Revisiting the Asymptotic Optimality of RRT*

RRT* is one of the most widely used sampling-based algorithms for asymptotically-optimal motion planning. RRT* 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 by 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 $\gamma {\left( {\frac{{\log n}}{n}} \right)^{1/d}}$ to $\gamma ^{\prime}{\left( {\frac{{\log n}}{n}} \right)^{1/\left( {d + 1} \right)}}$ 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.