Efficient Maintenance and Self-Collision Testing for Kinematic Chains

1. Efficient Maintenance and Self-Collision Testing for Kinematic Chains Itay Lotan Fabian Schwarzer Dan Halperin Jean-Claude Latombe

2. Motivation • Robotics: hyper-redundant snake-like robots (Mark Yim) (Joel Burdick)

3. Motivation – cont. • Molecular Simulation – exploring the conformational space of polypeptides (4PTI backbone)

4. Motivation – cont. • Both types of applications require: • Efficient updating of the chain after changes to its DOFs • Efficient detection of self-collisions caused by these changes

5. Outline of Talk • Description of the Problem • Previous approaches • Our chain representation • Detecting self-collisions • Experimental results • Analysis and theoretical results • Conclusions

6. Outline of Talk • Description of the Problem • Previous approaches • The chain representation • Detecting self-collisions • Experimental results • Analysis and theoretical results • Conclusions

7. Problem Description Given a chain of N links which deforms over time through changes to its DOFs: • Update the chain at each time-step to reflect the changes • Assuming no self-collisions at previous time-step, find any/all self-collisions caused by latest changes

8. Outline of Talk • Description of the Problem • Previous approaches • The chain representation • Detecting self-collisions • Experimental results • Analysis and theoretical results • Conclusions

9. I-COLLIDE (Cohen et al ’95) • Project objects onto each axis • Sort intervals • Find pairs of intervals that overlap on all axes • Reuse previous order to speed up sorting at next time-step O(N) update time O(N) detection time

10. Grid (e.g. Halperin and Overmars ’98) • Insert objects into grid one at a time • Check inserted objects for collisions with objects in neighboring cells O(N) update time O(N) detection time

11. Bounding Volume Hierarchies • Spheres(Quinlan ’94) • OBBs(Gottschalk et al ’96) • AABBs(van den Bergen ’97) • k-DOPs(Klosowski et al ’98) • Build a BV hierarchy • Test the hierarchy against itself to find collisions • At each time step update the BVs O(N logN) update time Ω(N) detection time

12. Previous Approaches - Drawbacks • In a kinematic chain – local changes have global effects. One change may cause O(N) links to move • When few changes are applied to the chain, large pieces of it remain rigid.

13. Maintenance of Kinematic Structures (Halperin et al ’96) • Algorithm for dynamic maintenance of kinematic chains • Does not support self-collision testing • update time

14. Outline of Talk • Description of the Problem • Previous approaches • The chain representation • Detecting self-collisions • Experimental results • Analysis and theoretical results • Conclusions

15. 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 connected by rigid-body transformations Hierarchy of shortcut transformations

16. Bounding Volume Hierarchy • Chain-aligned: bottom-up, along the chain • Each BV encloses its two children in the hierarchy • Each BV is stored in the reference frame of its left-most link. • At each time step only BVs that contain the changed joints need to be recomputed

17. Outline of Talk • Description of the Problem • Previous approaches • The chain representation • Detecting self-collisions • Experimental results • Analysis and theoretical results • Conclusions

18. 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.

19. Self-Collision: Example

20. Outline of Talk • Previous approaches • Description of the Problem • The chain representation • Detecting self-collisions • Experimental results • Analysis and theoretical results • Conclusions

21. 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 move an OBBTree is built for each rigid piece of the chain. Each pair of OBBTrees is tested for collisions.

22. Results – Compact Chain

23. Results – Protein Backbones

24. Outline of Talk • Previous approaches • Description of the Problem • The chain representation • Detecting self-collisions • Experimental results • Analysis and theoretical results • Conclusions

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

26. Analysis – Collision Detection in the worst case • This bound is very stable - holds for “not so tight” hierarchies like ours • Lower bound holds for any convex BV. • Slightly worse than Θ(N) we show for a regular hierarchy • If topology of regular hierarchy is not updated, can deteriorate to Θ(N2)

27. Outline of Talk • Previous approaches • Description of the Problem • The chain representation • Detecting self-collisions • Analysis and theoretical results • Experimental results • Conclusions

28. Conclusions • We presented an algorithm for efficient maintenance and self-collision detection of kinematic chains • update time and detection time in the worst case • It is very fast in practice • Most efficient when k << N