Reach-Avoid Problems via Sum-of-Squares Optimization and Dynamic Programming

Reach-avoid problems involve driving a system to a set of desirable configurations while keeping it away from undesirable ones. Providing mathematical guarantees for such scenarios is challenging but have numerous potential practical applications. Due to the challenges, analysis of reach-avoid problems involves making trade-offs between generality of system dynamics, generality of problem setups, optimality of solutions, and computational complexity. In this paper, we combine sum-of-squares optimization and dynamic programming to address the reach-avoid problem, and provide a conservative solution that maintains reaching and avoidance guarantees. Our method is applicable to polynomial system dynamics and to general problem setups, and is more computationally scalable than previous related methods. Through a numerical example involving two single integrators, we validate our proposed theory and compare our method to Hamilton-Jacobi reachability. Having validated our theory, we demonstrate the computational scalability of our method by computing the reach-avoid set of a system involving two kinematic cars.Comment: International Conference on Intelligent Robots & Systems (IROS), 201

Task Assignment for Multiplayer Reach-Avoid Games in Convex Domains via Analytical Barriers

This work considers a multiplayer reach-avoid game between two adversarial teams in a general convex domain which consists of a target region and a play region. The evasion team, initially lying in the play region, aims to send as many its team members into the target region as possible, while the pursuit team with its team members initially distributed in both play region and target region, strives to prevent that by capturing the evaders. We aim at investigating a task assignment about the pursuer-evader matching, which can maximize the number of the evaders who can be captured before reaching the target region safely when both teams play optimally. To address this, two winning regions for a group of pursuers to intercept an evader are determined by constructing an analytical barrier which divides these two parts. Then, a task assignment to guarantee the most evaders intercepted is provided by solving a simplified 0-1 integer programming instead of a non-deterministic polynomial problem, easing the computation burden dramatically. It is worth noting that except the task assignment, the whole analysis is analytical. Finally, simulation results are also presented

Matching-Based Capture Strategies for 3D Heterogeneous Multiplayer Reach-Avoid Differential Games

This paper studies a 3D multiplayer reach-avoid differential game with a goal region and a play region. Multiple pursuers defend the goal region by consecutively capturing multiple evaders in the play region. The players have heterogeneous moving speeds and the pursuers have heterogeneous capture radii. Since this game is hard to analyze directly, we decompose the whole game as many subgames which involve multiple pursuers and only one evader. Then, these subgames are used as a building block for the pursuer-evader matching. First, for multiple pursuers and one evader, we introduce an evasion space (ES) method characterized by a potential function to construct a guaranteed pursuer winning strategy. Then, based on this strategy, we develop conditions to determine whether a pursuit team can guard the goal region against one evader. It is shown that in 3D, if a pursuit team is able to defend the goal region against an evader, then at most three pursuers in the team are necessarily needed. We also compute the value function of the Hamilton-Jacobi-Isaacs (HJI) equation for a special subgame of degree. To capture the maximum number of evaders in the open-loop sense, we formulate a maximum bipartite matching problem with conflict graph (MBMC). We show that the MBMC is NP-hard and design a polynomial-time constant-factor approximation algorithm to solve it. Finally, we propose a receding horizon strategy for the pursuit team where in each horizon an MBMC is solved and the strategies of the pursuers are given. We also extend our results to the case of a bounded convex play region where the evaders escape through an exit. Two numerical examples are provided to demonstrate the obtained results.Comment: 17 pages, 8 figure