A simplified model of RRT coverage for kinematic systems

It has been shown that the Rapidly Exploring Random Tree algorithm is complete - both probabilistically and in the sense of resolution; however little analysis exists on the rate of convergence. We present a model of state space coverage as a function of the number of nodes in the tree, for holonomic systems in expansive configuration spaces. Based on two simplifying assumptions, we develop a stochastic difference equation, whose expected value exponentially converges to one as the number of nodes increases. The convergence rate is related through a closed form expression to the step size and a Lipschitz constant. Using a grid-based coverage measurement, we present experimental evidence supporting the model across a range of dimensions, obstacle densities and parameter choices.