State-Continuity Approximation of Markov Decision Processes via Finite Element Analysis for Autonomous System Planning
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Motion planning under uncertainty for an autonomous system can be formulated as a Markov Decision Process. In this paper, we propose a solution to this decision theoretic planning problem using a continuous approximation of the underlying discrete value function and leveraging finite element methods. This approach allows us to obtain an accurate and continuous form of value function even with a small number of states from a very low resolution of state space. We achieve this by taking advantage of the second order Taylor expansion to approximate the value function, where the value function is modeled as a boundary-conditioned partial differential equation which can be naturally solved using a finite element method. We have validated our approach via extensive simulations, and the evaluations reveal that our solution provides continuous value functions, leading to better path results in terms of path smoothness, travel distance and time costs, even with a smaller state space.
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