Asif إ abanoviؤ‡and Kouhei Ohnishi Motion Control Systems 2011-05-09آ  12 1 x W F x Fdx...

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Transcript of Asif إ abanoviؤ‡and Kouhei Ohnishi Motion Control Systems 2011-05-09آ  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 Systems Chapter 1

    Asif Šabanović and Kouhei Ohnishi

  • 2Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

    Basic Quantities Mechanical Systems

    x vx =& θ ωθ =&

    F ∫= 2112 x xF FdxW

    ( ) pxx,F && =

    2 2 1

    2 1

    2 1 mvvmvxmxT === &&

    mvxmp == &

    xFxpdt dT &&& ==

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

    tUtUddttTtT qq Ut

    t dt dT

    =+=+

    −=∫−=∫=− ∂ ∂

    2211

    1221 2

    1 2

    1 qq

    q

    ( )xU ( ) q qUF ∂

    ∂−=

    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 Quantities Electrical Systems

    Charge and current

    Flux linkage and voltage

    Magnetic energy

    Potential energy & Electric potential

    Linear systems

    ( )eQQ = idtdQ =

    ( ) ∫∫∫ === QQtt ddQtte edQdeeidU 000 ξξ ξ ( )dQQdUee =

    ( )iφφ = edtd =φ

    ( ) ∫∫∫ === φφξφ φξξ 000 iddiiedT t

    t d dt

    te ( ) φ φ

    d dTei =

    Power eiP =

    CeQ = Li=φ

  • 4Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

    Fundamental Concepts Principle 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

    q L

    q L

    dt d

    ∂ ∂=⎟⎟

    ⎞ ⎜⎜ ⎝

    ⎛ ∂ ∂ &

  • 5Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

    Fundamental Concepts Dynamics - Example

    Comparison of (*) and (**)

    The solution is Euler-Lagrange equation

    Linear motion of body in Potential field

    q L

    q L

    dt d

    ∂ ∂=⎟⎟

    ⎞ ⎜⎜ ⎝

    ⎛ ∂ ∂ &

    ( ) ( ) x xUxm

    dt d

    ∂ ∂−=&

    xm x L

    x U

    x L

    & &

    = ∂ ∂

    ∂ ∂−=

    ∂ ∂

    ( )xcxmL += 221 & ( ) ( ) ( )xUxc

    x U

    x xc

    x L −=⇒

    ∂ ∂−=

    ∂ ∂=

    ∂ ∂

    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= ∂ ∂−⎟⎟

    ⎞ ⎜⎜ ⎝

    ⎛ ∂ ∂

    q L

    q L

    dt d

    &

    UTL −=

    2 2 1 φ&LT = 2121 φLU =

    2 2 1 QLT &= 2121 QU C=

    Mechanical systems xmxT && 2 1= ( )xU

  • 7Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

    Fundamental Concepts Non-potential and Dissipative Forces

    Non-potential forces

    Dissipative forces

    niF q L

    q L

    dt d

    i ii

    ,...,2,1 , == ∂ ∂−⎟⎟

    ⎞ ⎜⎜ ⎝

    ⎛ ∂ ∂ &

    niF q R

    q L

    q L

    dt d

    i iii

    ,...,2,1 , == ∂ ∂+

    ∂ ∂−⎟⎟

    ⎞ ⎜⎜ ⎝

    ⎛ ∂ ∂

    &&

    i iR

    n

    r

    n

    s srrs q

    RFqqbR &

    && ∂ ∂−=∑ ∑=

    = = ;

    2 1

    1 1

    Raleigh fct.

  • 8Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

    Fundamental Concepts Constraints

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

    mj q h

    F j m

    j i

    jcst i ,...,1 ,

    1 =∑

    ∂ ∂

    −= =

    λ

    niFF q R

    q L

    q L

    dt d cst

    ji iii

    ,...,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 Concepts Equations of Motion – Planar manipulator

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

    ∑∑ ==

    +=

    =+= 2

    1 2 1

    2

    1 2 1

    2 1

    2 1 2,1 ,

    i mii

    T mi

    i mii

    T mi

    mii T mimii

    T mii

    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 Concepts Equations of Motion – Planar manipulator

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

    Velocities of the Centre of the mass

    ( ) ( ) qJv &&

    & v

    m

    m

    ym

    xm m q

    q ql ql

    v v

    1 2

    1

    11

    11

    1

    1 1 0cos

    0sin =⎥

    ⎤ ⎢ ⎣

    ⎡ ⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡− =⎥

    ⎤ ⎢ ⎣

    ⎡ =

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

    & v

    mm

    mm

    ym

    xm m q

    q qqlqqlql qqlqqlql

    v v

    2 2

    1

    21221211

    21221211

    2

    2 2 coscoscos

    sinsinsin =⎥

    ⎤ ⎢ ⎣

    ⎡ ⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ +++ +−+−−

    =⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ =

  • 11Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

    Fundamental Concepts Equations of Motion – Planar manipulator

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

    Translational motion kinetic energy

    ( ) ( ) ( )

    ( ) ( )( )qJJJJq qJJqvv

    qJJqqJqJvv

    &&

    &&

    &&&&

    vvTvvTT v

    i

    v ii

    vT i

    T

    i mii

    T miv

    v ii

    vT i

    Tv ii

    Tv imii

    T mivi

    mmT

    mmT

    immmT

    2221112 1

    2

    1 2 1

    2

    1 2 1

    2 1

    2 1

    2 1 2,1 ,

    +=

    ==

    ====

    ∑∑ ==

  • 12Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

    Fundamental Concepts Equations of Motion – Planar manipulator

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

    Rotational motion kinetic energy

    [ ] ;01 1 2

    1 1 qJ &&

    & ω=⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ =

    q q

    ω [ ] qJ & &

    & ω 2

    2

    1 2 11 =⎥

    ⎤ ⎢ ⎣

    ⎡ =

    q q

    ω

    ( ) ( ) ( ) ( )

    ( ) ( )( )qJJJJq qJJq

    qJJqqJqJ

    &&

    &&

    &&&&

    ωωωω

    ωω

    ωωωω

    2221112 1

    2

    1 2 1

    2

    1 2 1

    2 1

    2 1

    2 1

    IIT

    IωIωT

    IIωIωT

    TTT r

    i ii

    T i

    T

    i ii

    T ir

    ii T

    i T

    ii T

    iii T iri

    +=

    ==

    ===

    ∑∑ ==

  • 13Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

    Fundamental Concepts Equations of Motion – Planar manipulator

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

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

    ( )

    ( ) ( )∑ =

    +=

    =

    ++

    ++=+=

    2

    1

    2 1

    2221112 1

    2221112 1

    i ii

    T i

    v ii

    vT i

    T

    TTT

    vvTvvTT rv

    Im

    II

    mmTTT

    ωω

    ωωωω

    JJJJqA

    qqAq

    qJJJJq

    qJJJJq

    &&

    &&

    &&

    ( ) [ ] ⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ ⎥ ⎦

    ⎤ ⎢ ⎣

    ⎡ ==

    2

    1

    2221

    1211 212

    1 2 1

    q q

    aa aa

    qqT T &

    & &&&& qqAq

  • 14Motion Control Systems Asif Šabanović and Kouhei Ohnishi© 2011 John Wiley & Sons (Asia) Pte Ltd

    Fundamental Concepts Equations of Motion – Planar manipulator

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