Closed-Loop Nominal and Abort Atmospheric Ascent - SMARTech

193 Pages·2005·1.49 MB·English

Closed-Loop Nominal and Abort Atmospheric Ascent Guidance for Rocket-Powered
Launch Vehicles

A Dissertation
By
Greg A. Dukeman

Submitted to
School of Aerospace Engineering

In Partial Fulfillment of the
Requirements for the Degree
Doctor of Philosophy in Aerospace Engineering

Georgia Institute of Technology

May 2005
Closed-Loop Nominal and Abort Atmospheric Ascent Guidance for
Rocket-Powered Launch Vehicles

Approved by:

Dr. Anthony J. Calise,Advisor Dr. Aldo Ferri
School of Aerospace Engineering School of Mechanical Engineering
Georgia Institute of Technology Georgia Institute of Technology

Dr. Dewey H. Hodges Dr. John M. Hanson
School of Aerospace Engineering NASA Marshall Flight Center
Georgia Institute of Technology

Dr. Panagiotas Tsiotras
School of Aerospace Engineering Date Approved: January 11, 2005
Georgia Institute of Technology
ACKNOWLEDGEMENT

Many people contributed to the research and results reported in this thesis. My colleagues in
the Guidance, Navigation and Control Systems Group at the NASA Marshall Space Flight Center
contributed in substantial ways. Dr. John Hanson has been a constant source of encouragement,
advice and motivation to me for many years. Thanks to my friend Jim McCarter for developing
one of the country’s most capable trajectory simulations and helping with simulation questions.
A sincere thank you goes to my thesis advisor, Dr. Anthony Calise, for his assistance and
patience and for demanding nothing less than high-quality work. Thanks to each of my Final
Committee members who collectively provided a very thorough review of the thesis: Dr. Dewey
H. Hodges and Dr. Panagiotis Tsiotras of the School of Aerospace Engineering, Dr. Aldo Ferri of
the School of Mechanical Engineering, and Dr. John M. Hanson of the NASA Marshall Space
Flight Center. Thanks to NASA and the Marshall Space Flight Center for providing so many
great opportunities for continuing engineering education. Much of this research was supported in
particular by the Next Generation Launch Technology project at NASA-Marshall – I express my
gratitude.
I would like to thank my wife Margaret for her love and support over the years and for
tolerating many late evenings. Reducing my hours somewhat sounds like a winning proposition
to me. Thanks to my kids, Anton and Alex, just for being the beautiful human beings that you
are. The world is a better place with you in it. Thanks to my parents, Bud and Eileen (deceased)
for instilling in me a love of learning.
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TABLE OF CONTENTS
Acknowledgement.............................................................................................................iii

List of Tables.....................................................................................................................vi
List of Figures...................................................................................................................vii
Nomenclature.....................................................................................................................xi
Summary..........................................................................................................................xiv
1. Introduction..................................................................................................................1
1.1 Background..........................................................................................................1
1.2 Previous Research................................................................................................4
1.3 Main Contributions..............................................................................................7

2. Atmospheric Ascent Guidance Formulation..............................................................11
2.1 Atmospheric Equations of Motion.....................................................................12
2.2 Optimization Problem........................................................................................15
2.3 Candidate Solution Approaches.........................................................................20
2.4 Necessary Conditions of Optimality..................................................................21
2.4.1 Costate Differential Equations...................................................................21
2.4.2 Optimality Condition.................................................................................23
2.4.3 Transversality Conditions..........................................................................29
2.5 Numerical Solution Methods.............................................................................30
3. Vacuum Guidance Formulation.................................................................................33
4. Single-Stage-To-Orbit Vehicle..................................................................................41
5. Transversality Conditions..........................................................................................56
5.1 Analytical Treatment.........................................................................................57
5.2 Numerical Treatment.........................................................................................64
6. Free Final Time Multi-Arc Optimization Formulation..............................................67
6.1 General Development........................................................................................69
6.2 Multiple Shooting Method Applied to Multi-Arc Optimization.......................73
6.3 Numerical Results..............................................................................................76
7. Multi-Arc Closed Loop Guidance.............................................................................83
7.1 Closed-Loop Guidance Logic for Burn-Coast-Burn Ascent.............................84
7.2 Examination of Various Terminal State Constraints.........................................85
8. Abort Trajectory Generation......................................................................................96
8.1 General Development........................................................................................98
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8.1.1 Construction of Target Position and Velocity Vectors............................100
8.1.2 Determination of ∆V Required.................................................................103
8.1.3 Feasible Trajectory Search.......................................................................106
8.1.4 Justification for Cubic Position................................................................109
8.2 Open-Loop Trajectories...................................................................................111
9. Closed-Loop Abort Guidance..................................................................................122
9.1 General Development......................................................................................122
9.2 Choice of Independent Variable......................................................................125
9.3 Guided Trajectory Results...............................................................................126
10. Conclusions and Future Research........................................................................143
Appendix A: Miscellaneous Guidance Relations......................................................148
Appendix B: Constraint on Final Hamiltonian is Ignorable......................................150
Appendix C: Conditions Under Which H (t) is Guaranteed To Vanish.................154
NT f
Appendix D: Partial Derivatives Expressions...........................................................156
Appendix E: Vehicle Properties................................................................................159
Appendix F: Treatment Of Gimballed Engine Vehicles...........................................171
References........................................................................................................................175

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LIST OF TABLES
7-1: Results of Perigee-Raise Burn Using Various Constraint Sets.................................91
E-1: X-33 Composite Mass Properties vs. Propellant Remaining..................................160
E-2: X-33 Axial Force Coefficient vs. Mach, α.............................................................161
E-3: X-33 Normal Force Coefficient vs. Alpha, Mach...................................................163
E-4: Generic RLV 1st Stage Composite Mass Properties vs. Propellant Remaining......166
E-5: Generic RLV 2nd Stage Composite Mass Properties vs. Propellant Remaining.....166
E-6: Generic RLV Lift Coefficient vs. Mach, α.............................................................167
E-7: Generic RLV Drag Coefficient vs. Mach, α...........................................................169

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LIST OF FIGURES
2-1: Guidance Reference Frame.......................................................................................15
2-2: Angles Pertaining To Optimality Condition.............................................................28
2-3: Second Variation Test: Hamiltonian vs. δ................................................................28
4-1: Hamiltonian vs. Time................................................................................................44
4-2: Altitude vs Speed, POST/MAVERIC Comparison, Sub-orbital Mission.................45
4-3: Dynamic Pressure vs Speed, POST/MAVERIC Comparison, Sub-orbital Mission.45
4-4: qα, qβ vs Speed, POST/MAVERIC Comparison, Sub-orbital Mission...................46
4-5: α, β vs Speed, POST/MAVERIC Comparison, Sub-orbital Mission.......................46
4-6: Ground Track, POST/MAVERIC Comparison, Sub-orbital Mission.......................47
4-7: Mass vs Time, POST/MAVERIC Comparison, Sub-orbital Mission.......................47
4-8: Euler Angles vs Speed, POST/MAVERIC Comparison, Sub-orbital Mission.........48
4-9: Altitude vs Speed, POST/MAVERIC Comparison, Circular Insertion....................48
4-10: Dynamic Pressure vs Speed, POST/MAVERIC Comparison, Circular Insertion..49
4-11: qα, qβ vs Speed, POST/MAVERIC Comparison, Circular Insertion.....................49
4-12: α, β vs Speed, POST/MAVERIC Comparison, Circular Insertion.........................50
4-13: Ground Track, POST/MAVERIC Comparison, Circular Insertion........................50
4-14: Vehicle Mass, POST/MAVERIC Comparison, Circular Insertion.........................51
4-15: Euler Angles, POST/MAVERIC Comparison, Circular Insertion.........................51
4-16: Altitude vs Speed, POST/MAVERIC Comparison, 51.6 deg mission....................52
4-17: Dynamic Pressure vs Speed, POST/MAVERIC Comparison, 51.6 deg mission...52
4-18: qα, qβ vs Speed, POST/MAVERIC Comparison, 51.6 deg Mission.....................53
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4-19: α, β vs Speed, POST/MAVERIC Comparison, 51.6 deg Mission.........................53
4-20: Ground Track, POST/MAVERIC Comparison, 51.6 deg Mission.........................54
4-21: Mass vs Time, POST/MAVERIC Comparison, 51.6 deg Mission.........................54
4-22: Euler Angles vs Speed, POST/MAVERIC Comparison, 51.6 deg Mission...........55
6-1: Base-10 Logarithm of the Absolute Value of Boundary Condition Residuals.........79
6-2: Speed vs Time on Optimized Burn-Coast-Burn Trajectory......................................79
6-3: Altitude vs Time on Optimized Burn-Coast-Burn Trajectory...................................80
6-4: Apogee/Perigee vs Time on Optimized Burn-Coast-Burn Trajectory......................80
6-5: Flight Path Angle vs Time on Optimized Burn-Coast-Burn Trajectory...................81
6-6: Vehicle Mass vs Time on Optimized Burn-Coast-Burn Trajectory.........................81
6-7: Switching Function vs Time on Optimized Burn-Coast-Burn Trajectory................82
6-8: Switching Function vs Time on Optimized Burn-Coast-Burn Trajectory................82
7-1: Number of Iterations Per Guidance Cycle vs Time..................................................93
7-1: Angle of Attack vs Time During Perigee-Raise Burn...............................................94
7-3: Angle of Sideslip vs Time During Perigee-Raise Burn............................................94
7-4: Vehicle Angular Velocity Magnitude During Perigee-Raise Burn...........................95
8-1: Entry altitude vs range-to-go on a typical entry trajectory......................................113
8-2: Entry speed vs range-to-go on a typical entry trajectory........................................113
8-3: ∆V Required vs range-to-go for RTLS Trajectories................................................114
8-4: Altitude vs time on Open-loop RTLS Trajectories.................................................114
8-5: Speed vs time on Open-loop RTLS Trajectories.....................................................115
8-6: Ground Tracks on Open-loop RTLS Trajectories...................................................115
8-7: Down range vs time on Open-loop RTLS Trajectories...........................................116
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8-8: Mass vs time on Open-loop RTLS Trajectories......................................................116
8-9: Thrust vs time on Open-loop RTLS Trajectories....................................................117
8-10: Dynamic pressure vs time on Open-loop RTLS Trajectories...............................117
8-11: Altitude vs time on Open-loop ATDS Trajectories...............................................118
8-12: Speed vs time on Open-loop ATDS Trajectories..................................................118
8-13: Ground Tracks on Open-loop ATDS Trajectories................................................119
8-14: Down range vs time on Open-loop ATDS Trajectories........................................119
8-15: Mass vs time on Open-loop ATDS Trajectories...................................................120
8-16: Thrust vs time on Open-loop ATDS Trajectories.................................................120
8-17: Dynamic pressure vs time on Open-loop ATDS Trajectories...............................121
9-1: Altitude vs time on Guided RTLS Trajectories.......................................................130
9-2: Speed vs time on Guided RTLS Trajectories..........................................................130
9-3: Down range vs time on Guided RTLS Trajectories................................................131
9-4: Mass vs time on RTLS Guided Trajectories...........................................................131
9-5: Thrust vs time on Guided RTLS Trajectories.........................................................132
9-6: Dynamic pressure vs time on Guided RTLS Trajectories.......................................132
9-7: qα, qβ vs time on Guided RTLS Trajectories.........................................................133
9-8: α, β vs time on Guided RTLS Trajectories.............................................................133
9-9: Time-to-go vs time on Guided RTLS Trajectories.................................................134
9-10: Heading error vs time on Guided RTLS Trajectories...........................................134
9-11: Altitude error vs time on Guided RTLS Trajectories............................................135
9-12: Speed error vs time on Guided RTLS Trajectories...............................................135
9-13: Altitude vs time on Guided ATDS Trajectories....................................................136
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9-14: Speed vs time on Guided ATDS Trajectories.......................................................136
9-15: Down range vs time on Guided ATDS Trajectories.............................................137
9-16: Mass vs time on Guided ATDS Trajectories.........................................................137
9-17: Thrust vs time on Guided ATDS Trajectories.......................................................138
9-18: Dynamic pressure vs time on Guided ATDS Trajectories....................................138
9-19: qα, qβ vs time on Guided ATDS Trajectories......................................................139
9-20: α vs time on Guided ATDS Trajectories...............................................................139
9-21: β vs time on Guided ATDS Trajectories...............................................................140
9-22: Time-to-go vs time on Guided ATDS Trajectories...............................................140
9-23: Heading error vs time on Guided ATDS Trajectories...........................................141
9-24: Altitude Error vs time on Guided ATDS Trajectories..........................................141
9-25: Speed error vs time on Guided ATDS Trajectories..............................................142
F-1: Typical First Stage Pitch Gimbal Angle.................................................................174

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