Reward Machines for Signal Temporal Logic

We propose a Reinforcement Learning (RL) based control design framework for handling complex tasks. The approach extends the concept of Reward Machines (RM) with Signal Temporal Logic (STL) formulas that can be used for event generation. The use of STL allows not only a more efficient representation of rewards for complex tasks but also guiding the training process to converge towards behaviors satisfying specified requirements. We also propose an implementation of the framework that leverages the STL online monitoring algorithms. We illustrate the framework with three case studies (minigrid, cart-pole and high-way environments) with non-trivial tasks.

This paper addresses the control problem for robotic systems, including non-holonomic and underactuated platforms operating under unknown dynamics and strict actuator limits to satisfy complex high-level specifications. We denote these high-level specifications using Signal Temporal Logic (STL) and propose a novel time-aware Reinforcement Learning (RL) framework that leverages the geometric properties of Spatiotemporal Tubes (STTs). While traditional analytical STT controllers often struggle to enforce input constraints, and existing RL approaches rely on memory-intensive state history, our method natively overcomes both limitations. By mapping the logical and temporal complexities of the full class of STL into time-varying geometric boundaries, we directly constrain the multidimensional system state without relying on scalar robustness metrics. Augmenting the state space with time, we train a time-aware Soft Actor-Critic (SAC) agent using a continuous, geometry-aware reward function that eliminates the need to explicitly evaluate complex logical semantics during execution. The proposed framework offers a history-free, computationally efficient approach to learn continuous control policies that ensure robust satisfaction of specifications while strictly adhering to system input constraints.

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.