Lagrangian Approximations for Stochastic Reachability of a Target Tube

TL;DR

This paper introduces Lagrangian set-theoretic methods to efficiently compute stochastic reachability sets for nonlinear systems without convexity assumptions, demonstrated on various complex examples.

Contribution

It presents a scalable Lagrangian approach for stochastic reachability that avoids state space gridding, applicable to general nonlinear systems with conservative approximations.

Findings

Methods successfully applied to double-integrator and chain of integrators.

Extended to a 4D space vehicle rendezvous problem.

Achieved faster computation with increased scalability.

Abstract

In this paper we examine how Lagrangian techniques can be used to compute underapproximations and overapproximation of the finite-time horizon, stochastic reach-avoid level sets for discrete-time, nonlinear systems. This approach is applicable for a generic nonlinear system without any convexity assumptions on the safe and target sets. We examine and apply our methods on the reachability of a target tube problem, a more generalized version of the finite-time horizon reach-avoid problem. Because these methods utilize a Lagrangian (set theoretic) approach, we eliminate the necessity to grid the state, input, and disturbance spaces allowing for increased scalability and faster computation. The methods scalability are currently limited by the computational requirements for performing the necessary set operations by current computational geometry tools. The primary trade-off for this improved extensibility is conservative approximations of actual stochastic reach set. We demonstrate these methods on several examples including the standard double-integrator, a chain of integrators, and a 4-dimensional space vehicle rendezvous docking problem.