Robust Trajectory Planning for Unmanned Aerial Vehicles in Uncertain Environments Brandon ...

153 Pages·2008·54.4 MB·English

Robust Trajectory Planning for Unmanned Aerial
Vehicles in Uncertain Environments
by
Brandon Luders
B.S., Aerospace Engineering
Georgia Institute of Technology (2006)
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
Master of Science in Aeronautics and Astronautics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2008
(cid:13)c Massachusetts Institute of Technology 2008. All rights reserved.
Author ..............................................................
Department of Aeronautics and Astronautics
August 28, 2008
Certified by..........................................................
Jonathan How
Professor
Thesis Supervisor
Accepted by.........................................................
Prof. David L. Darmofal
Associate Department Head
Chair, Committee on Graduate Students
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Robust Trajectory Planning for Unmanned Aerial
Vehicles in Uncertain Environments
by
Brandon Luders
Submitted to the Department of Aeronautics and Astronautics
on August 28, 2008, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics and Astronautics
Abstract
As unmanned aerial vehicles (UAVs) take on more prominent roles in aerial missions,
it becomes necessary to increase the level of autonomy available to them within the
mission planner. In order to complete realistic mission scenarios, the UAV must be
capable of operating within a complex environment, which may include obstacles and
other no-fly zones. Additionally, the UAV must be able to overcome environmental
uncertainties such as modeling errors, external disturbances, and an incomplete situa-
tional awareness. By utilizing planners which can autonomously navigate within such
environments, the cost-effectiveness of UAV missions can be dramatically improved.
This thesis develops a UAV trajectory planner to efficiently identify and execute
trajectories which are robust to a complex, uncertain environment. This planner,
namedEfficientRSBK,integratespreviousmixed-integerlinearprogramming(MILP)
pathplanningalgorithmswithseveralimplementationinnovationstoachieveprovably
robuston-linetrajectoryoptimization. Usingtheproposedinnovations, theplanneris
abletodesignintelligentlong-termplansusingaminimalnumberofdecisionvariables.
The effectiveness of this planner is demonstrated with both simulation results and
flight experiments on a quadrotor testbed.
Two major components of the Efficient RSBK framework are the robust model
predictive control (RMPC) scheme and the low-level planner. This thesis develops
a generalized framework to investigate RMPC affine feedback policies on the distur-
bance, identify relative strengths and weaknesses, and assess suitability for the UAV
trajectory planning problem. A simple example demonstrates that even with a con-
ventional problem setup, the closed-loop performance may not always improve with
additional decision variables, despite the resulting increase in computational com-
plexity. A compatible low-level troller is also introduced which significantly improves
trajectory-following accuracy, as demonstrated by additional flight experiments.
Thesis Supervisor: Jonathan How
Title: Professor
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Acknowledgments
First, I would like to thank Professor How for being a constant source of guidance
and wisdom in my research pursuits at MIT, leading to this thesis. Regardless of the
problem I encounter in my work, he always several handfuls of ideas to try. Professor
How has played a critical role in helping me identify my personal research interests
and apply them to projects such as this thesis.
This research has been funded by the Air Force Office of Scientific Research,
USAF, under grant FA9550-08-1-0086.
Muchoftheworkrelatedtorobustmodelpredictivecontrol(RMPC)inthisthesis
was inspired by the prior work and discussions of Professor Arthur Richards and Dr.
Yoshiaki Kuwata, both of whom have provided valuable insights. Through frequent
report revisions and e-mail exchanges, they (along with Professor How) have been
essential in refining the state of our current RMPC research.
The members of the Aerospace Controls Laboratory have been extremely support-
ive throughout my work on this thesis, whether through proofreading this document,
helping me reason through a problem, or simply offering helpful advice. I would like
to offer a special thanks to Brett Bethke and Josh Redding, who graciously offered to
read this document and suggest revisions. I would also like to thank Spencer Ahrens,
Luc Brunet, Cameron Fraser, and Frant Sobolic for the specific ways in which they
have assisted me. Above all, I would like to thank Kathryn Fischer for her support
and resourcefulness throughout my time here.
Finally, my family has been a constant source of love and inspiration for me
throughout my education. So, to my parents, Brian and Lori, and my brother Chris,
thank you for always being there for me and supporting my aspirations.
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Contents
1 Introduction 17
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.1 Robust Model Predictive Control . . . . . . . . . . . . . . . . 21
1.2.2 Mixed-Integer Linear Programming . . . . . . . . . . . . . . . 24
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.2 Success Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Affine Feedback Policies in Robust Model Predictive Control 29
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Nominal MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Disturbance RMPC Policies . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Constraint Tightening . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Affine Feedback Parametrization . . . . . . . . . . . . . . . . 38
2.3.3 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.1 Disturbance Free Cost . . . . . . . . . . . . . . . . . . . . . . 43
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2.4.2 Expected Cost . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.3 Worst Case Cost . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Analysis and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5.2 Terminal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5.3 Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5.4 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Robust Trajectory Planning using Mixed-Integer Linear Program-
ming 67
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 CT-Based Trajectory Planning Formulation . . . . . . . . . . . . . . 70
3.4 Robust Safe but Knowledgeable (RSBK) Algorithm . . . . . . . . . . 73
3.4.1 Trajectory Safety . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.2 Cost-to-Go Map . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5.1 Selective CT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5.2 Linear Interpolation Points . . . . . . . . . . . . . . . . . . . . 82
3.5.3 Detection Radius . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.4 Variable-Density Constraint Selection . . . . . . . . . . . . . . 83
3.6 Model-Specific Refinements . . . . . . . . . . . . . . . . . . . . . . . 88
3.6.1 Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6.2 Obstacle Expansion . . . . . . . . . . . . . . . . . . . . . . . . 91
3.6.3 Obstacle Reachable Horizon . . . . . . . . . . . . . . . . . . . 92
3.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
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4 Experimental Results 111
4.1 The Quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 RAVEN Testbed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2.1 User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2.2 Vehicle Manager . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.3 Low-Level Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.3.1 Termination Criteria . . . . . . . . . . . . . . . . . . . . . . . 118
4.3.2 Trajectory Generator . . . . . . . . . . . . . . . . . . . . . . . 118
4.3.3 Reference Controller . . . . . . . . . . . . . . . . . . . . . . . 120
4.4 Flight Results: Low-Level Controller . . . . . . . . . . . . . . . . . . 124
4.4.1 Stationary Hover . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.4.2 Single-Dimension Tests . . . . . . . . . . . . . . . . . . . . . . 126
4.4.3 Advanced Trajectories . . . . . . . . . . . . . . . . . . . . . . 130
4.5 Flight Results: Efficient RSBK . . . . . . . . . . . . . . . . . . . . . 134
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Conclusion 139
5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1.1 Disturbance-Aware Cost-to-go . . . . . . . . . . . . . . . . . . 141
5.1.2 Expansion of RMPC Analysis . . . . . . . . . . . . . . . . . . 141
5.1.3 Intelligent Selection of Cost-to-Go Nodes . . . . . . . . . . . . 142
5.1.4 Aerobatic Quadrotor Control . . . . . . . . . . . . . . . . . . 142
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