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
2Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
3Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
4Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
∂∂=⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂&
5Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
(*)
(**)
(***)
6Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Fundamental Concepts
Electromagnetic systems
Euler-Lagrange equation
Lagrangian
0=∂∂−⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
qL
qL
dtd
&
UTL −=
221 φ< = 21
21 φLU =
221 QLT &= 21
21 QU C=
Mechanical systems xmxT &&21= ( )xU
7Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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.
8Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
9Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
10Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Fundamental ConceptsEquations of Motion – Planar manipulator
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=⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡++++−+−−
=⎥⎦
⎤⎢⎣
⎡=
11Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Fundamental ConceptsEquations of Motion – Planar manipulator
2,1 , == iq ii θ 21211 , qqq &&& +== ωωCoordinates
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 ,
+=
==
====
∑∑==
12Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Fundamental ConceptsEquations of Motion – Planar manipulator
2,1 , == iq ii θ 21211 , qqq &&& +== ωωCoordinates
Rotational motion kinetic energy
[ ] ;01 12
11 qJ &
&
& ω=⎥⎦
⎤⎢⎣
⎡=
ω [ ] qJ &&
& ω2
2
12 11 =⎥
⎦
⎤⎢⎣
⎡=
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
+=
==
===
∑∑==
13Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Fundamental ConceptsEquations of Motion – Planar manipulator
2,1 , == iq ii θ 21211 , qqq &&& +== ωωCoordinates
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
aaaa
qqT T
&
&&&&& qqAq
14Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Fundamental ConceptsEquations of Motion – Planar manipulator
2,1 , == iq ii θ 21211 , qqq &&& +== ωωCoordinates
( ) ( )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
15Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Fundamental ConceptsEquations of Motion – Planar manipulator
2,1 , == iq ii θ 21211 , qqq &&& +== ωωCoordinates
( ) ( )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
+∂∂+
∂∂++=
∂∂−
∂∂
∂∂+
∂∂++=
∂∂
+=∂∂
−=+−=∂∂−=
∂∂
&&&&&&&&
&&&&&&&&
&&
16Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Fundamental ConceptsEquations of Motion – Planar manipulator
2,1 , == iq ii θ 21211 , qqq &&& +== ωωCoordinates
( )
( ) 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
17Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Fundamental ConceptsEquations of Motion – Planar manipulator
( ) ( ) ( ) 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
18Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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 , =
19Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Equations of MotionProperties
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
20Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Equations of MotionProperties
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
21Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Equations of MotionProperties
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
22Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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 ,
23Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
24Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Equations of MotionOperational Space Dynamics
( ) ( ) ( )
( ) 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
25Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Equations of MotionOperational Space Dynamics
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
26Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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.
27Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
=++
=&
28Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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 ==⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂ *
&
29Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Electromechanical SystemsElectromagnetic Levitation
( )
( ) ( )( ) ei
TdtdixL
dtdRitv
fx
TxxLimgxm
e
e
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂==−
=∂
∂=∂
∂=−
*
*2
2&&
Equations of motion
30Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
31Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
32Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Electrical MachineLagrangian andRaleigh fct.
( ) ( ) ( )
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
33Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
Electrical MachineLagrangian andRaleigh fct.
( ) ( ) ( )
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
34Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
35Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd
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
Top Related