Equations of motion - Universiteit Twente · PDF file• Linearised equations of motion ......

Efficient dynamic closed-loop simulationsof flexible manipulators

• Linearised equations of motion (for e.g. closed-loop simulations)

• Superposition of rigid link motion and small elastic deformations

• Perturbation method

• Mode-Acceleration Method / Adaptive Modal Integration

• Examples:One-link manipulator with constrained motionSpatial two-link flexible manipulator with PID control

• Conclusions

FMSA4CP-LP / 1Ronald Aarts

§ 12.6 or paper LP-1

Equations of motion

Flexible manipulator withem: large relative displacements and rotations,εm: flexible deformation parameters.

Equations of motion, adapted from slide DvM/93:[

Mee Meε

Mεe Mεε

] [

em

εm

]

+

[

DemFT

DεmFT

]

[

M(D2F ·(em, εm))·(em, εm)− f]

= −

[

σem

σεm

]

Components of the reduced mass matrix

Mee = DemFTMDemF , Meε = DemF

TMDεmF ,

Mεe = DεmFTMDemF , Mεε = DεmF

TMDεmF .


Equations of motion (2)With generalised coordinates (D.O.F.) q =

[

em

εm

]

M(q)q + C(q, q)q + g(q) + Kq = Bu,

• M(q) is the reduced mass matrix M = DFTMDF

• C(q, q)q represents the Coriolis and centrifugal forces• g(q) is the vector of external nodal forces, including gravity,• σem are the driving forces and torques, i.e. control input vector −u (note

the sign), and B =

[

I

0

]

• σεm is the stress resultant vector of flexible elements, characterised byHooke’s law: Symmetric stiffness matrix Kεε with the elastic constants and

K =

[

0 0

0Kεε

]

The direct solution of the non-linear equations of motion is rather timeconsuming (both low and high frequent behaviour).


Perturbation method

Model the vibrational motion of the manipulator as a first-order perturbation δq

of the nominal rigid link motion q0

q = q0 + δq or

[

em

εm

]

=

[

em00

]

+

[

δem

δεm

]

;u = u0 + δu

The perturbation method involves two steps:

1. Compute nominal rigid link motion q0 from the non-linear equations of motionwith the rigidified model, i.e. all εm ≡ 0.

Mee0 em0 +DemF

T0

[

M0(D2F0·(e

m0,0))·(e

m0,0)− f

]

= −σem0 = u0,

Mεe0 em0 +DεmF

T0

[

M0(D2F0·(e

m0,0))·(e

m0,0)− f

]

= −σεm0 .

For a known nominal trajectory em0 , em0 , em0 the generalised stress resultantsσem0 = −u0 and σεm0 are obtained.


2. Compute the (small) vibrational motion δq from linearised equations ofmotion:

M0δq + C0δq + K0δq =

[

δu

σεm0

]

.

σεm0 are the generalised stress resultants applied as internal excitation forces.

M0 is the system mass matrix,

C0 is the velocity sensitivity matrix,

K0 is the combined stiffness matrix defined as

K0 =

[

0 0

0Kεε0

]

+G0 +N0,

including the structural stiffness matrix Kεε0 , the geometric stiffening matrix G0

and the dynamic stiffening matrix N0.


Perturbation method: Applications

M0

[

δem

εm

]

+ C0

[

δem

εm

]

+ K0

[

δem

εm

]

=

[

−δσem0σεm0

]

.

1. Constrained motion (§ 12.6 or paper LP-1): em = em0 , so δem ≡ 0.

→ Solve differential equation for εm and compute δσem0 .

→ Example of one-link manipulator.

2. Prescribed forces and torques (§ 12.6): σem = σem0 , so δσem0 ≡ 0.

→ Solve differential equation for em and εm.

3. Controlled trajectory motion (paper LP-1): δσem0 = −δu from controlsystem.

→ Solve differential equation for em and εm.

→ Example of two-link manipulator.


Constrained motion: One-link flexible manipulator

[

mee0 Meε

0Mεe

0 Mεε0

] [

δem

εm

]

+

[

0 0

0 Kεε0 + Nεε

0 + Gεε0

] [

δem

εm

]

=

[

−δσem

σεm0

]

Constrained motion: em = em(t) ; δem(t) = 0.

em(t) =

0 t < 0,

Ω

T

[

1

2t2 +

T2

4π2(cos

2πt

T− 1)

]

0 ≤ t ≤ T,

Ω(t−1

2T) t > T.

Flexible motion: Mεε0 εm +

[

Kεε0 + Nεε

0 + Gεε0

]

εm = σεm0


Tip deflection δy using different superposition approximations

0 5 10 15 20 25 30−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

t [s]

δy [m

]

1) M0, K

0, N

0 and G

02) K

0, N

0 and G

0

3) M0, K

0

4) K0

5) M

0, K

0, N

0

1 2 3 4 5


Comparison of the maximum tip deflection

Model number of elements Max. deflection [m]SPACAR, non-linear 4 0.536

Wu & Haug 4 substructures 0.556idem 6 substructures 0.543

superposition, case 1 4 0.537idem, case 2 4 0.531idem, case 3 4 0.569idem, case 4 4 0.569idem, case 5 4 ∞

case 1 M εε0 εm+

[

Kεε0 + N εε

0 + Gεε0

]

εm = σεm0

case 2[

Kεε0 + N εε

0 + Gεε0

]

εm = σεm0

case 3 M εε0 εm+

[

Kεε0

]

εm = σεm0

case 4[

Kεε0

]

εm = σεm0

case 5 M εε0 εm+

[

Kεε0 + N εε

0

]

εm = σεm0


Natural frequency of the first constrained bending mode

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

t [s]

ω1 [r

ad/s

]

1) M0, K

0, N

0 and G

05) M

0, K

0, N

0

Frequency equation: det(−ω2i M

εε0 + Kεε

0 + Nεε0 + Gεε

0 ) = 0


Perturbation method for controlled trajectory motion

As before, model the vibrational motion of the manipulator as a first-orderperturbation δq of the nominal rigid link motion q0

q = q0 + δq,

Apply the two steps of the perturbation method:1. Compute nominal rigid link motion q0 from the non-linear equations of motionwith all εm ≡ 0, i.e. all vibrational modes set to zero.2. Compute the vibrational motion δq from linearised equations of motion:

M0δq + C0δq + K0δq = σ0, σ0 =

[

δudσεm0

]

.

σεm0 are the generalized stress resultants applied as internal excitation forces.δud is the control input vector (minus u0) and ...


δud is the control input vector δu = u− u0, possibly minus the proportionalaction of the controller represented by a matrix Kp

δu = −Kpδem + δud.

M0 is the system mass matrix,

C0 is the velocity sensitivity matrix,

K0 is the combined stiffness matrix defined as

K0 =

[

0 0

0Kεε0

]

+G0 +N0 +

[

Kp 0

0 0

]

.

It includes the structural stiffness matrix Kεε0 , the geometric stiffening matrix

G0, the dynamic stiffening matrix N0 and the matrix Kp of the proportionalcontrol action.

Note that one can also take δud = δu in which case Kp = 0 and theproportional control action is not included in the linearised equation of motion.Using a realistic Kp is particular beneficial for the modal analysis to bediscussed next.


Mode-superposition method

Equations of motion in n principal coordinates η

δq = Φη Φ =[

φ1, φ2, . . . , φn]

δq = Φη + Φη (KS0 − ω2

i M0)φi = 0, i = 1,2, ..., n,

δq = Φη +2Φη + Φη Symmetric KS0 = 1

2(K0 + KT0 )

K0 includes Kp.

M η + Cη + Kη = σ,

where

M =ΦTM0Φ modal mass matrix,C =ΦTC0Φ+ 2ΦTM0Φ modal damping matrix,K =ΦT K0Φ+ΦTC0Φ +ΦTM0Φ modal stiffness matrix,σ=ΦTσ0 modal force vector.

“Adaptive Modal Integration” (AMI): Time-varying nature ofmodal matrix Φ is taken into account.


Mode-Displacement Method (MDM)

Solution δq using only n < n modes

δq = Φη, Φ =[

φ1, φ2, . . . , φn

]

.

Mode-Acceleration Method (MAM)

Improved convergence after rewriting the equations of motion

δq = K−10 (σ0 − M0δq − C0δq),

and substitution the MDM solution δq in the right hand side

δq = K−10 σ0 − K−1

0 (M0δ¨q + C0δ ˙q).

First term in the expression of the MAM solution δq represents a pseudo staticresponse of the system.


Controlled trajectory motion

A (0.536, 0, 0)

B (0, 1.300, 0)

x

y

z

Motion trajectory

0.0 0.2 0.8 1.00

1.76

t [s]

Tip

vel

ocity

[m/s

]

Velocity profile ofthe manipulator tip

0.0 0.5 1.0

−200

0

200

400

600

t [s]

u 0 [N/m

]

σ1

σ2

σ3

Nominal torques u0 forthe three actuators

MIMO PID feedback control: δu = −Mee0 H(s) δem.

Mee0 : mass matrix for decoupling between the actuators.

H(s): controller with three SISO PID controllers on the diagonal.


Block diagram for simulations in Simulink

Non-linear simulation:

ltv

y−Reference

ltv

times M0

dYtip

ltv

Setpoint U0

Selector

Selector Ytipref

Selector

Selector Ytip

Selector

Selector Eref

Selector

Selector E

spasim

SPASIM

In1 Out1

MIMO lead

In1 Out1

MIMO PI

Output y = y0 + δy

Nominal torques u0

Referenceoutput y0

SPASIM block for (non-linear) mechanism simulation.

Nominal torques u0 (applied as feedforward) and reference output y0 are readfrom files.


Perturbation method with modal analysis (“AMI”):

ltv

times M0

dYtip

ltv

Setpoint Sigma

Selector

Selector Ytip

Selector

Selector E

In1 Out1

MIMO lead

In1 Out1

MIMO PIltv

LTV

omega*omega

Kp

Output δy

Reference is 0

LTV-block for simulation of Linear Time-Varying state space system

xss =Ass xss +Bss uss

yss =Css xss

xss =

[

δq

δq

]

or

[

η

η

]

, uss =

[

δudσεm0

]

,

yss: User defined outputs.

Proportional controller part Kp is included in the LTV block and has to beexcluded in the controller.


Analysis 1: Natural frequencies along

the trajectory

0.0 0.2 0.4 0.6 0.8 1.00

50

100

150

t [s]

Nat

. con

s. fr

eq. [

rad/

s]

0.0 0.2 0.4 0.6 0.8 1.0

10

12

14

t [s]

Nat

. sym

. fre

q. [r

ad/s

]

Four lowest natural frequencies ωc,i

for a constrained manipulator.The bandwidth of the PID controllersis set to ωb = 12 rad/s.

Three lowest closed-loop naturalfrequencies ωi during the controlledtrajectory motion of the manipulatorcomputed with a symmetric KS

0 .


Analysis 2: Open loop Bode plots for

actuator 1 + controller (SISO)

Frequency (rad/sec)Pha

se (

deg)

; Mag

nitu

de (

dB)

−40−20

02040

100

101

102

103

−270−180

−900

90

Initial configuration

Frequency (rad/sec)Pha

se (

deg)

; Mag

nitu

de (

dB)

−40−20

02040

100

101

102

103

−270−180

−900

90

Final configuration

The solid lines are with the PID controller.The dashed lines are for a controller with an additional pole: Unstablebehaviour is expected from the graph near 100 rad/s, which can be confirmedby simulations.


Simulations 1a: Deviations from the nominal

trajectory

0 0.5 1 1.5 2

−20

−10

0

10

t [s]

δe [m

rad]

δe1

δe2

δe3

Actuator rotations

non−linear methodperturbation methodAMI, 4 modes

0 0.5 1 1.5 2

−20

−15

−10

−5

0

5

t [s]

δxyz

tip [m

m]

δxtip

δytip

δztip

Tip co-ordinates

Deviations from the nominal trajectory according to three simulation methods:No differences, so the AMI method performs well with only 4 modes (3 modifiedrigid link modes + 1 additional mode).


Simulations 2a: Errors and CPU time

0 2 4 6 8 10 12 PM0.001

0.01

0.1

1

10

Number of modes

Max

imum

err

or [m

m]

xtip

ytip

ztip

Maximum errors tip co-ordinates

0 2 4 6 8 10 12 PM NL10

1

102

103

104

Number of modes

Number oftime steps

CPU time [s]

CPU time

• The maximum error in the tip co-ordinates found with the AMI method withonly 3 modes is comparable to the difference between the perturbationmethod and the non-linear simulation (indicated with “PM”).

• With more modes the accuracy hardly improves at the expense of slowersimulations. A significant reduction in CPU time is obtained in comparisonthe perturbation method (“PM”) and the non-linear simulation (“NL”).


Simulations 1b: Deviations from the nominal

trajectory

Second case: A stiffer manipulator with a controller bandwidth of approximately60 rad/s = 9.5 Hz.

0 0.5 1 1.5 2

−0.1

0

0.1

t [s]

δe [m

rad]

δe1

δe2

δe3

Actuator rotations

non−linear methodperturbation methodAMI, 4 modes

0 0.5 1 1.5 2−0.8

−0.6

−0.4

−0.2

0

t [s]

δxyz

tip [m

m]

δxtip

δytip

δztip

Tip co-ordinates

• Conclusions as before.


Simulations 2b: Errors and CPU time

0 2 4 6 8 10 12 PM10

−6

10−5

10−4

10−3

10−2

10−1

Number of modes

Max

imum

err

or [m

m]

xtipytipztip

Maximum errors tip co-ordinates

0 2 4 6 8 10 12 PM NL10

1

102

103

104

Number of modes

Number oftime steps

CPU time [s]

CPU time (faster PC)

• For the stiffer manipulator the approach works as well. As before the AMImethod with only 3 modes gives accurate results for e.g. the tip position.


Conclusions

• The presented perturbation method allows an efficient numerical simulationof the controlled trajectory motion of a flexible manipulator as well as astraightforward vibration control formulation.

• A further reduction of the simulation time was obtained by applying a modalreduction technique, which we refer to as the Adaptive Modal Integration(AMI) method.

• For the spatial flexible two-link manipulator, results of both the perturbationmethod and the AMI method agree well with the results obtained from a fullnon-linear analysis. In the AMI method only three (modified rigid link) orfour degrees of freedom are needed to reach a satisfying accuracy.

• Crucial elements in the AMI method are the availability of accuratelylinearized equations and a careful modal analysis in which the time-varyingnature of the mode shape functions and the proportional feedback gainsare taken into account.


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