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

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

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

4. Generalized MAPRM Framework PRM with uniform sampling MAPRM

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

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

7. MAPRM, MAPRM  and MAPM  l Clearance and penetration depth computation –Exact methods –Approximate methods Algorithm Clearance Computation Penetration Computation MAPRMexact MAPRM  exactapproximate MAPRM  approximate Applied to Convex rigid body General rigid body Rigid/articulated body l Clearance and Penetration depth: distance to the closest contact configuration.

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

9. MAPRM for a Rigid Body in 3D [ Wilmarth, Amato, Stiller. SoCG’99 ] l Clearance –The closest pair of points on the boundary of two polyhedra l Penetration depth –If both polyhedra are convex l 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. Approximate Variants of MAPRM l Clearance and penetration depth –Both clearance and penetration depth are approximated –Following N random directions until collision status changes approximate MAPRM  approximateexact MAPRM  exact MAPRM Penetration Computation Clearance Computation Algorithm Rigid/articulated body General rigid body Convex rigid body Applied to Obstacle

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

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

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

14. Experiment: Time S-tunnel Environment

15. Experiment: Time Hook Environment

16. Experiment: Time Serial Wall Environment

17. Experiment: Approximation Study Accuracy and Computation Time l 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 l 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. l 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. l Low numbers of approximate directions can result in good estimate of clearance and penetration depth.