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

TL;DR

This paper introduces a scalable method combining sum-of-squares optimization and dynamic programming to solve reach-avoid problems with guarantees, applicable to polynomial systems and validated through numerical examples.

Contribution

It presents a conservative, computationally scalable approach for reach-avoid problems using sum-of-squares and dynamic programming, applicable to polynomial dynamics.

Findings

Validated on two single integrators with comparison to Hamilton-Jacobi reachability

Demonstrated scalability on a system of two kinematic cars

Provided theoretical guarantees for reach-avoid solutions

Abstract

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.