# Improved Performance of Asymptotically Optimal Rapidly Exploring Random Trees

## Abstract

Three algorithms that improve the performance of the asymptotically optimal Rapidly exploring Random Tree (RRT*) are presented here. First, we introduce the Goal Tree (GT) algorithm for motion planning in dynamic environments where unexpected obstacles appear sporadically. The GT reuses the previous RRT* by pruning the affected area and then extending the tree by drawing samples from a shadow set. The shadow is the subset of the free configuration space containing all configurations that have geodesics ending at the goal and are in conflict with the new obstacle. Smaller, well defined, sampling regions are considered for Euclidean metric spaces and Dubins' vehicles. Next, the Focused-Refinement (FR) algorithm, which samples with some probability around the first path found by an RRT*, is defined. The third improvement is the Grandparent-Connection (GP) algorithm, which attempts to connect an added vertex directly to its grandparent vertex instead of parent. The GT and GP algorithms are both proven to be asymptotically optimal. Finally, the three algorithms are simulated and compared for a Euclidean metric robot, a Dubins' vehicle, and a seven degrees-of-freedom manipulator.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. of California, San Diego, CA (United States). Dept. of Mechanical Engineering

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1467259

- Report Number(s):
- [LA-UR-17-24020]

[Journal ID: ISSN 0022-0434]

- Grant/Contract Number:
- [AC52-06NA25396]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Dynamic Systems, Measurement, and Control

- Additional Journal Information:
- [ Journal Volume: 141; Journal Issue: 1]; Journal ID: ISSN 0022-0434

- Publisher:
- ASME

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 42 ENGINEERING; Motion Planning, Robotics

### Citation Formats

```
Boardman, Beth, Harden, Troy, and Martínez, Sonia. Improved Performance of Asymptotically Optimal Rapidly Exploring Random Trees. United States: N. p., 2018.
Web. doi:10.1115/1.4040970.
```

```
Boardman, Beth, Harden, Troy, & Martínez, Sonia. Improved Performance of Asymptotically Optimal Rapidly Exploring Random Trees. United States. doi:10.1115/1.4040970.
```

```
Boardman, Beth, Harden, Troy, and Martínez, Sonia. Mon .
"Improved Performance of Asymptotically Optimal Rapidly Exploring Random Trees". United States. doi:10.1115/1.4040970. https://www.osti.gov/servlets/purl/1467259.
```

```
@article{osti_1467259,
```

title = {Improved Performance of Asymptotically Optimal Rapidly Exploring Random Trees},

author = {Boardman, Beth and Harden, Troy and Martínez, Sonia},

abstractNote = {Three algorithms that improve the performance of the asymptotically optimal Rapidly exploring Random Tree (RRT*) are presented here. First, we introduce the Goal Tree (GT) algorithm for motion planning in dynamic environments where unexpected obstacles appear sporadically. The GT reuses the previous RRT* by pruning the affected area and then extending the tree by drawing samples from a shadow set. The shadow is the subset of the free configuration space containing all configurations that have geodesics ending at the goal and are in conflict with the new obstacle. Smaller, well defined, sampling regions are considered for Euclidean metric spaces and Dubins' vehicles. Next, the Focused-Refinement (FR) algorithm, which samples with some probability around the first path found by an RRT*, is defined. The third improvement is the Grandparent-Connection (GP) algorithm, which attempts to connect an added vertex directly to its grandparent vertex instead of parent. The GT and GP algorithms are both proven to be asymptotically optimal. Finally, the three algorithms are simulated and compared for a Euclidean metric robot, a Dubins' vehicle, and a seven degrees-of-freedom manipulator.},

doi = {10.1115/1.4040970},

journal = {Journal of Dynamic Systems, Measurement, and Control},

number = [1],

volume = [141],

place = {United States},

year = {2018},

month = {8}

}

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Probabilistic roadmaps for path planning in high-dimensional configuration spaces

journal, January 1996

- Kavraki, L. E.; Svestka, P.; Latombe, J. -C.
- IEEE Transactions on Robotics and Automation, Vol. 12, Issue 4

##
Motion planning using adaptive random walks

journal, February 2005

- Carpin, S.; Pillonetto, G.
- IEEE Transactions on Robotics, Vol. 21, Issue 1

##
Sampling-based algorithms for optimal motion planning

journal, June 2011

- Karaman, Sertac; Frazzoli, Emilio
- The International Journal of Robotics Research, Vol. 30, Issue 7

##
An elementary approach to generic properties of plane curves

journal, March 1984

- Bruce, J. W.; Giblin, P. J.
- Proceedings of the American Mathematical Society, Vol. 90, Issue 3

##
On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents

journal, July 1957

- Dubins, L. E.
- American Journal of Mathematics, Vol. 79, Issue 3

##
Anytime Motion Planning using the RRT*

conference, May 2011

- Karaman, Sertac; Walter, Matthew R.; Perez, Alejandro
- 2011 IEEE International Conference on Robotics and Automation (ICRA)

##
Use of relaxation methods in sampling-based algorithms for optimal motion planning

conference, May 2013

- Arslan, Oktay; Tsiotras, Panagiotis
- 2013 IEEE International Conference on Robotics and Automation (ICRA)

##
Sampling heuristics for optimal motion planning in high dimensions

conference, September 2011

- Akgun, Baris; Stilman, Mike
- 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2011)

##
RRT^{∗}-Smart: Rapid convergence implementation of RRT^{∗} towards optimal solution

conference, August 2012

- Islam, Fahad; Nasir, Jauwairia; Malik, Usman
- 2012 IEEE International Conference on Mechatronics and Automation (ICMA)

##
Replanning with RRTs

conference, January 2006

- Ferguson, D.; Kalra, N.; Stentz, A.
- Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006.

##
Multipartite RRTs for Rapid Replanning in Dynamic Environments

conference, April 2007

- Zucker, Matt; Kuffner, James; Branicky, Michael
- Proceedings 2007 IEEE International Conference on Robotics and Automation

##
Real-time randomized path planning for robot navigation

conference, January 2002

- Bruce, J.; Veloso, M.
- IROS 2002: IEEE/RSJ International Conference on Intelligent Robots and Systems

##
Lazy Reconfiguration Forest (LRF) - An Approach for Motion Planning with Multiple Tasks in Dynamic Environments

conference, April 2007

- Gayle, Russell; Klingler, Kristopher R.; Xavier, Patrick G.
- Proceedings 2007 IEEE International Conference on Robotics and Automation

##
RRT-connect: An efficient approach to single-query path planning

conference, January 2000

- Kuffner, J. J.; LaValle, S. M.
- 2000 ICRA. IEEE International Conference on Robotics and Automation, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)

##
Optimal kinodynamic motion planning in environments with unexpected obstacles

conference, September 2014

- Boardman, Beth; Harden, Troy; Martinez, Sonia
- 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)

##
Optimal kinodynamic motion planning using incremental sampling-based methods

conference, December 2010

- Karaman, Sertac; Frazzoli, Emilio
- 2010 49th IEEE Conference on Decision and Control (CDC)