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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

References

1.Simeon, T., Yershova, A., Jaillet, L. and LaValle, S. M., “Dynamic-Domain RRTs: Efficient Exploration by Controlling the Sampling Domain,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Barcelona, Spain (2005) pp. 3856–3861.Google Scholar
2.Amato, N. M. and Wu, Y., “A Randomized Roadmap Method for Path and Manipulation Planning,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Minneapolis, MN, USA (May 1996) pp. 113–120.CrossRefGoogle Scholar
3.Bohlin, R. and Kavraki, L. E., “Path Planning Using Lazy PRM,” In: Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, CA (2000) pp. 521–528.Google Scholar
4.Branicky, M. S., LaValle, S. M., Olson, K. and Yang, L., “Quasi-Randomized Path Planning,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Seoul, Korea (May 2001) pp. 1481–1487.Google Scholar
5.Burns, B. and Brock, O., “Single-Query Motion Planning with Utility-Guided Random Trees,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Rome, Italy (2007) pp. 3307–3312.Google Scholar
6.Canny, J. F., The Complexity of Robot Motion Planning (MIT Press, Cambridge, Massachusetts, 1988).Google Scholar
7.Cheng, P. and LaValle, S. M., “Resolution complete rapidly-exploring random trees,” In: Proceedings IEEE International Conference on Robotics and Automation, Washington DC, USA (2002) pp. 267–272.Google Scholar
8.Frazzoli, E., Dahleh, M. A. and Feron, E., “Real-time motion planning for agile autonomus vehicles,” J. Guid. Control Dyn. 25 (1), 116–129 (Jan–Feb, 2002).CrossRefGoogle Scholar
9.Hall, P., Introduction to the Theory of Coverage Processes (Wiley, New York, 1988).Google Scholar
10.Halton, J. H., “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,” Numer. Math. 2 (1), 84–90 (1960).CrossRefGoogle Scholar
11.Hammersley, J. M., “Monte-Carlo methods for solving multivariable problems,” Ann. New York Acad. Sci. 86, 844–874 (1960).CrossRefGoogle Scholar
12.Hsu, D., “Randomized Single-Query Motion Planning in Expansive Spaces,” Ph.D. thesis (Stanford University, California, 2000).Google Scholar
13.Hsu, D., Kavraki, L. E., Latombe, J. C., Motwani, R. and Sorkin, S., “On Finding Narrow Passages with Probabalistic Roadmap Planners,” In: Proceedings of the 3rd Workshop on the Algorithmic Foundations of Robotics (WAFR), Houston, TX (1998) pp. 141–153.Google Scholar
14.Hsu, D., Latombe, J.-C. and Kurniawati, H., “On the probabilistic foundations of probabilistic roadmap planning,” International Syposium on Robotics Research, San Francisco, CA, USA (2005).Google Scholar
15.Karaman, S. and Frazzoli, E., “Sampling-based algorithms for optimal motion planning,” Int. J. Robot. Res. 30 (7), 846–894 (June 2011).CrossRefGoogle Scholar
16.Kavraki, L. E., Kolountzakis, M. N. and Latombe, J. C., “Analysis of probabilistic roadmaps for path planning,” IEEE Trans. Robot. Autom. 14 (1), 166–171 (1998).CrossRefGoogle Scholar
17.Kavraki, L. E., Svestka, P., Latombe, J. C. and Overmars, M. H., “Probabilistic roadmaps for path planning in high dimensional configuration space,” IEEE Trans. Robot. Autom. 12, 566–580 (1996).CrossRefGoogle Scholar
18.Kim, J., Keller, J. and Kumar, V., “Design and Verification of Controllers for Airships,” Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Las Vegas, NV, USA (Oct. 2003).Google Scholar
19.Kim, J. W., Esposito, J. M. and Kumar, V., “RRT enhancements,” Int. J. Robot. Res. 15 (12), 1257–1272 (2006).CrossRefGoogle Scholar
20.Ladd, A. and Kavraki, L. E., “Measure theoretic analysis of probabilistic path planning,” IEEE Trans. Robot. 20 (2), 229–242 (2004).CrossRefGoogle Scholar
21.Ladd, A. M. and Kavraki, L. E., “Fast Tree-Based Exploration of State Space for Robots with Dynamics,” In: Proceedings of the Workshop on the Algorithmic Foundations of Robotics, Utrecht, Netherlands (2005) pp. 297–312.Google Scholar
22.LaValle, S. M., Branicky, M. S. and Lindemann, S. R., “On the relationship between classical grid search and probabilistic roadmaps,” Int. J. Robot. Res. 23, 673–692 (2004).CrossRefGoogle Scholar
23.LaValle, S. M., Planning Algorithms (Cambridge University Press, Cambridge, UK, 2006).CrossRefGoogle Scholar
24.LaValle, S. M. and Kuffner, J. J., “Randomized kinodynamic planning,” Int. J. Robot. Res. 20 (5), 378–400 (2001).CrossRefGoogle Scholar
25.LaValle, S. M. and Kuffner, J. J., “Rapidly exploring random trees: Progress and prospects,” In: Algorithmic and Computational Robotics: New Directions (Donald, B., Lynch, K. and Rus, D., eds.) (A.K. Peters, Wellesley, MA, 2001) pp. 293–308.Google Scholar
26.Lindemann, S. R. and LaValle, S. M., “Incrementally Reducing Dispersion by Increasing Voronoi Bias in RRTs,” In: Proceedings of the IEEE International Conference on Robotics and Automation, New Orleans, LA, USA (2004) pp. 3251–3257.Google Scholar
27.von Hundelshausen, F., Himmelsbach, M., Hecker, F., Mueller, A. and Wuensche, H., “Driving with tentacles: Integral structures for sensing and motion,” J. Field Robot. 29 (9), 640–665 (2008).CrossRefGoogle Scholar
28.Wedge, N. A. and Branicky, M. S., “On Heavy-Tailed Runtimes and Restarts in Rapidly Exploring Random Trees,” Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence, Chicago, Illinois (2008).Google Scholar