Probabilistic Roadmaps

Sujay BhattacharjeeCarnegie Mellon University

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Overview

What is a Probabilistic Roadmap (PRM)? Why PRM? Algorithm

Proof Accuracy Examples

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PRM

A planning method which computes collision-free paths for robots of virtually any type moving among stationary obstacles.

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Previous Approaches

Visibility Graphs Voronoi Diagrams Cellular Decomposition Potential Fields

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Visibility Graphs

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Voronoi Diagrams

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Exact Cellular Decomposition

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Approximate Cellular Decomposition

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Potential Fields

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Problems before PRMs

Hard to plan for many d.o.f robots Computation complexity for high dimensional

configuration spaces would grow exponentially Potential fields run into local minima Complete, general purpose algorithms are at best

exponential and have not been implemented

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Example

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Advantages of PRM

Fast collision checking techniques Avoid computing an explicit representation of the

configuration space Great for many degrees of freedom

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Phases of PRM

Learning Phase• Construction Step• Expansion Step

Query Phase

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The Learning Phase

Construct a probabilistic roadmap by generating random free configurations of the robot and connecting them using a simple, but very fast motion planner, also know as a local planner

Store as a graph whose nodes are the configurations and whose edges are the paths computed by the local planner

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The Query Phase

Find a path from the start and goal configurations to two nodes of the roadmap

Search the graph to find a sequence of edges connecting those nodes in the roadmap

Concatenating the successive segments gives a feasible path for the robot

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Learning Phase (Construction Step) Initially, the graph R = (N, E) is empty Then, repeatedly, a random free configuration

is generated and added to N For every new node c, select a number of

nodes from N and try to connect c to each of them using the local planner.

If a path is found between c and the selected node n, the edge (c,n) is added to E. The path itself is not memorized.

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How do we determine a random free configuration? We want the nodes of R to be a rather uniform

sampling of CS Draw each of its coordinates from the

interval of values of the corresponding degrees of freedom. (Use the uniform probability distribution over the interval)

Check for collision both with robot itself and with obstacles

If collision free, add to N, otherwise discard

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What’s the local path planner?

Can pick different ones Nondeterministic – have to store local paths

in roadmap Powerful - slower but could take fewer nodes

but takes more time Fast - less powerful, needs more nodes

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Go with the fast one

Need to make sure start and goal configurations can connect to graph, which requires a somewhat dense roadmap

Can reuse local planner at query time to connect start and goal configurations

Don’t need to memorize local paths

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Create random configurations

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Update Neighboring Nodes’ Edges

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End of Construction Step

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Expansion Sometimes R consists of several large and small

components which do not effectively capture the connectivity of Cfree

The graph can be disconnected at some narrow region

Assign a positive weight w(c) to each node c in N

w(c) is a heuristic measure of the “difficulty” of the region around c. So w(c) is large when c is considered to be in a difficult region. We normalize w so that all weights together add up to one. The higher the weight, the higher the chances the node will get selected for expansion.

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How to choose w(c) ? Can pick different heuristics

Count number of nodes of N lying within some predefined distance of c.

Check distance dc from c to nearest connected component not containing c.

Use information collected by the local planner. (If the planner often fails to connect a node to

others, then this indicates the node is in a difficult area).

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One Example: At the end of the construction step, for each node

c, compute the failure ratio rf(c) defined by:

where n(c) is the total number of times the local planner tried to connect c to another node and f(c) is the number of times it failed.

How to choose w(c) ?

1)()()(

cncfcrf

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How to choose w(c) ? At the beginning of the expansion step, for every node c in N, compute w(c) proportional

to the failure ratio.

Na

f

f

arcrcw

)()()(

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Now that we have weights… To expand a node c, we compute a short random-bounce

walk starting from c.This means Repeatedly pick at random a direction of motion in C-

space and move in this direction until an obstacle is hit. When a collision occurs, choose a new random direction. The final configuration n and the edge (c,n) are inserted

into R and the path is memorized. Try to connect n to the other connected components like

in the construction step. Weights are only computed once at the beginning and

not modified as nodes are added to R.

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Expansion Step

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End of Expansion Step

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The Query Phase

Objective:

To find a path between an arbitrary start and goal configuration, using the roadmap constructed in the learning phase.

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Select start and goal

Start Goal

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The Query Phase (contd.) Let start configuration be s Let goal configuration be g Try to connect s and g to Roadmap R at two nodes

ŝ and ĝ, with feasible paths Ps and Pg. If this fails, the query fails.

Compute a path P in R connecting ŝ to ĝ. Concatenate Ps ,P and reversed Pg to get the

final path.

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How do we find Ps, Pg, and P?

Consider nodes in R in order of increasing distance from s (according to D) and try to connect s to each of them with the local planner, until one succeeds.

Random-bounce walks: Try to connect to R with the local planner.

Re-compute paths found during the learning phase Selective collision check.

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Final Step If there are multiple connected components,

sometimes the free C-space is not itself connected or the roadmap is not dense enough.

In this case, we try to connect s and g to nodes in the same component. If we can’t do this, we end in failure. This is usually detected pretty quickly.

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Select start and goal

Start Goal

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Connect Start and Goal to Roadmap

Start Goal

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Find the Path from Start to Goal

Start Goal

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What if we fail?

Maybe the roadmap was not adequate. Could spend more time in the Learning Phase Could do another Learning Phase and reuse R

constructed in the first Learning Phase. In fact, Learning and Query Phases don’t have to be executed sequentially.

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Theoretical Analysis

Analyze a simple PRM model and attempt to find factors which affect and control the performance of the PRM.

Define -goodness, -Lookout and expansive space. Derive more relations linking probability of

failure to controlling factors Finding Narrow passages with PRM

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The Simplified Probabilistic Roadmap Planner (s-PRM)The parameters of our 2D model are: Free Space F: An arbitrary open subset of the

unit square W=[0,1]2

The Robot: A point free to move in F The Local Connector: It takes the robot from

point a to point b along a straight line and succeeds if the straight line segment ab is contained in F

The collection of Random Configurations:Collection of N independent points uniform in F

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s-PRM: Probability of FailureTake any two pointsWe assume that they can be connected by a continuous path such that

where and

We will try to find upper bounds for the probabilityof failure to find such a path between a and bwhen we assume a) minimum distance from obstacles b) varying (mean) distance from obstacles

Fba ,

FL ]:0[:

a)0(

bL )(

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Theorem 1(Upper Bound Involving Minimum Distance) Let be a path of (Euclidean) length L.

Then the probability that s-PRM will fail to connectthe points a and b is at most

Where is a constant.Here, we assume minimum distance from theobstacles.

FL ]:0[:

NRRL )1(2 2

||4 F

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Theorem 1: Proof

Pr[FAILURE] Pr[Some Ball is Empty] b d Pr [The j-th ball is empty] c

xj+1

xj R/2

In 2D, Area of the ball with radius R/2 = R2/4

R a

i.e. Pr[FAILURE]

1

1

n

j

N

R

FArea

RL

||||112 2/

NRRL )1(2 2

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Theorem 2(A bound that exploits varying distance)

b

(tk+1)

(tk)

a

Again, the idea of the proofis to cover the curve withnot-too-many balls thatoverlap to a certain extent.

We already know from Theorem 1 that

Pr[FAILURE] (1-rk2)N …(1)

And now define the integral

I = …(2)

1

1

n

j

L N

dttr

tr

0

2

)())()4/(1(

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Theorem 2(A bound that exploits varying distance)

Let be a path of (Euclidean) length L.

Distance of (t) from the obstacles isr(t)

Then the probability of failure of s-PRM is at most

FL ]:0[:

|)(|inf xtx

L N

dttr

tr

0

2

)())()4/(1(6

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What the results tell usRearrange the inequalities derived from Theorems 1 and 2using the inequality , for

1) Bound of Theorem 1 becomes

Pr[FAILURE]

2) Bound of Theorem 2 becomes

Pr[FAILURE]

xex 1 0x

)exp(2 2NRRL

dttr

NtrL

0

2

)())()4/(exp(6

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What the results tell usThe results give us an idea of what factors control failure: Dependence on N is exponential Dependence on L is linear Tweaking these factors can give us desired success rate PRM avoids creating large number of nodes in

areas where connections are obtained easily

However, The bounds computed here are not very easy to use as

they depend on the properties of postulated connecting path (t) from a to b (difficult to measure a-priori)

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Some Definitions - goodnessA configuration q F is – good if

For any subset denotes its volume.F is the Free Space.

denotes the set of all free configurations seen by q.

)())(( Fq

)(q

)(, SCS

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Some Definitions - goodnessA narrow passage in a -good space.

F1 F2

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Definitions (contd.)-LOOKOUT

Let be a constant in [0,1] and S be a subset of acomponent E of the free space F.

The -LOOKOUT of S is the set

)}\()\)((|{)( SESqSqSLookout

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Definitions (contd.)expansive spacesLet , , be constants in [0, 1]The free space is (, , ) expansive if it is -goodand, for every connected subset S, we have

We can abbreviate “(, , ) expansive”by simply “expansive”

)())(( SSLOOKOUT

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Definitions (contd.)Example:An expansive free space where , , ~ /W

w w w

W

W

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Roadmap CoverageAssume that F is - good. Let be a constant in [0,1]Let k is a positive real such that for any x [0,1]

If N is chosen such that

Then the roadmap generates a set of milestones thatadequately covers F, with probability at least 1-

This still does not allow us to compute N.However, it tells us that the although adequate coverage isis not guaranteed, the probability that this happensDecreases exponentially with the number of milestones N.

2/)1( )/2log()/( xx xxk

)2log1(log

KN

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Roadmap ConnectivityAssume that F is (, , ) expansive .Let be a constant in [0,1]. If N is chosen such that

Then with the probability 1 - , the roadmap generates a roadmap such that no two of its components lie on the same component of F.

This tells us that the probability that a roadmap does notadequately represent F decreases exponentially with thenumber of milestones N. Also, number of milestones neededincreases moderately when , and increase.

468log16

N

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Finding Narrow PassagesAlgorithm: Dilate free space F to F’ by allowing penetration

distance into the obstacles Construct Roadmap R’ in F’ Push milestones and links that do not lie in F

into F by performing local re-sampling operations. The outcome is a Roadmap R in F.

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Finding Narrow PassagesExample:

F F -> Free Space

P -> Narrow Passage

P

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Finding Narrow PassagesExample:

F’ F

P

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Finding Narrow PassagesExample:

F’ F

P

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Finding Narrow PassagesExample:

F’ F

2’ 2 P 1

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Example – A Planar Articulated Robot(Customized Implementation)

Consider and arbitrary number of revolute joints Links, which can be any polygons, are denoted L1 to Lq (q=

5 for our example) Points J1 through Jq are the revolute joints, Jq+1 is the

endpoint. J1 is the base, which may or may not be fixed A self-collision configuration exists when two nonadjacent

links of the robot intersect each other, otherwise the range of each joint prevents collision with the adjacent link

So, free C-space is constrained by the obstacles and self-collisions.

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Example – Results This is a fixed-based

articulated robot with 7 revolute degrees of freedom.

Each configuration is tested with a set of 30 goals with different learning times.

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With expansion

Withoutexpansion

Results

With General Planner

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PRM - Other Applications Can be used on nonholonomic robots

Need to use special local planners that take limitations into account

Can be used for robots with almost infinite dof, like robots made of flexible material

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PRM – Other Work Spends a lot of time planning paths that

will never get used Heavily reliant on fast collision checking

An attempt to solve these is made with Lazy PRMs Tries to minimize collision checks Tries to reuse information gathered by queries

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References On Finding Narrow Passages with Probabilistic Roadmap Planners.

Hsu, Kavraki, Latombe, Motwani, Sorkin: Proceeds of the 3rd Workshop on Algorithmic Foundations of Robotics, 1998.

Analysis of Probabilistic Roadmaps for Path Planning. Kavraki, Kolountzakis, Latombe: IEEE Transactions on Robotics and Automation, Vol. 14, No. 1, Feb. 1998

Probabilistic Roadmaps for path planning in high dimensional configuration spaces. Kavraki, Svestka, Latombe, Overmars, IEEE Transactions on Robotics and Automation, Vol. 12, No. 4, Aug. 1996

Probabilistic Roadmaps for Robotic Path Planning. Kavraki and Latombe, In K. Gupta and P. del Pobil, editors, Practical Motion Planning in Robotics: Current Approaches and Future Directions, pages 33--53. John Wiley, 1998.

Latombe, Robot Motion Planning, Kluwer Academic Pub, 1991