Signal Temporal Logic Motion Planning via Graphs of Convex Sets

2026-05-22

AI Summary

This work addresses continuous-time motion planning under Signal Temporal Logic (STL) specifications by proposing a novel approach that integrates timed automata with Graph of Convex Sets (GCS) optimization. The method translates STL specifications into compact timed automata and constructs a joint transition system by combining these automata with a convex decomposition of the configuration space, thereby reformulating the original problem as a shortest-path problem in a GCS framework. This enables the generation of smooth Bézier spline trajectories that satisfy both high-level temporal constraints and low-level dynamical limits. To the best of our knowledge, this is the first integration of timed automata with the GCS framework, supporting expressive fragments of STL while guaranteeing polynomial scalability for fixed automata and decompositions. The approach demonstrates efficiency and practicality across diverse benchmarks, including low-dimensional systems, a 3D quadrotor, a 30-degree-of-freedom humanoid robot, and a UR-3 manipulator.

Abstract

This paper investigates continuous-time motion planning under Signal Temporal Logic (STL) specifications. The goal is to generate smooth robot trajectories that satisfy high-level logical and timing requirements while respecting low-level motion constraints. To this end, we propose an efficient framework that combines timed-automata reasoning with graphs of convex sets (GCS). An STL specification is first represented by a timed automaton, which is then coupled with a convex decomposition of the configuration space to form a joint transition system encoding both task progress and region occupancy. Based on this joint transition system, the STL motion-planning problem is reformulated as a shortest-path problem over a GCS, whose solution induces a smooth Bézier-spline trajectory satisfying the STL specification, smoothness requirements, and velocity bounds. We establish the soundness of the proposed formulation and analyze its computational complexity, showing that, once the timed automaton and convex decomposition are fixed, the convex relaxation scales polynomially with the configuration-space dimension and the Bézier degree. We further develop a compact timed-automaton construction for an expressive STL fragment using dedicated templates and Boolean composition. Numerical experiments on low-dimensional benchmarks, a $3$-D quadrotor, a $30$-DoF humanoid, and a hardware experiment on a UR-3 robot arm demonstrate that the proposed method efficiently solves complex STL motion-planning problems and produces smooth executable trajectories.

Problem

• Signal Temporal Logic
• motion planning
• smooth trajectories
• timed automata
• robotics

Innovation

• Signal Temporal Logic
• Graphs of Convex Sets
• Timed Automata
• Bezier-spline Trajectory
• Motion Planning