Integrated design methodology of ball-screw driven...

Mechatronics 16 (2006) 491–502

Integrated design methodology of ball-screw driven servomechanismswith discrete controllers. Part I: Modelling and performance analysis

Min-Seok Kim a, Sung-Chong Chung b,*

a Department of Mechanical Design and Production Engineering, Hanyang University, 17 HaengdangDong, SungdongGu, Seoul 133-791, South Koreab School of Mechanical Engineering, Hanyang University, 17 HaengdangDong, SungdongGu, Seoul 133-791, South Korea

Received 29 March 2005; accepted 22 January 2006

Abstract

In order to ensure high-speed and high-precision specifications in ball-screw driven servomechanisms, an integrated design method-ology in which driving mechanisms and motion controllers are designed simultaneously is required. As a prior study of the integrateddesign procedure, it is necessary to obtain not only mathematical models of servomechanisms but also proper formulation of the inte-grated design problem. In this paper, the feedback and feedforward controllers described in discrete-time domain are incorporated in themotion controller. Design requirements of the servomechanism such as stability, geometric errors, resonance of the driving mechanism,deformation of the structure, actuator saturation and so on are described in detail. Numerical simulations of the servomechanism per-formance according to design and operating parameters are performed based on the developed mathematical model. An accurate iden-tification process of the driving mechanism is introduced to verify the mathematical subsystem model. Circular motion experiments areconducted to investigate interactions between parameters of the driving mechanism and controller gains, as well as analyze the influenceof the interactions on the servomechanism performance. Results of the analysis and experiments let us understand accurate dynamiccharacteristics of the ball-screw driven servomechanism and render an integrated design possible.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Abbe error; Ball-screw driven servomechanism; Discrete controller; Optimization; Radius error; Stability; System identification; z-transform

1. Introduction

A servomechanism is an automatic device for control-ling the position, velocity and force (or acceleration) of adriving mechanism by means of feedback information.There are various types of servomechanisms, and typicalservomechanisms give linear motion by using ball-screwsor linear motors. In this paper, a ball-screw driven servo-mechanism is considered. Ball-screw driven servomecha-nisms are widely used in the industries such assemiconductor, precision machinery, manufacturing andso on [1–3].

A ball-screw driven servomechanism is composed ofdriving mechanisms such as ball-screws and ball-nuts, feed-back sensors, motion controllers and so on. It is divided

0957-4158/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechatronics.2006.01.008

* Corresponding author. Tel.: +82 2 2220 0444; fax: +82 2 2298 4634.E-mail address: [email protected] (S.-C. Chung).

into two parts. One is a mechanical subsystem composedof the mechanical structure and driving mechanism. Theother is a control subsystem consisting of feedback sensorsand motion controllers. Performance of servomechanismslargely depends on interaction between the subsystems[4,5]. For instance, it is supposed that a designer wants toreduce table weight so as to obtain high-speed characteris-tics. Then, although the natural frequency of the mechani-cal structure is increased, the static stiffness of theservomechanism and the range of allowable controller gainsare decreased. Therefore, dynamic characteristics of themotion controller deteriorate and the resonance frequencyof the controller comes close to the one of the mechanicalsubsystem. Consequently, stability of the servomechanismdwindles. Although the diameter of a ball-screw is opti-mally determined in the mechanical aspect, it may not bean optimum value from the viewpoint of control perfor-mance. Therefore, in order to design high-performance

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Nomenclature

Am gain margin of a feed servomechanism (dB)amax maximum acceleration of a table (m/s2)Bm viscous damping coefficient of a motor shaft

(N m s/rad)Bt viscous damping coefficient of a table (N s/m)Btb width of a table (m)Da Abbe offset (m)Dbs ball-screw diameter (m)Dc(z) characteristic polynomial of a servomechanismDcp coupling diameter (m)Ebs elastic modulus of a ball-screw (Pa)Elg elastic modulus of a linear guide (Pa)Er contour error ratioEtb elastic modulus of a table (Pa)F(z) discrete transfer function of a feedforward con-

trollerFc external force vector (N)Fd driving force (N)Fw load due to table and workpiece weight (N)Gbs shear modulus of a ball-screw (Pa)Gc(z) closed loop discrete transfer function of a ser-

vomechanismGcp shear modulus of a coupling (Pa)Glg shear modulus of a linear guide (Pa)Gm(s) transfer function of a mechanical subsystem in

continuous domainGm(z) Gm(s)/s in discrete domainbGmðzÞ input transfer function of the ARMAX modelGo(z) open loop discrete transfer function of a servo-

mechanismGpc(z) discrete transfer function of a position controllerGsat(z) discrete saturation transfer function of a servo-

mechanismGvc(z) discrete transfer function of a velocity control-

lerbH mðzÞ noise transfer function of the ARMAX modelHtb height of a table (m)Ibs moment of inertia of a ball-screw (m4)Ilg moment of inertia of linear guides (m4)Itb moment of inertia of a table (m4)J cost function in the identification processJeq equivalent inertia of a servomechanism (kg m2)Jm inertia of rotating elements (kg m2)Kemf back-emf constant of a motor (Vrpm/rpm)Keq equivalent stiffness of a servomechanism (N/m)Kf velocity feedforward controller gain (V/V)Ki integral gain of a position controller (V/V)Kl equivalent stiffness in the axial direction (N/m)Knt nut stiffness (N/m)Kp proportional gain of a position controller (V/V)Ksb stiffness of a support bearing (N/m)Kt torque constant of a motor (N m/Arms)Kv proportional gain of a velocity controller (V/V)

Kh equivalent stiffness in the radial direction (N m/rad)

Lbs ball-screw length (m)Lcp coupling length (m)Llg length of a linear guide (m)Lm inductance of a motor amplifier (H)Lsp distance between linear guides (m)Lstr stroke of a driving mechanism(m)Ltb table length (m)l ball-screw lead (m)Mt table mass (kg)n order of a closed loop systemPb axial buckling load of a ball-screw (N)R l

2p

� �, conversion ratio of linear-to-rotational

motionRi radius of a circular motion command (m)Rm resistance of a motor amplifier (X)Ro radius of a circular motion output (m)Ti sampling period of a reference input (s)Ts sampling period (s)Vc critical velocity of a ball-screw (m/s)Vs reference voltage referring to torque reference s

(V)vt table velocity (m/s)v̂t estimated velocity of a table (m/s)xc position command (m)xs transverse distance of a nut (m)xt transverse distance of a table (m)aa angular error (rad)dc structural deformation error (m)e Abbe error (m)g efficiency of a driving mechanismhm rotational angle of a motor shaft (rad)hs rotational angle of a ball-screw (rad)n white noiseqbs ball-screw density (kg/m2)qtb table density (kg/m2)s torque reference (N m)smax

c maximum magnitude of a control input (N m)sF load torque on a ball-screw (N m)si time constant of an integral controller (s)sm driving torque of a motor (N m)smax

m maximum torque applied to the motor (N m)u vector of a regressor/m phase margin of a servomechanism (deg.)xB bandwidth of a servomechanism (rad/s)xbs natural frequency of a ball-screw shaft (rad/s)xg gain crossover frequency (rad/s)xi rotational speed of the reference input (rad/s)xp phase crossover frequency (rad/s)xs sampling frequency (rad/s)xwB bandwidth of a servomechanism in w-domain

(rad/s)

492 M.-S. Kim, S.-C. Chung / Mechatronics 16 (2006) 491–502

mJsθmθ

sx

Kθ

lK

mτ

tx

tM dF

R

Fig. 1. Free-body diagram of the driving mechanism.

M.-S. Kim, S.-C. Chung / Mechatronics 16 (2006) 491–502 493

servomechanisms, an integrated design methodology whichmeans that both the mechanical and control subsystems aresimultaneously incorporated in the design processes isrequired for an optimum design [6–8]. The integrated designof a servomechanism must be preceded through the investi-gation of interactions between the mechanical and controlsubsystems.

To investigate the interactions, accurate and reliablemathematical models of servomechanisms are requiredfirst. A number of mathematical models of servomecha-nisms have been proposed according to design objectives[9–12]. When a designer focuses on a motion controllerdesign, a mechanical subsystem has been assumed as atime-delay component with infinite stiffness. In case ofdesigning driving mechanisms, on the other hand, theyhave been described as high-order systems and controllershave been represented as a series of proportional gains.However, accurate models of both the mechanical and con-trol subsystems are indispensable to perform the integrateddesign satisfactorily.

In addition, a trial and error method [13] and heuristicmethods [14,15] have been used for optimal tuning of ser-vomechanisms. In order to conduct an optimal tuning ofservomechanisms, interactions among mechanical and con-trol parameters, and influence of the interactions on theservomechanism performance should be analyzed mathe-matically as well.

In this paper, unlike the previous study [8] in whichthe motion controller was composed of conventionalcascade-structure controllers in continuous-time domain,the feedback and feedforward controllers satisfying bothrobustness and tracking performance are incorporated inthe control subsystem in discrete-time domain. Perfor-mance-index functions, constraints and design variablesincluding parameters of both the mechanical and controlsubsystems are defined based on the mathematical models.An accurate identification process of the mechanical sub-system is conducted to verify the obtained mathematicalmodel by using a weighted least square method and a pairof biased-square wave signals. Parametric studies andcircular motion experiments on a ball-screw driven servo-

Fig. 2. Assembly of the

mechanism are conducted to investigate interactionsbetween parameters of the mechanical and control subsys-tems, and to analyze the influence of interactions on theservomechanism performance. Results of this paper let usunderstand accurate dynamic behavior of a ball-screwdriven servomechanism and render an integrated designpossible. The integrated design procedure of the ball-screwdriven servomechanism will be described as a Part II of thispaper.

2. Modelling of servomechanisms

Mechanical characteristics of a servomechanism such asoverall flexibility, stiffness, natural frequency and inertiaaffect significant influence on the design optimization. Itis important to consider those characteristics togetherthrough the integrated design procedure. Mathematicalmodels of mechanical subsystems are generally constructedby developing equations of motion between a motor andother mechanical components of a servomechanism. Figs.1 and 2 show a free-body diagram and an assembly ofthe driving mechanism, respectively.

A driving torque of the motor sm is transformed into arotational motion of the ball-screw hs and an elastic defor-mation dh due to the equivalent torsional stiffness Kh exist-ing between a motor and a ball-screw

sm ¼ J m

d

dtxm þ Bmxm þ sF ð1:aÞ

hs � hm ¼ dh ¼sm

Khð1:bÞ

where Jm is the inertia of rotating elements composed ofthe motor rotor, ball-screw and coupling, Bm the damping

driving mechanism.

eqKR1

t tM s B+

1/s

tK mτ sθ sx

tx

tvdF

1s

1m mJ s B+

Vτ

/R η

emfK

1m mL s R+

sθ·

Fτ

Fig. 3. Block diagram of the mechanical subsystem.

sT

tkx( )pcG z ( )vcG z ( )mG zckx ke

1s

zT z–

sT

( )F z

tkx·

kτ

Fig. 4. Block diagram of the motion controller.


coefficient of the motor shaft, hm and xm the rotationalangle and velocity of the motor, respectively. A load torquesF on the ball-screw to overcome an axial drive force Fd isgiven by

sF ¼l

2pgF d ¼

Rg

F d ð2Þ

where l is the ball-screw lead, R the conversion ratio of lin-ear-to-rotational motion and g the efficiency of the drivingmechanism. The equivalent torsional stiffness is the sum oftorsional stiffnesses of the ball-screw Kbsh and couplingKcph

Kh ¼1

Kbshþ 1

Kcph

� ��1

¼ 32Lbs

pGbsD4bs

þ 32Lcp

pGcpD4cp

!�1

ð3Þ

where Dbs and Lbs are the diameter and length of the ball-screw, respectively. Lcp and Dcp are the length and diameterof the coupling, respectively. Gbs and Gcp are the elasticmoduli of the ball-screw and coupling, respectively. Therotational motion of the ball-screw is converted to a linearmotion of the ball-nut xs, and an elastic deformation dx dueto the equivalent longitudinal stiffness Kl existing betweenthe ball-nut and table

xs ¼l

2phs ¼ R � hs ð4:aÞ

F d ¼ M t

d

dtvt þ Btvt ð4:bÞ

xs � xt ¼ dx ¼F d

K l

ð4:cÞ

where xt is a transverse distance of the table, Mt the tablemass, vt a table velocity and Bt the damping coefficientexisting in the ball-screw and ball-nut as well as the linearguide. Kl is the sum of the ball-screw, ball-nut and supportbearing stiffnesses, Kbs, Knt and Ksb, respectively

K l ¼1

Kbs

þ 1

Knt

þ 1

2Ksb

� ��1

¼ 4Lbs

pEbsD2bs

þ 1

Knt

þ 1

2Ksb

� ��1

ð5Þ

where Ebs is the elastic shear modulus of the ball-screw. Weassume that stiffnesses of the support bearings Ksb and theball-nut Knt are proportional to the ball-screw diameterDbs. From Eqs. (1)–(5), the equivalent stiffness Keq of thedriving mechanism is given by

Keq ¼R2

g� 1

Khþ 1

K l

� ��1

ð6Þ

Fig. 3 shows the block diagram of the mechanical sub-system. In Fig. 4, Lm and Rm are the inductance and resis-tance of the motor amplifier, respectively. Kt and Kemf arethe motor torque and back-emf constants, respectively.(Lms + Rm)�1 represents the amplifier dynamics. As Rm/Lm is much smaller than Bm/Jm, it is negligible in the

mechanical subsystem model. Therefore, the transfer func-tion of the mechanical subsystem Gm(s) between the refer-ence voltage Vs referring to a torque reference s and thetable velocity vt is given by

GmðsÞ ¼b1

s3 þ a1s2 þ a2sþ a3

a1 ¼J mBt þ ðK tKemf þ BmÞM t

J mM t

a2 ¼ðK tKemf þ BmÞBt

J mM t

þ ðJ mgþ R2M tÞKeq

J mM tg

a3 ¼ðK tKemf þ BmÞKeq

J mM t

þ R2BtKeq

J mM tg; b1 ¼ RK tKeqg

ð7Þ

Feedback and feedforward elements are used as motioncontrollers in this paper. The feedback controllers are com-posed of a position P controller Gpc and velocity PI con-troller Gvc to reject both steady-state error anddisturbance force. The velocity feedforward controller F

that adds velocity command to the velocity feedback loopis used to improve tracking performance.

In general, controllers are designed in continuous-timedomain, s-domain, and implemented in discrete-timedomain, z-domain, through appropriate transformationmethods. However, discrepancies between the two control-lers exist whatever transformation methods are used.Therefore, although performance of a controller designedin continuous-time domain is proper, there is practical lim-itation to apply the continuous controller to the digitalcontroller. In this paper, a control subsystem is designedand implemented in discrete-time domain to address thisproblem. When a controller is designed directly in z-domain, the sampling frequency of a control subsystemcan be reduced with less degradation in performance of aservomechanism. Furthermore, quantization errors of thecoefficients are able to be reduced, because the controllerto be implemented has poles and zeros that are no longercrowded near z = 1. Fig. 4 shows the block diagram ofthe motion controller. Transfer functions of the feedbackand feedforward controllers are given by


GpcðzÞ ¼ Kp

GvcðzÞ ¼ Kv 1þ T s

svi

� K izz� e�T s=svi

� �ð8:aÞ

F ðzÞ ¼ K fz� 1

T szð8:bÞ

where Kf is the velocity feedforward controller gain, Kp theproportional gain of the position controller, Kv the propor-tional gain of the velocity controller, Ki the gain and svi thetime constant of the integral controller, and Ts the sam-pling period of the discrete controller. The discrete transferfunction Gm(z) in Fig. 4 is obtained by applying a zero-order hold equivalent method to the integration of themechanical subsystem model Gm(s), Eq. (7), as follows:

GmðzÞ ¼ ð1� z�1ÞZ GmðsÞs2

� �ð9Þ

From Eqs. (7)–(9), as well as Figs. 3 and 4, the open looptransfer function Go(z) and the closed loop transfer func-tion Gc(z) of the ball-screw driven servomechanism are gi-ven by Eqs. (10) and (11), respectively.

GoðzÞ ¼GvcðzÞGmðzÞ½GpcðzÞ þ F ðzÞ�T sz

ð1� F ðzÞ � GvcðzÞ � GmðzÞÞT szþ GvcðzÞGmðzÞðz� 1Þð10Þ

GcðzÞ ¼N cðzÞDcðzÞ

¼ GvcðzÞGmðzÞ½GpcðzÞ þ F ðzÞ�T szð1þ GpcðzÞ � GvcðzÞ � GmðzÞÞT szþ GvcðzÞGmðzÞðz� 1Þ

ð11Þ

3. Design requirements of the servomechanism

3.1. Stability and system response analysis

If roots of the characteristic equation described in Eq.(12) locate inside a unit circle, the servomechanism willbe stable. This induces a following nominal stabilitycriterion:

jzij < 1; zi ¼ fz : DcðzÞ ¼ 0g for i ¼ 1; . . . ; n ð12Þwhere n is the order of the closed loop system in Eq. (11).In order to secure stability of the designed servomechanismeven if there are uncertainties in the modelling process, rel-ative stability criteria defined by the gain and phase mar-gins must be considered in the design process as follows:

Am ¼ 20log10

1

jGoðejxpT sÞjxp ¼ minfx : \GoðejxT sÞ ¼ �pg ð13:aÞ/m ¼ \fGoðejxgT sÞg þ p;

xg ¼ minfx : jGoðejxT sÞj ¼ 1g ð13:bÞ

It is common to use bandwidth as a qualitative criterionof the system response. An explicit representation of thebandwidth is required in the design procedure. However,

it is difficult to represent the bandwidth as an explicit formin z-domain because z is related to jx through the relationejxT s in frequency domain. The most convenient way offinding the bandwidth of a discrete system is to describeit graphically [16]. In order to obtain an explicit expressionof the bandwidth in z-domain, a bilinear transformation inw-plane is introduced as follows:

w ¼ 2

T s

z� 1

zþ 1

� �¼ j

xs

ptan

pxxs

ð14Þ

where xs is the sampling frequency. The w-transformationtransforms the unit circle back to the left-hand side of com-plex plane. The imaginary axis of w-plane closely resemblesjx axis of s-plane. If w-transformation is used, the band-width obtained in terms of Gc(w) is described as xwB

xwB ¼ xw : j½GcðwÞ�w¼jxwj ¼ 1ffiffiffi

2p

ð15Þ

The bandwidth of the servomechanism xB in w-domain isgiven by

xB ¼xs

ptan�1 pxwB

xs

ð16Þ

3.2. Contour and geometric errors

In order to minimize the contour error originated from acircular motion, characteristics of both the mechanical andcontrol subsystems must be considered simultaneously.The contour error is not only proportional to the frequencycommand of signals but also closely related to the dynamiccharacteristics of a servomechanism. For a circular motionwith a command radius Ri, amplitude of the sinusoidalinput signal is Ri. Amplitude of the corresponding har-monic response xt(z) is given by the radius of an actual cir-cular motion as

Ro ¼ jGcðejxiT iÞjRi ð17ÞThe contour error ratio of a servomechanism related to acircular motion is given by

Er ¼Ri � Ro

Ri

¼ 1� Ro

Ri

¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRefGcðejxiT iÞg2 þ ImfGcðejxiT iÞg2

qð18Þ

In Eq. (18), Er < 0 means that the radius of an actual circu-lar motion is larger than the command radius. Er > 0means that the actual radius is smaller than the commandradius.

In general, driving mechanisms have moving pairs thatmove relative each other. There is an angular error aa whenthere is clearance between the ball-screw and ball-nut asshown in Fig. 5. The distance Da, Abbe offset, proportion-ally amplifies the angular error [17]. Based on the configu-ration of the driving mechanism shown in Fig. 2, Abbeoffset is given by

Da ¼ H tb þDbs

2ð19Þ

Fig. 5. Abbe offset and Abbe error of a servomechanism.

Fig. 6. Plate model on the elastic foundation.


where Htb is the height of the table. This results in an axialpositioning error called Abbe error e as shown in Fig. 5. Inorder to design a high-precision servomechanism, the Abbeerror should be minimized in the design process.

3.3. Resonance and deformation analysis

In order to derive critical velocity of a ball-screw shaft, abeam model for the ball-screw shaft of which both ends areclamped is considered. Since critical velocity of a servo-mechanism leads to a resonance, the critical velocity ofthe ball-screw shaft must be included in the integrateddesign procedure. The natural frequency of the beammodel is given by [8]

xbs ¼4:732

L2bs

ffiffiffiffiffiffiffiffiffiffiffiffiEbsIbs

qbsAbs

sð20Þ

where Ibs, qbs and Abs is the moment of inertia, density andcross-sectional area of the ball-screw shaft, respectively.The critical speed Vc is given by

V c ¼l

2pxbs ¼ 0:89

Dbsl

L2bs

ffiffiffiffiffiffiffiEbs

qbs

sð21Þ

Buckling of the ball-screw shaft due to external distur-bance and cutting force is also incorporated in the designof a servomechanism. The buckling load Pb of the beammodel of which both ends are clamped with bearings isgiven by [8]

P b ¼ 4p2EbsIbs

L2bs

¼ p2EbsD4bs

16L2bs

ð22Þ

A plate model on the elastic foundation [18] is consideredto derive a structural deformation error as shown in Fig. 6.

In Fig. 6, F zc is the external force acting on the mechan-

ical component in z-direction, Fw the external force due tothe table and workpiece mass, Lstr the stroke of the servo-mechanism, Etb the elastic modulus of the table, Btb thetable width, Ltb the table length, Itb the moment of inertiaof the table, and Elg and Ilg the elastic modulus andmoment of inertia of the linear guide, respectively. Weassume that Ilg is constant during the design procedures,and dimensions of the linear guides are dependent on thetable dimension and the stroke of the driving mechanismas follows:

Lsp ¼2

3Btb; Llg ¼ 2Lstr ð23Þ

where Lsp is the distance between linear guides. From Ref.[18], the spring constant ke of the linear guides on elasticfoundation is given by

ke ¼48ElgI lg

LspL3lg

¼ 9ElgI lg

BtbL3str

ð24Þ

Using the mechanical dimensions and the external loadingcondition shown in Fig. 6, the structural deformation errorin z-direction is derived from a plate deflection model onthe elastic foundation [18] as follows:

dc ¼b

2ke

2þ cos bBtb þ cosh bBtb

sin bBtb þ sinh bBtb

� �ðF z

c þ F wÞ ð25:aÞ

b ¼ ke

4EtbI tb

� �14

¼ 9ElgI lg

4BtbEtbI tbL3str

� �14

ð25:bÞ

From the definition of Abbe offset (refer to Eq. (19)), theheight of a table Htb can be represented in terms of Abbeoffset Da and the ball-screw diameter Dbs. Therefore, themoment of inertia of the table Itb is calculated as follows:

I tb ¼LtbH 3

tb

12¼ Ltbð2Da � DbsÞ3

96ð26Þ

Consequently, the structural deformation error dc of a driv-ing mechanism equipped with a table over two linearguides in the vertical direction including Abbe offset andthe ball-screw diameter is given by Eq. (27)

dc ¼BtbL3

str

3ElgI lg

ElgI lg

6EtbLtbBtbL3strð2Da � DbsÞ3

!14

� 2þ cos bBtb þ cosh bBtb

sin bBtb þ sinh bBtb

� �ðF z

c þ F wÞ ð27:aÞ

b ¼ 9ElgI lg

4BtbEtbI tbL3str

� �14

ð27:bÞ

020

4060

80

05

1015

20-3

-2

-1

0

1

D bs (mm)

l (mm)

Er

Fig. 7. Effects of mechanical design variables on the contour error ratio.

60-0.2

0

0.2

0.4E

r


3.4. Actuator saturation

In general, it is known that the best tracking and distur-bance rejection performance of a servomechanism isachieved by the selection of allowable maximum controllergains. However, actuators should fall into saturation whenthe large gain is used, which renders the system nonlinearand the linear analysis invalid. Therefore, saturated condi-tions of the motion controller and the maximum allowabletorque fed to the actuator must be considered in the designprocedure. The maximum value of the control signal smax

c

fed to the motor amplifier is calculated from Eqs. (10)and (11) as well as Fig. 4 as follows:

smaxc ¼ maxfjsðzÞjz¼ejxT s g¼ max jGsatðejxT sÞ � xcðejxT sÞj

� �ð28:aÞ

GsatðzÞ ¼sðzÞxcðzÞ

¼ GvcðzÞ � ½GpcðzÞ þ F ðzÞ�T szð1þ GpcðzÞ � GvcðzÞ � GmðzÞÞT szþ GvcðzÞ � GmðzÞðz� 1Þ

ð28:bÞThe maximum torque applied to the motor smax

m due to theinertia of mechanical components is given by

smaxm ¼ max J eq

d2

dt2h ¼ J eq

Rd2

dt2xt

¼ 1

2R½2J m þ qtbBtbLtbð2Da � DbsÞR2�amax ð29Þ

02

46

810

020

40

Kv (V/ V)

Kp (V/ V)

Fig. 8. Effects of control design variables on the contour error ratio.

0 20

4060

80

020

4060-4

-2

0

2

D bs (mm)

Kp (V/ V)

Er

Fig. 9. Effects of mechanical–control interaction on the contour errorratio.

4. Performance analysis

Parameters of both the mechanical and control subsys-tems are selected as design variables for the integrated designprocedure. Since interactions between the subsystems areconsidered simultaneously through those design variables,quality of the design process applicable to high-performanceservomechanisms is improved and a systematic design pro-cedure is realized. As a prior study of the integrated design,which will be presented as Part II of this paper, analysis ofthe servomechanism performance and dynamic characteris-tics according to design variables is conducted.

4.1. Analysis of contour errors

Fig. 7 shows the contour error ratio (Er) according todesign variables of the mechanical subsystem such as theball-screw diameter and lead. Er is slightly increased whenthe ball-screw diameter increases. There are design variablepairs that cause severe decrease of Er.

Fig. 8 shows Er according to design variables of the con-trol subsystem. As widely known, radius errors decreaseaccording as controller gains increase. Especially, actualradius becomes smaller than the command radius whenthe proportional gain of the position controller becomessmaller. However, small proportional gains of the velocitycontroller generate bigger actual radii than command radii.

Fig. 9 shows Er according to design variables of themechanical and control subsystems. Similarly to Fig. 7,there are design variable pairs between the mechanical

010

2030

0 5

1015

0

30

60

90

120

l (mm)D

bs (mm)

Am

(dB

)

0 10

2030

0 5

1015

0

1

2

3

l (mm)D

bs (mm)

Am

(dB

)

b

a

Fig. 11. Effects of mechanical–control variation on stability: (a) lowintegral gain of the velocity controller; (b) high integral gain of the velocitycontroller.

80


and control subsystems, which cause severe increase ofradius errors. It is difficult to consider these interactionsin design processes through the conventional componentdesign methodology. It is the integrated design methodol-ogy that reflects these interactions in the design processof a servomechanism.

4.2. Analysis of system stability

Fig. 10 shows relative stability (Am) according to designvariables of the control subsystem. Contrary to the resultsof radius error analysis in Section 4.1, stability is reducedproportional to increase of controller gains. Therefore, ifa designer increases controller gains so as to reduce radiuserrors, the stability of the servomechanism is notguaranteed.

Fig. 11 shows Am according to design variables of themechanical and control subsystems. As shown in Fig. 11,mechanical design variables have various effects on the sta-bility according to integral gains of the velocity controller.Gain margin is increased proportional to the ball-screwlead when the integral gain of the velocity controller is rel-atively small. As ball-screw lead is increased, however, thegain margin is decreased when the integral gain of thevelocity controller exceeds a certain level. These interac-tions show importance of the integrated designmethodology.

4.3. Analysis of system response

Fig. 12 shows system response (xB) of the servomecha-nism according to design variables of the mechanical andcontrol subsystems. In general, response characteristicsare improved according as dimensions of the mechanicalsubsystem are decreased and magnitude of controller gainsare increased.

Fig. 13 shows xB according to the proportional gain ofthe velocity controller with the same design variable pairsof the mechanical subsystem. As widely known, response

02

46

810

010

2030

4050

0

20

40

60

K v (V

/V)K

p (V/V)

Am

(dB

)

Fig. 10. Effects of control design variables on stability.

020

4060

0 20

4060

0

20

40

60

K p (V/V)

Dbs (mm)

ωB (

rad/

s)

Fig. 12. Effects of mechanical–control interaction on response.

of the servomechanism is improved when the proportionalgain of the velocity controller is increased. However, it isshown in Fig. 13 that changes of mechanical design vari-ables generate different effects on the system responseaccording to the proportional gain of the velocity controller.

0 20

4060

0 10

2030

0

20

40

60

80

l (m)D

bs (mm)

ωB (

rad/

s)

0 20

4060

0 10

2030

0

100

200

300

l (m)D

bs (mm)

ωB (

rad/

s)

b

a

Fig. 13. Effects of mechanical–control variation on response: (a) lowproportional gain of the velocity controller; (b) high proportional gain ofthe velocity controller.

Fig. 14. Configuration of the x–y table.

0 0.5 1 1.5 2 2.5 3-2.0

0.0

2.0

4.0

6.0

8.0

Spe

ed (

m/m

in)

Output signal

0 0.5 1 1.5 2 2.5 3-0.2

0.0

0.2

0.4

0.6

0.8Input signal

Time (sec)

Tor

que

(N.m

)

Fig. 15. Signals used in the system identification.


5. Experiments

In order to investigate the influence of interactionbetween the mechanical and control subsystems on the per-formance of a servomechanism, circular motion experi-ments on the x–y table are performed. To verify theobtained mathematical model described in Eq. (7), an accu-rate identification of the mechanical subsystem is con-ducted first.

5.1. Identification of the servomechanism

The identification process applies to the one-axis of anx–y table. The x–y table is equipped with a ball-screw,AC motor and amplifier, encoder, digital I/O interface,and PC-based controller. The table is constrained to movein x-direction by linear rolling guides. Fig. 14 shows the x–y table.

In reality, accurate discrete-time transfer functions formechanical subsystems of servomechanisms are difficultto obtain, especially when small sampling times are used.Since Nyquist frequency is inversely proportional to thesampling interval, higher order effects of amplifier dynam-

ics corrupt signals used for identification processes. More-over, inherent nonlinearities such as Coulomb friction,stiction and backlash degrade the accuracy of identificationprocesses.

To avoid these effects, a pair of biased square wave inputsignals are used for the excitation [19]. If unidirectional sig-nals having a proper magnitude are used as torque com-mands, stiction and backlash will be eliminated becausethere are no zero-velocity commands at all through theidentification process. This identification method does notsuffer from the problems due to the nonlinear effects and,therefore, is able to provide accurate results. Two sets ofinput signals composed of Gaussian pseudo-random bin-ary sequences are used as torque commands while the rota-tional motor speeds are synchronously acquired as outputsignals. After collecting the two sets of input and outputsignals, one set of unidirectional input and output signalsis obtained as shown in Fig. 15.

Power spectrum of reference trajectories fed to servo-mechanisms is primarily composed of low frequency

101 102 103-40

-20

0

20

Mag

nitu

de (

dB)

101 102 103

-400

-200

0

Frequency (Hz)

Pha

se (

degr

ee)

Fig. 16. Comparisons of Bode diagrams of the mechanical subsystem (—mechanical subsystem model, � � � spectral estimate, � ARMAX model).


signals. Therefore, a weighted least squares penalty func-tion in frequency domain [20], of which cost function isdescribed by Eq. (30), is used. Through the frequency-

-0.04

-0.02

0

0.02 [mm]

30

210

60

240

90

270

120

300

150

330

180 0

-0.04

-0.02

0

0.02 [mm]

30

210

60

240

90

270

120

300

150

330

180 0

a

c

Fig. 17. Experimental results (Mt = 80 kg): (a) case A (Kp = 100); (b)

dependent weighting function, the identification is biasedto emphasize the fit in a particular frequency range.

JðuÞ ¼Z p

�pjGmðejxT sÞ � bGmðejxT s ;uÞj2Qxðx;uÞdx ð30Þ

where Qx is a weighting function in frequency domain, u a

parameter vector and bGm the input transfer function of an

Auto-Regressive Moving Average with eXogenous (AR-

MAX) model described in Eq. (31).

v̂tðkÞ ¼ bGmðzÞ � sðkÞ þ bH mðzÞ � nðkÞ ð31:aÞ

bGmðzÞ ¼b1zþ b2z2 þ � � � þ bnb

znb

1þ a1zþ � � � þ ana znað31:bÞ

u ¼ ½ b1b2 � � � bnba1a2 � � � ana �

T ð31:cÞ

where v̂t is the estimated table velocity, bH m a noise transferfunction and n the white noise fed to a system. By using theinput–output signal set and Eqs. (30) and (31), the transferfunction of the mechanical subsystem is obtained. Fig. 16shows comparisons of Bode diagrams. The bold line meansBode diagram of the mechanical subsystem when

-0.04

-0.02

0

0.02 [mm]

30

210

60

240

90

270

120

300

150

330

180 0

-0.04

0.02

0

0.02 [mm]

30

210

60

240

90

270

120

300

150

330

180 0

b

d

case B (Kp = 170); (c) case C (Kp = 200); (d) case D (Kp = 250).


Bm = 0.00001 and Bt = 0.0001 in Eq. (7). The dot line andsymbol ‘‘*’’ are identified systems obtained through thespectral estimation and the ARMAX model, respectively.It is confirmed that the mechanical subsystem model foran integrated design has been reliably derived.

5.2. Performance analysis

Circle radius of 25 mm and circular motion speed of3000 mm/min are used for the experiment. In manufactur-ing of aerospace components, weight of the workpiecechanges substantially during the machining process. Theweight change of the workpiece (or table) represents atime-varying characteristics of the servomechanism [1,2].Therefore, it is important to investigate the influence ofinertia variations on the servomechanism performance. Inthe experiment, axis inertias (table mass) and proportionalgains of the position controller are used to investigate theinfluence of interactions between the mechanical and con-trol subsystems on the servomechanism.

Fig. 17 shows experimental results of circular motionswith the constant axis inertia and various proportional

-0.04

-0.02

0

0.02 [mm]

30

210

60

240

90

270

120

300

150

330

180 0

-0.04

-0.02

0

0.02 [mm]

30

210

60

240

90

270

120

300

150

330

180 0

a b

c

Fig. 18. Experimental results (Mt = 25 kg): (a) case A (Kp = 100); (b)

gains. It is confirmed that radius errors are decreasedaccording as the proportional gain of the position control-ler is increased. However, there is little difference betweenthe circular motion profiles (Fig. 17(b) and (c)) in whichthe proportional gain changes to a relatively large value.However, stability of the servomechanism cannot be guar-anteed when the proportional gain of the position control-ler is excessively large as shown in Fig. 17(d), wherevibrations in the circular motion occur at accelerationand deceleration motions of each axis.

Fig. 18 shows experimental results of circular motionswith decreased axis inertia and the same proportional gainsused in the first experiment. As shown in Fig. 18(a) and17(a), there is little difference between the circular motionprofiles when the position controller of the servomecha-nism has small proportional gains. However, contrary toFig. 17(c) in which the axis inertia is larger than the caseof Fig. 18(c), there is unacceptable vibrations inFig. 18(c) when the proportional gain is over 170. There-fore, it is impossible to attain high-speed characteristicsof the servomechanism by decreasing axis inertia only with-out considering the controller gain.

-0.04

-0.02

0

0.02 [mm]

30

210

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180 0

-0.1

0

0.1

0.2 [mm]

30

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90

270

120

300

150

330

180 0

d

case B (Kp = 170); (c) case C (Kp = 200); (d) case D (Kp = 250).


Figs. 17 and 18 show that contour errors are sensitive tochanges in the controller gain, and the range of allowablecontroller gain is decreased when the axis inertia isdecreased. From the experiment results, it is conformedthat the limitation of performance depends on characteris-tics of both the mechanical and control subsystems.Changes of control design variables have various effectson the servomechanism performance according to designconditions of the mechanical subsystem. Therefore, it isimpossible to satisfy the desired performance of servo-mechanisms through a component design methodology,and an advanced design methodology is required, whereinteractions between the mechanical and control subsys-tems should be considered in the design process.

6. Conclusions

In this study, a reliable modelling and identification pro-cess of a ball-screw driven servomechanism, which is com-posed of the mechanical and control subsystems, areconducted. Parametric studies and circular motion experi-ments on an x–y table are performed in order to investigateinteractions between the mechanical and control subsys-tems, as well as the influence of the interactions on servo-mechanism performance. Main results are described asfollows:

– A strict mathematical model of the mechanical subsys-tem is derived from the detail modelling of the subsys-tem, and feedback and feedforward controllerssatisfying both robustness and tracking performanceare applied as the control subsystem in discrete-timedomain.

– In order to verify the obtained mathematical model, anaccurate identification process of the mechanical subsys-tem has been conducted, where a pair of biased signalsand a weighted least squares penalty function in fre-quency domain are used.

– Performance functions, constraints and design variablesincluding parameters of both the mechanical and con-trol subsystems are derived. Analysis of the system per-formance according to the design variables has beenconducted as a prior study of the integrated design.

– There are design variable pairs between the mechanicaland control subsystems that cause severe increase ofradius errors. It is confirmed that those interactionsshould be included in the design process through anintegrated design methodology.

– Mechanical design variables, such as ball-screw diame-ter and lead, have various effects on servomechanismstability according to integral gains of the velocity con-troller. And changes of mechanical design variablesaffect different effects on servomechanism response withrespect to the proportional gain of the velocitycontroller.

– From the results of circular motion experiments, it isconfirmed that contour errors are sensitive to changesof controller gains, and the range of allowable controllergains is decreased when axis inertia is decreased. Limita-tions of system performance depend on characteristicsof both the mechanical and control subsystems. Interac-tions between the mechanical and control subsystemsmust be included in the design process.

Acknowledgement

This work was supported by the research fund of Hany-ang University (HY-2005-I).

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