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Visibility Graphs

May 2012

Shmuel WimerBar-Ilan Univ., Eng. Faculty

Technion, EE Faculty

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We would like to find a good path, rather than any feasible path as done in motion planning.Shortest path is usually good, but not necessarily the best. Number of turns is important, speed is important, etc.

We consider only translating planar robot and seek shortest path.May 2012

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Roadmap Reminder

Get trapezoids of free area

Free area obtained Minkowski sum

Roadmap construction

Find shortest path in roadmap

This is not necessarily the shortest pathMay 2012

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startP

goalP

Roadmap

When BFS is used, path of minimum number of vertices is found.

For weighted roadmap Dijkstra’s algorithm can be used.

Shortest path of roadmap

Shortest path

May 2012

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start goal Any shortest path between and

among the set of polygonl objects is a polygonalpath whose inner vertices are of .

P P:Lemma

SS

If the shortest path was not polygonal, it would

have a point interior of freespace (not a vertex of ), andnot contained in any linesegment of (otherwise must be polygonal).

p

:Proof

S

May 2012

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A disc around can be found and a short cut canbe established, yielding a paths shorter than .

p

May 2012

is polygonal, and assume it has a vertex interiorof free area. As before, a disc and a shortcut can beestablished, yielding a shorter path.

Similarly, cannot lie on the interior of an obstacle'sedge. Otherwise there was a disc partially intersectingthe interior of the free area and a shortacut couls befound.

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Visibility Graph: Nodes are the vertices in S. Edge is defined

for each vertices visible to each other.

May 2012

start goal

*vis

*start goal

The shortest path between  and

among a set  of disjoint polygonal obstacles

consists of arcs of the visibility graph ,

where  , .

P P

G

P P

Corollary

S

S

S S

:

startPgoalP

Shortest path

May 2012 8

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start goal

vis start goal

vis

start goal

( , , ) {

VisibilityGraph ( , ) ;

Assign weigh // Euclidet to each , ; Find shortest path from to with

an length Dijkstr

uv

void p p

G p p

e u v Gp p

ShortestPath

S

S

a's algorithm ;

}

With n polygonal objects and O(n) nodes, naïve computation

of Gvis would take O(n3) time.

Dijkstra’s algorithm finds shortest path between two nodes of

n nodes and k arcs graph in O(nlogn + k) time.

It is possible to compute Gvis in O(n2logn) time.

May 2012

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Computing the Visibility Graph

( ) { Initialize , , is all vertices of , ;

{ VisibleVertices( , ) ; { add , to }

} , ; }

graphG V E V E

u VW u

v W e u v E

return G V E

VisibilityGraph S S

S

Checking the visibility between two vertices, not much can be done. All objects must be tested for intersection, taking O(n) time. By ordering the vertices it is possible to accelerate tests by using info obtained in previous tests.

May 2012

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1

2

3

4

5

6

7

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1112

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p

Scanning Ray – Rotational Planar Sweep

May 2012

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p

w

The intersection status of edges with the scanning ray is stored

in a balanced binary tree T, were edges are stored at leaves. It

is possible by bi-section to locate the vertex w with respect to

the edges. It is visible from p if it falls left to the leftmost

(nearest to p) leaf.

If p and w are of the same

polygon w is invisible.

May 2012

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1e

2e3e 4e

5e6e

1e

2e

3e

4e

5e 6e

1e

2e

3e

4e

5e

Scanning ray maintains a binary balances tree T. Vertices are

first sorted in clockwise order. Vertices are checked for visibility

and then incident edges are inserted and / or deleted to / from

T, depending on their position with respect to the scanning ray.

May 2012

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All such operations take O(logn) time per vertex.

T is initialized by inserting all the edges intersecting the ray

of negative X axis (angle 0), an operation taking 0(nlogn)

time.

Vertices are sorted clockwise starting from angle 0.

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1w

2w 4w

3w

p

5w

6w8w

7w

a

b

c

d

e

f h

g

initialize : , , ,h f d bT

1 : , , ,w h g d bT

2 : , , ,w h g d cT

3 : ,w d cT

4 : w T

5 : ,w h eT

6 : , , ,w h e d aT

7 : , , ,w h f d aT

8 : , , ,w h f d bT

May 2012

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1

( , ) { 1. Sort vertices of clockwise w.r.t . In case of tie closer precedes farther. Let , , be the

points ; 2. Construct balanced search tree sto

n

points pp

w w

VisibleVertices

SS

T

// initialize visible point

ring edge intersections with scanning ray ; Initialize ; ; ( 1

s// traverse all points to ) {

Visible ( , ) // visibility tes{ } ti i

Wfor i n

if p w W W w

T

add to startin / / edges / at delete from end . }

g }

g in;

iwreturn W

T

May 2012

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1iw

p

iw

1iw

p

iw1iw

p

iw

1iw

p

iw

wi is not visiblewi is visible

All special cases can be detected at visibility test of wi

May 2012

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1

// ordinary hiding tes

( , ) ( intersects 's object, locally at ) {return }

(( 1 ) || ( is not on )) { find in the leftmost edge

t

i

i i i

i i

boolean p wif pw w w false

else if i w pw

Visible

T

1

1

// colinea

; (( exists) && (intersects )) {return } {return } }

( is not visible) {return } {

r , ,

find in e

i i

i

i

eif

p w

e pw e falseelse true

else if w falsee

w

lse

T 1dge intersecting ; ( exists) {return } {return }}}

i ie w wif e falseelse true

May 2012

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Probabilistic Roadmaps

Planar problems with translational robot (2 degrees of freedom)

are simple. Finding visibility graph explodes with degrees of

freedom increase.

- configurations space (coordinates, rotation angle, etc.) - a configuration

- collision free configurations

: a local planner which for every ,

returns eithe

free

free free free free

Qq QQ

Q Q q q Q Q

r a collision free path from to , or if such one not exists.

: 0 is a distance function

q q

d Q Q

R

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(# nodes: , # closest neighbors ) { ; ; while

// initialization// find nodes i( ) {

randomly dra

n

w ; if ( ) { }

} { g

free

free

graph n kV E

V n

q Q q Q V V

n

q

q V

Q

ConstructRoadMap

et , the closest neighbors according to }

{

if ( ( , ) && ( , ) ) { , }

}}

q

q

N V k d

q N

q q E q q E E q q

Random roadmap construction

May 2012

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Sampling Strategies

Random sampling works well in many practical cases involving

robots with large number of degrees of freedom. It is not well

performing when essential narrow passage exists, or when

more intensive sampling is required near obstacles.

Quasirandom is a deterministic alternative to random sampling,

ensuring better uniformity.

Let be a sample of points in a space , .

How uniformly covers ?

P X P N

P X

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is a collection of axis-alined rectangular subsetsof . is a rectangle. is a measure.X R R

R

May 2012

The of with respect to range space

over is defined as , sup .R

PP RR

X D PX N

R

R

R

discrepancy

Discrepancy provides a measure of how uniformly points

are distributed over a space X

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The largest portion of that contains no points of is called the of point set with

respect to the metric , sup min , .p Px X

XP P

P x p

dispersion

July 2011

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For a sample of points in -dimentional unit cube1there exists , .

2 d

P N d

PN

**

1So to obtain dispersion at least .2

To minimize dispersion grid sampling is the best.

d

N

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For 0,1 sequence gives a

set of samples that minimizes both dispersion anddiscrepancy.

X Van der Corput

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Its sample is generated by using the binary

representation 2 , 0,1 .

The element, , is defined as

2 .

th

ii ii

th

iii

n

n a a

n n

n a

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1 1 3 1 5The first 16 samples are 0, , , , , ,2 4 4 8 8

3 7 1 1 9 5 13 3 11 7 15, , , , , , , , , , 8 8 8 16 16 16 16 16 16 16 16

May 2012

Van det Corput sequence can be used only for thereal line. generalizes it to dimensions.

dHalton sequence

Given a sequence = 2,3,5,7, of prime

numbers, an integer is represented in base

by , 0,1, , 1 .

i

j

iij j ij ji

b

n b

n a b a b

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1

1 2

is defined as .

The sample is then defined by thecoordinates

, , , .

iij ji

th

n

j j

d

b b

b b b

n a b

n

p n n n

May 2012