Asif Šabanovićand Kouhei Ohnishi Motion Control Systems … · 2011-05-09 · 12 1 x W F x Fdx...

1Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

Electromechanical Systems Dynamics

Motion Control SystemsChapter 1

Asif Šabanović and Kouhei Ohnishi


Basic QuantitiesMechanical Systems

x vx =&θ ωθ =&

F ∫= 2112xxF FdxW

( ) pxx,F && =

221

21

21 mvvmvxmxT === &&

mvxmp == &

xFxpdtdT &&& ==

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) EtUtTtUtT

tUtUddttTtT qq

Utt dt

dT

=+=+

−=∫−=∫=− ∂∂

2211

12212

12

1qq

q

( )xU( )qqUF ∂

∂−=

Position and velocity

Force and work

Momentum

Kinetic energy

Potential energy

Total energy


Basic QuantitiesElectrical Systems

Charge and current

Flux linkage and voltage

Magnetic energy

Potential energy &Electric potential

Linear systems

( )eQQ = idtdQ =

( ) ∫∫∫ === Q

Q

t

t ddQt

te edQdeeidU000

ξξ ξ( )

dQQdUee =

( )iφφ = edtd =φ

( ) ∫∫∫ === φφξ

φ φξξ000iddiiedT t

t ddt

te( )φφ

ddTei =

Power eiP =

CeQ = Li=φ


Fundamental ConceptsPrinciple of Least Action for Conservative Systems

Travel between fixed points

The actual path taken by the system is an extreme of

The solution is Euler-Lagrange equation

Action along path

( )11 tq ( )22 tq

( ) ( ) ( )( )∫=Γ Γ dtttqtqLS ,, &

( ) ( )( )ttqtqL ,, &Lagrangian

Γ

( )ΓS

qL

qL

dtd

∂∂=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂&


Fundamental ConceptsDynamics - Example

Comparison of (*) and (**)

The solution is Euler-Lagrange equation

Linear motion of body in Potential field

qL

qL

dtd

∂∂=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂&

( ) ( )xxUxm

dtd

∂∂−=&

xmxL

xU

xL

&&

=∂∂

∂∂−=

∂∂

( )xcxmL += 221 &

( ) ( ) ( )xUxcxU

xxc

xL −=⇒

∂∂−=

∂∂=

∂∂

From (***) follws

(*)

(**)

(***)


Fundamental Concepts

Electromagnetic systems

Euler-Lagrange equation

Lagrangian

0=∂∂−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

qL

qL

dtd

&

UTL −=

221 φ&LT = 21

21 φLU =

221 QLT &= 21

21 QU C=

Mechanical systems xmxT &&21= ( )xU


Fundamental ConceptsNon-potential and Dissipative Forces

Non-potential forces

Dissipative forces

niFqL

qL

dtd

iii

,...,2,1 , ==∂∂−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂&

niFqR

qL

qL

dtd

iiii

,...,2,1 , ==∂∂+

∂∂−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

&&

iiR

n

r

n

ssrrs q

RFqqbR&

&&∂∂−=∑ ∑=

= = ;

21

1 1

Raleigh fct.


Fundamental ConceptsConstraints

Holonomics constraints ( ) mjqqh nj ,...,1 ,0,...,1 ==

mjqh

F j

m

j i

jcsti ,...,1 ,

1=∑

∂∂

−==

λ

niFFqR

qL

qL

dtd cst

jiiii

,...,2,1, =+=∂∂+

∂∂−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

&&

Interaction forces

mjj ,...,1 , =λLagrange multipliers

Euler-Lagrange equations


Fundamental ConceptsEquations of Motion – Planar manipulator

2,1 , == iq ii θ 21211 , qqq &&& +== ωω

∑∑==

+=

=+=2

121

2

121

21

21 2,1 ,

imii

Tmi

imii

Tmi

miiTmimii

Tmii

ImT

iImT

ωω

ωω

vv

vv

Coordinates

Kinetic energy



2,1 , == iq ii θ 21211 , qqq &&& +== ωωCoordinates

Velocities of the Centre of the mass

( )( ) qJv &

&

& v

m

m

ym

xmm q

qqlql

vv

12

1

11

11

1

11 0cos

0sin=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡=

( ) ( ) ( )( ) ( ) ( ) qJv &

&

& v

mm

mm

ym

xmm q

qqqlqqlqlqqlqqlql

vv

22

1

21221211

21221211

2

22 coscoscos

sinsinsin=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡++++−+−−

=⎥⎦

⎤⎢⎣

⎡=




Translational motion kinetic energy

( ) ( )( )

( ) ( )( )qJJJJq

qJJqvv

qJJqqJqJvv

&&

&&

&&&&

vvTvvTTv

i

vii

vTi

T

imii

Tmiv

vii

vTi

Tvii

Tvimii

Tmivi

mmT

mmT

immmT

22211121

2

121

2

121

21

21

21 2,1 ,

+=

==

====

∑∑==




Rotational motion kinetic energy

[ ] ;01 12

11 qJ &

&

& ω=⎥⎦

⎤⎢⎣

⎡=

qq

ω [ ] qJ &&

& ω2

2

12 11 =⎥

⎦

⎤⎢⎣

⎡=

qq

ω

( ) ( ) ( )( )

( ) ( )( )qJJJJq

qJJq

qJJqqJqJ

&&

&&

&&&&

ωωωω

ωω

ωωωω

22211121

2

121

2

121

21

21

21

IIT

IωIωT

IIωIωT

TTTr

iii

Ti

T

iii

Tir

iiT

iT

iiT

iiiTiri

+=

==

===

∑∑==




Total kinetic energy ( ) ( )( )( ) ( )( )

( )

( ) ( )∑=

+=

=

++

++=+=

2

1

21

22211121

22211121

iii

Ti

vii

vTi

T

TTT

vvTvvTTrv

Im

II

mmTTT

ωω

ωωωω

JJJJqA

qqAq

qJJJJq

qJJJJq

&&

&&

&&

( ) [ ] ⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

2

1

2221

1211212

121

qq

aaaa

qqT T

&

&&&&& qqAq




( ) ( )qqqAq UL T −= &&21

( ) ( ) ( )( ) ( )

( )11222

1212

2

11212111

11

222

1212

2

11212111

1

2121111

2111211211111

,coscoscos

qgqqqaqq

qaqaqa

qL

qL

dtd

qqqaqq

qaqaqa

qL

dtd

qaqaqL

qqgqlqlgmqglmqU

qL

mm

+∂∂+

∂∂++=

∂∂−

∂∂

∂∂+

∂∂++=

∂∂

+=∂∂

−=+−−=∂∂−=

∂∂

&&&&&&&&

&&&&&&&&

&&

Lagrangian

1st joint




( ) ( )qqqAq UL T −= &&21Lagrangian

2nd joint ( ) ( )

( )212222

2212

2

21222121

22

222

2212

2

21222121

1

2221212

212212212

,

,cos

qqgqqqaqq

qaqaqa

qL

qL

dtd

qqqaqq

qaqaqa

qL

dtd

qaqaqL

qqgqqglmqU

qL

m

+∂∂+

∂∂++=

∂∂−

∂∂

∂∂+

∂∂++=

∂∂

+=∂∂

−=+−=∂∂−=

∂∂

&&&&&&&&

&&&&&&&&

&&




( )

( ) 2212222

2212

2

21222121

111222

1212

2

11212111

, τ

τ

=+∂∂+

∂∂++

=+∂∂+

∂∂++

qqgqqqaqq

qaqaqa

qgqqqaqq

qaqaqa

&&&&&&&&

&&&&&&&&

( )

( ) 222

2212

2

212

222

1212

2

111

qqqaqq

qab

qqqaqq

qab

&&&&&

&&&&&

∂∂+

∂∂=

∂∂+

∂∂=

qq,

qq,

Equations of motion



( ) ( ) ( ) τqgqq,bqqA =++ &&&

( ) 22×ℜ∈qA

( ) ( ) ( )[ ]qq,qq,qq,b &&& 21 bbT =

( ) ( ) ( )[ ]qqqg 21 ggT =

[ ]21 ττ=Tτ

Equations of motion

Input forces

Kinetic energy (inertia) matrix


Equations of MotionProperties

11,τθ x

y

22 ,τθ33 ,τθ

44 ,τθ

f

mir

P

iiTimii

Tmiivii mTTT ωIωvv 2

121 +=+= ω

niTTn

ii ,..,2,1 ,

1=∑=

=

nimi ,..,2,1 , =v

iI

Kinetic energy

Translation velocities

Rotational speeds

Link masses

Moment of inertia

nimi ,..,2,1 , =ω

nimi ,..,2,1 , =



Kinetic energy

Velocitiesn

iii

nvi

vimi

×

×

ℜ∈=

ℜ∈=3

3

,

,ωω JqJω

JqJv

&

&

( ) ( )( ) ( )( ) 1

21

21

1∑ +=∑ +===

n

iii

Ti

Tvii

vTi

Tn

iivi mTTT qJIJqqJJq &&&& ωω

ω

( ) ( )∑∑= =

==n

i

n

jjiij

T qqaT1 1

21

21

&&&& qqqAq

( ) ( ) ( ) nnn

iii

Ti

vii

vTi m ×

=ℜ∈∑ += qAJIJJJqA ,

1

ωωKinetic energymatrix



Componentsof equations of motion

niqdt

daqaqa

dtd

qT

dtd n

jj

ijn

jjij

n

jjij

i

,...,2,1 ,111

=∑+∑=⎟⎠⎞

⎜⎝⎛∑=

∂∂

===&&&&

&

niqqqa

qqaqq

TqL n

j

n

kkj

i

jkn

j

n

kkjjk

iii

,...,2,1 , 21

21

1 11 1=∑ ∑

∂∂

=⎟⎠⎞

⎜⎝⎛

∑ ∑∂∂=

∂∂=

∂∂

= == =&&&&

( ) ( )

( ) niqqqa

qqqa

b

b

qqqa

qqqa

qaqT

qT

dtd

n

j

n

kkj

i

jkn

jj

n

kk

k

iji

iTi

n

j

n

kkj

i

jkn

jj

n

kk

k

ijn

jjij

ii

,...,2,1 ,21

21

1 11 1

1 11 11

=∑ ∑∂∂

−∑ ∑∂∂

=

+=

∑ ∑∂∂

−∑ ∑∂∂

∑ +=∂∂−

∂∂

= == =

= == ==

&&&&&

&&&

&&&&&&

qq,

qq,qqa



Componentsof equations of motion

( )∑==

n

i

miTimU

10 qrg

( ) ( ) ( ) nicolmq

mqUg

n

i

vji

Ti

n

j i

mjT

ji

i ,...,2,1 ,11

0 =∑=∑∂

∂=

∂∂=

==Jgqrgq


Equations of Motion

( ) ( ) ( ) cstFτqgqq,bqqA +=++ &&&

( ) ( )i

q

qT

qT

qi

n

∂∂=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−= AAqAq

qAqqqAqq,b ,......

21 1

&&

&&

&&&

( ) ( ) 1×ℜ∈∂

∂−= nUqqqg

( ) ( ) mnh

Th

Tcon ×ℜ∈

∂∂=−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂−=qqhJλJλ

qqhF ,


Equations of MotionOperational Space Dynamics

[ ]mT xx ....1=x ( ) m×ℜ∈= 1qfx

( ) ( ) nm×ℜ∈∂

∂=∂

∂=qqfJq

qqfx ,&&

( )xxΛx && TT 21=

( ) ( )

Fxx

xxx,

=∂∂−⎟

⎠⎞

⎜⎝⎛

∂∂

−=

LLdtd

UTL

&

&

( )

( ) ( )xxΛxxΛx

xxΛx

&&&&&

&&

+=⎟⎠⎞

⎜⎝⎛

∂∂

=∂∂

Ldtd

L

Configuration

Velocity

Kinetic energy

Lagrangian



( ) ( ) ( )

( ) mix

UTL

ixi

xmT

xT

,..,2,1 ,

......

21

121

=∂

∂=

−⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=∂

∂−∂

∂=∂∂

xΛΛ

xpxΛx

xΛx

xx

xxx,

x&&

&&&

( ) ( ) ( )

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

=++

xΛx

xΛxxxΛxxμ

FxpxxμxxΛ

&&

&&

&&&

&&&

xm

x

21

121

.....,

,Equations of motion



Kinetic energyinvariance

( ) ( )xxΛxqqAq &&&& TTT 21

21 ==

( ) ( )( ) ( )qΛJJqqΛJqJ

xxΛxqqAq

&&&&

&&&&

TTT

TT

21

21

21

21

=

=

( ) ( ) ( )( ) ( ) ( ) ( ) τqgqq,bqqAFJxpxxμxxΛJ =++==++ &&&&&& TT ,

FJτ

qJxT=

= &&Velocity and Forcerelationship

Configuration space Operational space


Electrical Circuits

( )tv

R L

CQ

Q&

( ) ( )tvRiCQLi

dtd

iQ

=++

=&

( ) ( ) ( )

22

*2

2

21

21

21

21,

RiQR

QUQTC

QQLQQL

e

eee

==ℜ

−=−=

&

&&&

( )tQQ

Li

Ldtd

eeee ℑ=

∂∂ℜ

+∂∂

−∂

∂&

Circuit dynamics

Lagrangian and Raleigh fct.

Euler-Lagrange ect.


Electrical Circuits

( )tv

R L

CQ

Q&

( )

( ) ( )tvt

RiQ

CQ

CQLi

QQL

Lidtd

CQLi

idtd

iL

dtd

e

e

e

e

=ℑ

=∂∂ℜ

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂−=

∂∂−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂=

∂∂

&

22

22

21

21

21

21

Circuit dynamics

( ) ( )tvRiCQLi

dtd

iQ

=++

=&


Electromechanical SystemsElectromagnetic Levitation

( ) ( ) 222

21 ,

21

21,,, QRQxLmgxxmQQxxL e

&&&&& =ℜ++=

( ) ( ) 02

2

=∂

∂−−⇒ℑ=∂∂ℜ+

∂∂−

∂∂

xxLimgxmt

xxL

xL

dtd

x &&&&

( ) ( ) ( )tvRiixLdtdt

QQL

QL

dtd

Q =+⇒ℑ=∂∂ℜ+

∂∂−

∂∂

&&

( ) ( )x

xTxxLif e

∂∂−=

∂∂−=

*2

2

( )( ) eixLdtd

iT

dtd

QL

dtd e ==⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ *

&


Electromechanical SystemsElectromagnetic Levitation

( )

( ) ( )( ) ei

TdtdixL

dtdRitv

fx

TxxLimgxm

e

e

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂==−

=∂

∂=∂

∂=−

*

*2

2&&

Equations of motion


Electrical Machine

++

+

+

+

+ ++++

+++

+

+

Stator “a” axis

Rotor “a” axis

rsaa,,θ

[ ]sn

ssT vv ....1=v [ ]rn

rrT vv ....1=v

[ ]sn

ssT ii ....1=i [ ]rn

rrT ii ....1=i

( )θcos,, srrsaa LL =

⎟⎠⎞

⎜⎝⎛ ±=

nkLL sr

rska

πθcos,,

( ) ( )( ) ⎥

⎦

⎤⎢⎣

⎡= rrsr

rsss

,,

,,

LLLLL

θθθ

( )iLi θTeT

21* =

Variables

Inductances

Inductance matrix

Magneticcoenergy


Electrical MachineLagrangian andRaleigh fct.

Mechanical system

( ) ( ) ( )

Rii

iLi

Tem

Tem

B

aiQL

212

21

2

21

21,,,

+=ℜ

+=

θ

θθθθθ

&

&&

( ) ( )

( ) ( )

( ) ( )tt

B

aQ

L

dtdaaa

dtda

idtdL

dtd

exte

em

TTem

Tem

τ

θθ

θθθθ

θ

θθθθθθ

−=ℑ

=∂

∂ℜ

∂∂−=⎟

⎠⎞

⎜⎝⎛ +

∂∂−=

∂∂

−

+==⎟⎠⎞

⎜⎝⎛ +

∂∂=

∂∂

&&

&

&&&&&&

iLiiLi

iLi

21

21

21

21

21

2

2



( ) ( ) ( )

Rii

iLi

Tem

Tem

B

aiQL

212

21

2

21

21,,,

+=ℜ

+=

θ

θθθθθ

&

&&

( )

( )( ) ( ) ( )

( )

( ) ( )ttQ

aQQ

Ldt

ddtd

dtd

aidt

di

Ldtd

e

em

Tem

Tem

v

Ri

iLi

iLiLiL

iLi

=ℑ

=∂

∂ℜ

=⎟⎠⎞

⎜⎝⎛ +

∂∂−=

∂∂

−

+==

⎟⎠⎞

⎜⎝⎛ +

∂∂=

∂∂

&

&

&

021

21

21

21

2

2

θθ

θθθ

θθElectrical system



( ) ( ) ( )

Rii

iLi

Tem

Tem

B

aiQL

212

21

2

21

21,,,

+=ℜ

+=

θ

θθθθθ

&

&&

Dynamics

( )

( ) ( ) ( )tdtd

Bdtdaa ext

T

vRiiLiL

iLi

=+∂

∂+

−=∂

∂−⎟⎠⎞

⎜⎝⎛ ++

θθθθ

τθθθθ

&

&&&2

( ) ii Lθθτ ∂

∂= T21

( ) iLθθθ ∂

∂&

Force

EMF


Electrical Machine

Dynamics

( )

( ) ( ) ( )tdtd

Bdtdaa ext

T

vRiiLiL

iLi

=+∂

∂+

−=∂

∂−⎟⎠⎞

⎜⎝⎛ ++

θθθθ

τθθθθ

&

&&&2

( ) ii Lθθτ ∂

∂= T21Force

[ ] ( )

( ) ⎥⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

∂∂

∂∂

r

srTsT

sr

rs

iiii

L

L

00,

,

θθ

θθ

τ

( )sr

f

srf

srf

ik

k

k

⊥=

=

×=

Φ

iΦ

iΦ

sin ϕ

τ


Electrical Machine

Dynamics

( )

( ) ( ) ( )tdtd

Bdtdaa ext

T

vRiiLiL

iLi

=+∂

∂+

−=∂

∂−⎟⎠⎞

⎜⎝⎛ ++

θθθθ

τθθθθ

&

&&&2

Force srf ik ⊥= Φτ

( ) ( )( ) ( ) ( ) s

Ts

T

ext

iKiKt

tqqbqa

ΦΦ ==

=++

⊥τ

ττ&&& ,Dynamics

Asif Šabanovićand Kouhei Ohnishi Motion Control Systems … · 2011-05-09 · 12 1 x W F x Fdx...

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Transcript of Asif Šabanovićand Kouhei Ohnishi Motion Control Systems … · 2011-05-09 · 12 1 x W F x Fdx...