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### Algorithmic Roboticsand Motion Planning

### Based on the papers:

Fall 2006/7

Dynamic Maintenance and Self-Collision Testing for Large Kinematic Chains

Dan Halperin

Tel Aviv University

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

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)

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

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]

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

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

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 representationA Sequence of reference frames (links) connected by rigid-body transformations (joints)

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

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

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 changestime, but never more than

Previous BV hierarchies required O(N log N) updating time

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

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

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

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

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

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

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)

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

Upper bound, cont’d

finally we choose

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

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

- 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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