A General Framework for Sampling on the Medial Axis of the Free Space

1. A General Framework forSampling on the Medial Axis of the Free Space Jyh-Ming Lien, Shawna Thomas, Nancy Amato {neilien, sthomas,amato}@cs.tamu.edu

2. Obstacle based PRM [Amato, Bayazit, Dale, Jones, Vallejo.’98] • Gaussian PRM [Boor and Overmars.’99] • RBB PRM [Hsu, Jiang, Reif, Sun.’03] • Medial Axis based PRM (MAPRM)[Wilmarth, Amato, Stiller.’99] Narrow Passage s obstacle g Probabilistic Roadmaps and the Narrow Passage Problem • Probabilistic roadmap (PRM) [Kavraki, Svestka, Latombe, Overmars.’96]

3. p is collision-free p is in collision q = NearestContactCfg_Clearance(p) V = p- q q = NearestContactCfg_Penetration(p) V = q- p Retractpto the Medial Axis of the free C-space in direction V samples < N Connect sampled configurations Generalized MAPRM Framework Sample a Configuration, p

4. Generalized MAPRM Framework PRM with uniform sampling MAPRM

5. Sampling is increased in Narrow Corridors • In-collision configurations are retracted to free C-space • The volume of the narrow passage is increased Vol(S )+Vol(B’ ) Pro( Sampling in S ) = Vol(C )

6. Sample a Configuration, p p is collision-free p is in collision q = NearestContactCfg_Clearance(p) V = p- q q = NearestContactCfg_Penetration(p) V = q- p Retractpto the Medial Axis of the free C-space in direction V < N Connect sampled configurations The Limitation of MAPRM • Can only be applied to problems with low (<6) dimensional C-space of rigid objects.

7. MAPRM, MAPRM and MAPM • Clearance and Penetration depth: distance to the closest contact configuration. • Clearance and penetration depth computation • Exact methods • Approximate methods

8. clearance penetration MAPRM for Point Robot in 2D[Wilmarth, Amato, Stiller. ICRA’99] • Clearance and penetration depth • The closest point on the polygon boundary

9. MAPRM for a Rigid Body in 3D [Wilmarth, Amato, Stiller. SoCG’99] • Clearance • The closest pair of points on the boundary of two polyhedra • Penetration depth • If both polyhedra are convex • Use Lin-Canny closest features algorithm [Lin and Canny ICRA’99] • Otherwise • Use brute force method [Wilmarth, Amato, Stiller. SoCG’99] (test all possible pairs of features)

10. Algorithm Clearance Computation Penetration Computation Applied to MAPRM exact exact Convex rigid body MAPRM exact approximate General rigid body MAPRM approximate approximate Rigid/articulated body Approximate Variants of MAPRM • Clearance and penetration depth • Both clearance and penetration depth are approximated • Following N random directions until collision status changes Obstacle

11. Sampling is Increased in Narrow Passage [Wilmarth, Amato, Stiller.’99]

12. Experiments • PRM with uniform sampling,MAPRM, MAPRM and MAPRM. • Solution time • Number of approximate directions, N, for MAPRM and MAPRM • Map node generation time • Accuracy of sampled map nodes • Solution time

13. Serial Walls Hook rigid body rigid body rigid body articulated body Experiment Environments S-tunnel

14. Experiment: Time S-tunnel Environment

15. Experiment: Time Hook Environment

16. Experiment: Time Serial Wall Environment

17. Experiment: Approximation StudyAccuracy and Computation Time • Study accuracy and computation time by varying N for clearance and penetration depth.

18. Approximation Study S-tunnel Environment MAPRM MAPRM

19. Approximation Study Hook Environment MAPRM MAPRM

20. Approximation Study Serial Wall Environment MAPRM MAPRM

21. Conclusion • A general framework for sampling configurations on the Medial Axis of free C-space. • Exact and approximate computation of clearance and penetration depth. • Approximate clearance and penetration depth computation is applied to general C-space. • PRM, MAPRM, MAPRM and MAPM • MAPRM is the most efficient among all. • MAPRM and MAPM are slightly slower than MAPRM but can handle more general problems. • Low numbers of approximate directions can result in good estimate of clearance and penetration depth.