Table Of ContentCAMTOS - A Software Suite Combining
Direct and Indirect Trajectory Optimization
Methods
Von der Fakultät Luft- und Raumfahrttechnik und Geodäsie der Universität Stuttgart
zur Erlangung der Würde eines Doktor-Ingenieurs (Dr.-Ing.)
genehmigte Abhandlung
Vorgelegt von
Peter Friedrich Gath
aus Waiblingen
Hauptberichter: Prof. Klaus H. Well, Ph.D.
Mitberichter: Prof. Anthony J. Calise, Ph.D.
Prof. Dr.-Ing. Arnold Kistner
Honorarprof. Dr.-Ing. Eveline Gottzein
Tag der mündlichen Prüfung: 29. November 2002
Institut für Flugmechanik und Flugregelung
Universität Stuttgart
2002
Abstract
Many engineering applications involve the solution of optimal control problems. Such prob-
lems cover a wide variety of different disciplines, starting from typical aerospace applications,
e.g. a fuel minimal launcher ascent trajectory, up to more exotic applications such as the opti-
mal motion of the human body in different sport disciplines.
This thesis presents a new method for solving a general class of multi-phase trajectory
optimization problems. This new method uses a combination of direct multiple shooting and
direct collocation. Depending on the specific demands of the problem to solve, a different kind
of transcription method can be used for each phase. In addition, the method can be combined
with the indirect method. Once the adjoint differential equations are available in addition to the
dynamic system, one or more phases can be modeled as indirect phases. Since the transversal-
ity conditions that are usually required for solving the multi-point boundary value problem are
contained in the Karush-Kuhn-Tucker conditions of the NLP-solver, they need not be formu-
lated explicitly.
The benefits of combining direct and indirect methods is demonstrated on an Ariane 5
dual payload mission. While the atmospheric parts of this mission are modelled with the direct
multiple-shooting technique, the upper stage is using the indirect method. An analysis of the
switching function, which is generated as a by-product of the indirect method, yields an
improved mission profile. This profile involves an additional coast-arc and improves the pay-
load performance by 66%.
An additional example demonstrates the advantages of using direct collocation and
direct multiple shooting to solve a complex branched trajectory optimization problem. The
ascent and return of the suborbital Hopper vehicle is optimized such that the payload delivered
to the geostationary transfer orbit by an upper stage is maximized. The robustness and large
convergence radius of the direct collocation method significantly simplifies the generation of
an approximate solution of the problem which is then refined by changing the transcription
method of the atmospheric parts of the trajectory to the direct multiple shooting technique.
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CAMTOS - A Software Suite Combining Direct and Indirect Trajectory Optimization Methods 4
Zusammenfassung
Viele technische Anwendungen beinhalten die Lösung von Optimalsteuerungsproblemen.
Derartige Probleme decken einen weiten Bereich verschiedener Fachrichtungen ab, angefan-
gen bei typischen Raumfahrtanwendungen, wie z.B. der treibstoffoptimale Aufstieg einer
Rakete, bis hin zu exotischeren Gebieten wie z.B. der optimalen Bewegung des menschlichen
Körpers in verschiedenen Sportdisziplinen.
Diese Arbeit stellt eine neue Methode zur Lösung einer allgemeinen Klasse von mehr-
phasigen Bahnoptimierungsproblemen vor. Dieses Verfahren verwendet eine Kombination aus
direktem Mehrzielverfahren und direkter Kollokation. In Abhängigkeit von den spezifischen
Anforderungen des zu lösenden Problems kann für jede Phase die geeignete Transkriptionsme-
thode ausgewählt werden. Außerdem kann das Verfahren mit der indirekten Methode kombi-
niert werden. Sobald neben der Systemdynamik auch die Adjungierten-Differentialgleichun-
gen verfügbar sind, können eine oder mehrere Phasen auch mit Hilfe der indirekten Methode
modelliert werden. Da die Transversalitätsbedingungen bereits in den Karush-Kuhn-Tucker
Bedingungen des NLP-Verfahrens enthalten sind, müssen sie nicht explizit formuliert werden.
Die Vorteile einer Kombination aus direktem und indirektem Verfahren wird am Bei-
spiel einer Ariane 5 Mission mit zwei Nutzlasten demonstriert. Dabei werden die Missionsab-
schnitte innerhalb der Atmosphäre mit Hilfe des direkten Mehrzielverfahrens modelliert,
während das indirekte Verfahren in der Oberstufe zum Einsatz kommt. Eine Analyse der
Schaltfunktion, die sich als Nebenprodukt des indirekten Verfahrens ergibt, führt dabei zu
einer Verbesserung des Missionsablaufes. Durch Hinzufügen einer zusätzlichen Freiflugphase
kann die Nutzlast um 66% verbessert werden.
Ein weiteres Beispiel zeigt die Vorteile einer Kombination aus direkter Kollokation und
direktem Mehrzielverfahren für die Lösung eines verzweigten Bahnoptimierungsproblems.
Die Aufstiegs- und Rückkehrbahn des suborbitalen Hopper-Fahrzeuges wird derart optimiert,
dass die durch eine Oberstufe in den geostationären Transferorbit beförderte Nutzlast maxi-
miert wird. Dabei vereinfacht die Robustheit und der große Konvergenzradius des direkten
Kollokationsverfahrens die Berechnung einer Näherungslösung erheblich. Durch eine
Umschaltung der Transkription auf die direkte Mehrzielmethode in den atmosphärischen
Abschnitten der Trajektorie ermöglicht schließlich die Berechnung einer genauen Lösung.
Acknowledgements
The dream of yesterday
is the hope of today
and the reality of tomorrow.
Robert H. Goddard, 1904
This work was carried out during my time as a research scientist at the Institute of Flight
Mechanics and Control (IFR) at the University of Stuttgart.
I greatly appreciate and value the guidance and support of my advisor and director of
IFR, Professor Klaus H. Well. His experience and passion for trajectory optimization always
encouraged and motivated me during the past few years. In addition, he always gave me a lot
of freedom to carry out my ideas.
The software that was developed in the framework of this thesis is part of the HISTOS
project that was financed by the European Space Research and Technology Center ESTEC. I
would like to acknowledge this support and thank our project monitor, Dr. Klaus H. Mehlem
from ESTEC, for many useful discussions and giving me the required freedom within this
project to write this thesis. I was always looking forward to our interesting progress meetings
during the course of the HISTOS project.
My sincerest thanks also go to Professor Anthony J. Calise who has introduced me to the
methods of indirect trajectory optimization during my stay at the Georgia Institute of Technol-
ogy in 1998 and who was member of the reading committee. Ever since, I always look forward
to meeting him again and I deeply admire his tremendous engineering expertise and knowl-
edge.
I would also like to express my gratitude to Professor Arnold Kistner and Professor Eve-
line Gottzein who were also members of the reading committee.
Dr. Werner Grimm did an excellent job in proof-reading one of the first versions of this
thesis report, and I thank him very much for many discussions about mathematical problems
and details.
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CAMTOS - A Software Suite Combining Direct and Indirect Trajectory Optimization Methods 6
I would also like to thank Prof. Philip E. Gill for kindly providing his SQP-solver
SNOPT which serves as a fundamental basis of the new trajectory optimization method.
Moreover, I would like to express my special thanks to three individuals from The Boe-
ing Company: Dr. John T. Betts and Steve Paris for valuable discussions about direct colloca-
tion methods, optimal control and some numerical “tricks”, and Rocky Nelson for many
discussions about launcher trajectory optimization and the application of SNOPT.
Andreas Wiegand did an outstanding job in enhancing GESOP’s graphical user interface,
implementing branching into ASTOS, updating PROMIS and TROPIC to the latest version of
SNOPT, and assisting me during the implementation of the indirect method into the ASTOS
library. I always appreciated our discussions about several software engineering problems and
the enhancement of the ASTOS model library, as well as the software training sessions and
presentations we performed together for our GESOP and ASTOS customers.
Working within the GESOP/ASTOS team, together with Denis Fischer, Jörg Kindler,
Dr. Albert Markl, and Andreas Wiegand, was a great pleasure and experience for me. The
many discussions we had with numerous customers have proved to be very beneficial also for
the development and implementation of the methods presented in this thesis.
I always enjoyed the excellent work atmosphere at IFR and the very good team spirit, not
only within the GESOP/ASTOS team, but also between the other teams at IFR. I would like to
thank all of my colleagues and our students for many technical and not-so-technical discus-
sions we had in the past few years. My special thanks go to our system administrator Dirk
Wimmer, who took over the burden of taking care of our Windows NT environment with great
passion in addition to his own research project.
Last but not least, I would like to thank my parents, my friends, and especially Ms. Bar-
bara Nucera for their ever-lasting support and encouragement during this endeavour.
Markdorf, December 2002 Peter F. Gath
Table of Contents
Abstract 3
Zusammenfassung 4
Acknowledgements 5
Table of Contents 7
Table of Figures 13
List of Tables 19
Nomenclature 21
1 Introduction 25
1.1 Motivation ...................................................................................................... 25
1.2 General Problem Formulation ........................................................................ 27
1.3 Indirect Trajectory Optimization Methods ..................................................... 28
1.4 Direct Trajectory Optimization Methods ....................................................... 30
1.4.1 Direct Shooting Methods ................................................................. 31
1.4.2 Direct Collocation Methods ............................................................. 32
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CAMTOS - A Software Suite Combining Direct and Indirect Trajectory Optimization Methods 8
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1.5 Problems with the Current Methods ............................................................... 33
1.6 Design of a New Trajectory Optimization Method ........................................ 34
2 The Direct Trajectory Optimization Method 37
2.1 The Direct Multiple Shooting Method ........................................................... 37
2.1.1 Discretization of a Multiple Shooting Phase .................................... 38
2.1.2 Integration Methods ......................................................................... 40
2.1.3 Comparison of the Integration Methods .......................................... 42
2.2 The Direct Collocation Method ...................................................................... 43
2.2.1 Discretization of a Collocation Phase .............................................. 43
2.2.2 State Approximation Using Hermite-Simpson Polynomials ........... 44
2.2.3 Other Collocation Schemes .............................................................. 46
2.2.3.1 Trapezoidal Scheme ........................................................................... 47
2.2.3.2 Runge-Kutta 4 Scheme ...................................................................... 47
2.2.4 Handling Path Constraints ................................................................ 47
2.3 Software Interfaces ......................................................................................... 48
2.3.1 General User Model Interface .......................................................... 49
2.3.2 General Interface to NLP Solvers .................................................... 49
2.3.3 General Interface to Integration Methods ........................................ 52
2.3.4 General Interface to Collocation Schemes ....................................... 54
2.4 Implementation Considerations ...................................................................... 57
2.4.1 Roundoff and Truncation Errors ...................................................... 57
2.4.2 Parameter and Function Scaling ....................................................... 59
2.4.2.1 Parameter Scaling .............................................................................. 60
2.4.2.2 Scaling of the Performance Index ...................................................... 61
2.4.2.3 Default Scaling of Constraints ........................................................... 61
2.4.2.4 Scaling of Continuity Conditions in Multiple Shooting Phases ........ 61
2.4.2.5 Scaling of Defects in Collocation Phases .......................................... 61
2.4.2.6 Scaling of Phase Connect Conditions ................................................ 61
2.4.3 Selecting the Parameter Perturbation Size for Gradient Calculation 62
2.5 Setup of Multi-Phase Problems ...................................................................... 62
CAMTOS - A Software Suite Combining Direct and Indirect Trajectory Optimization Methods 9
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2.6 NLP Solvers ................................................................................................... 63
2.6.1 SNOPT 6.1 ....................................................................................... 63
2.6.2 Other Methods .................................................................................. 64
2.7 Performance of the Direct Optimization Method ........................................... 64
2.7.1 Performance of the Direct Multiple Shooting Method .................... 64
2.7.2 Performance of the Direct Collocation Method and its Combination with
Direct Multiple Shooting ..................................................................66
3 Indirect Method for Optimal Thrust Arcs in Vacuum 69
3.1 Formulation of Dynamics ............................................................................... 69
3.1.1 Gravity Field with J terms .............................................................. 70
2
3.1.2 Altitude over an Oblate Planet ......................................................... 71
3.1.3 Including Backpressure in the Thrust Calculation ........................... 72
3.2 Adjoint Differential Equations ....................................................................... 73
3.2.1 Adjoint Differential Equations including J Terms ......................... 73
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3.2.2 Correction for Backpressure Term ................................................... 74
3.3 Optimal Control .............................................................................................. 74
3.4 Boundary Conditions ...................................................................................... 75
3.4.1 Boundary Conditions for Circular Orbits with a Fixed Inclination . 76
3.4.2 Boundary Conditions for Equatorial Orbits ..................................... 78
3.4.3 Boundary Conditions for Elliptical Orbits with a Fixed Argument of Peri-
gee ....................................................................................................79
3.4.4 Boundary Conditions for Elliptical Orbits with a Fixed Inclination 79
3.5 Optimal Coast Arcs and Switching Function ................................................. 81
3.6 Generating an Initial Guess for the Initial Costates ....................................... 82
3.6.1 Ad-Hoc Initial Guess for P and Q .................................................... 82
3.6.2 Estimate based on Thrust-Acceleration and Target Radius ............. 82
3.7 Solution of the Two-Point Boundary Value Problem .................................... 83
3.8 Solution in the Framework of an SQP Method .............................................. 84
3.9 Verification of the Indirect Method ................................................................ 86
CAMTOS - A Software Suite Combining Direct and Indirect Trajectory Optimization Methods 10
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4 Example 1: Ariane 5 Dual Payload Mission Analysis 91
4.1 Reference Mission Scenario ........................................................................... 92
4.1.1 Generating the Initial Guess ............................................................. 93
4.1.2 Optimization using CAMTOS ......................................................... 94
4.2 Improved Mission Profile with Two Coast Arcs ............................................ 98
4.2.1 Generating the Initial Guess ............................................................. 99
4.2.2 Optimization using CAMTOS ......................................................... 99
4.3 Summary ........................................................................................................ 103
5 Example 2: Branched Mission Scenario 109
5.1 Mission Scenario ............................................................................................ 110
5.2 Initial Guess Generation for the Hopper Mission .......................................... 113
5.3 Optimization of the Branched Mission Scenario ........................................... 115
5.4 Optimization Results ...................................................................................... 117
6 Conclusions and Future Research 123
6.1 Conclusions .................................................................................................... 123
6.2 Future Research .............................................................................................. 125
References 127
A Integration Methods for Direct Multiple Shooting 135
A.1 Runge-Kutta 2/3 Method ................................................................................ 135
A.2 Runge-Kutta 4/5 Method ................................................................................ 137
A.3 Integration Method of Paus ............................................................................ 138
B Derivation of Important Derivatives 143
B.1 Derivatives of Inclination w.r.t. Radius and Velocity Vectors ...................... 143
B.2 Derivatives of Orbital Momentum w.r.t. Radius and Velocity Vectors ......... 144
B.3 Derivatives of Semimajor Axis w.r.t. Radius and Velocity Vectors .............. 145
B.4 Derivatives of Orbital Energy w.r.t. Radius and Velocity Vectors ................ 145
