STL-SVPIO: Signal Temporal Logic guided Stein Variational Path Integral Optimization

Abstract

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.

Figures (9)

• Figure 1: Overview of the STL-SVPIO Optimization Algorithm.
• Figure 2: Selected trajectories of each method through iterations.
• Figure 3: Aggregated average robustness and satisfaction rate of methods. All algorithms besides MILP were tested across 100 different random sampling seeds. Each solution with positive robustness counts as a satisfaction. Multiagent Button corresponds to the scenario in Section \ref{['sec:exp_button']}, Multiagent Corridor corresponds to the scenario in Section \ref{['sec:exp_corridor']}, Multiagent Sync Goals corresponds to the scenario in Section \ref{['sec:exp_sync']}, and Long Horizon corresponds to the scenario in Section \ref{['sec:exp_long_horizon']}. Gurobi was given a 10-hour time limit to solve the MILP.
• Figure 4: Long horizon trajectory synthesized by STL-SVPIO through a cluttered obstacle field. Dark green and red markers indicate the start and final states, respectively.
• Figure 5: STL-SVPIO synthesizes a multi-agent trajectory that satisfies strict ordering constraints between task events.