Efficient Probabilistic Planning with Maximum-Coverage Distributionally Robust Backward Reachable Trees

TL;DR

This work addresses robust multi-query motion planning for linear Gaussian systems by introducing ball-shaped and ellipsoidal ambiguity sets to capture distributional uncertainty. It develops distributionally robust backward belief roadmaps (BRTs), notably MAX-COV-BALL and MAX-ELL-BALL, that maximize the size of ambiguity sets and demonstrate provable coverage guarantees alongside practical performance. Theoretical results show maximal coverage in noise-free settings and dominance over prior methods under mild conditions, while empirical tests on a double-integrator with obstacles and a 2D quadrotor validate improved roadmap connectivity and robustness. Overall, the paper advances scalable, provably robust multi-query planning by tightening uncertainty representations and leveraging convex relaxations and mixed-integer strategies to guarantee or improve coverage.

Abstract

This paper presents a new multi-query motion planning algorithm for linear Gaussian systems with the goal of reaching a Euclidean ball with high probability. We develop a new formulation for ball-shaped ambiguity sets of Gaussian distributions and leverage it to develop a distributionally robust belief roadmap construction algorithm. This algorithm synthe- sizes robust controllers which are certified to be safe for maximal size ball-shaped ambiguity sets of Gaussian distributions. Our algorithm achieves better coverage than the maximal coverage algorithm for planning over Gaussian distributions [1], and we identify mild conditions under which our algorithm achieves strictly better coverage. For the special case of no process noise or state constraints, we formally prove that our algorithm achieves maximal coverage. In addition, we present a second multi-query motion planning algorithm for linear Gaussian systems with the goal of reaching a region parameterized by the Minkowski sum of an ellipsoid and a Euclidean ball with high probability. This algorithm plans over ellipsoidal sets of maximal size ball-shaped ambiguity sets of Gaussian distributions, and provably achieves equal or better coverage than the best-known algorithm for planning over ellipsoidal ambiguity sets of Gaussian distributions [2]. We demonstrate the efficacy of both methods in a wide range of conditions via extensive simulation experiments.

Figures (3)

• Figure 1: a) MAXELLIPSOID aggarwal2025tac decouples first- and second-order moment uncertainty, requiring the mean of each distribution to fall inside the ellipsoid $\mathcal{E}$. $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ represent $(1-\epsilon)$ percentile covariance contours for distributions that satisfy the goal constraint. b) When the goal region $\mathcal{G}$ is parameterized by the Minkowski sum of $\mathcal{E}$ and a Euclidean ball, some distributions where the mean falls outside $\mathcal{E}$ can still satisfy the goal constraint. $\mathcal{D}$ represents the $(1-\epsilon)$ percentile covariance contour for one such valid distribution.
• Figure 2: Obstacles and goal region used in the double integrator planning experiment plotted against 99th percentile covariance contours for a backward reachable tree generated with MAX-ELL-BALL (left) and a backward reachable tree generated with MAX-COV-BALL (right).
• Figure 3: The color of each cell corresponds to the increase in coverage achieved by MAX-COV-BALL relative to MAXCOVAR, as a fraction of the coverage achieved by MAX-COV-BALL. The number in each cell corresponds to the fraction of initial distributions covered by MAX-COV-BALL (out of 250).

Theorems & Definitions (14)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Definition 4.1
• Definition 4.2
• Theorem 4.1
• Lemma 4.1
• Lemma 4.2
• Theorem 4.2