Feedback Motion Planning for Stochastic Nonlinear Systems with Signal Temporal Logic Specifications

2026-05-04

AI Summary

This work addresses the problem of feedback motion planning for continuous-time stochastic nonlinear systems under Signal Temporal Logic (STL) specifications by proposing a novel framework that integrates predicate erosion with probabilistic reachable tubes. Predicate erosion is employed to transform stochastic STL constraints into tightened deterministic ones, while probabilistic reachable tubes quantify the deviation of stochastic trajectories from their nominal counterparts. Leveraging contraction theory, a tracking controller is designed to establish a closed-loop planning pipeline. The proposed approach significantly reduces the conservatism inherent in conventional methods, achieving high STL satisfaction probability without compromising planning performance. Simulations and real-world experiments on a quadrupedal robot demonstrate that the method outperforms baseline approaches in both STL satisfaction rate and computational efficiency.

Abstract

We study feedback motion planning for continuous-time stochastic nonlinear systems under signal temporal logic (STL) specifications. We propose a framework that synthesizes control policies for chance-constrained STL trajectory optimization problems, with the goal of ensuring that the closed-loop stochastic system satisfies a given STL formula with high probability (e.g., 99.99\%). Our approach is based on a predicate erosion strategy that transforms the intractable stochastic problem into a deterministic STL trajectory optimization problem with tightened STL formula constraints. The amount of erosion is determined by a probabilistic reachable tube (PRT) that bounds the deviation between the stochastic trajectory and an associated nominal trajectory. To compute such bounds, we leverage contraction theory and feedback design, and develop several tracking controllers. This yields a complete feedback motion planning pipeline which can be implemented by numerical optimizations. We demonstrate the efficacy and versatility of the proposed framework through simulations on several robotic systems and through experiments on a real-world quadrupedal robot, and show that it is less conservative and achieves higher specification satisfaction probability than representative baselines.

Problem

• stochastic nonlinear systems
• signal temporal logic
• feedback motion planning
• chance-constrained optimization
• trajectory satisfaction

Innovation

• Signal Temporal Logic
• Stochastic Nonlinear Systems
• Predicate Erosion
• Probabilistic Reachable Tube
• Contraction Theory