Linear Vernier actuator with two movers
 Authors: Heya A.^{1}, Hirata K.^{1}, Niguchi N.^{1}

Affiliations:
 Osaka University
 Issue: Vol 6, No 1 (2020)
 Pages: 6379
 Section: Original papers
 URL: https://journals.ecovector.com/transsyst/article/view/21195
 DOI: https://doi.org/10.17816/transsyst2020616379
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Abstract
Background: Linear motion devices for industrial machines and robots are expected to realize their high efficiency drive and simple structure. Usually, a feed screw mechanism composed of a rotary motor and a ballscrew or slidescrew is employed. However, it has some problems such as the decrease of the drive efficiency, flexibility against external forces, noise, etc. Various linear actuators and motors have been developed utilizing the feature of a direct drive.
Aim: In this paper, we propose a novel linear actuator which 2 movers can be independently controlled using 3phase and 6phase superimposed currents for decreasing the size and weight of the system. The proposed linear actuator is driven by the operating principle of a vernier motor which is expected to achieve a high thrust force density per permanent magnet volume.
Methods: The operating principle and the static thrust force characteristics of the proposed linear actuator are verified by an electromagnetic field analysis using 3D finite element method, and the back electromotive force characteristics are also analyzed. In addition, the dynamic characteristics under position feedback control are analyzed. The control system uses a vector control using PID controller, and the control input is given by the 3phase and 6phase superimposed currents.
Results: The static force characteristics were investigated. From the analyzed results, the force interference between the two movers was small. Moreover, the interference of the back electromotive force of the 3phase and 6phase movers were not observed. The movers could be independently driven under position feedback control using 3phase and 6phase superimposed currents. The dynamic characteristics analyses showed that the mover well followed a target position. From a step response, the time constant and the response of the position feedback system were investigated.
Conclusion: This paper presents a linear vernier actuator with two movers. The basic structure and operating principle of the actuator were described. Moreover, the static characteristics and the dynamic characteristics under position feedback control were analyzed. It was found that the movers can be independently driven.
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INTRODUCTION
Linear motion devices for industrial machines and robots are expected to realize their high efficiency drive and simple structure. Usually, a feed screw mechanism composed of a rotary motor and a ballscrew or slidescrew is employed. The mechanism utilizes frictional forces. Therefore, it has some problems such as the decrease of the drive efficiency, flexibility against external forces, noise, etc. In order to solve the problems, various linear actuators and motors have been developed utilizing the feature of a direct drive [1 3]. In particular, a linear electromagnetic actuator (LEA) has many advantages such as a high drive efficiency, reduction of a noise, maintenancefree operation, and highprecision positioning.
The LEAs can drive flexibly against external forces due to a magnetic spring effect. Therefore, the LEAs are expected to apply an artificial muscle for a humanoid robot. Nakata et al. developed upper and lower extremity robots using LEAs [4]. Fujimoto et al. developed a musculoskeletal biped robot driven by LEAs [5]. These systems need a lot of linear actuators and links, and have some problems such as the increase in size and weight.
In order to solve these problems, we propose a novel linear actuator with two movers which can be independently controlled. The proposed actuator is driven by the operating principle of a vernier motor which is expected to achieve a high thrust force density per permanent magnet volume. The movers can be independently driven using 3phase and 6phase superimposed currents.
In this paper, the force characteristics and dynamic performance of the proposed linear actuator are investigated. The force characteristics are computed by an electromagnetic field analysis using 3D finite element method. The dynamic characteristics under position feedback control are also verified.
LINEAR VERNIER ACTUATOR WITH TWO MOVERS
The basic structure of the proposed actuator is shown in Fig. 1 (a). The stator consists of coils, permanent magnets, and yokes. The permanent magnets and teeth are arranged at even intervals. All permanent magnets are magnetized in the yaxis direction. The winding direction are shown in Fig. (b). The movers shown in Fig. (c) consist of only a yoke. Therefore, the amount of permanent magnet does not increase. Moreover, the number of teeth of the movers is different from each other.
The operating principle is shown in Fig. 2. The proposed actuator is driven by the operational principle of a vienier motor [6]. A magnetomotive force is modulated due to a permeance distribution, and a modulated magnetic flux is generated.
Fig. 1. Basic structure of the proposed linear actuator
The magnetomotive force of the stator is defined as follows:
$F=\text{sin}\frac{2\pi {Z}_{s}}{L}p$ (1)
where Z_{s} is the number of pole pairs of the stator, L is the stator length, and p is the position of the mover. The fundamental component of the mover permeance distribution is defined as follows:
${R}_{p}=a+\text{sin}\frac{2\pi {Z}_{m}}{L}p$ (2)
where Z_{m} is the number of pole pairs of the mover, and a is the average permeance of the mover. The modulated magnetic flux in the air gap is defined as follows:
$\text{\phi}=F\xb7{R}_{p}=a\text{sin}\frac{2\pi {Z}_{s}}{L}p\frac{1}{2}\mathrm{cos}\left({Z}_{s}+{Z}_{m}\right)\frac{2\text{\pi}}{L}p+\frac{1}{2}\mathrm{cos}\left({Z}_{s}{Z}_{m}\right)\frac{2\text{\pi}}{L}p$ (3)
The proposed actuator is driven by synchronizing a moving magnetic field generated by the coils with a modulated magnetic flux. Therefore, the relationship of the number of teeth and order of the moving magnetic field generated by the coils is defined as follows:
$\left\{\begin{array}{c}{Z}_{s}+{Z}_{m}=\pm {O}_{c}\\ {Z}_{s}{Z}_{m}=\pm {O}_{c}\end{array}\right.$ (4)
Fig. 2. Operating principle of the proposed linear actuator
where O_{c} is the order of the moving magnetic field generated by the coils.
The armature current of the proposed actuator has 3phase and 6phase alternating currents (ACs), and each mover utilizes either of them to move. The 3phase AC is defined as follows:
$\left\{\begin{array}{c}\begin{array}{c}{I}_{U}=\sqrt{2}I\text{sin}\left(\frac{2\text{\pi}{Z}_{1}{p}_{1}}{L}\right)\\ {I}_{V}=\sqrt{2}I\text{sin}\left(\frac{2\text{\pi}{Z}_{1}{p}_{1}}{L}\frac{2\text{\pi}}{3}\right)\\ {I}_{W}=\sqrt{2}I\text{sin}\left(\frac{2\text{\pi}{Z}_{1}{p}_{1}}{L}\frac{4\text{\pi}}{3}\right)\end{array}\end{array}\right.$ (5)
where Z_{1} and p_{1} is the number of salient pole pairs and position of the mover driven by the 3phase AC, respectively, and I is the effective current. Similarly, The 6phase AC is defined as follows:
$\left\{\begin{array}{c}\begin{array}{c}{I}_{A}=\sqrt{2}I\text{sin}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\right)\\ {I}_{B}=\sqrt{2}I\text{sin}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\frac{\pi}{3}\right)\end{array}\\ {I}_{C}=\sqrt{2}Isin\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\frac{2\pi}{3}\right)\\ \begin{array}{c}{I}_{D}=\sqrt{2}I\text{sin}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\pi \right)\\ {I}_{E}=\sqrt{2}I\text{sin}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\frac{4\pi}{3}\right)\\ {I}_{F}=\sqrt{2}I\text{sin}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\frac{5\pi}{3}\right)\end{array}\end{array}\right.$ (6)
where Z_{2} and p_{2} is the number of salient pole pairs and position of the mover driven by the 6phase AC, respectively. The interlinkage flux of the 3phase coils is defined as follows:
$\left\{\begin{array}{c}\begin{array}{c}{\varphi}_{U}=\varphi \text{cos}\left(\frac{2\pi {Z}_{1}{p}_{1}}{L}\right)\\ {\varphi}_{V}=\varphi \text{cos}\left(\frac{2\pi {Z}_{1}{p}_{1}}{L}\frac{2\pi}{3}\right)\\ {\varphi}_{W}=\varphi \text{cos}\left(\frac{2\pi {Z}_{1}{p}_{1}}{L}\frac{4\pi}{3}\right)\end{array}\end{array}\right.$ (5)
where Φ is the amplitude of the interlinkage flux. Simiraly, the interlinkage flux of the 6phase coils is defined as follows:
$\left\{\begin{array}{c}\begin{array}{c}{\varphi}_{A}=\varphi \text{cos}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\right)\\ {\varphi}_{B}=\varphi \text{cos}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\frac{\pi}{3}\right)\end{array}\\ {\varphi}_{C}=\varphi cos\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\frac{2\pi}{3}\right)\\ \begin{array}{c}{\varphi}_{D}=\varphi \text{cos}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\pi \right)\\ \text{}{\varphi}_{E}=\varphi \text{cos}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\frac{4\pi}{3}\right)\\ {\varphi}_{F}=\varphi \text{cos}\left(\frac{2\pi {Z}_{2}{p}_{2}}{L}\frac{5\pi}{3}\right)\end{array}\end{array}\right.$ (6)
The force of the mover due to the 6phase magnetic flux and 3phase AC is defined as follows:
${F}_{36}=\text{}{I}_{U}{\varphi}_{A}+{I}_{V}{\varphi}_{B}+{I}_{W}{\varphi}_{C}+{I}_{U}{\varphi}_{D}+{I}_{V}{\varphi}_{E}+{I}_{W}{\varphi}_{F}=0$ (7)
The force of the mover due to the 3phase magnetic flux and 6phase AC is defined as follows:
${F}_{63}=\text{}{I}_{A}{\varphi}_{U}+{I}_{B}{\varphi}_{V}+{I}_{c}{\varphi}_{W}+{I}_{D}{\varphi}_{U}+{I}_{E}{\varphi}_{V}+{I}_{F}{\varphi}_{W}=0$(8)
From (7) and (8), the interference of the force is zero. Thus, the movers can be independently driven [7, 8].
The proposed actuator is controlled using vector control, and 3phase and 6phase ACs are transformed into dq axis. The transformation matrix of the currents from the UVW to dq coordinate systems is defined as follows:
$\left[\begin{array}{c}{I}_{d3}\\ {I}_{q3}\end{array}\right]={D}_{3}\left[\begin{array}{c}{I}_{U}\\ {I}_{V}\\ {I}_{W}\end{array}\right]$ (9)
${D}_{3}\text{}=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\text{cos\theta}& \text{cos}\left(\text{\theta}\frac{2}{3}\text{\pi}\right)& \text{cos}\left(\text{\theta}\frac{4}{3}\text{\pi}\right)\\ \text{sin\theta}& \text{sin}\left(\text{\theta}\frac{2}{3}\text{\pi}\right)& \text{sin}\left(\text{\theta}\frac{4}{3}\text{\pi}\right)\end{array}\right]$ (10)
where θ is the electrical angle of the position defined as follows:
$\theta =\frac{2\pi Zp}{L}$ (11)
where Z is the number of salient pole pairs of the mover. The transformation matrix of the currents from the ABCDEF to dq coordinate systems is defined as follows:
$\left[\begin{array}{c}{I}_{d6}\\ {I}_{q6}\end{array}\right]={D}_{6}\left[\begin{array}{c}{I}_{A}\\ {I}_{B}\\ {I}_{C}\\ {I}_{D}\\ {I}_{E}\\ {I}_{F}\end{array}\right]$ (12)
${D}_{6}\text{}=\sqrt{\frac{1}{3}}\left[\begin{array}{cccccc}\text{cos\theta}& \text{cos}\left(\text{\theta}\frac{1}{3}\text{\pi}\right)& \text{cos}\left(\text{\theta}\frac{2}{3}\text{\pi}\right)& \text{cos}\left(\text{\theta}\text{\pi}\right)& \text{cos}\left(\text{\theta}\frac{4}{3}\text{\pi}\right)& \text{cos}\left(\text{\theta}\frac{5}{3}\text{\pi}\right)\\ \text{sin\theta}& \text{sin}\left(\text{\theta}\frac{1}{3}\text{\pi}\right)& \text{sin}\left(\text{\theta}\frac{2}{3}\text{\pi}\right)& \text{sin}\left(\text{\theta}\text{\pi}\right)& \text{sin}\left(\text{\theta}\frac{4}{3}\text{\pi}\right)& \text{sin}\left(\text{\theta}\frac{5}{3}\text{\pi}\right)\end{array}\right]$ (13)
The inverse dq transformation of the voltage from dq to UVW coordinate systems is defined as follows:
$\left[\begin{array}{c}{V}_{U}\\ {V}_{V}\\ {V}_{W}\end{array}\right]={{D}_{3}}^{1}\left[\begin{array}{c}{V}_{d3}\\ {V}_{q3}\end{array}\right]\text{}$ (14)
${{D}_{3}}^{1}=\sqrt{\frac{2}{3}}\left[\begin{array}{cc}\text{cos\theta}& \text{sin\theta}\\ \text{cos}\left(\text{\theta}\frac{2}{3}\text{\pi}\right)& \text{sin}\left(\text{\theta}\frac{2}{3}\text{\pi}\right)\\ \text{cos}\left(\text{\theta}\frac{4}{3}\text{\pi}\right)& \text{sin}\left(\text{\theta}\frac{4}{3}\text{\pi}\right)\end{array}\right]$ (15)
The inverse dq transformation of the voltage from dq to ABCDEF coordinate systems is defined as follows:
$\left[\begin{array}{c}{V}_{A}\\ {V}_{B}\\ {V}_{C}\\ {V}_{D}\\ {V}_{E}\\ {V}_{F}\end{array}\right]={{D}_{6}}^{1}\left[\begin{array}{c}{V}_{d6}\\ {V}_{q6}\end{array}\right]$ (16)
${{D}_{6}}^{1}=\sqrt{\frac{1}{3}}\left[\begin{array}{cc}\text{cos\theta}& \text{sin\theta}\\ \mathrm{cos}\left(\text{\theta}\frac{1}{3}\text{\pi}\right)& \mathrm{sin}\left(\text{\theta}\frac{1}{3}\text{\pi}\right)\\ \mathrm{cos}\left(\text{\theta}\frac{2}{3}\text{\pi}\right)& \mathrm{sin}\left(\text{\theta}\frac{2}{3}\text{\pi}\right)\\ \mathrm{cos}\left(\text{\theta}\text{\pi}\right)& \mathrm{sin}\left(\text{\theta}\text{\pi}\right)\\ \mathrm{cos}\left(\text{\theta}\frac{4}{3}\text{\pi}\right)& \mathrm{sin}\left(\text{\theta}\frac{4}{3}\text{\pi}\right)\\ \mathrm{cos}\left(\text{\theta}\frac{5}{3}\text{\pi}\right)& \mathrm{sin}\left(\text{\theta}\frac{5}{3}\text{\pi}\right)\end{array}\right]$ (17)
The block diagram of the control system is shown in Fig. 3. In this paper, I_{d} = 0 control is used. The position feedback control is achieved using a PID control. A PI control is used for the current control. The 3phase AC is extracted from the superimposed current as follows:
Fig. 3. Block diagram of the control system
$\left\{\begin{array}{c}\begin{array}{c}{I}_{a}=\text{}{I}_{A}+{I}_{U}\\ {I}_{b}=\text{}{I}_{B}+{I}_{V}\end{array}\\ {I}_{c}={I}_{C}+{I}_{W}\\ \begin{array}{c}{I}_{d}=\text{}{I}_{D}+{I}_{U}\\ {I}_{e}=\text{}{I}_{E}+{I}_{V}\\ {I}_{f}=\text{}{I}_{F}+{I}_{W}\end{array}\end{array}\right.$ (18)
From (18), the 3phase AC is defined as follows:
$\left\{\begin{array}{c}\begin{array}{c}{I}_{U}=\text{}\frac{{I}_{a}+{I}_{d}({I}_{A}+{I}_{D})}{2}\\ {I}_{V}=\text{}\frac{{I}_{a}+{I}_{d}({I}_{A}+{I}_{D})}{2}\\ {I}_{W}=\text{}\frac{{I}_{a}+{I}_{d}({I}_{A}+{I}_{D})}{2}\end{array}\end{array}\right.$ (19)
The relationship of the 6phase ACs is defined as follows:
$\left\{\begin{array}{c}\begin{array}{c}{I}_{A}+{I}_{D}=\text{}0\\ {I}_{B}+{I}_{E}=\text{}0\\ {I}_{C}+{I}_{F}=\text{}0\end{array}\end{array}\right.$ (20)
From (19) and (20), the component of the 3phase AC can be extracted. The 6phase AC is extracted from the superimposed current as follows:
$\left\{\begin{array}{c}\begin{array}{c}{I}_{A}=\text{}{I}_{a}{I}_{U}\\ {I}_{B}=\text{}{I}_{b}{I}_{V}\end{array}\\ {I}_{C}={I}_{c}{I}_{W}\\ \begin{array}{c}{I}_{D}=\text{}{I}_{d}{I}_{U}\\ {I}_{E}=\text{}{I}_{e}{I}_{V}\\ {I}_{F}=\text{}{I}_{f}{I}_{W}\end{array}\end{array}\right.$ (21)
From (19), the component of the 6phase AC can also be extracted.
RESULTS OF CHARACTERISTICS ANALYSIS
The specification of the analysis model is shown in Table 1. The dimensions of the stator and mover are shown in Tables 2 and 3. The length of one period of the 3phase and 6phase movers is 15 mm and 16.15 mm, respectively. The positional relationship of the movers and stator is shown in Fig. 4.
Table 1. Specification of the analysis model
Number of coils  6 
Number of pole pairs of the stator  12 
Air gap length [mm]  1 
Residual magnetic flux density [Br]  1.4 
Distance between movers [mm]  1.5 
Number of coil turns  388 
Coil space factor [%]  63 
Maximum amplitude of current [A]  0.6 
Table 2. Dimensions of the movers
 3phase  6phase 
Fundamental order of permeance  14  13 
Total height [mm]  8  8 
Pole height [mm]  3  3 
Pole width [mm]  7.5  8.08 
Table 3. Dimensions of the movers
Height [mm]  22.5 
Width [mm]  210 
Depth [mm]  16 
Back yoke height [mm]  5.5 
Permanent magnet height [mm]  5 
Permanent magnet width [mm]  8.75 
Permanent magnet depth [mm]  16 
Fig. 4. Positional relationship of the movers and stator
For verifying the static force characteristics, a 3D finite element method is used [9, 10]. The 3D mesh model except for the air region is shown in Fig. 5. The analysis conditions are shown in Table 4. In the static force analysis, the amplitude of the 3phase and 6phase ACs is 0.6 A. The current density is 5 A/mm^{2}.
The analysis patterns are defined as follows:
Pattern 11: The 3phase mover is moved under 3phase AC, and the 6phase mover is fixed.
Pattern 12: The 6phase mover is moved under 6phase AC, and the 3phase mover is fixed.
Pattern 13: Both movers are moved under 3phase and 6phase superimposed currents.
The static force analysis results of each pattern are shown in Figs. 6, 7, and 8. The average forces are shown in Table 4. From the results of pattern 11, the average force of the 3phase and 6phase movers is 6.47 N and 0.24 N, respectively. It is found that the 3phase mover can be driven by the 3phase AC. From the results of pattern 12, the average force of the 3phase and 6phase movers is 0.007 N and 5.70 N, respectively. It shows that the 6phase mover can be driven by the 6phase AC. From the results of pattern 13, the average force of the 3phase and 6phase movers is 5.89 N and 5.13 N, respectively. It is observed that the 3phase and 6phase movers can be driven by only the 3phase and 6phase AC, respectively.
Fig. 5. 3D mesh model except for the air region
Table 4. Analysis conditions
Number of elements  4,730,229 
Number of nodes  801,229 
Number of edges  5,532,118 
CPU time [h]  12.5 
Fig. 6. Analysis results of the force characteristics (Pattern 11)
Fig. 7. Analysis results of the force characteristics (Pattern 12)
Fig. 8. Analysis results of the force characteristics (Pattern 13)
Table 4. Average force of the proposed linear actuator
 Average force of the 3phase mover [N]  Average force of the 6phase mover [N] 
Pattern 1  6.47  0.24 
Pattern 2  0.007  5.70 
Pattern 3  5.89  5.13 
For verifying the independency of both movers, the back electromotive force is analyzed. In this study, the analysis patterns are shown as follows:
Pattern 21: Only the 3phase mover is moved.
Pattern 22: Only the 6phase mover is moved.
Pattern 23: Both movers are moved
The moving velocity is 300 mm/s, and the operating time is 100 ms.
The analysis results are shown in Fig. 9. From these results, the back electromotive force of pattern 23 is the composition of that of pattern 1 and pattern 2. It shows that the movers can be independently driven.
For verifying the dynamic performance under position feedback control, the dynamic characteristics analysis is conducted. The target trajectory is given using a trapezoidal velocity waveform. The analysis patterns using the target trajectory are shown as follows:
Pattern 31: The 3phase mover is controlled for the target trajectory, and the 6phase mover is controlled to be fixed.
Pattern 32: The 6phase mover is controlled for the target trajectory, and the 3phase mover is controlled to be fixed.
Pattern 33: Both movers are controlled for the target trajectory
The target trajectory is shown in Fig. 10. The maximum velocity and acceleration is 0.4 m/s and 0.2 m/s^{2}, respectively The PID controller gain is shown in Table 5. In this control, the control period is 2.5 ms. The mass of each mover is 3 kg.
Fig. 9. Analysis results of the back electromotive force
Table 5. PID controller gain

 Current controller  Position controller 
3phase  Kp  4.85  300 
Ki  0.24  10  
Kd    150  
6phase  Kp  4.85  300 
Ki  0.24  10  
Kd    150 
The computed dynamic characteristics are shown in Figs. 11, 12, and 13. From the results, the steadystate error is very small, and both movers can be controlled following the target trajectory.
Fig. 10. Target trajectory
Fig. 11. Analysis results of the position feedback control (Pattern 31)
Fig. 12. Analysis results of the position feedback control (Pattern 32)
Fig. 13. Analysis results of the position feedback control (Pattern 33)
The step response of the control system is also analyzed. The analysis patterns are shown as follows:
Pattern 41: The step target trajectory is applied to the 3phase mover, and the 6phase mover is controlled to be fixed.
Pattern 42: The step target trajectory is applied to the 6phase mover, and the 3phase mover is controlled to be fixed.
The analysis results of the step response are shown in Fig. 14. The time constant of the mover of the 3phase and 6phase movers is 0.115 s and 0.125 s, respectively. From the results, the mover well followed the target trajectory, and the movers can be independently controlled.
Fig. 14. Analysis results of the step response
CONCLUSION
In this paper, we proposed a linear vernier actuator with two movers. The basic structure, the operating principle, and the control method of the two movers were described. The force characteristics were investigated by the magnetic field analysis using 3D finite element method. The dynamic characteristics under position feedback control were verified, and it was found that the movers can be independently controlled. In future works, we will verify the control performance on a prototype.
About the authors
Akira Heya
Osaka University
Author for correspondence.
Email: Akira.heya@ams.eng.osakau.ac.jp
ORCID iD: 0000000159664387
Ph.D. Student, Department of Adaptive Machine Systems, Graduate School of Engineering
Japan, Yamadaoka, Suita, Osaka 5650871Katsuhiro Hirata
Osaka University
Email: khirata@ams.eng.osakau.ac.jp
ORCID iD: 0000000255975265
Professor, Department of Adaptive Machine Systems, Graduate School of Engineering
Japan, Yamadaoka, Suita, Osaka 5650871Noboru Niguchi
Osaka University
Email: noboru.niguchi@ams.eng.osakau.ac.jp
ORCID iD: 0000000210057946
Assistant professor, Department of Adaptive Machine Systems, Graduate School of Engineering
Japan, Yamadaoka, Suita, Osaka 5650871References
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