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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

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Sampling From the Medial Axis

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  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

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