The objective of this paper is to present a methodology for designing cooperative control laws for individual agents that guarantee collision avoidance in multiagent systems. The proposed avoidance control laws are easy to design and implement, and may be directly appended to the optimal control laws of the individual agents within the cooperation framework. The avoidance control laws are computed using value functions that resemble the behavior of barrier functions in the static optimization theory. The most attractive feature of the proposed optimization scheme is the fact that the avoidance laws are active only in the bounded sensing regions of each individual agent, and they do not interfere with the agents’ individual optimal control laws outside of these regions.

1.
Leitmann
,
G.
, and
Skowronski
,
J.
, 1977, “
Avoidance Control
,”
J. Optim. Theory Appl.
0022-3239,
23
, pp.
581
591
.
2.
Getz
,
W. M.
, and
Leitmann
,
G.
, 1979, “
Qualitative Differential Games With Two Targets
,”
J. Math. Anal. Appl.
0022-247X,
68
, pp.
421
430
.
3.
Leitmann
,
G.
, 1980, “
Guaranteed Avoidance Strategies
,”
J. Optim. Theory Appl.
0022-3239,
32
, pp.
569
576
.
4.
Leitmann
,
G.
, and
Skowronski
,
J.
, 1983, “
A Note on Avoidance Control
,”
Opt. Control Appl. Methods
0143-2087,
4
, pp.
335
342
.
5.
Corless
,
M.
,
Leitmann
,
G.
, and
Skowronski
,
J.
, 1987, “
Adaptive Control for Avoidance or Evasion in an Uncertain Environment
,”
Comput. Math. Appl.
0898-1221,
13
, pp.
1
11
.
6.
Corless
,
M.
, and
Leitmann
,
G.
, 1989, “
Adaptive Controllers for Avoidance or Evasion in an Uncertain Environment: Some Examples
,”
Comput. Math. Appl.
0898-1221,
18
, pp.
161
170
.
7.
Stipanović
,
D. M.
,
Sriram
,
S.
, and
Tomlin
,
C. J.
, 2005, “
Multi-Agent Avoidance Control Using an M-Matrix Property
,”
Electron. J. Linear Algebra
1081-3810,
12
, pp.
64
72
.
8.
Mitchell
,
I.
,
Bayen
,
A. M.
, and
Tomlin
,
C. J.
, 2005, “
A Time-Dependent Hamilton-Jacobi Formulation of Reachable Sets for Continuous Dynamic Games
,”
IEEE Trans. Autom. Control
0018-9286,
50
, pp.
947
957
.
9.
Tomlin
,
C.
,
Lygeros
,
J.
, and
Sastry
,
S.
, 2000, “
A Game Theoretic Approach to Controller Design for Hybrid Systems
,”
Proc. IEEE
0018-9219,
88
, pp.
949
970
.
10.
Osher
,
S.
, and
Sethian
,
J. A.
, 1988, “
Fronts Propagating With Curvature-Dependent Speed: Algorithms Based on Hamilton–Jacobi Formulations
,”
J. Comput. Phys.
0021-9991,
79
, pp.
12
49
.
11.
Sethian
,
J. A.
, 2002,
Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science
, reprinted 2nd ed.,
Cambridge University Press
,
Cambridge
.
12.
Crandall
,
M. G.
, and
Lions
,
P.-L.
, 1983, “
Viscosity Solutions of Hamilton-Jacobi Equations
,”
Trans. Am. Math. Soc.
0002-9947,
277
, pp.
1
42
.
13.
Evans
,
L. C.
, 1998,
Partial Differential Equations
,
Graduate Studies in Mathematics
Vol.
19
,
American Mathematical Society
,
Providence, RI
.
14.
Hwang
,
I.
,
Stipanović
,
D. M.
,
, and
Tomlin
,
C. J.
, 2005, “
Polytopic Approximations of Reachable Sets Applied to Linear Dynamic Games and to a Class of Nonlinear Systems
,”
Advances in Control, Communication Networks, and Transportation Systems: In Honor of Pravin Varaiya
,
Systems & Control: Foundations & Applications
,
E.
Abed
, ed.,
Birkhäuser
,
Boston, MA
, pp.
1
20
.
15.
Stipanović
,
D. M.
,
,
Hwang
,
I.
, and
Tomlin
,
C. J.
, 2004, “
Computation of an Over-Approximation of the Backward Reachable Set Using Subsystem Level Set Functions
,”
Dyn. Contin. Discrete Impulsive Syst.: Ser. A - Math. Anal.
1201-3390,
11
, pp.
399
411
.
16.
İnalhan
,
G.
,
Stipanović
,
D. M.
, and
Tomlin
,
C. J.
, 2002, “
Decentralized Optimization, With Application to Multiple Aircraft Coordination
,”
Proceedings of the 2002 IEEE Conference on Decision and Control
, Las Vegas, Nevada, pp.
1147
1155
.
17.
Kim
,
Y.
,
Mesbahi
,
M.
, and
Hadaegh
,
F. Y.
, 2004, “
Multiple-Spacecraft Reconfiguration Through Collision Avoidance, Bouncing, and Stalemate
,”
J. Optim. Theory Appl.
0022-3239,
122
, pp.
323
343
.
18.
Hu
,
J.
,
Prandini
,
M.
, and
Sastry
,
S.
, 2003, “
Optimal Coordinated Motions of Multiple Agents Moving on a Plane
,”
SIAM J. Control Optim.
0363-0129,
42
, pp.
637
668
.
19.
Koditschek
,
D. E.
, and
Rimon
,
E.
, 1990, “
Robot Navigation Functions on Manifolds With Boundary
,”
Adv. Appl. Math.
0196-8858,
11
, pp.
412
442
.
20.
Rimon
,
E.
, and
Koditschek
,
D. E.
, 1991, “
The Construction of Analytic Diffeomorphisms for Exact Robot Navigation on Star Worlds
,”
Trans. Am. Math. Soc.
0002-9947,
327
, pp.
71
116
.
21.
Rimon
,
E.
, and
Koditschek
,
D. E.
, 1992, “
Exact Robot Navigation Using Artificial Potential Functions
,”
IEEE Trans. Rob. Autom.
1042-296X,
8
, pp.
501
518
.
22.
Dimarogonas
,
D. V.
,
Loizou
,
S. G.
,
Kyriakopoulos
,
K. J.
, and
Zavlanos
,
M. M.
, 2006, “
A Feedback Stabilization and Collision Avoidance Scheme for Multiple Independent Non-Point Agents
,”
Automatica
0005-1098,
42
, pp.
229
243
.
23.
Chang
,
D. E.
,
Shadden
,
S.
,
Marsden
,
J.
, and
Olfati-Saber
,
R.
, 2003, “
Collision Avoidance for Multiple Agent Systems
,”
Proceedings of the 42nd IEEE Conference on Decision and Control
, Maui, Hawaii, pp.
539
543
.
24.
Olfati-Saber
,
R.
, 2006, “
Flocking for Multi-Agent Dynamic Systems: Algorithms and Theory
,”
IEEE Trans. Autom. Control
0018-9286,
51
, pp.
401
420
.
25.
Bertsekas
,
D. P.
, 2000,
Dynamic Programming and Optimal Control
, 2nd ed.,
Athena Scientific
,
Belmont, MA
, Vol.
1
.
26.
Luenberger
,
D. G.
, 2003,
Linear and Nonlinear Programming
, 2nd ed.
Kluwer Academic
,
Boston, MA
.
27.
Vorotnikov
,
V. I.
, 2005, “
Partial Stability and Control: The State-of-the-Art and Development Prospects
,”
Autom. Remote Control (Engl. Transl.)
0005-1179,
66
, pp.
511
561
.
28.
Matrosov
,
V. M.
, 1962, “
On the Theory of Stability of Motion
,”
Prikl. Mat. Mekh.
0032-8235,
26
, pp.
992
1000
.
29.
Bellman
,
R.
, 1962, “
Vector Lyapunov Functions
,”
SIAM J. Control
0036-1402,
1
, pp.
32
34
.
30.
Michel
,
A. N.
, and
Miller
,
R. K.
, 1977,
Qualitative Analysis of Large-Scale Dynamical Systems
,
Academic
,
New York, NY
.
31.
Šiljak
,
D. D.
, 1978,
Large-Scale Dynamic Systems: Stability and Structure
,
North-Holland
,
New York, NY
.
32.
Šiljak
,
D. D.
, 1991,
Decentralized Control of Complex Systems
,
Academic
,
Boston, MA
.
33.
Lakshmikantham
,
V.
,
Matrosov
,
V. M.
, and
Sivasundaram
,
S.
, 1991,
Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems
,
Kluwer
,
Dordrecht
.
34.
Khalil
,
H. K.
, 2002,
Nonlinear Systems
, 3rd ed.,
Prentice-Hall
,
Upper Saddle River, NJ
.
35.
Anderson
,
B. D. O.
, and
Moore
,
J. B.
, 1989,
Optimal Control: Linear Quadratic Methods
,
Prentice-Hall
,
Upper Saddle River, NJ
.
You do not currently have access to this content.