Design of Trajectory Optimization Approach for Space Maneuver Vehicle Skip Entry Problems

216 Pages·2020·11.725 MB·English

by  Runqi Chai, Al Savvaris, Antonios Tsourdos, Senchun Chai

Springer Aerospace Technology
Runqi Chai
Al Savvaris
Antonios Tsourdos
Senchun Chai
Design of Trajectory
Optimization
Approach for Space
Maneuver Vehicle
Skip Entry Problems
Springer Aerospace Technology
The Springer Aerospace Technology series isdevoted tothe technology of aircraft
and spacecraft including design, construction, control and the science. The books
present the fundamentals and applications in all fields related to aerospace
engineering. The topics include aircraft, missiles, space vehicles, aircraft engines,
propulsion units and related subjects.
More information about this series at http://www.springer.com/series/8613
Runqi Chai Al Savvaris
(cid:129) (cid:129)
Antonios Tsourdos Senchun Chai
(cid:129)
Design of Trajectory
Optimization Approach
for Space Maneuver Vehicle
Skip Entry Problems
123
RunqiChai AlSavvaris
Cranfield University Cranfield University
Cranfield,Bedford, UK Cranfield,Bedford, UK
AntoniosTsourdos Senchun Chai
Cranfield University Schoolof Automation
Cranfield,Bedford, UK Beijing Institute of Technology
Beijing,China
ISSN 1869-1730 ISSN 1869-1749 (electronic)
SpringerAerospace Technology
ISBN978-981-13-9844-5 ISBN978-981-13-9845-2 (eBook)
https://doi.org/10.1007/978-981-13-9845-2
©SpringerNatureSingaporePteLtd.2020
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
Preface
Space vehicle trajectory planning has become increasingly important due to its
extensive applications in industry and military fields. A well-designed trajectory is
usuallyakeyforstableflightandforimprovedguidanceandcontrolofthevehicle.
Hence, the aim of this research is usually to work on trajectory optimization, and
then improve on one of the existing trajectory optimization methods in order to
circumvent the limitations brought by the classic techniques.
Thisbookpresentsthedesignofoptimaltrajectoryforspacemaneuvervehicles
using optimal control-based techniques. It starts from a broad introduction and
overviewtothreemainapproachestotrajectoryoptimization.Itthenfocusesonthe
design of a novel hybrid optimization strategy, which incorporates an initial guess
generatorwithanimprovedgradient-basedinneroptimizer.Further,ithighlightsthe
developmentofmulti-objectivespacecraft trajectory optimizationproblems,witha
particular focus on multi-objective transcription methods, and multi-objective evo-
lutionary algorithms. Finally, the spacecraft flight scenario with noise-perturbed
dynamics and probabilistic constraints is studied. New chance-constrained optimal
controlframeworksaredesignedandvalidated.Thecomprehensiveandsystematic
treatmentofpracticalissuesinspacecrafttrajectoryoptimizationisoneofthemajor
features of the book, which is particularly suited for readers who are interested to
learn practical solutions in spacecraft trajectory optimization. The book can also
benefit researchers, engineers, and graduate students in fields of GNC systems,
engineering optimization,applied optimal controltheory, etc.
Theauthorshavecarefullyreviewedthecontentofthisbookbeforetheprinting
stage. However, it does not mean that this book is completely free from any
possibleerrors.Consequently,theauthorswouldbegratefultoreaderswhowillcall
out attention on mistakes as they might discover.
Cranfield, UK Runqi Chai
Cranfield, UK Al Savvaris
Cranfield, UK Antonios Tsourdos
Beijing, China Senchun Chai
May 2019
v
Acknowledgements
TheauthorswouldliketoexpresstheirsincereappreciationstoProf.YuanqingXia,
Prof.GuopingLiu,andProf.PengShifortheirconstructivecommentsandhelpful
suggestions with regards to the theoretical part of this book.
The first author would like to thank other colleagues from center of
cyber-physics systems, Cranfield University, for providing valuable comments.
Without their support, the writing of the book would not have been a success.
Also, we would like to thank all the staff in the autonomous systems research
group,CranfieldUniversity,foreverythingtheyhavedonetomakethingseasierfor
us throughout the preparation of this work.
Finally, the authors would like to thank Cranfield University, School of
Aerospace, Transport, and Manufacturing, and Beijing Institute of Technology,
School of Automation, for giving us the support to make the work a reality.
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mission Scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Skip Reentry Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Regional Reconnaissance . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Book Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Chapter Layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Overview of Trajectory Optimization Techniques. . . . . . . . . . . . . . . 7
2.1 Spacecraft Trajectory Optimization Problems and Optimal
Control Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Optimization Techniques and Applications. . . . . . . . . . . . . . . . . . 9
2.2.1 Gradient-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Evolutionary-Based Methods . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Convexification-Based Methods . . . . . . . . . . . . . . . . . . . . 13
2.2.4 Dynamic Programming-Based Methods. . . . . . . . . . . . . . . 14
2.3 Multi-objective Trajectory Optimization Overview . . . . . . . . . . . . 14
2.3.1 Multi-objective Evolutionary Algorithms. . . . . . . . . . . . . . 15
2.3.2 Multi-objective Transcription Methods . . . . . . . . . . . . . . . 17
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Modeling of the Trajectory Optimization Problems . . . . . . . . . . . . . 27
3.1 Mathematical Formulation of the Problem . . . . . . . . . . . . . . . . . . 27
3.1.1 Continuous Dynamical Systems . . . . . . . . . . . . . . . . . . . . 28
3.1.2 Variable/Path Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.3 Mission Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.4 Overall Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.5 Numerical Solution Approach. . . . . . . . . . . . . . . . . . . . . . 30
ix
x Contents
3.2 SMV Trajectory Optimization Formulation . . . . . . . . . . . . . . . . . 31
3.2.1 Dynamic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 SMV Initial and Terminal Constraints. . . . . . . . . . . . . . . . 34
3.2.3 Box Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.4 Trajectory Event Sequence. . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.5 Interior-Point Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.6 Skip Entry Path Constraints . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.7 Objective Functions for the Skip Entry Problem . . . . . . . . 36
3.2.8 Overall Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Discretization of the SMV Skip Entry Problem . . . . . . . . . . . . . . 37
3.3.1 Pseudospectral Discretization . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 Mesh Refinement Strategy . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Initial Simulation Results and Case Studies . . . . . . . . . . . . . . . . . 40
3.4.1 Parameters Specification. . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.2 Optimal Skip Hopping Results . . . . . . . . . . . . . . . . . . . . . 41
3.4.3 Analysis of Different Skip Hopping Scenarios. . . . . . . . . . 44
3.4.4 Sensitivity with Respect to Path Constraint . . . . . . . . . . . . 45
3.4.5 Finding Solution for n[4 Scenarios . . . . . . . . . . . . . . . . 48
3.4.6 Optimal Results for a Multiple Regional Reconnaissance
Mission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Performance Analysis of Different Optimization Strategies. . . . . . . . 55
4.1 General NLP Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Applying Gradient-Based Optimization Methods . . . . . . . . . . . . . 56
4.2.1 Sequential Quadratic Programming. . . . . . . . . . . . . . . . . . 56
4.2.2 Interior Point Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Heuristic Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Constraint-Handling Method. . . . . . . . . . . . . . . . . . . . . . . 60
4.3.2 Genetic Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.3 Differential Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.4 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . 62
4.3.5 Artificial Bee Colony. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 Problem Modification and Parameter Specification. . . . . . . 63
4.4.2 Combine Global Methods with Discretization Scheme. . . . 64
4.4.3 Optimal Solutions Obtained via Different Optimization
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Analysis of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5.1 Characteristic Arcs of the Trajectory. . . . . . . . . . . . . . . . . 66
4.5.2 Performance of Different Optimization Methods . . . . . . . . 68
Contents xi
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 Hybrid Optimization Methods with Enhanced Convergence
Ability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1 Initial Guess Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.1 Violation Learning Differential Evolution Algorithm . . . . . 74
5.2 Inner Optimization Solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.1 An Improved Gradient-Based Optimization Strategy . . . . . 77
5.2.2 Mesh Refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.3 Overall Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Solution Optimality Verification . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.1 First-Order Necessary Conditions . . . . . . . . . . . . . . . . . . . 81
5.3.2 Terminal Transversality Conditions. . . . . . . . . . . . . . . . . . 82
5.3.3 Hamiltonian Function Condition. . . . . . . . . . . . . . . . . . . . 83
5.3.4 Properties of the Control Variable. . . . . . . . . . . . . . . . . . . 84
5.3.5 Bellman’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Simulation Results for a Time-Optimal Entry Scenario . . . . . . . . . 85
5.4.1 Optimal Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.2 Verification of Optimality . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4.3 Comparison with Existing Evolutionary Solvers . . . . . . . . 90
5.4.4 Dispersion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4.5 Comparison Against Other Optimal Control Solvers . . . . . 94
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6 Multi-objective Trajectory Optimization Problem. . . . . . . . . . . . . . . 99
6.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.1.1 General Formulation of Multi-objective Optimization
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1.2 Multi-objective Optimal Control Problems . . . . . . . . . . . . 101
6.2 An Improved Multi-objective Evolutionary Algorithms. . . . . . . . . 101
6.2.1 Extended NSGA-II Algorithm . . . . . . . . . . . . . . . . . . . . . 101
6.2.2 Superiority of Feasible Solution Method . . . . . . . . . . . . . . 103
6.2.3 Penalty Function Based Method . . . . . . . . . . . . . . . . . . . . 105
6.2.4 Multi-objective Constraint-Handling Technique . . . . . . . . . 105
6.2.5 Computational Complexity Analysis . . . . . . . . . . . . . . . . . 106
6.3 Multi-objective Transcription Methods. . . . . . . . . . . . . . . . . . . . . 106
6.3.1 Fuzzy Physical Programming . . . . . . . . . . . . . . . . . . . . . . 107
6.3.2 Interactive Fuzzy Physical Programming. . . . . . . . . . . . . . 109
6.3.3 Fuzzy Goal Programming Method . . . . . . . . . . . . . . . . . . 111
6.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4.1 Multi-objective SMV Trajectory Planning. . . . . . . . . . . . . 114
6.4.2 Pareto Front Results Obtained Using MOEAs . . . . . . . . . . 115