636 J.SPACECRAFT,VOL. 39,NO. 4: ENGINEERING NOTES

2Crimi,P .,andSeigelman, D., “ Analysisof Mechanical andGas Dy- namic LoadingsDuring Sabot Discard forGun-Launched Projectiles,” U.S. Army Research Lab.,ARBRL-CR-341, Aberdeen Proving Ground, MD, June 1977. 3Seigelman,D., W ang,J., and Crimi, P .,“Computationof Sabot Discard,” U.S.Army Ballistic Research Lab.,ARBRL-CR-505, Aberdeen Proving Ground,MD, Feb.1983. 4Nusca, M.,“ ComputationalFluid Dynamics Applicationto the Aerody- namics ofSymmetric SabotDiscard,” AIAA Paper90-2246, Aug. 1990. 5Nusca, M.,“Numerical Simulationof Sabot Discard Aerodynamics,” AIAA Paper91-3255, Sept. 1991. 6Ferry,E., Sahu,J., andHeavey, K., “Navier –StokesComputations ofSabot Discard UsingChimera Scheme,”U.S. Army Research Lab., ARL-TR-1525,Aberdeen Proving Ground, MD, Oct. 1997. 7Cox,R. N.,and Crabtree, L.F., Elements ofHypersonicAerodynamics , EnglishUnivs. Press, London,1965, pp. 70, 71. 8Lees, L.,“ HypersonicFlow,” Proceedingof the 5th International AerospaceConference ,Inst.of Aerospace Society,1953, pp. 241 –276. 9“Equations,Tables, and Charts for Compressible Flow,” NACA Rept.1135, Ames AeronauticalLab., Moffett Field, CA, 1953. 10Clippinger,R. F.,Giese, J.H.,and Carter, W.C.,“ Tables ofSupersonic Fig.11 Comparative front bourrelet normalforce results forsabot FlowAbout Cone Cylinders, Part I: Surface Data,”U.S.Army Ballistic models 1–7. Research Lab.,Rept. 729, Aberdeen Proving Ground, MD, July 1950. 11Sahu,J., Pressel, D.,Heavey, K., and Dinavahi, S., “Parallel Numerical Theanalysis, being overpredictive but simple and fast, was then Computationsof Projectile FlowŽ elds,” AIAA Paper99-3138, June 1999. appliedto six other sabot models, as given in Figs. 9 and10, to 12Edge,H. L.,Sahu,J., Sturek, W. B., Pressel, D.M.,Heavey,K. R., examinetheir relative lifting capability. These sabot models and Weinacht,P .,Zoltani,C. K.,Clarke,J., andBehr, M., “ComputationalFluid dimensionswere generated by thePRODAS code.13 Thesesabots Dynamics (CFD) Computationswith Zonal Navier –StokesFlow Solver (ZNSFLOW) CommonHigh Performance ComputingScalable Software arefor both the 120- and 105-mm calibers, as provided in T able2. Initiative (CHSSI) Software,”U.S. Army Research Lab.,ARL-TR-1987, Notethat sabot model 3 hasfourpetals instead of themore common AberdeenProving Ground, MD, June 1999. three-petaldesign. All calculations were made at Mach 4.5 and the 13“PRODASCode, V ersion3.6,” Arrow Tech Associates, Inc.,South sameconditions. The resulting front bourrelet normal force on each Burlington,VT, 1991. petalmodel is shown in Fig. 11. Sabot models and their correspond- ingnormal forces are provided in Table2. These predicted forces R.M.Cummings areto be considered overestimated, possibly by a factorof about AssociateEditor 1.6 (basedon theCFD resultsfor model 1 ).Anactual test is rec- ommendedto be made to provide a validityfor the CFD results Atmospheric Reentry aswell.Meanwhile, the comparative results for the different sabot conŽgurations provide some insight about the relative effectiveness Modeling and Simulation ofdifferentfront bourrelet designs in producinglift.

Conclusions † ‡ R. R. Costa,¤ J. A. Silva, S. F. Wu, Aerodynamiclifting force on a closed,three-petal sabot conŽ g- § ¶ urationmounted on its host projectile is analyzed. The modiŽ ed Q. P. Chu, andJ. A.Mulder Newtoniantheorem was used to estimatea Žrst-ordervalue for the DelftUniversity of T echnology, liftingforce on thefront bourrelet (during ightin closeproximity 2629HS Delft,The Netherlands ofthemuzzle ).CFD calculationswere made, and theresult suggests thatthe estimate obtained by the Newtonian method overpredicted theCFD valueby afactorof 1.6. The present investigation provides Introduction newinsight about the aerodynamic forces on the sabot, their rela- TMOSPHERIC reentrypresents challenges in several do- tivemagnitudes, and the role of thedrag force in thesabot opening A mainsof engineering and science, being one of the principal motion.The modiŽ ed Newtonianmethod is appliedto amultitude researchŽ eldsin spacetechnology. T oprovidean environmentto ofdifferentsabot models for 120 and 105 mm calibers,to provide designcontrol laws for reentry vehicles, Delft University of Tech- comparativevalues for the front bourrelet lifting force. nology (TuDelft) isdeveloping a simulationtool for atmospheric Thefollowing conclusions are supported by the results ob- reentry.The simulation tool is called general simulator for atmo- tainedfor the two-bourrelet, three-petal closed sabot conŽ guration sphericreentry dynamics (GESARED) andwas implemented in mountedon its host projectile. The conclusions apply to a com- MATLAB®/SIMULINK. Thesimulation tool is meant to work on pletelyclosed sabot, which occurs only during the Ž rstfew meters apersonalcomputer. GESARED wasinitially developed to be the fromthe muzzle. First, the front bourrelet generates a liftingforce (upward),butthe total normal force on thepetal may still be down- ward.Second, the drag force on thesabot petal studied was about Received 25September2000; revision received 15February 2002; ac- ceptedfor publication 22 February2002. Copyright c 2002by the American seventimes the value of thenet normal force, for the conŽ guration ° considered.Third, the drag force on a petalis a majorcontributor Instituteof Aeronauticsand Astronautics, Inc. All rights reserved. Copies tothelifting up ofthe sabot front end, in theinitial opening stage. ofthispaper may bemade forpersonal or internaluse, on conditionthat the copierpay the $10.00 per-copy fee totheCopyright Clearance Center,Inc., Fourth,anatomy of the drag on thesabot is given, indicating that 222Rosewood Drive, Danvers, MA 01923;include the code 0022-4650/ 02 thefront cup contributed 87% of the total drag, whereas the second $10.00in correspondence with the CCC. contributed11%. Note that the outer ring of the front cup contributes ¤ Research Engineer,Faculty of Aerospace Engineering;Rodrigo. 30%ofthetotal sabot drag. Fifth, the modiŽ ed Newtoniantheorem [email protected] AIAA. overestimatedthe CFD valueby a factorof about1.6 for the cur- †Graduate Student,Faculty of Aerospace Engineering;J.G. rentsabot conŽ guration at Mach 4.5. Use of such a methodmust be [email protected]. accompaniedwith that realization. ‡Research Fellow,Facultyof Aerospace Engineering;S.F.Wu@LR. TUDelft.NL. References §Assistant Professor,Faculty of Aerospace Engineering;Q.P .Chu@ 1Schmidt,E. M.,“WindTunnel Measurements ofSabot Discard Aerody- LR.TUDelft.NL. namics,”U.S. Army Ballistic Research Lab.,ARBRL-TR-2246, Aberdeen ¶ Professorand Chairman of the Control and Simulation Division, Faculty ProvingGround, MD, July1980. ofAerospace Engineering;[email protected]. J.SPACECRAFT,VOL. 39,NO. 4: ENGINEERING NOTES 637 designand test bed for guidance, navigation, and control (GN&C) EnvironmentModeling systemsfor representative reentry vehicles. Its primary goal was Theenvironment included in the simulation tool is that of the tobetheopen-loop plant for reentry simulation, to be closedby a Earth.The model of theenvironment comprises three submodels: GN&C algorithm.In this Note, focus is given to the modeling of atmosphere,planet shape, and gravity. T wodifferent models were thelifting body reentry vehicle (LBRV) andthe atmospheric reentry usedto simulate the atmosphere: the standard atmosphere 1962 and capsule (ARC),butmodels of other reentry vehicles are also dis- theglobal reference atmospheric model of 1999 (GRAM-99). The cussed.Currently GESARED isthe simulation environment used in standardatmosphere of 1962 assumes the division of the atmosphere thedesign of GN&C systemsfor the LBR VandtheARC. 1 The entry into11 layers and linear variation of temperature with altitude. 2 phasein Earth’ s atmospherecorresponds to an altituderange from Assumingthe model of spheric Earth (anassumption used only for 120km to nearlyzero. This phase is characterized by large varia- theatmospheric computations ),therelations between pressure, den- tionsof environmentalconditions. Therefore, it isnecessary to have sity,and altitude can be extrapolated. 2 Thismodel lacks accuracy, realisticmodels of theenvironment. Because of largevariations of butit is simple and allows fast computations. GRAM-99, an empir- atmosphericconditions and aerodynamic angles, the model of the icalmodel of NASA MarshallSpace Flight Center, is accurate,but vehicle’s aerodynamicshas major importance. The main objective becauseof its complexity and detail computations with this model isthedesign of controllaws, and thusthe modelsof thesensors and arerelatively slow. The Earth shape model is used in the computation actuatorsshould also be realistic,in order to obtain a reliableoverall ofaltitudegiven the distance toward the Earth’ s center.The model performanceevaluation of the closed-loop system. usedin GESARED considersthe Earth as athree-dimensionalŽ g- ureobtained from the rotation of anellipseover its small axis. The EntryDynamics Earthgravitation Ž eldmodel 2 implementedin GESARED usesthe Theplanetary entry motion is modeledas ofsix degrees of free- constantsfrom the world geodetic system of 1984 (WGS-84) and ( ) dom DOF ,whereuncoupled translational and angular motions are theJeffery constants until J4.Consequentlythe gravity force is de- considered.The translational motion represents the point-mass tra- pendentnot only on altitude but also on latitude. The gravity model jectorymotion. The generic kinematics and dynamics equations 2 providesadequate accuracy. thatdescribe this motion are d2r dr LiftingBody Reentry V ehicle F m cm ; V cm (1) E D dt 2 D dt TheLBR Visa conceptualsmall reentry vehicle, based on a real-worldexample. The vehicle creates lift by  yingat high an- in which FE denotesexternal force, rcm denotesthe positionvector, glesof attack. The model of the spacecraft is implemented in m isthe vehicle mass, and V denotesthe velocity vector in the inertial MATLAB®/SIMULINK andconsistsof three models: aerodynam- frame.The choice of the coordinate system is of majorimportance ics,sensors, and actuators. becauseit will in uence the effectiveness of the simulation tool. Theavailable aerodynamic database is nonlinear, and it is pre- Theobjective is tohave a minimumnumber of constraints and maxi- sentedin the form of look-up tables. These tables were gener- mumgenerality, while avoiding unduly the complexity of equations. atedmerging results from computational  uiddynamics (CFD) Thereforevelocity is expressedin Cartesian coordinates of north, withresults from wind-tunnel tests. 4 Theaerodynamic computa- east,and down. Position is expressed in spherical coordinates of tionsare divided in twospeed regions: subsonic/ supersonic (Mach latitude,longitude, and distance toward the Earth’ s center.The re- number 4.6) andhypersonic (Mach number 4).Theaerody- sultingequations of translational motion comprise three equations · ¸ 3 namiccoefŽ cients for Mach numbers between 4 and4.6 are com- forvelocity and three more for position, theexternal force vector putedusing a “bridging”function. The resulting subsonic, super- components Fx , Fy , and Fz beingexpressed in the vertical frame. sonic,and hypersonic aerodynamic force and moment coefŽ cients Theonly singularity of these equations of translational motion is arepresented depending on thecontrol surface de ections, aerody- whenthe latitude is ¼/2.The consequence of this singularity is § namicangles, angular rates, Mach number, Knudsen number, air aconstraintin thesimulation tool of polar reentry trajectories. As density,and air temperature. suchtrajectories are very uncommon, this is not thought to limitthe TheLBR Vhastwo types of sensorsthat are used in the reentry versatilityof the tool in practice.The forces acting on the vehicle phase:the  ushair data system (FADS) andthe space-integrated arethe aerodynamic forces, the gravity force and the thrust force, globalpositioning system –inertialnavigation system (SIGI). The resultingin thetotal external force FE .Theaerodynamic and thrust FADS measuresthe  owvelocity, angle of attack, and angle of forcesare computed from both the vehicle and the environment sideslip.This sensor is representative of a state-of-the-artsensor models,given a speciŽc ightcondition. The gravity force is com- system,developed for advanced airplanes  yingin extreme condi- putedfrom the environment model, given the mass and the altitude tions.The sensor consists of aparabolicprobe that is mountedon ofthevehicle. thenose of thespacecraft and a ightair data computer. The probe Theangular motion represents the rigid-body attitude motion. 2 hasseveral holes, connected to pressure transducers. The relation Thismotion is describedby thefollowing equations : betweenthe real pressure and the measured pressure is modeled dH dµ asasecond-ordertransfer function. The relation between pressure ME ; ! (2) measurementsand velocity, angle of attack and angle of sideslipis D dt D dt highlynonlinear, and can be solvedusing estimation algorithms; in in which ME denotesan external moment, H isthe angular momen- thiscase we used the least-squares algorithm. The model used for tum, ! isthe angular velocity vector, and µ isthe attitude vector. theF ADS wasa TUDelftmodel developed for the EuroŽ ghter. The Onceagain, it is necessaryto choosethe reference frames in which SIGI providesthree navigation solutions: inertial navigation, global theattitude rates and attitude position are expressed. The angular positioningsystem (GPS) navigation,and combined GPS inertial velocitiesare conveniently expressed in the vehicle’ s body-Žxed navigation.The choice of the solution used depends on the  ight referenceframe, represented by thecommon notation of p, q, and phase.During the blackout phase (rangingfrom 70 to 30 km ), GPS r.Theattitude position is chosenin order to avoid singularities. If isnotavailable, and the inertial solution is used.In therest of the theEulerean attitude angles of roll,pitch, and yaw are used, there ight,the combined solution is used. T wokinds of actuators are willbe asingularitywhenever one particular angle is ¼/2, depend- availableduring reentry: aerodynamic surfaces and reaction control ingon theorder of rotation.This constraint might be §unacceptable system (RCS).Thevehicle has two types of controlsurfaces, that whendealing with planetary entries in general, and thereforea four- is,  aps (leftand right ) and elevons (leftand right ).Thesurfaces are elementquaternion is used instead. The external moments acting on drivenindependently by using electromechanical actuators. Actu- thevehicle are the aerodynamic moments and the moments result- atordynamics are represented by second-ordertransfer functions. ingfrom the activation of the thrusters. The resulting equations for TheRCS isa pressure-regulatednitrogen system and includes a to- angularmotion 3 comprisethree equations for angular rates and four talof 22cold-gasthrusters. Sixteen thrusters are located in the aft forthe attitude quaternion, where the external moment (vector Mx portionof thespacecraft, and the other six are located in the nose My Mz ) isexpressedin the body-Ž xed reference frame. ofthespacecraft. 638 J.SPACECRAFT,VOL. 39,NO. 4: ENGINEERING NOTES

AtmosphericReentry Capsule Thetests performed with the ARC modelsaimed to compare TheARC isan Apollo-like shape guided unmanned capsule, GESARED simulationswith real  ightdata and, therefore, validate whoseobjective is todemonstrateGN&C technology.The ARC’ s thesimulator translational and angular motions models, the ARC Žrst ightwas successfully completed in 1998 and consisted of aerodynamicdatabase, and the thrusters models. Having this goal in launch,suborbital ballistic  ight,reentry, and descent. For the reen- mindand aimingto assessthe origin of the mismatches, two different tryphase the modeling of the vehicle was divided in three parts: testswere performed. The Ž rsttest setup is of sixDOF ,makinguse of aerodynamics,sensors, and actuators. theforces computed through the aerodynamicdatabase. This simu- Thenavigation system provides velocity and position informa- lationwas performed in a purelyopen-loop conŽ guration, implying tionto the guidance and control systems to allow a landingaccuracy thatthe inputs to GESARED duringthe entire simulation came from ofatleast100 km. The navigation system comprises two sensors: ightmeasurements (includingatmospheric data, aerodynamic an- astrapdowninertial measuring unit (IMU) andaGPS receiver.The gles,altitude, and  ight-pathspeed ).Theaerodynamic forces and IMUoutputsthe position vector, the absolute velocity vector, and momentswere computed from the ARC aerodynamics,having the thespacecraft attitude. The GPS receiver,used to update the IMU inputfrom  ightdata fed totheaerodynamic database. These forces measurements,outputs position (altitude,longitude, and latitude ) andmoments, together with the gravity force, were inserted in the andrelative velocity (north,west, vertical ) referencedto the WGS six-DOFequations of motion. In this conŽ guration error propaga- 84geoid.In caseof GPS failureor during blackout phase, the drag- tionwas avoided, and the atmospheric model was eliminated. This derivedaltitude (DDA) isused instead. The DDA isa meansof testsetup enabled an evaluation of boththe aerodynamic modeling estimatingthe altitude by combiningthe measurements of aerody- andangular motion modeling. The results obtained were compared namic (ornongravitational) accelerations,aerodynamic and atmo- withthe  ight-testdata, and for the attitude, the simulations and sphericmodels. The measurement accuracy is bounded by 176and ightproved to matchalmost perfectly; the total velocity curve also 400m forposition, whereas velocity is bounded by 2and2.5 m/ s. matcheswith the  ightmeasurements, whereas the  ight-pathan- Theaerodynamic database was obtained via CFD andis divided glematching is degraded. The second test setup is also of six DOF , intwoparts according to the owregime: the continuum and the butin this case error accumulation was not avoided as it wasin the freemolecular. T oensurea smoothtransition from free molecular tocontinuum ow,a bridgingfunction is used.The  owregime de- pendson theKnudsen number, which represents the rarefaction of theatmosphere. The aerodynamic tables only provide coefŽ cients forthe longitudinal motion. However, because the vehicle has an axis-symmetricshape the data tables can be consideredvalid for a totalangle of attack, that is, the angle between the longitudinal body axis X B andthe velocity. Thus, from any combination of ® and ¯ threeangles can be computed (oneofthembeing the total angle of at- tack),withwhich a totallift force can be determined.This force will thenbe decomposed in “purelift” (X B Z B plane) and“ sideforce” (X B YB plane).TheARC ismaneuveredand stabilized by means ofanRCS. TheRCS forthe ARC iscomposedof sevenhydrazine thrusterspositioned to generatepure moments in the three axes.

SoftwareV alidation Thevalidation of GESARED wasperformed in a stepwise approach.First, the aerodynamic, sensor, actuator, and environ- mentmodels of both vehicles were separately tested, as well as theequations of motion.After successfully validating all individual GESARED blocks,we performed tests to thecomplete simulation tool.Thus, with the two different vehicles implemented and making useof three-DOF simulated trajectory for the LBR Vandsix-DOF ighttrajectory for the ARC, wehave data to run complementary testsnot only to validateGESARED butalso to allowthe identiŽ - cationof thediscrepancy sources. Thevalidation of GESARED withthe LBR Vmodelin theloop wasperformed by means of comparisonbetween open-loop simu- lationresults and a referencereentry trajectory. The reference tra- jectorywas computed by thecommercial validated software pack- agefor reentry trajectory optimization ASTOS, 5 whichgenerates anoptimal drag acceleration vs velocity proŽ le that respects the boundaryconstraints (maximumheat  ux,dynamic pressure, and normal load).Toperformthe test, a simpliŽed GESARED setup wasimplemented, consisting of themodel of translationalmotion, theLBR Vmodels,and the environment models. The inputs of the systemwere the initial state and the time history of theaerodynamic angles (angleof attack,sideslip angle, and bank angle ) fromthe ref- erencetrajectory. These inputs were fed to the aerodynamic database implementation,and the resulting lift and drag forces, summed withthe gravitational force obtained from the Earth model, were insertedinto the equations of translationalmotion. The GESARED outputs (altitude,latitude, longitude,  ight-pathspeed, azimuth and ight-pathangles )werecompared with the corresponding ones from thereference trajectory. The comparison was satisfactory, and thus GESARED’s three-DOFtranslational motion model and the LBR V models (sensors,actuators, and aerodynamic database ) were vali- dated.Furthermore, the atmospheric and gravitational models were Fig.1 ARC reentry trajectory: ,ightdata; – – –,Žrst conŽgura- alsovalidated. tionsimulation; and — —,second¢ ¢conŽ ¢ ¢ guration simulation. J.SPACECRAFT,VOL. 39,NO. 4: ENGINEERING NOTES 639 earliertest conŽ guration. In this case the inputs of thesystem were 4Wu,S. F.,and Chu, Q. P.,“TheAtmospheric Re-Entry Spacecraft theinitial state at entry interface and theangular rates  ighthistory. CRV/X-38—Aerodynamic Modeling and Analysis,” ESA –EuropeanSpace (Thereis nothruster activity data from  ight. ) Thus,the system state TechnologyCenter, ESTEC Rept., Noordwijk, The Netherlands, July 2000. atagiveninstant results from the state history since the beginning 5 ofsimulation.This simulator conŽ guration is moreprecise because Wiegand,A., Markl, A., Well, K. H.,Mehlem, K., Ortega, G., and Steinkopf,M., “AL-TOS—ESA ’sTrajectoryOptimization Tool Applied to theatmospheric model is inthe loop. The test allows evaluation of Re-EntryV ehicle TrajectoryDesign,” Proceedingsof the50th International theARC modelsand validation of GESARED. Moreover,this test AstronauticalF ederationCongress ,InternationalAstronautical Federation, canalso be used for post ight analysis, and the results obtained Amsterdam, 1999. werecompared with  ightdata. It can beconcludedthat there is an increasein error, mainly resulting from error propagation. Figure 1 C.A.Kluever showsthe results of bothtests, compared with the real  ightdata. Al- AssociateEditor thougherrors are neglectable in the beginning of thesimulation, they increasetoward the end,being more signiŽ cative for the second test conŽguration. The justiŽ cation for this error lies in the aerodynamic database (bothsimulations ) andin the error accumulation (second New ReliabilityConŽ guration for simulation).Becausein the last part of thesimulation the total ve- Large Planetary Descent Vehicles locitydecreases, the relative importance of forces increase, making thediscrepancies between real and predicted aerodynamics more visible.These results are considered satisfactory as GESARED six- † Huseyin¨ Sarper ¤ andW olfangJ. Sauer DOF simulationsusing attitude as referencetrajectory are shown to Universityof SouthernColorado, Pueblo, Colorado 81001 closelymatch real  ightmeasurements.

Conclusions Introduction Anewreentry simulation tool, GESARED, wasimplemented in 1;2 ® HIS Noteis basedon presentations atthepast two confer- the MATLAB /SIMULINK environment.The primary objective of T encesof the Mars Society. Several successive large vehicles thissimulator is to beused as adesignenvironment of GN&C sys- mustland in an attempt to establish the Ž rsthuman outpost. A temsfor reentry vehicles. As a testbed, GESARED isalso designed singledescent engine, as in the lunar module, is not under consid- tofacilitateperformance evaluations against requirements in both erationbecause it would not provide enough reverse thrust to slow nominaland off-nominal conditions, allowing the entire GN&C thevehicle and it wouldcause balance problems during landing. It systemdevelopment process to be condensed in one tool based wasstated that, if one of the even number of engines fails while onacompatible,versatile, and easy-to-useenvironment. Moreover, landing,the opposite engine would have to be shutoff to maintain GESARED isalso designed to makeopen-loop simulations possi- vehiclebalance. The need to shut off the opposite engine presents a blefor model veriŽ cation or postight analysis. GESARED, more uniquereliability problem, not readily found in theliterature. T wo thanusing the potentialities of thesimulation environment, has itself newengine conŽ gurations are considered: a balancedfour-engine anopenstructure, allowing extensibility and model modiŽ cations. (BFE) vehicleand a larger,balanced six-engine (BSE) vehicle. ( Theversatility is exempliŽ ed inthe set of reentry vehicles such as Figures1 and2 showthese conŽ gurations, respectively. Both vehi- ) theLBR Vorthe ARC thatcan be simulatedusing GESARED. In clescan land as long as eachexperiences at mostone engine pair ( ) addition,other entry environments suchas Marsor Titan can be unavailability. (Oneengine fails, and the opposite engine is shut simulated. off.) Theproblem, at Ž rst,appears to bethat of classic k-out-of-n Thevalidation of GESARED wasperformed by comparingsim- (Ref. 3) reliabilitystructure where any (random) k of the n (k n) ( ) ulationresults with real  ightdata for the ARC andprecomputed enginesare sufŽ cient, but this is not thecase in thenew problem. · In ( ) reentrytrajectory for the LBRV .Thecomparison between sim- fact,this problem has never been discussed in the literaturebefore. ulationresults with the LBR Vmodelsin the loop and a reference Thesesystems were proposed 1;2 astheoreticalfuture powered ve- trajectoryallowed the validation of thethree-DOF translational mo- hicles,but they resemble Delta Clipper (or DC-X) vehicle.Visual tionmodel, the atmospheric model, the Earth model, and the LBR V inspection‡ showsfour symmetrical thrust sources or engines.It ap- models.Comparison with ARC ightdata allowed wider valida- pearsthat DC-X vehicleis relevant,and it shouldhelp to framethe tion,in the sense that it waspossible to comparethis data with the stateof theart. The purpose of thisNote is not to performreliability aerodynamicmodel of theARC, theenvironment models, and the analysisof DC-X oranothersimilar vehicle. The purpose is to pro- six-DOFmotion models. Thus, combining the three tests performed videreliability tools that can be usedin makingrisk or probability ( ) twowith the ARC modeland another with the LBR Vmodel it statementson the performance of these new reliability structures, waspossible to fully validate GESARED andperform the ARC whichcan be usedin determiningthe overall reliability of alarge postight analysis. From the second test with the ARC modelin descent (or ascent) vehicle. theloop, it can be concludedthat there is a neglectablemismatch betweenthe aerodynamicmodel and the real vehicle. The two ARC softwarevalidation tests gave insight on theimportance of errorac- ProblemDescription and Solution cumulationin a processas reentry, characterized by largeduration Discrete RandomV ariableApproach andsigniŽcant variation of both environment and  ightconditions. Eachengine’ s performanceis actually a randomvariable. A dis- Flightcontroller design and post ight analysis for two representa- creterandom variable is suitable if the consideration is whetheran tivereentry vehicles were successfully conducted using the simula- tiontool discussed in this paper. GESARED alsoshows potentials inGN&C designand analysis for different reentry vehicles and Received 4October2001; revision received 8April2002; accepted for missions. publication8 April2002. Copyright c 2002by theAmerican Instituteof ° Aeronauticsand Astronautics, Inc. All rights reserved. Copies of this paper References may bemade forpersonal or internaluse, on conditionthat the copier pay 1Costa,R. R.,Chu, Q. P.,andMulder, J. A.,“ReentryFlight Controller the$10.00 per-copy fee tothe Copyright Clearance Center,Inc., 222 Rose- DesignUsing Non-Linear Dynamic Inversion,” Proceedingsof the Guid- woodDrive, Danvers, MA 01923;include the code 0022-4650/ 02$10.00 in ance,Navigation and Control Conference [CD-ROM],AIAA, Reston,V A, correspondencewith the CCC. 2001. ¤ Professorof Engineering and Faculty Coordinator of Colorado Space 2Regan,F .J.,and Anandakrishnan, S. M., Dynamics ofAtmospheric Re- GrantConsortium at USC,2200 Bonforte Boulevard. Entry,1sted., AIAA EducationSeries, AIAA, WashingtonDC, 1993, pp. †Chairof Engineering Technology and Director ofColorado Space Grant 111–135. Consortiumat USC,2200 Bonforte Boulevard. 3Mooij, E., TheMotion of aVehicle ina PlanetaryAtmosphere , 1st ed., ‡Data availableonline at http://antwrp.gsfc.nasa.gov/apod/ap951028. Delft Univ.Press, Delft,The Netherlands, 1994, pp. 14 –43. html[cited 20 March2002].