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.
TIllS TECIINICAL REPORT HIAS BEEN REVIEWED AND IS APPROVED FOR PUB-
LICATION.
Harry A. Karasopoulos Kevin J. Langan
Aerospace Engineer Aerospace EngineerFlight Mechanics Research Section Aerodynamics & Performance Section
Thomas R. Sieron David R. Selegan
Technical Manager, Chief,Flight Mechanics Research Section Aeromechanics Division
Aeromechanics Division Flight Dynamics Directorate
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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
WL-TR-92-31 129. Sponsoring / Monitoring Agency Name(s) and Address(es) 10. Sponsoring / Monitoring Agency
Report No.Same as Block 7.
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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
A General Optimization Approach
[ Initial |SParameteri-r Reference | . J , ...M Vector, X0 Spac Solution Cmn .oun Ovi Pagerna 23 M[I I•Solution?ye
Varilation ' NoCompute of So, uAn •.VJ.V. VCl
Iterate NZSOL, ..
Mc~onnell~~ -oga SpStSsenBCmanye Hoestor liln f2
Forms of Constraints
"( 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
220-
200 Maneuver 2
S160
~140 ''
120. __ __ _
120_- _ Mans Nver I
100]
Phase Angle (deg) •OMAT Unconstrained- - -OMAT Constrained
McDonnell Douglas Space Systems Company. Houston Division Page 10 of 22
Demonstration 2 (Cont'd)
0 Results"* OMAT handles perigee constraint for unperturbed orbits"* Optimum unconstrained trajectory required 461 ifs,
constrained trajectory required 603 ft/s
M•Donnel Douglas Spae Syem. Company. Houston Dfvin Page 11 of 22
6
Perturbations
"O Primer Vector Theory (Presently) Requires Perturbations to have Form- 9t(R,t)
"O Largest Geopotential Perturbation for Earth & Mars Is J2 , Has Form
9tJ2 jrR +JkK
• Where K is a Unit Vector Normal to the Equator and Jr and Jkare Functions of the Position Vector.
O Presently Incorporating NxM Geopotential Model
"O Need to Extend Theory to Functions 9t(R,Vt)
"o Need to Develop Theory for Third-Body Effects (Libration Point Rendezvous)
McDonnell Douglas Space Syatems Company. Houston Division Page 12 of 22
Demonstration 4:Mars MEV Post-Ascent Rendezvous with MTV
"o MEV (Chaser)"* 117 x 135 nmi altitude"* Inclination = 164.264662""* Longitude of ascending node = 194.39079""* Argument of perigee = 159.820571"* True anomaly = 0'
"o MTV (Target) Targt
a 135 x 18,294 nmi altitude Chasr
- Inclination = 164.2"* Longitude of ascending node = 195"a Argument of perigee = 16'a True anomaly =0
McDonnell Douglas Space Systems Company - Houston Divlon Pg 13 Of 22
7
Demonstration 4 (Concl'd)
Demonstration 4: Mars MEV Post-Ascent Rendezvous with MTVOMAT J2 Perturbed Soluiorn
P
SManeuve 2
M oeuver 1- - I� -neuver 3
-140 -20 -10 -80 -60 -40 -20 0 20Phase Angle (deg)
McDonnell Douglas Space Syatema Company - Houston DIvWion Page 14 of 22
Perturbed Lambert Problem
o A Definition of Lambert's Problem:* What is the Initial Velocity Vector,V1 , that Generates a Trajectory that
Passes between Two Radii Separated by a Given Angle in aSpecified Time Interval ?
V1
Mc~onnea Dovmia Spec SytMW COWpMY - Houston DMIWW Pag 15 of 22
8
Perturbed Lambert Problem (cont'd)
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
Mcoonmwi Cougiee _%"w Sysftms Conmay.- Housto Divisn Page i9 Of 22
10
Outline of Optimal Solution Approach (concl'd)
Q Perturbed Problem"* Force Field (Perturbed Inverse-Square)
V = -g(R/r 3)+9t"* State and Variation in State Obtained by Numerical Integration of
Differential Equations* Variable-Step Runge-Kutta-Fehlberg 7/8 Algorithm• Maximum of 42 Linear & Non-Linear Differential Equations
"* Boundary Value Problem Satisfied by Solving* Perturbed Lambert Problem
"* Constraint Targeting Evaluated Using Non-Periodic Elements"* Constraints and Variation of Constraints Evaluated on Perturbed
Trajectory"* NZSOL Solves Perturbed, Constrained Optimal Problem"* Solutions Obtained (Presently) in Minutes
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
Mconne 0ouglas Ape. Syse Compauy. Ho-ston Mm~Won Paep 1 of 2
11
Concluding Remarks
C1 Integration of Theoretical & Operational Aspects Of Optimal Rendezvous
"o Development of an Autonomous Optimal Rendezvous Solution
"o Primer Vector Theory Basis
"o Approach Based on Solution to Lambert's Problem and it's Extention in aPerturbed Force Field
"o Premise is Made that Perturbed Problem is an e away from Unperturbed
"O Constraints are Adjoined to Objective Function by Lagrange Multipliers
" General Mapping for Constraints from State to Parameter Space
"o NZSOL Used to Solve Constrained Non-Linear Programming Problem
"o Program is Fast, Reliable, Robust, Flexible, and Readily Extended
"0 Solution Time: Unperturbed (sec.), Perturbed (min.)
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)
* Mr. Raymond Moszee, Branch Chief
- Capt. Tim Middendorf
* 1 Lt. Paul Castro. 2 Lt. Naftali Dratman
* Mr. Francis McDougall
- Mr. Gerry Sayles
- Ms. Pamela Tanck, SPARTA
- Mr. James Eckmann, SPARTA
* Maj. Leo Matuszak, AF Reservest
21
PRKSA - Applications Branch
RKSA Simulation Environment
* Integrated Tools
* Ethernet Network(TCP/IP)
* Connectivity SoftwareMAcnl (NFS, Versaterm Pro)
' #RKSA - Applications Branch
Integrated Analysis
Persuasion * POST *
• Trajectories, Vehicle Analysis SPIRAL
e Sizing, Geometry* Structures and Weights
• Presentation MODEC
CONSIZ
..... . APAS EVA
SMART
22
W , RKSA - Applications Branch
Summary of POST Models"* Atlas II
"* Delta
- Titan IV *
"* Space Shuttle *
"• Pegasus *
"* Ground Based Interceptors
"* Small ICBM
"* Minuteman III
"* National Launch System (NLS)
"* Single Stage To Orbit (SSTO)
"* Delta Clipper - VTVL Concept *"* RASV - HTHL Concept
"* RST - VTHL Concept *
RKSA - Applications Branch
Applications of POST Models"* Atlas II
- Monopropellants; Hfigh Energy Density Mater (HEDM) propellants; Composite shroud
"• Delta" Titan IV
- Soviet RD-170 strap-on LRB's to replace SRB's
"* Space Shuttle* Clean propellant SRMs
"* Pegasus* Advanced Liquid Axial Stage (ALAS) as 4th stage; Potential booster for NASP program
flight test experiment"* Ground Based Interceptors
o Single-stage, two-stage, three-stage, and dual-pulse motor boosters; Standard Missile andSRAM 2 boosters for LEAP tests
" SICBM* Advanced ICBM studies baseline
23
"•I RKSA -Applications Branch
Minuteman III Model* APPLICATION:
The ICBM system of the future. Baseline for assess~ag advancedtechnology payoffs.
"* Clean Propellant Trade Studies
"• Impact of Reducing the Number of Warheads
"* Two-stage Missile Studies
* CONSTRUCTION:
"* Objective Function: Maximum range for fixed payload or Maximumpayload for fixed range
"* Constraints (2): Maximum dynamic pressure, Minimum re-entry angle
"* Control Variables (6): Pitch rate at motor ignition for each of 3 stages;Tune at which inertial attitude is held constant for each of the 3 stages
"• Phasing Events (16): 3 motor firings, 3 stage seperations, initial pitchover, 3 constant attitude segments, 3 ballistic flight segments, payloadshroud seperation, atmospheric re-entry, ground impact.
"••t RKSA - Applications Branch
Sample Minuteman Ill Results
TAGE 2 &3 PROPELANTM DL.H435 (CLEAN)
...... . .......24
RKSA - Applications Branch
National Launch System (NLS) Vehicles
Trade Studies Performed: 7
Castor 120's on NLS-3
Mixture Ratio -
Tank Sizing
Thrust Level
Engine Out
Throttling Effects l
Staging Algorithms
Upper Stages .... . . .. mat
GLOW, RM U *A 1.9 4.3ItF nvghg.N% I3x• 152mf hXum 17un
40P7 RKSA -Applications Branch
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
t CExcel by MicrosoftData Conditioning NwCd
rNew Code -
: Object OrientedProgram Execution • • Software Packages
Technologies* Sparse Matrix Methods
Results Interpretation • Defect Formulation• Singular Arcs* Node Placement
[Off-the-Shelf Software
* AGPS - Ribbon Plots_AgileVu - Animated Trajectories
BoeingDense & SummarySpace Group
"• Boeing Continues OTIS Development
"* Focus on Speed & Usability
"* Exploit Off-the-Shelf Software
"* Goal is a "Better" OTIS
31
-AFS
OTIS ACTIVITIES
AT
MCDONNELL DOUGLAS SPACE SYSTEMS COMPANY
R. L. NELSON
10 AUGUST 1992
McDonnell Douglas Space Systems Company -A. Ne"Ibv2
33
OTIS ACTIVITIES
-AFS
"* Applications
"* OTIS Project Development (PD 1-301) at MDSSC-HB
"* Launch Vehicle Sizing: ELVIS/OTIS
"* OTIS upgrades for Wright Labs-AFB
McDonnell Douglas Space Systems Company -N. NelsorV2
ADVANCED APPLICATIONSPD 1-301 OPTIMIZATION TECHNIQUES FOR
ADVANCED SPACE MISSIONS1 MDSSC-HB
L Theater High Altitude Area Defense (THAAD)
U ENDO/EXO Atmospheric Interceptor (E21)
L) HEDI
"U DELTA
" National Aerospace Plane (NASP)
"U SSRT
"U Aerobrakes
" Hypersonic Advanced Weapon (HAW)
"U Fighter Aircraft
"* Evasive maneuvers"* Agility
"U Military Space
" Space Transfer Vehicles
PO 1-,o
34 5SROCKwJ•Lf.
TASK FLOWPD 1-301 OPTIMIZATION TECHNIQUES FOR
ADVANCED SPACE MISSIONS"MDSSC-HB
PD 1-301
ITask I Task 2 Task 3
NLP Theory Numerical Low ThrustAlgorithms/ Transfers
Dense/Sparse optimizers OTIS
NZSOL/MINOS/NZSPARSE
TASK 1-APPROACH (1992-1993)-NLP THEORYPD 1-301 OPTIMIZATION TECHNIQUES FOR
ADVANCED SPACE MISSIONSMDSSC-HB
U Develop strong robust globally convergent nonlinear optimizers
"* Dense and sparse optimizers
"* NZSOL, a dense optimizer- Initial feasibility algorithms- MIn - Max optimizer
"* NZSPARSE, a sparse optimizer
"* Dr. Philip E. Gill, Professor, University of California, San DiegoDr. Michael Saunders, Research Professor, Stanford University
PO 1-,o
35
TASK 1 - PROGRESS-NLP THEORYPD 1-301 OPTIMIZATION TECHNIQUES FOR
ADVANCED SPACE MISSIONS- MDSSC-HB
U Developed State of the Art optimizer, NZSOL ( dense)* NPOOL 2.1* NPSOL 4.02"* Continual testing"* Tuned NZSOL for OTIS type problems
U BREAKTHROUGH ALGORITHM: NZSPARSE (sparse)* Theoretical formulation"* Development and checkout"* MINOS
U Modified OTIS structure to accept dense / sparse optimizers
PD 1-301
TASK 2 - APPROACH (1992-1993)-NUMERICAL ALGORITHM:PD 1-301 OPTIMIZATION TECHNIQUES FOR
ADVANCED SPACE MISSIONS-m MDSSC-HB
U Algorithms for OTIS
"* Automatic scaling
"* Automatic node placement ( University of Illinois)
"* Automatic tabular data smoothing
"* Lagrange multiplier Interpretation (Continuous / discrete)
"* Minimum curvature cubic control splines
36
TASK 2 - PROGRESS - ALGORITHMS-AFS ,
e Tabular Data Smoothing
e Enhanced Velocity Loss Model for Launch Vehicles
* Generalized Stage - Phase Concept for Sizing
a Automatic Node Placement
TASK 3 - APPROACH (1992-1993)-LOW THRUSTPD 1-301 OPTIMIZATION TECHNIQUES FOR
ADVANCED SPACE MISSIONSMDSSC-HB
"Li Develop restricted and general 3-body equations of motion
"Li Boundary conditions and coordinate systems
"Li Quantify the transition region for earth-moon low thrust / weighttransfers
"Li SECKSPOT / NASA Code - COSMIC Library
* Strong gravity field
* Orbit averaging techniques
"Li QT2 Interplanetary code
"Li Dr. Richard Shi, MDSSC-HB
PD 1.-01
37
TASK 3 - PROGRESS-LOW THRUSTPO 1-301 OPTIMIZATION TECHNIQUES FOR
ADVANCED SPACE MISSIONS-- MDSSC-HS
"Q Key Idea to solve problem
"* The existence of the Jacobi (energy) integral for the restrictedthree-body problem will aid us In the general three-bodyproblem.
Zero velocity or zero energy curves are the regions wherethe low thrust earth-moon transfers are possible
"e No Integral available for the general 3-body problem
"i Develop for OTIS
"e 3-body equations of motion
"* boundary conditions
"* coordinate systems
PD 1-301
583fiCK=XY21RLHAW
PD 1-301 OPTIMIZATION TECHNIQUES FORADVANCED SPACE MISSIONS VKSI5S
- MaDSSC-SSD
ma-.th• .."Moon
TransitionRlegion
Earth-Moon Transfer
38
CONTOURS OF CONSTANT POTENTIAL ENERGY (REFERRED TOROTATING SYSTEM) IN PLANE Z =0 WITH ~i=0.01213 a"I
MODSSC-SSD0. gin of C~oactuname stC~i. of Eanht. CG of Earm.Vm-
vSBaien @I (OI.01120) WhIUA is Within Emili
Al oA LI F 22 2 AA3 63 40396 Ag
PDD.SC-0B _ __DIUES FORAIMCWH
MDC PROOMe
Q TacWalWW id
" HAIM Low TII*low VrAl a Bmnold opMlastion* we"" . ag -
beese P 1-87 P I AUS- muIUvshlals
"* NAGP f"0 tdbe eabrse
Q NASA LAW NO0@ 1 swft08R
bfrqch (OTIS /IVYOTS) AL0O Low thrus I weight0 icraft G&C LN yetvaSibrefbranch ("Ope- O f
39
-*AF S
ELVIS /OTIS
ARCHITECTURE
MeongwiIf Douglas Spsc. Systems Con Wan
Legend:Information flow Ee
*Top leve cnhrftModule Norm. Nulwo~ldn
* Daa ManagemrentModule function Exec menus
a Spawn stanrd atonepro~an (Aft)
Pidlam module is a Alow Luur to LVS 1.5amuted(de) an APO w/ Ui&fWIS*Qm Iaawe
------V41:11C10i wind stg massu I rc~
"* LV paranmoc =Wt ai gaOpUII mumhacio(W/O ORs Cog. Junil S2) dwkwW Wi"* LVPC Meum ac*veIn Exec e2-0 dagram duplay and________Mae ftmwwan
* * LVS minus actve in Exec
OTItSen (Vax)(U
" Op~mxncw~v40
OTIS UPGRADES FOR WRIGHT LABS-AFS
* TASK 1: NZSOL
* TASK 2: Automatic Variable Scaling
* TASK 3: Variable Names for NZSOL Output
* TASK 4: Minny Heating Model
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
ET= [a, e, i, Q, co, f]
* Geometry of the problem
rt r& targlet
rif 'trajectoryTrajectory•,
Earth' i quatorial Plane
Continuing Research - Problems
1. Automatic node placement. (A. Herman)
2. Optimal very-low-thrust trajectories (W. Scheel)
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
solutions ?
• Analytical Approach
estimate adjoint variables, examine discretized necessaryconditions
Engineering Approach
numerical comparison of collocation solution to exactsolution for a representative problem
52
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
solutions ?
* Analytical Approach
estimate adjoint variables, examine discretized necessaryconditions
* Engineering Approach
numerical comparison of collocation solution to exactsolution for a representative problem
52
Example : ALS Ascent to Orbit
Example Problem :
Steer a two stage launch vehicle from a given initial condition to aspecified target orbit in minimum fuel.
" Exact or TPBVP solution available in literature ( Ref. HansSeywald and E. M. Cliff )
" Care was taken to keep the vehicle/atmosphere/planet modelssame in OTIS
" Only solution methodologies were different
Comnarlson of Optimal ALS Aseant with OTI saolutmoan
OTIS Solutions TPBVP solution
12 nodes 22 nodes 32 nodes 40 nodes
tj sew 477.2 477.2 477.2 477.2 477.2
Mas( tf ) Kgs 149.881 149.877 149.895 149.891 149.900
Velocity (t1 ) 7855 7855 7857 7857 7857
Altitude (tf ) km 146.68 148.74 147.80 147.96 148.2
Apogee (kin) 274.58 274.11 279.52 278.77 277.81
PoftNe (kin) 146.53 148.66 147.6 147.9 146.16
CPU Tsimesea,) 74 377 935 1675 ?
53
1.6 10Oe
1.2 1 0'.-
21.0 a -- - - - ...-- --8.0 105*-_ _
2.010 5
4.0 10' -t .-2.0 10'
0 100 200 300 400 500TmW (s0c)
Figure [l]. Comparison of optimal ALS ascent with OTISsolution, mass (kg) vs. time.
1 5 0 0 0 0 -1
1 .00000a -OTIS
Iooo- ... . .. .. 0--... - --M
50 - - - ----- -
0 100 200 300 400 500Tim (sec)
Figure [2]. Comparison of optimal ALS ascent with OTISsolution, altitude above spherical Earth vs. time.
54
8000-7 0 0 0 . ..................... i ....... .......... .... " ............ ....... ............. ......... i. .... . - -
:00 --...---
6000 TPBVP ...... . ........i• °° I ... .... OT IS I ..........7 ......
5000 ........... ......... ..3000-:----
2000- ..- ......1000 .... ..-..
0-0 100 200 300 400 500
"lime (see)
Figure [3]. Comparison of optimal ALS ascent with OTISsolution, Earth relative velocity (in/sec) vs. time.
7 0 -- - • --..- , - -: ... .T ........ .... ....... ..... ... .. ......OTIS
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
65
Overview
Motivation
Regular Perturbation Analysis
The Method of Collocation
Hybrid Collocation / Regular Perturbation Analysis Approach
Examples
Duffing Equation
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
- consider first order polynomials
x(i)= xZI1 +p1 (i-d j4l) ; .(i)=Xj_1 +qj(i-ijj) ;J=1, 2, .. , N
- enforce the derivative constraints at the mid point of each element
S~aH|t]- tj_ = '*=(ij +i_ )I2; x=(zj +xj! )12; ),=O.j +•LJ--.j )12
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-
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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
oli) = Xoij. +Pi(l-i -j.g) ; v(i) = Voj.. + PvJ(i- ij..) ; j= 1, .. , N
X,(i)=)Xzvj-j +qi-i j-j) ; ).,Goi)=Xvl-n +qvj(i-ijj)
- 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.
----- O 1 %8 ----- ith
%1 0
0..
""°o \ I 23 0 , 2-Woi -Wi
I o 1 £ $
Time Time
Fgre 4.16. LeM2 NowOrderReia n4 o-2N.. Ape4.14. Lavl2 Hi•eOrdr esu inv for P-
Simple Example (continued)Level 3 Formulationt
. enhanced level 2 formulation by fully utilizing analytically tradtableportion of the necessary conditions
. decompose the dynamics as:
1=v
# = - -).T + p,+ E(-ax.3- pvJ) ;J=It 29 .. , Ni0 .v +qxj + 2(3a vx 3 2 2 )
- .imilar to Level 0 except for additional unknown constants Pvje qes- ue piecewse constant terms to approximate the nonlinear parts
. both zero and lirat order result; for hL- are superior than theLevel 0 cae (fi'. 4.17 - 4.20)
- Level 2 and 3level demlatraio the use ofntilziant tratable
74
0 ..... h 4, eO
£ q" -
'i • ,,,• e.* •
- L0 "* /
1.0
2 1 3Time Tim*
Figure 4.17. Level 3 Higher Order Results Fipm 4.19. L&M 3Higba duResulsin forN-1.
in x for N=.% Sem2
1s't'
0"• m£l %
- %% %' * . aa% .
% %
-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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dating the bunching operuati" samouin the e0arnpThi macliid f hsmibig rwimatemilbooimpof Pasaieslo fre.Iomm the catapula engie sa dhe walecafTheametod f hethag equiit iiiNl e~tP o eleough CMe beach he. bridl or pIaaIe A tentp. a-.
the litidi. at Iedn A sad the boidhech by the caumpult twead so the. catapul tow fitting. amsah. Ath sift-it antdech crew after the sitatig him hess stated into pot. rol amd two, fitting. For heidle~stuaghed eifcd.theilisa -0 the CRtPuLt wbee the aircraft mauben the end of bobkdub telm prit, ideanmes - a nchorag painm"th COtaPOlN P~own sand the mete feret decaya time foe the hoidhat an4 release, assembly. This dech claw.beidh or edtI dtofm from thme steanit maw finting iscme Mt , .o. mod on the ceemetiae af "h catpuland it brought ise &,mop on the fliiMe dack by the bridle tect The he idisterese mtopa sad mitass the bridle awaso e syst""N. p.-n sline it is Shadl fron the OirceaL. Finally, the Ait
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441
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LW Aerospace and Defense CompanyAircraft Division PROGRAM OVERVIEW
THE CATAPULT LAUNCH SIMULATION
CONSISTS OF FIVE PHASES
s STATIC BALANCE
"* HOLDBACK
"* CATAPULT STROKE
"* DECK RUN
s FLYAWAY
LTV Aerospace and Defense CompanyAircraft Division PROGRAM OVERVIEW
THE CATAPULT LAUNCH SIMULATION INCLUDES:
"* CATAPULT FORCES
"* HOLDBACK FORCES
"* HIGH FIDELITY LANDING GEAR MODEL
"* AERODYNAMIC DATA AS A FUNCTION OF
- ANGLE OF ATTACK OR LIFT COEFFICIENT
- NOZZLE DEFLECTION
- THRUST COEFFICIENT OR NOZZLE PRESSURE RATIO (NPR)
- FLAP DEFLECTION- PITCH TRIM SURFACE DEFLECTION
" GENERIC FLIGHT CONTROL AND STABILITY AUGMENTATION
SYSTEM
" LONGITUDINAL THRUST VECTORING
83
LTV Aerospace and Defense CompanyAircraft Division PROGRAM OPTIONS
PROGRAM OPTIONS
"* AUTOMATIC WIND OVER DECK SOLUTION
"* SOLUTION TERMINATION
"* POWERSETTING
"* FLAP DEFLECTION
"* FLIGHT CONTROL AND STABILITY AUGMENTATION SYSTEM
"* THRUST VECTORING CONTROL SYSTEM
"* LANDING GEAR
"* ENGINE FAILURE
"* STORE JETTISON
"* LANDING GEAR RETRACTION
LTV Aerospace and Defense CompanyAircraft Division PROGRAM OPTIONS
AUTOMATIC WIND OVER DECK SOLUTION
TWO CONSTRAINTS:
"* MAXIMUM SINK
"* MAXIMUM ANGLE OF ATTACKOR
MAXIMUM PITCH RATEOR
MINIMUM LONGITUDINAL ACCELERATION
TWO VALUES DETERMINED:
"* WIND OVER DECK
"* STICK DISPLACEMENT (OR TAIL DEFLECTION)
S84 ,m,
LTV Aerospace and Defense CompanyAircraft DIvslion PROGRAM OPTIONS
SOLUTION TERMINATION
"* POSITIVE TMAX
- TIME HISTORIES STOP AT THE SPECIFIED TMAX
"* NEGATIVE TMAX
- TIME HISTORIES STOP WHEN A POSITIVE RATE OF CLIMB HASBEEN ACHIEVED AND ANGLE OF ATTACK HAS PEAKED.
- IF A POSITIVE RATE OF CLIMB IS NOT ACHIEVED QB ANGLE OFATTACK IS CONTINUOUSLY INCREASING. THE TIME HISTORY WILLSTOP AT THE ABSOLUTE VALUE OF TMAX.
LTV Aerospace and Defense CompanyAircraft Division PROGRAM OVERVIEW
AERODYNAMIC DATA INCLUDES
PROPULSION-INDUCED EFFECTS
COEFFICIENTS ARE FUNCTIONS OF:
"* ANGLE OF ATTACK OR LIFT COEFFICIENT
* NOZZLE DEFLECTION
"* THRUST COEFFICIENT OR NPR
"* FLAP DEFLECTION
"* TRIM SURFACE DEFLECTION
mco0Mal
85
LTV Aerospace and Defense CompanyAircaft Division PROGRAM OVERVIEW
TYPICAL AERODYNAMIC DATAFOR A THRUST VECTORING CONFIGURATION
11 1FLAPAERODYNAMIC COEFFICIENTS HAVE aTHE DIRECT PROPULSION EFFECTS SFA l.REMOVED AND ARE FUNCTIONS OF. 0:4
"* ANGLE OF ATTACK ORULFT W a~,E* 8COEFFICIENT ;.8
"* THRUST COEFFICIENT Oft.8NPR NOZL
"* NOZZLE DEFLECTION ..so- '0
"* FLAP DEFLECTION
"* TRIM SURFACE DEFLECTION
0 sa ANGL OF ATTACK
~ 4 ANGLE OF ATTACKOR CL
ANGLE OF ATTACK -C GRMS THRUST (TOTALIOR CLR*SO
LTV Aerospace and Defense Company PORMOEVEAircraft Divsion PORMOEVE
FORCES ACTING ON THE AIRCRAFT
AEIAIVE
I WE01 M I
simm 86 Fi
INERTIAL AXES X. z
BODY CENTERLIE AXES X. Z 0WINO AXES Xw. Zw 0NOSE GEAR AXES
-4 ZMAIN GEAR AXES
c 2, --4AIRCRAFT REF. SYSTEM XF4. ZIS ZW
(A0Q
WITH ~ ~ ~ ~ ~ IRRF CEOEC LAUNCHRTA RAEOFRFEEC
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1. 2. 3. 4. - . ý - 7. '. 0 0
-10.-
-20.
-30.
-40.
FIGURE 9a. CONTROL SURFACE DEFLECTION AS A FUNCTION OF TIME
87
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FIGURE 9b - PITCH RATE AS A FUNCTION OF TIME
24. -
20.
* 16.
I
412.
.. 7. . 9 0.
-4
FIGURE k-ANGLE OF ATTACK AS A FUNCTION OF TIME
88
140. = l G-_E3
120.
100.
! 80..I
60.
40.
20.
0 I. 2. 3. 4. 9. 6. 7. I. 9. 10.TW -- a•moads
FIGURE 3d. TRUE AIRSPEED AS A FUNCTION OF TiME
THREE )EGRE OF FREOED CATAPULTOUTPUT hoME "ISTORY
-. , -. o-.- -ti I a
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jew- 90
AIRBREATHING BOOSTER PERFORMANCE OPTIMIZATION
USING MICROCOMPUTERS
Ron Oglevie
Irvine Aerospace Systems Co.2001 Calle Candela,Fullerton, CA 92633
(714) 526-6642
AIAA Astrodynamics ConferenceWorkshop on Trajectory Optimization
10 August, 1992
91
M .Irvine0'... OVERVIEW Aerospace
0S,...f SystemsTS...-. Company
"* MICROCOMPUTER-BASED OPTIMIZING SIMULATION OFTRAJECTORIES (MOST)1
"* MOTIVATION - Fill void in preliminary design tools
"* Easy to use fast running modes
"* TPBV solution for truth model
"* OTIS PROGRAM OPERATION ON PC
"* LOW-THRUST TRAJECTORY OPTIMIZATION PROGRAM (MICROTOP)
Work performed under Air Force Contract F33615-91-C-2100.
M'I I Irvineo,-. I MOST TECHNICAL OBJECTIVES Aerospaces...I BEING MET I Systems
T,,, Company
"* SUITABLE FOR RAPID PRELIMINARY DESIGN
"* MICROCOMPUTER OPERATION - Run time less than 5 mins. on PC AT
"* AIRBREATHING & ROCKET PROPULSION VIA TABLES ANDEQUATIONS - Including realistic flight constraints
"* EARTH-TO-ORBIT (ETO) FLIGHT
"* PLANAR FLIGHT - Simple rotating earth model facilitates 3-D typeresults with minimal complexity
"* EASE-OF-USE - Easy input, good graphics, & robust convergence
92
I ~Irvineo,,-,.. HYBRID APPROACH OFFERS AI o
Sw.m SPEED AND PRECISION Systw$T. Company
N1ROWNPUTER-WED OPTMIZING SIMULATION OF TRAJECTORIES (MOST). A HYDRID APROAC
inw i- mm m ares- im
* MIm3ni maul u *iHN 2I.nuk t*mmlnums
oe ? S IMIMS oME
:411N OWARE *I1M D off nme. 301EUNIUM. Rmii* .. ....... inmm Ui U
ISlJrAS14Aillt orn"IE
o NmiW Sim s a ll
eU11 MIMIES KSEEMAOWI
00,SINGLE-STAGE-TO-ORBIT TechnologyOPTIMIZATION GOALS ACHIEVED ON PC Group, Inc.
MOST versus OTIS H-V Flight Profile Comparison
Test Case No. 2. Single-Stage-to-Orbit
Altitude, h (feet)200,000
150,000 - --
100,000
SO0.000
OI
0
0 5.000 10,000 15.000 20,000 25.000 30.000
Relative Velocity, Vr (fps)Plie. =2mA.NVDUW
93
2STAGES TO ORBIT TechnologySh.ko. of OPTIMIZATION GOALS ACHIEVED ON PC IGroup. Inc.Oimo
HOST ve MTIS Toot Case I fl-V Prof ile Comwarimon
Liftoff-to-Rocket Burnout
3h.0's te I 01.0. GO
V &s000"
Al,4
-4 LOGO**o
0 So" i1e.. 11166 200001 210400 30000
Relative V*loCity (fps)
Irvkw.0 ~ GRAPHICS ENHANCE UNDERSTANDING TchnalMg
Soo~.o.OF PERFORMANCE OPTIMIZATION Group. Inc.
CONTOURS OF CONSTANT MANEUVERING EFFICIENCY
o-a
0A
00
to 03 0 o----
Dyt1"o
25 31 2 60.0 88.7 117.5 146.2 1750 203 7 2325
MISSION TIME (SEC) .101
94
FA CONCLUSIONS - Aerospace
ETO PERFORMANCE OPTIMIZATION SystemT ACHIEVED ON PERSONAL COMPUTER Company
* MOST - LOW COST, RAPID RESPONSE TOOL FOR PRELIM. DESIGNSUCCESSFULLY ACHIEVED
o ADVANTAGES OF PC DEMONSTRATED - Low Cost, portability, and goodgraphics and support software (LOTUS, Harvard Graphics, Freelance, etc.)
* USER-FRIENDLY ENVIRONMENT FOR PREPARATION OF INPUT FILES & OUTPUTDATA - Facilitates OTIS input file preparation
o MOST FAST RUNNING MODES DEMONSTRATED - Good agreement with OTISresults. Eady engineering model delivered
* PRELIMINARY RESULTS FROM 2-D NLP/COLLOCATION ALGORITHMS (MINI-OTIS) ARE ENCOURAGING
* FAST RUNNING MODES FACILITATE NEW APPLICATION - Trajectory optimizersimple and fast enough to imbed in vehicle design optimization code
0 OTIS HOSTED ON PC - ETO flight achievable with large RAM (-40KBYTES)
7
95
An Algorithm for Trajectory Optimization on aDistributed-Memory Parallel Processor
Mark L. Psiaki and Kihong ParkSibley School of Mechanical and Aerospace Engineering
Cornell University
Acknowledgement:
Work supported by NASA/LaRC
97
Continuous-Time Problem to be Solved
find: u(t) and x(t) for to < t < tftf
to minimize: J = f L[x(t),u(t),t] dt + V[x(tf)]to
subject to: x(to) given1 =I[x(t),u(t),t]
ae1x(t),u(t),t] = 0
ai[x(t),u(t),t] !5 0
aefx(tf)] = 0
aif[x(tf)] < 0
Approach
"* Use zero-order-hold control parameterization
"* Model as a multi-stage parameter optimization problem
"* Retain state variables and dynamic constraints explicitly
"* Solve using a nonlinear programming algorithm that ...
... has fast local and robust global convergence
... allows infeasible intermediate results
... parallelizes function, gradient, etc. evaluations at differenttime steps
... exploits dynamic structure and parallelism to get searchdirections
98
Multi-Stage Nonlinear ProgrammingProblem
find: x- U 4, Ul ,..., UN1 ,X
N-Ito minimize: J = • Lk(Xk,Uk) + V[XNI
k=O
subject to: xo given
Xk+I = fk(Xk,Uk) for k = 0 ... N-I
aek(Xk,Uk) = 0 for k = 0 ... N-I
aik(Xk,Uk) - 0 for k = 0 ... N-i
aeN(XN) = 0
aiN(XN) < 0
A Static/Dense Nonlinear ProgrammingProblem
find: x
to minimize: J (x)
subject to: Ce(X) = 0
Ci(X) !5 0
99
Status of Project
Parallel search direction algorithm:
FORTRAN version tested on 32-node INTEL iPSC/2
General static NP algorithm:
FORTRAN version tested on 1 node of INTEL iPSC/2
Compared to NPSOL version 4.02 on static problems
Full parallel trajectory optimization algorithm:
A "next generation" of the NP algorithm that exploitsparallelism and dynamic problem structure
FORTRAN components currently being tested on 32-nodeINTEL iPSC/860
CY
-JWU Z4
Z ~0* ~(A
0 0 DC C N
o 0a( COa. 00 V- 32 nodes [Ref. 121
> N stages on N nodes
0 (extrapolated [Ref. 12])o . .0 . .................. . .E 100 101 102 103
Number of Problem Stages
100
Plans(the LORD willing)
"* Finish component and full algorithm testing (Present-Oct. '92?)
"* Model and solve guidance problems for NASP and generichypersonic vehicles (Oct. '92 - Dec. '93)
"* Compare to existing codes (199?)
"* Evaluate suitability for real-time guidance updates (199?)
"* Make code user-friendly and disseminate (199?)
Distrtbidlon of Problem Stages onPlralelProces:,egQ'
24 Son, Pinoblm on 8 ftcmsa
0,1,2
1213,.14
9,10,11 3,4,5 15,16,17 21,22,23
101
Divide-and-Conquer TrajectoryOptimization
Iteration I x is fixed atx these times
- t
Iteration 2 x is fixed at
X& these times
! : t
Iteration 3 x is fixed atthese times
V It
Iteration 4 x is fixed atthis timeXI It
Test Problem 2 FoR SrA T.X C NP ACLGoAI'T-rtM
find: X9, x2
to minimize: J = -X2
2 2 2subject to: (xI- I)2 + x2 + 1000 (X2 + X2 1)2 - .062 5 -< 0
Test Results
"* Augmented Lag.: 77 iterations
(52 feasibility and 25 optimality)
"* NPSOL 4.02:
102
Tat Probbu S FOR STAT(ATrCP ALEýE
find. m81 AV,-AV, &V2
to mmnimize: J =moo
subject to: Newton's laws for a spherical Earth
Fixed fuel specific impulse
rum-• - er < 5rLm+ rV~i. -G <y: Vf -,I Var. + ev
*eqxy :5Y 5+C28o -i F,: if :5 28o + e
MCMS • mf
" Augmoed Lag.: ft/n iterta.,;o,, s
"* NPSOL 4.02: 9 iterations
Aero-Assisted Orbital ManeuveringExample
(taken from Miele, 1989 ACC)
Problem: Minimize Fuel for GEO to LEO transfer with +280 inc!ination
change
x [V, 0,, r, ,
U=[Vcqyc, Wc]T or [CL, a, 9]T
Constraint: Heating rate < 100 watts/cm 2
LOR-like problem derivation: Linear-quadraticize about guessed solution
103
LEOGEO 1.f•-- : = • - -..---- )-. .
2 . nAtmosphere.. ..
,rU.S. Governmefft Printirig Off"c: 19093-750-113/80106 104
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