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Algorithmic Robotics and Motion Planning

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Algorithmic Roboticsand Motion Planning

Fall 2006/7

Dynamic Maintenance and Self-Collision Testing for Large Kinematic Chains

Dan Halperin

Tel Aviv University

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

- 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)

links

joints

chain, tree, graph

http://www.youtube.com/watch?v=k-VgI4wNyTo

- 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 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]

- 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

- 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)

T(R,t)

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

Hierarchy of “shortcut” transformations

- 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

- 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

- 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

Single Joint Change

100 Joint Changes

1LOX

(1941 atoms)

1B4E

(969 atoms)

1SHG

(171 atoms)

Single Joint Change

10 Joint Changes

- 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

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

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

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

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

- 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

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

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)

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

finally we choose

d

parameter d

- a copy of a unit tranalted by
(2r,-2r,0)

- a layer:
d/8 units

- a copy of a layer tranalted by
(0,-2r,2r)

- overall:
d/8 layers

- a unit consists of
cn1/3 links

- 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

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.

THE END