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A General Framework for
Sampling on the Medial Axis of the Free Space
Jyh-Ming Lien, Shawna Thomas, Nancy Amato
{neilien, sthomas,amato}@cs.tamu.edu
Probabilistic Roadmaps and the Narrow Passage Problem
obstacles g
Narrow Passage
Probabilistic roadmap (PRM) [Kavraki, Svestka, Latombe, Overmars.’96]
Obstacle based PRM [Amato, Bayazit, Dale, Jones, Vallejo.’98] Gaussian PRM [Boor and Overmars.’99] RBB PRM [Hsu, Jiang, Reif, Sun.’03] Medial Axis based PRM (MAPRM) [Wilmarth, Amato,
Stiller.’99]
Generalized MAPRM Framework
Sample a Configuration, p
p is in collision
q = NearestContactCfg_Penetration(p)
V = q - p
q = NearestContactCfg_Clearance(p)
V = p - q
p is collision-free
Retract p to the Medial Axis of the free C-space in
direction V
samples < N
Connect sampled configurations
Generalized MAPRM Framework
PRM with uniform sampling MAPRM
Sampling is increased in Narrow Corridors
In-collision configurations are retracted to free C-space The volume of the narrow passage is increased
Vol(S )+Vol(B’ )
Vol(C )Pro( Sampling in S ) =
The Limitation of MAPRM
Can only be applied to problems with low (<6) dimensional C-space of rigid objects.
Sample a Configuration, p
p is in collision
q = NearestContactCfg_Penetration(p)
V = q - p
q = NearestContactCfg_Clearance(p)
V = p - q
p is collision-free
Retract p to the Medial Axis of the free C-
space in direction V
< N
Connect sampled configurations
MAPRM, MAPRM and MAPM
Clearance and penetration depth computation– Exact methods– Approximate methods
AlgorithmClearance Computation
Penetration Computation
MAPRM exact exact
MAPRM exact approximate
MAPRM approximate
approximate
Applied to
Convex rigid body
General rigid body
Rigid/articulated body
Clearance and Penetration depth: distance to the closest contact configuration.
MAPRM for Point Robot in 2D[Wilmarth, Amato, Stiller. ICRA’99]
Clearance and penetration depth– The closest point on the polygon boundary
clearance
penetration
MAPRM for a Rigid Body in 3D [Wilmarth, Amato, Stiller. SoCG’99]
Clearance– The closest pair of points on the boundary
of two polyhedra Penetration depth
– If both polyhedra are convex Use Lin-Canny closest features algorithm [Lin and Canny ICRA’99]
– Otherwise Use brute force method [Wilmarth, Amato, Stiller. SoCG’99]
(test all possible pairs of features)
Approximate Variants of MAPRM
Clearance and penetration depth– Both clearance and penetration depth are
approximated– Following N random directions until collision status
changes
approximateapproximateMAPRM
approximateexactMAPRM
exactexactMAPRM
Penetration Computation
Clearance Computation
Algorithm
Rigid/articulated body
General rigid body
Convex rigid body
Applied to
Obstacle
Sampling is Increased in Narrow Passage
[Wilmarth, Amato, Stiller.’99]
Experiments
PRM with uniform sampling, MAPRM, MAPRM and MAPRM.– Solution time
Number of approximate directions, N, for MAPRM and MAPRM – Map node generation time– Accuracy of sampled map nodes– Solution time
rigid body
S-tunnel
Experiment Environments
articulated body
rigid body
Serial Walls
Hook
rigid body
Experiment: Time S-tunnel Environment
Experiment: Time Hook Environment
Experiment: Time Serial Wall Environment
Experiment: Approximation StudyAccuracy and Computation Time
Study accuracy and computation time by varying N for clearance and penetration depth.
Approximation Study S-tunnel Environment
MAPRM MAPRM
Approximation Study Hook Environment
MAPRM MAPRM
Approximation Study Serial Wall Environment
MAPRM MAPRM
Conclusion
A general framework for sampling configurations on the Medial Axis of free C-space.– Exact and approximate computation of clearance and
penetration depth.– Approximate clearance and penetration depth computation is
applied to general C-space. PRM, MAPRM, MAPRM and MAPM
– MAPRM is the most efficient among all.– MAPRM and MAPM are slightly slower than MAPRM but can
handle more general problems. Low numbers of approximate directions can
result in good estimate of clearance and penetration depth.