AD-A259 761 * (2 WL-TR-92-3112 Workshop • n Trajectory Optimization Methods and Apphat ions Presentations from the 1992 AIAA Atmospheric Flight Mechanics Conference Compiled by: Harry A. Karasopoulos Flight Mechanics Research Section Kevin J. Langan Aerodynamics & Performance Section FINAL REPORT FOR 10 AUGUST 1992 November 1992 DTI v L __ S'FS 02 19193 Approved for Public Release; Distribution Unlimited. AEROMECHANICS DIVISION FLIGHT DYNAMICS DIRECTORATE WRIGHT LABORATORY AIR FORCE MATERIEL COMMAND WRIGHT-PATTERSON AIR FORCE BASE, OH 45433-6553 't 93-01785 I l ll iN\\i/
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AD-A259 761 * (2
WL-TR-92-3112
Workshop • n TrajectoryOptimization Methodsand Apphat ions
Presentations from the 1992 AIAAAtmospheric Flight Mechanics Conference
Compiled by:
Harry A. KarasopoulosFlight Mechanics Research Section
Kevin J. LanganAerodynamics & Performance Section
FINAL REPORT FOR 10 AUGUST 1992
November 1992 DTIv L __S'FS 02 19193
Approved for Public Release; Distribution Unlimited.
AEROMECHANICS DIVISIONFLIGHT DYNAMICS DIRECTORATEWRIGHT LABORATORYAIR FORCE MATERIEL COMMANDWRIGHT-PATTERSON AIR FORCE BASE, OH 45433-6553
't 93-01785
I l ll iN\\i/
NOTICE
WHEN GOVERNMENT DRAWINGS, SPECIFICATIONS OR OTHER DATA ARE USED FORANY PURPOSE OTHER THAN IN CONNECTION WITH A DEFINITELY GOVERNMENT-RELATED PROCUREMENT, THE UNITED STATES GOVERNMENT INCURS NO RESPON-SIBILITY OR ANY OBLIGATION WHATSOEVER. THE FACT THAT THE GOVERNMENTMAY HAVE FORMULATED OR IN ANY WAY SUPPLIED THE SAID DRAWINGS, SPECI-FICATIONS, OR OTHER DATA, IS NOT TO BE REGARDED BY IMPLICATION, OR OTII-ERWISE IN ANY MANNER CONSTRUED, AS LICENSING THE HOLDER, OR ANY OTHERPERSON OR CORP(5RATION; OR AS CONVEYING ANY RIGHTS OR PERMISSION TOMANUFACTURE, USE, OR SELL ANY PATENTED INVENTION THAT MAY IN ANY WAYBE RELATED THERETO.
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ii
Report Documentation Page
Pablic roportiag bard*& for this coiiectios of isfornotion io estimated to average I beat per reap...., Including she time for rovi-iag instruction..soarchiag *aiotiag data sources.gatoodaig and masfataimaeg th, data seeded. &8d Completing &ad revienirag the collectlon of iawormatioa. 5eed contineasregording this bard#. estimate* or any other eapegt of this collection of isformatiom, isladudig suggestion$ for redectug this bard.&. to W~ehabgtoa,He~adquarters Service.. Directorate for Iaformatitos Operations and Reports, 1218 Jeffersoa Davis Highway, Suets 1204. Astlaglom. VA 231102-4302, god t.the O1ffice of Management and Budget, Paperwork Rteduction Project (0704.0144), Washington, DC 20502.
1. Agency Use Only (leave blank) 2. Report Date 13. Report Type and Dates Covered
SN ovember 1992 IFinal Report for 10 August 19924. Title and Subtitle 5. Flunding Numbers
Workshop on Trajectory Optimization Methods andApplications, Presentations from the 1992 AIAA Atmospheric FlightMechanics ConferenceWU20 79
o 6. Author(s)
Harry A. Karasopoulos, Kevin J. Langan
7. Performing Organization Name(s) and Address(es) 8. Perforating OrganizationReport No.
WL/FIMWright- Patterson Air Force Base, OH 45433-6553(513) 255-6578
1 2a. Distribution / Availability Statement 12b. Distribution Code
Approved for Public Release; Distribution Unlimited
13. Abstract (mnamximu 200 words)
This report is a compilation of presentations from the Workshop on Trajectory Optimization Methods and Appli-cations, at the AIAA Atmospheric Flight Mechanics Conference, Hilton Head, South Carolina, on 10 Aug 1992.
14. Subject Terms 15. Number of PagesTrajectory Optimization, Optimization Methods and 107Applications, Trajectory Simulation. 1.PieCd
17. Security Classification 18. Security Classification 19. Security Classification 20. Limitation of Abstractof Report of page of AbstractI
Unclassified Unclassified Unclassified I UnlimitedNSN 7540-01-280-5500 Standard Formu 298 (Rev~. 2-59)
Prescribed by ANSI Std. Z39-l8"ll6.102
Preface
This report is a compilation of presentations given at the "Workshop on Trajec-
tory Optimization Methods and Applications", held at the 1992 AIAA Atmospheric
Flight Mechanics Conference in Hilton Head, South Carolina. This workshop was
co-chaired by Harry Karasopoulos and Kevin J. Langan, both of the former Flight
Performance Group of the High Speed Aero Performance Branch in Wright Itbora-
tory.
It is hoped that this document will help the attendees retain some of the ideas
presented in the workshop, in addition to providing useful information to those who
were unable to attend. Appreciation is expressed to the presenters and attendecs.
For the third year in a row, this workshop has proved to be a successful folrm
for highlighting current work in trajectory optimization. Thanks are also due to
the American Institute of Aeronautics and Astronautics fir making this workshop
possible.
The following was the workshop schedule:
SCHEDULE
e 1:00 - Introduction - Harry Karasopoulos, Wright Laboratory.
e 1:00 - OMAT: An Autonomous Optimal Solution to Rendezvous Problemswith Operational Constraints - Don Jezewski, McDonnell Douglas Space Systems
Company, Houston, TX.
* 1:15 - MULIMP: Multi-Impulse Trajectory and Mass Optimization Pro-gram - Darla German, Science Applications International Corporation, Sclia,,imirg,IL.
@ 1:30 - Phillips Laboratory Applications of POST - Jim Eckmann, SPARTAInc., Edwards AFB, CA.
* 1:45 - OTIS Advances at the Boeing Company - Steve Paris, The BoeingCompany, Seattle, WA.
* 2:00 - OTIS Activities at McDonnell Douglas Space Systems Company -
Rocky Nelson, McDonnell Douglas Space Systems Company, Huntington Beach, CA.
nli
* 2:15 - Advances in Trajectory Optimization Using Collocation and Non-linear Programming - Dr. Bruce A. Conway, University of Illinois, Urbana, IL.
* 2:45 - BREAK
* 3:00 - Flight Path Optimization of Aerospace Vehicles Using OTIS - Dr.Rajiv S. Chowdhry, Lockheed Engineering and Sciences Company, NASA LaRC,Hampton, VA.
* 3:15 - Trajectory Optimization of Launch Vehicles at LeRC: Present andFuture - Dr. Koorosh Mirfakhraie, ANALEX Corp., NASA Lewis Research Center,Cleveland, OH.
e 3:30 - Collocation Methods in Regular Perturbation Analysis of OptimalControl Problems - Dr. Anthony J. Calise, Georgia Institute of Technology,Atlanta, GA.
e 3:45 - Automatic Solutions for Take-Off from Aircraft Carriers - Lloyd 1[.Johnson, AIR-53012D, Naval Air Systems Command, Washington, DC.
e 4:00 - Current Pratt-Whitney OTIS Applications - Russ Joyner 1 , UnitedTechnologies, Pratt-Whitney, West Palm Beach, FL.
* 4:15 - Airbreathing Booster Performance Optimization Using Microcom-puters - Ron Oglevie, Irvine Aerospace Systems Co., Fullerton, CA.
e 4:30 - Scheduled Session End
Unscheduled Speakers
* Dr. Klaus Well, University of Stuttgart. 2
* Dr. Mark L. Psiaki, Cornell University.
'Cancelled"2Presentation copy not available at the time of printing
iv
U2 m, A&I
OTIC TABUlxennounced 0JuSttficationi
By__ _ _018tr~bution.1
Contents AvQziabilityodes
Vall AW%1
DTIC QUAL r Nr I CTKD 3
Preface
1 OMAT: An Autonomous Optimal Solution to Rendezvous Problemswith Operational Constraints - D. J. Jezewski, J.P. Brazzel, B.R. Ilautlcr,and E.E. Prust
2 MULIMP: Multi-Impulse Trajectory and Mass Optimization Pro-gram - Darla German 13
3 Phillips Laboratory Applications of POST - James B. Eckmann 19
4 OTIS Advances at the Boeing Company - Steve Paris 27
5 OTIS Activities at McDonnell Douglas Space Systems Company -R.L. Nelson 33
6 Advances in Trajectory Optimization Using Collocation and Non-linear Programming - Dr. Bruce A. Conway 43
7 Flight Path Optimization of Aerospace Vehicles using OTIS - Dr.Rajiv S. Chowdhry 51
8 Trajectory Optimization of Launch Vehicles at LeRC: Present andFuture - Dr. Koorosh Mirfakhraie 59
V
9 Collocation Methods in Regular Perturbation Analysis of OptimalControl Problems - Dr. Anthony J. Calise and Martin S.K. Leung 65
10 Automatic Solutions for Take-Off from Aircraft Carriers - Lloyd H.Johnson 77
11 Airbreathing Booster Performance Optimization Using Microcom-puters - Ron Oglevie 91
12 An Algorithm for Trajectory Optimization on a Distributed-Memory P~krallel Processor - Dr. Mark L. Psiaki and Kihong Park 97
vi
OMATAn Autonomous Optimal Solution to
Rendezvous Problems withOperational Constraints
by
D. J. J.iwsk4 J. P. fraw,
A. R. Hhafle, ad E. E. Pftus
McDonnell Douglas Space Systems CompanyHouston Division
16055 Space Center Blvd.Houston, Texas 77062-6208
AAA Atmospheric Right Mehlacis ConfemaeFIGton lied South Carolia
August 10, 1992
McDonnell Douglas Space Systems Company - Houston Divtison Pge 1 0122
Sketch of Rendezvous Problem
z
((t), (S ) ST - (RWT)
C - Chaser VehicleX T - Target Vehicle
McDonnell Douglae Space Systems Company - Houston Divhlon Pge 2 of 22
General Approach for Solving Optimal Rendezvous Problems
Constraints Constraints
Paamte ParameterVector > Vector
Unperturbed PerturbedProblem Problem
Analylica Numerical
"McOonn Dow" * spae semW company, Hfmw Don Pe M, SoM2
2
Definition of an Optimal Rendezvous Problem
"o Given:"* Chaser and Target States: (Sc,tc), (STtT)"* Attracting Body"* Objective Function, i.e., Delta-V, Fuel, Time, etc.
"O Subject To:"* Force Field"a Perturbations"* Terminal & Inflight Constraints and Limits
U Define:* Optimal Sequence of Maneuvers (Impulses, Finite Bums)
- Number, Location, Magnitude or Duration, and Direction
U Such That:* Chaser & Target Vehicles Achieve a Relative Configuration at Some
Time
McDonnell Douglas Space Systems Company - Houston Division Page 4 of 22
"( Three Types of Constraints"* Parameter Constraints
. a<X<b"* Linear Constraints (Constant Jacobian Matrix)
- L(S,t) 0"* Non-Linear Constraints
* NL(St) > 0"c Bounds
* Constraints are Bounded by Upper and Lower Bounds (BLBu)* Equality Defined by BU - BL* Unbounded Defined by BL- - -, Bu= + cc
Mconnll Douglas Spame Syatems Company - Houston Divekn Pk. 6 of 22
Constraints are Defined by Five Integers
"0 I - Constraint Number
"0 J - What the the Constraint is Referenced to:"* An Impulse"* Another Constraint
"0 K - Reference Impulse or Constraint Number
" L - Condition that Triggers or Initiates ConstraintT lime from GMT or Reference Event (Impulse or Constraint)
* Phase Angle* Lighting Condition between Chaser and Target Vehicles* Delta-Angular Measurement in Chaser Orbit* Number of Revolutions* Chaser Vehicle's nth Periapsis, Apoapsis Crossing
Wevnn"l Douglas 4"SYa. 8heam COmpany - Houston Divsio PV* 7.of22
4
Constraints are Defined by Five Integers (concl'dd
0 N - Type of Constraint- Periapsis, Apoapsis@ Differential Height between Chaser Vehicle & Target Orbit
a Phase Angle- Sleep Cycle or Quiet Timea Chaser Orbit Coelliptic with Target Orbita Chaser Position Vector Relative to Target LVLH Frame- Chaser Velocity Vector Relative to Target LVLH Framew Bounded Delta-V in Chaser LVLH Framea Inertial Line-of-Sight Angular Ratem Wedge Angle
a_
McDonnell Douglas Space Systems Company -Houston Divon Pae 8 of 22
Demonstration 2:Typical Shuttle Rendezvous
"o Shuttle (Chaser)"* 110 circular altitude"* Inclination = 28.5""* Longitude of ascending node = 101"* Argument of perigee = 0""* True anomaly = 180" c
"o SSF (Target) t"* 190 nmi circular altitude"* Inclination = 28.5""* Longitude of ascending node = 100""* Argument of perigee = 0""* True anomaly = 0" (Not to Scab)
" Limited to 2 maneuvers
McDonneIl Douglas Space Systems Company - Houston Division Pg 0 9f122
5
Demonstration 2 (Concl'd)
Demonstration 2: Shuttle Rendezvous with SSF OJnperturbed)OMAT Unconstrained and Constrained So&jtion
U Standard Approach"• V, Obtained from Classical Lambert Problem"* BVI Obtained from Solution to Variational Equations, i.e.,
- 00(t".t') = 1 1 121* ' 12 [021022j
* Difficulties with this Approach* 8RF Must be mSmall" (Unear Approximation)* 012 Must be Well Conditioned* J2 Frequently Must be Reduced (Sub-Problem)* Excessive Number of Iterations and Integrations
McDonnell Douglas Space Systems Company - Houston Divhwon Pae 16 of 22
Perturbed Lambert Problem (concl'd)
Improved Approach• V, Obtained from First-Order Correction to the Inverse-Square
Problem Resulting from the J 2 Perturbation- Analytic Solution- Expressed in "ideal Reference Frame"- Regular, No Singularities- Solution Expressed in Terms of Elements & their
Variations8 6V, Obtained from Solution to Variational Equations, i.e.,6v,. = 06R,
• SRF 0 O(10-3) Smaller than Classical Approach* No Requirement for J 2 Reduction (Sub-Problems)• Solution to TPBVP Requires Only a Few Iterations and
Integrations
McAonnoll Douglas Spam Systems Company - Houston Dlvision Phg 17 Otf22
9
ComDarison of Final Position Vector Errors for the ClassicalLambert and Predictor/Corrector Solutions
"O Earth Centered
"O Transfer Angle = 170 degrees
S
r• • J2-Lwvbon
1,- 4-CR0
3-
o0 1000 20;00 30000 40000 50000
Transfer Time, sec.
McDonnell Douglas Space Systems Company - Houston Division Page 18 of 22
Outline of Optimal Solution Approach
Q Unperturbed Problem"* Force Field (Inverse-Square), i.e.,
V =- (R/r3)
"* State and Variation in State Obtained from Solution of- Kepler's Problem (Goodyear, Analytic)
"* Boundary Value Problem Satisfied by Solution of* Lambert's Problem (Gooding, Analytic)
"* Constraints and Variation of Constraints Evaluated Using• Keplerian Elements
"* Non-Linear Programming Algorithm, NZSOL, Solves ConstrainedOptimal Problem
"* Solution Obtained in seconds on Sun Sparcstation 2
AacaomwU Douglas Spam Systems Company- Hston Dlsion Pag 20 of 22
What Have We Done. Where Are We Going?
"1 Present Status of Development:"* Proof of Concept, Using Unperturbed Solution to Solve Perturbed
Problem"* Verify Solution Approach for Handling Perturbations and Constraints
and their Conditions"* Developed Solution Approach for Perturbed Lambert Problem"* Illustrate Initial Capability of the Algorithm, .OMAT), to Efficiently
Solve Optimal Rendezvous Problems with Operational Constraints"o Planned Future Development:
a Expand Perturbation and Constraint Modelsv Develop Multi-Rev. Capabilitya Develop Finite-Thrust Model0 Libration Point Rendezvousa Solve Advanced Problems
AMOonneI Dgous Sae Syatems Company. Houton Divson Pep 2 of 22
12
MULIMP
Multi-Impulse Trajectory and Mass Optimization Program
Darla German
Science Applications International Corporation
13
MULIMP GENERAL DESCRIPTION
"° DESIGNED TO COMPUTE A MULTI-TARGETED TRAJECTORY ASA SEQUENCE OF "TWO-BODYw SUBARCS IN A CENTRALGRAVITATIONAL FIELD USING KEPLER AND LAMBERTANALYTICAL SOLUTION ALGORITHMS
"* BODIES MAY BE PLANETS (ORBITAL ELEMENTS STOREDINTERNALLY), ASTEROIDS OR COMETS (ORBITAL ELEMENTSCONTAINED IN ASTCOM.ELM FILE), OR FICTITIOUS (ELEMENTSINPUT BY USER)
"* CENTRAL BODY MAY BE THE SUN, ANY OF THE 9 PLANETS ORAN ARBITRARY BODY (GRAVITATIONAL CONSTANT INPUT BYUSER)
* UP TO 19 SUBARCS MAY BE SPECIFIED
A -VARIABLES IN OPTIMIZATION SEARCH
"* TIMES (DATES) OF THE NODAL POINTSCONNECTING TRAJECTORY SUBARCS
"* POSITION COORDINATES OF MIDCOURSE AVPOINTS NOT MADE AT AN EPHEMERIS BODY
14
DEPARTURE CONDITONS
• RENDEZVOUS DEPARTURE IN WHICH CASE THE FIRSTIMPULSE AVi, IS EQUAL TO THE HYPERBOLIC EXCESSSPEED V.t
* PARKING ORBIT DEPARTURE IN WHICH CASE THE FIRSTIMPULSE IS THAT NECESSARY TO ATTAIN THEHYPERBOLIC EXCESS SPEED FROM THE PARKING ORBIT(rp, e) WITH THE MANEUVER ASSUMED TO BE COPLANAR
"* A 'FREE" DEPARTURE IN WHICH CASE THE FIRST AVIMPULSE IS EXCLUDED FROM THE PERFORMANCE INDEX
"* A GRAVITY-ASSIST DEPARTURE IN WHICH CASE THEAPPROACH HYPERBOLIC VELOCITY VECTOR MUST BESPECIFIED BY INPUT
w
INTERMEDIATE TARGET CONDITIONS
ARRIVAL• RENDEZVOUS
• ORBIT CAPTURE (ORBIT IS USER DEFINED)* UNCONSTRAINED FLYBY SPEED
• CONSTRAINED FLYBY SPEED (HYPERBOLIC FLYBY)
SPEED IS USER INPUT
"* RENDEZVOUS DEPARTURE"* ORBIT DEPARTURE (ORBIT IS USER-DEFINED)
OTHER
• GRAVITY-ASSISTED SWINGBY
15 0
GRAVITY-ASSISTED SWINGBY
"* MODEL IS FORMULATED WITH POWERED MANEUVER ASTHE GENERAL CASE
"* AV WILL OFTEN ITERATE TO ZERO VALUE IF THE PROBLEMIS NOT OVERLY CONSTRAINED BY SWINGBY DATE ANDDISTANCE
"* USER OPTION TO SPECIFY POWERED MANEUVER LOCATION
- INBOUND ASYMPTOTE
• PERIAPSIS
* OUTBOUND ASYMPTOTE
- BEST CHOICE
TERMINAL TARGET CONDITIONS
"* RENDEZVOUS
"* TARGET-BODY ORBIT CAPTURE
"• SATELLITE ORBIT CAPTURE
"* UNCONSTRAINED FLYBY
"* CONSTRAINED FLYBY
- SPECIFIED ORBIT ELEMENTS (a~e,I) RELATIVE TO CENTRALBODY; FINAL TARGET MUST PROVIDE GRAVITY-ASSIST
16
"FREE' MIDCOURSE AV POINTS
MIDCOURSE VELOCITY CHANGES MAY BE MADE ATINTERIOR IMPULSE POINTS NOT OCCURRING AT ANEPHEMERIS BODY. THESE MIDCOURSE AV POINTS MAYBE INCLUDED IN TWO WAYS:
* TIME AND POSITION COORDINATES MAY BEESTIMATED AND INPUT; ON USER OPTION,THE TIME ANDIOR POSITION COORDINATESWILL BE OPTIMIZED
• AUTOMATIC IMPULSE ADDITION MAY BEREQUESTED
MULTIPLE REVOLUTIONSAND
RETROGRADE MOT1ON
* MULTIPLE REVOLUTIONS AND/OR RETROGRADE MOTION ARESPECIFIED BY INPUT
* TWO OPTIONS ARE PROVIDED FOR HANDLING MULTIPLEREVOLUTION SOLUTIONS:
"* THE NUMBER OF COMPLETE REVOLUTIONS ANDENERGY CLASS MAY BE SPECIFIED
"* THE MAXIMUM NUMBER OF COMPLETEREVOLUTIONS TO CONSIDER MAY BE ENTERED INWHICH CASE ALL INCLUSIVE SOLUTIONS WILL BEEXAMINED AND THE "BEST ONE SELECTED ONTHE BASIS OF A VELOCITY CHANGE CRITERIA
17 6
IR&D ENHANCEMENTS
* ADDITION OF THE UNPOWERED SPECIFICATION FOR PLANETARY SWINGBYS
* A NEW DEPARTURE OPTION OF SPACE STATION LAUNCHES
* A NEW TERMINAL ARRIVAL OPTION OF 3-IMPULSE PLANET ORBIT CAPTURE TO AFINAL ORBIT DETERMINED BY USER INPUT rp. r., AND INCLINATION
* INCLUSION OF JPL SATELLITE EPHEMERIDES ROUTINES FOR MOST NATURALSATELLITES
* CONVERSION OF THE WORKING COORDINATE SYSTEM FROM EMOSO TO J2OO0
* ABILITY TO CONSTRAIN TOTAL TRANSIT TIME
18
U* RKSA - Applications Branch
Phillips LaboratoryApplications of POST
James B. EckmannSPARTA, Inc.
Phillips Laboratory SETAEdwards AFB, CA
10 Aug 92
19
RKSA - Applications Branch
Presentation Overview
"• Organization and Mission
"* Simulation Work Environment
"* Summary of POST Models
• Applications and Some Results
"• Future Plans
RKSA - Applications Branch
Organizational Hierarchy
INC IIII I- Ij I M
I I I I w II
I" PAA•a. & M I MOW_.__._m•UU~ 9. Lnm ADV ~AFO , W m.A HAMS~ &,.'• e IOU• ,n I
.N3U M SPN PLAN•&W I rNr1l1,vW.MWu•'.•,,., lAnAW•GN"a "'•"'l -- S I°' • l'C'e I I Su•'•'1
2OUINS)O
20
RKSA - Applications Branch
Propulsion Directorate (RK) Mission
"* Provide propulsion technology and expertise for U.S. space and missile systems.
"* Be a center of excellence in propulsion research and development.
"* Develop a broad, advanced technology base for future propulsion system designers.
"* Demonstrate propulsion concepts for current systems designers.
"• Assist in solving operational problems.
40 RKSA - Applications Branch
System Support Division,Applications Branch (RKSA)
Single-Stage-To-Orbit (SSTO) ModelsMcDonnell Douglas Vertical Rockwell International Vertical Boeing Horizontal TakeoffTakeoff Vertical Landing Takeoff Horizontal Landing Horizontal Landing (HTHL)(VTVL) Delta Clipper model (VTHL) Reusable Space Transport Reusable Aerodynamic Spacedeveloped and provided by (RST) model developed by Vehicle (RASV) modelNASA/Langley Rockwell and provided by developed in-house
NASA/Langley
25
W RKSA - Applications Branch
Future Plans
DEVELOPE A COMPLETE VEHICLE SIMULATION CAPABILITY
"* Apply SMART and CONSIZE to current analysis tasks
"• Complete integration of Silicon Graphics machines
"* Develop a cost analysis capability
"* Continually evaluate new analysis tools
26
27
Boin Optimal Trajectories by Implicit Simulation_V"c Group (OTIS) Development
Collocation based Ontimal Control Methods
Chebytop CTOP mmo
Indirect TrlmactCoytMethonsSimltson ofth
TOP As Vehicles
TOP53 NTO I SPOT
POST MM
BoeingDelo JP OTIS Modes
Space Grou
BoeingDefense &ou Next Wave of OTIS Advances
Problem Data Prram ResultsS.e~t.U Conditioning E xeution Interpreta.inl
Current Resources
Goal 4 fold overall reductionOTIS runtimes reduced by 10 fold
BoeingDeloGr A OTIS 3.0 Lunar Test CaseSpace Group
Explicit Trajectory Generation Optimize"* Launch Date"* A parking orbit"• tO (burn1)-A•V1"• tO (bumn2)
burn 2
lower limit
24 hour Orbit
Low Earth orbit to high polar Earth Orbit (24hr).
B9
BoeingDehens &Spac Grwp OTIS Elements
Tabular listingand printer plots
Namnefistfl Restart file...
77S plot fie
•' Trajectory
plots
fileTabular data plots
BoeingDeor,, • OTIS 3.0 Provides Extreme FlexibilitySp"c Group
*Global ConstraintsAnalYtical ArcsPhase Dependent
- Equations of Motion (EQM)- Control Variables- Quadrature Variables
COAS
BOOST RE-ENTRYF% Pat EON RMg Pam EGMPit1ch td Yaw conrol Anges of AlbIM& &an controls1
AdM "MS Load to Stga Vacts
U~rOwF\%
Age. of AMbIN* Conlrol
30
BoeingDefense & Future TrendsSpace Group
(X- Window Interface)
rOff-the-Shelf Software-"Problem Set-Up ITransform by Spyglass
McDonnell Douglas Space Systems CompanyinFL NeliV2
41
Advances in Trajectory
Optimization Using Collocationand Nonlinear Programming
Bruce A. ConwayDept. of Aeronautical & AstronauticalEngineeringUniversity of Illinois
Urbna,r ,L
August 1992
43
Outline
Introduction
Progress to Date - Theory
Progress to Date - Solved Problems
Continuing and Proposed Research - Problems
Progress - Theory
1. Use of costates to improve an optimal trajectory.
Lagrange multipliers for the discrete (NPSOL) solution are arepresentation of the Lagrange multipliers of the continuous case.(Enright & Conway, JGC&D 15, No. 4, 1992)
Knowledge of the Lagrange multipliers allows a posteriori determinationof the optimality of the solution, e.g., can examine the switching function.
2. Generalized defects
Can be used when the differential equation for a state variable isintegrable, e.g., on a coast arc.
May significantly reduce the number of NLP parameters and henceexecution time.
3. Coordinate transformation within the H-P structure
Necessary for orbit transfer when changing sphere of influence
Keeps state variables near one order of magnitude, as NPSOL prefers
44
4. Method of parallel shooting
Replaces single Hermite-Simpson "integration step" with multipleRunge-Kutta steps allowing use of larger intervals.
Results in smaller NLP problems for a given accuracy.
5. Automatic node placement
Computer solves a succession of NLP problems in which additionalnodes are inserted as needed to acheive a given accuracy.
More efficient than using a uniform distribution of nodes
6. Neighboring optimal feedback control
Determines gains for linear feedback controller to yieldneighboring optimal controller
Unnecessary to solve NLP problem for small change in initialor terminal conditions
Feedback gain history easily loaded into small memory
Illustration of Generalized Defects
TC
010 ý integrals Qj~%iterl Q *
If€rthrust thrustarc arc
segment segment
coast arc
Ql- Q t(z n) - Q l(a
no~4
noe node
45
Illustration of Parallel Shooting
R9 Step IW"
VA-I R,,stop K S ,
!I I
I vIz I
I li I-
lt ti-+ 13 ti-I 2h/3 U
Progress - Solved Problems
1. Optimal low-thrust escape trajectory (Enright Ph. D. thesis)
2. Optimal 2 and 3 burn circle-circle low-thrust rendezvous(Enright Ph. D. thesis)
3. Optimal low-thrust Earth-Moon transfer (Enright Ph. D. thesis)
4. Optimal spacecraft detumbling (A. Herman M.S. thesis)
5. Optimal low-thrust insertion- Mars Observer (Enright Ph. D. thesis)
6. Optimal 2D and 3D direct ascent time-bounded interception(J. Downey Ph. D. thesis)
7. Neighboring optimal feedback control for continuous-thrust ascentmaximizing horizontal velocity (F. Chen Ph. D. research)
46
pI I I I I III I I I + , , ,
Low-) hrust Minimum Fuel Escape
=-1
a, 0.0125tw= 16n
All in canonical units
Method VaybIAd CPUHermite/Sinpeon (60) 427 190aw
Parail shdofn (34 x 3) 365 95 an
Pamlael shoofi (5 x 20) 270 72 9W
Optimal 2 and 3 Burn Circle-Circle Rendezvous
.. ..... .... .-..
. . . . .. .....
P& 06w f, swm W6 6. MOM6.80
47
COPY AVAILABLE TO DTIC DOES NOT PERbIT FULLY LEGIBLE ,EPRO.,J.'WZOS
Optimal Low-Thrust Earth-Moon Transfer
Optimal Low-Thrust Earth-Moon Transfer, cont'd
25 . . .
20
~15
-0 :
-54 a a1
time (days)
48
Optimal Spacecraft Passivation (Detumbling)
0 ,II
View of OMV / Disabled Satellite System
Spacecraft Passivation, cont'd.
* . .. - - . -.. . . .
Results from the
TPBVP solver
.. .. .... . .. . .. .
*, 4.,
Externala Toq e Historie
I , U *W - i
NIP method
-sL ,. .|'
-3 * *8 a , _ .. _ ,. -
External Torque Histories
490CY, AVAILALE TO DTIC DOE8 NOT PERMIT FULLY LEGIBLE AEPRODbCTION
Optimal 2D & 3D Direct -Ascent Interception
* Tar= tis assumed to be in a general Keplerian orbit withorbital elements
3. Optimal Earth-Mars low-thrust transfer including escape andarrival spirals and coordinate transformations at sphere ofinfluence of each planet. (S. Tang)
4. Neighboring optimal feedback control for complex problems
Automation of NOFC using symbolic programming (F. Chen)
5. Optimal trajectories for interception of Earth-crossing asteroids(B. Conway)
50
FLIGHT PATH OPTIMIZATION OF
AEROSPACE VEHICLES USING
OTIS
Rajiv S. Chowdhry
Lockheed Engineering & Sciences CompanyMS 489
Aircraft Guidance & Control BranchNASA, LaRC, Hampton VA.
51
Outline
"* Accuracy of OTIS solutions.
"* Overview of OTIS applications at AGCB
Accuracy of OTIS Solutions
OTIS : Optimal control solutions via direct transcription
Combination of collocation and nonlinear programming
Question:
How do OTIS solutions compare to the "exact" or TPBVP
s o -.. . ... ... ....... ...... ............... .... !. ......... i ............ .......
0 100 200 300 400 500TkiM (s89)
Figure [4]. Comparison of optimal ALS ascent with OTISsolution, local horizontal flight path angle (deg) vs. time.
55
20-
Time (Sed)
Figure [5]. Comparison of optimal ALS ascent with OTISsolution, thrust vector angle (deg) vs. time.
OTIS Applications at AGCB
* Fuel efficient ascent for SSTO airbreathing hypersonic vehicle.
* fuel optimal path definition for G&C studies
HL-20 abort maneuvers : ELSA ( Efficient Launch Site Abort)
"* Parameter sensitivity studies to support design activities.
"* Guidance algorithm development & real time validation.
* ALS ascent for OTIS calibration.
* Optimal maneuvers for a high performance fighter aircraft (HARV)
in air combat situation.
56
Conclusions
For the ALS Ascent Problem :
"• Excellent match of the collocation solution to the TPBVPsolution
"* Relatively quick turnaround time for OTIS solutions
"* Very robust to Initial guesses
57
Trajectory Optimization of Launch Vehiclesat LeRC: Present and Future
Presented byKoorosh Mirfakhrale
atWorkshop on Trajectory Optimization Methods and Applications
Hilton Head, SCAugust 10, 1992
ANALEX NASA Lewis Research CenterCORPORATION Advanced Space Analysis Office
KM W1/92
59
Outline
Introduction
Present method of solution and code
Capabilities of the present code
Motivation for replacing the code (and method)
Examination of methods using collocation
Introduction
Trajectory optimization* of ELV's at the Advanced Space
Analysis Office at LeRC is performed for:
Mission design for approved programs
Feasibility and planning studies
Corroboration of contractors' data for NASA missionsflown on Atlas and Titan
Trajectory optimization: Maximizing the final payloadsubject to a set of intermediate and final constraints.
ANALEX NASA Lewis Research CenterCOMTMI Advanced Space Analysis Office
60
Introduction (Cont'd)
Mission profiles for launch vehicle systems with booster andupper stages include:
* Launches from ER and WR
* LEO, GTO, and GSO insertion
0 Interplanetary escape trajectories
0 Orbit transfers
Present Method of Solution and Code
Calculus of Variations is used to formulate the problem.The resulting two point boundary value problem is solvedusing a Newton-Raphson algorithm.
The computer program (DUKSUP) was written entirely atLeRC during 1960's and early 70's.
DUKSUP is a 3-D.O.F. code written for performanceanalysis of multi-stage high-thrust launch vehicles.
ANALEX NASIA Lewis Research CenterCORPORAnN Advanced Space Analysis Office
KM l I0/96
61
DUKSUP Features
Detailed modeling (e.g., propulsion and aerodynamic)of a launch vehicle is possible.
A variety of constraints can be imposed on the model.
They include:
- Instantaneous and total aerodynamic heating
- Maximum dynamic pressure
- Parking orbit parameters (e.g., radius of perigee,energy, velocity, etc.)
- G-limit staging
DUKSUP Features (Cont'd)
Several in-plane and out-of-plane final target conditionscan be specified (e.g., energy, radius, true anomaly,inclination, declination of outgoing asymptote, etc.).
Variables free for optimization include:
- Upper stage burn and coast times
- 'Kick angle'
- Payload fairing jettison time
- Thrust angle in the non-atmospheric flight
ANALEX NASA Lewis Research CenterCOOM Tm Advanced Space Analysis Office
62
Motivation for Replacing DUKSUP
Sensitivity to initial guesses
Difficulty in reformulating the C.O.\ problem whenadding new features and constraints to the code
Difficulty In modifying and expanding the code due tolack of documentaion and outdated programmingpractices
Examination of Methods Using Collocation
Two main features of collocation making it attractive are
- Lack of sensitivity to initial guesses
- Relative ease of formulation
Concerns about using collocation for ELV optimizatioin are
- Ability to handle complex modeling requirements andconstraints typical of ELV flight
- Computer run time
- Fidelity of the solution vis a vis C.O.V.
Evaluation of collocation uses DUKSUP as the benchmarkfor comparison.
ANALEX NASA Lewis Research CenterCAePRATIcON
KMAdvanced Spce Analysis Office
63
Using Collocation (Cont'd)
Available collocation codes are used as testbeds withnecessary modifications.
A simple LV model Is used first and moved progressivelyto a full DUKSUP model.
Enright's orbit transfer program was used for the firstsimple model comparison. Results matched those ofDUKSUP.
OTIS is used for the more sophisticatedcomparisons.
OTIS is currently used to model an Atlas II/Centaur toLEO.
ANALEX NASA Lewis Research CenterC mORATM Advanced Space Analysis OfficeKM Wilm6
64
Collocation Methods in Regular Perturbation Analysisof Optimal Control Problems*
August 10, 1992
Prepared for
Workshop on Trajectory Optimization Methods and ApplicationsAIAA Guidance, Navigation, and Control Conference
Hilton Head, SC
Anthony J. Calise** & Martin S.K. LeungGeorgia Institute of Technology
School of Aerospace EngineeringAtlanta, GA 30332
*See conference paper no. 92-4304. Research supported by NASA LAngley under grant No. NAG-I-939**Pbome: (404) 894-7145, Fax: (404) 894-2760, E-Mail: AE231TCGrIrVM1.GATECI.EDU
Launch Vehicle Guidance Application (presented at 1.GNC-1)
Motivation
Analytical Methods Humarical methods
ReglarFoturatonsMutipe bo66n
Advantages / Disadvantages
Analyt Methods
"Approximates solution by expansion in an asymptotic series in a smallparameter
Zero order problem is simpler to solve =, Insight
Higher order problems are linear
Zero order problem must reasonably approximate the full order problem
For practical applications, zero order problem must be analyticallytractable or reducible to a simple algebraic problem
Significant amount of analysis Is required for each problem formulation ofInterest
Advantages / Disadvantages (continued)
Numerical (Collocation) Methods
Finite element method that enforces interpolatory constraints at specificpoints within each element
Simple to use for a wide variety of optimization problems
Large dimensional nonlinear programming problem
No general guarantee of convergence
Note." Advantages of analytical and numerical methods are in many respectscomplimentary In the sense that If the advantages can be combined insome way, then most of the important disadvantages for real-timeapplications can be removed.
67
Regular Perturbations in Optimal Control
Given:
dildt = f(xut) + e g(x,u,t); x(to) = xe
Find the control that minimizes J subject to the terminal time constraints:
I(x, t) 0 --
HIu O assumingaHu >O u=U(x,04,t)
where:
H = XT~r + e g) ; H(tt) =-(Pt I tf; 4 + VT•f
d)Jdt =-H; utr) = Itr
Regular Perturbation Analysis
Based on a simplified model (when £ is set to zero)
- Treat neglected dynamics as perturbation- Define a normalized independent variable, r = (t - t) / T
where T = t~r- to. Compute zero order solution
Consider an asymptotic series in x, A, and T
Evaluate high order corrections from sets of nonhomogeneous,time-varying linear 0. D. E's.d[xk1=[Al AnlXk]+ -.C1I[,C I A)~)I~ ]+ [Plk)
d L~kJ A2, An ~]+T'o'-C2J P2k]
enforcing all boundary conditions to lIth order
Compute feedback control at current time (to) using x(to) and kth orderapproximation for Uto)
68
Regular Perturbation Analysis (continued)
- A's and C's depend only on the zero order (k = 0) values.- C's are the explicit correction term for free final time, T.
- P's are the forcing functions involving lower order (k-1, .. , 1, 0) terms.
Solution:
r rXk(to)1+_ i-t[i[X(t)]+ t , Plk(h)
Xk 1!)] =00 t kA (tto +k o is ()+ t,10 9'9P21k i%)-1L k(t)J ,k toJ T*L, (t L oki)JJto
Higher order correction involves simple operations of quadrature and solutionof linear algebraic equations
Can be easily modified to account for discontinuous dynamics
Solution of Optimal Control Problems by Collocation
Methodology- a finite element approach. approximates the solution with interpolating functions
qj = ix -iis1 -" i=(ij+ij--t /2; x=(zj+uj-l)/2;..X; )=()-J+)-/2
- N is the number of elements, xj ind Xj are nodai values
- control assumed to be eliminatet! using optimality condition
69
Hybrid Collocation / Regular Perturbation
A Regular Perturbation Formulation
- rewrite the actual dynamics as
aH ap, + l-•pj) ; .•qj + rl.-LH--qj)
& 0
- perturbation terms are zero at mid point of each element.- for cases that control cannot be eliminated explicitly, use an
analytic portion rl(x, x, u)
aH0=fl+e(5--l)
Carry out a Regular Perturbation Analysis
- expand about the zero order solution (derived from collocation)- provides higher order corrections to collocation solution- further exploitation of the analytically tractable portion of the dynamics
will result in more intelligent interpolating functions (see simple example)
A Simple Example
Dumng's equation in first order form:
I=v ; x(O) = x0
S= -x - ax3 + u ; v(O) = vo
J = S1 x20t6) + Sv v2(to) + Jo6t(1 + u2/2) dt
- hardening effect Is given by the nonlinear term, ax3
- the optimal control problem is a fourth order example- will demonstrate different lev,.Is of intelligent Interpolating functions
that enhance the approximation with fewer number of elements
70
Simple Example (continued)
Level 0 Formulation:- degenerate case, uses only regular perturbation with a completely
analytic zero order solution- let e = a = 0.4, and treat the nonlinear terms as perturbations- SK=Sw= 100
1=v ; x(O) = x0*= -x+ u-E e3 ; v(O) = vo
X,s +- +e3kvx2 ; •X (tt) = 2SIX(tt)
X', AI•' ; Xv (if ) = 2Svv(tf)
HU =u+XV. =0 ; [H = XvV+Xv(-x+u-ex3)+1+u2/2}Itr =0
- zero order problem is linear ani1 time-invariant- compute up to second order corrected solutions (FIg's. 4.1 and 4.2)- series not convergent, most accurate approximation is first order- nonlinear term ax3 is too large to be neglected In the zero order problem
Oth -- ftA .~ a m 2 ad
0.S • " • %1 0,.% " •
% %
o. - . - ,a-•-
1 2 3 4 3
.rTime Tom*Figure 4.1. Level 0 Result In x. Fig=43. Led0 Rcstdt W.
3-
0 .OP &I £ 0
%%
6.0.d 1 2 *2*a
O 2 3 4 TimeTime fTim 4.4. Lew 0 R.an t X,
Figure 4.2. Level 0 Result In v. 71
. .
a0 4.
X,
0.0- 0
A OS, -2 -0 1 2 •1 4 0 1 a a
Time TimeFqum 4.5. L&ve I Zlu Order Resb i x fbr Ddfm N. Filme 4.7. Level I Zern Order Resuls in A, for Diffaenm N.
a 2-
I S-
211% *.PC',: SW0 J04
0 169#.* 1
Fge4.. LeelIF4.,t
Time, Time
lipr 4.6. LzvtJ I Zo Order Resulhsmv fo ms Pt Fqn 4.8. Lav] I Z OrerRnduinAhr Difff X
Simple Example (continued)
Level 1 Formulation:
i-us hybrid approach, approximate all state and costates as piecewiselinear functions
- number of unmowns is 4N + 5- lit and 2ad order corrections are computed for N = 3 (Fig's. 43.4.12)* discontinuity in slope is smoothed as order of correction Increases
c correction by regular perturbation analysis allows use of crude numberof element representation in tht, zero order collocation solution
72
Gt. ----- tam
- IN 19d& Optimal a*%
LO' ,%'S a
2 3 0Time TimeFigure 4.9. Level Higher Order Results Fi=4. LevlICodAw5duin , for N-3
in x for N =3.
0.0 -"o 0 S
o i2 r 3 0 1 a ,
•optimal
.2, m Time
0 I Tie 3 Aipa• 4.11. LeMe I 1Hibw Otda Res~ulin X, for N-3.
Figure 4.10. Level 1 Higher Order ResultsIn v for N z 3.
Simple Example (continued)
Level :2 Formulation:
- eqhunced level I formulation by interpolating only those variables that
have nonlinear coupling. decompose the dynamics as:
dx/dt = v
dv/dt = pvj + e{-x "-.v .ax3 -Pvj) IX = ,2 .N
d)L./dt = qxj + e{ XYv(l + 3ax2).- qxj)
dli/dt = --.x- number of unknowns is 2N + 5
-both zero and first order results for N&I are superior than theN = .3 results for the Level 1 formulation (Fig's. 4.13 - 4.16)
73
'tlo lot. "
& I." %.%1.0
0.
** 3
- / - O.
Tim* Time
Fqlp 4.15. Lew'd 2 Mowin OrderlReadshin for N=2. Slpre4.13. .,vel 2 Wole Order Res~u] in a forN=s2.
-2 ,, Tim0 1 2 3 iigm 4.20. ee 31Hjer G~d Resuhtsin).,fo N =I.Time
Figure 4.18. Level 3 Higher Order ResultsIn v for N = 1.
Alternative Implementations
Repeat zero order solution and perform quadratures at each control updateinterval
Or
Compute zero order solution and quadratures off line, and store for in-flightuse
rjt; to) - -, 0 .,
.0M
to to + A
Improves reliability and computational effciency with some loss in accuracy
75
Summary
Benefits of Hybrid Approach:
Silgpi•ficantly improves a collocation solution
First and higher order corrections are obtained by quadrature
Intelligent interpolation functions obtained by retaining as muchof the analytically tractable portion of the solution as possible
Possible to Implement the control solution so that the zero ordersolution and quadratures are performed once off-line and stored
Signiricandy improve a regular perturbation solution
Retain more of the nonlinearities in the zero order problem byusing finite elements and collocation to construct an improvedzero order solution
Important Implications in real-time gujidance applicutions
Computational efficiency and reliability
76
AUTOMATIC SOLUTIONS FOR TAKE-OFFFROM AIRCRAFT CARRIERS
Lloyd H JohnsonAIR-53012D
77
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