Convex Hull Asymptotic Shape Evolution

Abstract

Summary. The asymptotic properties of Rapidly exploring Random Tree (RRT) growth in large spaces is studied both in simulation and analysis. The main phenomenon is that the convex hull of the RRT reliably evolves into an equilateral triangle when grown in a symmetric planar region (a disk). To characterize this and related phenomena from flocking and swarming, a family of dynamical systems based on incremental evolution in the space of shapes is introduced. Basins of attraction over the shape space explain why the number of hull vertices tends to reduce and the shape stabilizes to a regular polygon.

Probabilistic Analysis of RRT Trees

“…Most of the space is still far away from the tree, so most steps give a large amount of progress. This is the phase most similar to the problem considered by Arnold et al [1].…”

“…We begin by studying how quickly the rrt spreads. Arnold et al [1] consider a modification to the rrt where the space is the plane R 2 , the step size is 1, and points are drawn at infinity by picking a ray from the origin with a uniformly random direction. They prove in this model that the perimeter of the convex hull of tree grows linearly in time which implies that the maximum distance from the tree to the origin also grows linearly in time.…”

“…While a number of such regions can be chosen, for ease of calculation we consider the case that we have an rrt on [0, 1] 2 and the region we want to reach is the right half of the square, 1 2 , 1 × [0, 1] when X 0 = (0, 0). Similar arguments extend to the general convex region.…”

“…In their analysis however, they commented that they had no result on the speed of this convergence. Further analysis has been done by Arnold et al [1] on the speed at which the rrt spreads out by studying an idealized version of the algorithm on an infinite disc. Work has also been done by Karaman and Frazzoli [9] on the complexity of the update step in the algorithm and the optimality of the path, while Dalibard and Laumond [3] have studied the effects of narrow passages on the performance of the algorithm.…”

A multi-directional rapidly exploring random graph (mRRG) for protein folding

“…T-RRT is efficient at finding a single pathway quickly but only expands a parent node in a single direction, namely towards q rand , causing it to focus on a small number of routes in the energy landscape funnel [3]. PRM is slower but covers the energy landscape better.…”