Revisiting the Asymptotic Optimality of RRT^*

RRT* is one of the most widely used sampling-based algorithms for asymptotically-optimal motion planning. This algorithm laid the foundations for optimality in motion planning as a whole, and inspired the development of numerous new algorithms in the field, many of which build upon RRT* itself. In this paper, we first identify a logical gap in the optimality proof of RRT*, which was developed in Karaman and Frazzoli (2011). Then, we present an alternative and mathematically-rigorous proof for asymptotic optimality. Our proof suggests that the connection radius used by RRT* should be increased from γ left(fraclog nnright)1/d to γ' left(fraclog nnright)1/(d+1) in order to account for the additional dimension of time that dictates the samples' ordering. Here γ, γ', are constants, and n, d, are the number of samples and the dimension of the problem, respectively.