SDP Synthesis of Distributionally Robust Backward Reachable Trees for Probabilistic Planning

TL;DR

This paper introduces MAXELLIPSOID BRT, a novel multi-query probabilistic planning algorithm that constructs a roadmap of ambiguity sets of distributions, enabling efficient and robust backward reachability analysis under stochastic uncertainties.

Contribution

It proposes a framework for building a maximal ambiguity set backward reachable tree using convex relaxations, improving planning efficiency over existing methods.

Findings

Efficient planning with smaller roadmap sizes.

Construction of maximal ambiguity sets via convex semidefinite relaxation.

Formal proof of maximum coverage for the proposed technique.

Abstract

The paper presents Maximal Ellipsoid Backward Reachable Trees MAXELLIPSOID BRT, which is a multi-query algorithm for planning of dynamic systems under stochastic motion uncertainty and constraints on the control input. In contrast to existing probabilistic planning methods that grow a roadmap of distributions, our proposed method introduces a framework to construct a roadmap of ambiguity sets of distributions such that each edge in our proposed roadmap provides a feasible control sequence for a family of distributions at once leading to efficient multi-query planning. Specifically, we construct a backward reachable tree of maximal size ambiguity sets and the corresponding distributionally robust edge controllers. Experiments show that the computation of these sets of distributions, in a backwards fashion from the goal, leads to efficient planning at a fraction of the size of the roadmap required for state-of-the-art methods. The computation of these maximal ambiguity sets and edges is carried out via a convex semidefinite relaxation to a novel nonlinear program. We also formally prove a theorem on maximum coverage for a technique proposed in our prior work.