Optimal Probabilistic Motion Planning with Partially Infeasible LTL Constraints

This paper studies optimal probabilistic motion planning of a mobile agent in an uncertain environment where pre-specified tasks might not be fully realized. The agent's motion is modeled by a probabilistic labeled Markov decision process (MDP). A relaxed product MDP is developed, which allows the agent to revise its motion plan to not strictly follow the desired LTL constraints whenever the task is found to be infeasible. To evaluate the revised motion plan, a utility function composed of violation and implementation cost is developed, where the violation cost function is designed to quantify the differences between the revised and the desired motion plan, and the implementation cost are designed to bias the selection towards cost-efficient plans. Based on the developed utility function, a multi-objective optimization problem is formulated to jointly consider the implementation cost, the violation cost, and the satisfaction probability of tasks. Cost optimization in both prefix and suffix of the agent trajectory is then solved via coupled linear programs. Simulation results are provided to demonstrate its effectiveness.