motion system design of a thin and compact linear switched

Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.12, No.1, 2018

Paper No.17-00443© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0025]

Motion system design of a thin and compact linear switched

reluctance motor with disposable-film mover

Mohd Nazmin MASLAN*,** and Kaiji SATO***

* Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology

4259 Nagatsuta, Midori-ku, Yokohama, Kanagawa 226-8503, Japan

** Faculty of Manufacturing Engineering, Universiti Teknikal Malaysia Melaka

Hang Tuah Jaya, 76100 Durian Tunggal, Melaka, Malaysia

*** Department of Mechanical Engineering, Toyohashi University of Technology

1-1 Hibarigaoka, Tempaku-cho, Toyohashi, Aichi 441-8580, Japan

E-mail: [email protected]

Abstract

This paper describes a suitable motion system design that utilizes a thin and compact linear switched

reluctance motor (LSRM) with a disposable-film mover. The motor features a simple structure that is easy to

fabricate and install into existing and newly designed instruments. Made using readily available materials, the

mover is considered disposable. To meet the objectives of a motion system, i.e., simplicity in terms of use and

mover exchangeability, the motion performance should remain the same even when the mover is exchanged.

To meet these objectives, it is desirable to provide controllers that are robust to changes to the movers, and it is

important to clarify the limitations of the motion performance resulting from the different motor

characteristics. Thus, using a controller designed for precise tracking, experiments were carried out to verify

the robustness of the motion system against the influences of changes in the length and mass of the movers.

The limitations of the motion performance were then formulated for systems with a small effective thrust force

such as those used in the developed LSRM, and validated. Based on the results achieved, the range of

additional mass specifically applied to maintain the same motion performance is clarified in the present paper.

Keywords : Linear switched reluctance motor, Thin, Compact, Disposable mover, Motion, System design,

Easy fabrication, Precision, Tracking

1. Introduction

The demand for linear-drive mechanisms is increasing because such mechanisms help reduce the reliance on

rotary-to-linear motion converters, which are prone to excessive vibration. Electromagnetic linear motors have

properties suitable for high-speed and high-precision systems (Kurisaki et al., 2010; Mori et al., 2010; Sato, 2015). To

provide a high-speed and high-precision performance, linear motors depend on permanent magnets (PMs), which are

bulky and have powerful attractive forces, thereby making their assembly and disassembly difficult.

Sato (2013) proposed a linear motor based on a linear switched reluctance motor (LSRM), which does not require

the use of powerful PMs and is therefore free from their negative effects. Hence, the components can be easily

assembled, disassembled, and recycled; moreover, a further reduction in cost can be expected. Without the use of PMs,

the resulting shape of the LSRM is simple, which is a suitable characteristic for a thin and compact basic structure,

thereby saving space for easy installation in existing and newly designed instruments. An LSRM with active stator and

passive mover topologies can help in the design of a mover with a configuration independent from that of the stator

because no coils are attached. Hence, the low-cost mover can be replaced and easily discarded for hazardous

applications. Figure 1 shows a practical example of the developed LSRM. The film-like mover helps handle objects

easily and can be used as a disposable sheet for mounting micro-parts and test bodies, the masses of which are much

lighter than the mover.

1

Received: 1 September 2017; Revised: 28 December 2017; Accepted: 18 February 2018

2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0025]

Maslan and Sato, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

In recent years, the demand for high-precision performance has increased (Oiwa et al., 2011). However, owing to

the three-dimensional relationship between the thrust force, applied current, and mover position, the LSRM is nonlinear

in nature (Krishnan R., 2001); hence, achieving precision control is difficult. Moreover, a high force ripple is observed

(Masoudi et al., 2016). In addition, the motor exhibits a nonlinear friction that changes with respect to the applied

current and mover position, thereby reducing the effective thrust force. Maslan et al. (2017) reported the precision

positioning results for the developed LSRM. As shown in Fig. 1, application of the developed LSRM requires the

motion performance to be maintained even if the mover is exchanged for easier use. For the motion system, the length

of the mover is easy to adjust depending on the intended working range. Thus, the controller should be robust to

changes to the movers, which may vary in their length and mass, and it is important to clarify the limitation of the

robustness to such changes.

The objective of this work is to clarify a suitable motion system design method for the developed LSRM that has a

small effective thrust force and validate its usefulness. To meet the objectives of the motion system, i.e., simplicity of

use and mover exchangeability, the motion performance should remain the same even when the mover is exchanged.

The motion system design method includes the controller design for precision motion and the clarification of

limitations of the motion performance based on the motor characteristics. First, the control system used for precise

tracking was determined and examined experimentally. Then, because the robustness of the motion controller is critical,

the performance effects from changing the length and mass of the movers were also experimentally investigated.

Further examinations were then conducted on the limitations of the motion performance for systems with a small

effective thrust force such as those used in the developed LSRM. Finally, the range of additional mass specifically

applied to maintain the same level of performance was clarified.

2. LSRM Prototype

Figure 2 shows the experiment setup, which includes the LSRM prototype. The motor comprises an active stator

and a passive mover, both of which are made from a single-layer magnetic core. The basic structure (Fig. 2(a)) and

driving principle are the same as those of the LSRM described elsewhere (Sato, 2013). Maslan et al. (2017) increased

the width of the mover core to reduce the magnetic resistance, which resulted in a higher thrust force characteristic than

that in the previous study. The authors also provided the simple fabrication process of the mover.

Figure 2(b) shows the overall view of the experiment setup. The mover is placed on the sliding surface at the center

of the stator between the coils. A polytetrafluoroethylene (PTFE) film is bonded to the sliding surface to reduce the

frictional effect. The surface is supported using a linear sliding guide made from PTFE. The displacement of the mover

is measured using a linear encoder (Mercury II 5800, GSI Group, Inc.) with a resolution of 0.1 μm, which is mounted

above the motor. The stator core has multiple slots and core teeth with a pitch of 2.0 mm made from a high

permeability permalloy B. Both sides of the core are wound with 12 (30-turn) coils, which are divided into three phases

(phases A, B, and C). Each phase of the coil is driven using a separate commercial current amplifier (maximum supply

current, ± 3.33 A). The driving signals applied to these current amplifiers are provided using a digital signal-processing

unit, which obtains the data from the sensors.

Fig. 1 Practical example of the developed LSRM.

Mover

Micro-part

Stator

CleaningManufacturing Assembly Testing

StorageDisposal

2



(b) Overall view of the experiment setup

(a) 3-D view of the basic structure (c) Fabricated disposable-film mover

x

yz

Mover

Stator

Traveling direction

Stator

core teeth

Coil

Non-magnetic

material

Mover core

Stator

core

Mover

Stator Linear sliding guides

Linear encoder head

(resolution: 0.1 μm)

z

y

x

Magnetic material

for mover coreNon-magnetic

material film

Sensor scale

(a) Applied current signal waveforms

(b) Measured static thrust force characteristic

0.0 0.5 1.0 1.5 2.0-303 Phase C

Position (mm)

-303 Phase B

Curr

ent

(A)

-303 Phase A

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

Position (mm)

Thru

st f

orc

e (m

N)

The structure of the fabricated mover is thin and simple, as shown in Fig. 2(c). The mover comprises an array of

cores arranged uniformly with a pitch of 2.4 mm. The core is made from a sheet of silicon steel. To complete the

arrangement, the cores are laminated using two different types of non-magnetic films. The films on the top and bottom

of the core are transparent polyethylene (PET) and PTFE films, respectively, both of which are respectively having

good adhesion and low frictional properties. The mover is attached with a sensor scale compatible with the linear

encoder to detect its position. Unlike the width of the mover, the length can be adjusted according to the required

working range. In this study, several movers are used. One of the fabricated movers has a length of 66 mm and a total

weight of 0.56 g, which includes the sensor scale.

The effective thrust force in this LSRM is not proportional to the position of the mover or applied current, which is

a similar observation for most LSRMs. Figure 3 shows the measured static thrust force characteristic at a maximum

current of 3.33 A using the driving signal waveforms obtained (Sato, 2013). As the figure indicates, the measured thrust

force fluctuates depending on the position of the mover and is considered small because of the friction force. Maslan et

al. (2017) has clarified the friction force behavior (ranging from 5×10-3

to 12×10-3

N) in a comprehensive dynamic

model of the LSRM.

Fig. 2 Experiment setup including the LSRM prototype (Maslan et al., 2017).

Fig. 3 Static thrust force characteristic in terms of the mover position at 3.33 A.

3



3. Control system for precise tracking and its motion performance

3.1 Motion control system structure

Figure 4 shows the structure of the control system for precise tracking based on the positioning controller (Maslan

et al., 2017) with additional control elements. The control system for precise positioning employs a

proportional-integral-derivative (PID) compensator that uses an anti-windup scheme and a linearizer unit to suppress

the strong nonlinearity of the driving characteristics. Meanwhile, the additional control elements comprise a feed

forward (FF) element to compensate the dynamic characteristic and a disturbance observer (DOB) to reduce the

negative influence of the unknown disturbance force. In contrast to the positioning controller, the motion control

system does not have the feedback control element for compensating the damping characteristic since the FF has the

component for compensating it. The design procedure of the controller is simple and practical. However, they require a

dynamic model that can be represented using a simple linear mass–damper system. The combination of an FF element

with a feedback compensator results in a two degree-of-freedom (DOF) controller, which is effective in the tracking

control of high-precision motion systems (Hama and Sato, 2015). As mentioned in section 1, the robustness of the

motion controller with respect to changes in the mover is important to avoid a redesign of the controller. In addition to

reducing the negative influence of the disturbance force, a disturbance observer was introduced to achieve a robust

performance (Kurihara, 2010).

The design method used for the control system, shown in Fig. 4, is as follows:

(i) Design of PID controller with the linearizer unit: The PID controller with the linearizer unit is designed using the

same procedure and has the same parameters as the positioning controller (Maslan et al., 2017). The linearizer unit

is expressed as a function of the required force and mover position. Its output is determined based on the generated

thrust force and the friction force resulting from the normal force.

(ii) Design of FF element for the dynamic characteristic compensation: The FF element is added to the motion

controller. The FF element used in a conventional two-DOF controller is designed to compensate the dynamic

characteristic. An inverse model of the LSRM represented as a linear mass–damper system is used, and is described

below.

zT

zB

zT

zMPinv

112

2

(1)

where Pinv is the inverse model of the plant, T is the sampling time, M is the mass of mover, and B is the damping

coefficient, which is determined to be 1.1×10-2

(N∙s/m) from the open-loop dynamic responses of the mover. To

avoid the system to be over-compensated, the compensator for damping characteristic used in the positioning

controller is not adopted.

4

Signal

generator

Plant

Position

x

Saturation

unit

P(s)i

Linearizer

unit

imax at 3.33 A

effF

Conditional

integrator

Two-DOF controller

refx

pK

zT

zKd

1

1

z

zTK i

FF

Disturbance

observer

Fig. 4 Block diagram of the control system for precise tracking.



(a) Tracking error comparison of four controllers for

sinusoidal response: 3 mm, 0.25 Hz with respect to time

(b) Tracking error comparison of four controllers for

sinusoidal response: 3 mm, 0.25 Hz with respect to

position for forward motion

(c) Sinusoidal response: 3 mm, 2 Hz with respect to time (d) Ramp response: 3.75 mm/s with respect to position

0 1 2 3 4

-20

-10

0

10

20

Tracking error (Positioning controller)

Tracking error (Motion controller without DOB)

Tracking error (Motion controller without FF)

Tracking error (Motion controller)

Displacement

Time (s)

Tra

ckin

g e

rror

(m

)

-3

-2

-1

0

1

2

3

Dis

pla

cem

ent

(mm

)

-20-10

01020

Tra

ckin

g e

rror

(m

)

-3 -2 -1 0 1 2 3-303 Phase C

Position (mm)

-303 Phase B

Curr

ent

(A)

-303 Phase A

0.00 0.25 0.50 0.75 1.00

-8

-6

-4

-2

0

2

4

6

8

Time (s)

Tra

ckin

g e

rror

(m

)

-3

-2

-1

0

1

2

3

Dis

pla

cem

ent

(mm

)

0 1 2 3 4 5

-8

-6

-4

-2

0

2

4

6

8

Position (mm)

Tra

ckin

g e

rror

(m

)

(iii) Design a disturbance observer: To reduce the negative influence of the model errors and unknown disturbance

force applied to the mover, a disturbance observer is designed. This is a minimum-order discrete observer

(Friedland, 2005) based on the dynamic model expressed in Eq. (1) with the same parameter values as used in the

FF element. The pole of the observer is located at − 5000 ± 2j for fast convergence and minimum vibration.

It is expected that the friction and inertia forces would change with respect to the changes in the movers, and thus,

the motion performance was examined experimentally to determine the usefulness of this controller.

3.2 Experiment motion performance

Tracking experiments were conducted to evaluate the effectiveness of the control system described in section 3.1,

which is named the motion controller. This system was designed to achieve a precise motion performance. The control

sampling time was 0.1 ms, and the resolution of the linear encoder was set to 0.1 μm.

Figure 5 shows the tracking response of the motion controller. In order to examine the usefulness of the design

method, responses of the positioning controller, motion controller without DOB, and the motion controller without FF

were also measured with the sinusoidal input shown in Figs. 5(a) and (b). The gains of the PID element in all the

controllers are the same. The comparison of the results indicates that the absolute maximum tracking error of the

Fig. 5 Tracking response of the motion control system for precise tracking.

5



positioning controller and motion controller without DOB reach up to 20 μm. Comparing the tracking error in Fig. 5(b)

and the driving signal taken from Fig. 3(a), it can be seen that errors exist at the position where the phases are switched

owing to a high rate of change of the current, which induces vibrations (Krishnan R., 2001). The tracking error also

increases when the direction of motion changes. These results suggest that the motion was under the negative

influences of the thrust force ripple and disturbance force applied to the mover. In contrast, the maximum absolute

tracking error by the motion controller without FF and motion controller that include the DOB is drastically reduced to

approximately 5.9 and 2.8 μm, respectively. Although the errors caused by the high rate of change of current cannot be

completely eliminated by the controllers, the errors are reduced to a minimum using the DOB. Furthermore, the

tracking error of the motion controller without FF oscillates due to the inertial force that was not compensated. This

oscillation was eliminated using the FF element in the motion controller, showing the usefulness of the FF element.

Overall, the experimental results suggest the effectiveness of using both additional control elements to reduce the

tracking error.

Next, the responses to other reference motions were analyzed. Figure 5(c) shows that the tracking error is less than

2.7 μm for a sinusoidal input of 3 mm amplitude at 2 Hz, which is a considerably higher speed than those shown in Fig.

5(a). With a ramp input of 3.75 mm/s, the tracking error is less than 2 μm, as shown in Fig. 5(d). The repeated values of

the experimental results indicate that the maximum absolute tracking error is within 5 μm (data not shown). These

results indicate that the control system realizes a precise motion performance for the developed LSRM.

3.3 Influences of the changes in the movers

Further examinations were conducted on the robust characteristic of the motion controller against the influences of

the changes in the movers, which have different lengths and masses. As mentioned in section 2, the length of the mover

can be easily adjusted. A long mover (Fig. 6) was fabricated with the intention of achieving a wider working range than

that of the mover shown in Fig. 2(c). The length, l, and mass, M, of the mover are 80 mm and 0.67 g, respectively,

which is equivalent to a change in mass, ΔM, of 20%. Table 1 lists the specifications of the movers. In addition, a mass

of 0.55 g was added to the mover, shown in Fig. 2(c), which is equivalent to a change in mass, ΔM, of 100%. The

tracking experiments were conducted using these movers. Because the changes of the movers based on their different

lengths affect the change in mass and friction, for labeling purposes, the robustness of the controller was validated

simultaneously based on the respective mass-change percentage.

Prior to the tracking experiments, the change in friction force resulting from the mass change is discussed. The

coefficient of friction, μ, increases with the increase in the length of the sliding surface, which in turn increases the

frictional effect (Katano et al., 2014). Notably, the changes in mass incur a variation of μ. Figure 7 shows the measured

dependence of the coefficient of friction on the normal force using the movers listed in Table 1. This dependency was

Parameter Mover shown in

Fig. 2(c)

Long mover Unit

Mass of mover (including scale) 0.56 0.67 g

Mover length 66 80 mm

Mover width 10 10 mm

Mover thickness 0.128 0.128 mm

Mover pole width 1.20 1.20 mm

Mover pole pitch 2.40 2.40 mm

Table 1 Specifications of the movers.

Fig. 6 Fabricated long mover.

Magnetic material

for mover coreNon-magnetic

material film

Sensor scale

6



mathematically modelled by matching the analytical thrust force with the measured results in a comprehensive dynamic

model of the LSRM (Maslan et al., 2017). In this figure, the longer mover of length 80 mm for a 20% change in mass

shows an increase in μ, and the maximum change rate is approximately 15%. The maximum change rate is observed at

a low normal force. Overall, the variation of μ decreases with the increase in normal force. It is important to note that

these variations affect the change in friction at high normal force where the effective thrust force is at minimum.

However, our experiment results show that the maximum μ at a high normal force is smaller than the 3% increase with

respect to each change in mass. Meanwhile, validation regarding the change in the inertial force is further discussed in

section 4.1.

Figure 8 shows the comparative tracking response of the motion controller under the influences of the changes in

length and mass of the movers. Table 2 lists the maximum tracking errors with respect to the designated reference

motion. In general, the tracking errors increase as the change in mass increases. In all cases, the tracking errors were

observed to increase when the motion direction changed owing to the change in frictional and inertial effects that occur

with a change in mass.

The tracking results at the same amplitude of 3 mm but different frequencies show that the precision motion

performance remains the same, with the occurrence of tracking errors within 5 μm except at the frequency of 2 Hz for a

100% change in mass (Figs. 8(a)–(c)). There is a clear possibility that the change in mass at this particular reference

motion exceeds the limitation threshold, causing the maximum tracking error in Fig. 8(c) to be much larger than those

in Figs. 8(a) and (b). These results indicate the importance of clarifying the limitations of the performance for systems

with a small effective thrust force, such as that shown here. To support the claim regarding the cause of such errors

described in section 3.2, tracking experiments were conducted for a short distance of 0.1 mm amplitude, which is

shorter than at the position where the phases are switched. The maximum tracking error, shown in Fig. 8(d), is 1.20 μm,

which is considerably smaller than the error resulting from a longer travel distance. A smaller tracking error occurs

because the reference amplitude, determined in Fig. 8(d), avoids any vibration during the switching of phases, and

therefore strongly confirms the cause of errors as previously described.

Table 2 Tracking errors resulting from changes in length and mass of the movers.

Reference

motion

Maximum absolute tracking error (μm)

l = 66 mm

∆M = 0%

l = 80 mm

∆M = 20%

l = 66 mm

∆M = 100%

Sinusoidal

3 mm, 0.25 Hz 2.79 4.11 4.51

Sinusoidal

3 mm, 1 Hz 3.13 4.18 4.86

Sinusoidal

3 mm, 2 Hz 2.69 3.12 9.85

Sinusoidal

0.1 mm, 2 Hz 0.78 0.88 1.20

Fig. 7 Dependence of the coefficient of friction on the normal force.

0 10 20 30 40 50 60

0.2

0.3

0.4

0.5 Data Polynomial fit

M = 0% M = 0%

M = 20% M = 20%

M = 100% M = 100%

Normal force (mN)

7



(a) Sinusoidal response: 3 mm, 0.25 Hz (b) Sinusoidal response: 3 mm, 1 Hz

(c) Sinusoidal response: 3 mm, 2 Hz (d) Sinusoidal response: 0.1 mm, 2 Hz

0 1 2 3 4

-8

-6

-4

-2

0

2

4

6

8

Tracking error (M = 0%)



Displacement

Time (s)

Tra

ckin

g e

rror

(m

)

-3

-2

-1

0

1

2

3

Dis

pla

cem

ent

(mm

)

0.0 0.5 1.0 1.5 2.0

-8

-6

-4

-2

0

2

4

6

8

Time (s)

Tra

ckin

g e

rror

(m

)

-3

-2

-1

0

1

2

3

Dis

pla

cem

ent

(mm

)

0.00 0.25 0.50 0.75 1.00

-10

-5

0

5

10

Time (s)

Tra

ckin

g e

rror

(m

)

-3

-2

-1

0

1

2

3

Dis

pla

cem

ent

(mm

)

0.00 0.25 0.50 0.75 1.00

-8

-6

-4

-2

0

2

4

6

8

Time (s)

Tra

ckin

g e

rror

(m

)

-0.1

0.0

0.1

Dis

pla

cem

ent

(mm

)

4. Limitations of the motion performance

4.1 Analytical method based on the motor characteristics

To proceed with the discussion regarding the above limitations, the system should satisfy the condition in which

the friction variation can be predicted. This condition is met by evaluating the friction variation (as evident from

section 3.3), and the variation can be predicted through the variation of μ from Fig. 7. It should be noted that the

prototype of the LSRM has a fluctuating thrust force characteristic, as observed in Fig. 3(b), and the thrust force, which

can be provided independently of the position is limited. Hence, it is necessary to clarify the limitations of the motion

performance resulting from the motor characteristics. An additional mass for the mover is included to represent objects

being handled as the intended application. When the mass changes, ΔM is considered. Accordingly, the motor

characteristics described by Maslan et al. (2017) can be modified as follows.

frth FFxBxMM (2)

where x is the position of the mover. In addition, Fth and Ffr are the generated thrust force and the frictional force,

respectively. The generated forces, thrust force Fth and normal force Fnormal, depend on the position of the mover and

applied current and are calculated (Maslan et al., 2017) using a commercial 3D finite element analysis program

(Maxwell 3D, ANSYS, Inc.). Considering Fnormal, the frictional forces are given as follows.

Fig. 8 Comparative tracking response of the motion controller under the influences of the changes in the length and mass

of the movers.

8



vx

FgMMFF offsetsnormalfr

,max,

(3)

and

vx

FgMMFx

xF offsetknormalfr

,

(4)

where μ is the coefficient of friction, g is the acceleration due to gravity, and ∆v is the threshold of the static condition

set at 1.0×10-9

(m/s). The respective offsets of the frictional force are included in the equations, which are

characteristics of systems with nonlinear friction behavior (Do et al., 2015). Both the offsets of the static friction Fs,offset

(Fs,offset = 5.62×10-5

N) and the kinetic friction Fk,offset (Fk,offset = 2.26×10-5

N) are obtained from the friction-versus-load

behavior graph for the static and kinetic conditions, respectively. The values of these offsets are relatively small

compared with other parameters of the frictional forces (ranging from 5×10-3

to 12×10-3

N), calculated from the

difference obtained between the generated and effective thrust forces (Maslan et al., 2017). Because these offsets can be

assumed to be negligible, only the effect of change in μ depending on the normal force is discussed in this paper.

In this section, the limitations of the motion performance from the motor characteristic are examined and clarified

based on the responses to sinusoidal inputs because the responses are generally used for evaluating the system

performance and the sinusoidal inputs are basic signals for discussing it. To investigate the limitation using a sinusoidal

input, the motion equation for the simple harmonic motion is given as follows.

0tsin xAx (5)

Equation (5) can then be derived and substituted into Eq. (2). To drive the mover to within the limitation using a

sinusoidal input, the motor characteristic should satisfy the following condition.

xiFxiFABAMM fr ,, maxthmaxmax,

2

max

22 (6)

Here, Ffr,max μ is the friction force at maximum μ that is used in the analytical result, and imax is the maximum supply

current at 3.33 A. To find the maximum μ, the value is determined from the high normal force where the effective thrust

force is at minimum with respect to each change in mass. When the system satisfies Eq. (6), the system is able to

follow the reference motion. The condition in Eq. (6) is considered to include not only the change in the frictional

effect, but also the inertial effect depending on the change in mass. To represent the limitation threshold for the

precision motion performance, the maximum damping coefficient, Bmax, is used at 1.67×10-2

(N∙s/m), as determined

from the open-loop dynamic responses of the mover. Table 3 provides a summary of the analytical results from Eq. (6),

in which the mass of the mover is varied at different sinusoidal reference motion frequencies. As mentioned previously

in this section, the analytical results already consider the friction variation by predicting the variation from Fig. 7. For

all changes in mass, this result indicates that the maximum μ increases up to 5%, thereby satisfying the condition in Eq.

(6), and is much larger than the experimental maximum value of μ under a high normal force. By observing Table 3, it

is clear that the condition in Eq. (6) is satisfied (S) under the motion of the sinusoidal response with 3 mm amplitude at

a frequency of 2 Hz for a change in mass of up to 50%. This percentage of change in mass represents the limitation

threshold for the precision motion performance. Any change in mass exceeding this limitation threshold with respect to

the same reference motion is considered to not satisfy (NS) the condition in Eq. (6).

Table 3 Analytical results of Eq. (6) at different sinusoidal reference motion frequencies.

Reference motion ∆M = 0% ∆M = 50% ∆M = 60% ∆M = 100%

Sinusoidal 3 mm, 0.25 Hz S S S S

Sinusoidal 3 mm, 1 Hz S S S S

Sinusoidal 3 mm, 2 Hz S S NS NS

S Satisfied NS Not satisfied

9



(a) Sinusoidal response: 3 mm, 2 Hz with respect to

time

(b) Sinusoidal response: 3 mm, 2 Hz with respect

to position for the first forward motion

0.00 0.25 0.50 0.75 1.00

-10

-5

0

5

10



Displacement

Time (s)

Tra

ckin

g e

rror

(m

)

-3

-2

-1

0

1

2

3

Dis

pla

cem

ent

(mm

)

-3 -2 -1 0 1 2 3

-8

-6

-4

-2

0

2

4

6

8

Position (mm)

Tra

ckin

g e

rror

(m

)

Moreover, to drive the mover slowly, within the limitation threshold, the condition shown in Eq. (6) can be

simplified as follows.

xiFxiFfr ,, maxthmaxmax, (7)

It was found that, by solving Eq. (7) analytically for a slow ramp-reference motion, the change in mass should not

exceed 100%. This condition in Eq. (7) also considers the variation in friction by predicting the variation shown in Fig.

7. Analytical results not satisfying Eqs. (6) and (7) were found at the position where the effective thrust force is at

minimum (Fig. 3(b)).

4.2 Experimental validation

Figure 9 shows the comparative tracking responses of the motion controller under performance limitations using a

sinusoidal input. These responses were measured to verify the limitation described in section 4.1. For the motion of the

sinusoidal response of 3 mm amplitude at a frequency of 2 Hz, the performance was maintained with up to a 50%

change in mass as determined analytically. A slight increase in the overall mass resulting from the additional mass was

included in the comparison to visualize the effects of the limitation, as shown in Fig. 9(a). The experiment results

indicate that the motion performance could not be maintained if the change in mass exceeded the limitation threshold,

which did not satisfy the condition in Eq. (6) as determined analytically in section 4.1. A comparison between Figs.

3(b) and 9(b) shows that the tracking error peaks at the position where the effective thrust force is at minimum,

regardless of whether the motion is accelerating or decelerating.

Moreover, the tracking responses were measured to analyze the limitation of the motion performance for slow

movement, as shown in Fig. 10. To validate this limitation, a ramp input was applied. As determined analytically in

section 4.1, the mover tracks the slow ramp input for up to a 100% change in mass. Accordingly, a slight increase in the

overall mass resulting from the additional mass was also compared. From Fig. 10(a), it can be seen that similar tracking

errors were observed throughout the course of the motion, but a sudden tracking error peaked at the midpoint for a

mass change of 110%. To show the results of the motion performance under this condition, the accuracy of the repeated

tracking is included in Fig. 10(b). Of the ten measured responses, the mover stopped abruptly for four responses. It is

evident that the effective thrust force could not support a change in mass exceeding 100%.

Fig. 9 Comparative tracking responses of the motion controller under the limitation of the motion performance for a

sinusoidal input.

10



(a) Ramp response: 3.75 mm/s with respect to position (b) Accuracy of the repeated tracking

0 1 2 3 4 5

-5

0

5

10

15

M = 100% M = 110%

Position (mm)

Tra

ckin

g e

rror

(m

)

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

M = 100% M = 110%

Iterations

Max

imum

abso

lute

trac

kin

g e

rror

(m

)

∆M = 110% stops abruptly

These experiment results are in good agreement with the analytical results obtained in section 4.1. Hence, for a

limited range of additional mass, the precision motion performance can be effectively maintained. Under the

performance limitations, the maximum percentages of the changes in mass for a sinusoidal input (3 mm, 2 Hz) and for

slow movement are 50% and 100%, respectively.

5. Conclusion

To summarize, a motion system design for precise tracking of the developed LSRM was presented. The motor used

is easy to fabricate and the mover is disposable. To increase the usability of this motion system, a control system for

precise tracking was introduced. Two control elements were added to the developed positioning controller: a

feedforward element and a disturbance observer. The experiment results indicate that the control system has a tracking

error of less than 5 μm. Even when the length and mass of the movers change, the precision motion performance

remains the same, given that the additional mass does not exceed the limitation threshold. Hence, the motion system

design can be initially presented by providing a controller that is robust to changes of the movers. Useful analytical

conditions that consider the influence of friction force including its variation were derived for the limitation threshold

determined from both the variation of the damping and the frictional effects. The experiment results regarding the

limitations of the motion performance are in excellent agreement with the formulated analytical results using the

limitation threshold. Under these limitations, the maximum percentages of the changes in mass for a sinusoidal input (3

mm, 2 Hz) and for slow movement are 50% and 100%, respectively. Overall, the tracking results indicate that the

performance is maintained under the conditions determined based on the motor characteristics even when the length of

the mover is changed. Using the knowledge from these results, the mover of the developed LSRM can be changed in

terms of length and mass while maintaining the same motion performance, thereby meeting the required objectives.

Acknowledgments

This work was supported by a fund from the Mikiya Science and Technology Foundation. One of the authors,

Mohd Nazmin Maslan, would like to extend his gratitude to Universiti Teknikal Malaysia Melaka (UTeM) and the

Ministry of Higher Education Malaysia for funding his studies.

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Fig. 10 Comparative tracking responses of the motion controller under the limitation of the motion performance for slow

movement.

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motion system design of a thin and compact linear switched

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