liang-jun zhang course project, comp 790-072 dec 13, 2006
DESCRIPTION
C-DIST : Distance Computation for Rigid and Articulated Models in Configuration Space. Liang-Jun Zhang Course Project, COMP 790-072 Dec 13, 2006. Distance Metric in Euclidean Space. Euclidean metric Manhattan metric L p Metric. q 1 =< x 1 ,y 1 , θ 1 >. q 0 =< x 0 ,y 0 , θ 0 >. - PowerPoint PPT PresentationTRANSCRIPT
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Liang-Jun Zhang Course Project, COMP 790-072
Dec 13, 2006
C-DIST: Distance Computation for Rigid and Articulated Models in Configuration Space
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Distance Metric in Euclidean Space
Euclidean metric
Manhattan metric
Lp Metric
1 1( , )x y 2 2( , )x y
2 22 1 2 1( ) ( )d x x y y
2 1 2 1( ) ( )p ppd x x y y
1 1( , )x y2 2( , )x y
2 1 2 1d x x y y
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Distance Metric in Configuration Space (C-space)
Workspace Configuration Space
X
Y
X
Y θ
q1=<x1,y1,θ1>
A q0=<x0,y0,θ0>
?
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Motivation
Sampling Based Motion Planning
Only connect the nearest neighbor(s) Evaluate the dispersion of samplesMeasured by a distance metric• [Amato 00 et al], [Kuffner 04], [Plaku and Kavraki
06]
free space
milestone
local path
PRM
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Motivation
Penetration depthProximity queryIdentify the easiest way to separate A from BMeasured by a distance metric
B
A
B
A
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Challenge
Without rotational motionEuclidean metric and Lp are directly applicable.
With rotational motionChallenge to naturally combine the translational and rotational component
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Previous Works
Model-independentDistance metric on SE(3)• No bi-invariant metric exists• [Loncaric 85], [Park 95], [Tchon & Duleba 94]
Weighting rot. and trans. components• Left-invariant, [Park 95]
Model-dependentBased on displacement vector• DISP [Latombe 91] , [LaValle 06]• Object Norm: [Kazerounian Rastegar 92]
Based on trajectory length• Generalized Distance Metric [Zhang 06]
Based swept volume• [Xavier 97], [Choset et al. 2005]
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DISP Metric
0 1 0 1( , ) max ( ) ( )x A
DISP q q x q x q
A
q0
q1
[Latobme 91] [LaValle 06]
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Properties
Naturally combine the translational and rotational component
Need not any scalar
Invariant w.r.t to both body and world frames
Independent from the representation of the rotation
Rotation matrix, quaternion
Easily extended for articulated body
Aq0 q1
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DISP Formulation
Can be proved by screw motion
Largely simplify the computationEnough to check the vertices on the convex hull
Theorem: DISP can only be realized by the vertex on its convex hull.
V=3,024 T=1,008
V=311
q0 q1 q0 q1
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DISP Computation
V=3,024 T=1,008
V=311
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DISP Computation Optimization
Walking on the convex hull
Accelerating using Bounding Volume Hierarchy (BVH)
Swept sphere volume (SSV)
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Results (Demo)
Triangle Soup
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Performance
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Extend for Articulated Models
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Performance
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Applications
Sampling Based Motion Planning
Choose the nearest neighbours
Continuous Collision DetectionReplace the motion bound with displacement bound
Generalized Penetration Depth
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Conclusion
A novel algorithm to compute DISP: C-DISTConvex realization theorem
A straightforward theorem
Computation optimizationWalking on the convex hull Accelerating using BVH
Extend for articulated modelsDiscuss the potential applications
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Future Work
Geodesic in C-space under the DISP metricOther useful metrics and properties in C-space
Area, VolumeConvexity
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Appendix
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Outline
Previous WorkDefinitionFormulationOptimizationExtended for Articulated BodiesResultsApplications
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Distance for an Object at Two Configurations
d
θ
θ
d + θ ?
d + rθ ?
d
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Sampling Based Motion Planning
K-nearest neighborsHow to quantify the ‘near’?Use DISP metric
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Continuous Collision Detection
If Separation Distance > Max Displacement, there is no collision.
AObstacle
q(t)
q(0)
q(1)
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min({ ( , ) | int( ( )) })ggPD D A B 0q q q
Generalized Penetration Depth
The minimum DISP distance over all possible collision-free configurationsA
B
PDg
Search for nearest collision-free configuration