advanced motion control -...

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/department of mechanical engineering

Advanced Motion Control

Maarten SteinbuchControl Systems Technology

Department of Mechanical EngineeringEindhoven University of Technology

APC workshop, Vancouver, may 2005

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Contents• background, motion systems• control for dummies• advanced motion control challenges• embedded dynamical systems

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Eindhoven University of Technology

• 9 scientific departments, 10 academic Bachelor programmes, 19 Master programmes, 3000 employees, 220 professors, 6800 students, 200 postgraduate students, 450 PhD students• located in the Eindhoven-Aachen-Leuven triangle• ‘mechatronics’ high tech industry:• Philips, ASML, FEI, Assembleon

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Mechanical Engineering Department

• 9 full prof., 60 senior research staff, • 18 Post Docs, 105 PhD students, • 700 BSc and MSc students

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Structure of Mechanical Engineering

Thermo Fluids EngineeringComputational and Experimental MechanicsDynamical Systems Design

2 ‘theme Mastertracks’:• Automotive Engineering Science• Micro and Nano Technology

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Mechanical Engineering Department

DynamicalSystemsDesign(DSD)

Thermo FluidsEngineering

(TFE)

Computational & Experimental

Mechanics(CEM)

Automotive Engineering Science (2001)

(Sub)-Micron Technology (2003)

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TU/e - W : Full Chairs in DSD Division

• Dynamics and Control:Prof.Dr. Henk Nijmeijer

• Control Systems Technology:Prof.Dr.Ir. Maarten Steinbuch

• Systems Engineering :Prof.Dr.Ir. Koos Rooda

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from Industry …

… to Academia

(1999)

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TU Eindhoven

Philips

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Isles of AcademiaIsles of Academia

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…

in theory there is no difference between theory and practice

in practice there is

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…2

simulation is like masturbation:the more you do it the more you think it is the real thing!

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bridging the gap

• education: merge classic & modern • bring in real industrial systems• confront PhDs with other disciplines• learn from experimental experience

how to proceed with theory

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Control Systems Technology

• 1 full prof• 2 part-time prof• 7 associate and assistant prof• 4 technical staff members• 20 PhD students• 40 MSc students/year

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• Motion Systems– industrial applications (pick-and-place,

(bio)-robots)– consumer applications (storage systems)– hydraulic servo systems

• Automotive– power trains (in particular CVT)– (passive) car safety systems– vehicle electrical power management

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DaVinci surgery robot © Intuitive Surgery Inc.

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Automotive Safety Restraint Systems

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Zero-Inertia

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Continously Variable Transmission

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1/1,000 mm

Source: intel, ICE

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Stepper Exposure Method

Reticle

Lens

Reticle

Wafer

Design

Source: ASML

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Illumination

Aberrations Scatter

Sourceoptimization

Doseoptimization Mask optimization software

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Mechanical Electronic / Electrical Pneumatic Hydraulic Chemical

AcousticalThermal Optical Software

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Motion Control for

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Motion Systems

m F

x

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x

Servo force ?Mass M

Disturbance F d

Fs

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Mechanical solution:

Mkf

π21

=

xdxkF ⋅−⋅−=

x

Mass M

k

d

F Disturbance Fd

Force spring damper

Eigenfrequency

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x

Mass M

Mk

f p

π21

=

xkxkF vps ⋅−⋅−=

Fservo

damping servo:stiffness servo:

v

p

kk

Eigenfrequency

Servo force

Servo analogon:

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x

Mass M

Disturbance F d

Fs

Example:

Slide: mass = 5 kgRequired accuracy 10 µm at all timesDisturbance (f.e. friction) = 3 N

1. Required servo stiffness?2. Eigenfrequency?

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h or xsx

)()(:damper-Spring

xhdxhkF −+−=

How to move to / follow a setpoint:

)()( :Controller

xxkxxkF svsps −+−=

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dtdkk vp +

controller

Fservo

process

MassxFxs

+

-

e

xsx

Fd

Kp/kv-controller or PD-controller

Fdisturbance

+

+

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?

?

=

=

v

p

k

k

controllerxs

Fservo

processxe+

-

v

p

k

kerror⇒ stability⇔

Trade off

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xsxSetpoints:

What should xs look like as a function of time, when moving the mass?

(first order, second order, third order,….?)

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x

F M

Apply a force F (step profile):

⇓

=

)()( txMtF

x(t) is second order, when F constant

Second order profile requires following information:

- maximum acceleration- maximum velocity- travel distance

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Example

2max

max

sec/36.1500sec/6320

3.62

radeAccradVel

radPos

≈=≈=≈=

πππ

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Concluding remarks time domain tuning

xsx

Mkv

kp

A control system, consisting of only a single mass m and a kp/kv controller (as depicted below), is always stable.kp will act as a spring; kv will act as a damper

As a result of this: when a control system is unstable, it cannot be a pure single mass + kp/kv controller(With positive parameters m, kp and kv)

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Time domain: Monday and Thursday at 22:10

Frequency domain:twice a week

Frequency domain

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/department of mechanical engineering0 0.5 1 1.5 2 2.5 3

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

time in sec

excitation force in N and displacement in m

excitation force (offset 0.1 &scaling 1e-4)

response

F

x

Mweak spring M = 5 kg

(f = 2.5 Hz)

F t t( ) sin( )= 400 2 7π

H Hz e m NH Hz

( ) . / /( )

7 0 045 400 1 47 1800

≈ = −

∠ ≈ −

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measurement mechanics stage

102 103-80

-60

-40

-20

frequency in Hz

amplitude in dB

102 103-200

0

200phase in deg

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Transfer function:

kdsMssFsxsH

++== 2

1)()()(

1Ms2 + ds + k

F x

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Bode plot of the PD-controller:

100 101 10260

80

100

frequency in Hz

amplitude in dB

100 101 1020

50

100phase in deg

kp = 1500 N/m; kv = 20 Ns/m

+1

+0

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H(s)exs +

- Fs

xC(s)

Fd

++

Four important transfer functions

)()()( sHsCsL =1. Open loop

)()(1)()()()(sHsC

sHsCsxxsTs +

==

S s ex

sC s H ss

( ) ( )( ) ( )

= =+

11

2. Closed loop

3. Sensitivity

4. Proces Sensitivity H s xF

s H sC s H sps

d

( ) ( ) ( )( ) ( )

= =+1

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Mimo

Filters

•Integral action•Differential action•Low-pass•High-pass•Band-pass•Notch filter

PeeDeePeeEye

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Integral action :

X(t) Y(t)siτ

1

τI integral time constant τI =1/ki

-1

ω=2πf0°

-90°

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+90°

0°

ω

+1Differential action

ωε

ωε

kujsuksH ==== ;;

ks uε

ω

ddd

fπωτ

21==

+90°

0°

1+=

sksu

dτε“tamme” differentiator :

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“lead” filter

s

sssuH

d

d

γττ

ττ

ε +

+=

++

==1

111

2

1

γ>1

0°

+90°

ω

11

1τ

ω =2

21τ

ω =

1

522

1 −≈ττγ

2121

1ττ

ωωω ==c

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P+I+D

ω

+90°

0°

-90°

ii τ

ω 1=

dd τ

ω 1=

-1 +1

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+

+⎟⎟⎠

⎞⎜⎜⎝

⎛+==

γττ

τε ss

skuH

d

d

i 1

111

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2nd order filter

121

21

2

++==

ωβ

ωε ss

kuH

ε

u(t)- s

1ωs

1ω

β2

-180°1ωω =

β21

0 °

-90 °

-2

1

Top: .1 21 βωω −=o

ε(t)

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General 2nd order filters

12

12

222

2

21

121

2

++

++==

ωβ

ω

ωβ

ωε ss

ssuH

+2

0°

+180°

1

2

1

2⎟⎟⎠

⎞⎜⎜⎝

⎛ωωω1≠ω2General:

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21 ωω ≥

0°

1

+2

-2⎟⎟⎠

⎞⎜⎜⎝

⎛2

1

22

ωω

ω

-180°ω2 ω1

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“Notch”-filter :ω1= ω2

-180°

ampl.

2

1

ββ

fase 0°

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Loop shaping procedure for motion systems1. stabilize the plant:

add lead/lag with zero = bandwidth/3 and pole = bandwidth*3, adjust gain to get stability; or add a pure PD with break point at the bandwidth

2. add low-pass filter:choose poles = bandwidth*6

3. add notch if necessary, or apply any other kind of first or second order filter and shape the loop

4. add integral action:choose zero = bandwidth/5

5. increase bandwidth:increase gain and zero/poles of integral action, lead/lag and other filters

during steps 2-5: check all relevant transfer functions, and relate to disturbance spectrum

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Implementation issues

1. sampling = delay: linear phase lag

for example: sampling at 4 kHz gives phase lag due to Zero-Order-Hold of:

180º @ 4 kHz18º @ 400 Hz9º @ 200 Hz

2. Delay due to calculations3. Quantization (sensors, digital representation)

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Feedforward design

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Feedforward based on inverse model

sx2ms

1sKK vp +

2ms

x

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Example: m=5 [kg], b=1 [Ns/m],

2nd degree setpoint

0.5

1

xs[m

]

0

0.5

1

1.5

vs[m

s-1]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-4

-2

0

2

4

0

t [s]

as

[ms-

2]

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Example: tracking error, no feedforward

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10-3

t [s]

erro

r [m

]

viscous damping effect

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Example: tracking error, with feedforward

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10-3

t [s]

= 0.9, = 0faKfvK

= 0.9, = 4.5 faKfvK

erro

r [m

]

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feedforward structure

sxH(s)

fvK

x

faK

C(s)

sx

sx

)sxsign(fcK

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3rd degree setpoint trajectory

xs[m

]

0

0.5

1

1.5

vs[m

s-1]

0

0.5

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-4

-2

0

2

4

as

[ms-

2]

t [s]

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Parasitic Dynamics

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Three Types of Dynamic Effects

- Actuator flexibility

- Guidance flexibility

- Limited mass and stiffness of frame

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1. Actuator flexibility

k

x

d

SensorMotorFs

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100 101 102-150

-100

-50

frequency in Hz

amplitude in dB

100 101 102-200

0

200phase in deg

x1F

-2

-2

c

d

M1

x1

M2F

100 101 102-150

-100

-50

frequency in Hz

amplitude in dB

100 10110

2-200

0

200phase in deg

H =xF

2

xF

1

xF

2

c

d

M1

x2

M2F

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2. Guidance flexibility

k

x

Fs

M, J

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3. Limited mass and stiffness of frame

x

FsMotor

Frame

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Motion Control Properties

• experimentation is ‘cheap’Disturbance Design Cycle: 7 min FRF measurement,

model, loopshape, implementation• plant decoupling, i.e. SISO• feedforward: low-order model-based• feedback: loopshaping• key limitation: bode gain/phase - sensitivity integral

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Gunter Stein’s Bode Lecture, CDC 1989

IEEE Control Systems Magazine, 23 (2003), pp 12- 25

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Motion Control Challenge:

how to cope with Bode sensitivity limitation?

0)(log0

=∫∞

ωω djS

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directions of motion control research

• MIMO loopshaping• nonlinear control of linear systems (reset…) • disturbance-based modelling and control• data-driven control

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Sensitivity dirt

‘Loop’ 1

‘Loop’ 2

… ‘Loop’ n

MIMO integral constraints…

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• MIMO loopshaping• nonlinear control of linear systems (reset…)• disturbance-based modelling and control• data-driven control

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Problem formulation

• Do there exist nonlinear feedback controllers that give better ‘performance’ for linear motionsystems than linear solutions?

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Approach

• Performance measures?• Plant is linear, but• disturbances and specifications ‘change’• Use LPV for synthesis?• How about non-smooth (reset) filters

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Bode Gain/Phase relation

Slope = n means

phase = n*90 degrees

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SPAN- filter

rectifier low-pass

X

sign lead

be creative with control!

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disturbance-based modelling and control

• disc errors vs shocks optical storage • stochastic vs deterministic disturbances• repetitive vs a-periodic setpoints or disturbances

Internal model principle….

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Iterative Learning Control (ILC)

( ) 11 <− PSLQkk ee <+1

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Machine-in-the-loop design

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Data-driven control• Examples:

– data-based LQG control– iterative feedback tuning– virtual reference feedback tuning– unfalsified control

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Problem statement

• Design a SISO LTI controller C for LTI plant P

r e u yPC+

-

• Control objective: realize the desired So and To

,1

1o

o PCS

+= .

1 o

oo PC

PCT+

=

• Ideal controller Co:

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Data-based controller design

• The controller class: }.)()({)},({ p θβθ zzCzC T=

• Cp(z) is directly prescribed by the designer: notches,

integrators, etc.• Basis functions: .)]()()([)( 10

Tn zβzβzβz …=β

• Tuning parameters: .]θθθ[ 10T

n…=θ

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)()()()( ooo zPzSzCzT =• Constraint on Co:

• Model-based cost function:

( ) 22ooMB )()()(),()()( zWzPzSzCzTJ θθ −=

• Processing the measurements:

)()(),()()()()()(),()()( oooo tyzSzCtuzTtuzPzSzCtuzT θθ =⇒=

• Data-based cost function:

( ) 2

1ooDB ])()(),()()()([1)( ∑ −=

=

N

t

N tyzSzCtuzTzLN

J θθ

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/department of mechanical engineering )ˆ,( DB1NzC θ

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Unfalsified control

Given a set of controllers, implement one, use the I/O data, and check which part of the set is not feasible, then change the set and iterate

Safanov, Tsao 1997

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Explore Motion Control Properties

• experimentation is ‘cheap’Disturbance Design Cycle: 7 min FRF measurement,

model, loopshape, implementation Data based• plant decoupling, i.e. SISO MIMO disturbances• feedforward: low-order model-based Learning control• feedback: loopshaping nonlinear control• key limitation: bode gain/phase - sensitivity integral

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advanced motion control -...

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