Fast Funnel Computation Using Multivariate Bernstein Polynomial

Abstract

Guaranteeing safety of robotic systems is essential for a reliable operation. A funnel, the set of states that can be reached from the initial set of states in the presence of disturbances, can be used as a means of safety verification. This letter presents a tractable algorithm for computing the funnel of nonlinear systems that are polynomial of state and disturbance. Based on the Hamilton-Jacobi reachability analysis, we propose a polynomial approximation of the value function of the Hamilton-Jacobi-Bellman equation. We conservatively approximate the value function so that the funnel certainly contains the forward reachable set of the system. By using the property of Bernstein polynomials, we show that the conservativeness condition can be expressed as linear inequality constraints. Consequently, we compute the funnel by iteratively solving linear programs. The simulation results validate the computational efficiency of the algorithm.