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 Algorithm Given

7. Finding points on C-objects 1.Determine a point o (the origin) inside s 2.Select m rays with origin o and directions uniformly distributed in C-space 3.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 (π B )around a point P, which has been chosen as first end- point. Combining Bridge Builder and Uniform Sampling –π =(1-w). π B +w.π v –π B : probability density induced by the Bridge Builder –π v : probability density induced by uniform sampling

17. Results NmilNclear Ncon

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 x o (nearest boundary point) for each point x in Free Space. Search along the ray x o x and find arbitrarily close points x a and x b s.t. x o is the nearest point on the boundary for x a but not x b. 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 C obstacle 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?