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Conditional Density Growth (CDG) model: a simplified model of RRT coverage for kinematic systems

Published online by Cambridge University Press:  25 January 2013

Joel M. Esposito
Joel M. Esposito*
Affiliation:
Department of Systems Engineering, United States Naval Academy, Annapolis, MD 21402, USA
*
*Corresponding author. E-mail: esposito@usna.edu
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Summary

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It has been shown before that the Rapidly Exploring Random Tree (RRT) algorithm is probabilistically and resolution complete; and that the probability of finding a particular solution path can be related to the number of nodes. However, little analysis exists on the rate at which the tree covers the configuration space. In this paper, we present a stochastic difference equation which models how the tree covers the configuration space as a function of the number of nodes in the tree. Using two simplifying assumptions, appropriate for holonomic, kinematic systems in expansive configuration spaces, we derive closed-form solutions for the expected value and variance of configuration space coverage, which only depend on two easily computable parameters. Using a grid-based coverage measurement, we present experimental evidence supporting this model across a range of dimensions, obstacle densities, and parameter choices. Collecting data from 1000 RRTs, we provide evidence that configuration space coverage concentrates tightly around the expected coverage predicted by the model; and the results of the Chi-squared test suggest that the distribution of coverage across these runs is highly Gaussian. Together these results enable one to predict the expected coverage, along with a confidence interval, after a certain number of nodes have been added to the tree. We also applied the model to an example with extremely narrow passages and to a system with non-holonomic kinematics. The expected value prediction is still qualitatively accurate; but the rate constant is reduced and the variance is higher. Overall, in addition to its theoretical value, the model may find future application as an online measure of search-progress and problem difficulty, useful for adaptive variants of the basic RRT algorithm.

Keywords

Path planning for manipulatorsMotion controlMotion planningRandomized methods
Type
Articles
Information
Robotica , Volume 31 , Issue 5 , August 2013 , pp. 733 - 746
Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States.
Copyright
Copyright © Cambridge University Press 2013

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