1. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Fast C-obstacle Query Computation for Motion Planning Liang-Jun Zhang 12/13/2005 Liang-Jun Zhang 1 Young J. Kim 2 Gokul Varadhan 1 Dinesh Manocha 1 1: University of North Carolina - Chapel Hill, USA 2: Ewha Womans University, Korea, http://gamma.cs.unc.edu/cobstacle

2. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 2 Configuration space Free space C-obstacle Do they intersect? Is its configuration in C-obstacle or free space? Free space and C-obstacle Robot Obstacle

3. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 3 C-obstacle query in C-space C-obstacle query ♦ Does a primitive lie completely inside C-obstacle? ♦ The primitive usually is a cell. Goal ♦ Design an efficient C-obstacle query algorithm Free space C-obstacle

4. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 4 Motivation- An important query for Motion Planning Cell Decomposition Method ♦ Label Cells as FULL and EMPTY Complete motion planning ♦ Able to find a path or report path non-existence EMPTY init goal FULL

5. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 5 Previous work Computing the boundary of C- obstacle ♦ Exponential complexity [Sacks 99, Sharir 97] ♦ Degeneracy and floating point error Contact surface constraints ♦ [Latombe 91, Zhu 91] ♦ Complexity of contact surface enumeration ♦ To deal with non-linear contact surfaces

6. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 6 Interpretation of C-obstacle query C-obstacle Free space Does the cell lie inside C-obstacle? Do the robot and obstacle intersect at all configurations? Can the robot ‘escape’ from the obstacle at some moment? Obstacle

7. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 7 Algorithm overview Penetration Depth ♦ How much does the robot penetrate into the obstacle at a configuration q ? Bounding Motion ♦ How much motion the robot can undergo, when its configuration changes from q but within the query primitive? Query criterion ♦ If Penetration Depth > Bounding Motion the robot can not escape ♦ the query primitive lies inside C-obstacle q A(q)

8. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 8 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 easier. B A A’ A B

9. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 9 Generalized Penetration Depth: PD g Consider both translation and rotation ♦ [Zhang, Kim, Varadhan, Manocha: UNC-CS TR05] Difficult for non-convex objects Convex A, B PD g (A,B)=PD t (A,B)

10. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 10 Algorithm-Lower bound on PD g 1.Convex decomposition 2.Eliminate non-overlapping pairs 3.PD t over overlapping pairs 4.LB(PD g ) = Max over all PD t s A B A2A2 A1A1 B1B1 B2B2 A2A2 A1A1 B2B2 B1B1

11. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 11 C-obstacle Query Criterion If Lower Bound (PD g (A(q a ), B))> Bounding Motion, the cell C is in C-obstacle. A(q a ): set A’s config as q a C

12. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 12 Upper bound of Motion for line segment ♦ Configurations q a and q b [Schwarzer,Saha,Latombe 04] Max trajectory length over points on the moving robot

13. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 13 Upper bound of Motion for cell Any diagonal line segment yields maximum bounding motion. q a is the center of the cell C q b is any point on the boundary of the cell. C

14. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 14 C-obstacle Query Criterion If Lower Bound (PD g (A(q a ), B))> the cell C is in C-obstacle. A(q a ): set A’s config as q a C

15. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 15 Application Star-shaped roadmap: a complete motion planning approach ♦ [Varadhan and Manocha 05] To identify cells which lie inside C- obstacle ♦ No subdivisions are applied for them

16. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 16 Results-`gear’: 2T+1R video

17. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 17 Results ‘Piano’ ‘World map’

18. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 18 Effective and Performance Query timing: 0.04ms to 0.12 ms Culled C-obstacle Cells Cell Culling Ratio = All C-obstacle Cells

19. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 19 Speedup For Star-shaped roadmap method

20. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 20 Conclusion A fast C-obstacle query algorithm for rigid robots Based on generalized penetration depth and bounding motion computation. ♦ Need not explicit computation of the boundary of free space. ♦ Robust and efficient Applied for accelerating a complete motion planning approach for 2D rigid robot.

21. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 21 Ongoing and Future work A Simple Path Non-Existence Algorithm for low DOF robots ♦ [L. Zhang, Y.J. Kim, D. Manocha] WAFR2006 Apply for 3D rigid robots Handle articulated robots

22. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 22 Acknowledgements Army Research Office DARPA/REDCOM NSF ONR Intel Corporation KRF, STAR program of MOST, Ewha SMBA consortium, the ITRC program (Korea)

23. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 23 Thanks Any Questions?

24. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 24 Appendix

25. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 25 Generalized Penetration Depth: PD g Consider both translation and rotation ♦ [Zhang, Kim, Varadhan, Manocha et al. 05] Trajectory length Separating path Robot Obstacle Robot

26. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 26 PD g Difficult for non-convex objects Convex A, B PD g (A,B)=PD t (A,B) MIN over all possible separating paths MAX of the trajectory length over all on the moving robot Generalized Penetration Depth: PD g

27. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 27 Upper bound of Motion for line segment [Schwarzer,Saha,Latombe 04] ♦ Max trajectory length over points on the moving robot ♦ The weighted sum of difference for x, y, components between q a and q b R: ‘radius’ of the object