1 / 23

Sampling From the Medial Axis

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

bibiana
Download Presentation

Sampling From the Medial Axis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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, andPeter 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. Sample Nearest Neighbor Retraction Illustration

  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,p1),(R,p2) 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(nU*nV*log(nUnV) + nU*nV*tcd(nU,nV)) • tcd(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 2.5 20

  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

More Related