Sampling Techniques for Probabilistic Roadmap Planners Roland Geraerts and Mark Overmars IAS-8 March 2004

Overview Probabilistic roadmaps Test scenes PRM choices Sampling Variation of running times Conclusions Important choices Collision checking Node adding Sampling Uniform Advanced

Probabilistic Roadmap Method Free space Construction (G =V,E ) Loop c  a free sample add c to the vertices V Nc  a set of nodes for all c’ in Nc in increasing distance if c’ and c are not connected in G then if local path between c and c’ exists then add the edge c’c to E Forbidden space Sample Local path Colliding path c c’ c c c’ c c c’

Probabilistic Roadmap Method Free space Construction (G =V,E ) Loop c  a free sample add c to the vertices V Nc  a set of nodes for all c’ in Nc in increasing distance if c’ and c are not connected in G then if local path between c and c’ exists then add the edge c’c to E Query connect sample s and g to roadmap Dijkstra’s shortest path Forbidden space Sample Local path Start / goal Shortest path

Test Scenes Comparison of techniques

Experimental Setup Environment Preprocessing method Single system SAMPLE Same computer Pentium IV 2.66GHz, 1GB memory Same test scenes Cage, clutter, hole, house, rooms, wrench Preprocessing method but single query Average time (s) over 30 runs

PRM choices Collision checking on local path Neighbor selecting Incremental Binary Rotate-at-s Neighbor selecting Nearest-k Component-k Visibility

Collision Checking Rotate-at-s Incremental Binary

Collision Checking – Results incremental binary rotate-at-s cage 2.4 1.9 2.7 clutter 1.8 1.3 3.1 hole 431.9 422.3 1206.7 house 6.4 4.9 47.1 rooms 0.5 0.4 1.1 wrench 0.9

Neighbor Selecting Nearest-k Component-k Component Visibility

Neighbor Strategy Nearest-k Component Component-k Visibility Nearest k neighbors Component Nearest 1 neighbor in each connected component Component-k Nearest k neighbors in each connected component Visibility Only useful nodes, add node/connection only if: Node cannot be connected to other nodes Node can be connected to 2 (or more) connected components

Neighbor Strategy – Results nearest-k comp. comp.-k visibility cage 1.9 3.4 1.6 3.0 clutter 1.3 1.4 2.3 hole 409.5 7428.2 7554.4 102.5 house 4.9 13.0 45.5 rooms 0.4 0.2 6.3 wrench 0.5

Uniform Sampling Random Grid

Uniform Sampling Cell Halton (variants)

Uniform Sampling – Results halton, seed: random grid cell cage 2.3 3.2 1.9 2.0 2.8 clutter 1.2 3.4 1.3 1.5 1.4 hole 433.9 370.4 422.3 201.4 279.5 house 78.7 4.9 10.4 10.0 rooms 0.7 0.8 0.4 wrench 0.5

Advanced Sampling Gaussian Bridge test

Advanced Sampling Obstacle based Nearest contact

Advanced Sampling Medial axis

Advanced Sampling – Results gaussian bridge obstacle NC MA rnd halton cage 7.3 3.1 5.3 5.4 215.9 2.0 clutter 2.8 2.6 7.1 620.4 1.5 hole 8.8 47.5 2.3 7.7 201.4 house 18.0 20.7 13.0 15.7 199.3 10.4 rooms 0.5 0.4 3.5 0.7 wrench 2.7 0.9 1.9 3.8 11.0

Variation of Running Times min avg max st dev without restart 0.03 1.01 60.81 2.32 with restart 0.05 0.88 8.20 1.00

Conclusions Many claims could not be confirmed Local planner Rotate-at-s is worse than binary collision checking Node adding Component-k seems to work best Visibility sampling didn’t perform as well as expected

Conclusions Sampling Small difference between uniform methods Halton didn’t outperform uniform sampling but may be preferred (deterministic) Advanced sampling is only useful in scenes with large open areas and some narrow passages It is not necessarily true that a combination of good techniques is good Reduction of variance is desirable

Questions Home page: www.cs.uu.nl/~roland Mail: roland@cs.uu.nl