Robust Stochastic Shortest-Path Planning via Risk-Sensitive Incremental Sampling

TL;DR

This work proposes an alternative risk-aware approach inspired by the asymptoticallyoptimal Rapidy-Exploring Random Trees (RRT*) planning algorithm, which selects nodes along path segments with minimal Conditional Value-at-Risk (CVaR) based on the step-wise coherence of the CVaR risk measure and the optimal substructure of the SSP problem.

Abstract

With the pervasiveness of Stochastic Shortest-Path (SSP) problems in high-risk industries, such as last-mile autonomous delivery and supply chain management, robust planning algorithms are crucial for ensuring successful task completion while mitigating hazardous outcomes. Mainstream chance-constrained incremental sampling techniques for solving SSP problems tend to be overly conservative and typically do not consider the likelihood of undesirable tail events. We propose an alternative risk-aware approach inspired by the asymptotically-optimal Rapidly-Exploring Random Trees (RRT*) planning algorithm, which selects nodes along path segments with minimal Conditional Value-at-Risk (CVaR). Our motivation rests on the step-wise coherence of the CVaR risk measure and the optimal substructure of the SSP problem. Thus, optimizing with respect to the CVaR at each sampling iteration necessarily leads to an optimal path in the limit of the sample size. We validate our approach via numerical path planning experiments in a two-dimensional grid world with obstacles and stochastic path-segment lengths. Our simulation results show that incorporating risk into the tree growth process yields paths with lengths that are significantly less sensitive to variations in the noise parameter, or equivalently, paths that are more robust to environmental uncertainty. Algorithmic analyses reveal similar query time and memory space complexity to the baseline RRT* procedure, with only a marginal increase in processing time. This increase is offset by significantly lower noise sensitivity and reduced planner failure rates.

Figures (3)

• Figure 1: Planning environment: An illustration of the grid-world environment capturing important objects used throughout this article. Tree edges are depicted in green color and path segments in red color, while the gray-filled circle denotes an obstacle. A magnified inset highlights the grid's resolution along the abscissa and ordinate axes.
• Figure 2: Visualizing the planned paths: Example shortest paths returned by the RRT* (top) and RA-RRT* (bottom) algorithms for a fixed value of risk-sensitivity ($\alpha = 0.9$) parameter and increasing stochasticity ($\sigma_{C_k}$, left-to-right). Here, we see that, as the noise parameter is increased, the RA-RRT* planner's performance degrades gracefully with increasing uncertainty (evidenced by the slowly-increasing optimal path length, $L^\star$), while the RRT* planner returns infeasible paths (of greater lengths than the RA-RRT*) connecting configurations on a disconnected tree (see the magnified insets).
• Figure 3: Mean and variance of the shortest path lengths obtained for different $\alpha$ and $\sigma_{C_k}$ values. The values of the mean path length and variance for the RA-RRT* algorithm ( red $\circ$) are either equal or better (lower) than that of the RRT* ( blue $\lozenge$).

Theorems & Definitions (7)

• Definition 1: Free Space Representation & SSP Problem Particulars
• Definition 2: Path
• Definition 3: Worst-Case Path Length
• Lemma 1
• proof
• Proposition 1: Probabilistic Guarantee on $L_{\mathrm{worst}}^\star$
• proof