Sampling Strategies for PRMs

1. Sampling Strategies for PRMs modified from slides of T.V.N. Sri Ram

2. Basic PRM algorithm

3. Issue • Narrow passages

4. OBPRMs A randomized roadmap method for path and manipulation planning (Amato,Wu ICRA’96) OBPRM: An obstacle-based PRM for 3D workspaces (Amato,Bayazit, Dale, Jones and Vallejo)

5. Roadmap candidate points chosen on C-obstacle surfaces

6. Basic Ideas Given Algorithm

7. Finding points on C-objects • Determine a point o (the origin) inside s • Select m rays with origin o and directions uniformly distributed in C-space • For each ray identified above, use binary search to determine a point on s

8. Issues • Selection of o in C-obstacle is crucial • To obtain uniform distribution of samples on the surface, would like to place origin somewhere near the center of C-object. • Still skewed objects would present a problem

9. Issues (contd) • Paths touch C-obstacle

10. Main Advantage • Useful in manipulation planning where the robot has to move along contact surfaces • Useful when C-space is very cluttered.

11. Results

12. Bridge Test The Bridge Test for Sampling Narrow Passages with Probabilistic Roadmap Planners (Hsu, Jiang, Reif, Sun ICRA’03)

13. Main Idea • Accept mid-point as a new node in roadmap graph if two end-points are in collision and mid-point is free • Constrain the length of the bridge: Favourable to build these in narrow passages

14. Algorithm

15. Contribution over previous Obstacle–Based Methods • Avoids sampling “uninteresting” obstacle boundaries. • Local Approach: Does not need to “capture” the C-obstacle in any sense • Complementary to the Uniform Sampling Approach

16. Issues • Deciding the probability density (πB )around a point P, which has been chosen as first end-point. • Combining Bridge Builder and Uniform Sampling • π =(1-w). πB +w.πv • πB : probability density induced by the Bridge Builder • πv : probability density induced by uniform sampling

17. Results Ncon Nmil Nclear

18. Medial-Axis Based PRM MAPRM: A Probabilistic Roadmap Planner with Sampling on the Medial Axis of the Free Space (Wilmarth, Amato, Stiller ICRA’99)

20. Definitions

21. Main Ideas • Beneficial to have samples on the medial axis; however, computation of medial axis itself is costly. • Retraction : takes nodes from free and obstacle space onto the medial axis w/o explicit computation of the medial axis. • This method increases the number of nodes found in a narrow corridor • independent of the volume of corridor • Depends on obstacles bounding it

22. Approach for Free-Space • Find xo (nearest boundary point) for each point x in Free Space. • Search along the ray xox and find arbitrarily close points xa and xb s.t. xo is the nearest point on the boundary for xa but not xb. • Called canonical retraction map

23. Extended Retraction Map • Doing only for Free-Space => Only more clearance. Doesn’t increase samples in Narrow Passages • Retract points that fall in Cobstacle also. • Retract points in the direction of the nearest boundary point

24. Results for 2D case • LEFT: Helpful: obstacle-space that retracts to narrow passage is large • RIGHT: Not Helpful: Obstacle-space seeping into medial axis in narrow corridor is very low

25. MAPRM for 3D rigid bodies

26. Example 2

27. Example 3

28. Main Results • Demonstrates an approach to use medial axis based ideas with random sampling • Advantages: • Useful in cluttered environments. Where a great volume of obstacle space is adjacent to narrow spaces • Above Environment: Bisection technique for evaluating point on medial axis???

29. Limitations • Additional primitive: “Nearest Contact Configuration”. For larger (complex) problems, this time may become significant…. • Extension to higher dimensions difficult. Which direction to search for nearest contact?