Vibration Control of a Three-Leg Optical Table by Mechatronic Inerter Networks

Yu-Chuan Chen, Sheng-Yao Wu and Fu-Cheng Wang

Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan

(Tel : +886-2-3366-2680; E-mail: fcw@ntu.edu.tw)

Abstract: This paper develops a three-leg optical table, and applies a newly-developed mechatronic inerter network to

suppress vibrations of the table. Optical tables can insulate precision machines from two types of disturbances: ground

disturbances from the environment and load disturbances from the equipment. Using disturbance response decoupling

(DRD) techniques, we can effectively isolate the ground disturbances by soft passive suspensions and improve the load

responses by active control. This paper further applies mechatronic inerter networks to a three-leg optical table, and

optimizes the ground responses by connecting the networks to suitable electric circuits. We then apply DRD techniques

to improve the load responses without influencing the ground responses, and implement the optical table for

experimental verification. Based on the results, the proposed mechatronic inerter networks and DRD structures are

deemed effective in improving system responses.

Keywords: mechatronic inerter, disturbance response decoupling, optical table, vibration control, optimization.

1. INTRODUCTION

Optical tables are usually applied to insulate

vibrations for precision machinery. Traditional optical

tables, such as a pneumatic leg Newport I-2000 [1],

apply passive elements, e.g. dampers and springs, to

isolate machines from disturbances. Because passive

structures cannot fully satisfy the increasing precision

requirements, active optical tables are developed for

further performance improvement. For example, Kato et

al. [2] applied model-following control to pneumatic

actuators to improve table responses. Oh et al. [7]

combined passive rubbers and piezoelectric actuators to

suppress disturbance responses. Wang et al. [14, 16]

designed robust controllers to a two-layer isolation

platform with voice coil motors.

In general, an optical table needs to suppress two

main vibration sources: ground disturbances and load

disturbances. However, the suspension settings for

restraining these two disturbances are conflicting; i.e.,

the suspensions need to be soft for isolating ground

disturbances and stiff to suppressing load disturbances.

That is, the suspension design is a compromise between

these two settings. Therefore, Wang et al. [22,23]

proposed a double-layer optical table, as shown in Fig. 1,

and applied disturbance response decoupling (DRD)

techniques [11] to independently control the two

disturbances: the ground disturbance rz is controlled

passively by passive elements 1 ,

2 and 3 , while

the system responses to the load disturbance sF is

improved by the actuator A without influencing the

ground responses. Because the passive elements cannot

be easily adjusted according to the operating conditions,

this paper further extends these ideas by applying the

mechatronic inerter network and designing the connected

electrical circuits to suppress particular ground

disturbances.

The inerter have been applied to vehicle suspensions

[5–6, 13, 15, 18, 20–21] and building structures [17] to

suppress vibrations. Until now, there have been four

inerter realisations: the rack-pinion inerter [12], the

ball-screw inerter [13], the hydraulic inerter [19], and

the mechatronic inerter network [20]. The former three

are mechanical structures whose parameters cannot be

easily adjusted according to the operating conditions.

The mechatronic inerter network consists of two parts:

the ball-screw inerter and permanent magnet electric

machinery (PMEM). Therefore, we can design the

system impedance by connecting the PMEM to suitable

electrical circuits to optimize the system’s responses to

particular ground disturbances.

This paper is arranged as follows: Section 2

introduces the mechatronic inerter network. Section 3

derives the dynamics of a three-leg optical table and its

DRD structure. Section 4 identifies the transfer

functions of system components by experiments.

Section 5 improves the ground disturbances by

designing an optimal circuit that connects to the

mechatronic inerter networks. Section 6 applies robust

control to design active suspensions that improve the

load responses without affecting the ground responses.

The designed DRD control structure and active control

are then implemented for experimental verification.

Lastly, we draw conclusions in Section 7.

Fig. 1. The generalized DRD structure [21].

SICE Annual Conference 2014September 9-12, 2014, Hokkaido University, Sapporo, Japan

978-4-907764-45-6 PR0001/14 ¥400 © 2014 SICE 426

2. MECHATRONIC INERTER NETWORKS

The mechatronic inerter network, as shown in Fig.

2, is composed of a ball-screw inerter and a PMEM with

the following system impedance [20]:

ˆ ( ),

ˆ( ) ( )

m

m m

a a e

KF sb s c

v s R sL Z s

(1)

where v is the relative velocity between the two

terminals and F is the corresponding force. Equation (1)

can be considered as two parts: the first part m mb s c

is the mechanical inerter and damper, while the second

part / ( ( ))m a a eK R sL Z s represents PMEM and the

connected electrical circuit Ze. The equivalent

impedance of the mechatronic inerter network is shown

in Fig. 3, where Km , Ra and La are the motor gain

constant, the armature resistance, and the armature

inductance, respectively. Therefore, we can adjust the

electric circuits Ze to change the impedance of the

mechatronic inerter network.

Fig. 2. The mechatronic inerter networks [20].

Fig. 3. An ideal network model [20].

3. SYSTEM DYNAMICS AND DRD

FILTERS

A three-leg optical table is shown in Fig. 4. The

dynamics of the table can be expressed as [22, 23]:

1 2 3 ,s s s p p pm z F u u u

1 2 3 ,f p f p r pI z T l u l u l u

1 2 ,f p f pI z T t u t u

1 1 1 ,u u p rm z u F

2 2 2 ,u u p rm z u F

3 3 3 ,u u p rm z u F

where

2 3 ( ),i i i i ip i a uu D z z

1 ( ),i i i ir u rF z z

for 1 3i .sm , I and I represent the mass, and

inertias of the table, respectively. sF , T and T

are

the force, and torque disturbances to the table. jA are

active actuators, while jrz are the ground disturbances

for 1 3j .ji are corresponding passive suspension

elements for 1 3i and 1 3j . The table

displacements 1 3~D D are derived as follows:

11 ,s f f uD z l z t z z

22 ,s f f uD z l z t z z

33 ,s r uD z l z z

Fig. 4. The full-optical table model with 3-legs.

We further assume1 2 3u u u um m m m ,

1 2 3k k k k for 1, 2, 3k , and define the

following symmetric transformation matrix ,1fL

and ,2fL :

,1

1 1 01

0 0 2 ,2

1 1 0

fL

,2

1

2fL

t ,

which decouple the full-table model into one half-table

and one quarter-table models, such that

,1 1 2 3( ) ( ) ( ) ,f r f

T T

b b fx x x L x x x

(2)

,2( ) ( ) ,froll fx L x (3)

in which x can be substituted by the following

SICE Annual Conference 2014September 9-12, 2014, Sapporo, Japan

427

parameter: strut displacement D , displacement rz

and uz , or control signals u , while the subscripts

,f rb b represent the front and rear bounce modes, ,f

represent the front roll modes.

We can represent the control structure of the table as

the linear fractional transformation (LFT) structure of

Fig. 5 with:

,f

z wP

y u

(4)

where

1 2 3 ,

T

s r r rw F T T z z z

1 2 3 ,

T

s u u uz z z z z z z

2 1 1 2 3 T

sy f f z z z D D D

4 3 1 2 3 T

u f f u u u ,

diag( , ),bp rollK K K

in which the 1 4~ f f are as follows:

1 ,1 diag(1, 1, 1, ),ff L

2 ,2 diag(1, -1, 1, 1, 1, ),ff L

1

3 ,2 diag(1, 1, ),ff L

1

4 ,1 .ff L

Fig. 5. The LFT structure.

Applying the DRD algorithms, we can design a suitable

filter 2U for the plant

fP P , such that the actuator

control signal u can only be activated by the load

disturbances T

sF T T , but not the ground

disturbances1 2 3

T

r r rz z z . The DRD filter

2U is

designed as follows:

2 3 2 3

2

2 3 2 3

2 3

02( )( ) 2( )( )

0 ,2( )( ) 2( )( )

0 02( ) ( )

s r

r f r f

s f

r f r f

f r f

m l I

l l l l

m l IU

l l l l

I

t t t

4. IDENTIFICATION OF SUSPENSION

ELEMENTS

In this section, we derive the transfer functions of

the suspension elements 1 , 2 , 3 , and the

actuator dynamics . For the mechatronic inerter

networks 1 , we consider its nonlinear factors, as

shown in Fig. 6(a), in which k is a parallel spring,

is the backlash , , s sk c are the elastic effects, and f

is the Stribeck friction model with the following

dynamics: /

( ) ( (1 ) ) sgn( ) ,sv v

static mf v F e v c v

, 0 1.c staticF F

where cF is the Coulomb friction and

staticF is the

static friction, v is the relative velocity between the

terminals, sv is the Stribeck velocity, and

mc is the

damping constant. We built the model in

Matlab/SimulinkTM

, and tuned the model parameters

by comparing the experimental and simulation data, as

illustrated in Fig. 6(b) and Table 1. Note that is

actually zero; this is because the backlash of a

ball-screw can normally be eliminated by preloading in

the manufacturing processes.

(a) The nonlinear network.

(b) The responses.

Fig. 6. Identification for1 .

Table 1. Model parameters for Fig. 6.

Parameters value Units

sF 30.5468 N

0.9577 -

sv 9.1195 mm/s

mc 589.1063 Ns/m

mb 47.0146 kg

sk 142.4507 kN/m

sc 3.6794 kNs/m

k 29.3750 kN/m

For the spring-damper sets 2 and

3 , we

SICE Annual Conference 2014September 9-12, 2014, Sapporo, Japan

428

considered the structure shown in Fig. 7, where can

be replaced by 2 or

3 . We generated a swept

sinusoidal input signal rz and measured the output

signalsz , and derived the transfer function from

rz to

sz by applying the subspace system identification

method [8]. We repeated the experiments six times to

account for system variation and uncertainties, and

denoted the transfer functions as ( ), 1, ,6iG s i . On

the other hand, we assumed 2 cs k and derived

the theoretical transfer function r sz zT as:

2.

r sz z

s

cs kT

m s cs k

Thus, we can calculate the values of c and k that

minimize the gap between r sz zT and ( )iG s , as in the

following:

,

, arg min max ( , ), ,r sz z i

c k ic k T G i

in which ( , )r sz z iT G

represents the gap [23] between

iG andr sz zT . The result gives:

2 96 21200.s

3 57289.

Fig. 7. Identification for

2 and 3 .

Lastly, we gave the piezoelectric transducer (PZT) a

swept sinusoidal signal input voltage inv and measured

the corresponding output displacement outd , and

derived the PZT dynamics as:

0.002616( ) .

628.7

out

in

ds

v s

5. OPTIMIZATION OF CIRCUIT Ze

In this section we apply the mechatronic inerter

network to 1i of Fig. 4, and design a circuit Ze to

optimize the ground responses. We simplified the design

procedures by regarding the friction and elastic as

model uncertainties and consider 1i as follows:

2

1 1 ,( )i

m

m m

a a e

Kb s c s k

R sL Z s

in which 21.0975aR , 1.5maL ,

7011VNs / A / mmmK , and mb , mc , k are given in

Table 1.

Regarding table responses, we considered the

following performance index:

0 ( ) ( ) ( ) ,2 2

s

l tJ z t z t z t (5)

to derive the optimal electrical circuits Ze,0 that

minimizes 0J as follows:

,0 0arg min .eZ J

We limited Ze as the following second-order impedance: 2

2 1 0

,0 2

1 0

,e

b s b s bZ

s a s a

(6)

because realization of higher-order impedance is costly

and difficult, and only slightly improves the system

performance that can be achieved by a second-order one

[9]. We used a full optical table with the following

parameters: ( ) 0.002616 ( 628.7)s s , 140.16 kgsm ,

15.5 kgum 217.27kg mI

29.92kg m ,I

0.6 mr fl l l , 0.45 mr ft t t , 3 57289,

2 96 21200s , , and optimized the networks of [4]

to obtain the optimal Ze,0 as follows:

1 6,0

5 2

1 2

2.484 10 0.004813 1

2.616 10 0.0092 2.107 10e

s s

s sZ

.

Jiang and Smith [4] showed that regular bi-quadratic

transfer function of (6) can be realized as a five-element

network, as shown Fig. 8 with 5

1 5.2308 10R ,

2 0.5232R ,5

3 9.4940 10R , 0.0092FC ,

0.0027L . The comparison in Fig. 9 and Table 2

indicate that Ze,0 can effectively reduce the

time-domain responses, and frequency responses

( )sZ at the frequency between 0.2Hz and 10Hz.

Table 2 illustrates an improvement of 3.62% for the

time responses, and an improvement of 32.96% for the

frequency responses at the concerned frequency range.

Fig. 8. The electric circuit structure[4].

(a) The time-domain load responses.

0 1 2 3 4 5 6 7 8 9 10-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Am

plit

ude (

m/s

2)

time (sec)

without Ze

with Ze

SICE Annual Conference 2014September 9-12, 2014, Sapporo, Japan

429

(b) The frequency-domain load responses.

Fig. 9. Experimental responses.

Table 2. Statistic data from Fig. 9

Without

,0eZ With

,0eZ Improvement

(0, 10)tJ 1.7541 1.6906 3.6159%

(0.2, 10)J

0.0679 0.0455 32.9622%

6. ACTIVE CONTROLLER DESIGN

Consider the LFT structure of Fig. 5; all stabilizing

controllers can be parameterized as [11]:

1

1 1 2 22 1( ) .K I QU P Q

Therefore, we can design a proper controller K1 for the

filtered plant 2 22U P . We applied the standard H

loop shaping procedures [23–25] with the following

weighting functions:

1 2

200( 5000), 1,

100

sW W

s

(7)

and designed the controllers as follows:

1 diag( , , ),bp warp rollK K K K

where 2 6 10

2 4

779.3 7.096 10 1.6 10

216.3 1.163 1,

0roll

s sK

s s

0,warpK

0,

0bp

c

c

K

KK

2 6 10

2 4

779.3 7.096 10 1.6 10

216.3 1 10.

.163c

s s

s sK

Note that we regarded the table as a rigid body, and

ignored table’s warp mode. The experimental results of

the active control are shown in Fig. 11 and Table 3. First,

we note that the ground disturbance did not activate the

control signals iu ; i.e., the DRD structure is effective.

Second, the time-domain and frequency-domain load

responses are effectively suppressed by the designed

robust controller. Table 3 illustrates an improvement of

3.97% for the time responses, and an improvement of

16.89% for the frequency responses at the concerned

frequency range. In the practical application, we can

adjust the concerned frequency range by tuning the

weighting functions of (7).

Fig. 10. The three lags system

(a) The DRD effects

(b) The time-domain load responses.

(c) The frequency-domain load responses.

Fig. 11. Experimental responses of the active control.

10-1

100

101

102

103

10-6

10-5

10-4

10-3

10-2

Frequency (Hz)

Am

plit

ude (

m/s

2)

without Ze

with Ze

0 0.5 1 1.5 2 2.5 3 3.5 4-5

0

5

Fs

0 0.5 1 1.5 2 2.5 3 3.5 4-1

01

zr

0 0.5 1 1.5 2 2.5 3 3.5 4-2

0

2x 10

-5

ub

0 0.5 1 1.5 2 2.5 3 3.5 4-2

02

x 10-5

uth

0 0.5 1 1.5 2 2.5 3 3.5 4-5

0

5x 10

-5

uphi

time (sec)

0 1 2 3 4 5 6 7 8-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

time (sec)

ddot(

zs)

(m/s

2)

open-loop

active control

10-1

100

101

102

103

10-6

10-5

10-4

10-3

frequency (Hz)

ddot(

zs)

(m/s

2)

open-loop

active control

SICE Annual Conference 2014September 9-12, 2014, Sapporo, Japan

430

Table 3. Statistic data from Fig. 11.

Open

loop

Active

control

Improve

ment 1/2

26

0( )sz t dt

0.8799 0.8450 3.968%

1/2216

2

0.33( )sz d

0.0112 0.0093 16.89%

7. CONCLUSION

This paper has proposed a three-leg optical table, and

applied mechatronic inerter networks and DRD

techniques to control vibrations of the table. First, we

introduced the mechatronic inerter network, whose

impedance can be adjusted by connecting suitable

electrical circuits. For example, we designed a

second-order circuit to optimize the ground responses.

Second, we applied the DRD structure and designed

robust loop-shaping controllers to improve the system’s

load responses without influencing the ground responses.

Lastly, the designed mechatronic inerter network and

active controllers were implemented for experimental

verification. Based on the results, the proposed design

was deemed effective.

REFERENCES

[1] Newport 2011 Vibration Isolators, “http://www.newport.com/”.

[2] Kato T, Kawashima K, Funaki T, Tadano K and Kagawa T, “A new, high precision, quick response pressure regulator for active control of pneumatic vibraiotn isolation tables”. Precision Engineering Vol. 34, No. 1,43–48, 2010

[3] Chen MZQ, Hu Y, Huang L, Chen G , “Influence of inerter on natural frequencies of vibration systems”. Journal of Sound and Vibration, Vol. 333, No. 7, pp. 1874–1887, 2014.

[4] Jiang JZ and Smith MC , “Regular positive-real function and five-element network synthesis for electrical and mechanical networks”. IEEE Transactions on Automatic Control , Vol. 56, No. 6, pp. 1275–1290, 2011.

[5] Jiang JZ, Matamoros-Sanchez AZ, Goodall RM and Smith MC, “Passive Suspensions Incorporating Inerters for Railway Vehicles” Vehicle System Dynamics, Vol. 50, No. 1, pp. 263–276, 2012.

[6] Jiang JZ, Matamoros-Sanchez AZ, Zolotas A, Goodall R and Smith MC , “Passive suspensions for ride quality improvement of two-axle railway vehicles”, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, DOI: 10.1177/0954409713511592, 2013.

[7] Oh JS, Han YM, Choi SB, Nguyen VQ and Moon SJ, “Design of a one-chip board microcontrol unit for active vibration control of a naval ship mounting system”, Smart Mater Struct, Vol. 21, No. 8, pp. 1-15, 2012 .

[8] Overschee PV and Moor BD , “N4SID-subspace algorithms for the identification of combined deterministic-stochastic system”, Automatica Volume,

Vol. 30, No. 1, pp. 75–93, 1994 [9] Papageorgiou C and Smith MC , “Positive real

synthesis using matrix inequalities for mechanical networks: application to vehicle suspension”. IEEE Transactions on Control Systems Technology , Vol. 14, No. 3, pp. 423–435, 2006.

[10] Smith MC , “Synthesis of mechanical networks: The inerter”. IEEE Transactions on Automatic Control ,Vol. 47, No. 10, pp. 1648–1662, 2002.

[11] Smith MC and Wang FC, “Controller parametrisation for disturbance response decoupling: application to vehicle active suspension control”, IEEE Transactions on Control Systems Technology, Vol. 10, No. 3, pp. 393–407, 2002.

[12] Wang FC and Smith MC, “Performance benefits in passive vehicle suspensions employing inerters”, Vehicle System Dynamics, Vol. 42, No. 4, pp. 235–257, 2004.

[13] Wang FC and Su WJ , “The impact of inerter nonlinearities on vehicle suspension control”. Vehicle System Dynamics, Vol. 47, No. 7, pp. 575–595, 2008.

[14] Wang FC, Tsao YC and Yen JY , “The application of disturbance response decoupling to the vibration control of an electron beam lithography system”, Japanese Journal of Applied Physics, Vol. 48, No. 6, pp. 1–5, 2009.

[15] Wang FC, Liao MK, Liao BH, Su WJ and Chan HA, “The performance improvements of train suspension systems with mechanical networks employing inerters”, Vehicle System Dynamics Vol. 47, No. 7, pp. 805–830, 2009.

[16] Wang FC, Hong MF and Yen JY , “Robust Control Design for Vibration Isolation of an Electron Beam Projection Lithography System”, Japanese Journal of Applied Physics, Vol. 49, No. 6, pp. 1–7, 2010.

[17] Wang FC, Hong MF and Chen CW, “Building suspensions with inerters”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 224, No. 8, pp. 1605–1616, 2010.

[18] Wang FC and Liao MK, “The lateral stability of train suspension systems employing inerters”, Vehicle System Dynamics, Vol. 48, No. 5, pp. 619–643, 2010.

[19] Wang FC, Hong MF and Lin TC , “Designing and testing a hydraulic inerter”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 225, No. 1, pp. 66–72, 2011.

[20] Wang FC and Chan HA, “Vehicle suspensions with a mechatronic network strut” Vehicle System Dynamics, Vol. 49, No. 5, pp. 811–830, 2011.

[21] Wang FC, Hsieh MR and Chen HJ, “Stability and performance analysis of a full-train system with inerters”, Vehicle System Dynamics, Vol. 50, No. 4, pp. 545–571, 2012.

[22] Wang FC, Yu CH, Tsai Jeff TH and Yang SH ,”Decoupled robust vibration control of an optical table”, Journal of Vibration and Control , Vol. 20, No. 1, pp. 38–50, 2014.

[23] Zhou K, Doyle JC and Glover K, Robust and Optimal Control. Prentice Hall, New Jersey, 1996.

[24] Wang FC, Chen LS, Tsai YC, Hsieh CH and Yen JY, “Robust Loop Shaping Control for a Nano-Positioning Stage”, Journal of Vibration and Control, Vol. 20, No.6, pp. 885–900, 2014.

[25] Wang FC, Yu CH, Tsai JTH, and Yang SH, “Decoupled Robust Vibration Control of an Optical Table”, Journal of Vibration and Control, Vol. 20, no.1, pp. 38–50, 2014.

SICE Annual Conference 2014September 9-12, 2014, Sapporo, Japan

431