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Precision Engineering 40 (2015) 151–159

Contents lists available at ScienceDirect

Precision Engineering

journal homepage: www.elsevier .com/ locate /precis ion

High-speed and high-precision tracking control of

ultrahigh-acceleration moving-permanent-magnet

linear synchronous motor

Tadashi Hama, Kaiji Sato∗

Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, 4259Nagatsuta, Midori-ku, Yokohama 226-8502, Japan

a r t i c l e i n f o

Article history:

Received 30 May 2014

Received in revised form 7 October 2014

Accepted 30 October 2014

Available online 4 December 2014

Keywords:

Linear motor

Precision

Positioning

Control

Learning

High acceleration

High velocity

a b s t r a c t

This paper describes the high-speed and high-precision tracking control of an ultrahigh-acceleration,

high-velocity linear synchronous motor (LSM). The linear motor can produce a thrust force of more than

3000 N, an acceleration greater than70 G (=686 m/s2), and move at a velocity of over 10 m/s. However, it

has highly nonlinear characteristics, and it is difficult to provide an exact dynamic model for the controller

design. Thus, a suitable controller that does not require a dynamic model in the design was selected and

used for the high-precision tracking control of the linear motor. The design procedure for the suitable

controller consists of two steps. In the first step, a two-degree-of-freedom controller with additional

control elements was designed, and its performance was examined. The additional elements were used to

suppress thenegative influences characterizingpermanent-magnetLSMs with cored electromagnets. The

controller showed high tracking accuracy at low speed, but not at high speed. To overcome this problem,

the controller was improved with a learning control element in the second step. The learning control

element does not require a dynamic model in the design, and it is effective at reducing reproducible errors

at high speed. The effectiveness of the controller was examined and demonstrated experimentally. The

improved controller with the learning control element reduced the maximum tracking error to 1.62m

in the sinusoidal reference motion at a frequency of 20Hz and an amplitude of 10mm.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

Linear motors are widely used in industrial applications such as

precisionmachine tools, photolithography machines,and assembly

equipment. Linearmotors have fewvibration factors andan advan-

tage in their response because they are directly driven without

power transmissions. Such characteristics are suitable for high-

speed machines [1,2] and precision motion mechanisms [3–8], and

the use of linear motors has increased in recent years [9]. Many

conventional high-speed machines have power transmissions, and

their acceleration and velocity are usually limited to less than 2 G

(=19.6 m/s2) and 2m/s, respectively. However, linear motors are

generallyused when higher acceleration or velocity are demanded.

The required acceleration and velocity have increased yearly,

and many high-speed linear motors have been developed. A lin-

ear motor called a tunnel actuator, which has peculiar-shape cored

electromagnets, achieves an acceleration of 38G [10]. The tubular

∗ Corresponding author. Tel.: +81 45 924 5045; fax: +81 45 924 5483.

E-mail address: [email protected] (K. Sato).

actuator developed for pick and place can move at an acceleration

of 20 G and a velocity of 1.5 m /s [11]. A large-thrust linear motor

that can move at an acceleration of over 27G and at a velocity of

4m/s is commercially available [12].

Permanent-magnet linear synchronous motors (PM LSMs) have

advantages of high controllability and high responsiveness. How-

ever, linear motion mechanisms with PM LSMs also have negative

characteristics that decrease their motion accuracy, for example,

thrust ripple, cogging forces, and friction force. To suppress the

negative characteristics, some controllers have been implemented

with PM LSMs, and their tracking performance has been discussed.

A controller with state feedback elements and a feedforward ripple

compensator provideda tracking performance witha tracking error

within 10m at a maximum velocity and acceleration of 0.5 m/s

and 20m/s2, respectively [13]. The responses of the control system

were examined with desired compensation adaptive robust control

[14]. The tracking error was within 5m at a velocityof 2m/s.

The author’s research group has studied ultrahigh-acceleration

and high-velocity linear motors [15] and have fabricated a moving-

permanent-magnet LSM (MPM LSM) having a thrust-to-mover

mass ratioof 908N/kgwithlow thrustrippleand lowcogging forces

http://dx.doi.org/10.1016/j.precisioneng.2014.11.005

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152 T. Hama, K. Sato / PrecisionEngineering 40 (2015) 151–159

Fig. 1. Experimental setup includingthe prototype MPM LSM. (a) Overall view of theMPM LSM, and (b)measuring system.

[16]. In this study, high-speed and precision tracking control with

an MPM LSM is discussed. As described later, the MPM LSM in this

study is different from the other MPM LSMs introduced in [15,16].

It hasbeen adjusted forease of assembly,and themaximum thrust-to-movermass ratio is smaller than theotherMPM LSMs.However,

the MPM LSM used has a much larger thrust-to-mover mass ratio

than the other linear motors previously reported, and it has the

potential to achieve both high-speed and high-precision motion

with a suitable controller.

This paper describes the high-speed and high-precision track-

ing control of an ultrahigh-acceleration, high-velocity linear motor.

The linear motor has highly nonlinear characteristics, and it is dif-

ficult to use an exact dynamic model in the controller design. Thus,

a controller that does not require a dynamic model in its design

is selected and used for high-precision tracking control of the lin-

ear motor. The design procedure for a suitable controller consists

of two steps. In the first step, a two-degree-of-freedom controller

with additional control elements is designed, and its performance

is examined. The additional elements are used for the suppression

of the negative influences characterizing the PM LSMs with cored

electromagnets.The controllershows high tracking accuracy at low

speed,but notat high speed.For overcoming this problem, thecon-

troller is improved with a learning element in the second step. The

effectiveness of the control system is examined experimentally.

The remainder of this paper is organized as follows. Section 2

explains the structure and the driving principle of the MPM LSM.

In Section 3, the basic causes of the MPM LSM trajectory errors are

described. Then, the controller design based on the simple model

and its control results are described. In Section 4, the controller for

high-speed motion using learning control is explained. Section 5

demonstrates the tracking control results with the learning control

element. Finally, concluding remarks are presented in Section 6.

2. Prototypeof the MPM LSM

Fig. 1 shows a prototype MPM LSM that has been designed to

have an ultrahigh thrust-to-mover mass ratio for ultrahigh accel-

eration and high velocity. The stator of the prototype MPM LSM

consists of two pairs of electromagnet (EM) lines, and its mover is

located between the EM lines. The structures of the stator and the

mover were designed so that the MPM LSM has a high thrust-to-

movermass ratioand a smallthrust ripple. Thebasic structureis the

same asthatof the MPM LSMdescribed in [15,16]. However, theair

gapis widened foreasyadjustment, anddifferentpartsof themover

are used for weight savings. Thelength ofthe EMlineis 2.03m, and

the working range is longer than 1 m. The mover is supported by a

Fig. 2. Structureof themover.

Phase A Phase B

Phase C Phase D

24 48 72 96

18 42 66 90

r r e n t A

12 36 60 84 C u r

6 30 54 78

Position mmFig. 3. Current command signal waveforms.

linearballguideconsisting of four carriagesand twoguiderails. The

displacement of the mover is measured using a laser interferom-

eter with a resolution of 79.1nm (ZLM series, JENAer Metechnik

GmbH).

Each of the EM lines has many core teeth with 100-turn coils,

and the core teeth are arranged at an 18mm pitch. The coils of the

EM are divided into eight phase coils (phase A, B, C, D, and their

reverse phases). Each phase coil is driven by different PWM ampli-

fiers (maximum supply voltage: 280V, maximum supply current:

20A).



T. Hama, K. Sato / PrecisionEngineering 40 (2015) 151–159 153

4000

3000 N

2000

u s t f o r c e N

1000 T h r u

0 5 10 15 200

Current amplitude A

Fig. 4. Measured static thrust characteristic.

Fig. 2 shows the structure of the mover. In the mover, 18 PMs

(NEOMAX-48BH) are arranged at a 24mm pitch. Spacers made of

engineering plastic are inserted between them. The mover is fabri-

cated using light-weight materials such as carbon-fiber-reinforced

plastic (CFRP) and extra super duralumin. The length of the mover

is 440 mm, and the total weight including the carriages and the

sensor target is 4.68kg.

Fig. 3 shows the waveform of the current command signal

applied to the amplifiers for driving the mover. Rectangular sig-nals are used as command signals to generate a large thrust force.

The sign of the current is switched by using the measured position.

Fig. 4 shows the measured static thrust characteristic of the

MPM LSM. Thethrust gain decreaseswith an increase in theapplied

current because of the magnetic nonlinearity.

3. Basic controller design for PM LSMs with cored EMs and

its control performance

3.1. Basic controller structure for the MPM LSM

Two-degree-of-freedom (two-DOF) controllers including a

feedback compensator and a feedforward element are widely used

forhigh response andhigh motion accuracy [17,18]. They are effec-tive for the tracking control of linear motors. Linear motors have

peculiar factors that deteriorate the motion accuracy, and suitable

compensators are needed to decrease the negative influences of

these factors. The MPM LSMs have cored electromagnets in which

the cogging force causes tracking errors. The friction caused in the

linear ball guides also increases the tracking error. Two-DOF con-

trollers with feedforward (FF) elements for cogging and friction

compensation are often used for overcoming these problems in PM

LSMs with cored electromagnets [19,20]. Thus, as a first controller

forthe tracking control of theMPM LSM, a two-DOF controller with

two compensators is designed, and its usefulness is examined.

Fig. 5 shows a block diagram of the adopted control system. The

controllerincludes a conventional two-DOF controller for compen-

sating the basic dynamic characteristics of the MPM LSM and theFF elements for compensating the cogging and friction forces. The

conventional two-DOF controller consists of a PID compensator as

a serial compensator and an FF element based on a simple inverse

model of the mechanism. The MPM LSM has high nonlinearity in

the thrust characteristic, and the linearizer for statically canceling

itsnegativeinfluence is inserted between thePID compensator and

the control object. Moreover, a disturbance observer is integrated

in order to compensate for the negative effects caused by model

errors and unknown disturbance forces.

In the control design, a dynamic model is necessary for the dis-

turbance observer andthe FF element in theconventionaltwo-DOF

controller. A detailed dynamic model is generally effective for the

improvement of the tracking performance. However, the MPM LSM

has many nonlinear factors, and it is difficult to model the MPM

LSM accurately. Particularly, it is unrealistic to construct a model

forthe control systemdesign. Thus, a simpledynamic model repre-

sented as a linear mass-damper system is used. The elements and

compensator of the controllers are explained individually.

(a) Nonlinear PID compensator

A nonlinearPIDcompensatoris adopted asa tractableand eas-

ily adjustable series compensator. The compensator includes a

conditional integrator, which operates under the rule described

in Eq. (3.1).

ui =

0,

|uo + ui| > us

and

e · ui ≥ 0

e, otherwise

(3.1)

Here, ui is the integrated value, ui is a variation of the inte-

grated value, e is the error, uo is the sum of the proportional

value and the derivative value, andus is the maximum value of

the control input.

The proportional and derivative elements are gain scheduled

sothatthe movervibration is reduced effectively.Theirgains are

expressedas functionsof an error signaland a time derivative of

an error signal, respectively. The gain functions used are shownin Fig. 6. These were experimentally decided by trial and error

on low speed motions.

(b) FF element for dynamic characteristic compensation

The FF element in the conventional two-DOF controller is

designed based on the simple linear inverse model of the MPM

LSM for dynamic characteristic compensation. Eq. (3.2) repre-

sents the function based on the inverse model,

P inv = m ·( z − 1)2

(T · z )2 + c ·

z − 1

T · z (3.2)

where m is the mass of the mover (m= 4.68 kg), c is the

viscous coefficient (c =30N s/m), and T is the sampling time

(T = 1/16,000s). Eq. (3.2) is derived from the discretization using

the backward differential rule. The rule sometimes makes thesystem sensitive to high frequency noises. However, the elec-

tromagnets of this linear motor have large inductance and work

as noise filter whose cut-off frequencyis lower than 8.5Hz. They

make the system insensitive to high frequency noises.

(c) Linearizer for compensation of nonlinear static thrust charac-

teristic

The MPM LSM used in this study produces a large thrust

force with strong PMs and a large current in the coils. Thus,

the influence of the nonlinearity caused by magnetic saturation

is prominent when a large thrust force is produced. In order to

statically compensate for the nonlinearity of the thrust gain, a

linearizeris implemented. The linearizerwas designed basedon

the measured static thrust characteristic.

(d) FF element for cogging compensationA combination of cored electromagnets and neodymium per-

manent magnets is used for a large thrust force and causes a

large cogging force that deteriorates the tracking performance.

In order to cancel the negative influence of the cogging force,

an FF element is used. The FF element was determined from the

estimatedcogging force as a function of the mover position. The

disturbance observermentionedbelow wasused to estimatethe

cogging force. The cogging force is estimated on the low speed

motion (25mm/s) which does not demand large current. The

large applied current to the linear motor may result in the mag-

netic saturation and influence the cogging force. However, the

influence is ignored in the design of the FF element for cogging

compensation.

(e) FF element for coulomb friction compensation




FF element for cogging compensation

FF element for coulomb frictioncompensation

FF element

for

cogging

compensation

+

PID

FF

compensation

Obj t

+ + +

+

+

Linearizer

PID

Disturbance

ect−

−

Two-DOF

controller

Fig. 5. Block diagram of thebasic control system forthe MPM LSM with cored EMs.

Fig. 6. Gain-scheduling functions. (a) Proportional gain, and (b) differential gain.

The coulomb friction force is often modeled by a static fric-

tion force anda kinetic friction force [21]. However, the detailed

characteristics of the coulomb friction depend on the mech-

anism. Then, the characteristics of the coulomb friction are

examined experimentally with the disturbance observer men-

tioned below. Fig.7 shows the disturbance force fora low-speed

motion (with sinusoidal references 10 m m and 100 m m in

amplitude at a frequency of 0.5 Hz). This result indicates that

the characteristics of coulomb friction in the linear motor are

different from the normalmodel of coulomb friction. In the nor-

mal model, the amplitude of the force is constant, and only the

sign changes; however, the amplitude of the actual force does

change. The inverting speed of the force depends on the input

20 At the point, the motion

N

10mm 100mm

10

.

u r b a n c e N

-10

m a t e d d i s t u

0.0 0.1 0.2 0.3 0.4

-20 E s t i m

Time s

Fig. 7. Behavior of estimated disturbance with the reversal of movement (with

sinusoidal references in twodifferent amplitudes at a frequencyof 0.5Hz).

trajectory. The coulomb friction force with sinusoidal refer-

ence motion inputs is measured in an off-line estimation by the

disturbance observer. Then, the coulomb friction F c is approxi-

mately expressed as:

F c =

F cn · sgn(v )

(v < 0, a ≥ 0)

sgn(v ){F cp − (F cn + F cp) · exp(k|v |)}

(v ≥ 0, a ≥ 0)

F cp · sgn(v )

F cp · sgn(v )

sgn(v ){F cn − (F cn + F cp) · exp(k|v |)}

(v < 0, a < 0)

(3.3)

where v and a are the velocity and acceleration of the mover,

respectively. The amplitude of the coulomb friction at the pos-

itive velocity F cp and the negative velocity F cn are expressed

as:

F cp = 23.5 N, F cn = 31.5 N (3.4)

The amplitude of the coulomb friction force is different

depending on the direction of the movement. The occurrence

mechanism of the difference is not clear. However, the manu-

facturing and assembly errors may cause the change in contact

condition of the linear guide, which results in the difference of




Fig. 8. Friction model for the coulomb friction compensator.

Fig. 9. Experimental tracking error for the sinusoidal reference motion with the

control system in Fig. 5 (f requency: 2 Hz, amplitude: 50mm).

the friction force. The coefficient k is approximately expressed

as Eq. (3.5),

k = −3

(2fA)0.9

−15

f 1.25 (3.5)

where f is the input frequency, and A is the input amplitude.

The coulomb friction is approximately expressed in Eqs.(3.3)–(3.5), andthe coulomb friction compensator outputs a sig-

nal for canceling the force expressed in these equations. Fig. 8

shows an outline of the output value of this compensator.

(f) Disturbance observer

In order to observe thedisturbance force applied to themover

and relieve its negative influence, the disturbance observer was

designed based on a simple linear dynamics model expressed

by Eq. (3.2). The pole of the observer is located at −1100±2 j.

The input and outputof the disturbance observer are connected

as shown in Fig. 5.

3.2. Experimental tracking performance

The responses of the designed control system to the sinusoidaland ramp reference inputs were measured, and the control per-

formance was examined. Fig. 9 shows the tracking error with a

sinusoidal reference 50mm in amplitude at a frequency of 2Hz,

and the tracking error is less than 1.10m. With a ramp reference

of 50mm/s, the maximum tracking error is less than 0.64m, as

shown in Fig.10. These results showthat the control system realizes

high precision motion at low speeds.

Then, the responses to high-acceleration and high-velocity

(rapid) reference motion inputs were examined. Fig. 11 shows the

tracking error with a sinusoidal reference 10mm in amplitude at

a frequency of 20Hz. For the motion, the tracking performance

is greatly degraded. The tracking error decreases just before the

direction of motion reverses and then significantly increases. It

is considered that these phenomena are caused by certain factors

Fig.10. Experimental tracking error for theramp reference motion withthe control

system in Fig.5 (velocity: 50 mm/s).

Fig. 11. Experimental tracking error for the sinusoidal reference motion with the

control system in Fig. 5 (f requency: 20Hz, amplitude: 10mm).

such as thrust ripple, current delay due to the inductance, and the

incompleteness of the FF element for coulomb friction compensa-

tion. Thus, a control systemin which theinaccuracy ofthe modeling

does not deteriorate the control performance is necessary for the

reduction of the tracking error during rapid motion.

4. Learning control system for rapid motions

The control system described in Section 3 provides high preci-

sion motion at low speed, but the motion accuracy decreases with

an increase in mover acceleration and velocity. The linear motor

includes many factors that deteriorate the tracking performance,

such as varying friction force, thrust ripple, and current delay. It is

difficult to exactly model these factors, and a control design based

on the exact dynamic model is impractical. Learning controllers

do not require an exact dynamic model of the control object and

are often effective at reducing the negative effect of unknown but

systematic factors on the tracking performance. Thus, the learning

FF control element shown in [22,23] is added to the previous con-

trol system for high-accuracy tracking during rapid motion. In thissection, the method is briefly introduced, and the learning control

element is regulated. Then, themodificationof thepreviouscontrol

system for using the element is explained.

4.1. Basic structure of the learning control system

Fig. 12 shows a conceptual block diagram of the control system

with the learning FF control element. The output signal of the feed-

back compensator is used to improve the learning control element.

In learning, the signal is filtered to avoid vibrating and to ensure

stability in the high-frequency domain.

The learning control element improves with repetitive motion.

The cycle period of the repetitive motion is referred to as the




Feedback

element

+ +

Pl tFeedbackcompensator

− +Plant

Feedforward

Fig. 12. Conceptual block diagram of thelearning control.

0

Fig. 13. Basis functionsof thelearningcontrol.

“movement period.” The basic algorithm for the learning control

is as follows:

u jF

(kh) = u j−1F

(kh) + v j−1(kh) (4.1)

where j denotes the jth repetitive operation, h is the sampling

period, k is the time index, u jF

(kh) is the output of the learning con-

trol element at the time of kh, is the learning gain, and v j−1(kh) is

a value calculated from the next algorithm.

The output of the feedback compensator is filtered by the func-

tion called the “basis function,” and v j(kh) is calculated. Fig. 13

illustrates the basis function used in the learning control. One basis

function has one triangular part in a certain period, and the func-

tion in theother periodequals zero. There arepluralbasisfunctions,

which are named “b1, b2,” according to the emergence order of thetriangular parts in each movement period. Two triangular parts of

the basis functions having adjacent numbers overlap each other.

The output of the feedback compensator is u jc(t ), the function eval-

uation of the basis function bi is defined as i(t ), and the span of

one movement period isN p times the sampling period. Then, v j(kh)

is given by:

v j(kh) =

N i=1

i(kh)

N pl=0

i(lh)u jC

(lh)N pl=0

i(lh)(4.2)

The period of one triangular part in a basis function is 2m times

the sampling period, and the width m determines the width of the

basis function, which influences the bandwidth of the filter.

4.2. Regulation of the learning controller

The learning control element has two significant parameters

that determine its characteristics. One parameter is m, which

decides the width of the basis function. The other is the learning

gain defined in Eq. (4.1). The widthm determines the bandwidth

of the filter. If the width m is small, the control element can also

learn a high-frequency component, but the stability in the high-

frequency domain is reduced. If the width m is large, the stability

of the high-frequency domain is increased, butthe control element

cannot relieve the fast-varying disturbances, and the tracking error

increases with an increase in the input frequency. Fig. 14 shows

the influence of the widthm on the tracking error with a sinusoidal

reference 10mm in amplitude at a frequency of 20Hz. When the

width m is small (m= 15), the amplitude of vibration is larger than

in the case where m is equal to 25. When the width m is large

(m= 40), the fast-varying tracking errors originally caused by the

mechanism are not greatly reduced. The width m was regulated

experimentally, and m=25 was selected as an adequate value. It

is confirmed that the optimum value of the width m is constant

regardless of the frequency of the reference trajectory.

The learning gain determines the speed of learning and influ-

ences the time required for learning and the stability of the control

system. If the gain is large, the time required for learning is

reduced, but the stability of the control system tends to decrease.

The optimum learning gain for the control performance was exam-

ined experimentally, and = 0 .8 was selected regardless of the

reference trajectory.

In order to suppress the negative influence of the current delay

caused by the current amplifiers, the output of the learning control

element is advanced. The optimum time was examined experi-

mentally, and it was decided to advance the output by six times

sampling period.

4.3. Removing the delay element

This control system is required to have a high response perfor-

mance, and a controller including delay elements can deteriorate

the tracking performance. Thus, the integrator is removed. In addi-

tion, the disturbance observer has a delay element and may have

a negative influence on the control performance, although it has a

higherresponseperformance thana simple integrator.Both the dis-

turbance observer and the learning controller have the function of

suppressing effects caused by model errors, butthe combination of

them may also have a negative influence. Then, the influence of the

disturbance observer on the learning control system is examined

experimentally. Fig. 15 shows the response of the learning control

system to a sinusoidal input 10mm in amplitude at a frequency of

20Hz, and the displayed tracking error is measured in the move-

ment period in which the error amplitude is no longer reduced by

the learning control. The gray line represents the tracking error of

Fig. 14. Effect of theparameter m of thelearning control element on thetracking error forthe sinusoidalreference motion (frequency: 20Hz, amplitude: 10mm).




Fig. 15. Effect of the disturbance observer on the tracking error of the learning

control system for the sinusoidal reference motion (frequency: 20Hz, amplitude:

10 mm).

the control system with both the learning control element and the

disturbance observer, and the black line represents the tracking

error without the disturbance observer. Both the control systems

can greatly reduce the tracking error and are useful for high-speed

and high-precision tracking control. However, the amplitude of the

tracking error with the disturbance observer is larger than that

of the tracking error without it, and it is shown that the disturb-

ance observer deteriorates the tracking performance in this case.

This study places great importance on the high precision tracking

motion. Thus, it is decided that the disturbance observer is to be

removed although the use of the disturbance observer is expected

to increase the robustness of the control system against sudden

and accidental disturbance force. The learning control system for

therapid motionafterthe modifications mentionedaboveis shown

in Fig. 16.

5. Control results of the learning control system for rapid

motions

The responses of the control system to sinusoidal and ramp

inputs are examined with the learning control system described in

Section 4. In this section, the learning control system is compared

with the control system based on the simple model described in

Section 3 (in this section, it is called “the basic control system”),

and the usability of the controller is discussed. The control results

are shown with regard to the “movement period” (defined in Sec-

tion 4), in which the amplitude of the error is no longer reduced by

the learning control.

5.1. Tracking performance of the sinusoidal referencemotion

Fig. 17 shows the tracking errors with the four different sinu-

soidal references. Table 1 shows thespecifications of theeach inputtrajectory and the maximum tracking errors. The gray and black

lines represent the tracking errors of the basic control system and

the learning control system, respectively.

The learning controller reduces the tracking error in all condi-

tions. In order to discuss the tracking performance, the tracking

errors with two kinds of reference motions are compared. One

motion has a sinusoidal reference 10 mm in amplitude at a

frequency of 20Hz (maximum acceleration: 16.1G, maximum

velocity:1.26m/s);this motion is definedas “motion (a)” in Table 1.

The other motion has a sinusoidal reference 50mm in amplitude

at a frequency of 6 Hz (maximum acceleration: 7.25G, maximum

velocity:1.88m/s); this motionis definedas “motion (d)” in Table 1.

The gray line in Fig. 17(a) shows the tracking error of the former

motion (“motion (a)”), and the gray line in Fig. 17(d) shows the

tracking error of the latter motion (“motion (d)”) with the basic

control system. The maximum error of theformer motion(“motion

(a)”) is 2.9m larger than that of the latter motion (“motion (d)”).

However, with the learning control system, the maximum tracking

error of the latter motion (“motion (d)”) is larger than that of the

formermotion (“motion (a)”) (comparing theblackline in Fig. 17(a)

and the black linein Fig. 17(d)). The maximum acceleration is larger

in the former motion (“motion (a)”), and the maximum veloc-

ity is larger in the latter motion (“motion (d)”). In both motions,

the learning control system reduces the tracking error more effec-

tively than the basic control system. However, in the latter motion

(“motion (d)”), the tracking error becomes large especially around

thepoint of peak velocity,and the performance of thelearningcon-

trol system tends to be affected by the velocity rather than the

acceleration. It is assumed that the change speed of the disturb-

ance force including thrust ripple increases with an increase in the

velocity,and the learning control system cannot reduce thetracking

error effectively. The change in acceleration causes the same effect

as other disturbance forces, but its frequency is as low as that of

the input signal. Thus, the effect of the acceleration is assumed to

be reduced easily by the learning control system, even if the effect

is too large for the basic controller to compensate.

FF element for cogging compensation

FF element for coulomb friction

compensation+

+Learning +

Inverse model of thedynamic characteristic

PD Object+

−

+

+

+

near zercontro er+

Fig. 16. Block diagram of thelearningcontrol systemfor rapid motions.




(a) (b)

(c) (d)

Fig. 17. Comparison of experimental tracking error results with the basic system and the learning control system for sinusoidal reference motions. (a) Frequency: 20Hz,

amplitude: 10 mm, (b) frequency: 10Hz, amplitude: 20mm, (c) frequency: 8 Hz, amplitude: 40mm, and (d) frequency: 6 Hz, amplitude: 50mm.

5.2. Tracking performance of the ramp reference motion

The responses of the control system to ramp inputs are exam-

ined with the learning control system. Fig. 18 shows the tracking

performance results with the ramp references at velocities of

0.6m/s, 1m/s, and 2m/s. The gray line represents the results of

the basic control system, and the black line represents the results

of the learning control system. At velocities of 0.6 m/s and 1.0m/s,

the learning control system reduces the tracking error more effec-

tivelythan thebasiccontrol system. On theotherhand,at a velocity

(a) (b)

(c)

Fig. 18. Comparison of experimental tracking error results with the basic system and the learning control system for ramp reference motions. (a) Velocity: 0.6m/s, (b)

velocity: 1.0m/s, and (c) velocity: 2.0m/s.




Table 1

Specifications of the reference motions and the maximum tracking errors.

Name of motion Reference motion

(frequency and

amplitude)

Maximum

acceleration

Maximum

velocity

Maximum error of

basic control system

Maximum error of

learning control system

(a) 20 Hz, 10 mm 16.1 G 1.26 m/s 5.62m 1.62m

(b) 10 Hz, 20 mm 8.06 G 1.26 m/s 1.86m 1.28m

(c) 8 Hz, 40 mm 10.3 G 2.01 m/s 3.20m 1.68m

(d) 6 Hz, 50 mm 7.25 G 1.88 m/s 2.72m 1.86m

of2.0 m/s, theimprovement in tracking performance is notso effec-

tive in terms of the maximum tracking error.

It is assumed that fast-varying disturbance forces including

thrust ripple mainly deteriorate the tracking performance for the

ramp reference motion. For high-speed motion, the tracking error

with the learning controller is oscillatory due to fast-varying dis-

turbance forces, although it is certain that the learning controller

reduces the amplitude of the tracking error. Meanwhile, for low-

speed motion, the amplitude of the disturbance forces is originally

small. Thus, for a limited velocity range, the learning control sys-

tem shows a large improvement in motion with a ramp reference

compared with the basic control system.

6. Conclusion

In this paper, a controller for high-speed and high-precision

tracking control of an ultrahigh-acceleration and high-velocity

linear motor was designed,and its tracking performance was exam-

ined. First, the tracking control system based on a simple model of

the linear motor was designed. In this control system, the tracking

performance was satisfactory at low speed, and the tracking error

was less than 1.10m with a sinusoidal reference50 mm in ampli-

tude at a frequency of 2Hz. However, the tracking performance

deteriorated for high-acceleration and high-velocity motion. Then,

in order to reduce the tracking error for the motion, a learning

controller that improved the feedforward element in the repetitive

motion was implemented into the control system. This improved

control system greatly reduced the repetitive errors without anaccurate model. For example, the maximum tracking error was

reduced to 1.62m with a sinusoidal reference 10mm in ampli-

tude at a frequencyof 20Hz. The experimentalcontrol results show

that the learning controller can effectively suppress the position-

ing errors that have high-frequency components, and the designed

control system can realize high precision motion performance.

Acknowledgements

The authors would like to thank students T. Imanishi and T.

Taguchi for assisting with the design of the basic control system.

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