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CS 326 A: Motion PlanningCS 326 A: Motion Planninghttp://robotics.stanford.edu/~latombe/cs326/2002

Configuration Space –Configuration Space –Basic Path-Planning MethodsBasic Path-Planning Methods

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What is a Path?What is a Path?

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ool: Configuration Spaceool: Configuration Space!C-Space C"!C-Space C"

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q=(q1,…,qn)

Configuration SpaceConfiguration Space

qq11

qq22

qq33

qqnn

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#efinition#efinition

A robot configuration is a specificationof the positions of all robot points

relative to a fixed coordinate syste

!sually a configuration is expressed as

a "vector# of position$orientationparaeters

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reference point

$igid $o%ot &'a(ple$igid $o%ot &'a(ple

% 3&paraeter representation' q = (x,y,θ)% n a 3& *or+space q *ould be of the for

(x,y,,α,β,γ )

'

) θ

robotreference direction

*or+space

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Articulated $o%ot &'a(pleArticulated $o%ot &'a(ple

qq11

qq22

q = (q1,q2,…,q1-)

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Configuration Space of a $o%otConfiguration Space of a $o%ot

.pace of all its possible configurations/ut the topology of this space is usuallynot that of a 0artesian space

C = S1 x S1

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C = S1 x S1

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Structure of Configuration SpaceStructure of Configuration Space

t is a anifoldor each point q, there is a 1&to&1 apbet*een a neighborhood of q and a

0artesian space R n, *here n is thediension of 0

his ap is a local coordinate syste

called a chart0 can al*ays be covered by a finite nuberof charts .uch a set is called an atlas

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&'a(ple&'a(ple

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reference point

Case of a Planar $igid $o%otCase of a Planar $igid $o%ot

% 3&paraeter representation' q = (x,y,θ) *ith

θ ∈ 4-,2π) *o charts are needed% 5ther representation' q = (x,y,cosθ,sinθ)c&space is a 3& cylinder $2 x .1

ebedded in a 6& space

'

) θ

robotreference direction

*or+space

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$igid $o%ot in 3-# Wor*space$igid $o%ot in 3-# Wor*space

% q = (x,y,,α,β,γ )

% 5ther representation' q = (x,y,,r11,r12,…,r33) *herer11, r12, …, r33 are the eleents of rotation atrix7' r11 r12 r13

r21 r22 r23

r31 r32 r33

*ith' 8 ri129ri229ri32 = 18 ri1r :1 9 ri2r2: 9 ri3r :3 = -8 det(7) = 91

he c&space is a ;& space (anifold) ebeddedin a 12& 0artesian space t is denoted by 73x.5(3)

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Para(eteri+ation of S,!3"Para(eteri+ation of S,!3"

% <uler angles' (φ,θ,ψ)

% !nit quaternion' (cos θ$2, n1 sin θ$2, n2 sin θ$2, n3 sin θ$2)

xx

y

zz

xxyy

zz

φφ

x

y

z

θ

xx

yy

zz

ψ ψ

2 3 .

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Metric in Configuration SpaceMetric in Configuration Space

A etric or distance function d in 0 is a apd' (q1,q2) ∈ 02 d(q1,q2) -

such that'

8 d(q1,q2) = - if and only if q1 = q2

8 d(q1,q2) = d (q2,q1)

8 d(q1,q2) > d(q1,q3) 9 d(q3,q2)

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Metric in Configuration SpaceMetric in Configuration Space

&'a(ple:% 7obot A and point x of A

% x(q)' location of x in the *or+space *hen A isat configuration q

% A distance d in 0 is defined by'd(q,q?) = axx∈A @@x(q)&x(q?)@@

*here @@a & b@@ denotes the <uclidean distancebet*een points a and b in the *or+space

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Specific &'a(ples in $Specific &'a(ples in $22 ' S' S

q = (x,y,θ), q? = (x?,y?,θ?) *ith θ, θ? ∈ 4-,2π)

α = in|θ−θ’| , 2π−|θ−θ’|B

d(q,q?) = sqrt4(x&x?)2 9 (y&y?)2 9 α2C

d(q,q?) = sqrt4(x&x?)2 9 (y&y?)2 9 (αρ)2C*here ρ is the axial distance bet*een thereference point and a robot point

θθ’’

θθ

α

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/otion of a Path/otion of a Path

A path in 0 is a piece of continuous curveconnecting t*o configurations q and q?'τ

' s ∈ 4-,1C τ (s) ∈ 0 s? → s ⇒ d(

τ

(s),τ

(s?)) → -

q1

q3

q0

qn

q4

q2

τ(s)

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,ther Possi%le Constraints on Path,ther Possi%le Constraints on Path

inite length, soothness, curvature, etc… A tra:ectory is a path paraeteried by tie'

τ

' t ∈ 4-,C τ

(t) ∈ 0

q1

q3

q0

qn

q4

q2

τ(s)

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,%stacles in C-Space,%stacles in C-Space

A configuration q is collision&free, or free, if therobot placed at q has null intersection *ith theobstacles in the *or+space

he free space is the set of freeconfigurations

A 0&obstacle is the set of configurations *herethe robot collides *ith a given *or+space

obstacleA configuration is sei&free if the robot at thisconfiguration touches obstacles *ithout overlap

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#isc $o%ot in 2-# Wor*space#isc $o%ot in 2-# Wor*space

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$igid $o%ot ranslating in 2-#$igid $o%ot ranslating in 2-#

CB 0 B A 0 1%-a a∈A %∈B4

a1

b1

b1&a1

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5inear-i(e Co(putation of5inear-i(e Co(putation ofC-,%stacle in 2-#C-,%stacle in 2-#

(convex polygons)

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$igid $o%ot ranslating and$igid $o%ot ranslating and$otating in 2-#$otating in 2-#

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C-,%stacle for Articulated $o%otC-,%stacle for Articulated $o%ot

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ree and .ei&ree Dathsree and .ei&ree Daths

A free path lies entirely in the freespace

A sei&free path lies entirely in thesei&free space

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$e(ar* on ree-Space opolog)$e(ar* on ree-Space opolog)

% he robot and the obstacles are odeled asclosed subsets, eaning that they contain theirboundaries

% 5ne can sho* that the 0&obstacles are closed

subsets of the configuration space 0 as *ell% 0onsequently, the free space is an open subset

of 0 Eence, each free configuration is the centerof a ball of non&ero radius entirely contained in

% he sei&free space is a closed subset of 0 tsboundary is a superset of the boundary of

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/otion of 7o(otopic Paths/otion of 7o(otopic Paths

*o paths *ith the sae endpoints arehootopic if one can be continuously deforedinto the other

7 x .1 exaple'

τ

1 and τ2 are hootopicτ1 and

τ3 are not hootopic

n this exaple, infinity of hootopy classes

8

89

τ

τ

τ

3

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Connectedness of C-SpaceConnectedness of C-Space

0 is connected if every t*o configurations can beconnected by a path

0 is siply&connected if any t*o paths

connecting the sae endpoints are hootopic<xaples' $2 or $3

5ther*ise 0 is ultiply&connected<xaples' .1 and .5(3) are ultiply& connected'& n .1, infinity of hootopy classes& n .5(3), only t*o hootopy classes

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C-,%stacle for Articulated $o%otC-,%stacle for Articulated $o%ot

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Classes of 7o(otopic ree PathsClasses of 7o(otopic ree Paths

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Motion-Planning ra(eor*Motion-Planning ra(eor*

0ontinuous representation(configuration space forulation)

iscretiation

Fraph searching(blind, best&first, AG)

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Path-Planning ApproachesPath-Planning Approaches

; $oad(ap7epresent the connectivity of the free spaceby a net*or+ of 1& curves

2; Cell deco(positionecopose the free space into siple cellsand represent the connectivity of the freespace by the ad:acency graph of these cells

3; Potential fieldefine a function over the free space thathas a global iniu at the goal configurationand follo* its steepest descent

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$oad(ap Methods$oad(ap Methods

<isi%ilit) graphntroduced in the.ha+ey pro:ect at.7 in the late ;-s

0an produceshortest paths in 2& configurationspaces

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<isi%ilit) graph <oronoi diagra(

ntroduced by0oputational Feoetry

researchers Feneratepaths that axiiesclearance Applicableostly to 2&configuration spaces

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<isi%ilit) graph <oronoi diagra( Silhouette

irst coplete general ethod that applies tospaces of any diension and is singly exponentialin H of diensions 40anny, IJC

Pro%a%ilistic road(aps

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Cell-#eco(position MethodsCell-#eco(position Methods

*o failies of ethods' <xact cell decoposition

he free space is represented by a

collection of non&overlapping cells *hoseunion is exactly <xaples' trapeoidal and cylindrical

decopositions

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rape+oidal deco(positionrape+oidal deco(position

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Cell-#eco(position MethodsCell-#eco(position Methods

*o failies of ethods' <xact cell decoposition Approxiate cell decoposition

is represented by a collection of non&overlapping cells *hose union is containedin

<xaples' quadtree, octree, 2n

&tree

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,ctree #eco(position,ctree #eco(position

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Potential ield MethodsPotential ield Methods

Approach initially proposed for real&tiecollision avoidance 4Khatib, I;C Eundreds ofpapers published on this topic

Dotential field' .calar function over the freespace deal field (navigation function)' .ooth, global

iniu at the goal, no local inia, gro*s to

infinity near obstacles orce applied to robot' Legated gradient of the

potential field Al*ays ove along that force

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Goal

Robot

Attracti=e>$epulsi=e ieldsAttracti=e>$epulsi=e ields

)( Goal pGoal x xk F −−=

>

≤∂∂ −=

0

02

0

0

,111

ρ ρ

ρ ρ ρ ρ ρ ρ

η

if

if x F Obstacle

Khatib, 1MI;

oal

$o%ot

Dath planning'

& 7egular grid F is placed over 0&space& F is searched using a best&first algorith *ith potential field as the heuristic function

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