D A P Output 4 The drecton of the drag force vector s opposte to the apparent velocty vector F D = F D V F g =gravtatonal force vector and s assumed to be F g =,, Mg T where g s found usng Eq. 27. Monte-Carlo Process Random Flght Program Fg. 1. Code block dagram for the Monte Carlo wrapper Stochastc Smulatons Usng the Monte Carlo Method Up to ths pont, we have descrbed a smulator for rocket flghts whch s entrely determnstc. In ths secton, we descrbe how to perform stochastc smulatons and how to quantfy the uncertanty n the rocket s landng poston by usng a Monte Carlo wrapper around the determnstc smulator. Archtecture Many of the nputs to the smulatons such as the aerodynamc propertes of the rocket and the atmospherc condtons wll have uncertan values. If quanttatve measures of the uncertantes n these nputs can be known or estmated, then the Monte Carlo method can be used to obtan an estmaton of the uncertanty n the rocket s flght path and landng poston. The block dagram for the Monte Carlo wrapper s shown n Fg. 1. The parameters and ntal state are frst passed to the Monte Carlo process block. Here, uncertan values from these No J. Aerosp. Eng :31-45.

9 standard devatons landng probablty. The rght hand plot shows the same Gaussan ellpses plotted on three dmensonal axes together wth the mean rocket flght path. Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Fg. 11. a Scatter plot of the smulated landng locatons after 5 teratons of the Monte Carlo smulator, ncludng Gaussan ellpses markng 1 and 2 probablty; b the same Gaussan ellpses plotted together wth the mean flght path nputs have random nose added to them generatng the random nput data. A detaled dscusson on ths step s presented n the next subsecton. The random nput data are then passed to the rocket flght block and a full flght s smulated. Ths process s repeated for a predefned number of teratons and thus generates a cluster of flght paths and a scatter of landng postons. Fg. 11 shows two plots, the left hand plot shows the scatter of 5 smulated landng ponts usng the Monte Carlo wrapper. Also plotted are Gaussan ellpses markng one and two Monte Carlo Process The Monte Carlo process ntroduced n the prevous secton nvolves takng nput parameters wth values that are uncertan and makng them stochastc by addng random nose. The method employed for dong ths n our model s dfferent for the rocket data and the atmospherc data. We descrbe the two methods below. Uncertanty n the Examples of rocket parameters whch may be made stochastc are: the aerodynamc coeffcents of the rocket and the parachutes, the locaton of the rocket s center of pressure, the rocket s mass and center of mass and the launch tower angles. Ths lst s not exhaustve and n fact all of the parameters can be made stochastc f requred. The method for makng a parameter stochastc s to add a random nose term whch s sampled from a zero-mean Gaussan dstrbuton. In practce, ths s done by multplyng the parameter by a nose coeffcent whch s drawn from a Gaussan dstrbuton wth mean 1 and varance 2. Eq. 43 shows an example usng the coeffcent of axal force C A C A + = C A, N 1, 2 43 where C A + =updated value of C A. If the stochastc parameter has multple values n the database, as C A does, then s sampled once for each smulated flght and used for all values. The varance of the dstrbuton 2 must be carefully chosen to reflect the uncertanty of the correspondng parameter. Ths varance can be determned expermentally, or estmated from the uncertanty n the measurement or estmaton technque that was used to determne the parameter n the frst place. Uncertanty n the Atmospherc In modelng the atmospherc data, the approach used n the prevous secton would be too smplstc. Smply samplng from a one-dmensonal Gaussan for wnd speed and drecton at each alttude ncrement would gnore the obvous correlaton between values at adjacent alttudes. Smlarly, samplng once for a wnd speed error and then addng that error at every alttude would gnore the fact that perfect knowledge of the wnd at one alttude would stll leave uncertanty n ts value at other alttudes. We must, therefore, sample an entre profle of wnd speeds from the dstrbuton of such profles. The samplng of functons s more complex than the samplng of values. If we assume Gaussan uncertanty, then we must sample from a Gaussan process Bshop 26. Here, we adopt a relatvely smple approach to Gaussan process samplng based on a lnear expanson n a set of fxed bass functons. In order to model the uncertanty n the atmospherc forecast data, we use an approach whch s based on a maxmum lkelhood estmaton usng hstorcal forecasts and correspondng hstorcal measurements of the atmospherc condtons. There are known uncertantes correspondng to both the forecast and measurement data. However these uncertanty data are not readly avalable to the researcher. Therefore the methodology JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 / 39 J. Aerosp. Eng :31-45.

10 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Fg. 12. a Dfference between forecast and measured easterly component of wndspeed; b raw Gaussan bass functons supermposed on a ; and c sum of weghted bass functons gvng a least-squares ft to the dfference profle descrbed here s based on the assumpton that the dfference between the forecast and measured data are a useful ndcator of the uncertanty n the forecast data. Lnearze the Dfference Profle. Fg. 12 a shows a sample plot of the dfference between a forecast and measured easterly component of wndspeed. A useful method for makng a lnear approxmaton of the data n Fg. 12 a s to use bass functons. Fg. 12 b shows a number of Gaussan bass functons of the form n Eq. 44 supermposed on the dfference profle j z = exp z 2 j / JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 where =mean of the bass functon and 2 =varance. By multplyng each of the J bass functons by a scalar w j and then summng over all J the resultng functon can approxmate the dfference profle as shown n Fg. 12 c. Thus, for a vector of dscrete alttudes z the dfference profle d can be approxmately descrbed usng Eq. 45 whch s a lnear functon of the vector of scalar weghts w d w j j z = w T z 45 j J where w takes the values that mnmze the square error between the real wndspeed data d and the rght hand sde of Eq. 45. Maxmum Lkelhood Formulaton. In a large data set of N dfference profles the probablty of each weghts vector w n s assumed to be Gaussan wth mean and covarance as n Eq. 46 p w n = N w n,, n N 46 where s a J 1 vector and s a J J matrx. Usng the dscrete formulaton for an M 1 vector of alttudes z, the correspondng vector of dfference profle ponts d n s modeled as zero-mean Gaussan whte nose added to the functon descrbed by the sum of our weghted bass functons.e., p d n w n = N d n w n, 1 I, n N 47 where s a M J matrx wth elements m,j = j z m from Eq. 44 ; 1 =varance of the whte nose; and I s a J J dentty matrx. From Bayes theorem for Gaussan varables Bshop 26, the margnal dstrbuton of d n s gven by p d n = N d n, 1 I + T, n N 48 and the condtonal dstrbuton of w n gven d n s where p w n d n = N w n S T d n + 1,S, n N 49 S = 1 + T 1 5 The lkelhood functon L, whch represents the probablty of the data gven the parameters and vewed as a functon of those parameters, s gven by L = p d n w n p w n 51 n N By maxmzng the lkelhood functon, we can determne the values of the parameters for whch the probablty of the observed data are maxmzed. Equvalently, we can mnmze the log of the lkelhood functon ths s more convenent both analytcally and numercally. It can be shown from Eqs. 51, 46, and 47 that the log of the lkelhood functon s ln L = d T 2 n d n 2d T n w n +Tr T w n w T n + NJ n N 2 ln N 2 ln 1 2 n N Tr 1 T 2 w n T + w n w n T 52 The lkelhood s maxmzed by mnmzng Eq. 52 wth respect to,, and. The maxmum lkelhood can be found analytcally by maxmzng the product over N of Eq. 48. Ths gves J. Aerosp. Eng :31-45.

11 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Wndspeed dfference (ms 1 ) Alttude (m) Fg. 13. Samples of wndspeed dfference profles drawn from Eq. 46 ˆ = 1 N n N d n 53 where denotes the Moore-Penrose nverse. Because t s unlkely that ths approach wll uncover any systematc error between measurement and forecast data, you would expect that the elements of ˆ would all be close to zero. Lkelhood maxmzaton wth respect to and 1 can be done numercally usng the expectaton-maxmzaton EM algorthm. The expected log-lkelhood E ln L s gven by Eq. 52 where the terms w n and w n w T n take ther expected values. From Eq. 49 the expected values are gven by E w n = S T d n E w n w n T = S + E w n E w n T 55 Mnmzng the expected log-lkelhood wth respect to and, respectvely, gves the followng expressons for ˆ and ˆ 1 ˆ = 1 d T NJ n d n 2d T n E w n +Tr T E w n w T n 56 n N ˆ = 1 T 2 E w T N n + E w n w T n n N Eqs can be solved teratvely for ˆ and ˆ. 57 Samplng Wnd Dfference Profles. Usng the learned values of ˆ and ˆ wnd speed dfference profles can be generated Motor Pston Electroncs Bay randomly by samplng from the dstrbuton n Eq. 46. Fg. 13 shows examples of sampled wndspeed dfference profles. These were generated usng the same atmospherc data as descrbed later n demonstraton of the smulator. For more realstc generated dfference profles, random nose wth varance ˆ 1 can optonally be added to the profles. As mentoned at the begnnng of ths secton, ths method s desgned to capture the varaton of uncertanty wth alttude and the correlaton of speeds at adjacent alttudes. The effects of ths can be seen n the profles n Fg. 13, where there s a trend of greater uncertanty around 1, m. Ths corresponds to the alttude range whch generally has the hghest wnd speeds the jet stream. Also, the smooth varaton of the profles suggests a correlaton n values at adjacent alttudes. The method descrbed n ths secton relates to a sngle component of the measured and forecast wndspeed easterly. Inthe applcaton of ths method, the same procedure must be carred out for the northerly component of wndpeed, and f requred the vertcal component of wnd speed. Furthermore that entre procedure must be carred out separately for forecast data wth dfferent lead tmes. For example, data for 4 and 8 h forecasts should not be used n the same data set but should be consdered separately gvng dfferent values of ˆ, ˆ, and ˆ 1. At each step of the Monte Carlo smulaton, sampled dfference profles are added to the forecast wnd profle to generate stochastc wnd data. Demonstraton of the A demonstraton of the approach descrbed n ths paper was carred out by flyng a hgh-power rocket wth some nstrumentaton on board. Here we present a comparson between measured data obtaned durng the flght and data generated from a smulated flght. Flght Demonstraton A schematc of the rocket used n ths demonstraton s shown n Fg. 14. Ths s a 2.6-m long 76-mm dameter rocket wth a glassrenforced plastc fuselage. The thrust s provded by a Cesaron L73 sold fuel motor Fg. 15. The rocket uses a dual deploy recovery system. At apogee, the forward pyrotechnc charges are fred whch causes the nose cone to detach. A Kevlar shock cord connects the nose cone to the rest of the rocket after separaton. Ths causes a sgnfcant ncrease n the drag of the rocket although no parachute s deployed at ths stage ths s nevertheless modeled as a parachute descent. When the rocket reaches an alttude of 3 m durng the descent, then the rear pyrotechnc charges are fred causng the tal secton to detach and a pston pushes out the parachute. Break ponts Parachute Pyro charges Shock Cord m Fg. 14. Schematc llustraton of the rocket used n the flght demonstraton JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 / 41 J. Aerosp. Eng :31-45.

12 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Thrust (N) Tme (s) Fg. 15. Thrust curve of the Cesaron L73 sold fuel rocket motor The electroncs bay contans a number of commercally avalable rocket avoncs devces whch consst of varous sensors ncludng three-axs accelerometers, two-axs Hall effect sensors, GPS recever and pressure transducers. Mcroprocessors control the frng of the pyrotechnc devces and log the sensor data. A lst of the devces s gven n Table 6. The system for detectng alttude and detonatng the charges s dual redundant for safety. Flght Smulaton The flght path of the rocket descrbed above was smulated usng the method lad out n ths paper. The aerodynamc coeffcents and the dynamc propertes of the rocket that are requred as nputs for the smulaton were estmated from the geometry of the rocket followng the method lad out n Box et al. 29. Here we have assumed that C N s ndependent of R. The aerodynamc coeffcents for the parachutes were estmated from the descent rate data recorded durng prevous flghts. To estmate the uncertanty n the atmospherc forecast we used data kndly suppled by the Brtsh Atmospherc Centre These conssted of 5 years of measurement and forecast data coverng a perod from 21 to 26. The measurement data came from the mesospherestratosphere-troposphere MST radar at the Unversty of Wales, Aberystwyth and also from soundng balloons that are launched from the same ste. Radar and soundng measurements are recorded four tmes a day at Aberystwyth, U.K. The forecast data came from the Met Offces s Numercal Weather Predcton NWP model. Dfference profles between the 1 h forecast data for the Aberystwyth, U.K. ste and the correspondng measurements were used as descrbed n ths paper to Table 6. Avoncs Devces Used on Board the Devce Name Manufacturer Flght computer ncludng pressure transducer and accelerometer RDAS AED electroncs Waalre, The Netherlands 2-axs accelerometer RDAS 2-axs AED electroncs 2-axs Hall effect sensor Magnetosensor Aerocon San Jose, Calf. GPS recever RDAS GPS AED electroncs Redundant flght computer ncludng pressure transducer and accelerometer mnalt PerfectFlte Andover, N.H. Table 7. Values of the Varance n the Nose Added to the Stochastc Parameters Stochastc parameter C A.2 C N.1 X cp.5 C D.1.87 generate stochastc dfference profles. These were added to the NWP 1-h forecast data for the rocket launch ste to generate stochastc wnd profles for the smulaton. Other parameters whch were made stochastc for ths demonstraton were the rocket aerodynamc coeffcents C A and C N, the center of pressure X cp and the parachute coeffcent of drag C D. These were made stochastc by multplyng by a random nose coeffcent as descrbed earler. Although the rocket s not desgned to roll durng flght mperfectons n the buld may nduce some roll. Ths s modeled by makng fn cant angle a stochastc parameter, because s zero-mean t s modeled as N, 2. The varances of the stochastc rocket parameters were estmated n an ad hoc manner. The method for estmatng the values of the rocket parameters from Box et al. 29 was used to examne the senstvty of the parameters to small changes n rocket or parachute geometry. The results were used to nform an expert guess at the varances, the values of used are shown n Table 7. Demonstraton Results Unfortunately the GPS sensor faled to log any poston data durng the flght so the only accurate poston data for the rocket are at the launchpad and the landng ste. Table 8 shows some selected statstcs from the flght together wth ther smulated values for a run where all the stochastc parameters are set to ther mean value. The measured veloctes alttudes and tmes are as reported by the R-DAS avoncs system. Fg. 16 shows a plot of the smulated flght path. The cross marks the launchpad locaton and the mean flght path s plotted. The end of the flght path marks the mean smulated landng poston. The two ellpses show, respectvely, one and two standard devatons n the probablty of the landng poston as calculated from 5 Monte Carlo smulaton flghts. The damond symbol marks the landng locaton of the actual rocket at the end of the demonstraton flght. In Fg. 17, the lateral acceleraton of the rocket s plotted. Specfcally, ths s the lateral acceleraton at the pont n the rocket where the accelerometers are located and both measured Table 8. Comparson between Smulated and Measured Flght Statstcs Smulated Measured Launch tower clearance velocty 4 ms 1 37 ms 1 Maxmum velocty ms ms 1 Apogee alttude 3,539 m 3,594 m Tme to apogee 24.5 s 24.5 s Total flght tme 17 s 182 s Landng poston E,N 135 m, 936 m 71 m, 142 m Dfference n landng postons 125 m 42 / JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 J. Aerosp. Eng :31-45.

13 35 3 velocty (m/s) Yaw Ptch Roll Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved. Alttude (m) Fg. 16. Mean smulated flght path wth 1 and 2 landng probabltes. The measured landng poston s marked by a damond. and smulated acceleraton s shown. The smulated data were generated wth all stochastc parameters at ther mean value. The fgure focuses on the short perod of the rocket flght just after the rocket has cleared the launch tower, when weather cockng occurs. The damped oscllaton can be seen n both the smulated and measured data and t can be seen that there s broad agreement n the ampltude, wavelength, phase and rate of decay of the oscllaton. The fgure shows a sgnfcant ncrease n the nose on the accelerometer sgnal both before and after the weather cockng event. The wrters are not certan why ths s the case although Acceleraton (ms 2 ) tme (t) Measured Predcted Fg. 17. Comparson between the smulated and measured lateral acceleraton of the rocket durng the weather cockng event angular velocty (rad/s) tme (s) Yaw Ptch Roll (.1) tme (s) Fg. 18. Outputs from a sample stochastc smulaton showng the Earth relatve lnear and angular veloctes of the rocket n reference axes yaw, ptch, and roll. The angular velocty about the roll axs s dvded by 1 to mprove scalng. one possblty s that there s a mode of resonance affectng the board where the accelerometers were mounted that was not excted durng weather cockng. Fg. 18 shows plots of lnear and angular velocty from a sample stochastc smulaton. The lnear veloctes are n the rocket s ptch, yaw and roll axes and the angular veloctes are about these axes. These data show the sx DOF n the smulaton. It can be seen that the angular roll velocty s strongly correlated wth the rocket s forward velocty as expected. The damped oscllatons durng the weather cockng event can be seen n the ptch and yaw angular veloctes and some further oscllatons occur as the rocket ptches over at apogee. Effect of Varyng Uncertanty To generate the stochastc data n Fg. 16, best estmates of the varance n the stochastc parameters were used. However, t s also nterestng to examne the effect that changng the varance has on the results of the smulatons. A full senstvty analyss coverng all stochastc parameters s beyond the scope of ths paper but below we present some results from addtonal experments where the standard devatons of some of the stochastc parameters were ncreased systematcally. Varyng the Uncertanty n C A The Monte Carlo smulatons that generated the results shown n Fg. 16 were repeated twce. Once wth the standard devaton of C A ncreased to double ts default value =.4 and once wth t doubled agan =.8. The standard devatons of all other stochastc parameters retaned ther default values gven n Table 7. Fg. 19 shows Gaussan ellpses markng the two standard devatons area of confdence n the landng poston of the rocket. These were generated usng the data from the orgnal experment and the two addtonal experments descrbed above. It s nterestng to note the drectonalty of the ncrease of the landng area. There s very lttle ncrease n the northwestsoutheast drecton, but sgnfcant ncrease n the northeastsouthwest drecton. JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 / 43 J. Aerosp. Eng :31-45.

14 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved σ =.2 σ =.4 σ =.8 Fg. 19. Gaussan ellpses showng the two standard devaton areas of confdence n landng poston. C A s stochastc wth varyng.all other stochastc parameters have ther default values of. Fg. 2 shows a scatter plot of smulated apogee ponts from the above experments. To avod clutter, only 1 apogees from each of the three experments are shown. Ths plot shows the spread of the apogee ponts ncreasng wth the ncrease n. Wth =.8, the varance s hgh enough that the drag force on the rocket can drop to zero, or even become negatve. Ths s why some of the apogee ponts are very hgh. In ths case, the varance n C A s unrealstcally large. Varyng the Uncertanty n C N A smlar test to the one descrbed above was carred out for the rocket s coeffcent of normal force C N. Here the standard devaton of C A was returned to ts default value =.2 and the standard devaton of C N was doubled, once to =.2 and agan to =.4. The results for landng poston are shown n Fg. 21. As standard devaton s ncreased to =.2, the landng area ncreases slghtly n the northwest-southeast drecton, but actually reduces n the northeast-southwest drecton. As the standard devaton s ncreased agan to =.4 the area contnues to grow n the northwest-southeast drecton wth no further reducton n the other drecton. Fg. 22 shows a scatter plot of smulated apogee ponts from the C N experments. Agan, only 1 apogees from each of the three experments are shown. Ths plot shows the spread of the apogee ponts ncreasng wth the ncrease of the uncertanty n C N but n general the growth n the spread of the apogee scatter s less than that observed when varyng C A. Varyng the Uncertanty n C D Varaton of the uncertanty n the drag coeffcents of the parachutes C D was nvestgated followng the same procedure as the prevous two tests. The standard devaton of C D for both the drogue and man parachutes was ncreased to =.2 and then =.4. The plots of landng poston are shown n Fg. 23. As wth the prevous examples, the ncrease n the landng area s hghly drectonal. In ths case the drectonalty s approxmately algned wth the prevalng wnd drecton, whch s predomnantly blowng from the southwest n these experments. 44 / JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 Alttude (m) Conclusons σ =.2 σ =.4 σ = Fg. 2. Scatter plot showng the spread of apogee ponts for 3 smulated flghts. 1 for each value of relatng to C A. The accuracy of the smulaton method presented n ths paper wll depend upon the valdty of the assumptons used n the rocket model e.g., axsymmetrc, rgd body, the error tolerance n the numercal ntegraton and, to a sgnfcant extent, the accuracy of the user suppled parameters descrbng the rocket dynamcs, aerodynamcs and the atmospherc condtons. There s a doman of uncertanty n the rocket s trajectory that arses from the naccuraces n these nput parameters. The approach that we have used of stochastc smulaton usng the Monte Carlo method allows us to explore ths doman usng knowledge of the naccuraces n the estmaton of these parameters. We have presented some results from a demonstraton where the smulator was used to predct the flght path of a real rocket. The results of ths demonstraton produced encouragng evdence that the determnstc smulaton method can be effectve. Of course a sngle comparson can tell us nothng about the effcacy of our method to estmate quanttatvely the uncertanty n the flght path. Valdatng ths method would requre data from many more test flghts and ths s a goal for future work. J. Aerosp. Eng :31-45.

15 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13. Copyrght ASCE. For personal use only; all rghts reserved σ =.1 σ =.2 σ = Fg. 21. Gaussan ellpses showng the two standard devatons areas of confdence n landng poston. C N s stochastc wth varyng.all other stochastc parameters have ther default values of. Alttude (m) σ =.1 σ =.2 σ = Fg. 22. Scatter plot showng the spread of apogee ponts for 3 smulated flghts. One hundred for each value of relatng to C N Acknowledgments Many thanks to Phl Charlesworth and Brett Saunders for numerous valuable dscussons, and to the Brtsh Atmospherc Centre for grantng access to ther of data. References σ =.1 σ =.2 σ =.4 Fg. 23. Gaussan ellpses showng the two standard devatons areas of confdence n landng poston. C D s stochastc wth varyng.all other stochastc parameters have ther default values of. Baraff, D Rgd body smulaton. Proc., Sggraph 97 24th Int. Conf. on Computer Graphcs and Interactve Technques Course Notes, Sggraph, Los Angeles. Bshop, C. 26. Pattern recognton and machne learnng, Sprnger, New York. Box, S., Bshop, C., and Hunt, H. 29. Estmatng the dynamc and aerodynamc propertes of small rockets for flght smulatons. Techncal Rep., sourceforge.net/ Feb. 29. Dobrokhodov, V. N., Yakmenko, O. A., and Junge, C. J. 23. Sxdegree-of-freedom model of a controlled crcular parachute. J. Arcr., 4 3, Duncan, L. D., and Ensey, R. J Sx degree of freedom dgtal smulaton model for unguded fn-stablzed rockets. U.S. Army electroncs research & development actvty, Stromng Meda, Washngton, D.C. Km, Y., and Peskn, C. S d parachute smulaton by the mmersed boundary method. SIAM J. Sc. Comput. (USA), 28 6, Nassr, N., Roushanan, J., and Haghgat, S. 24. Stochastc flght smulaton appled to a soundng rocket. Proc., 55th Int. Astronautcal Congress 24, Internatonal Astronautcal Congress, Vancouver, Canada. Natonal Fre Protecton Assocaton NFPA. 22. Report of the Commttee on Pyrotechncs, NFPA 1122/1127, Boston. Press, W., Teukolsky, S., Vetterlng, W., and Flannery, B. 27. Numercal recpes, Cambrdge Unversty Press, Cambrdge, Mass. Weatherll, S. 29. Hgh power rocketry records. Canadan Assocaton of ry, Feb. 29. JOURNAL OF AEROSPACE ENGINEERING ASCE / JANUARY 211 / 45 J. Aerosp. Eng :31-45.