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Finding “Narrow Passages” withProbabilistic Roadmaps:

The Small-Step Retraction Method

Mitul Saha and Jean-Claude Latombe

Research supported byNSF, ABB and GM

Artificial Intelligence LabStanford University

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Probabilistic Roadmaps (PRM)

[Kavraki, Svetska, Latombe, Overmars, 1996]

startconfiguration

goalconfiguration

free-spacec-obstacleConfiguration-space

components

milestonelocal path

Roadmapcomponents

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PRM planners solve complicated problems

Complex geometries:obstacles: 43530 polygonsRobot: 4053 polygons

High dimensional

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Main Issue: “Narrow Passages”

free samples

colliding samples colliding local path

narrow passagelow density of free samples

high density of free samples

The efficiency of PRM planners drops dramatically in spaces with narrow passages

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• Problems with “narrow passages” are commonly encountered

Main Issue: “Narrow Passages”

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Main Issue: “Narrow Passages”

?

Proposed strategies:

Filtering strategies, e.g., Gaussian sampling [Boor et al. ‘99] and bridge test [Hsu et al. ‘03] rely heavily on rejection sampling

Retraction strategies, e.g., [Wilmart et al. ‘99][Lien et al. ‘03] waste time moving many configurations out of collision

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Motivating Observation

decreasing width of the narrow passage

planningtime

easy narrow

passages

difficult narrow passages

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Roadmap constructio

nand repair

fattened free space

widened passage

Fattening

free spacec-obstacle

start

goal

Small-Step Retraction Method

1. Slightly fatten the robot’s free space2. Construct a roadmap in fattened free space3. Repair the roadmap into original free space

(1) (2 & 3)

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Small-Step Retraction Method

Roadmap constructio

nand repair

fattened free space

widened passage

Fattening

free spacec-obstacle

start

goal

-Free space can be “indirectly” fattened by reducing the scale of the geometries (usually of the robot) in the 3D workcell with respect to their medial axis

-This can be pushed into the pre-processing phase

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Small-Step Retraction Method

Roadmap constructio

nand repair

fattened free space

widened passage

Fattening

free spacec-obstacle

start

goal

Repair during construction

Repair after construction

goal

PessimistStrategy

OptimistStrategy

fattenedfree space

startstart

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Small-Step Retraction Method

Roadmap constructio

nand repair

fattened free space

widened passage

Fattening

free spacec-obstacle

start

goal

Repair during Repair during constructionconstruction

Repair after Repair after constructionconstruction

fattenedfree space

goal

PessimistStrategy

OptimistStrategy

- Optimist may fail due to “false passages” but Pessimist is probabilistically complete

- Hence Optimist is less reliable, but much faster due to its lazy strategy

start

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Small-Step Retraction Method

Roadmap constructio

nand repair

fattened free space

widened passage

Fattening

free spacec-obstacle

start

goal

Repair during construction

Repair after construction

goal

PessimistStrategy

OptimistStrategy

Integrated planner:

1. Try Optimist for N time. 2. If Optimist fails,

then run Pessimist

fattenedfree space

start

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Quantitative Results• Fattening “preserves” topology/

connectivity of the free space• Fattening “alters” the topology/

connectivity of the free space

TimeSSRP(secs)

TimeSBL

(secs)

(a) 9.4 12295

(b) 32 5955

(c) 2.1 41

(d) 492 863

(e) 65 631

(f) 13588 >100000

TimeSSRP(secs)

TimeSBL

(secs)

(g) 386 572

(h) 3365 >100000

(a) (b) (c)

(d) (e)(f)

(g) (h)Alpha 1.0

Alpha 1.1

Upto 3 orders of magnitude improvement in the planning time

was observed

Our planner

A recent PRM planner

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Quantitative Results

• Test environments “without” narrow passages– SSRP and SBL have similar performance

TimeSSRP

TimeSBL

(i) 1.68 1.60

(j) 2.59 2.40

(i) (j)

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Conclusion

• SSRP is very efficient at finding narrow passages and still works well when there is none

• The main drawback is that there is an additional pre-computation step

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Finding “Narrow Passages” withProbabilistic Roadmaps:

The Small-Step Retraction Method