2. Rapidly Exploring Random Trees Data structure/algorithm to facilitate path planning Developed by Steven M. La Valle (1998) Originally designed to handle problems with nonholonomic constraints and high degrees of freedom

3. Rapidly Exploring Random Trees Algorithm: BUILD_RRT(q init ) 1 T.init(q init ); 2 for k=1 to K do 3 q rand ← RANDOM_CONFIG(); 4EXTEND(T, q rand ); 5 Return T;

4. Rapidly Exploring Random Trees EXTEND(T,q) 1 q near ← Nearest_NEIGHBOUR(q,T); 2 if NEW_CONFIG(q, q near, q new ) then 3 T.add_vertex(q new ); 4T.add_edge(q near, q new ); 5if q new = q then 6Return Reached; 7else 8Return Advanced; 9 Return Trapped;

5. Rapidly Exploring Random Trees

6. Main advantage: biased towards unexplored regions Probability node selected for extension proportional to size voronoi region

7. Rapidly Exploring Random Trees Nice properties: Expansion heavily biased towards unexplored areas of state space Distribution nodes approaches the sampling distribution (important for consistency), usually uniform but not necessarily! Relatively simple algorithm Always remains connected

8. RRT-Connect Path Planner How to use RRT in a path planner? RRT-Connect: Single shot method Grow two RRTs one from the start position and one from the goal position After every extension try to connect the trees If the trees connect a path has been found

9. RRT-Connect Path Planner Algorithm: RRT_CONNECT_PLANNER(q init, q goal ) 1 T a.init(q init ); T b.init(q goal ); 2 for k=1 to K do 3 q rand ← RND_CFG(); 4if not(EXTEND(T a, q rand )= Trapped) then 5if(CONNECT(T b, q new ) = Reached) then 6Return PATH(T a, T b ); 7 SWAP(T a, T b ); 8 Return Failure;

10. RRT-Connect Path Planner CONNECT(T, q) 1 repeat 2S ← EXTEND(T, q) 3 until not (S = Advanced) 4 Return S;

11. RRT-Connect Path Planner

12. Variations: RRT_EXTEND_EXTEND RRT_CONNECT_CONNECT In CONNECT step only add last vertex to graph Grow more RRTs

13. RRT-Connect Path Planner Analysis: RRT-Connect is probabilistic complete Distribution vertices in RRT converges toward sampling distribution No theoretical characterization of the rate of converge!

14. RRT-Connect Path Planner Experiments: Average 100 trials Scene 1: 0.228s Scene 2: 5,94s Scene 3: 2.92s Using 3D non incremental collision checking algorithm

15. RRT-Connect Path Planner Experiments: In uncluttered scenes connect heuristic 3 to 4 times faster than other RRT-based variants Useful for complicated 3D scenes 7-DOF kinematic chain, over 8000 triangle primitives average 2 s for each motion to reach, grasp and move

16. RRT-Connect Path Planner Experiments: over 13.000 triangles 80 s SGI Indigo2 15 s high-end SGI

17. RRT-Connect Path Planner Conclusion: Randomised approach that yields good experimental performances with no parameter tuning no pre-processing simple and consistent behaviour balance between greedy searching and uniform exploration well suited for incremental distance computation and fast nearest neighbour algorithms

18. RRT-Connect Path Planner To do: optimise distance travelled during each step by using the radius of a collision free ball use approximate nearest neighbour methods use incremental collision detection algorithm compare performance to other path planning approaches identify conditions that lead to poor performance

19. RRT-Connect Path Planner Issues: Randomness can cause great variance in runtime due to ‘unlucky instance’

20. RRT-Connect Path Planner Issues: Ugly paths

21. References RRT-connect: An efficient approach to single-query path planning. J. J. Kuffner and S. M. LaValle. In Proceedings IEEE International Conference on Robotics and Automation, pages 995--1001, 2000 On Heavy-tailed Runtimes and Restarts in Rapidly-exploring Random Trees. Nathan A. Wedge and Michael S. Branicky. 2008 Chapter 5: Sampling-Based Motion Planning, Planning Algorithms. S. M. LaValle. Cambridge University Press, Cambridge, U.K., Rapidly-exploring random trees: A new tool for path planning. S. M. LaValle. TR 98-11, Computer Science Dept., Iowa State University, October 1998 2006. http://msl.cs.uiuc.edu/rrt/gallery.html Rapidly-Exploring Random Trees in Highly Constrained Environments. Amjad Almahairi. 2010.