1. the University of North Carolina at CHAPEL HILL A Simple Path Non-Existence Algorithm using C-obstacle Query http://gamma.cs.unc.edu/nopath Liang-Jun Zhang University of North Carolina - Chapel Hill Young J. Kim EWHA Womans University, Korea Dinesh Manocha University of North Carolina - Chapel Hill

2. the University of North Carolina at CHAPEL HILL Motion Planning Initial Goal Obstacle To find a path Robot 72 DOF Courtesy of P. Isto and M. Saha, 2006 Goal Initial Obstacle To report no path Robot

3. the University of North Carolina at CHAPEL HILL Path Non-existence Problem Obstacle Goal Initial Robot

4. the University of North Carolina at CHAPEL HILL Previous Work Exact Motion Planning ♦ Exact cell decomposition [Schwartz et al. 83] ♦ Roadmap [Canny 88] ♦ Criticality based method [Latombe 99] ♦ Implementation challenges ♦ Special and simple objects Ladders, sphere, convex shapes

5. the University of North Carolina at CHAPEL HILL Previous Work Approximation Cell Decomposition ♦ [Lozano-Pérez 83], [Zhu et al. 91], [Latombe 91] ♦ Relatively easy to implement ♦ Combinatorial complexity of cell decomposition ♦ Computational issue for labelling the cells during cell decomposition

6. the University of North Carolina at CHAPEL HILL Previous Work Probabilistic Sampling Based Approach ♦ [Kavarki et al. 96] [LaValle et al. 98], [Choset et al. 05], [LaValle 06] ♦ Simple and widely used ♦ May not be terminated when non-path exists ♦ Difficult for narrow passage

7. the University of North Carolina at CHAPEL HILL Previous Work Path non-existence for special cases ♦ Planar section, [Basch et al. 01], [Bretl et al. 04]

8. the University of North Carolina at CHAPEL HILL Main Results Efficient cell labelling algorithm ♦ Workspace-based ♦ C-obstacle query using generalized penetration depth Improved cell decomposition algorithm ♦ Simple ♦ Efficient for path non-existence

9. the University of North Carolina at CHAPEL HILL Path Non-existence Problem q goal More difficult than finding a path ♦ To check all possible paths Identify a region in C-obstacle ♦ separating q init and q goal q init Configuration space

10. the University of North Carolina at CHAPEL HILL C-obstacle Query Whether a primitive lies entirely in C-obstacle? ♦ Usually a cell Useful for path non-existence q goal q init

11. the University of North Carolina at CHAPEL HILL fullmixed empty [Lozano-Pérez 83] [Zhu et al. 91] [Latombe 91] Cell Decomposition for Path Non-existence

12. the University of North Carolina at CHAPEL HILL Cell Decomposition for Path Non-existence Connectivity Graph Guiding Path

13. the University of North Carolina at CHAPEL HILL Cell Decomposition for Path Non-existence Connectivity graph is not connected No path!

14. the University of North Carolina at CHAPEL HILL Previous Work on C-obstacle Query Explicit free space computation ♦ Exponential complexity [Sacks 99, Sharir 97] ♦ Hard in practice: degeneracy Check against every C-surface ♦ [Latombe 91, Zhu et al. 91] ♦ C-surface enumeration ♦ To deal with non-linear C-surfaces Workspace distance computation ♦ [Paden 89] ♦ Overly conservative

15. the University of North Carolina at CHAPEL HILL C-obstacle Query A Collision Detection Problem Does the cell lie inside C-obstacle? Do robot and obstacle intersect at all configurations? Obstacle Workspace Configuration space ? Robot

16. the University of North Carolina at CHAPEL HILL Clearance VS ‘Forbiddance’ Separation distance Clearance Penetration Depth ‘Forbiddance’ PD d

17. the University of North Carolina at CHAPEL HILL C-obstacle Query Algorithm Penetration Depth ♦ Extent of interpenetration between robot and obstacle Motion Bound ♦ Extent of the motion that robot can make. Is Penetration Depth > Motion Bound? Robot A(q) PD Cell Obstacle

18. the University of North Carolina at CHAPEL HILL Translational Penetration Depth: PD t Minimum translation to separate A, B ♦ [Dobkin 93, Agarwal 00, Bergen 01, Kim 02] PD t : not applicable ♦ The robot is allowed to both translate and rotate. ♦ Undergoing rotation, A may ‘escape’ from B more easily B A A’ A B

19. the University of North Carolina at CHAPEL HILL Generalized Penetration Depth: PD g Take into account translational and rotational motion ♦ [L. Zhang, Y. Kim, G. Varadhan, D. Manocha, ACM Solid and Physical Modeling 06] ♦ Trajectory length ♦ Distance metric D g ♦ Min/Max operations Trajectory length A(q 0 ) A(q 1 )

20. the University of North Carolina at CHAPEL HILL PD g Computation Difficult for non-convex objects Theorem: for convex objects, PD g = PD t Convex/Convex ♦ Known efficient PD t algorithms directly applicable ♦ [Dobkin 93, Agarwal 00, Bergen 01, Kim 02] Non-Convex / Non-Convex ♦ A lower bound on PD g based on convex decomposition

21. the University of North Carolina at CHAPEL HILL C-obstacle Query Is Penetration Depth > Motion Bound?

22. the University of North Carolina at CHAPEL HILL Motion Bound ♦ [Schwarzer, Saha, Latombe 04] ♦ Achieved by any diagonal line segment, e.g. q a,c Cell Configuration space qaqa

23. the University of North Carolina at CHAPEL HILL Free Cell Query Separation distance describes the clearance If Separation Distance ≥ Motion Bound the robot can not intersect with the obstacle ♦ The cell lies inside free space d

24. the University of North Carolina at CHAPEL HILL Experimental Results C-obstacle Query Computation Faces # of A288,452304 Faces # of B1,692336304 Per C-obstacle query 1.901 (ms) 6.127 (ms) 4.112 (ms)

25. the University of North Carolina at CHAPEL HILL Experimental Results Path Non-existence 2D rigid robots with 3-DOF ♦ 2 translational DOF and 1 rotational DOF B1B1 B2B2 B3B3 B4B4 A A'A'

26. the University of North Carolina at CHAPEL HILL Two-gear Example no path! Cells in C-obstacle Initial Goal Roadmap in F Vide o 3.356s

27. the University of North Carolina at CHAPEL HILL Performance of Two-gear Example # of C-obstacle queries30K # of free cell queries32K # of iterations41 Total timing3.356s Free cell queries0.858s C-obstacle queries0.827s Graph searching0.466s Subdivision1.205s # of total cells28K # of total free cells2K # of total c-obstacle cells12K # of total mixed cells14K

28. the University of North Carolina at CHAPEL HILL Five-gear Example Cells in C-obstacle Initial Goal Roadmap in F 6.317s No path!

29. the University of North Carolina at CHAPEL HILL Narrow Passage Modified Five-gear Example Total timing85s # of C-obstacle queries176K # of total cells168K Vide o Initial Goal roadmap in free space

30. the University of North Carolina at CHAPEL HILL 2D Puzzle No path! 7.9s Narrow passage 15.8s B1B1 B2B2 B3B3 B4B4 Initial Goal B1B1 B2B2 B4B4 A A'A' Vide o Removed

31. the University of North Carolina at CHAPEL HILL Conclusion C-obstacle query is essential for deciding path non-existence Efficient C-obstacle and free cell queries ♦ Workspace-based ♦ Using generalized penetration depth and separation distance computation Improved cell decomposition algorithm ♦ Simple ♦ Efficient for path non-existence

32. the University of North Carolina at CHAPEL HILL Limitations C-obstacle & free cell queries are conservative Can not deal with compliant motion planning Current implementation of cell decomposition is limited to 3-DOF robots

33. the University of North Carolina at CHAPEL HILL Future Work Higher DOF motion planning ♦ 6 DOF rigid robot ♦ C-obstacle & free cell queries are applicable ♦ Combinatorial complexity of cell decomposition Hybrid planner ♦ To combine with sampling based approach

34. the University of North Carolina at CHAPEL HILL Acknowledgements Army Research Office, DARPA/REDCOM, NSF, ONR, Intel Corporation KRF, STAR program of MOST, Ewha SMBA consortium, ITRC program, Korea Mink2D, Tel Aviv University GAMMA Group, UNC Chapel Hill

35. the University of North Carolina at CHAPEL HILL Thank you! Any Questions? http://gamma.cs.unc.edu/nopath

36. the University of North Carolina at CHAPEL HILL Min over every path connecting q 0 and q 1 Max trajectory length for distinct points D g (q 0, q 1 ) = D g Metric in C-space X Y θ q1q1 q0q0 l1l1 l2l2 Motion Paths in C-Space Trajectory length A(q 0 ) A(q 1 ) D g (q 0,q 1 )

37. the University of North Carolina at CHAPEL HILL PD g definition The minimum D g distance over all possible collision- free configurations A B PD g

38. the University of North Carolina at CHAPEL HILL Lower Bound on PD g 1.Convex decomposition 2.Eliminate non-overlapping pairs 3.PD t for overlapping pairs 4.LB(PD g ) = Max over all PD t s PD t

39. the University of North Carolina at CHAPEL HILL Performance of Five-gear Example Total timing6.317s # of total cells39K # of C-obstacle queries41K

40. the University of North Carolina at CHAPEL HILL Compared with Star-shaped roadmap Pros ♦ Simpler than the star-shaped test ♦ Need not capture the intra-connectivity ♦ More likely to be extended for higher DOF Cons ♦ More conservative