Sampling Strategies

Sampling Strategies By David Johnson

Roadmap components local path milestone Probabilistic Roadmaps (PRM) goal configuration start configuration free-space c-obstacle Configuration-space components [Kavraki, Svetska, Latombe, Overmars, 1996]

PRM planners solve complicated problems Complex geometries: obstacles: 43530 polygons Robot: 4053 polygons High dimensional

low density of free samples high density of free samples Main Issue: “Narrow Passages” narrow passage free samples colliding local path colliding samples The efficiency of PRM planners drops dramatically in spaces with narrow passages

Main Issue: “Narrow Passages” • Problems with “narrow passages” are commonly encountered

PRM Sampling • Random • Free points can see a “lot” of free space • Somewhat invariant with dimensionality • Not true for narrow passages • Can we better characterize a narrow passge/difficult problem?

Coverage and Connectivity • Coverage • How well does the roadmap approximate the configuration space • Connectivity • Are regions of C-space connected that can be connected

Bad Good Coverage

Bad Good Coverage almost any point of the configuration space can be connected by a straight line segment to some milestone

Bad Good Connectivity

Bad Good Connectivity 1-1 correspondence between the connected components of the roadmap and those of F

a narrow passage is difficult to define. Easy Difficult Narrow Passages • Connectivity is difficult to capture when there are narrow passages. How to characterize coverage/connectivity? Expansiveness

Visibility All configurations in Free Space that can be seen by a free configuration p p

1-good 0.5-good The domain is only as good as its worst member F is 0.5-good Є-good Every free configuration “sees” at least a єfraction of the free space, є is in (0,1].

0.4-lookout of S 0.4-lookout of S S F\S β-lookout of a subspace S • Subset of points in S that can see at least a β fraction of F\S, β is in (0,1]. S F\S This area is about 40% of F\S

F is ε-good ε=0.5 S F\S Volume(β-lookout) Volume(S)  α=0.2 Definition: (ε,α,β)-expansive • The free space F is (ε,α,β)-expansive if • Free space F isε-good • For each subspace S of F, its β-lookout is at least α fraction of S. ε,α,β are in (0,1] β-lookout β=0.4 F is (ε, α,β)-expansive, where ε=0.5, α=0.2,β=0.4.

Why Expansive? • ε, α, and β measure the expansiveness of a free space. • Bigger ε, α, and β • Easier to construct a roadmap with good connectivity/coverage.

p2 p3 q pn Pn+1 Linking sequence Lookout of V(p) Visibility of p p1 p Pn+1 is chosen from the lookout of the subset seen by p, p1,…,pn

Theorem 1 • Probability of achieving good connectivity increases exponentially with the number of milestones (in an expansive space). • As (ε, α,β) decrease  the number of milestones needs to be increased(to maintain good connectivity).

Theorem 2 • Probability of achieving good coverage, increases exponentially with the number of milestones (in an expansive space).

Probabilistic Completeness In an expansive space, the probability that a PRM planner fails to find a path when one exists goes to 0 exponentiallyin the number of milestones (~ running time). [Hsu, Latombe, Motwani, 97]

Summary • Main result • If a C-space is expansive, then a roadmap can be constructed efficiently with good connectivity and coverage. • Limitation in implementation • No theoretical guidance about the stopping time. • A planner stops when either a path is found or Max steps have been taken.

Guided Sampling 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 Running time: T = nt Tt + na Ta nt– milestones tried Tt<<Ta na– milestones added to graph • Idea: selectively pick milestones • Try more samples (nt ) • Keep fewer samples (na ) by filtering out non-promising samples

OBPRMs A randomized roadmap method for path and manipulation planning (Amato,Wu ICRA’96) OBPRM: An obstacle-based PRM for 3D workspaces (Amato,Bayazit, Dale, Jones and Vallejo)

Roadmap candidate points chosen on C-obstacle surfaces

Finding points on C-objects • Determine a point o inside s • Select m rays with origin o and directions uniformly distributed in C-space • For each ray identified above, use binary search to determine a point on s

Issues • Selection of o in C-obstacle is crucial • To obtain uniform distribution of samples on the surface, would like to place origin somewhere near the center of C-object. • Still skewed objects would present a problem

Main Advantage • Useful in manipulation planning where the robot has to move along contact surfaces • Useful when C-space is very cluttered. • On to the next ideas…

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 = more samples near obstacles

 /2 Parameters, Mixing Methods • indicates how close configurations are to obstacles • Hybrid strategy: mix uniform sampling w/ Gaussian

Algorithm I hence the name • 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 C1 d C2

Some Sampling Issues • How to choose the offset point • Make a random vector with components chosen from a Gaussian distribution • Bias towards the hypercube corners

Narrow Passage uniform sampling took 60 times longer than algorithm 1

Difficult Twist uniform sampling took 13 times longerthan algorithm 1

Twisty Track uniform sampling took 4 times longer than algorithm 1

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 p c2 c1

Clover

Two Squares Gaussian Sampler RBB Sampler

Depression, Zigzags

Bridge vs. Uniform RBB = Bridge

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

Medial-Axis Based PRM MAPRM: A Probabilistic Roadmap Planner with Sampling on the Medial Axis of the Free Space (Wilmarth, Amato, Stiller ICRA’99)

Main Ideas • Beneficial to have samples on the medial axis; however, computation of medial axis itself is costly. • Retraction : takes nodes from free and obstacle space onto the medial axis w/o explicit computation of the medial axis. • This method increases the number of nodes found in a narrow corridor • independent of the volume of corridor • Depends on obstacles bounding it

Approach for Free-Space • Find xo (nearest boundary point) for each point x in Free Space. • Search along the ray xox and find arbitrarily close points xa and xb s.t. xo is the nearest point on the boundary for xa but not xb. • Called canonical retraction map

Extended Retraction Map • Doing only for Free-Space => Only more clearance. Doesn’t increase samples in Narrow Passages • Retract points that fall in Cobstacle also. • Retract points in the direction of the nearest boundary point

Results for 2D case • LEFT: Helpful: obstacle-space that retracts to narrow passage is large • RIGHT: Not Helpful: Obstacle-space seeping into medial axis in narrow corridor is very low

Sampling Strategies