


The rapidlyexploring random tree and its many variants have become an essential component of randomized motion planning algorithm. This efficiency is achieved in part by settling for any feasible path that is obtained, regardless of the final quality. In this research, a new samplingbased optimal motion planning algorithm, which operates in the vector field, is proposed.
First, a new criterion for path quality is defined. We called it the upstream criterion, that measures the extent to which the path goes upstream against the optimal vector field. The criterion, expressed as a path integral, is invariant with respect to the parametrization of the path (e.g., arclength, time), and also defined continuously with respect to the path as long as the vector field is continuous.
A new motion planning algorithm retains the efficiency characteristic of RRT algorithms, and at the same time randomly samples nodes such that, with only minimal additional overhead, attempts to minimize the upstream criterion. The main distinguish feature of it is that trees tend to extend toward vector field direction. By using inversion method, it can find the optimal path in a short time.
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