Anytime computation algorithms for approach-evasion differential games

This paper studies a class of approach-evasion differential games, in which one player aims to steer the state of a dynamic system to the given target set in minimum time, while avoiding some set of disallowed states, and the other player desires to achieve the opposite. We propose a class of novel anytime computation algorithms, analyze their convergence properties and verify their performance via a number of numerical simulations. Our algorithms significantly outperform the multi-grid method for the approach-evasion differential games both theoretically and numerically. Our technical approach leverages incremental sampling in robotic motion planning and viability theory.

[1]  Emilio Frazzoli,et al.  Sampling-based algorithms for optimal motion planning , 2011, Int. J. Robotics Res..

[2]  P. Saint-Pierre,et al.  Set-Valued Numerical Analysis for Optimal Control and Differential Games , 1999 .

[3]  Alexandre M. Bayen,et al.  A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games , 2005, IEEE Transactions on Automatic Control.

[4]  Maurizio Falcone,et al.  Numerical Methods for differential Games Based on Partial differential equations , 2006, IGTR.

[5]  S. Resnick.  A Probability Path , 1999 .

[6]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[7]  M. Falcone,et al.  Numerical Methods for Pursuit-Evasion Games via Viscosity Solutions , 1999 .

[8]  J. Sethian.  Level set methods : evolving interfaces in geometry, fluid mechanics, computer vision, and materials science , 1996 .

[9]  John Lygeros,et al.  On reachability and minimum cost optimal control , 2004, Autom..

[10]  Jonathan P. How,et al.  Real-Time Motion Planning With Applications to Autonomous Urban Driving , 2009, IEEE Transactions on Control Systems Technology.

[11]  A. I. Subbotin,et al.  Game-Theoretical Control Problems , 1987 .

[12]  Alexandre M. Bayen,et al.  Validating a Hamilton-Jacobi Approximation to Hybrid System Reachable Sets , 2001, HSCC.

[13]  John H. Reif,et al.  Complexity of the mover's problem and generalizations , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[14]  P. Souganidis.  Two-Player, Zero-Sum Differential Games and Viscosity Solutions , 1999 .

[15]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[16]  R. Elliott,et al.  The Existence Of Value In Differential Games , 1972 .

[17]  Jean-Pierre Aubin,et al.  Viability Theory: New Directions , 2011 .

[18]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[19]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[20]  J. Tsitsiklis,et al.  An optimal one-way multigrid algorithm for discrete-time stochastic control , 1991 .

[21]  S. LaValle,et al.  Randomized Kinodynamic Planning , 2001 .

[22]  Emilio Frazzoli,et al.  An incremental sampling-based algorithm for stochastic optimal control , 2012, 2012 IEEE International Conference on Robotics and Automation.