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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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

WL-TR-92-31 129. Sponsoring / Monitoring Agency Name(s) and Address(es) 10. Sponsoring / Monitoring Agency

Report No.Same as Block 7.

11. Supplementary Notes

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

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-

211% *.PC',: SW0 J04

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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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cataplta war fitting ibeoughl the slated cyliadet walls.The heidhak device med with beidle/pendenet launch is Reweetaise, is aceomplisbed by a sepoms, hydro.wife tape. Chain. at W mtlinkhs. The Wkwmm Imn - 0 peemamaic resactieg eamtse.either ring ar a tensi- hue. The boidhac anseuhbly attacb.ameat paine on the aimcmaf is well aft mi she dack and at

tociss;si he hodhathdecl himee. Soction VIII descebee diAl DCKKEQUIPMENT. Thecamtpet iaauipptypical huilliack mad -. i. asebis trialh a shiued which moves alo dng h caaput trach

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

edg coet awl- pros metis the primmry central pai.4.1 LAIIMCIIII EGUIPMENT. fator a ang th $e catapult.

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

X5:

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40. F FI.ii T-EST

30.

20.

0

1. 2. 3. 4. - . ý - 7. '. 0 0

-10.-

-20.

-30.

-40.

FIGURE 9a. CONTROL SURFACE DEFLECTION AS A FUNCTION OF TIME

87

12O. -- --V --C5T

S ,......... . .. " 31ý ý= . .

10.

a.

-- 2

L

4 -o

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

_ I, I I _ _ _

-l •• ' i , I ' " • I l I l C -- c.- " I'"

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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