Narrow Passage Problem in PRM

Narrow Passage Problemin PRM

Amirhossein HabibianRobotic Lab, University of Tehran

Advanced Robotic Presentation

• Outline– A brief overview on PRM– What’s Narrow Passage Problem?– Solutions of Narrow Passage

Problem• Workspace-guided strategies• Filtering strategies• Adaptive strategies• Deformation strategies

A brief overview on PRM

• Basic idea of PRMs is:– Compute a very simplified representation

of the free space by sampling configurations at random



• Basic PRM Algorithm:Input: geometry of the moving object & obstaclesOutput: roadmap G = (V, E)

1: V and E .2: repeat3: q a configuration sampled uniformly at random from C.4: if CLEAR(q)then5: Add q to V.6: Nq a set of nodes in V that are close to q.6: for each q’ Nq

7: if LINK(q’,q)then8: Add an edge between q and q’ to E.

CLEAR(q)Is configuration q collision free or not?

LINK(q, q’) Is the straight-line path between q and q’ collision-free?


• PRM planner ignores the exact shape of Free Space. So, it acts like a robot building a map of an unknown environment with limited sensors •The goal is to minimize the expected number of remaining iterations to connect source to goal

Why is PRM planning probabilistic?

A brief overview on PRMA brief overview on PRM


Experimental Convergence Rate of Basic PRM:

The graph plots the percentage of unsuccessful outcomes out of 100 independent runs for the same query



But sometimes Basic PRM doesn’t converge

The plot shows the average running time for Basic PRM to connect the two query configurations q1 and q2, as the corridor width decreases.



Why?

• If Free Space is• expansive, then Basic PRM answers

planning queries correctly with high probability.

• poorly expansive, then there exist queries for which we cannot expect Basic PRM to work well.

Now we’ll see definition of expansiveness



• The visibility set of q є F is the set V(q),V(q) = {q’ є F | FreePath(q, q’) is true}

• Intuitively, є-good free space is a space in which every configuration q has a relatively large visibility set.

• Let є be a constant in (0, 1]. A point q є F is є-good if it sees at least an є-fraction of F.

Some definitions:



Some definitions:• β-LOOKOUT:

Let F’ be a connected component of F and G be any subset of F’. Let β be a constant in (0, 1].The β-LOOKOUT of G is the set of all points in G such that each point sees at least a β-fraction of the complement of G:

β-LOOKOUT(G) = {q є G | μ(V(q)\G) ≥ β*μ(F’\G)}.



Some definitions:• Let є, α, and β be constants in (0, 1]. A connected component F’ of F is (є, α, β)-expansive, if

• every point q є F’ is є-good• for any set M of points in F’

μ(β-LOOKOUT(V(M))) ≥ α*μ(V(M)).

•The free space F is (є, α, β)-expansive, if its connected components are all (є, α, β)-expansive.




What’s Narrow Passage Problem?

Loosely speaking, narrow passages are small regions in free space, which are crucial and whose removal will changes the overall connectivity of the free space

qinit

qgoal

2F

3F

1F

Configuration Space



• As mentioned, Narrow Passage occurs when the free space is poorly expansive.

• In Narrow Passages:• For relatively large amount of є, most

of the configurations aren’t є-good.• For relatively large amount of α, β the

below relation isn’t consistent:μ(β-LOOKOUT(V(M))) ≥

α*μ(V(M)). • So, when the free space is poorly expansive, we are faced with Narrow Passage Problem.




Solutions to Narrow Passage Problem

•Probabilistic Roadmap Methods• Uniform Sampling (original) [Kavraki, Latombe, Overmars, Svestka, 92, 94, 96]• Obstacle-based PRM (OBPRM) [Amato et al, 98]• PRM Roadmaps in Dilated Free space [Hsu et al, 98]• Gaussian Sampling PRMs [Boor/Overmars/van der Steppen 99]• PRM for Closed Chain Systems [Lavalle/Yakey/Kavraki 99, Han/Amato 00]• PRM for Flexible/Deformable Objects [Kavraki et al 98, Bayazit/Lien/Amato 01]• Visibility Roadmaps [Laumond et al 99]• Using Medial Axis [Kavraki et al 99, Lien/Thomas/Wilmarth/Amato/Stiller 99, 03, Lin et al 00]• Generating Contact Configurations [Xiao et al 99] • Single Shot [Vallejo/Remmler/Amato 01] • Bio-Applications: Protein Folding [Song/Thomas/Amato 01,02,03, Apaydin et al 01,02]• Lazy Evaluation Methods: [Nielsen/Kavraki 00 Bohlin/Kavraki 00, Song/Miller/Amato 01, 03]

• Related Methods• Ariadnes Clew Algorithm [Ahuactzin et al, 92]• RRT (Rapidly Exploring Random Trees) [Lavalle/Kuffner 99]

Previous Works Index



• Basic PRM planner employ uniform distribution for sample point generation.

• Most PRM planners employ non-uniform measures that dramatically improve Performance.

How Important is the Sampling Method?



Comparison of three strategies with different sampling measures. The plot shows the average running time over 30 runs.

How Important is the Sampling Method?



• What is the source of information?• Robot and environment geometry • Previous generated Samples

• How to exploit it?• Workspace-guided strategies• Filtering strategies• Adaptive strategies• Deformation strategies

How to find the Sampling Method?






• Narrow passages in Configuration space, are often caused by narrow passages in the workspace• So, we can find Narrow Passages by searching in workspace

• Cell decomposition• Medial-axis transform• [J.P. van den Berg and M. H. Overmars. Using Workspace

Information as a Guide to Non-Uniform Sampling in Probabilistic Roadmap Planners. IJRR, 24(12):1055-1071, Dec. 2005]

• [H. Kurniawati and D. Hsu. Workspace importance sampling for probabilistic roadmap planning. In Proc. IEEE/RSJ Int. Conf. on Intelligent Robots & Systems, pp. 1618–1623, 2004]

Workspace-guided strategies


• Workspace-guided strategies


Cell decomposition: •In this method, the environment is decomposed into black and white cells

• Narrow Passages can be found regarding to this cells, e.g. by image processing algorithms

• It is only applicable to configuration spaces with few dimensions





Cell decomposition:


A cell decomposition of the workspace of a two dimensional example scene



• Cell decomposition


• After finding Narrow Passages in workspace we can simply find them in Configuration Space,e.g. Inverse Kinematic • Use this information to generate more sample points near to Narrow Passages





Input: geometry of the moving object & obstaclesOutput: roadmap G = (V, E)

1: V and E .2: repeat3: NP = FIND //Find Narrow Passages in workspace4: NP_Q = CONVERT(NP)//Convert NPs to Configuration Space5: GENERATE(NP_Q) //Generate Sample points regarding to NP_Q 6: CONTINUE //Similar to BASIC_PRM







Filtering strategies• FreePath, which checks the connection between two configurations, has much higher computational cost than FreeConf

• Filtering increases the number of calls to FreeConf, but yields a smaller set of better placed roadmap nodes and thus reduces the number of calls to FreePath

• It often leads to significant savings in computational time


• Filtering strategies


Filtering strategies1: WHILE TRUE2: Sample a Configuration in the same region3: IF generated sample follows desired pattern4: Retain generated sample as a milestone5: ELSE6: Discard generated sample7: END WHILE

•More sampling work, but better distribution of nodes•Less time is wasted in connecting nodes




Filtering strategies• Some Filtering Strategy Sampling method:

• Gaussian Sampling• Bridge Test• Hybrid




Gaussian Sampling1: Sample a configuration q uniformly at random from

configuration space

2: Sample a real number x at random with Gaussian distribution N[0,s](x)

3: Sample a configuration q’ in the ball B(q,|x|) uniformly at random

4: IF only one of q and q’ is in free space5: retain the one in free space as a node6: ELSE7: retain none

The gain is not in sampling fewer milestones, but in connecting fewer pairs of milestones


• Filtering strategies• Gaussian

Sampling


Uniform vs. Gaussian Sampling

13,000 Milestones created by uniform sampling before the narrow passage was adequately sampled

150 Milestones created by Gaussian sampling



Sampling


Gaussian Sampling• This strategy tries to locate the boundary of Free Space and sample more densely there

• The rationale is that configurations with poor visibility often lie close to the boundary of Free Space



Sampling

http://en.wikipedia.org/wiki/Image:Normal_distribution_pdf.png


Bridge Test1: Sample two conformations q and q’ using Gaussian

sampling technique2: If none is in free space 3: IF qm = (q+q’)/2 is in free space4: retain qm as a milestone5: ELSE6: retain none


• Filtering strategies• Bridge Test


Bridge Test• Main idea is: “Building short bridges is much easier in narrow passages than in wide-open free space”

Gaussian Bridge test




Bridge Test•Bridge Test tries to capture a different kind of geometric pattern




Hybrid1: Sample two configurations q and q’

using Gaussian sampling technique2: IF both are in free space 3: retain one (any of the two) as a

node with low probability (e.g., 0.1)

4: ELSE IF only one is in free space5: retain it as a node with

intermediate probability (e.g., 0.5)6: ELSE IF qm = (q+q’)/2 is in free space7: retain it as a node with high

probability (e.g., 1)


• Filtering strategies• Hybrid


Hybrid

Uniform Sampling Bridge Test

Hybrid Sampling


• Filtering strategies• Hybrid




Adaptive strategies• Use intermediate sampling results to identify regions of the free space whose connectivity is more difficult to capture

• Greater density of milestones in “difficult” regions of the feasible space


• Adaptive strategies


Adaptive strategies• Two-phase connectivity expansion• Tree expansion• Unsupervised on-line learning


• Adaptive strategies


Adaptive strategiesTwo-phase connectivity expansion:

1: Construct initial PRM with uniform sampling

2: Identify milestones that have few connections to their close neighbors\* Performed by counting the numbers of successful and unsuccessful connections of nodes while generation of random samples*\

3: Sample more configurations around them


• Adaptive strategies• Two-phase


Adaptive strategiesTwo-phase connectivity expansion:

(a) the Gaussian strategy(b) the two-phase connectivity expansion strategy


• Adaptive strategies• Two-phase


Adaptive strategies

• Try to expand each portion of Free Space, independently

• hypothesize the location of the boundary of the portion of Free Space represented by the current roadmap

• In each sampling step, try to expand this boundary by sampling new configurations around a node of the roadmap believed to be close to the boundary

•Tree expansion:


• Adaptive strategies• Tree Expansion


Adaptive strategies

• Probability measure for sampling a new configuration is conditioned on the existing roadmap nodes

• Automatically adapts over time

• Do not intentionally try to sample more densely in regions with poor visibility

•Tree expansion:


• Adaptive strategies• Tree Expansion


Adaptive strategies•Unsupervised on-line learning:

• Closest form to the strategy for constructing optimal sampling measures

• Creates and updates an approximate model of Free Space in the form of a collection of Gaussian functions

• Uses this model to sample configurations so that the expected value of a utility function is maximized


• Adaptive strategies• Unsupervised


Adaptive strategies•Unsupervised on-line learning:

•Works as Expectation Maximization Optimization algorithm

•Has a quasi static approach


• Adaptive strategies• Unsupervised




Deformation strategies

We observed that slightly widening difficult narrow passages dramatically improves the efficiency of PRM planning

decreasing width of the narrow passage

planningtime

easy narrow passages

difficult narrow passages


• Deformation strategies



Main Idea:• Deform the free space to make it more expansive

Method:• Free space dilatation





free spacec-obstacle

start

goal

fattened free space

widened passage

Fattening

(1)

Roadmap constructionand repair

(2 & 3)

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



• Free space dilatation



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

• This can be pushed into the pre-processing phase






• Deformation strategies are 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

F

A milestone with small penetration




Questions

?

References

• All references and other useful materials are available in:Http://khorshid.ece.ut.ac.ir/~a.habibian/PRM

• Main references:• 1. On the Probabilistic Foundations of Probabilistic Roadmap

Planning, David Hsu, Jean-Claude Latombe, Hanna Kurniawati

• 2. Using Workspace Information as a Guide to Non-uniform Sampling in Probabilistic Roadmap Planners, Jur P. van den Berg, Mark H. Overmars

• 3. The Gaussian Sampling Strategy for Probabilistic Roadmap Planners, Valkrie Boor, Mark H. Overmars, A. Frank van der Stappen

• 4. The Bridge Test for Sampling Narrow Passages with Probabilistic Roadmap Planners

• 5. Smoothed Analysis of Probabilistic Roadmaps, Siddhartha Chaudhuri,Vladlen Koltun

http://khorshid.ece.ut.ac.ir/~a.habibian/PRM

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Narrow Passage Problem in PRM