aRCIZ Local Distributed Algorithms for Multi-Robot Systems Alejandro Cornejo

173 Pages·2013·12.95 MB·English

-aRCIZ
Local Distributed Algorithms
for Multi-Robot Systems
by
Alejandro Cornejo
Submitted to the Department of Electrical Engineering and
Computer Science
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2013
@ Massachusetts Institute of Technology 2013. All rights reserved.
A u th or ..............................................................
Department of Electrical Engineering and Computer Science
September 28, 2012
Certified by I....*1. .....
Nancy Lynch
Professor of Electrical Engineering and Computer Science
Thesis Supervisor
A ccepted by ................. . ..................
Leslie Kolodziejski
Chairman, Department Committee on Graduate Students
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Local Distributed Algorithms
for Multi-Robot Systems
by
Alejandro Cornejo
Submitted to the Department of Electrical Engineering and Computer Science
on September 28, 2012, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
The field of swarm robotics focuses on controlling large populations of simple robots
to accomplish tasks more effectively than what is possible using a single robot. This
thesis develops distributed algorithms tailored for multi-robot systems with large
populations. Specifically we focus on local distributed algorithms since their perfor-
mance depends primarily on local parameters on the system and are guaranteed to
scale with the number of robots in the system.
The first part of this thesis considers and solves the problem of finding a trajectory
for each robot which is guaranteed to preserve the connectivity of the communication
graph, and when feasible it also guarantees the robots advanced towards a goal
defined by an arbitrary motion planner. We also describe how to extend our proposed
approach to preserve the k-connectivity of a communication graph. Finally, we show
how our connectivity-preserving algorithm can be combined with standard averaging
procedures to yield a provably correct flocking algorithm.
The second part of this thesis considers and solves the problem of having each
robot localize an arbitrary subset of robots in a multi-robot system relying only on
sensors at each robot that measure the angle, relative to the orientation of each robot,
towards neighboring robots in the communication graph. We propose a distributed
localization algorithm that computes the relative orientations and relative positions,
up to scale, of an arbitrary subset of robots. For the case when the robots move in
between rounds we show how to use odometry information to allow each robot to
compute the relative positions complete with scale, of an arbitrary subset of robots.
Finally we describe how to use the our localization algorithm to design a variety of
multi-robot tasks.
Thesis Supervisor: Nancy Lynch
Title: Professor of Electrical Engineering and Computer Science
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Acknowledgments
First I would like to thank my advisor Nancy Lynch, for her support, advice, and
encouragement, not only during the writing of this thesis, but overall during my
time at MIT. From Nancy I have learned how to think as a researcher and how
to communicate ideas with clarity and precision. I would also like to thank Erik
Demaine, Leslie Kaelbling and Daniela Rus for sitting on my thesis committee and
providing valuable feedback on this work.
A lot of the work on this thesis was a result of different collaborations with Fabian
Kuhn, James McLurkin and Majid Khabbazian, whom I would like to thank as well.
It has been my pleasure to work with you.
I also thank Prasant Gopal and Mohsen Ghaffari, my current office mates, for
making my time at the office more pleasant and for being an endless source of con-
versations and discussions, both work and mostly non-work related.
Finally, I would like to dedicate this thesis to my wife, Patty, who has endured
with me this entire process and made me feel like its all worth it, and to my parents,
who have always supported me and encouraged me to pursue my dreams.
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Contents
List of Figures 9
List of Algorithms 14
1 Introduction 17
1.1 Connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.1.1 k-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.1.2 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.1 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3 Additional Related Work . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.1 Distributed Computing. . . . . . . . . . . . . . . . . . . . . . 30
1.3.2 Multi-Robot Systems . . . . . . . . . . . . . . . . . . . . . . . 31
2 Model 33
2.1 Graph Theory Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Geometry Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Modeling a Multi-Robot System . . . . . . . . . . . . . . . . . . . . . 36
I Connectivity 39
3 Selecting Edges 43
3.1 The Edge Selection Problem . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Sparse Connectivity-Preserving Sets of Edges . . . . . . . . . . . . . 45
3.2.1 Gabriel graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.2 Relative Neighbor graph . . . . . . . . . . . . . . . . . . . . . 47
3.2.3 Cone-Based Topology Control graph . . . . . . . . . . . . . . 48
3.2.4 Local Minimum Spanning graph . . . . . . . . . . . . . . . . . 50
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3.2.5 Local Minimum Spanning Graphs With Few Edges .5 53
3.3 Optimal Local Minimum Spanning Graphs ..... . . . . . . 55
4 Distributed Connectivity-Preserving Algorithm 57
4.1 The Connectivity-Preserving Problem . . . . . . . . 59
4.2 The Connectivity-Preserving Algorithm . . . . . . . 62
4.3 Safety . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Progress . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.1 Unconditional Progress . . . . . . . . . . . . 67
4.4.2 Robust Progress . . . . . . . . . . . . . . . . 68
4.4.3 Weak Progress . . . . . . . . . . . . . . . . 69
4.5 Multi-Round Executions . . . . . . . . . . . . . . . 92
5 Preserving a k-Connected Graph 97
5.1 From 1-Connected to k-Connected . . . . . . . . . 98
5.2 Preserving a k-Connected Graph . . . . . . . . . . 99
6 Applications of Connectivity 101
6.1 What is flocking? . . . . . . . . . . . . . . . . . . . 101
6.2 Alignment and Agreement . . . . . . . . . . . . . . 102
6.3 Flocking Algorithm . . . . . . . . . . . . . . . . . . 105
II Localization 109
7 Relative Orientations 113
7.1 Defining Angle-Constraints . . . . . . . . . . . . . . 114
7.2 Computing an Angle-Constraint . . . . . . . . . . . 116
8 Relative Positions 119
8.1 Defining Unique Realizations . . . . . . . . . . . . . 120
8.2 Satisfying Realizations . . . . . . . . . . . . . . . . 123
8.2.1 Basic Facts . . . . . . . . . . . . . . . . . . 124
8.2.2 Trees . . . . . . . . . . . . . . . . . . . . . . 125
8.2.3 Facts about Cycles . . . . . . . . . . . . . . 127
8.2.4 Graphs with Cycles . . . . . . . . . . . . . . 128
8.2.5 Computing Satisfying Realizations . . . . . 130
8.3 Unique Subset Realizations . . . . . . . . . . . . . 134
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9 Distributed Localization Algorithm 137
9.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
9.2 A lgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.3 Correctness and Optimality . . . . . . . . . . . . . . . . . . . . . . . 142
9.4 Recovering Scale Through Odometry . . . . . . . . . . . . . . . . . . 144
10 Applications of Localization 151
10.1 Localization for Static Applications . . . . . . . . . . . . . . . . . . . 152
10.2 Localization for Motion Control Applications . . . . . . . . . . . . . . 154
11 Conclusion 157
11.1 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
11.1.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . 157
11.1.2 Future Work and Open Questions . . . . . . . . . . . . . . . . 158
11.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
11.2.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . 160
11.2.2 Future Work and Open Questions . . . . . . . . . . . . . . . . 160
11.3 Other Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Index 163
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