Calculation and optimization of a permanent magnetic hybrid driven 3-DOF motor

Abstract

In this paper, a new type of permanent magnet type 3-DOF hybrid driven motor is proposed in view of the poor precision of existing deflection movement control and low speed of multi-degree-of-freedom motor rotation. The motor contains two part named lateral rotation module and internal deflection modules. The output torque of the motor, the rotation speed and control precision are improved by the coordinated control of rotation movement and deflection movement. The magnetic field shielding and optimization of two modules inside and outside the motor are realized by using Halbach permanent magnet array, and the magnetic field characteristics of the motor are improved. First, using analytical method and finite element method, the magnetic field characteristics and distribution of the permanent magnet of the motor are analyzed, and the electromagnetic characteristics of load and no load condition and output torque of the motor is analyzed by finite element method. Then, the loss of this motor is analyzed and calculated, the motor response surface equation is built by using taguchi method and central composite design method. Finally, the optimal structure parameters of motor is proposed by the numerical model based on maximizing the motor output torque of the target, and it is compared with finite element and experiment to verify the accuracy of the structural parameters. It provides a new theoretical basis for the follow-up optimization of the motor.

Keywords

3-DOF motor Halbach array permanent magnet magnetic field calculation torque structure optimization taguchi method

1 Introduction

The traditional multi-degree-of-freedom motor adopts the same set of winding control for rotation and deflection motion. The winding is generally a centralized winding; the motor rotates at a slower speed and the control accuracy of the yaw movement often fails to meet the requirements for industrial production.

Therefore, this paper proposes a novel permanent magnet hybrid-driven three-degree-of-freedom motor. The three-degree-of-freedom movement of the motor is accomplished by the external rotation module and the internal deflection module [1,2]. The internal deflection module is located in the rotor yoke of the external rotation module. The external rotation module is the inner rotor structure and it is responsible for the rotation movement. The inner deflection module is the inner rotor structure and is responsible for the yaw movement. Rotating and deflecting permanent magnets use inner rotor Halbach array permanent magnets, which have good magnetic properties outside the permanent magnets. In this paper, analytical modeling and finite element method modeling analysis of the air gap magnetic field outside the rotation and deflection of permanent magnets and electromagnetic field of rotation module are carried out [3–6]. The finite element method is used to analyze the rotation and yaw torque characteristics of the three-degree-of-freedom motor [7–10]. The loss of this motor is analysed and calculated [12–14]. The three-degree-of-freedom motor structural parameters are optimized by using the Taguchi algorithm to maximize the output torque of the motor [15].

2 Novel permanent magnet hybrid driven three-degree-of-freedom motor structure and control mechanism

The permanent magnet hybrid driven three-degree-of-freedom motor consists of an external rotation module and an internal deflection module. The schematic diagram of the overall structure of the motor is shown in Fig. 1, and the overall appearance is shown in Fig. 2.

The outer rotation module of the motor is the inner rotor structure, the stator windings are 36 slots 2 laminated windings, the permanent magnets of the rotor are 4-pole Halbach arrays arranged permanent magnets, each pole is divided into six parts, each part of permanent magnet is magnetized in a sinusoidal direction law. The overall structure of the outside rotation module is shown in Fig. 3. The inner deflection module is located inside the rotor yoke of the outer rotation module, as shown in Fig. 4, the rotor yoke of the outer rotation module is the stator shell of the inner deflection module. The inner deflection module stator windings are concentrated windings, and the four rows of stator windings and winding yokes are distributed in the vertical direction on the spherical wall of the stator spherical inner cavity. Each row of stator windings is composed of nine winding coils. The four-row rotor permanent magnets correspond to the stator windings and are fixed inside the spherical cavity of the stator spherical shell. The deflection permanent magnets are six-pole Halbach array permanent magnets, and each pole is divided into three parts. The permanent magnet yoke of the rotor fix with the joint bearing and the stator shell. The output shaft of the motor fixed at the geometric center of the yoke of the inner deflection module protrudes through the active opening at the center of the top cover of the motor, and outputs the rotation and deflection torque with three degrees of freedom movement.

Three-phase alternating current is applied to the stator windings of the outer rotation module to generate a rotating magnetic field, interact with the rotor Halbach array permanent magnets to drive the rotation module rotor to rotate, the rotation speed is the synchronous rotation speed. The applied current of the internal deflection module windings are DC to generate a N-pole or S-pole magnetic field that interacts with the rotor Halbach array permanent magnet to drive the internal deflection module rotor to move. The parameters of this motor are shown as Table 1.

3 Halbach array permanent magnet structure and magnetic field characteristics

3.1 Analysis of magnetic field of Halbach array permanent magnet by analytic method

3.1.1 Analytical result of magnetic field of rotation permanent magnet

Halbach array magnetic ring has two kinds of structure, named outer rotor structure and inner rotor structure; the magnetic lines of external rotor structure permanent magnet are concentrated in the inner side of the permanent magnet magnetic ring, the outer magnetic field of this permanent magnet magnetic ring is shielded; the magnetic lines of inner rotor structure permanent magnet are concentrated on the outside of the permanent magnets, and the magnetic fields inside the permanent magnet rings are shielded. Since the outer rotation module and the inner deflection module both have an inner rotor and outer stator structure, the inner rotor Halbach array permanent magnet ring is used as the permanent magnet in this motor.

The rotation permanent magnet is a quadrupole Halbach array permanent magnet. Each permanent magnet is divided into six segments to magnetize, and the segmented magnetization of the rotation permanent magnet is shown in Fig. 5 as follows:

The solution domain is placed in the polar coordinate system. The Laplace equation of the air gap magnetic field of the rotating permanent magnet is solved by the discrete variable method, and the analysis results of the air gap flux density on the outer side of the permanent magnet are obtained.

In order to distinguish and calculate the characteristics of magnetic flux density in the outer air gap of the rotating permanent magnet, the solution domain of the rotating permanent magnet is divided into three regions. The magnetic field outside the permanent magnet is region 1, the magnetic field on the permanent magnet is region 2, and the magnetic field inside the permanent magnet is the region 3. The division of the solution domain of the permanent magnet of the rotation is shown in Fig. 6.

The general solution of the Laplace equation in polar coordinates is as follows: $\begin{eqnarray}\displaystyle {\varphi}(r,{\theta})=\left[\begin{array}{@{}l@{}}\displaystyle \mathop{\sum }_{k}(A_{k}r^{k}+B_{k}r^{-k})\cos (k{\theta})+\displaystyle \mathop{\sum }_{k}(C_{k}r^{k}+D_{k}r^{-k})\sin (k{\theta})\\ \quad +∼(A_{0}\ln r+B_{0})(C_{0}{\theta}+D_{0})\end{array}\right] & & \displaystyle\end{eqnarray}.$ (1)

Because the Halbach array permanent magnet adopts segmented magnetization, the magnetization angles of each segment of the magnet are different from each other. In order to further explain the influence of the number of permanent magnets on the magnetic field characteristics of the Halbach array permanent magnet, the nth block is now used. The relationship between the residual magnetization of the magnet and the parameter n is expressed in the Cartesian coordinate system as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M_{x}n=M\cos \left[(1-p)\frac{360(n-1)}{2pn}\right]\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M_{y}n=M\sin \left[(1-p)\frac{360(n-1)}{2pn}\right]\end{eqnarray}$ (3) where M is the residual magnetization of the permanent magnet, M _x is the X-axis component of the residual magnetization of the nth permanent magnet of the Halbach array permanent magnet, and M _y is the y-axis component of the residual magnetization of the nth permanent magnet of the Halbach array permanent magnet.

In the polar coordinate system, the expression of the residual magnetization of the nth permanent magnet is: $\begin{eqnarray}\displaystyle M_{n}=\left[\begin{array}{@{}c@{}}M_{r,n}\\ M_{{\theta},n}\end{array}\right]=M\left[\begin{array}{@{}l@{}}\cos ({\theta}-{\theta}_{m,n})\\ -\sin ({\theta}-{\theta}_{m,n})\end{array}\right]. & & \displaystyle\end{eqnarray}$ (4)

The expression of the magnetization angle θ_{m, n} of the nth permanent magnet of the Halbach array permanent magnet is: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\theta}_{m,n}=(1\pm p){\theta}_{n}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\theta}_{n}=\frac{n-1}{pl}{\pi}\quad (n=1,2,3\ldots 2pl)\end{eqnarray}$ (6) where p is the number of pole pairs of the Halbach array permanent magnet, “+” sign corresponds to the outer rotor structure, “−” sign corresponds to the inner rotor structure, l is the number of permanent magnet segments per pole, and θ_n is the angle between the centerline of nth piece of permanent magnet and horizontal line of 0 angle.

The Fourier series expansion of M _r and M _θ is: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M_{r}=\frac{M_{r0}}{2}+\mathop{\sum }_{n}M_{rn}\cos (np{\theta})\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M_{{\theta}}=\mathop{\sum }_{n}M_{{\theta}n}\sin (np{\theta}).\end{eqnarray}$ (8) The Fourier series expansion is: $\begin{eqnarray}\displaystyle M_{r0}=\frac{1}{{\pi}/p}\int _{-{\pi}/p}^{{\pi}/p}M_{r}d{\theta}=\frac{2p}{{\pi}}\int _{0}^{{\pi}/p}M_{r}d{\theta}=0 & & \displaystyle\end{eqnarray}$ (9) while np ≠1: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M_{rn}=\left\{\begin{array}{@{}l@{}}{\displaystyle \frac{2pl}{{\pi}}}M\sin \left({\displaystyle \frac{1\pm p}{2pl}}{\pi}\right){\displaystyle \frac{(-1)^{\frac{n\pm 1}{2l}}}{np+1}}(n=2lj\mp 1)\\ -∼{\displaystyle \frac{2pl}{{\pi}}}M\sin \left({\displaystyle \frac{1\pm p}{2pl}}{\pi}\right)\frac{(-1)^{\frac{n\pm 1}{2l}}}{np-1}(n=2lj\mp 1)\\ 0\quad (else)\end{array}\right\}\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M_{{\theta}n}=\left\{\begin{array}{@{}l@{}}{\displaystyle \frac{2pl}{{\pi}}}M\sin \left({\displaystyle \frac{1\pm p}{2pl}}{\pi}\right)\frac{(-1)^{\frac{n\pm 1}{2l}}}{np+1}(n=2lj\mp 1)\\ -∼{\displaystyle \frac{2pl}{{\pi}}}M\sin \left({\displaystyle \frac{1\pm p}{2pl}}{\pi}\right)\frac{(-1)^{\frac{n\pm 1}{2l}}}{np-1}(n=2lj\mp 1)\\ 0\quad (else)\end{array}\right\}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{while }np=1:M_{r1}=M_{{\theta}1}=M\frac{\sin \frac{{\pi}}{l}}{\frac{{\pi}}{l}}.\end{eqnarray}$ (12)

The expression of B _{1 r}, the magnetic field density radial component of region 1, is shown as below: $\begin{eqnarray}\displaystyle B_{1r}=-\mathop{\sum }_{n}G_{1}\times \left[\left(\displaystyle \frac{r}{r_{2}}\right)^{np-1}\left(\displaystyle \frac{r_{m2}}{r_{2}}\right)^{np+1}+\left(\displaystyle \frac{r_{m2}}{r}\right)^{np+1}\right]\cos (np{\theta}). & & \displaystyle\end{eqnarray}$ (13)

Because the rotation module of permanent magnet hybrid drive three-degree-of-freedom motor is an inner rotor structure, the magnetic field of the Halbach array permanent magnet is region one, and since the rotor yoke of the rotation module is the stator shell of the deflection module, the ferromagnetic material is used as the material of rotor yoke. Therefore, in order to analyze the air gap magnetic field in area 1, the expression of G ₁ in the above formula is obtained as:

While np ≠1: $\begin{eqnarray}\displaystyle G_{1}=\displaystyle \frac{{\mu}_{0}np\left\{\displaystyle \frac{M_{rn}-M_{{\theta}n}}{np+1}-\displaystyle \frac{M_{rn}+M_{{\theta}n}}{np-1}\left(\frac{r_{m1}}{r_{m2}}\right)^{2np}+\frac{2(M_{rn}+npM_{{\theta}n})}{(np)^{2}-1}\left(\frac{r_{m1}}{r_{m2}}\right)^{np+1}\right\}}{\left\{\left[({\mu}_{r}+1)\left(\displaystyle \frac{r_{m1}}{r_{2}}\right)^{2np}-({\mu}_{r}-1)\left(\displaystyle \frac{r_{m1}}{r_{m2}}\right)^{2np}\right]-\left[({\mu}_{r}+1)-({\mu}_{r}-1)\left(\frac{r_{m2}}{r_{2}}\right)^{2np}\right]\right\}}. & & \displaystyle\end{eqnarray}$ (14)

While np =1: $\begin{eqnarray}\displaystyle G_{1}=\displaystyle \frac{{\mu}_{0}\left\{\displaystyle \frac{M_{r1}-M_{{\theta}1}}{2}\left[1-\left(\frac{r_{m1}}{r_{m2}}\right)^{2}\right]-(M_{r1}+M_{{\theta}1})\left(\frac{r_{m1}}{r_{m2}}\right)^{2}\ln \left(\displaystyle \frac{r_{m1}}{r_{m2}}\right)\right\}}{\left\{\left[({\mu}_{r}+1)\left(\displaystyle \frac{r_{m1}}{r_{2}}\right)^{2}-\left({\mu}_{r}-1\right)\left(\frac{r_{m1}}{r_{m2}}\right)^{2}\right]-\left[({\mu}_{r}+1)-({\mu}_{r}-1)\left(\displaystyle \frac{r_{m2}}{r_{2}}\right)^{2}\right]\right\}}. & & \displaystyle\end{eqnarray}$ (15) Figure 7 is the B _{1 r} of rotation permanent magnet changes with the spatial angle 𝜑 and θ calculated by analytical method.

3.1.2 Magnetic field analysis result of deflection module

As shown in Fig. 8, the magnetic field of deflection permanent magnets is also divided into three regions: Zone 1 is the outer air gap magnetic field of the permanent magnet, Zone 2 is the inner magnetic field of the permanent magnet, and Zone 3 is the air gap magnetic field inside the permanent magnet. Since the deflection permanent magnet is the inner rotor structure, the magnetic characteristics of the area 1 are analyzed with emphasis. The analysis process is similar to the rotation module, so it’s not going to be covered in detail.

The scalar magnetic field of the deflection permanent magnet is as shown in the following formula: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\Phi}_{1}=\displaystyle \mathop{\sum }_{n=0}^{\infty }\displaystyle \mathop{\sum }_{m=n}^{n}B_{n1}^{m}\cdot \frac{1}{r^{n+1}}Y_{n}^{m}({\theta},{\varphi})\\ {\Phi}_{2}=\displaystyle \mathop{\sum }_{n=0}^{\infty }\displaystyle \mathop{\sum }_{m=-n}^{n}\left(A_{n2}^{m}\cdot r^{n}+B_{n2}^{m}\cdot \frac{1}{r^{n+1}}\right)Y_{n}^{m}({\theta},{\varphi})\\ {\Phi}_{3}=\displaystyle \mathop{\sum }_{n=0}^{\infty }\displaystyle \mathop{\sum }_{m=-n}^{n}A_{n3}^{m}\cdot r^{n}Y_{n}^{m}({\theta},{\varphi}).\end{array}\right. & & \displaystyle\end{eqnarray}$ (16)

The magnetic flux density on the outer side of the deflection permanent magnet can be expressed as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{1,r}=\mathop{\sum }_{n=0,\ldots }^{\infty }\mathop{\sum }_{m=-n}^{n}\left((n+1){\mu}_{r}B_{n}^{m}\frac{1}{r^{n+2}}\right)Y_{n}^{m}({\theta},{\varphi})\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{1,{\theta}}=\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{m=-n}^{n}\left(-{\mu}_{r}B_{n}^{m}\frac{1}{r^{n+2}}\right)\frac{\partial Y_{n}^{m}({\theta},{\varphi})}{\partial {\theta}}\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{1,{\varphi}}=\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{m=-n}^{n}\left(-{\mu}_{r}B_{n}^{m}\frac{1}{r^{n+2}}\right)\frac{1}{\sin {\theta}}\frac{\partial Y_{n}^{m}({\theta},{\varphi})}{\partial {\theta}}.\end{eqnarray}$ (19)

The magnetic field of region 1 is the main part of focus, because the air gap magnetic field of deflection PM (permanent magnet) is outside the deflection permanent magnet. So B _{1 r} of deflection permanent magnet changes with spatial angle 𝜑 and θ is shown as Fig. 9.

3.2 Finite element method analysis

An electromagnetic analysis software is used to apply the approximate segmentation principle to discretize the three-dimensional motor model elements. Then the model is solved and reorganized in each subunit. The PM hybrid-driven three-degree-of-freedom motor outer rotation module and inner deflection module were analyzed. The Halbach array PM and its excellent characteristics compared with parallel magnetization and radial magnetization PM were further explained. The necessity of applying the Halbach structure to the permanent magnets of the rotation module and the deflection module is verified.

3.2.1 Permanent magnet structure and magnetic field characteristics of rotation module

The structure of rotation permanent magnet is shown in Fig. 10: rotation permanent magnets contain four-pole ring-shaped permanent magnets, per-pole permanent magnet consists of six pieces of permanent magnets, N-S poles alternately arranged. The rotating permanent magnet is fixed on the outer side of the stator yoke of the rotating module, and the axial height of the permanent magnet is equal to the stator yoke of the rotating module.

The conventional magnetization method of permanent magnets are radial magnetization and parallel magnetization. The magnetic flux distribution of permanent magnet hybrid-driven three-degree-of freedom motor permanent magnet using parallel magnetizing method is shown in Fig. 11.

The magnetic flux distribution of the rotation permanent magnets which using radial magnetization is shown in Fig. 12:

Comparing Fig. 11 with Fig. 12, it can be obtained that the magnetic properties of radial magnetized permanent magnets are significantly better than that of parallel magnetized permanent magnets, and the strength of the magnetic field inside the magnetic ring is strong both of them. Since the permanent magnet hybrid driven three-degree-of-freedom motor is composed of internal and external modules, in order to prevent the occurrence of magnetic interference of the external rotation PM to the internal deflection PM, parallel magnetization PM or Radial magnetization PM is not appropriate to be selected as the permanent magnet of this motor.

The magnetization of Halbach array permanent magnets are divided into two types: ideal magnetization method and dividing one pole with pieces magnetization method. There is no physical segmental gap between each pole of the ideal magnetization permanent magnet ring, and the four poles PM is a whole part. The permanent magnet is magnetized by a sinusoidal function, and the distribution of the magnetic lines of the Halbach ideal magnetized permanent magnet of the outer rotation module rotor is shown in Fig. 13:

From Fig. 13, it can be seen that ideal Halbach array permanent magnet has no magnetic field lines inside the permanent magnets, and the internal closed loop of the magnetic lines mainly concentrates on the permanent magnet magnetic rings; the external magnetic lines of permanent magnet are radially and parallely magnetized and distributed more uniformly, and there are more flux lines. So using the outer rotor module type as Halbach array magnetization permanent magnet, not only optimize the sinusoidal characteristics of the air gap magnetic field, but also achieve the magnetic shielding of the internal deflection module.

Using ideal Halbach magnetization as the permanent magnet magnetization method is difficult and the process is more complicated. In order to reduce the production cost and ensure the sinusoidal of the external air gap magnetic field of the Halbach permanent magnet is not greatly affected, the Halbach array magnetization which divided one pole into six segments is used, and each part of permanent magnet is magnetized in a sine law in the relative coordinate system, which simulates the ideal Halbach permanent magnet array better.

Figure 14 is a magnetic flux distribution diagram of a segmented magnetized permanent magnet in a Halbach array, and Fig. 15 is a schematic diagram of flux density vector of the permanent magnet in Fig. 14. As shown in Figs 14 and 15, the magnetic field of the permanent magnet of the rotor is concentrated on the periphery of the permanent magnet. The magnetic field strength inside the magnet is almost zero, the flux density vector enters the permanent magnet from the S pole, the magnetic flux path does not include the rotor yoke, and the magnetic flux density vector flows from S pole to the N pole inside the permanent magnet block, and flows out of the N pole PM, and a closed loop of N pole to S pole is formed outside the permanent magnet.

The spatial three-dimensional distribution of magnetic flux density B changes along the spatial angle 𝜑, θ of Halbach array permanent magnets (six-segment structure per pole) is shown in Fig. 16: air-gap flux density does not change significantly along θ direction. It varies periodically along the 𝜑 direction. As the 𝜑 angle ranges from 0 to 360 degrees, the air gap flux density changes by 180 degrees, and the peak value is at 𝜑 = 0, 50, 250, and 300 degrees which is close to 0.5 T; the change cycle of the air gap magnetic flux edge along the 𝜑 angle corresponds to the quadrupole structure of the rotor permanent magnet; the air gap magnetic flux density along the θ angle does not change and it corresponds to the up and down vertical structure of the rotor columnar permanent magnet, in the vertical direction on a cylindrical surface, the air gap magnetic field does not change significantly.

Figure 17 is a spatial three-dimensional distribution diagram of the r, 𝜑, and θ components of the rotating Halbach array permanent magnet (each pole is divided into six segments) magnetic flux density in the spherical coordinate system with space angle 𝜑, θ (FEM).

As shown in Fig. 17, the radial compoment Br of the outer air gap magnetic field of the permanent magnet has no significant change in the direction of θ angle, and it shows a periodic distribution similar to the sine function in the range of 𝜑 angle from 0 to 360 degrees, the period is 180 degrees, corresponding to the quadrupole structure of the rotating permanent magnet; the peak is located at 𝜑 = kπ∕2 and the value is approximately 0.3 T, smaller than the overall magnetic peak amplitude of the air gap magnetic field. For the 𝜑 component B𝜑, there is no significant change in the θ angle direction, and it changes periodically within the 𝜑 angle change interval with a period of 180 degrees. The peak is at 𝜑 =50 degrees and 250 degrees and is close to 0.25 T, which is slightly less than the radial magnetic density Br; can be seen from the figure that the θ component B _θ amplitude is very small, the peak value is about 0.06T, the distribution trend is not obvious, the rotation segmented permanent magnet outer air gap magnetic field mainly has a radial component and a 𝜑 component. The three-dimensional distribution of r, 𝜑, and θ components of the air-gap flux density in Fig. 14 corresponds to the analytical formulas (13) and (14), which verifies the accuracy of the finite element model.

The space three-dimensional diagram of rotational Halbach array permanent magnet (ideal magnetization) outside air gap magnetic field flux density along the spatial angle 𝜑, θ is shown in Fig. 18: the permanent magnet air gap magnetic flux sweep in 𝜑 direction is distributed as sine function, the function cycle is 180 degrees, ideal sinusoidal permanent magnet air gap magnetic field sinusoidal is better than the segmented permanent magnet, and it can be seen from the figure, the ideal magnetic field strength is greater than that of Segmented magnetized permanent magnet.

Figure 19 shows the air gap radial magnetic flux density of the 3-DOF motor.

It can be seen from Fig. 19 that the radial magnetic density waveform of the Halbach permanent magnet array is a sine wave, and the harmonic content is very small. The radial magnetic density reaches its maximum point at the junction of the NS poles, and the maximum value is 0.3 T. In a change cycle of the 𝜑 angle from 0 degrees to 360 degrees, the radial flux density change waveform has two peaks, two troughs corresponding to the quadrupole structure of the Halbach array permanent magnet of the rotation module.

Figure 20 shows the amplitude of each harmonic component of the radial magnetic density waveform of the air-gap field at a distance of 5 mm from the outer edge of the permanent magnet in a Halbach array (six segments per pole). It can be seen from the figure that the amplitude of the fundamental wave component of the air gap magnetic field waveform is the largest, and the amplitude of the fundamental wave is much larger than the other harmonic amplitudes, except for the fundamental wave, the 2th harmonic and 7th harmonic amplitude are also relatively large, which is the main factor affecting the sinusoidal radial magnetic density waveform of the external air gap magnetic field of the permanent magnet. Therefore, in order to further optimize the air gap magnetic field characteristics, weaking the 2th harmonic and 7th harmonic can be the main means, and the rest of the harmonic amplitude has little influence on the fundamental wave and can be basically ignored.

3.2.2 Deflection module permanent magnet structure and magnetic field characteristics

The structure of internal deflection permanent magnet is shown in Fig. 21, due to Halbach array permanent magnets increase the outer air gap magnetic density sinusoidal and strength compared with conventional magnetized permanent magnets, the Halbach array permanent magnet structure is chosen as the internal deflection permanent magnet; deflection permanent magnet is divided into four groups, each group of permanent magnets is composed of NS poles which alternately arranged six-pole permanent magnets, taking into account Halbach array permanent magnet manufacturing process and production costs, each pole is divided into three segments.

Figure 22 is a magnetized magnetic density vector diagram of a Halbach array deflecting permanent magnet. Figure 23 shows the distribution of magnetic lines in the Halbach array deflection permanent magnet. In these figures, the red magnetic block represents the N pole and the black magnetic block represents the S pole. The magnetic field lines flow out from the N pole and flow in from the S pole. The outer gap magnetic field intensity of the permanent magnet is large, the inner magnetic field strength is almost 0, and the magnetic flux density vector and magnetic field lines are almost zero distribution. As can be seen from Fig. 22, each segment of permanent magnet has a different magnetization direction and changes sinusoidally in the relative coordinate system, which corresponds to the sinusoidal nature of the radial magnetic gap of the outer magnetic field magnetization.

Figure 24 shows the scalar cloud image of the deflection permanent magnet. As shown in the figure, the Halbach permanent magnet array does not have a magnetic field abrupt change phenomenon. The magnetic flux density of each pole is the same, indicates that the air gap magnetic field outside the deflection permanent magnet is uniformly distributed.

Figure 25 is a spatial three-dimensional distribution diagram of the magnetic flux density of the air gap magnetic field outside the deflection permanent magnet; the spatial angle 𝜑 ranges from 0 to 360 degrees, and the spatial angle θ ranges from −50 to 50 degrees. It can be seen from Fig. 25 that the magnetic flux density of the air gap magnetic field outside the deflection permanent magnet is periodically distributed with 𝜑, θ, and the variation curve with 𝜑 angle is approximated as a sine function variation curve with a period of 90 degrees and the peak value is approximately for 0.6 T, the peak appears at θ = kπ∕4; the change trend of the air gap flux density with θ changes is approximately cosine function, the function period is 120 degrees, and the peak value appears at θ = 60 + kπ∕4.

Figure 26 is a three-dimensional map of the spatial distribution of magnetic flux density along the r, 𝜑, and θ directions of the magnetic field along the outside of the deflection permanent magnet in the spherical coordinate system. The value of 𝜑 ranges from 0 to 360 degrees and θ ranges from −45 to 45 degrees.

4 Analysis of electromagnetic characteristics

4.1 Analysis of electromagnetic and torque characteristics of load condition of rotation module

Three-phase alternating current is applied to the three-phase windings of the stator of the rotation module. The distribution of magnetic lines of the rotation module is shown in Fig. 27. The distribution of magnetic lines is uniform and the magnetic field characteristics meet the requirements.

The scalar flux density distribution of the rotation module is shown in Fig. 28.

It can be seen from the Fig. 28 that under the load condition of the autorotation module, the magnetic field strength of the spinning winding part is weak, and the overall magnetic field distribution of the motor is uniform, which corresponds to the distribution of magnetic lines in Fig. 27.

Figure 29 shows the distribution of torque variation with time in transient mode. The transient model of motor rotation was established. The initial rotation speed and inertia of the motor are determined, and the rotor’s load torque and damping parameter are also set. The motion domain of the rotor is established and the above parameters are applied to the motion domain, and the starting and ending time of the motor is set with 0 to 2 s. Therefore, the change rule of the motor’s spin-torque in time between 0 and 2s can be obtained by using finite element analysis.

When the 20A current is applied to the stator winding, the motor rotation torque sweep with time is show in Fig. 29. As shown in this figure, the amplitude of torque curve is damping until 1.4 s, torque tends to stabilize over time, and the stable value is 5 N ⋅ m. The deflection torque characteristic is shown in Fig. 30. The maximum point of deflection torque is at 5° when the motor deflecting, and when the deflection angle approaching the point of 7°, the excitation strategy should be changed.

4.2 Analysis of no-load characteristics of rotation module

When the rotation module is unloaded and the rotating speed is 1500 r∕min, the curve of the back-EMF of the three-phase winding of the stator is shown in Fig. 31. From the figure, it can be seen that the sinusoidal of the back-EMF of the stator winding is high and the amplitude is 300 V. It shows that the sinusoidal of the air gap magnetic field of the rotor Halbach array permanent magnet is higher and the characteristics of rotation torque are better. The larger the no-load back electromotive force amplitude is, the larger the output torque of the motor is.

The trend of flux linkage is shown in Fig. 32. Since no-load back EMF is obtained by deriving the flux of an open-winding winding versus time, the change trend of no-load back EMF corresponds to the trend of the flux curve. Due to the low distortion rate of the no-load flux linkage waveform, Halbach permanent magnet utilization rate of magnetization force in the rotation module is high and is suitable for high-speed autorotation motion.

5 Analysis and calculation of loss of permanent magnet hybrid drive type 3-DOF motor

5.1 Analysis of motor loss by analytical method

5.1.1 Calculation of eddy current loss by analytical method

Since the stator windings of the rotation module are distributed windings, the current amplitude of the stator windings is large and there are a large number of harmonics, and the main magnetic field waveform generated by the stator excitation current will be greatly distorted and it contains a large amount of time and space harmonics. These harmonics’ velocities are different from the rotation speed of the permanent magnets of the rotor and thus generate relative motion with the permanent magnets of the rotor. In the permanent magnets of the rotor, a closed current line called eddy current is induces to generate eddy current loss; eddy current losses easily cause the temperature of the permanent magnets of the rotor to rise, so the permanent magnets magnetic permeability is weakened; in addition, the poor thermal stability of material NdFeB may lead to irreversible demagnetization, which directly affects the motor performance and service life. Therefore, weakening the permanent magnet eddy current loss is of great significance for improving the magnetic field characteristics of the motor.

According to the characteristic equation of the motor, the magnetic flux density of the rotor permanent magnet can be expressed as a superposition of time and space harmonics. The sub harmonics of the magnetic flux density in the permanent magnet poles which induces electromotive forces generated in the permanent magnet poles and the air gap magnetic fluxes passing through the permanent magnet poles are: $\begin{eqnarray}\displaystyle e_{mn} & = & \displaystyle {\omega}_{mn}{\phi}_{mn}\end{eqnarray}$ (20) $\begin{eqnarray}\displaystyle {\phi}_{mn} & = & \displaystyle 1/{\pi}B_{mn}{\tau}_{mn}h_{PM}\end{eqnarray}$ (21) $\begin{eqnarray}\displaystyle {\tau}_{mn} & = & \displaystyle {\tau}_{r}/[2p(m+n)]\end{eqnarray}.$ (22)

In the above formula, τ_mn is the pole pitch of the permanent magnet air gap magnetic field (unit is meter); h _PM is the length (unite is meter) of the magnetization direction of each permanent magnet of the Halbach permanent magnet array.

The induced eddy current density W _e is: $\begin{eqnarray}\displaystyle W_{e}={\sigma}\frac{de_{mn}}{dL} & & \displaystyle\end{eqnarray}$ (23) where σ is the conductivity of the Halbach permanent magnet, de _mn is the fluctuation value of the electromotive force between the cell nodes, and dL is the distance between the nodes.

The dynamic flux B in the permanent magnet is represented by the Fourier series decomposition as follows: $\begin{eqnarray}\displaystyle B=\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=1}^{\infty }[a_{mn}\cos ({\omega}_{mn}t)+b_{mn}\sin ({\omega}_{mn}t)]. & & \displaystyle\end{eqnarray}$ (24)

Therefore, the eddy current loss density of a Halbach permanent magnet can be expressed as: $\begin{eqnarray}\displaystyle W_{e}={\pi}^{2}{\sigma}h_{PM}^{2}\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=1}^{\infty }f_{mn}m^{2}n^{2}{\displaystyle \frac{a_{mn}^{2}+b_{mn}^{2}}{2}}. & & \displaystyle\end{eqnarray}$ (25)

The eddy current losses induced by Halbach permanent magnets can be expressed as: $\begin{eqnarray}\displaystyle P_{w}=\displaystyle \int _{0}^{2{\pi}}\int _{{\varphi}_{1}}^{{\varphi}_{2}}\int _{{\theta}_{1}}^{{\theta}_{2}}{\pi}^{2}{\sigma}h_{PM}^{2}\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=1}^{\infty }f_{mn}m^{2}n^{2}\frac{a_{mn}^{2}+b_{mn}^{2}}{2}{\rho}d{\rho}d{\varphi}dzdt. & & \displaystyle\end{eqnarray}$ (26)

Halbach permanent magnet eddy current loss can be expressed as a linear superposition of eddy current loss of per unit permanent magnet: $\begin{eqnarray}\displaystyle P_{w_{total}}=\displaystyle \mathop{\sum }_{j=1}^{40}P_{j}=\mathop{\sum }_{j=1}^{40}\int _{0}^{2{\pi}}\displaystyle \int _{{\varphi}_{1}}^{{\varphi}_{2}}\int _{{\theta}_{1}}^{{\theta}_{2}}{\pi}^{2}{\sigma}h_{PM}^{2}\mathop{\sum }_{m=1}^{\infty }\displaystyle \mathop{\sum }_{n=1}^{\infty }f_{mn}m^{2}n^{2}\frac{a_{mn}^{2}+b_{mn}^{2}}{2}{\rho}d{\rho}d{\varphi}dzdt. & & \displaystyle\end{eqnarray}$ (27)

5.1.2 Calculation of stator iron loss by analytical method

The stator yoke core loss of the permanent magnet hybrid-driven three-degree-of-freedom motor is composed of eddy current loss $P_{PU}^{class}$ , hysteresis loss $P_{PU}^{hys}$ and additional loss $P_{PU}^{exc}$ . Core loss formula is as follow: $\begin{eqnarray}\displaystyle P_{PU}^{core}=P_{PU}^{hys}+P_{PU}^{class}+P_{PU}^{exc}. & & \displaystyle\end{eqnarray}$ (28)

Considering the influence of alternating magnetic field and rotating magnetic field synthetically, in a stator core splitting unit, it is: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{pu}^{(hys)}=\mathop{\sum }_{K=1}^{N}C_{h}kf(B_{kmaj}^{n}+B_{k\min }^{n})\end{eqnarray}$ (29) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{pu}^{(class)}={\displaystyle \frac{{\pi}^{2}{\sigma}d^{2}}{6{\rho}}}f^{2}\mathop{\sum }_{k=1}^{n}k^{2}(B_{kmaj}^{2}+B_{kman}^{2})\end{eqnarray}$ (30) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{pu}^{(exc)}=C_{n}\frac{1}{T}\int _{0}^{T}\left[\left({\displaystyle \frac{dB_{x}(t)}{dt}}\right)^{2}+\left({\displaystyle \frac{dB_{y}(t)}{dt}}\right)^{2}\right]^{3/4}dt.\end{eqnarray}$ (31)

In the formula, C _h is the hysteresis loss coefficient of the stator silicon steel sheet material, n is an empirical coefficient, and generally has a value of 2, σ is the conductivity of the silicon steel sheet, d is the thickness of the iron core, 𝜌 is the mass density of the silicon steel sheet, and C _n is the additional loss coefficient.

In summary, the total stator core loss of a permanent magnet hybrid-driven three-degree-of-freedom motor is: $\begin{eqnarray}\displaystyle P_{Fe}=\int _{s}P_{pu}^{(core)}L_{Fe}{\rho}_{Fe}(x,y)ds=\mathop{\sum }_{e=1}^{Ne}P_{pu}^{(core)}{\rho}_{Fe}L_{Fe}{\rm\Delta}_{e}. & & \displaystyle\end{eqnarray}$ (32)

In the formula, N _e is the stator core split number of the permanent magnet hybrid-driven three-degree-of-freedom motor, L _Fe is the stator height of the motor, 𝜌_Fe is the mass density of the stator silicon steel, Δ_e is the stator split element area, P _Fe is the stator core loss value.

5.1.3 Calculation of stator winding copper loss by analytical method

According to Joule’s law, it is known that the winding loss is related to the work heat generated by the current passing through the winding, and the stator winding is generally made of copper material, so the expression of the winding copper consumption is: $\begin{eqnarray}\displaystyle P_{cu}=mI^{2}R & & \displaystyle\end{eqnarray}$ (33) where m is the number of motor winding phases, I is the value of the current through the winding, and R is the value of the equivalent resistance of the single-phase winding. Since the winding temperature rise will cause the winding resistance value to change, the influence of temperature on the winding resistance cannot be ignored: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R=R_{a}[1+{\beta}_{a}({\eta}-{\eta}_{a})]\end{eqnarray}$ (34) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{a}={\rho}\frac{2L_{av}N}{{\pi}a\left[N_{1}\left(\frac{d}{2}\right)\right]^{2}}.\end{eqnarray}$ (35)

Among them, R is the resistance value when the temperature is 𝜂_a, 𝜂_a is the initial temperature of the experiment, and 𝛽_a is the temperature coefficient of the winding. 𝜌 is the conductivity of the copper material, L _av is a coil length of a half turn, N is the number of series windings per phase of the winding, N ₁ is the number of parallel connections of per phase winding, a is the number of parallel branches, and d is the diameter of the copper wire winding.

5.2 Finite element analysis and calculation of motor loss

Permanent magnets eddy current losses, stator core iron losses and stator winding copper loss under Halbach array magnetization and radial magnetization methods are shown as Figs 24–26. The curves in the figure show the Halbach magnetization and radial magnetization modes respectively.

From Fig. 33, the eddy current loss of the Halbach permanent magnet rapidly rises to the maximum point at the start phases of the motor. After the motor runs smoothly, it gradually decreases until it stabilizes at about 2 w, while the value of eddy current loss of the radial magnetized permanent magnets tends to stabilize around 4 w. During the start and running period of the motor, the eddy current loss value of Halbach magnetized permanent magnets is smaller than that of radial magnetized permanent magnets; the eddy current losses of permanent magnets of the two magnetizing methods are basically the same tendency sweeping with time.

As shown in Fig. 34, when the loss fluctuates with time and tends to be stable and the permanent magnet adopts the Halbach magnetization method, the stator yoke core loss is lower than the value when the permanent magnet adopts the radial magnetization method.

Halbach magnetized permanent magnet stator yoke iron consumption is stable at about 7.5 w, while radial magnetized permanent magnet stator yoke iron consumption is stable at about 8.5 w, the difference is about 1W, the reason is the sinusoidal characteristics of the gap magnetic field of Halbach magnets are better.

As shown in Fig. 35, after the motor runs smoothly, the copper windings of the Halbach magnetizing array permanent magnet motor have the smaller copper losses, with an average value of about 10 w. Radially-magnetized permanent magnet motor windings have larger copper losses, with an average value of about 15 w. It can be seen from the figure that different magnetization methods have no effect on the trend of the loss with time, they only affect the size of the loss value.

Figure 36 is a comparison of radial magnetic density of an air gap magnetic field of Halbach permanent magnet with radial magnetic permanent magnet.

As shown in Fig. 36, the magnetic field characteristics of the Halbach permanent magnet array are much better than those of the radial-magnetized permanent magnets. The sinusoidal radial magnetic density is much better than that of the radial-magnetized permanent magnets. Due to the loss of the motor is related with the harmonic distribution of the air-gap magnetic field and the peak value. So, Halbach permanent magnets have better air-gap magnetic field characteristics and less harmonic content than radial-magnetized permanent magnets, which in turn leads to less motor losses. Therefore, the Halbach array arrangement of the rotor permanent magnets can significantly reduce the 3-DOF motor losses.

6 Optimization of structural parameters of a permanent magnet hybrid driven three-degree-of-freedom motor based on response surface methodology

The Response Surface Method was used as an optimization method. The basic idea is to construct a polynomial with a clear expression, and fit a complex unknown function relationship, consider the experimental error, and display the results of the analysis of the data in the form of an image, the best value within the range is solved. The Response Surface Method can be easily combined with practical problems, it greatly reduces the time-consuming and computational burden. Therefore, it is widely applied to biology, chemistry, engineering and other fields.

In this section, the Response Surface Method is used to optimize the rotation torque and deflection torque of the permanent magnet hybrid drive three degree of freedom motor. The structural parameters of the motor are mainly analyzed. Considering that the research purpose of the motor is to enable the motor to achieve multi-degrees of freedom movement with load, the motor efficiency and power factor are not the primary considerations. Therefore, the motor is optimized for the torque characteristics.

6.1 Using Taguchi algorithm to determine motor structure optimization

Taguchi algorithm is a global optimization algorithm based on orthogonal experiment design. Compared with the traditional algorithm, it has excellent ability of parameter combination design. When there are many changes in the parameters, it greatly reduces the number of tests and saves time. The purpose of motor optimization is to adjust the motor’s structural dimensions in a limited range so that the output torque of the output shaft is optimal which based on constant motor size (the overall height of the motor is 135 mm and the outer radius of the stator of the rotation module is 135 mm). In the past, a multi-degree-of-freedom motor was difficult to achieve rotation and deflection at the same time. By controlling a set of coils, multi-degree-of-freedom motion of the motor is achieved. The permanent magnet hybrid driven three-degree-of-freedom motor adopts a nested structure design. Two sets of coils control the motor’s rotation and deflection respectively. According to the structural characteristics of the motor, the air gap length between the stator and the rotor, the thickness of the permanent magnet, the length of the permanent magnet and the outer diameter of the permanent magnet of the three-degree-of-freedom motor are initially selected as the parameters of the rotation. Besides, the distance between the coil and the permanent magnet σ, the permanent magnet thickness H ₁, the permanent magnet length L ₁ and the inner diameter R ₁ are the rotation parameter variables. The distance between the inner deflection module coil and the permanent magnet d, the permanent magnet thickness H ₂, the permanent magnet length L ₂ and the deflection inner diameter R ₂ are the deflection parameter variables. Figure 26 shows the motor optimization parameters.

Three levels of variables A, B, and C, were taken for each optimization parameter, and the values are shown in Table 2.

According to the Taguchi algorithm, L 9 (34) orthogonal tables were selected and an experimental matrix was established. Where L represents an orthogonal matrix, 9 is the number of trials, 3 is the number of values for each parameter variable, and 4 is the number of optimization parameters. According to the experimental matrix, the maximum torque T1 and the maximum deflection torque T2 of the motor are obtained by finite element analysis.

According to the results of the orthogonal experiment and finite element analysis designed in Table 3, the mean and variance analysis are further performed to estimate the influence of the change of the parameters on the torque and the importance of determining the change of the parameters to the torque. The proportion of each optimization parameter to the torque performance index can be calculated by the following formula: $\begin{eqnarray}YY=3\mathop{\sum }_{i=1}^{3}[m_{x_{i}}(T_{i})-m(T)]^{2}\end{eqnarray}$ where YY is the ratio of the various parameter variables to the performance indicators, x _i is the optimized parameter variable, m _xi(T _i) is the average value of the parameter variable x under the ith value variable, and m (T) is the average value of T.

From Table 4, it can be clearly seen that the change of each parameter variable has an effect on the motor torque. The proportions of rotation outside diameter and deflection inner diameter are 5.1% and 1.6% respectively, and the impact on the corresponding torque is very small. On the contrary, other structural parameters produce a relatively large impact rate. Therefore, when the selected optimization objectives are different, the degree of influence of each parameter variable on the optimization goal can be determined, and the best combination among each parameter variable can be obtained.

6.2 Response surface method CCD experiment design and results

Using Desidn Expert V8.0.6 software, the air gap length between the stator and rotor σ, the thickness of permanent magnet H ₁, the length of permanent magnet L ₁, the distance between the coil and the permanent magnet D, the thickness of permanent magnet H ₂ and the length of the permanent magnet L ₂ are taken as the parameter variables and the motor’s rotation torque T ₁ and deflection torque T ₂ are taken as the response value. The central composite design method (CCD) uses 3-variable 5-level and 6-center points, and in order to ensure that the model has rotation with the other structural parameters remain unchanged of the motor, the value of a is 1.682. The design of the parameter variables and the corresponding code conversion values are shown in Table 5, and the experimental results of the experimental points and finite element analysis are shown in Table 6.

6.3 Model error and response surface analysis

Based on the data fitting of the least squares method, through the analysis and processing of the above two sets of response surface experimental results, the regression equations with the rotation torque and the deflection torque as the response values are obtained respectively. $\begin{eqnarray}\displaystyle Y_{1} & = & \displaystyle -1246.49592+12.91747\cdot A-1.58653\cdot B+18.71231C+0.28884\cdot A\cdot B\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼0.070842\cdot A\cdot C-0.13421\cdot A^{2}-0.88318\cdot B^{2}-0.070596\cdot C^{2}\nonumber\end{eqnarray}$ where Y ₁ is the autorotation torque T ₁(Nm), A is the thickness of the permanent magnet H ₁(mm), B is the length of the permanent magnet L ₁(mm), and C is the length of the air gap between the stator and rotor σ(mm). $\begin{eqnarray}\displaystyle Y_{2} & = & \displaystyle -0.93313+0.35042\cdot A+0.070704\cdot B-0.14573\cdot C-0.056683\cdot A\cdot C\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle-6.14844\cdot 10^{-3}\cdot B\cdot C -0.010890\cdot A^{2}-6.851\cdot 10^{-4}\cdot B^{2}+0.072228\cdot C^{2}\nonumber\end{eqnarray}$ where Y ₂ is the deflection torque, A is the thickness of the permanent magnet H ₂(mm), B is the length of the permanent magnet L ₂(mm), and C is the distance between the coil and the permanent magnet d (mm).

The variance of the response surface model of the rotation torque deflection torque of a permanent magnet hybrid-drive three-degree-of-freedom motor is analyzed and the Table 7 is obtained.

From Table 7, the P values of both models are less than 0.01, indicating that the quadratic regression model is extremely significant and the test scheme is reliable. The P values of the missing items are all greater than 0.05, indicating that there is no missing factor, and the response model fitted well to the test results with less error. The fitting accuracy of the two models is 0.9978 and 0.9947 (both greater than 0.9000), indicating that more than 90% of the experimental data can be explained by the model, and the experimental data can be fully reflected by the model.

According to the quadratic regression equation, the three-dimensional response surface among the self-rotating torque and the yaw torque of the motor and each structural parameter variable are shown in Fig. 37.

From Figs 37(a) and 37(b), the magnitude of the response surface can be seen that the permanent magnet thickness of the motor rotation part has a greater influence on the response surface. As the length of the air gap between the stator and rotor increases, the motor’s rotation torque is reduced. The figure (c) shows that the interaction between the length of the permanent magnet and the air gap length between the stator and the rotor is not significant, and the length of the permanent magnet has little effect on the autorotation torque. From figures (d) and (e) it can be seen that the magnitude of the change in the response surface is large, indicating that the interaction between the thickness of the permanent magnet and the length of the permanent magnet in the deflection part of the motor has a significant effect on the response value. The figures (d) and (e) show that as the thickness of the permanent magnet increases, the deflection torque increases, and (d) and (f) show that the deflection torque decreases as the distance between the coil and the permanent magnet increases. The figure (f) shows that the interaction between the length of the permanent magnet and the distance between the coil and the permanent magnet is not significant, and the former has little influence on the deflection torque of the motor.

In order to obtain the optimal structure size of the permanent magnet hybrid-driven three-degree-of-freedom motor structure, Analytical software was used to perform the optimal solution to the response surface model. At the same time, taking into account the manufacturing cost of the motor, the manufacturing process level and the efficiency of the motor, the air gap length between the stator and rotor is 1.2 mm, the thickness of the permanent magnet is 13 mm, the length of the permanent magnet is 130 mm, and the length of the permanent magnet named d which is the distance between the coil and the permanent magnet is 0.8 mm, the thickness of the permanent magnet is 6.5 mm, and the length of the permanent magnet is 26 mm, these parameters optimize the structure of the three-degree-of-freedom motor and make the rotation torque and the deflection torque reach the maximum value.

7 Conclusion

In this paper, a novel permanent magnet type 3-DOF hybrid driven motor is proposed. Using analytical method and finite element method, the magnetic field characteristics and distribution of the permanent magnet of the motor are analyzed, the electromagnetic characteristics and output torque of the motor is analyzed by finite element method. The loss of this motor is analysed and calculated. With the application of taguchi method and central composite design method, the motor response surface equation was built. The optimal structure parameters of motor is proposed by the numerical model based on maximizing the motor output torque of the target, and it is compared with finite element and experiment to verify the accuracy of the structural parameters.

Footnotes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (nos. 51577048, 51637001, 51107031), the Natural Science Foundation of Hebei Province of China (no. E2018208155), the Overseas Students Science and Technology Activities Funding Project of Hebei Province (no. C2015003044), the Hebei Industrial Technology Research Institute of Additive Manufacturing (Hebei University of Science and Technology) open projects funding, the National Engineering Laboratory of Energy-saving Motor & Control Technique, Anhui University (no. KFKT201804), Key Project of Science and Technology Research in Hebei Provincial Colleges and Universities (ZD2018228).

References

Lee

H.-J.

Ahn

H.-W.

Lee

J.-K.

, Newly proposed hybrid type mutli-DOF operation motor for multi-copter UAV systems, IEEE Energy Conversion Congress and Exposition (2015), 2782–2790.

Lee

H.J.

Park

H.J.

Ryu

G.H.

, Performance improvement of operating three-degree-of-freedom spherical permanent-magnet motor, IEEE Transactions on Magnetics 48(11) (2012), 4654–4657.

Lun

Xing

and Gao

, Analysis and implementation of a 3-DOF deflection-type PM motor, IEEE Transactions on Magnetics 51(11) (2015), 1–4.

Xiao

and Zhao

, Design and analysis of a novel flexure-based XY micro-positioning stage driven by electromagnetic actuators, in: International Conference on Fluid Power and Mechatronics (FPM), Beijing, China, 2011, pp. 953–958.

Wang

and Jiang

, Magnetic field computation of a PM spherical stepper motor using integral equation method, IEEE Transactions on Magnetics 42(4) (2006), 731–734.

Zhang

Wang

and Yan

, Design and comparison of a novel 3-DOF deflection type permanent magnet actuator, International Journal of Applied Electromagnetics and Mechanics 46(1) (2014), 229–242.

Sun

and Liu

, Modeling and levitation control of a novel M-DOF actuator based on neural network, International Journal of Applied Electromagnetics and Mechanics 38(4) (2012), 217–230.

Oner

and Altinta

, Computer aided design and 3D magnetostatic analysis of a permanent magnet spherical motor, Journal of Applied Sciences 7(22) (2007), 3400–3409.

Davey

Vachtsevanos

and Powers

, The analysis of fields and torques in spherical induction motors, IEEE Transactions on Magnetics 23(1) (1987), 273–282.

10.

Zheng

, Robust control of PM spherical stepper motor based on neural networks, IEEE Transactions on Industrial Electronics 56(8) (2009), 2945–2954.

11.

Raye

and Lee

K.-M.

, Finite element torque modeling for the design of a spherical motor, in: Proceedings of the 7th International Conference on Control, Automation and Robotics and Vision, Singapore, 2002, pp. 390–395.

12.

Kawase

Ota

and Fukunaga

, 3-D eddy current analysis in permanent magnet of interior permanent magnet motors, IEEE Transactions on Magnetics 36(4) (2002), 1863–1866.

13.

Yamazaki

, Torque and efficiency calculation of an interior permanent magnet motor considering harmonic iron losses of both the stator and rotor, IEEE Transactions on Magnetics 39(3) (2003), 1460–1463.

14.

Al-Naemi

F.I.

and Moses

A.J.

, FEM modeling of rotor losses in PM motors, Journal of Magnetism & Magnetic Materials 304(2) (2006), 794–797.

15.

Lai

Y.S.

Lin

J.C.

and Wang

J.J.

, Direct torque control induction motor drives with self-commissioning based on Taguchi methodology, IEEE Transactions on Power Electronics 15(6) (2000), 1065–1071.