RRT$^η$: Sampling-based Motion Planning and Control from STL Specifications using Arithmetic-Geometric Mean Robustness

Signal Temporal Logic (STL) enables formal specification of complex spatiotemporal constraints for robotic task planning. However, synthesizing long-horizon continuous control trajectories from complex STL specifications is fundamentally challenging due to the nested structure of STL robustness objectives. Existing solver-based methods, such as Mixed-Integer Linear Programming (MILP), suffer from exponential scaling, whereas sampling methods, such as Model-Predictive Path Integral control (MPPI), struggle with sparse, long-horizon costs. We introduce Signal Temporal Logic guided Stein Variational Path Integral Optimization (STL-SVPIO), which reframes STL as a globally informative, differentiable reward-shaping mechanism. By leveraging Stein Variational Gradient Descent and differentiable physics engines, STL-SVPIO transports a mutually repulsive swarm of control particles toward high robustness regions. Our method transforms sparse logical satisfaction into tractable variational inference, mitigating the severe local minima traps of standard gradient-based methods. We demonstrate that STL-SVPIO significantly outperforms existing methods in both robustness and efficiency for traditional STL tasks. Moreover, it solves complex long-horizon tasks, including multi-agent coordination with synchronization and queuing while baselines either fail to discover feasible solutions, or become computationally intractable. Finally, we use STL-SVPIO in agile robotic motion planning tasks with nonlinear dynamics, such as 7-DoF manipulation and half cheetah back flips to show the generalizability of our algorithm.

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.

We address multi-robot motion planning under Signal Temporal Logic (STL) specifications with kinodynamic constraints. Exact approaches face scalability bottlenecks and limited adaptability, while conventional sampling-based methods require excessive samples to construct optimal trajectories. We propose a two-stage framework integrating sampling-based online learning with formal STL reasoning. At the single-robot level, our constrained Bayesian Optimization-based Tree search (cBOT) planner uses a Gaussian process as a surrogate model to learn local cost maps and feasibility constraints, generating shorter collision-free trajectories with fewer samples. At the multi-robot level, our STL-enhanced Kinodynamic Conflict-Based Search (STL-KCBS) algorithm incorporates STL monitoring into conflict detection and resolution, ensuring specification satisfaction while maintaining scalability and probabilistic completeness. Benchmarking demonstrates improved trajectory efficiency and safety over existing methods. Real-world experiments with autonomous surface vehicles validate robustness and practical applicability in uncertain environments. The STLcBOT Planner will be released as an open-source package, and videos of real-world and simulated experiments are available at https://stlbot.github.io/.