Theoretical Planning Limits using Markov Chains


The performance of a state lattice motion planning algorithm depends critically on the resolution of the lattice to ensure a balance between solution quality and computation time. There is currently no theoretical basis for selecting the resolution because of its dependence on the robot dynamics and the distribution of obstacles. In this paper, we examine the problem of motion planning on a resolution constrained lattice for a robot with non-linear dynamics operating in an environment with randomly generated disc shaped obstacles sampled from a homogeneous Poisson process. We present a unified framework for computing explicit solutions to two problems – i) the critical planning resolution which guarantees the existence of an infinite collision free trajectory in the search graph ii) the critical speed limit which guarantees infinite collision free motion. In contrast to techniques used by Karaman and Frazzoli, we use a novel approach that maps the problem to parameters of directed asymmetric hexagonal lattice bond percolation. Since standard percolation theory offers no results for this lattice, we map the lattice to an infinite absorbing Markov chain and use results pertaining to its survival to obtain bounds on the parameters. As a result, we are able to derive theoretical expressions that relate the non-linear dynamics of a robot, the resolution of the search graph and the density of the Poisson process. We validate the theoretical bounds using Monte-Carlo simulations for single integrator and curvature constrained systems and are able to validate the previous results presented by Karaman and Frazzoli independently using the novel connections introduced in this paper.

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