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.