Reach-avoid problems with time-varying dynamics, targets and constraints

By Jaime Fernández Fisac, Mo Chen, Claire Jennifer Tomlin, S. Sastry. International Conference on Hybrid Systems: Computation and Control, 2014

295 citations recorded by the source corpus.

We consider a reach-avoid differential game, in which one of the players aims to steer the system into a target set without violating a set of state constraints, while the other player tries to prevent the first from succeeding; the system dynamics, target set, and state constraints may all be time-varying. The analysis of this problem plays an important role in collision avoidance, motion planning and aircraft control, among other applications. Previous methods for computing the guaranteed winning initial conditions and strategies for each player have either required augmenting the state vector to include time, or have been limited to problems with either no state constraints or entirely static targets, constraints and dynamics. To incorporate time-varying dynamics, targets and constraints without the need for state augmentation, we propose a modified Hamilton-Jacobi-Isaacs equation in the form of a double-obstacle variational inequality, and prove that the zero sublevel set of its viscosity solution characterizes the capture basin for the target under the state constraints. Through this formulation, our method can compute the capture basin and winning strategies for time-varying games at virtually no additional computational cost relative to the time-invariant case. We provide an implementation of this method based on well-known numerical schemes and show its convergence through a simple example; we include a second example in which our method substantially outperforms the state augmentation approach.

Subfields: Aerospace Engineering, Computational Theory and Mathematics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/286e3bc65c2bd4a59ec6b93d9bf42778fa1e87dc

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Citations in the corpus

Hamilton-Jacobi reachability: A brief overview and recent advances (2017, methodology): “time-dependent constraints within the reachability framework [39], [51].”
A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems (2017, background): “[37], [38]; an alternative formulation involves a modified partial differential equation [18].”