Sampling Strategies for Narrow Passages

Presented by Irena PashchenkoCS326A, Winter 2004

Motivation Two major costs of PRMs:

FREE - Check if sample point is in free space JOIN – Check if path between milestones is in free space

JOIN is 10 to 100 times slower than FREE

Idea: selectively pick milestones Try more samples (nt ) Keep fewer samples (na ) by filtering out non-promising samples

Filtering occurs before creating edges

Running time: T = nt Tt + na Ta

nt – milestones tried Tt << Ta

na – milestones added to graph

Two Similar Approaches

The Gaussian Sampling Strategy for PRMs Valerie Boor, Mark H. Overmars, A. Frank van der

Stappen ICRA 1999

The Bridge Test for Sampling Narrow Passages with PRMs

David Hsu, Tingting Jiang, John Reit, Zheng Sun ICRA 2003

Overview Gaussian Strategy

Goals Two Proposed Algorithms Experimental Results

Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

Goal

More samples in hard regions

Sampling density of each point = Conv (N(c, ) , Obstacles)

= more samples near obstacles

Parameters, Mixing Methods indicates how close configurations are to

obstacles

/2

s.t. most configs are at length of the robot

Hybrid strategy: mix uniform sampling w/ Gaussian

Algorithm I loop

c1 = random config. d = distance sampled from Gaussian c2 = random config. distance d from c1

if Free(c1) and !Free(c2), add c1 to graph if Free(c2) and !Free(c1), add c2 to graph

Intuition: Pick free points near blocked points Avoid adding configurations in large empty

regions

hence the name

Algorithm II Motivated by the case of convex obstacles:

Difficult places lie close to at least two obstacles

loop c1 = random config. d1,d2 = distances sampled from Gaussian c2,c3 = random configs distance d1,d2 from c1

if Free(c1) and !Free(c2) and !Free(c3), add c1

if !Free(c1) and Free(c2) and !Free(c3), add c2

if !Free(c1) and !Free(c2) and Free(c3), add c3

Narrow Passage

uniformsampling

took60 timeslongerthan

algorithm 1

Narrow Passage

uniformsampling

tookless time

thanalgorithm II

Difficult Twist

uniformsampling

took13 timeslongerthan

algorithm 1

Twisty Track

uniformsampling

took4 timeslongerthan

algorithm 1

Is Gaussian Sampling Appropriate for Finding Narrow Passages? In n-dimensional configuration space,

the obstacle boundaries still form an (n-1)-D subspace!

Overview Gaussian Strategy

Goals Two Proposed Algorithms Experimental Results

Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

Bridge Test loop

c1 = random config. if Free(c1), continue (restart the loop) d = distance sampled from

Gaussian c2 = random config. distance d from c1

if Free(c2), continue (restart the loop) p = midpoint(c1,c2) if Free(p), add p

c1

p c2

Bridge vs. Gaussian Paper mentions Gaussian but no

comparison Want to compare:

Expected # of calls to free (lower is better) Expected # points generated (higher better) If points can be reused in a hybrid strategy Quality of sampled points

Let p be prior probability of Free

Bridge vs. Gaussian

Strategy Calls to Free per

iter

Samples per iter

Reuse Points

Point Quality

Gaussian 1

2 2p(1-p) yes OK

Gaussian 2

3 - p2 3p(1-p)2

yes Better

Bridge 1 + (1-p) + (1-p)2

p(1-p)2 yes Best

Smallest

Clover

Two Squares

RBB SamplerGaussian Sampler

Depression, Zigzags

Bridge vs. Uniform

RBB = Bridge

Combined Algorithm (JCL)Loop until N milestones have been generated Pick a configuration q1 uniformly at random in C

and a configuration q2 uniformly at random in a neighborhood of q1

If q1 and q2 are both in collision then If (q1+q2)/2 is collision-free, retain it as a milestone

Else if q1 (or q2) is in collision

Then retain q2 (or q1) as a milestone with high probability p’ Else retain q1 or q2 as a milestone with low probability p

Conclusion Better configurations

= fewer configurations= less edge computations= faster running time

Gaussian Points near obstacles Points near two obstacles

Bridge Points between parts of obstacles

Still un-tested in high-dimensional spaces