Analysis and Experimental Verification of Radial-Gap Two-Degree-of-Freedom Motor Based on a Magnetic Screw Structure

This article proposes a radial-gap two-degree-of-freedom motor based on a magnetic screw structure. The proposed motor is composed of stator parts, rotor parts, and a mover part, and can generate torque and thrust force simultaneously and independently. Additionally, the motor can generate a large thrust force with the magnetic screw structure. In the proposed motor, the rotor parts and mover part, which act as the nut and screw of ball screws, are coupled with each other by means of magnetic force. Therefore, there is no friction between the rotor parts and mover part. The proposed motor can not only contribute to the minimization of multidegree-of-freedom systems, such as industrial robots but also improve the energy efficiency of robots. This article examines the effectiveness of the proposed motor through analysis and experimental results.


I. INTRODUCTION
I N GENERAL, industrial robots have a multidegree-offreedom (multi-DOF) structure so that they can perform human tasks, and more DOFs are necessary as the robots perform more complex tasks. However, one DOF generally consists of one motor so that each DOF can be controlled independently. As a result, robots become larger and heavier as more complex tasks are necessary.
The increase in the number of motors signifies an increased number of parts that require maintenance. Any parts have their life spans and will be broken sometime. They need maintenance, such as part replacement when they are broken or before. Therefore, the increase in parts including motors, connectors between the motors and other parts, bolts to fix the connectors, and so on lead to the increase in the number of parts that need maintenance. Moreover, the number of robots, which are composed of many parts, is increasing in the industrial field. However, many kinds of maintenance are done by humans. The increase in robots causes an increase in manpower to maintain them, although the number of workers is expected to decrease in the future. Research on multi-DOF motors has attracted attention because these motors can reduce the total number of motors in robots, while retaining the number of DOFs. One type of multi-DOF motor is a spherical motor [1]- [3], which is composed of a spherical mover and a stator that covers the mover. This motor has two or three DOFs, and its mover rotates around multiple axes. Additionally, multi-DOF motors that perform not only rotational motion but also linear motion have been proposed [4]- [11]. Actuators which realize 2-D or 3-D motion with piezoelectric elements have been proposed [12]- [14].
Although the solution of size and weight are required of the industrial robots, the improvement of energy efficiency is also required. Each joint of industrial robots is composed of a motor and a gear so that the robots can generate a large force; however, gears generally cause energy loss. Therefore, it is important to improve not only the motors but also the power transmission, including the gears, to increase the energy efficiency of the robots.
Recently, to reduce power transmission loss, research on magnetic gears has been conducted [15]- [17]. In a magnetic gear, the input part is coupled to the output part by means of magnetic force. There is no friction between the input part and output part because they are connected without physical contact. Therefore, magnetic gears can reduce transmission loss [18].
Not only magnetic gears but also magnetic screws have been researched [19], [20]. Additionally, the radial-gap helical ROTLIN motor, which is a combination of a rotary motor and a magnetic screw, as shown in Fig. 1, has been proposed [21]. A This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0 /  TABLE I  COMPONENTS INCLUDED IN EACH PART OF RADIAL-GAP ROTLIN MACHINE AND RADIAL-GAP 2-DOF MOTOR magnetic screw has the structure of a ball screw and is composed of a screw part and nut part, which have spiral permanent magnets (PMs). They are coupled to each other by means of magnetic force. The helical force around the central axis of the screw part is generated in the screw part by the rotation of the nut part. By canceling the rotation of the screw part, the only linear component in the helical force is the output of the screw part. Therefore, a magnetic screw can convert rotational motion into linear motion. The reduction ratio of the rotational velocity to the linear velocity can be adjusted by the lead length of the magnetic screw. The magnetic screw can generate a large thrust force by shortening the lead length.
As mentioned above, a multi-DOF motor is an important element for addressing the decreasing birthrate and aging of the population. Therefore, the development of multi-DOF motors that generate a large force with low transmission loss can make a significant contribution to the industrial field. The authors thus propose a radial-gap two-degree-of-freedom (2-DOF) motor based on a magnetic screw structure [22].
The proposed motor is composed of two stator parts, two rotor parts, and a mover part. It can remove parts that a conventional 2-DOF device with a rotary motor and a linear actuator needs. The proposed motor does not have a connector that the conventional device needs between the rotary motor and the linear actuator. Additionally, it integrates several parts required for the linear actuator, composed of a rotary motor and a ball screw. The rotor part in the proposed motor combines the rotor of the rotary motor, the input part of the ball screw, and the coupling between them. Moreover, the proposed motor can generate a large thrust force because it has a magnetic screw structure. The authors evaluate the effectiveness of the proposed motor using finite element analysis (FEA).
This article also establishes a mathematical representation of the proposed radial-gap 2-DOF motor. Additionally, the effectiveness of the proposed motor is evaluated based on both analysis results and experimental results. Table I shows the summary of components in the proposed motor illustrated in Fig. 2 and the ROTLIN machine [21].   Table II presents the design parameters including the ones shown in Fig. 3. The ROTLIN machine comprises a stator, a rotor, and a mover that moves only in the z-direction. The structure integrates some components, including a rotary motor and a right-hand magnetic screw, into the ROTLIN machine. The rotation of the nut part, which is included in the rotor, as shown in Table I, generates a left-helical force in the mover. The mover moves in the z-direction by the left-helical force.

A. Structure of Proposed Motor
On the other hand, the proposed motor is composed of two stator parts, two rotor parts, and a mover part that moves in  the θ direction and the z-direction. Some parts have the special arrangements of PMs to integrate rotary motors, a righthand magnetic screw, and a left-hand magnetic screw into the proposed motor. The PM arrangement in rotor parts works as a rotor in the rotary motor and the nut part, as shown in Table I. The mover part has the PM arrangement which realizes both a right-hand screw and a left-hand screw. Therefore, the mover part can output left-helical and right-helical forces.
Additionally, the left-helical and right-helical forces are generated independently in the mover part because each of the rotor parts rotates independently. Therefore, torque in the θ direction and thrust force in the z-direction are generated independently by the combination of two kinds of helical-direction forces. In the section, the principle to generate the torque and thrust force is described. Fig. 4 displays the PM arrangement in the R-rotor part. The light-colored lines in Fig. 4(a) indicate the PM arrangement for a rotor of the rotary motor. The N and S poles of the PMs are arranged alternately along the θ direction and interact with the stator part. In the proposed motor, the stator parts and rotor parts are designed based on a nine-slot-eight-pole (9s8p) rotary motor, which contributes to the dispersion of radial forces and the decrease in noise and cogging torque. Therefore, four N poles and four S poles are arranged in the rotor parts.
Additionally, the PMs are formed so that the N and S poles are arranged alternately along the left helical direction, as the  light-colored lines illustrated in Fig. 4(b). In the figure, l p represents the lead length of the R-rotor part. The PM arrangement functions as the right-hand nut part with a double screw. In the mover part, the PMs have a 2-D arrangement, and the N and S poles are arranged alternately along the left helical direction, as the light-colored lines illustrated in Fig. 6(a). Additionally, the N and S poles are arranged alternately along the right helical direction, as the light-colored lines in Fig. 6(b). The mover part functions as the right-hand and left-hand screw parts, as shown in Table I.
The phase of the light-colored lines between Figs. 4(b) and 6(a) changes, when the R-rotor part rotates or the mover part moves in the left helical direction. The phase generates a magnetic force in the left helical direction in the mover part. In contrast, the phase of the light-colored lines between Figs. 5(b) and 6(b) changes, when the L-rotor part rotates or the mover part moves in the right helical direction. The phase generates a magnetic force in the right helical direction. Therefore, the proposed motor can generate a torque in the θ direction and a thrust force in the z-direction independently by the combination of the magnetic force in two helical directions.

B. Modeling the Relationship Between Rotor Parts and Mover Part
The proposed motor is a combination of rotary motors and magnetic screws. This subsection describes the magnetic screws, which are composed of rotor parts and a mover part.
First, it is assumed that rotor parts with a thickness of δr R are placed at r = r R , as illustrated in Fig. 3(a). The interlinkage flux interlinking with the R-rotor part is calculated as follows: where Φ RM 1 (r R ) represents the interlinkage flux, which is generated by the mover part and interlinks with the PMs of the R-rotor part; θ R1 indicates the position of the R-rotor part in the θ direction; and θ M and z M denote the positions of the mover part in the θ and z-directions, respectively. The magnetic flux density, which interlinks with the PMs of the R-rotor part, depends on r R and is defined as in Fig. 7, where B M (r R ) represents the amplitude of the magnetic flux density at r = r R . The interlinkage flux is defined with B M (r R ). Φ M 1 is calculated as , which is calculated with (1)-(8) and Table II. In the figure, α 1(θ R1 ) and α 2 (θ R1 ) are defined as follows: Here, α 1 (θ R1 ) denotes the motion of the R-rotor part toward the mover part in the left helical direction, while α 2 (θ R1 ) denotes the motion in the right helical direction. Fig. 8 indicates that is close to a cosine wave. Therefore, Φ RM 1 (r R ) approaches the following: where Φ aRM 1 (r R ) is a value depending on r R . The inductance matrix between the R-rotor part and mover part L RM 1 (r R ) is described by where L R1 and L M represent the self-inductance of the Rrotor part and mover part, respectively, and I mM indicates the equivalent magnetization current of the PMs in the mover part. Therefore, the torque and thrust force generated by the R-rotor part with a thickness of δr R are calculated as follows: δF M 1 = 1 2 Here, δT M 1 and δF M 1 represent the torque and thrust force generated in the mover part, and I mR is the equivalent magnetization current of the PMs in the rotor parts, which is defined as follows: where B r and μ m represent the residual magnetic flux density and magnetic permeability of the R-rotor part, respectively. The torque and thrust force generated by the entire PM in the R-rotor part are calculated as The PM arrangements in the R-rotor and L-rotor parts are symmetric. The interlinkage flux interlinking with the L-rotor part Φ RM 2 (r R ) is defined as where Φ aRM 2 (r R ) is a value depending on r R , like Φ aRM 1 (r R ).
The inductance matrix between the L-rotor part and mover part L RM 2 (r R ) is described by where L R2 represents the self-inductance of the L-rotor part. The torque and thrust force generated by the L-rotor part with a thickness of δr R are calculated as follows: From these formulas, the torque and thrust force generated by the entire PM in the L-rotor part are calculated as In the proposed motor, the R-rotor and L-rotor parts have a symmetric structure. The amplitude of the interlinkage flux is defined as Φ aRM (r R ) = Φ aRM i (r R ) (i ∈ 1, 2). In this article, suffixes 1 and 2 represent the components of the right-hand screw and left-hand screw, respectively. T Mi and F Mi are described as (30)-(33), respectively, The ratio of the thrust force to the torque is calculated as where G r is equivalent to the ratio of general ball screws. It turns out that the proposed motor can convert the torque input to the rotor parts into a large thrust force by adjusting G r . However, the maximum F Mi is F aM , as shown in (32) and (33). Because of that, it is necessary to consider not only G r but also B r and Φ aRM (r R ), which F aM depends on, to generate a large thrust force in the mover part.

C. Modeling the Relationship Between Rotor Parts and Stator Parts
This subsection describes the rotary motors, which are composed of rotor parts and stator parts. The stator parts are composed of three-phase coils and cores. Therefore, the voltage equation of each stator part is represented as follows: where I uSi , I vSi , and I wSi represent the current of each phase in the stator part, and I mRi indicates the equivalent magnetization current of the PMs in the rotor part. Each component of V Si is the voltage corresponding to I uSi , I vSi , I wSi , and I mRi . Furthermore, θ Ri is the electrical angle defined based on the period of the PM arrangement in the θ direction. In addition, Φ aSR = Φ aSRi (i ∈ 1, 2) is the amplitude of the interlinkage flux, which is generated by the rotor part and interlinks with the stator part. The torque generated by the current of the stator part is calculated as where I dSi , I qSi , and I 0Si represent the current in the dp-axis, and C indicates the transformation matrix from the three-phase static axis to the dq-axis. The torque in the rotor part T Ri is calculated using (38)-(41) as where K tR represents the torque constant. Therefore, the dynamics model of the rotor parts is described as follows: where J R represents the inertia of the rotor parts. From (30)-(34), T M and F M in the static condition, whereθ Ri = 0, are derived as (45) and (46), respectively, T M is controlled by the sum of I qS1 and I qS2 , while F M depends on the difference between I qS1 and I qS2 . It turns out that T M and  F M can be controlled independently. Additionally, the proposed motor can generate a large thrust force based on G r .

A. Analysis Environment
The proposed motor was analyzed using JMAG-Designer, which is FEA software provided by the JSOL Corporation. Figs. 9 and 10 present the 3-D analysis models, which are composed of the parameters listed in Table II. In FEA, the component of the right-hand screw was analyzed basically because the R-rotor part had the opposite structure of the L-rotor part. Therefore, the FEA analysis model was composed of the mover part, R-rotor part, and stator part for the R-rotor part.

B. Interlinkage Flux Interlinking With the R-Rotor Part and Calculation Based on the Mathematical Model
First, Φ RM 1 (r R ) was analyzed in the condition in which α 2 (θ R1 ) was kept 0, and Fig. 11 presents the FEA results. Φ RM 1 (r R ) was dependent on α 1 (θ R1 ) and was close to the cosine wave presented in (11). The amplitude of Φ RM 1 (r R ) decreased as r R increased.

C. Torque and Thrust Force in the Mover Part
T M 1 and F M 1 with different α 1 (θ R1 ) and α 2 (θ R1 ) were analyzed, and Figs. 13 and 14 present the FEA results. The results indicate that T M 1 and F M 1 were dependent on α 1 (θ R1 ) but independent of α 2 (θ R1 ). Additionally, T M 1 and F M 1 were close to sine waves depending on α 1 (θ R1 ).     lines represent the calculation results based on the mathematical model. T aM and F aM of the calculation results are presented in (48) and (49), respectively, and the approximation curves are close to the calculation results. Table III presents the amplitude of the approximation curves. The ratio of F M 1 to T M 1 is close to G r = 1.57 × 10 2 calculated by (34), and T M 1 and F M 1 are generated based on (30)-(34).
Additionally, T M 2 and F M 2 with different α 1 (θ R2 ) and α 2 (θ R2 ) were analyzed. Figs. 16 and 17 show the analysis results. T M 2 and F M 2 were dependent on α 2 (θ R2 ) but independent of α 1 (θ R2 ). It was confirmed that the L-rotor part had the opposite characteristics of the R-rotor part.

D. Torque Constant
T R1 , which is generated by the current in the stator parts, was analyzed. Fig. 18 presents the FEA results of the torque generated in the R-rotor part. In the FEA, torque with different θ R1 was analyzed, while maintaining α 1 (θ R1 ) = 0. Therefore, FEA was performed so that torque was not generated between the R-rotor part and mover part. It included only torque generated by the current of the stator part.
In the figure, the dots indicate the FEA results at each I qS1 , and the broken line is an approximate straight line calculated based on the least-squares method. The formula of the approximate straight line is expressed as Therefore, K tR calculated with the FEA results is 0.2381 Nm/A.

A. Experimental Environment
Figs. 19 and 20 display the experimental machine. The machine had rotary encoders, a linear encoder, and a force sensor. The rotary encoders were E80H by AUTONICS, which were used to measure θ R1 , θ R2 , and θ M . The linear encoder was QUANTiC by Renishaw to measure z M , and the force sensor was PFS by Leptrino to measure T R1 , T M , and F M . The force sensor is fixed to the base plate, where the stator parts are fixed. Additionally, the mover part is connected to the force sensor mover part through the connector. The experimental parameters are presented in Table IV.

B. Torque Constant of R-Rotor Part
First, T R1 was measured at different I qS1 . In the experiment, the measurement was performed in the condition, where the mover part was detached from the experimental machine so that torque was not generated between the R-rotor part and mover part. Fig. 21 presents the experimental results. At each I qS1 , T R1 was measured six times, and the average values are plotted as dots in the figure. The broken lines represent the approximate straight lines calculated with the measurement values. The formulas of both approximate straight lines are as follows: K tR was calculated based on the average of the proportional constant in both approximate straight lines. K tR calculated with the measurement values was 0.2201 Nm/A, which was close to the analysis result of K tR = 0.2381 Nm/A. The second term in the right side of the equation represents friction. The friction causes energy loss and changes with speed. Although the experimental system does not have the function to measure torque when rotor parts rotate, it is necessary to measure it in the future, to evaluate the energy efficiency of the proposed motor.

C. Torque and Thrust Force in the Mover Part
T M 1 and F M 1 were measured at different α 1 (θ R1 ). In the experiment, the measurement was performed in the condition where the mover part was fixed. Therefore, θ M and z M were kept constant. Figs. 22 and 23 present the experimental results. In the figures, the dots represent the average of the measurement values. The error bars were calculated based on standard deviation. The results indicate that T M 1 and F M 1 increased depending on α 1 (θ R1 ).
In the figures, the broken lines represent the analysis results, which are the broken lines in Fig. 15(a) and (b). The solid lines represent the calculation results, which are the solid lines in Fig. 15(a) and (b). The dots, which represent the experimental results, differed from the analysis and calculation results. The reason for the error is the difference in the magnetic flux density between the analysis model and experimental machine.
The FEA was conducted with the ideal parameter, including the residual magnetic density of the PMs. However, the actual    parameters are different from the ideal parameters because tolerance always exists in the actual machine. Table V presents the magnetic flux density of the analysis model and experimental machine. Magnetic Meter (MG-3000SD) provided by FUSO Corporation was used for the measurement. The magnetic force density on the PM's surface was measured by contacting the     Figs. 27 presents the ratio of the thrust force to the torque. In the figure, the dots indicate G r calculated with the measurement values presented in Figs. 25 and 26. The broken line represents G r = 2π l p . As α 1 (θ R1 ) decreased, the difference between the experimental results and 2π l p increased. The reason for this may be friction. As α 1 (θ R1 ) decreased, the ratio of friction to F M 1 increased. Therefore, G r was small when α 1 (θ R1 ) was small. In contrast, the experimental result approached G r = 2π l p as α 1 (θ R1 ) increased.

D. Torque and Thrust Force Generated by Current in Two Stator Parts
In the experiment, the current was input to both stator parts to investigate the independent generation of T M and F M . Two experimental cases were examined: Case 1 and Case 2. In each case, the current input to both stator parts was defined as follows: Case2 I q = − I qS1 = I qS2 .
Figs. 28 and 29 present the experimental results. In Case 1, T M increased depending on the increase of I q . In Case 2, F M increased depending on the increase of I q . In contrast, the generation of F M in Case 1 and of T M in Case 2 was suppressed. The results indicate that T M was generated by the sum of I qS1 and I qS2 , while F M was generated by the difference between I qS1 and I qS2 . Therefore, T M and F M were generated independently.
In the figures, broken lines indicate the approximate straight lines of T M in Case 1 and F M in Case 2. The formulas of both approximate straight lines are as follows: G r = 164.0.
The error between (56) and G r = 2π l p was less than 4%. The results thus confirm the effectiveness of (45) and (46).

V. CONCLUSION
This article proposes a radial-gap 2-DOF motor based on a magnetic screw structure. The motor generates a torque and thrust force simultaneously and independently. Additionally, the motor can generate a large thrust force due to the magnetic screw structure. In this article, the structure of the proposed motor and mathematical modeling are described, and the analysis and experimental results demonstrate the effectiveness of the mathematical modeling.
The proposed motor can contribute to the minimization of multi-DOF systems, such as industrial robots, because it has two DOFs. Furthermore, it can offer lower energy loss than a ball screw, in which a nut and screw are coupled with physical contact, because the rotor parts and mover part are coupled by means of magnetic force.
In future work, it is necessary to measure the energy loss of the proposed motor. In addition, the proposed motor has rotor parts and stator parts and is thus composed of a multimass model. It is necessary to design a controller based on the multimass model to control the proposed motor.
Additionally, the proposed motor potentially has both of different driving systems for the rotor parts. The usefulness of the combination of different driving systems has been indicated by the hybrid actuator system [23]. The hybrid actuator system for the proposed motor is worth considering.