algorithmic robotics and motion planning
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Algorithmic Robotics and Motion Planning. 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 - PowerPoint PPT PresentationTRANSCRIPT
Algorithmic Roboticsand Motion Planning
Dan HalperinTel Aviv University
Fall 2006/7
Dynamic Maintenance and Self-Collision Testing for Large Kinematic Chains
Kinematic structures
A collection of rigid bodies hinged together---motion along jointsLARGE structures:hyper-redundant robots [Burdick, Chirikjian, Rus, Yim and others],macro-molecules
The static model
n links of roughly the same sizepossibly slightly interpenetratingmany 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)
The kinematic model
linksjoints
chain, tree, graphhttp://www.youtube.com/watch?v=k-VgI4wNyTo
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
Collision testing, 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)
( )O N ( )O N
( )O N( )O N
( log )O N N ( )N
Self-collision testing, assumptions
a small number of joint values change from one step to the otherthe chain was self-collision free at the last step
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)
Hierarchy of “shortcut” transformations
Bounding Volume Hierarchy
Chain-aligned: bottom-up, along the chain Each BV encloses its two children in the hierarchyShortcuts allow to efficiently compute relative position of BVsAt 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
(log )O N
(log )O N( log )O k N( )O N
Upper bound holds for “not so tight” hierarchies like oursLower bound holds for any convex BVSlightly worse than bound we prove for a regular hierarchy If topology of regular hierarchy is not updated, can deteriorate toGuibas et al '02: bounds for spherical hierarchy
in the worst case
Analysis – collision detection
43( )N
( )N
2( )N
Upper boundwe 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’dStep 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
• 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?
Upper bound, cont’d
YES!
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
parameter d
d
Lower bound, one unit (3d links)
Lower bound, a layer
a copy of a unit tranalted by
(2r,-2r,0)
a layer: d/8 units
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
Based on the papers:
I. Lotan, F. Schwarzer, D. Halperin and J.-C. Latombe Algorithm and data structures for efficient energy maintenance during Monte Carlo simulation of proteins Journal of Computational Biology 11 (5), 2004, 902-932.
II. 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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