1. Sampling From the Medial Axis Presented by Rahul Biswas April 23, 2003 CS326A: Motion Planning

2. Citation Motion Planning for a Rigid Body Using Random Networks on the Medial Axis of the Free Space Steven A. Wilmarth, Nancy M. Amato, and Peter F. Stiller Proceedings of the 15th Annual ACM Symposium on Computational Geometry, 1999, pp. 173-180 Wilmarth is a Math Ph.D. from Texas A&M MAPRM = Medial Axis PRM

3. Voronoi Diagram Consists of points equidistant from obstacles Obstacles

4. Example Voronoi Diagram

5. Another Voronoi Diagram

6. Why Medial Axes are Useful Medial axis = lines in voronoi diagram Represent maximal clearance paths for robots Excellent vertices for PRM in narrow passages

7. Sampling from the Medial Axis Very difficult to compute medial axis explicitly Main idea: “retract a configuration, free or not, onto the medial axis of the free space without having to compute the medial axis explicitly” retract = map a point onto another point

8. Retraction to the Medial Axis Two types of points: simple point – one nearest neighbor multiple point – two nearest neighbors Want to retract simple points Find nearest neighbor of simple point Move away from nearest neighbor until additional nearest neighbor arises

9. Retraction Illustration Sample Nearest Neighbor

10. Retraction from Blocked Space Find nearest point on obstacle boundary Retract from that point as before

11. Retraction Maps

12. Sampling Narrow Passages

13. Uniform vs MAPRM Samples

14. Rigid Body Robots More complicated problem Not assuming convex robots, obstacles Collision checking more expensive Must account for both rotation and translation Robot

15. SE(3) Configuration Space SE(3) Translation: tx, ty, tz (T) Rotation:rx, ry, rz (R) 6-dim, as opposed to 3-dim point robot Collision checking transformed point q becomes Rq + p transforming robot yields set of points O(n) collision checking is now much more

16. Distance Metric Want Riemannian (distance) metric on SE(3) Two criteria: Shortest path between (R,p 1 ),(R,p 2 ) is wholly translational Shortest path from free configuration to contact configuration is also wholly translational Achieved by weighted sum of T and R R is weighted more so that movement via rotation is more expensive than translation

17. Algorithm

18. Algorithm 4.2

19. Complexity  Analysis for Algorithm 4.2 Must check all features of robot and all features of obstacles O(n U *n V *log(n U n V ) + n U *n V *t cd (n U,n V )) t cd (n,m) is collision detection time for objects of size m and n constant for polygonal robots and obstacles Finding nodes is substantially more expensive

20. Test Scenario Must pass block through narrow pipe Rest of the block is solid Two experiments 1: Cube Width = 2 2: Cube Width = 1.5 20 2.5

21. Experiment 1 Results Authors surmise problem with corners

22. Experiment 2 Results

23. Conclusion Technique to sample intelligently for PRMs Sample points from the medial axis without computing Voronoi diagram Works for both rigid bodies and point robots No extension to articulated robots Interesting concept but not useful in practice