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12.3 & paper ES A Linearised Input-Output Representation for Control Synthesis in Flexible Multibody System A Linearised Input-Output Representation for Dynamics Control Synthesis in Flexible Multibody System Layout Dynamics Finite

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SLIDE 1

A Linearised Input-Output Representation for Control Synthesis in Flexible Multibody System Dynamics

J.B. Jonker, J. van Dijk and R.G.K.M. Aarts

Department of Mechanical Automation and Mechatronics University of Twente The Netherlands

FMSA4CP / Input-Output / 1 Jonker/van Dijk/Aarts

§ 12.3 & paper ES

A Linearised Input-Output Representation for Control Synthesis in Flexible Multibody System Dynamics

Layout

  • Finite element representation of flexible multibody systems
  • Equations of motion and reaction
  • Linearised equations of motion and reaction
  • Linearised state-space equations
  • Stationary and equilibrium solutions
  • From state-space equations to transfer function(s)
  • Illustrative examples
  • Conclusions

FMSA4CP / Input-Output / 2 Jonker/van Dijk/Aarts

§ 2

Finite element representation of multibody systems

Physical description of a flexible multibody system Element k with set of nodal coordinates x(k) (Cartesian and rotational) in a fixed inertial coordinate system and deformation modes specified by a vector of deformation parameters e(k).

FMSA4CP / Input-Output / 3 Jonker/van Dijk/Aarts

Planar flexible beam element x y p q φp φq Rpnx Rpny Rqnx Rqny nx ny β(k) Rp ≡

  • cos φp

− sin φp sin φp cos φp

  • Rq ≡
  • cos φq

− sin φq sin φq cos φq

  • l(k) ≡ xq − xp

= [xq − xp, yq − yp]T Elongation: ε(k)

1

= D(k)

1

(x(k)) =

  • (xq − xp)2 + (yq − yp)21/2 − l(k)

Bending: ε(k)

2

= D(k)

2

(x(k)) = −(Rpny, l(k)) ε(k)

3

= D(k)

3

(x(k)) = (Rqny, l(k))

FMSA4CP / Input-Output / 4 Jonker/van Dijk/Aarts

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SLIDE 2

§ 3

Kinematic analysis

Deformation equations e = D(x) x: nodal coordinates ˙ e = ∂D ∂ ˙ x = DxD ˙ x e: deformation mode coordinates Partitioning: x =

   

x(o) x(c) x(m)

   

fixed coordinates dependent nodal coordinates absolute generalized / independent coordinates e =

   

e(o) e(m) e(c)

   

rigid / zero deformations relative generalized /independent coordinates dependent deformations Generalised coordinates x(m), e(m) collected in vector q with ndof kinematic degrees of freedom.

FMSA4CP / Input-Output / 5 Jonker/van Dijk/Aarts

Geometric transfer functions x = F(x)(q) q: generalised coordinates x(m) and e(m) e = F(e)(q) Velocities ˙ x = DqF(x) ˙ q DqF: first-order geometric transfer function ˙ e = DqF(e) ˙ q D2

qF: second-order geometric transfer function

Accelerations ¨ x = D2

qF(x) ˙

q ˙ q + DqF(x)¨ q ¨ e = D2

qF(e) ˙

q ˙ q + DqF(e)¨ q

FMSA4CP / Input-Output / 6 Jonker/van Dijk/Aarts

§ 4

Equations of motion expressed the kinematic degrees of freedom q:

¯ M(q)¨ q = DF(x)T (f − MD2F(x,c) ˙ q ˙ q) − DF(e)Tσ ¯ M = DF(x)TMDF(x) system mass matrix DF(x)Tf = DF(x,c)Tf(c) + DF(x,m)Tf(m) nodal forces DF(e)Tσ = DF(e,m)Tσ(m) + DF(e,c)Tσ(c) stress resultants

  • σ(m)

σ(c)

  • =

  σ(m) a

σ(c)

a  +

  • S(m,m) S(m,c)

S(c,m) S(c,c)

  • e(m)

e(c)

  • +

  S(m,m) d

S(m,c)

d

S(c,m)

d

S(c,c)

d  

  • ˙

e(m) ˙ e(c)

  • Elastic coefficients S(m,m), S(m,c) and S(c,c) (symmetric matrices)

Viscous damping coefficients S(m,m)

d

, S(m,c)

d

and D(c,c)

d

(symmetric matrices) Driving forces and torques σ(m)

a

and σ(c)

a .

FMSA4CP / Input-Output / 7 Jonker/van Dijk/Aarts

§ 6

Equations of reaction for unknown stress resultants and reaction forces

(DxD)Tσ = f −M¨ x with partitioning f =

   

f(o) f(c) f(m)

    and σ =    

σ(o) σ(m) σ(c)

       (D(o)D(o))T (D(o)D(m))T (D(o)D(c))T (D(c)D(o))T (D(c)D(m))T (D(c)D(c))T (D(m)D(o))T (D(m)D(m))T (D(m)D(c))T       σ(o) σ(m) σ(c)    =    f (o) − M (o,c) ¨ x(c) − M (o,m) ¨ x(m) f (c) − M (c,c) ¨ x(c) − M (c,m) ¨ x(m) f (m) − M (m,c)¨ x(c) − M (m,m)¨ x(m)   

If the square matrix [(D(c)D(o))T , (D(c)D(m))T ] is non-singular, then

  • σ(o)

σ(m)

  • = ˜

D1

  • f(c) − M(c,c)¨

x(c) − M(c,m)¨ x(m) − (D(c)D(c))Tσ(c) , with ˜ D1 =

  • (D(c)D(o))T , (D(c)D(m))T −1 .

Vector σ(c) is known from the previous slide, so the reaction forces f(o) and the driving forces f(m) are then determined as well.

FMSA4CP / Input-Output / 8 Jonker/van Dijk/Aarts

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SLIDE 3

§ 5

State equations

q =

  • qd

qr

  • qd :

dynamic degrees of freedom (to be computed) qr : rheonomic degrees of freedom (known)

¯

Mdd ¯ Mdr ¯ Mrd ¯ Mrr ¨ qd ¨ qr

  • =

  DqdF(x)T

DqrF(x)T

  (f−MD2F(x) ˙

q ˙ q)−

  DqdF(e)T

DqrF(e)T

  σ

¯ Mdd(q)¨ qd = ¯ fd(q, ˙ q, t) − ¯ Mdr¨ qr f: nodal forces ¯ Mdd = DqdF(x)TMDqdF(x) σ: stress resultants ¯ Mdr = DqdF(x)TMDqrF(x) M: mass matrix ¯ fd = DqdF(x)T (f − MD2F(x) ˙ q ˙ q) − DqdF(e)Tσ Non-linear state-space equations d dt

  • qd

˙ qd

  • =
  • ˙

qd ¯ M−1

dd (¯

fd − ¯ Mdr¨ qr)

  • with state vector z =
  • qd

˙ qd

  • FMSA4CP / Input-Output / 9

Jonker/van Dijk/Aarts

§ 7 & § 12.3

Linearised equations: prefix δ indicates small variations

x = x0 + δx q = q0 + δq so q =

  • qd

qr

  • =
  • qd

qr

  • +
  • δqd

δqr

  • ˙

x = ˙ x0 + δ ˙ x ˙ q = ˙ q0 + δ ˙ q so ˙ q =

  • ˙

qd ˙ qr

  • =
  • ˙

qd ˙ qr

  • +
  • δ ˙

qd δ ˙ qr

  • ¨

x = ¨ x0 + δ¨ x ¨ q = ¨ q0 + δ¨ q so ¨ q =

  • ¨

qd ¨ qr

  • =
  • ¨

qd ¨ qr

  • +
  • δ¨

qd δ¨ qr

  • Stresses σ = σ0 + δσa

and forces f = f0 + δf. Linearised equations of kinematics δx = DF(x)δq , δ ˙ x = DF(x)δ ˙ q + (D2F(x) ˙ q)δq , δ¨ x = DF(x)δ¨ q + 2(D2F(x) ˙ q)δ ˙ q + (D2F(x)¨ q + D3F(x) ˙ q ˙ q)δq with third-order geometric transfer function D3F(x).

FMSA4CP / Input-Output / 10 Jonker/van Dijk/Aarts

§ 7.2

Linearised equations of motion

¯ Mδ¨ q +

  • ¯

C + ¯ D

  • δ ˙

q +

  • ¯

K + ¯ N + ¯ G

  • δq = DF(x)T δf − DF(e)T δσa

with DF(x)T δf = DF(x,c)T δf(c) + DF(x,m)T δf(m) and DF(e)T δσa = DF(e,m)T δσ(m)

a

+ DF(e,c)Tδσ(c)

a .

¯ M = DF(x)TMDF(x) ¯ C = DF(x)T (D ˙

xfin)DF(x) + 2MD2F(x) ˙

q

  • ¯

D = DF(e)TSdDF(e) ¯ K = DF(e)TSDF(e) ¯ N = DF(x)T Dx(M ¨ x − fin)DF(x) + (D ˙

xfin)D2F(x) ˙

q +M

  • D2F(x)¨

q + (D3F(x) ˙ q)˙ q

  • + DF(e)TSdD2F(e) ˙

q ¯ G = −D2F(x)T [f − M¨ x] − D2F(e)Tσ

FMSA4CP / Input-Output / 11 Jonker/van Dijk/Aarts

§ 7.3

Linearised equations of reaction

First order terms in Taylor series expansion: (DxD)T δσ + ((D2

xD)Tσ)δx = δf + (Dxfin)δx + (D ˙ xfin)δ ˙

x − Dx(M¨ x)δx − Mδ¨ x

  • r (DxD)T δσ = δf + M(x)δ¨

q − C(x)δ ˙ q − (N(x) + G(x))δq M(x) = MDF(x) C(x) = (D ˙

xfin)DF(x) + 2MD2F(x) ˙

q N(x) = Dx(M ¨ x − fin)DF(x) + (D ˙

xfin)D2F(x) ˙

q + M(D2F(x)¨ q + D3F(x) ˙ q ˙ q) G(x) = ((D2

xD)Tσ)DF(x)

FMSA4CP / Input-Output / 12 Jonker/van Dijk/Aarts

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SLIDE 4

Partitioning:

   (D(o)D(o))T (D(o)D(m))T (D(c)D(o))T (D(c)D(m))T (D(m)D(o))T (D(m)D(m))T   

  • δσ(o)

δσ(m)

  • =

   δf (o) δf (c) δf (m)    −    (D(o)D(c))T (D(c)D(c))T (D(m)D(c))T   

  • δσ(c)

−    M (x,o) C(x,o) (N (x,o) + G(x,o)) M (x,c) C(x,c) (N (x,c) + G(x,c)) M (x,m) C(x,m) (N (x,m) + G(x,m))      δ¨ q δ ˙ q δq  

  • Stress resultants of redundant elastic elements δσ(c) can be expressed in

δσ(c)

a , δq and δ ˙

q, see Eq. (41).

  • If the square partitioned matrix [(D(c)D(o))T , (D(c)D(m))T ] is not

singular, then the generalised stress resultant components of δσ(o) and δσ(m) can be computed with Eq. (43) from δq, δ ˙ q, δ¨ q, δf(c) and δσ(c).

  • Next the reaction forces δf(o) and external driving forces δf(m) follow

from Eqs. (45) and (48).

FMSA4CP / Input-Output / 13 Jonker/van Dijk/Aarts

§ 8.1

Linearised state-space equations — state equations

δ ˙ z = Aδz + Bδu with state vector δz =

  • δqd

δ ˙ qd

  • and

input vector δu =

  • δf(c)T , δf(m)T , δσ(m)T

a

, δσ(c)T

a

, δ¨ qrT , δ ˙ qrT , δqrT

T

. d dt

  • δqd

δ ˙ qd

  • =
  • O

I A21 A22 δqd δ ˙ qd

  • +
  • O

B2

  • δu

A21 = − ¯ M−1

dd

  • ¯

Kdd + ¯ Ndd + ¯ Gdd

  • A22 = − ¯

M−1

dd

  • ¯

Cdd + ¯ Ddd

  • B2 = ¯

M −1

dd

  • DqdF (x,c)T

DqdF(x,m)T −DqdF(e,m)T −DqdF(e,c)T − ¯ M dr − ¯ Cdr − ¯ N dr + ¯ Gdr

T

FMSA4CP / Input-Output / 14 Jonker/van Dijk/Aarts

§ 8.2

Linearised state-space equations — output equations

δy = Cδz + Dδu ,

  • δy(kin)

δy(dyn)

  • =
  • C(kin)

C(dyn)

  • δz +
  • D(kin)

D(dyn)

  • δu

δy =

  • δy(kin)

δy(dyn)

  • ,

δy(kin) =

   

δx δ ˙ x δ¨ x

    ,

y(dyn) =

      

δσ(o) δσ(m) δf(o) δf(m)

      

FMSA4CP / Input-Output / 15 Jonker/van Dijk/Aarts

§ 8.3

Linearised state-space equations — kinematic output matrices

δy(kin) =

   

δx δ ˙ x δ¨ x

    (δqd) (δ ˙ qd) C(kin) =    DqdF(x) DqdDF (x) ˙ q DqdF(x)A21 + DqdDF(x)¨ q + DqdD2F (x) ˙ q ˙ q O DqdF(x) DqdF (x)A22 + 2DqdDF (x) ˙ q   (δu) D(kin) =   O O DqdF(x)B2   (δ¨ qr) (δ ˙ qr) (δqr) +   O O DqrF(x) O DqrF(x) 2DqrDF(x) ˙ q DqrF(x) DqrDF(x) ˙ q DqrDF (x)¨ q + DqrD2F (x) ˙ q ˙ q   

FMSA4CP / Input-Output / 16 Jonker/van Dijk/Aarts

slide-5
SLIDE 5

§ 8.4

Linearised state-space equations — dynamic output matrices

y(dyn) =

      

δσ(o) δσ(m) δf(o) δf(m)

      

(δqd) (δ ˙ qd) C(dyn) =

  • ˜

M(x)

d

A2

  • +
  • ˜

K(x)

d

+ ˜ N(x)

d

+ ˜ G(x)

d

| ˜ C(x)

d

+ ˜ D(x)

d

  • (δf(c))

(δf(m)) (δσ(m)

a

) (δσ(c)

a )

D(dyn) =

  • ˜

M(x)

d

B2

  • +
  • ˜

Df(c) | O | O | ˜ Dσ(c) | (δ¨ qr) (δ ˙ qr) (δqr) | ˜ M(x)

r

| ˜ C(x)

r

+ ˜ D(x)

r

| ˜ K(x)

r

+ ˜ N(x)

r

+ ˜ G(x)

r

  • where the matrices are given in § 8.4.

FMSA4CP / Input-Output / 17 Jonker/van Dijk/Aarts

§ 9

Stationary and equilibrium solutions

¯ Mdd(q)¨ qd = ¯ fd(q, ˙ q, t) − ¯ Mdr¨ qr ¨ qd = 0, ˙ qd = 0, ¨ qr = 0

  • ˙

qd ¯ fd(qd, ˙ qd)

  • =
  • Stability of stationary solution is determined by eigenvalues of state matrix A.

Frequency equation: det

  • −ω2

i ¯

Mdd + ¯ Kdd + ¯ Ndd + ¯ Gdd

  • = 0

Solutions of eigenvalue problem are natural frequencies ωi and mode shapes. Stability of equilibrium equation: det

  • ¯

Kdd + λi ¯ Gdd

  • = 0

λi = fi/f0, f0 is reference load and λi the critical mode multipliers

FMSA4CP / Input-Output / 18 Jonker/van Dijk/Aarts

§ 10

From state space equations to transfer function(s)

Linearised state-space equations with state vector δz =

  • δqd

δ ˙ qd

  • :

δ ˙ z = Aδz + Bδu δy = Cδz + Dδu with general output vector δy =

  • δy(kin)

δy(dyn)

  • and input vector

input vector δu =

  • δf(c)T , δf(m)T , δσ(m)T

a

, δσ(c)T

a

, δ¨ qrT, δ ˙ qrT , δqrT

T

The state space representation can be translated into a transfer function matrix with the standard expression ˜ G(s) = C(sI − A)−1B + D which is straightforward for the inputs δf(c), δf(m), δσ(m)

a

and δσ(c)

a .

Note however the occurrence of δqr and the time derivatives δ ˙ qr and δ¨ qr!!!

FMSA4CP / Input-Output / 19 Jonker/van Dijk/Aarts

A transfer function relates the Laplace transforms of the system’s input δu =

  • δf(c)T , δf(m)T , δσ(m)T

a

, δσ(c)T

a

, δ¨ qrT , δ ˙ qrT , δqrT

T

and output δy: L{δy(t)} = δy(s) = ˜ G(s)δu(s) = ˜ G(s)L{δu(t)} The Laplace transforms of δqr, δ ˙ qr and δ¨ qr are not independent as L{δ ˙ qr(t)} = sL{δqr(t)} and L{δ¨ qr(t)} = s2L{δqr(t)} and have to be taken into account to specify e.g. the transfer function (part) from δqr to δy(t). Case 1: Define the input vector as δu1 =

  • δf(c)T , δf(m)T , δσ(m)T

a

, δσ(c)T

a

, δqrT

T

and determine the transfer function in δy(s) = G1(s)δu1(s)

FMSA4CP / Input-Output / 20 Jonker/van Dijk/Aarts

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SLIDE 6

Case 1 (position input): Define the input vector as δu1 =

  • δf(c)T , δf(m)T , δσ(m)T

a

, δσ(c)T

a

, δqrT

T

then δu(s) =

          I I I I s2 I s I I          

δu1(s) and the transfer function from δu1 to δy is G1(s) =

  • C(sI − A)−1B + D

         I I I I s2 I s I I          

FMSA4CP / Input-Output / 21 Jonker/van Dijk/Aarts

Case 2 (velocity input): Define the input vector as δu2 =

  • δf(c)T , δf(m)T , δσ(m)T

a

, δσ(c)T

a

, δ ˙ qrT

T

then δu(s) =

          I I I I s I I 1/s I          

δu2(s) and the transfer function from δu2 to δy is G2(s) =

  • C(sI − A)−1B + D

         I I I I s I I 1/s I          

FMSA4CP / Input-Output / 22 Jonker/van Dijk/Aarts

Case 3 (acceleration input): Define the input vector as δu3 =

  • δf(c)T , δf(m)T , δσ(m)T

a

, δσ(c)T

a

, δ¨ qrT

T

then δu(s) =

          I I I I I 1/s I 1/s2 I          

δu3(s) and the transfer function from δu3 to δy is G3(s) =

  • C(sI − A)−1B + D

         I I I I I 1/s I 1/s2 I          

FMSA4CP / Input-Output / 23 Jonker/van Dijk/Aarts

Note 1: The three cases can be combined, e.g. to define an input δu with the position of one rheonomic degree of freedom and the acceleration of another rheonomic degree of freedom. Note 2: In the case only accelerations δ¨ qr and no velocities δ ˙ qr and positions δqr appear in the input, the transfer function matrix can also be obtained by adding the velocities and positions to the state vector δx.

FMSA4CP / Input-Output / 24 Jonker/van Dijk/Aarts

slide-7
SLIDE 7

§ 11.1

Spring-mass-damper system mounted on a cart

k mass m δx2 d δx1 massless cart

δq = δx2 − δx1 δu =

  • δ¨

x1, δ ˙ x1, δx1T

A =

  • 1

−m−1k −m−1d

  • B =
  • −1
  • C =
  • 1
  • D =
  • 1
  • Transfer functions
  • G(s) = δx2(s)

δu(s) = C(sI − A)−1B + D =

 

−1 s2 + d

ms + k m

1

 

FMSA4CP / Input-Output / 25 Jonker/van Dijk/Aarts

Single input δx1(s) to the output δx2(s) δu(s) =

  • s2

s 1

T δx1(s)

G1(s) = δx2(s) δx1(s) = G(s)

   

s2 s 1

    = d ms + k m

s2 + d

ms + k m

Single input δ ˙ x1(s) = δv1 to output δx2(s) δu(s) = [s 1 1/s]T δv1(s) G2(s) = δx2(s) δv1(s) = G(s)

   

s 1 1/s

    = d ms + k m

s(s2 + d

ms + k m)

FMSA4CP / Input-Output / 26 Jonker/van Dijk/Aarts

§ 11.2

Active vibration isolation of a metrology frame

x z y 3D view of lens suspension frame of a wafer stepper/scanner

FMSA4CP / Input-Output / 27 Jonker/van Dijk/Aarts

z lens center of gravity lens frame mount c 360 mm 200 mm y 45° y 1500 mm z x

Front view Top view

FMSA4CP / Input-Output / 28 Jonker/van Dijk/Aarts

slide-8
SLIDE 8

Detailed view of a mount

45° 45° frame plate acceleration sensor flexible beam force sensor piezo actuator floor plate

Piezo actuator (force σ(k)

a

) with parallel spring (stiffness s(k))

s(k) σ(k)

1

σ(k)

a

σ(k)

1

σ(k)

1

= σ(k)

a

+ s(k)e(k)

1

FMSA4CP / Input-Output / 29 Jonker/van Dijk/Aarts

Finite element model of metrology frame and floor

17 19 11 9 15 13 (5) (8) (9) (3) (4) (6) (7) (1) (2) (11) (10) 3 5 7 z y x

q(d) =

  • e(4)

1 , e(5) 1 , e(6) 1 , e(7) 1 , e(8) 1 , e(9) 1

T u(floor) =

  • ¨

x9, ¨ z11, ¨ y13, ¨ z15, ¨ z17, ¨ y19T u(actuator) =

  • σ(4)

a

, σ(5)

a

, σ(6)

a

, σ(7)

a

, σ(8)

a

, σ(9)

a

T y(frame) =

  • ¨

x3, ¨ z3, ¨ y5, ¨ z5, ¨ z7, ¨ y7T y(force) =

  • σ(4)

1 , σ(5) 1 , σ(6) 1 , σ(7) 1 , σ(8) 1 , σ(9) 1

T

FMSA4CP / Input-Output / 30 Jonker/van Dijk/Aarts

Generalised plant G with 12 inputs and 12 outputs and controller C with 6 inputs and 6 outputs

G11 C G21 G12 G22

Feedback control equations u(s)(actuator) = C(s)y(s)(force) C(s) = −

 K(P) + K(I)

s

 

K(P) =

  • ω2

dI ¯

Mdd

−1 ¯

Kdd − I K(I) = 2ζωd

  • I + K(P)

FMSA4CP / Input-Output / 31 Jonker/van Dijk/Aarts

Mode shapes and natural frequencies of the passive system

Mode 1: 13.9 Hz Mode 2: 13.9 Hz Mode 3: 20.0 Hz Mode 4: 31.3 Hz Mode 5: 31.3 Hz Mode 6: 49.0 Hz

FMSA4CP / Input-Output / 32 Jonker/van Dijk/Aarts

slide-9
SLIDE 9

Closed loop transfer function T T = G11 + G12 · C · (I − G22 · C)−1 · G21

10

−2

10

−1

10 10

1

10

2

10

3

−150 −100 −50 50 100 150 Singular Values Frequency (Hz) Singular Values (dB)

Singular values; dashed line is closed loop, solid line is open loop

FMSA4CP / Input-Output / 33 Jonker/van Dijk/Aarts

Conclusions

  • Linearised state-space formulation for flexible multibody

systems.

  • Arbitrary combination of positions, velocities, accelerations and

forces can be taken as input variables and as output variables.

  • Finite element based multibody concept enables a low

dimensional description of prototype models suitable for design purposes.

  • Insight into the relations between component properties and

dominant system behaviour.

FMSA4CP / Input-Output / 34 Jonker/van Dijk/Aarts