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

Motivation Robotics: Snake-like robots Biology: Motion of macro-molecules (Mark Yim) protein backbone

Properties of Kinematic Chains 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

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

Existing Techniques UpdatingSelf-collisions I-COLLIDE (Cohen et al ’95) GRID (e.g. Halperin and Overmars ’98) BV Hierarchies (Quinlan ’94, Gottschalk et al ’96, van den Bergen ’97, Klosowski et al ’98) Dynamic Kinematic Structures (Halperin et al ’96)

Our Algorithm updating per time-step worst-case self-collision detection per time-step. Much faster in practice Novel chain representation based on: Transformations hierarchy to approximate the kinematics at different resolutions OBB hierarchy to approximate the geometry at different resolutions

Chain Representation A Sequence of reference frames (links) connected by rigid-body transformations (joints) TT(R,t)TT(R,t) TT(R,t)TT(R,t) TT(R,t)TT(R,t) TT(R,t)TT(R,t) TT(R,t)TT(R,t) TT(R,t)TT(R,t) TT(R,t)TT(R,t) TT(R,t)TT(R,t) TT(R,t)TT(R,t) TT(R,t)TT(R,t) TT(R,t)TT(R,t) Hierarchy of “shortcut” transformations

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

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

Self-Collision: Example

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

Results: Extended Chain (1) Single Joint Change

Results: Extended Chain (2) 100 Joint Changes

Protein Backbones 1SHG (171 atoms) 1B4E (969 atoms) 1LOX (1941 atoms)

Results: Protein Backbones (1) Single Joint Change

Results: Protein Backbones (2) 10 Joint Changes

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

Upper bound is stable - holds for “not so tight” hierarchies like ours Lower bound is stable 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 in the worst case Analysis – Collision Detection

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

Proof of Lower Bound 3d links form a unit d/8 units shifted by 1 along X and -Y form a layer d/8 layers shifted by 1 along –Y and Z form a chain [chain] [layer] Convex hull of all units overlaps! P=[2(d-1),d-1,(d-1)/4]

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

Current Work Applying the algorithm to Monte-Carlo simulation of proteins Allow tree-like structure with short branches to model side-chains Replace collision detection with distance computation Efficiently compute internal energy by reusing unchanged terms