modeling of linear and belt object deformation based on

1OSAKA University, Department of Manufacturing ScienceOSAKA University, Department of Manufacturing Science

Advanced Manufacturing Systems Lab.Advanced Manufacturing Systems Lab.

Modeling of Linear and Belt Object Deformation Modeling of Linear and Belt Object Deformation Based on Differential GeometryBased on Differential Geometry

HidefumiHidefumi Wakamatsu, Wakamatsu, EijiEiji Arai, Arai, andand Shinichi HiraiShinichi Hirai

AgendaAgendaIntroductionIntroductionModeling ofModeling of

Linear Object DeformationLinear Object DeformationApplication to Application to

Linear Object StructureLinear Object StructureModeling ofModeling of

Belt Object DeformationBelt Object DeformationConclusionsConclusions

OSAKA University, Department of Manufacturing ScienceOSAKA University, Department of Manufacturing ScienceAdvanced Manufacturing Systems Lab.Advanced Manufacturing Systems Lab. 2

Manipulation of Flexible Linear/Belt ObjectsManipulation of Flexible Linear/Belt Objects

Wire harnessWire harness Flexible printed circuit boardFlexible printed circuit board

A modeling of linear/belt object deformation is required for plaA modeling of linear/belt object deformation is required for planning of nning of manipulative operations and their execution by a mechanical systmanipulative operations and their execution by a mechanical system.em.


FrenetFrenet--SerretSerret FormulasFormulas

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

′′′

ζηξ

ζηξ

00

0

ξη

ξζ

ηζ

ωωωωωω

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

′′′

bnt

bnt

000

00

ττκ

κ

Frenet-Serret formulas Angular verocities of rigid body

t

nb

ξ

η

ζ

ξω

ηω

ζω


Modeling of Linear Object DeformationModeling of Linear Object Deformation

x

y

zOP(s)

ξζ

η

η

x

y

zO

ξ ζP(s)

Assumption :Assumption :Deformation in any direction perpendicular to the central axis oDeformation in any direction perpendicular to the central axis of a linear f a linear

object is negligibleobject is negligible..

L

s


Rotation MatrixRotation Matrix

η

x

y

zO

ξ ζP(s)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−+−−−

=θψθψθφθψφψφθψφψφθφθψφψφθψφψφθ

cossin sincos sinsin sincos cos sin sin cossin cos cos sin coscos sincos sin sin cos cossin sin cos cos cos

A

Rotation matrix Rotation matrix ::


Rotation of Object Coordinate SystemRotation of Object Coordinate System

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

)()()(

00

0

)()()(

sss

sss

dsd

ζηξ

ζηξ

ξη

ξζ

ηζ

ωωωωωω

ξ

η

ζξω

ηω

ζω

( )sP

ζ

η

ξ

( )dss +P

dsd

dsd ψθφωζ += cos

ψθφψθωξ cossinsindsd

dsd

−=

ψθφψθωη sinsincosdsd

dsd

+=

Infinitesimal rotational anglesInfinitesimal rotational angles ::


Infinitesimal Rotational AnglesInfinitesimal Rotational Angles

ηω

ξω

ζω

Vertical bendVertical bend

TwistTwist

Horizontal bendHorizontal bend


Curvature, Curvature, TorsionalTorsional Ratio, and Normal Strain Ratio, and Normal Strain

θφθωωκ ηξ2

22222 sin

dsd

dsd

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=+=

222 cos ⎟

⎠⎞

⎜⎝⎛ +== θφψωω ζ ds

ddsd

Curvature Curvature ::

TorsionalTorsional ratio ratio ::

Normal strain Normal strain :: )(sε


Spatial CoordinatesSpatial Coordinates

η

x

y

zO

ξ ζP(s)

+= )0()( xx s

)0(x

The geometrical shape of a deformed linear object can be represeThe geometrical shape of a deformed linear object can be represented by nted by four functions :four functions : )(),(),(),( ssss εψθφ

{ }∫⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

s dss

ssss

s0)(cos

)(sin)(sin)(cos)(sin

)(1θ

φθφθ

ε

)(sx

)0P(


Potential EnergyPotential Energy

Uflex =120

L∫ Rfκ2ds

Utor =120

L∫ Rtω2ds

Uext =120

L∫ Reε2ds

Ugrav = Dgx0

L∫ ds

U =Uflex +Utor +Uext +UgravPotential energy :Potential energy :

Gravitational energy :Gravitational energy :

TorsionalTorsional energy :energy :

Flexural energy :Flexural energy :

Extensional energy :Extensional energy :

VariationalVariational principle in principle in staticsstatics ::The potential energy of a linear object attains its minimum valuThe potential energy of a linear object attains its minimum value in its e in its

stable deformed state under the imposed constraints. stable deformed state under the imposed constraints.

: Flexural rigidity: Flexural rigidity

: : TorsionalTorsional rigidityrigidity

: : ExtentionalExtentional rigidityrigidity

: Linear density: Linear density

fR

tR

eR

D


η

ξ

ζ)P( cs

( )( )( ) ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

c

c

c

c

c

c

sss

ψθφ

ψθφ

Positional/Positional/OrientationalOrientational ConstraintsConstraints

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=−

z

y

x

ab

lll

ss )()( xx

)P( bs

)P( as


( ) ( ) ( )( ) [ ]Lsszsysxf ,0,0,, ∈∀≤

Consideration of Contact with ObstaclesConsideration of Contact with Obstacles

( ) 0,, =zyxf

( ) 0,, >zyxf

Surface Surface ::

Inside Inside ::

Outside Outside :: ( ) 0,, <zyxf


Consideration of SelfConsideration of Self--interactioninteraction

[ ] rssLssrss jijiji 2s.t.,,0,,2)()( ≥−∈∀≥− xx

x(si )x(s j )

2r

The geometrical constraints imposed on a linear object are givenThe geometrical constraints imposed on a linear object are given by not by not only only equationalequational conditions but also inequality conditions.conditions but also inequality conditions.


Minimization ProblemMinimization Problem

[ ]εψθφ aaaaa =

,)()(1∑=

=n

iii seas φφ ,)()(

1∑=

=n

iii seas θθ

,)()(1∑=

=n

iii seas ψψ ∑

==

n

iii seas

1)()( εε

),()( ss ea ⋅= φφ ),()( ss ea ⋅= θθ ),()( ss ea ⋅= ψψ )()( ss ea ⋅= εε

)(aU

),,1(0)( Kkgk L=≤a

),,1(0)( Jjf j L==aMinimize potential energyMinimize potential energy

Subject toSubject to

Positional/Positional/orientationalorientational constraintsconstraints

Avoidance of (selfAvoidance of (self--)interference)interference


Computational ResultsComputational Results

Basis functions:Basis functions:

,,1 21 see ==

,sin12 Lise iπ

=+

)4,3,2,1(cos22 ==+ iLise iπ


Experimental VerificationExperimental Verification (1)(1)

][800 mmlz = ][600 mmlz =

1.01.0××1010--2 2 [N/m][N/m]Weight per unit lengthWeight per unit length22..33××1010--4 4 [Nm[Nm22]]TorsionalTorsional rigidityrigidity6.66.6××1010--4 4 [Nm[Nm22]]Flexural rigidityFlexural rigidity8.78.7××101022 [mm][mm]LengthLength


Experimental VerificationExperimental Verification (2)(2)

][400 mmlz = ][200 mmlz =


Knotted Shape of Linear ObjectKnotted Shape of Linear Object

,0)()(,0)()(,0)()( 332211 =−=−=− lululu szszszszszsz

ls1

us2

ls3

us1us3ls2 2rx

y z

,0)()(,0)()(,0)()( 332211 =−=−=− lululu sxsxsxsxsxsx

,2)()(,2)()(,2)()( 332211 rsysyrsysyrsysy lululu =−=−=−

Lssssss ululul ≤<<<<<≤ 3213210

[ ]lululu ssssss 332211ψθφ aaaa =


Computational Result of Overhand KnotComputational Result of Overhand Knot


Knitted Shape of Linear ObjectsKnitted Shape of Linear Objects

( )6,5,2,1,0)()( 2 ==− + iszsz ii

( )4,3,)()( 2 ==− − ilsxsx wii

( )6,5,)()( 2 ==− + ilsxsx wii

( )7,3,2)()( 2 ==− − irsysy ii

( )6,2,2)()( 2 ==− + irsysy ii

( )7,,1,0 1 L=≤<≤ + iLss ii1C

2C

3C

4C

8C

7C

6C

5C

wlAssumption :Assumption :The shape of the fabric can be represented by repetitions of theThe shape of the fabric can be represented by repetitions of the same same

shape of one loop.shape of one loop.


Computational Result of Plain Knitted FabricComputational Result of Plain Knitted Fabric


Modeling of Belt Object DeformationModeling of Belt Object Deformation

Assumptions:Assumptions:A belt object is rectangular.A belt object is rectangular.The width of the object is sufficiently small compared to its leThe width of the object is sufficiently small compared to its length.ngth.The object is The object is inextensibleinextensible. Namely, it can be bent and twisted but cannot . Namely, it can be bent and twisted but cannot be expanded or contracted.be expanded or contracted.Its both ends cannot be deformed because connectors are attachedIts both ends cannot be deformed because connectors are attached to to the ends.the ends.

x y

z

O

ξη

ζ

( )uPu

v

U

V0x


Infinitesimal Rotational AnglesInfinitesimal Rotational Angles

ζω

ξω

ηω

BendBend TwistTwist

Shape in Shape in uvuv--spacespace

Assumption:Assumption:A belt object is inextensible.A belt object is inextensible.

0≡ξωIn case of rectangular object: In case of rectangular object:


Developable SurfacesDevelopable Surfaces

Assumption:Assumption:A belt object is inextensible. Its surface is A belt object is inextensible. Its surface is developabledevelopable..Developable surface :Developable surface :

It can be generated by sweeping a straight line in 3D space.It can be generated by sweeping a straight line in 3D space.It includes straight lines.It includes straight lines.

Cylindrical surfaceCylindrical surface Conic surfaceConic surface


Fishbone ModelFishbone Model

)(uα

η

ζ

)(P u：：rib anglerib angle

The shape of a belt object:The shape of a belt object:Shape of the bent and twisted Shape of the bent and twisted spine linespine line

Direction of straight Direction of straight rib linesrib lines)(),(),( uuu ψθφ

)(uα

Central axisCentral axis Straight lineStraight lineRib lineRib lineSpine lineSpine line


Potential Energy of Belt ObjectPotential Energy of Belt Object

Cylindrical surfaceCylindrical surface

ζ

ηζ

ωωω

κ22

max+

−=

( )∫∫

+== U fU f du

Rdu

RI 0 2

222

02max 22 ζ

ηζ

ω

ωωκPotential energy :Potential energy :

fR : flexural rigidity along the spine line: flexural rigidity along the spine line


ConstraintsConstraints

[ ]UuVdu

dV

,0,cos2cos2 22

∈∀≤≤−ααα

[ ]Uu ,0,tan 1 ∈∀−= −

ζ

η

ωω

α

[ ]Uuu ,0,0 ∈∀=ξω

To prevent rib lines from intersecting To prevent rib lines from intersecting with themselves :with themselves :

Necessary constraints for Necessary constraints for developabilitydevelopabilityTo maintain initial shape in To maintain initial shape in uvuv--space :space :

α αα d+

)(P u du

)(P duu +

Geometric constraintsGeometric constraints

Relationship between the rib angle and Relationship between the rib angle and infinitesimal rotational angles:infinitesimal rotational angles:

ηωζω

α


(a) Computational result(a) Computational result (b) Experimental result(b) Experimental result

Experimental VerificationExperimental Verification

Obverse sideObverse side

Reverse sideReverse side

PolystyrenePolystyrene200[mm] long200[mm] long20[mm] wide20[mm] wide140[140[µµm] thickm] thick


Application to Curved/Bent Belt ObjectApplication to Curved/Bent Belt Object

cκωξ ≡

Curved belt objectCurved belt object Bent belt objectBent belt object

Assumption :Assumption :The rib line at the bent point The rib line at the bent point

coincides with the connecting coincides with the connecting line.line.

connecting lineconnecting line

Shape in Shape in uvuv--space:space:

cκ1


Deformed Shape of Curved Belt ObjectDeformed Shape of Curved Belt Object

(b) Computational result(b) Computational result (c) Experimental result(c) Experimental result

(a) Initial shape(a) Initial shape


Deformed Shape of Bent Belt ObjectDeformed Shape of Bent Belt Object

(b) Computational result(b) Computational result (c) Experimental result(c) Experimental result

(a) Initial shape(a) Initial shape


ConclusionsConclusions

A modeling method of linear/belt object deformation based on difA modeling method of linear/belt object deformation based on differential ferential geometry was proposed.geometry was proposed.

Differential geometry was extended to describe linear object Differential geometry was extended to describe linear object deformation including flexure, torsion, and extension.deformation including flexure, torsion, and extension.

The shape of a linear object can be described by The shape of a linear object can be described by four independent four independent variablesvariables if it is extensible and by three otherwise.if it is extensible and by three otherwise.

It was shown that more complex shapes such as knots and knittedIt was shown that more complex shapes such as knots and knittedfabrics also can be computed using our proposed approach.fabrics also can be computed using our proposed approach.

This approach was applied to deformation of an inextensible belThis approach was applied to deformation of an inextensible belt object. t object.

It was found that the belt object shape can be described by It was found that the belt object shape can be described by two two independent variables.independent variables.

modeling of linear and belt object deformation based on

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