Reinforcement learning with timed constraints for robotics motion planning

Abstract: Robotic systems operating in dynamic and uncertain environments increasingly require planners that satisfy complex task sequences while adhering to strict temporal constraints. Metric Interval Temporal Logic (MITL) offers a formal and expressive framework for specifying such time-bounded requirements; however, integrating MITL with reinforcement learning (RL) remains challenging due to stochastic dynamics and partial observability. This paper presents a unified automata-based RL framework for synthesizing policies in both Markov Decision Processes (MDPs) and Partially Observable Markov Decision Processes (POMDPs) under MITL specifications. MITL formulas are translated into Timed Limit-Deterministic Generalized Büchi Automata (Timed-LDGBA) and synchronized with the underlying decision process to construct product timed models suitable for Q-learning. A simple yet expressive reward structure enforces temporal correctness while allowing additional performance objectives. The approach is validated in three simulation studies: a $5 \times 5$ grid-world formulated as an MDP, a $10 \times 10$ grid-world formulated as a POMDP, and an office-like service-robot scenario. Results demonstrate that the proposed framework consistently learns policies that satisfy strict time-bounded requirements under stochastic transitions, scales to larger state spaces, and remains effective in partially observable environments, highlighting its potential for reliable robotic planning in time-critical and uncertain settings.