Algorithmic Robotics and Motion Planning

1. Algorithmic Roboticsand Motion Planning Fall 2006/7 Dynamic Maintenance and Self-Collision Testing for Large Kinematic Chains Dan Halperin Tel Aviv University

2. Kinematic structures • A collection of rigid bodies hinged together---motion along joints • LARGE structures:hyper-redundant robots[Burdick, Chirikjian, Rus, Yim and others],macro-molecules

3. The static model • n links of roughly the same size • possibly slightly interpenetrating • many favorable properties and simple algorithms (HSR, union boundary construction), in particular, data structures for intersection queries: O(n log n) preprocessing -> O(n) rand. O(n) space O(log n) query -> O(1)

4. The kinematic model links joints chain, tree, graph http://www.youtube.com/watch?v=k-VgI4wNyTo

5. Dynamic maintenance, self collision testing • the problem: Carry out a sequence of operations efficiently update of joint values the query is for self collision • sample motivation: monte carlo simulation of protein folding paths

6. Dynamic maintenance:what’s available dynamic spatial data structures insertions and deletions kinetic data structures [Basch, Guibas, Hershberger 97] independent movements robot motion planning small number of degrees of freedom dynamic maintenance for kinematic struct’s link-size queries [H-Latombe-Motwani 96,Charikar-H-Motwani 98]

7. Dynamic maintenance, self collision testing • the problem (reminder): Carry out a sequence of operations efficiently update of joint values the query is for self collision n: # of links ~ # of joints • theory, worst case: rebuild spatial structure at each update

8. Collision testing, existing techniques

9. Self-collision testing, assumptions • a small number of joint values change from one step to the other • the chain was self-collision free at the last step

10. T(R,t) T(R,t) T(R,t) T(R,t) T(R,t) T(R,t) T(R,t) T(R,t) T(R,t) T(R,t) T(R,t) Chain representation A Sequence of reference frames (links) connected by rigid-body transformations (joints) Hierarchy of “shortcut” transformations

11. Bounding Volume Hierarchy • Chain-aligned: bottom-up, along the chain • Each BV encloses its two children in the hierarchy • Shortcuts allow to efficiently compute relative position of BVs • At each time step only BVs that contain the changed joints need to be recomputed

12. Self-collision detection • Test the hierarchy against itself to find collisions. But … • Do not test inside BVs that were not updated after the last set of changes Benefits: • Many unnecessary overlap tests are avoided • No leaf node tested against itself

13. Self-collision: Example

14. Experimental results • We tested our algorithm (dubbed ChainTree) against three others: • Grid– Collisions detected by indexing into a 3D grid using a hash table • 1-OBBTree– An OBB hierarchy is created from scratch after each change and then tested against itself for collisions • K-OBBTree– After each change an OBB hierarchy is built for each rigid piece of the chain. Each pair of hierarchies is tested for collisions

15. Results: Extended chain (1) Single Joint Change

16. Results: Extended chain (2) 100 Joint Changes

17. Protein backbones 1LOX (1941 atoms) 1B4E (969 atoms) 1SHG (171 atoms)

18. Results: Protein backbones (1) Single Joint Change

19. Results: Protein backbones (2) 10 Joint Changes

20. Analysis – updating • For each joint change: • shortcut transformations need to be recomputed • BVs need to be recomputed • For k simultaneous changestime, but never more than Previous BV hierarchies required O(N log N) updating time

21. Analysis – collision detection in the worst case • Upper bound holds for “not so tight” hierarchies like ours • Lower bound holds for any convex BV • Slightly worse than bound we prove for a regular hierarchy • If topology of regular hierarchy is not updated, can deteriorate to • Guibas et al '02: bounds for spherical hierarchy

22. Upper bound we first show for tight spherical hierarchy, the extend to OBBs tight hierarchy: the bounding sphere is the minimal for the original links at each level Reminder, well-behaved chain, two constants: (1) the ratio between the biggest and smallest bounding sphere of a link (2) the minimum distance between the centers of two bounding sphere of links

23. Upper bound, cont’d Step 1: regularize chain all spheres of same radius r two successive spheres in the chain are not disjoint level i=0, tree leaves at level i there are gi = 2i each bounding volume, a bounding sphere of radius gir the number of bounding spheres at level i intersecting a single bounding sphere is

24. Upper bound, cont’d Mi can be as large as n/gi Max Mi is attained for the smallest i such that which, since gi = 2i, occurs when Ti denotes the number of sphere overlaps at level I, T is the overall number of sphere overlaps

25. Upper bound, cont’d

26. Upper bound, cont’d • Will the bound hold for a “not so tight” hierarchy like ours? YES! • OBBs are larger than tight bounding spheres by a constant factor at each level • This factor is fixed for all levels of the hierarchy

27. Upper bound, cont’d lemma: given two OBBs contained in a sphere D of radius R, the OBB bounding both of them is contained in a sphere of radius √3R concentric with D

28. Upper bound, cont’d lemma: at level I of an OBB hierarchy, each OBB is contained in a sphere of radius c2ir, where c is an absolutre constant Proof: C1 is chosen such that this is true for levels i = 0,1, …, 4 assume for i-1 (i>4) and prove for i S sphere of radius 2ir containing the subchain bounded by the 32 boxes at level i-5 S0 sphere concentric with S with radius 2ir(1+c/16)

29. Upper bound, cont’d Consider the OBB at level i-4 S1 sphere concentric with S0 with radius √3 times the radius of S0 contains all the OBBs at level i-4 Continuing up to level I we get sphere S5 of radius √352ir(1+c/16) that contains the OBB at this level that contains all the 32 OBBs of level i-5 in its subtree c must be such that

30. Upper bound, cont’d finally we choose

31. d Lower bound parameter d

32. Lower bound, one unit (3d links)

33. Lower bound, a layer • a copy of a unit tranalted by (2r,-2r,0) • a layer: d/8 units

34. Lower bound, overall construction • a copy of a layer tranalted by (0,-2r,2r) • overall: d/8 layers • a unit consists of cn1/3 links

35. Lower bound, overall construction, cont’d • there are c'n2/3 units at the level where the links of a unit are grouped together • the convex hull of each unit contains the point (2(d-1)r, (d-1)r, (d-1)r/4) • overall (n4/3) overlaps

36. Based on the papers: • Lotan, F. Schwarzer, D. Halperin and J.-C. Latombe Algorithm and data structures for efficient energy maintenance during Monte Carlo simulation of proteinsJournal of Computational Biology 11 (5), 2004, 902-932. • Efficient maintenance and self-collision testing for kinematic chains, Proc. 18th ACM Symposium on Computational Geometry, Barcelona, 2002, pp, 43-52.

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