Design, Analysis and Test of a Hyperbolic Magnetic Field Voice Coil Actuator for Magnetic Levitation Fine Positioning Stage

: The multi-degree-of-freedom high-precision positioning system (MHPS) is one of the key technologies in many advanced industrial applications. In this paper, a novel hyperbolic magnetic ﬁeld voice coil actuator using a rhombus magnet array (HMF-VCA) for MHPS is proposed. Beneﬁting from the especially designed rhombus magnet array, the proposed HMF-VCA has the advantage of excellent force uniformity, which makes it suitable for multi-degree-of-freedom high-precision positioning applications. First, the basic structure and operation principles of the HMF-VCA are presented. Second, the six-degree-of-freedom force and torque characteristic of the HMF-VCA is studied by three-dimensional ﬁnite element analysis (3-D FEA). Third, the inﬂuence of structural parameters on force density and force uniformity is investigated, which is conducive to the design and optimization of the HMF-VCA. Finally, a prototype is constructed, and the comparison between the HMF-VCA and conventional VCAs proves the advantage of the proposed topology.


Introduction
The multi-degree-of-freedom high-precision positioning system (MHPS) is one of the key technologies in many advanced industrial applications, such as semiconductor lithography, scanning tunneling microscope, chemistry and biomedical science [1][2][3][4][5][6][7]. Generally, the MHPS can be realized by three solutions: stacking linear actuators, planar actuators [1,8,9], and combining coarse positioning stage (CPS) and fine positioning stage (FPS) [10][11][12]. Benefiting from the two-stage structure, the third solution becomes a widely adopted scheme, which can simultaneously satisfy the requirements of long stroke and high accuracy. Among various FPSs, the magnetic levitation fine positioning stage (MLFPS) is considered to be the state of the art because of its feature of no mechanical contact. In the MLFPS, a stage that integrates multiple actuators and sensors is often adopted, in which the actuator is an electromagnet [13][14][15], or a voice coil actuator (VCA) [12,16,17]. As a key component of the MLFPS, the performance of actuators strongly determines the MLFPS's positioning ability. Compared with single-degree-of-freedom applications, the MLFPS has special technical requirements for the actuators because of its multi-degree-of-freedom motion. When an actuator works in multi-degree-of-freedom, not only will its output force in the driving direction be influenced by the motion in the other directions, but it will also produce parasitic force and torque, which will reduce the MLFPS's performance and increase the control system's complexity [18]. Therefore, the actuators used in the MLFPS should be designed to be able to output a force which is independent from the six-degree-of-freedom motion, and in this paper the ability is named as force uniformity. In this paper, we put forward a novel hyperbolic magnetic field voice coil actuator using a rhombus magnet array (HMF-VCA) to achieve this goal, and focus on the analysis of its complex six-degree-of-freedom force and torque characteristics. Figure 1 shows the structure of the proposed HMF-VCA and MLFPS. The HMF-VCA consists of two components, i.e., stator and mover. The stator consists of two rectangular stator coils with cooling duct, and a non-ferromagnetic stator frame. The mover consists of a rhombus magnet array, a non-ferromagnetic auxiliary frame, and a non-ferromagnetic mover frame. The rhombus magnet array consists of four cubic mover magnets which are arranged on the four edges of a rhombus, and the magnetization is shown as arrows. The MLFPS is made up of a load stage, a fixed base, three vertical HMF-VCAs, three horizontal HMF-VCAs, three magnetic springs, six relative position sensors, and the corresponding drive and control system. To reduce heat generation and mechanical coupling on the moving part, most devices with cables are installed on a fixed base. It should be noted that the liquid cooling system is optional, depending on the practical application.

Basic Structure
Energies 2019, 12, x FOR PEER REVIEW 2 of 14 uniformity. In this paper, we put forward a novel hyperbolic magnetic field voice coil actuator using a rhombus magnet array (HMF-VCA) to achieve this goal, and focus on the analysis of its complex six-degree-of-freedom force and torque characteristics. Figure 1 shows the structure of the proposed HMF-VCA and MLFPS. The HMF-VCA consists of two components, i.e., stator and mover. The stator consists of two rectangular stator coils with cooling duct, and a non-ferromagnetic stator frame. The mover consists of a rhombus magnet array, a non-ferromagnetic auxiliary frame, and a non-ferromagnetic mover frame. The rhombus magnet array consists of four cubic mover magnets which are arranged on the four edges of a rhombus, and the magnetization is shown as arrows. The MLFPS is made up of a load stage, a fixed base, three vertical HMF-VCAs, three horizontal HMF-VCAs, three magnetic springs, six relative position sensors, and the corresponding drive and control system. To reduce heat generation and mechanical coupling on the moving part, most devices with cables are installed on a fixed base. It should be noted that the liquid cooling system is optional, depending on the practical application.

Operating Principles
Similar to conventional VCAs, the HMF-VCA also works based on the Lorenz force law. The difference between the HMF-VCA and conventional VCAs is their different magnetic field distribution. Figure 2 shows the structure of four types of conventional VCAs which are used in high-precision applications, i.e., VCA with four rectangular magnets (FR-VCA), and VCA with one rectangular or square or circular magnet (OR/OS/OC-VCA) [18], and Figure 3 shows the magnetic field distribution and output force of these VCAs. The magnetic field distribution is calculated with Ansys Maxwell 2D. The materials of permanent magnets and coils are set to be NdFeB35 (Hc = 890 kA/m, Br = 1.23 T) and copper, respectively. The current density in the coils is set to be 5 A/mm 2 . The parameters in Figure 3 are set as follows: in Figure 3a, wm1 = 5.4 mm, tm1 = 2.539 mm, wc1 = 5.7 mm, tc1 = 3.6 mm, ws1 = 2.5 mm, ws2 = 2.2 mm, ts1 = 1 mm, model depth 64.64 mm; in Figure 3b, wm2 = 9.5 mm, tm2 = 5.12 mm, wc2 = 8.9 mm, tc2 = 6.3 mm, ws3 = 3.7 mm, ts2 = 1 mm, model depth 59.3 mm; in Figure 3c, wm

Operating Principles
Similar to conventional VCAs, the HMF-VCA also works based on the Lorenz force law. The difference between the HMF-VCA and conventional VCAs is their different magnetic field distribution. Figure 2 shows the structure of four types of conventional VCAs which are used in high-precision applications, i.e., VCA with four rectangular magnets (FR-VCA), and VCA with one rectangular or square or circular magnet (OR/OS/OC-VCA) [18], and Figure 3 shows the magnetic field distribution and output force of these VCAs. The magnetic field distribution is calculated with Ansys Maxwell 2D. The materials of permanent magnets and coils are set to be NdFeB35 (H c = 890 kA/m, B r = 1.23 T) and copper, respectively. The current density in the coils is set to be 5 A/mm 2 . The parameters in Figure 3 are set as follows: in Figure 3a, w m1 = 5.4 mm, t m1 = 2.539 mm, w c1 = 5.7 mm, t c1 = 3.6 mm, w s1 = 2.5 mm, w s2 = 2.2 mm, t s1 = 1 mm, model depth 64.64 mm; in , w m2 = 9.5 mm, t m2 = 5.12 mm, w c2 = 8.9 mm, t c2 = 6.3 mm, w s3 = 3.7 mm, t s2 = 1 mm, model depth 59.3 mm; in Figure 3c, w m = 18 mm, t m = 8 mm, w c = 6 mm, t c = 5 mm, w s = 2 mm, t s = 2 mm, y q = 8 mm, z q = 8 mm, model depth 36.86 mm. The stroke is set to be ±0.5 mm × ±0.5 mm.   As shown in Figure 3a,d, the magnetic field in the coil region of FR-VCA is relatively strong, which is beneficial to obtainig higher force density; however, the output force Fz will inevitably change when the coil moves along the z axis, because the magnetic field component By is approximately sinusoidal, as shown in Figure 3g.    As shown in Figure 3a,d, the magnetic field in the coil region of FR-VCA is relatively strong, which is beneficial to obtainig higher force density; however, the output force Fz will inevitably change when the coil moves along the z axis, because the magnetic field component By is approximately sinusoidal, as shown in Figure 3g. As shown in Figure 3a,d, the magnetic field in the coil region of FR-VCA is relatively strong, which is beneficial to obtainig higher force density; however, the output force F z will inevitably change when the coil moves along the z axis, because the magnetic field component B y is approximately sinusoidal, as shown in Figure 3g.  Figure 3b,e, it can be seen that the magnetic field component B y in the coil region of OR-VCA is rather inhomogeneous; although it is occasionally used because of its simple structure, it has the worst force characteristic, as shown in Figure 3h, and this conclusion is also applicable to OS-VCA and OC-VCA.
To obtain better force uniformity, the rhombus magnet array is designed and applied in the HMF-VCA. As shown in Figure 3c, the magnetic field in the HMF-VCA looks like a cluster of hyperbolas. With this hyperbolas-like magnetic field, the magnetic field component B y is proportional to the value of z, as shown in Figure 3f. In this case, when the mover moves relative to the stator, the increase (decrease) of the force acting on the upper part of the stator coils is equal to the decrease (increase) of the force acting on the lower part of the stator coils, thus the output force will remain constant, as shown in Figure 3i, and the HMF-VCA can achieve excellent force uniformity.
In addition, the MLFPS's load stage will be levitated and positioned with at least six HMF-VCAs, but this content is not discussed in this paper. We focus on the analysis of the HMF-VCA's six-degree-of-freedom force and torque characteristics.

Characteristics Analysis and Optimization
As discussed in Section 1, the mover of the HMF-VCA will move in six-degrees-of-freedom relative to the stator, i.e., it will translate along the x, y, z axis, and rotate around the x, y, z axis, as shown in Figure 4a. Besides the six-degree-of-freedom motion, the levitated mover is also subjected to six-degree-of-freedom force and torque, i.e., the force F x , F y , F z along the x, y, z axis and the torque T x , T y , T z around the x, y, z axis, as shown in Figure 4b. From Figure 3b,e, it can be seen that the magnetic field component By in the coil region of OR-VCA is rather inhomogeneous; although it is occasionally used because of its simple structure, it has the worst force characteristic, as shown in Figure 3h, and this conclusion is also applicable to OS-VCA and OC-VCA.
To obtain better force uniformity, the rhombus magnet array is designed and applied in the HMF-VCA. As shown in Figure 3c, the magnetic field in the HMF-VCA looks like a cluster of hyperbolas. With this hyperbolas-like magnetic field, the magnetic field component By is proportional to the value of z, as shown in Figure 3f. In this case, when the mover moves relative to the stator, the increase (decrease) of the force acting on the upper part of the stator coils is equal to the decrease (increase) of the force acting on the lower part of the stator coils, thus the output force will remain constant, as shown in Figure 3i, and the HMF-VCA can achieve excellent force uniformity.
In addition, the MLFPS's load stage will be levitated and positioned with at least six HMF-VCAs, but this content is not discussed in this paper. We focus on the analysis of the HMF-VCA's six-degree-of-freedom force and torque characteristics.

Characteristics Analysis and Optimization
As discussed in Section 1, the mover of the HMF-VCA will move in six-degrees-of-freedom relative to the stator, i.e., it will translate along the x, y, z axis, and rotate around the x, y, z axis, as shown in Figure 4a. Besides the six-degree-of-freedom motion, the levitated mover is also subjected to six-degree-of-freedom force and torque, i.e., the force Fx, Fy, Fz along the x, y, z axis and the torque Tx, Ty, Tz around the x, y, z axis, as shown in Figure 4b. Considering the influence of six-degree-of-freedom motion on the six-degree-of-freedom force and torque, a three-dimensional finite element analysis (3-D FEA) is employed to study the HMF-VCA's complex force and torque characteristic. The analytical method which has the fastest computing speed is another choice to study the characteristic of voice coil actuators, and the related modeling study can be found in [19]. However, there is a modeling error when the analytical method is used to calculate a permanent magnet the permeability of which is not 1 [20]. Therefore, the finite element method is chosen in this paper. The HMF-VCA's structural parameters are listed in Figure 5 and Table 1. The materials of permanent magnets and coils are set to be NdFeB35 (Hc = 890 kA/m, Br = 1.23T) and copper, respectively. The current density in the coils is set to be 5 A/mm 2 . As the two-dimensional magnetic field distribution of the HMF-VCA is given in Figure 3c, it is not repeated here. Considering the influence of six-degree-of-freedom motion on the six-degree-of-freedom force and torque, a three-dimensional finite element analysis (3-D FEA) is employed to study the HMF-VCA's complex force and torque characteristic. The analytical method which has the fastest computing speed is another choice to study the characteristic of voice coil actuators, and the related modeling study can be found in [19]. However, there is a modeling error when the analytical method is used to calculate a permanent magnet the permeability of which is not 1 [20]. Therefore, the finite element method is chosen in this paper. The HMF-VCA's structural parameters are listed in Figure 5 and Table 1. The materials of permanent magnets and coils are set to be NdFeB35 (H c = 890 kA/m, B r = 1.23T) and copper, respectively. The current density in the coils is set to be 5 A/mm 2 . As the two-dimensional magnetic field distribution of the HMF-VCA is given in Figure 3c, it is not repeated here.
According to the analysis of Figure 3c,f, the HMF-VCA's output force will remain constant in its stroke, and the 3-D FEA results shown in Figures 6-11 will prove this.  According to the analysis of Figure 3c,f, the HMF-VCA's output force will remain constant in its stroke, and the 3-D FEA results shown in Figures 6-11 will prove this. Figure 6 shows the characteristic of the HMF-VCA's output force Fz varied with the mover's motion. From Figure 6, the nominal output force Fz is about 3.055N, and it is almost constant in its full stroke. The variation of Fz caused by the translation or rotation in z, x, y, θx, θy, θz are as small as 0.059% mm -1 , 0.076% mm -1 , 0.121% mm -1 , 0.0044% deg -1 , 0.0022% deg -1 , 0.0092% deg -1 , respectively. This proves that the proposed structure is very effective at reducing the fluctuation of the output force Fz.

Parasitic Force
Besides the characteristic of the output force Fz, we are also concerned with the characteristics of the parasitic force Fx and Fy, because these two forces are undesirable and would increase the  From Figure 8, the parasitic force Fy is approximately proportional to z, θx, and it is almost insensitive to the variation of x, θy, θz. The variation of Fy caused by z, θx are 0.250% mm -1 , 1.71% deg -1 , respectively.
Similar to the analysis of parasitic force Fx, although the fluctuation of parasitic force Fy is much larger than the fluctuation of output force Fz, we can still say that it has good performance based on the same two reasons.   From Figure 8, the parasitic force Fy is approximately proportional to z, θx, and it is almost insensitive to the variation of x, θy, θz. The variation of Fy caused by z, θx are 0.250% mm -1 , 1.71% deg -1 , respectively.
Similar to the analysis of parasitic force Fx, although the fluctuation of parasitic force Fy is much larger than the fluctuation of output force Fz, we can still say that it has good performance based on the same two reasons.   The characteristics of the parasitic torque Tz, Tx and Ty are also taken into account because of their adverse effect on the system, similar to the parasitic force. Figures 9-11 show the characteristic of the HMF-VCA's parasitic torque Tz, Tx and Ty varied with the mover's motion.
Compared with the variation of force, the variation of torque is less obvious. Only when θz and z change simultaneously will the parasitic torque Tz change little, as shown in Figure 9, and the parasitic torque Tz is approximately proportional to θz, z. Thus, its adverse effect can also be   their adverse effect on the system, similar to the parasitic force. Figures 9-11 show the characteristic of the HMF-VCA's parasitic torque Tz, Tx and Ty varied with the mover's motion.
Compared with the variation of force, the variation of torque is less obvious. Only when θz and z change simultaneously will the parasitic torque Tz change little, as shown in Figure 9, and the parasitic torque Tz is approximately proportional to θz, z. Thus, its adverse effect can also be eliminated by adding a compensation coefficient in the control system easily.    At the same time, from Figure 10 and 11, the parasitic toque Tx is approximately proportional to the variation of y, z, θx, and the parasitic toque Ty is approximately proportional to the variation of x, z, θy. The most significant variations are the variation of Tx caused by y and the variation of Ty caused by x, which are in fact caused by the offset of the output force, and can be solved in the decoupling process of multi-degree-of-freedom system. Thus, they will not increase the control system's complexity. Consequently, the torque characteristic could be ignored in the design and optimization progress.

Parametric Analysis
Based on the above analysis of Figures 6-11, the HMF-VCA possesses excellent force uniformity as expected. Actually, its performance can be further improved. Several important or relatively large indexes are chosen to be further optimized, i.e., the variation of Fz caused by z, x, y, the variation of Fx caused by z, θy, and the variation of Fy caused by z, θx. In the following text, these force indexes are abbreviated as Sz,z, Sz,x, Sz,y, Sx,z, Sx,θy, Sy,z, Sy,θx, respectively. In addition, considering the volume limitation in some applications, the HMF-VCA's force density (output force per active volume ρA) should be also considered and optimized, and it is calculated with the following formula: where VA is the active volume of the HMF-VCA, and all the other variables are defined in Figure 5 and Table 1.
To find a useful optimization method for future design, the influence of structural parameters on force density and uniformity are analyzed. For maximum space utilization, there is a geometric relationship for the parameters of coils, wc + tc + 2 δ = yq + zq, and considering the HMF-VCA's stroke and manufacturability, δ, ws, ts are chosen to be 3 mm, 2 mm, 2 mm, respectively. Figures 12-14 show the HMF-VCA's force density and uniformity varied with important structural parameters. Figure 12 shows the HMF-VCA's force density and uniformity varied with yq and zq. From Figure 12, the increase of both yq and zq will lead to the increase of force density ρA; however, only when yq and zq are in particular intervals will the indexes Sz,z, Sz,y, Sy,z obtain small values. Thus, for the applications where the volume is not limited, yq and zq should be set to be the values where Sz,z,  Figure 6 shows the characteristic of the HMF-VCA's output force F z varied with the mover's motion. From Figure 6, the nominal output force F z is about 3.055N, and it is almost constant in its full stroke. The variation of F z caused by the translation or rotation in z, x, y, θ x , θ y , θ z are as small as 0.059% mm −1 , 0.076% mm −1 , 0.121% mm −1 , 0.0044% deg −1 , 0.0022% deg −1 , 0.0092% deg −1 , respectively. This proves that the proposed structure is very effective at reducing the fluctuation of the output force F z .

Parasitic Force
Besides the characteristic of the output force F z , we are also concerned with the characteristics of the parasitic force F x and F y , because these two forces are undesirable and would increase the complexity of the control system; thus, these two forces are expected to be as small as possible. Figures 7 and 8 show the characteristic of the HMF-VCA's parasitic force F x and F y varied with the mover's motion. From Figure 7, the parasitic force F x is approximately proportional to z, θ y , and it is almost insensitive to the variation of y, θ x , θ z . The variations of F x caused by z, θ y are 0.143% mm −1 , 1.84% deg −1 , respectively.
Compared with the fluctuation of output force F z , the fluctuation of parasitic force F x is much larger. However, we can still say that it has good performance based on the following two reasons. The first reason is that the parasitic force F x is approximately proportional to z, θ y . In this case, the adverse effect of parasitic force F x can be eliminated by adding a compensation coefficient in the control system easily. The second reason is that, in general, the maximum of θ y in an actual magnetic levitation fine positioning stage (MLFPS) is much smaller than the value set in the simulation. Thus, the parasitic force F x caused by the rotation around y axis will not be as large as in Figure 7d.
From Figure 8, the parasitic force F y is approximately proportional to z, θ x , and it is almost insensitive to the variation of x, θ y , θ z . The variation of F y caused by z, θ x are 0.250% mm −1 , 1.71% deg −1 , respectively.
Similar to the analysis of parasitic force F x , although the fluctuation of parasitic force F y is much larger than the fluctuation of output force F z , we can still say that it has good performance based on the same two reasons.

Parasitic Torque
The characteristics of the parasitic torque T z , T x and T y are also taken into account because of their adverse effect on the system, similar to the parasitic force. Figures 9-11 show the characteristic of the HMF-VCA's parasitic torque T z , T x and T y varied with the mover's motion.
Compared with the variation of force, the variation of torque is less obvious. Only when θ z and z change simultaneously will the parasitic torque T z change little, as shown in Figure 9, and the parasitic torque T z is approximately proportional to θ z , z. Thus, its adverse effect can also be eliminated by adding a compensation coefficient in the control system easily.
At the same time, from Figures 10 and 11, the parasitic toque T x is approximately proportional to the variation of y, z, θ x , and the parasitic toque T y is approximately proportional to the variation of x, z, θ y . The most significant variations are the variation of T x caused by y and the variation of T y caused by x, which are in fact caused by the offset of the output force, and can be solved in the decoupling process of multi-degree-of-freedom system. Thus, they will not increase the control system's complexity. Consequently, the torque characteristic could be ignored in the design and optimization progress.

Parametric Analysis
Based on the above analysis of Figures 6-11, the HMF-VCA possesses excellent force uniformity as expected. Actually, its performance can be further improved. Several important or relatively large indexes are chosen to be further optimized, i.e., the variation of F z caused by z, x, y, the variation of F x caused by z, θ y , and the variation of F y caused by z, θ x . In the following text, these force indexes are abbreviated as S z,z , S z,x , S z,y , S x,z , S x,θy , S y,z , S y,θx , respectively. In addition, considering the volume limitation in some applications, the HMF-VCA's force density (output force per active volume ρ A ) should be also considered and optimized, and it is calculated with the following formula: where V A is the active volume of the HMF-VCA, and all the other variables are defined in Figure 5 and Table 1.
To find a useful optimization method for future design, the influence of structural parameters on force density and uniformity are analyzed. For maximum space utilization, there is a geometric relationship for the parameters of coils, w c + t c + 2 δ = y q + z q , and considering the HMF-VCA's stroke and manufacturability, δ, w s , t s are chosen to be 3 mm, 2 mm, 2 mm, respectively. Figures 12-14 show the HMF-VCA's force density and uniformity varied with important structural parameters. Figure 12 shows the HMF-VCA's force density and uniformity varied with y q and z q . From Figure 12, the increase of both y q and z q will lead to the increase of force density ρ A ; however, only when y q and z q are in particular intervals will the indexes S z,z , S z,y , S y,z obtain small values. Thus, for the applications where the volume is not limited, y q and z q should be set to be the values where S z,z , S z,y , S y,z obtain small values as shown in Figure 12b,d,g,h, and for the applications where the volume is limited strictly, y q and z q should be set to be the values where ρ A is larger as according to Figure 12a. Figure 13 shows the HMF-VCA's force density and uniformity varied with w m and t m . From Figure 13, it is suggested that a small w m will be useful for obtaining high force density ρ A ; however, a larger w m will lead to better force uniformity. Furthermore, the influence of t m on force density and uniformity is very small. Thus, similar to the choice of y q and z q , w m should be designed carefully according to the requirements of volume limitation and force uniformity. Sz,y, Sy,z obtain small values as shown in Figure 12b,d,g,h, and for the applications where the volume is limited strictly, yq and zq should be set to be the values where ρA is larger as according to Figure  12a.  Figure 13 shows the HMF-VCA's force density and uniformity varied with wm and tm. From Figure 13, it is suggested that a small wm will be useful for obtaining high force density ρA; however, a larger wm will lead to better force uniformity. Furthermore, the influence of tm on force density and uniformity is very small. Thus, similar to the choice of yq and zq, wm should be designed carefully according to the requirements of volume limitation and force uniformity.    Figure 13 shows the HMF-VCA's force density and uniformity varied with wm and tm. From Figure 13, it is suggested that a small wm will be useful for obtaining high force density ρA; however, a larger wm will lead to better force uniformity. Furthermore, the influence of tm on force density and uniformity is very small. Thus, similar to the choice of yq and zq, wm should be designed carefully according to the requirements of volume limitation and force uniformity.   Figure 14 shows the HMF-VCA's force density and uniformity varied with w c and l e . From Figure 14, it is recommended that the parameter w c be about 6 mm, because the force density and uniformity are improved at the same time. In addition, the increase of parameter l s can decrease S z,x , S x,z , but this will also decrease ρ A and increase S x,θy .
In conclusion, the change of most of the structural parameters will bring advantages as well as disadvantages, and it is contradictory to reach highest force density and best force uniformity at the same time. Consequently, the researchers should choose reasonable structural parameters according to the actual requirements of force density and uniformity. Further optimization can be achieved through the multi-parameter optimization method. In this paper, we focus on proving the proposed new topology can achieve good force uniformity, and demonstrating that a topology with good force uniformity can be designed through field analysis and adjusting dimensions. Therefore, the content of multi-parameters optimization is not included in the paper.  Figure 14 shows the HMF-VCA's force density and uniformity varied with wc and le. From Figure 14, it is recommended that the parameter wc be about 6 mm, because the force density and uniformity are improved at the same time. In addition, the increase of parameter ls can decrease Sz,x, Sx,z, but this will also decrease ρA and increase Sx,θy. In conclusion, the change of most of the structural parameters will bring advantages as well as disadvantages, and it is contradictory to reach highest force density and best force uniformity at the same time. Consequently, the researchers should choose reasonable structural parameters according to the actual requirements of force density and uniformity. Further optimization can be achieved through the multi-parameter optimization method. In this paper, we focus on proving the proposed new topology can achieve good force uniformity, and demonstrating that a topology with good force uniformity can be designed through field analysis and adjusting dimensions. Therefore, the content of multi-parameters optimization is not included in the paper.

Performance Test and Comparison
A prototype is constructed and tested to validate the performance of the proposed topology. The test installation is made up of a position adjuster, a force sensor, a high-precision multimeter, and a support, as shown in Figure 15a.

Performance Test and Comparison
A prototype is constructed and tested to validate the performance of the proposed topology. The test installation is made up of a position adjuster, a force sensor, a high-precision multimeter, and a support, as shown in Figure 15a. The measured output force is 2.956 N, as shown in Figure 15b. Compared to the FEA value of 3.055 N, the deviation is about 3.24%, and the general trend is in good agreement with the FEA predications. The reasons for the deviation between the measured value and the FEA value are discussed below. The first reason may be that the coils are considered to completely fill the slots in The measured output force is 2.956 N, as shown in Figure 15b. Compared to the FEA value of 3.055 N, the deviation is about 3.24%, and the general trend is in good agreement with the FEA predications. The reasons for the deviation between the measured value and the FEA value are discussed below. The first reason may be that the coils are considered to completely fill the slots in the stator; however, the actual coils are a little smaller than those in the simulation because of the manufacturing process. The second reason may be that the magnets are not perfectly magnetized, in other words, the remanence may be slightly smaller than the simulation value. The third reason may be the deviation of geometric parameters.

Discussion
The HMF-VCA is compared with conventional VCAs [18], as listed in Table 2. It should be noted that the stroke of VCAs in [18] is set to be ±5 × 10 −4 m and ±5 × 10 −3 rad; thus, the HMF-VCA's performance is also calculated in the same stroke. From Table 2, the HMF-VCA's variation of F z caused by z, θ z and variation of F x caused by x, y are much lower than the other VCAs. Although the HMF-VCA's variation of F z caused by θ x or θ y is larger than the VCAs with one square or circular magnet, the value of 0.001% is small enough for high-precision applications. In addition, the HMF-VCA's variation of F x and F y caused by θ x and θ y is between the VCA with four rectangular magnets and the VCAs with one square or circular magnet. On the whole, the HMF-VCA features much better force uniformity than conventional VCAs.

Conclusions
In this paper, a novel HMF-VCA has been investigated. Its six-degree-of-freedom force and torque characteristics, as well as the influence of structural parameters on force density and uniformity, are studied by 3-D FEA. The results show that the proposed topology possesses the advantage of excellent force uniformity, and is suitable for MHPS applications.