Design of Small Permanent-Magnet Linear Motors and Drivers for Automation Applications with S-Curve Motion Trajectory Control and Solutions for End Effects and Cogging Force

Abstract

This paper designs and fabricates a small-type permanent-magnet linear motor and driver for automation applications. It covers structural design, magnetic circuit analysis, control strategies, and hardware development. Magnetic circuit analysis software JMAG is used for flux density distribution, back electromotive force (back-EMF), and electromagnetic force analysis. To address the lack of a complete closed magnetic circuit path at the ends of the linear motor, which causes magnetic field asymmetry, a phenomenon known as end effects, auxiliary core structures are proposed to compensate for the magnetic field at the ends. It successfully utilizes auxiliary cores to achieve the phase voltages of each phase, which are balanced at a phase voltage error of 0.02 V. To address the cogging force caused by variations in the magnetic reluctance of the core, this paper analyzes the relationship between electromagnetic force and mover position, conducting harmonic content analysis to obtain parameters. These parameters are applied to the designed cogging force control compensation strategy. It successfully achieves q-axis current compensation of around 1.05 A based on the mover’s position, ensuring that no jerking caused by cogging force occurs during closed-loop electromagnetic force control. The S-curve motion trajectory control is proposed to replace the traditional trapezoidal acceleration and deceleration, resulting in smoother position control of the linear motor. Simulations using JMAG-RT models in MATLAB/Simulink verified these control strategies. After verification, practical test results showed a maximum position error of approximately 5.0 μm. Practical tests show that the designed small-type permanent-magnet linear motor and its driver provide efficient, stable, and high-precision solutions for automation applications.

1. Introduction

The structure of a permanent-magnet linear motor mainly consists of a stator and a mover, corresponding to the coil side and the magnet side, respectively. The coil side is further divided into iron-core linear motors [1] and ironless linear motors [2]. Iron-core linear motors, due to the presence of an iron core in their structure, possess higher electromagnetic force and lower manufacturing costs, making them particularly favored in applications requiring high thrust. However, the disadvantages include heavier weight and the presence of cogging force [3], which leads to unsmooth motion, vibration, and reduced precision. Ironless linear motors, on the other hand, do not contain an iron core, thereby eliminating cogging force, resulting in smoother motion and reduced vibration. Additionally, the ironless design makes the motor lighter, suitable for applications requiring lightweight equipment. However, this design typically has lower electromagnetic force and higher manufacturing costs. Cogging force can be addressed by analyzing the electromagnetic force of the linear motor model and experimentally deriving a compensation function to be incorporated into the control system for improvement [4,5]. Furthermore, due to the asymmetrical magnetic field distribution at the ends of the linear motor, end effects [6,7] occur, which cause imbalance in the back-EMF and reduce electromagnetic force output while increasing jitter. To effectively address this issue, auxiliary teeth [8,9,10,11] can be added to both ends of the iron core on the coil side, which helps enhance the symmetry of the magnetic field and improve overall motion stability and efficiency.

The control of permanent-magnet linear motors can be performed through open-loop control by adjusting the voltage and frequency of the inverter [12]. Pure inverter open-loop control can essentially verify whether the linear motor is operating normally, but it has lower control accuracy, slower response, and higher power consumption, making it unsuitable for automated applications requiring high precision. Further improvement can be achieved by using sensorless field-oriented control (FOC), which enhances control accuracy [13,14,15,16]. However, sensorless FOC imposes certain requirements on speed estimation, and due to the smaller back-EMF during motor startup and at low speeds, speed estimation becomes more challenging, leading to significant differences in control stability compared to high-speed operation. This makes it unsuitable for automated applications with variable conditions. Therefore, the most widely used and highest-precision control method currently is FOC with position or speed feedback sensors [17,18].

To meet the stringent requirements for precise control in industrial automation, significant progress has been made in recent years in enhancing acceleration control. These innovative methods aim to mitigate the jitter caused by rapid changes in acceleration, thereby improving the stability and efficiency of linear motor operation. These methods are based on a core principle that by integrating an acceleration control layer into speed control, the changes in the speed curve can be managed more precisely. Similarly, to further stabilize acceleration control, a jerk control mechanism has been introduced [19]. A common approach involves pre-calculating the trajectory path and inputting it into the controller to control the motor [20,21], as well as using specific algorithms that account for particular curvatures [22,23]. This multi-layered control strategy not only enhances control precision and smoothness but also effectively reduces energy waste and mechanical wear during system operation, providing higher operational efficiency and reliability for automated control applications.

In this paper, Section 2 introduces the design and analysis results of a miniature permanent-magnet linear motor, where the end effect is improved by using auxiliary core. It also conducts harmonic analysis of the electromagnetic force to facilitate compensation in subsequent control, and measures the back-EMF of the physical motor. Section 3 proposes a control strategy to mitigate cogging force and introduces an S-curve motion trajectory control strategy suitable for automation applications. Finally, MATLAB/Simulink is used to simulate and verify the feasibility and effectiveness of the control strategy. Section 4 presents the system architecture of the motor driver and the encoder feedback system for the multi-pole magnetic strips, followed by experimental verification of the previously mentioned control strategies.

2. Design of Small-Type Permanent-Magnet Linear Motor

2.1. Structure of Permanent-Magnet Linear Motor

The motor selected in this document is a three-phase motor with six coils and 7-pole magnets. It features a fractional slot design, which produces a back-EMF waveform closer to an ideal sine wave, thereby reducing harmonic components and improving efficiency. The magnets and their cores are selected as the mover, while the coils and their cores are used as the stator, paired with an 8 mm high stainless steel miniature linear rail. The slide rail serves as the mover and is mounted on the magnet-side housing, while the two sliders act as the stator and are fixed together with the coil-side housing on the motor housing. The cross-sectional view of the motor structure is shown in Figure 1.

The electrical angle of each coil differs by 210°, and the electrical angle distribution is shown in

Table 1. Electrical angle distribution for each coil in the three-phase motor with 6 coils and 7-pole magnets.

Number of Coil	#1	#2	#3	#4	#5	#6
Electrical angle	0°	210° (30°)	60° (240°)	270°	120°	330° (150°)
Number of phases	a+	a−	b−	b+	c+	c−

. The winding phase sequence and direction are determined according to the electrical angle distribution, with #2, #3, and #6 being reverse-connected to reverse the current direction. The wiring and vector diagram are shown in Figure 2.

The permanent-magnet linear motor described in this document is designed with a focus on miniaturization, high speed, and high acceleration. The maximum movement speed is 1.0 m/s, and the total travel distance is 80 mm. After analysis, the specifications with the least back-EMF harmonic content were selected. The schematic diagram of the motor dimensions is shown in Figure 3, and detailed motor specifications and dimensions are provided in

Table 2. Motor material and dimensions.

Mover	trapezoidal magnet	material	sintered NdFeB magnet N35
		width of the long side $W{1}_{mag}$	8
		width of the short side $W{2}_{mag}$	7
		height $H_{mag}$	4
		pole pitch $z_p$	10
	core	material	silicon steel sheet 50CS600
		length $L_{mc}$	200
		width $W_{mc}$	17
		height $H_{mc}$	10
air gap $g_a$			1
Stator	coil	material	polyurethane enamelled copper wire
		wire diameter $d_w$	0.25
		length $L_{coil}$	23
		width $W_{coil}$	10.67
		height $H_{coil}$	10
	core	material	silicon steel sheet 50CS600
		length $L_{sc}$	86.33
		width $W_{sc}$	17
		height $H_{sc}$	18
		front iron height $H_{scf}$	2
		back iron height $H_{scb}$	6

The unit of length is millimeters (mm).

.

2.2. Auxiliary Core to Improve End Effects

To address the end effects, auxiliary core structures are proposed to compensate for the asymmetric magnetic field at the ends of the linear motor shown in Figure 3. Simulations and analyses of the back-EMF were conducted at a speed of 1.0 m/s with 80 turns per coil. The results, shown in

Table 3. Adding auxiliary core back-EMF analysis.

Auxiliary Core		Without	With
Phase a back-EMF	fundamental (V)	3.735	3.619
	THD (%)	3.93	0.58
Phase b back-EMF	fundamental (V)	3.929	3.621
	THD (%)	0.68	0.40
Peak back-EMF difference between phase a and b (V)		−0.194	−0.002

, indicate that the three-phase back-EMF is balanced, with a peak voltage difference of 0.002 V between phase a and phase b.

2.3. Electromagnetic Force Analysis and Improvement

A current control analysis was conducted with a peak current of 1 A per phase, resulting in an average electromagnetic force of 5.46 N, as shown in Figure 4. The harmonic components are predominantly even-order harmonics, and after improvement, the electromagnetic force increased and the harmonic content significantly improved. The harmonic amplitude of the electromagnetic force and its phase angle after improvement are shown in

Table 4. Harmonic amplitude of electromagnetic force and its phase angle.

Harmonic Order	Harmonic Amplitude (N)	Phase Angle (°)
2	6.05	119.7
4	0.42	238.4
6	0.21	198.7
8	0.08	−53.6

, and these data will serve as key parameters for cogging force improvement in the control strategy discussed in this document.

3. Control Strategy and Simulation of the Small-Type Permanent-Magnet Linear Motor

3.1. Electromagnetic Force Control Strategy After Cogging Force Improvement

As shown in Figure 5, cogging force improvement can be achieved by adding the ripple component of the cogging force into the q-axis current command $i_q^{*}$. The relationship between the electromagnetic force command $F_e^{*}$ and the electromagnetic force constant $k_F$ determines the dq-axis current commands $i_d^{*}$ and $i_q^{*}$:

$$i_d^{*}=0$$

(1)

$$i_q^{*}={\displaystyle \frac{1}{k_F}}(F_e^{*}-{\tilde{F}}_r)$$

(2)

where ${\tilde{F}}_r$ is the electromagnetic force harmonic function, and its function is given by:

$${\tilde{F}}_r=\sum {\tilde{F}}_{rn}sin(n{\theta }_r+{\theta }_{Frn})$$

(3)

${\tilde{F}}_{rn}$ is the electromagnetic force harmonic amplitude, where n is the harmonic order, ${\theta }_r$ is the mover’s magnetic pole angle and ${\theta }_{Frn}$ is the harmonic phase angle. Based on the data in Table 4, the second, fourth, and sixth harmonics are incorporated into the electromagnetic force ripple compensation ${\tilde{F}}_r({\widehat{\theta }}_r)$ as follows:

$$\begin{array}{c}\hfill {\tilde{F}}_r({\widehat{\theta }}_r)={\tilde{F}}_2+{\tilde{F}}_4+{\tilde{F}}_6\\ \hfill ={\tilde{F}}_{r2}sin(2{\widehat{\theta }}_r+{\theta }_{Fr2})\\ \hfill +{\tilde{F}}_{r4}sin(4{\widehat{\theta }}_r+{\theta }_{Fr4})\\ \hfill +{\tilde{F}}_{r6}sin(6{\widehat{\theta }}_r+{\theta }_{Fr6})\end{array}$$

(4)

Based on the analysis data in Table 4, ${\tilde{F}}_{r2}$ = 6.05 N, ${\tilde{F}}_{r4}$ = 0.42 N, ${\tilde{F}}_{r6}$ = 0.21 N, ${\theta }_{Fr2}$ = 119.7°, ${\theta }_{Fr4}$ = 238.4° and ${\theta }_{Fr6}$ = 198.7°. Using the three-phase current feedback ${\widehat{i}}_a$, ${\widehat{i}}_b$ and ${\widehat{i}}_c$, the dq-axis current feedback ${\widehat{i}}_d$ and ${\widehat{i}}_q$ are obtained through coordinate transformation, and the dq-axis current errors $\Delta i_d$ and $\Delta i_q$ in the mover coordinate system are calculated as follows:

$$\Delta i_d=i_d^{*}-{\widehat{i}}_d$$

(5)

$$\Delta i_q=i_q^{*}-{\widehat{i}}_q$$

(6)

The ideal functions $G_d(z)$ and $G_q(z)$ of the dq-axis current regulators in the mover coordinate system in the z-domain are as follows:

$$G_d(z)=k_{pd}(1+k_{id}{\displaystyle \frac{{\tau }_s}{z-1}})$$

(7)

$$G_q(z)=k_{pq}(1+k_{iq}{\displaystyle \frac{{\tau }_s}{z-1}})$$

(8)

where $k_{pd}$ and $k_{id}$ are the proportional–integral gains for the d-axis current controller, $k_{pq}$ and $k_{iq}$ are the proportional–integral gains for the q-axis current controller, and ${\tau }_s$ is the controller sampling time. The output of the dq-axis current regulators in the mover coordinate system is as follows:

$$u_d^{*}=G_d\circ \Delta i_d$$

(9)

$$u_q^{*}=G_q\circ \Delta i_q$$

(10)

where $u_d^{*}$ and $u_q^{*}$ are the outputs of the dq-axis current regulators. The dq-axis voltage commands $v_d^{*}$ and $v_q^{*}$ in the mover coordinate system are as follows:

$$v_d^{*}=u_d^{*}-{\omega }_r{L}_s{i}_q$$

(11)

$$v_q^{*}=u_q^{*}+{\omega }_r(L_s{i}_d+{\lambda }_m^{\prime })$$

(12)

where ${\omega }_r$ is the mover’s electrical angular frequency, $L_s$ is the equivalent inductance of each phase winding in the permanent-magnet linear motor and ${\lambda }_m^{\prime }$ is the flux linkage of the mover’s magnet equivalent to the stator. Finally, $v_d^{*}$ and $v_q^{*}$ are transformed into the stationary coordinate system’s abc-axis voltages $v_a^{*}$, $v_b^{*}$ and $v_c^{*}$, which are then output via VSVPWM to the inverter, thereby driving the linear motor.

3.2. S-Curve Motion Trajectory Control Strategy

The mover’s position trajectory $Z_{m(traj)}$, speed trajectory ${\upsilon }_{m(traj)}$, acceleration trajectory $a_{m(traj)}$, and jerk trajectory $j_{m(traj)}$ in the S-curve motion trajectory are shown in Figure 6a. The main principle is to add a constant jerk to the acceleration, forming a trapezoidal acceleration curve, which in turn creates an S-curve speed profile. This optimization prevents sudden jerks during acceleration or deceleration in traditional trapezoidal motion trajectory control shown in Figure 6b, resulting in smoother and more stable position control compared to traditional acceleration and deceleration methods.

The mover position command $Z_m^{*}$ first enters the S-curve motion trajectory to calculate the mover position trajectory command $Z_{m(traj)}^{*}$, with the trajectory control block diagram shown in Figure 7. This process involves first calculating the braking distance for the current speed, then sequentially calculating the maximum allowable speed and maximum allowable acceleration, before entering the acceleration limiter and speed limiter, and finally computing the position trajectory.

The output and judgment formula for the braking distance $Z_{m(brake)}$ are as follows:

$$Z_{m(brake)}=\left\{\begin{array}{cc}{\displaystyle \frac{{\upsilon }_{m(max)}(\frac{{\upsilon }_{m(max)}}{a_{m(max)}}+\frac{a_{m(max)}}{j_m})}{2}}\hfill & \mathrm{if}{\upsilon }_{m(max)}>{\displaystyle \frac{a_{m(max)}^2}{j_m}}\hfill \\ {\displaystyle \frac{{\upsilon }_{m(max)}\sqrt{j_m{\upsilon }_{m(max)}}}{j_m}}\hfill & \mathrm{if}{\upsilon }_{m(max)}\le {\displaystyle \frac{a_{m(max)}^2}{j_m}}\hfill \end{array}\right.$$

(13)

where ${\upsilon }_{m(max)}$ is the maximum mover speed, and $a_{m(max)}$ is the maximum mover acceleration. The output and judgment formula for the maximum allowable speed ${\upsilon }_{m(allow)}$ are as follows:

$${\upsilon }_{m(allow)}=\left\{\begin{array}{cc}\sqrt[3]{{\displaystyle \frac{j_m\Delta z_{m(error)}^2}{4}}}\hfill & \mathrm{if}|\Delta Z_{m(error)}|<Z_{m(brake)}\hfill \\ {\upsilon }_{m(max)}\hfill & \mathrm{if}|\Delta Z_{m(error)}|\ge Z_{m(brake)}\hfill \end{array}\right.$$

(14)

where $\Delta Z_{m(error)}$ is the mover position trajectory error. The current speed direction is then determined based on $\Delta Z_{m(error)}$:

$${\upsilon }_{m(allow)}=\left\{\begin{array}{cc}{\upsilon }_{m(allow)}\hfill & \mathrm{if}\Delta Z_{m(error)}>0\hfill \\ -{\upsilon }_{m(allow)}\hfill & \mathrm{if}\Delta Z_{m(error)}\le 0\hfill \end{array}\right.$$

(15)

The output of the maximum allowable acceleration $a_{m(allow)}$ is as follows:

$$a_{m(allow)}=\sqrt{2j_m\Delta {\upsilon }_{m(traj)}}$$

(16)

where $\Delta {\upsilon }_{m(traj)}$ is the speed trajectory error. The current acceleration direction is then determined based on $\Delta {\upsilon }_{m(traj)}$, and the comprehensive output and judgment formula are as follows:

$$a_{m(allow)}=\left\{\begin{array}{cc}\sqrt{2j_m\Delta {\upsilon }_{m(traj)}}\hfill & \mathrm{if}\Delta {\upsilon }_{m(traj)}>0\hfill \\ -\sqrt{2j_m\Delta {\upsilon }_{m(traj)}}\hfill & \mathrm{if}\Delta {\upsilon }_{m(traj)}\le 0\hfill \end{array}\right.$$

(17)

The position trajectory variation $\Delta Z_{m(traj)}$ is as follows:

$$\Delta Z_{m(traj)}={\upsilon }_{m(traj)}\Delta t$$

(18)

where $\Delta t$ is the controller operation cycle. The position trajectory $Z_{m(traj)}^{*}$ is as follows:

$$Z_{m(traj)}^{*}={\widehat{Z}}_{m(traj)}+\Delta Z_{m(traj)}$$

(19)

where ${\widehat{Z}}_{m(traj)}$ is the position trajectory feedback. The mover position error $\Delta Z_m$ is as follows:

$$\Delta Z_m=Z_{m(traj)}^{*}-{\widehat{Z}}_m$$

(20)

where ${\widehat{Z}}_m$ is the mover position feedback. The mover position regulator function $G_z$ is as follows:

$$G_z(z)=k_{pz}(1+k_{iz}{\displaystyle \frac{{\tau }_s}{z-1}}+k_{dz}{\displaystyle \frac{z-1}{{\tau }_s}})$$

(21)

where $k_{pz}$, $k_{iz}$ and $k_{dz}$ are the proportional–integral–derivative (PID) controller gains for mover position control. The output of the mover position regulator $F_e^{*}$ is as follows:

$$F_e^{*}=G_z\circ \Delta Z_m$$

(22)

where $F_e^{*}$ is the electromagnetic force command. By inputting $F_e^{*}$ into the current closed loop, the position closed-loop control is completed, as shown in Figure 8.

3.3. Control Strategy Simulation

3.3.1. Electromagnetic Force Control Simulation

Based on the electromagnetic force control strategy described in Section 3.1 (before compensation), a MATLAB/Simulink simulation model was established. When an external force drives the linear motor in this document at a speed of 1.0 m/s, and the electromagnetic force command is controlled at $F_e^{*}$ = 5.46 N, the current and magnetic force responses are shown in Figure 9. The q-axis current feedback ${\widehat{i}}_q$ = 1 A, and the electromagnetic force exhibits a stable ripple, consistent with the JMAG simulation, with a magnitude of approximately ±6 N.

3.3.2. Cogging Force Improvement Simulation

Based on the cogging force improvement strategy described in Section 3.1, a MATLAB/Simulink simulation model was established. When an external force drives the linear motor in this document at a speed of 1.0 m/s, and the electromagnetic force command is controlled at $F_e^{*}$ = 5.46 N, the current and magnetic force responses are shown in Figure 10. The electromagnetic force ripple is significantly reduced to approximately ±0.5 N.

3.3.3. S-Curve Motion Trajectory Control Simulation

Based on the S-curve motion trajectory control strategy described in Section 3.2, a MATLAB/Simulink simulation model was established. When the mover position command is controlled at $Z_m^{*}$ = 80 mm, the position and speed responses are shown in Figure 11. The speed trajectory exhibits an S-curve motion, the position feedback nearly overlaps with the position trajectory, and there is a slight deviation between the speed trajectory and the speed feedback curve.

4. System Testing

4.1. Drive System Structure

This paper uses the Texas Instruments (TI) C2000 series 32-bit microcontroller (MCU) TMS320F280 0045 as the control core and writes the control program in C using Code Composer Studio (CCS). The enhanced pulse width modulator (ePWM) within the MCU controls the switching circuits of the three-phase inverter. A 12-bit analog-to-digital converter (ADC) measures voltage and current feedback. The enhanced quadrature encoder pulse (eQEP) and serial peripheral interface (SPI) measure feedback from the multi-pole magnet strip, with the absolute position of the multi-pole magnet strip calibrated using the ADC. A controller area network (CAN) provides control for external devices. The configuration of the small-type permanent-magnet linear motor includes the stator, mover, and multi-pole magnet strip. The small-type permanent-magnet linear motor driver block diagram is shown in Figure 12.

The magnetic encoder paired with the multi-pole magnet strip outputs at 12-bit resolution, with a resolution of 0.488 μm/step and a maximum linear movement speed of 2345 mm/s. Additionally, the linear Hall effect sensor DRV5055A1 is used to sense the absolute position of the multi-pole magnet strip. The final physical implementation of the driver is shown in Figure 13.

4.2. End Effect Improvement Testing

Based on the specifications of the small-type permanent-magnet linear motor described earlier, the physical implementation is shown in Figure 14. The measured motor parameters are as follows. The equivalent resistance $R_s$ of each phase winding is 3.054, 2.985, and 2.976 $\mathrm{\Omega }$, with an average of approximately 3.0 $\mathrm{\Omega }$; the equivalent inductance $L_s$ of each phase winding is 1.992, 1.967, and 1.982 mH, with an average of approximately 1.98 mH.

The small-type permanent-magnet linear motor pull-test platform is shown in Figure 15. It is used to measure the motor’s back-EMF or motor loading. A pull-test was conducted at $f_e$ = 25 Hz (${\upsilon }_m$ = 0.5 m/s), and the back-EMF phase voltage of the generator side was measured as shown in Figure 16. The peak value of the back-EMF phase voltage $E_m$ = 1.64 V, the phase voltages of each phase are balanced at a phase voltage error of 0.02 V, and the flux linkage of the mover magnets equivalent to the stator ${\lambda }_m^{\prime }$ was

$${\lambda }_m^{\prime }={\displaystyle \frac{E_m}{{\omega }_r}}={\displaystyle \frac{E_m}{2\pi f_e}}$$

(23)

According to Equation (23), ${\lambda }_m^{\prime }$ is calculated to be 0.0104 V/(rad/s).

4.3. Cogging Force Improvement Testing

After removing the generator side from the small-type permanent-magnet linear motor pull-test platform, a fixed pulley was installed and secured to the platform. A steel wire with a diameter of 0.3 mm was used to connect one end to the tail of the linear motor, while the other end was passed through the fixed pulley and hung vertically downward. Weights were added as needed for the test. The electromagnetic force test platform for the linear motor is shown in Figure 17.

Before implementing the cogging force improvement strategy, an external load greater than the maximum cogging force was required to move the linear motor. After implementing the cogging force improvement strategy, the linear motor can move smoothly with the application of an external load. When the cogging force improvement strategy is applied with position closed-loop control, the maximum load is 4.90 N. When the cogging force is added and $F_e^{*}$ = 0.0 N is controlled, the q-axis current feedback ${\widehat{i}}_q$ for the externally applied push force adjusts with the mover position to provide approximately 1.05 A of current compensation, as shown in Figure 18, with a 0.07 A error from the simulation results.

4.4. S-Curve Motion Trajectory Control Testing

Based on the S-curve motion trajectory control strategy proposed in Section 3.2, linear motor position closed-loop control tests were conducted. With a maximum speed of 0.3 m/s, maximum acceleration of 3 m/s², and maximum jerk of 300 m/s³, the motor was moved from 20 mm to 100 mm, then back to 20 mm, as shown in Figure 19. After reaching steady-state, the maximum position error was about 5.0 μm, and when the speed reached stability, the speed fluctuation was about 0.03 m/s.

4.5. Simulation and Testing Comparison

According to the simulation and testing comparison in

Table 5. Simulation and testing comparison.

Improvement Items	Comparison Items	Simulation	Testing
End effect improvement	${\lambda }_m^{\prime }$ (V/(rad/s))	0.0115	0.0104
	a phase voltage balance error (V)	0.002	0.020
Cogging force improvement	q-axis current compensation (A)	±1.12	±1.05
	result	±0.5 N electromagnetic force ripple	Able to move smoothly with any load
S-curve motion trajectory control	steady-state speed error (m/s)	0.02	0.03
	steady-state position error (μm)	60	5

, for the improvement of end effects, although the error in the testing results is slightly larger than that in the simulation results, the value still demonstrates a significant improvement in end effects. Regarding cogging force improvement, both the simulation and testing results show a q-axis compensation current slightly above 1A. The smooth movement of the linear motor during no-load testing confirms that the proposed improvement strategy is effective. In the S-curve motion trajectory control, although there is a slight speed error, since the control in this paper is position closed-loop control, the position error is more critical. Therefore, the position error observed in the test was only 5 μm, showing that the proposed trajectory control is highly accurate.

5. Conclusions

This paper designed and fabricated a small-type permanent-magnet linear motor and its driver for automation applications. It successfully compensates for the magnetic field asymmetry caused by the end effects of the linear motor using auxiliary core structures, achieving a phase voltage balance error of 0.02 V. Based on the harmonic content analysis of the electromagnetic force and mover position before compensation, a cogging force control compensation strategy was designed, ultimately achieving q-axis current compensation of approximately 1.05 A, ensuring no jerking caused by cogging force occurs during closed-loop electromagnetic force control. The S-curve motion trajectory control successfully enhanced the smoothness and accuracy of the linear motor’s position control. Test results showed a maximum position error of approximately 5.0 μm.

The future research directions include two key points. First, to address the cogging effect, while this paper proposes control strategies to compensate for electromagnetic force ripples, this increases current consumption. Future work could explore using new magnetic materials to reduce the cogging effect, improving motor performance while reducing size and weight. Second, it is recommended to add a push–pull force gauge to the test platform to accurately measure the motor’s thrust and pull. Real-time data from the force gauge will help optimize electromagnetic force control strategies and further enhance system stability and reliability.