Magnetic field of mathematical modeling and simulation of 3D magnetic pole array spherical actuator

MENG Hong-jun(孟宏君), WANG Zhan-lin(王占林), JIAO Zhong-xia(焦中夏)

(School of Electrical Engineering and Automation, Bei Hang University, Beijing 100191, China)

?

Magnetic field of mathematical modeling and simulation of 3D magnetic pole array spherical actuator

MENG Hong-jun(孟宏君), WANG Zhan-lin(王占林), JIAO Zhong-xia(焦中夏)

(School of Electrical Engineering and Automation, Bei Hang University, Beijing 100191, China)

In this paper a new spherical actuator is designed and its advantages are compared to an existing spherical actuator, which function is limited by several design bottlenecks. First the output torque is too small. Second, the attitude is difficult to be accurately detected. The new three-dimensional magnetic pole array can solve these major problems. The new actuator features an outer rotor with multiple permanent magnet (PM) poles. Using an analytical solution and the finite element solution simulation, the feasibility of the approach is verified. A prototype was developed, tested, and experiments were conducted to obtain the practical value of the magnetic flux density.

3-D magnetic pole array; spherical actuator; simulation; halbach; motor design

The multiple degree-of-freedom (DOF) rotations widely exist in the world, such as the motions of shoulder, wrist, hip joints, and the eye balls. Conventional multi-DOF rotations are achieved with quite a few single-axis motors connected with linkages in serial or parallel, which inevitably leads to a complex, bulky and heavy structure of the system, and compromises the dynamic performance. Compared with the single-DOF motors, the size of multi-DOF motor is very compact, and it can be used for robotic joints, precision assembly tools, and satellite tracking antennas. The concept of spherical actuator refers to the electrical device that can achieve multi-DOF motions in a single joint. It was first proposed by Williams et al. in 1960s[1]. The rotor is made of ferromagnetic material with a spherical barrel shaped surface. A copper mesh is inlaid on the rotor surface to allow induction current to travel in longitudinal and latitudinal directions. The stator consists of two multiphase winding blocks that can be twisted with an angle about an axis perpendicular to the rotor’s axle, generating a rotational magnetic field. Including the rotation of the rotor and the twist of stator blocks, this motor has two-DOF motions, despite that the twist of the stator blocks was not accomplished by winding currents. Following that, spherical actuators with various structures have been proposed by researchers, such as the spherical stepper by Lee et al[2]. The hemispheric stator structure is designed to support coils, iron trimmings, bearings and the position measurement devices. A pair of permanent magnet (PM) poles is mounted on the rotor. The rotor is supported by six bearings on the stator. Energizing the stator coils sequentially, the rotor can be pulled to any desired position. Chirikjian et al. developed a PM spherical stepper motor[3-4]. 80 cylindrical rare-earth PMs were placed along the inside surface of a hollow plastic sphere with a twelve-inch diameter. Sixteen coils with iron cores are placed on the surface of a spherical cap and polarized to form the electromagnetic field within the stator. An adjustable magnet saddle holds the stator magnets so that the stator magnets can be repositioned and reoriented. Because the symmetry of the rotor-pole arrangement is different from that of the stator poles, the magnetic field created by energizing two or more stator coils generates a torque that can change the rotor orientation. The commutation has been studied in detail. Spherical PM actuators that can achieve 2/3 DOF motions has been developed by Wang et al[5]. Two or four PM poles and three or four windings are mounted in the system. The rotor is basically considered as fully magnetized rare earth materials (NdFeB) with high density (7.5 g/mm3), which increases the moment of inertia of the rotor in a certain degree. Wang Q et al. have proposed a PM spherical actuator with a multi-pole rotor and a multi-coil stator. The magnetic field is computed with an integral equation method[6]. Other studies of spherical actuator with multiple poles can be found in Refs.[7-8]. Multiple layers of PM poles and coils are mounted on the rotor and stator surfaces respectively, to generate torque in three orthogonal directions. Thus, the motions in three directions about the rotor center can be achieved accordingly. With respect to the multi-DOF spherical motions, various orientation measurement methods have also been proposed and investigated[9]. For all spherical actuators developed in the past, the electromagnetic poles are all mounted in a spherical-surface pattern or 2D pattern, although the PM and coil poles arrangement is different for particular designs. This design method undoubtedly constrains the design space of the electromagnetic poles, and thus reduces the force and torque output. Therefore, this objective of this paper is to propose a new type of concept for pole patterns in 3D space. Specifically, the PM and coil poles are not only arranged in a spherical surface. The new design concept purposes to increase the magnetic flux density in the actuator space, and thus the force/torque output performance. The magnetic field distribution of spherical actuator with 3D topological pole arrangement is studied in a numerical way in this paper. It will contribute to our subsequent study for actuator output performance.

1 Permanent magnet spherical Actuators structure and operation mechanism

Fig.1a shows the drive of the sphere in the sphere radial using two layers of the stator and rotor design. Inner rotor PM and outer rotor PM correspond to the coil even in Fig.1b. In order to facilitate the observation, outer rotor, outer stator aluminum frame and inner rotor, inner stator aluminum frame are made to hide. And along the latitudinal direction to do a bit split, each layer of the rotor permanent magnet spherical actuators are using a spherical structure on the surface. The rotor spherical uniform distribution of the eight fan spherical permanent magnets around the equator form a circle, eight permanent magnets are positive and negative staggered fixed in the stator, the permanent magnet material NdFeB N35H, its center is located on the equator, the stator spherical shell placed a three-tier independent control of the concentrated winding coil according to certain rules, the middle layer of the coil center is located in the equatorial position, another two-tier center and the equatorial plane was the distribution of plus or minus 30 degrees on each coil’s number 12 uniform, an stator a total of three and 36 coils. Rotor output shaft fitted with flanges, used to pass the electromagnetic torque shown in Fig.2.

We increase the inner and outer rotor and design the structure of a layer with soft iron. It has two purposes: internal and external rotor field are isolated, so that the inner and outer rotor field directly interact only with its corresponding stator coil, the magnetic field and the moment of independence stand modeling. The inner and outer rotor poles of soft iron, output poles and other components form a closed loop magnetic circuit, which can complement one another.

Fig.1 Schematics illustrating the mechanical structure of 3D magnetic pole array spherical actuators

Fig.2 Permanent magnet spherical actuators mechanical structure

According to a certain strategy, the stator coil is energized, the coil produces a magnetic field, fixed rotor magnetic field interaction. The drive Actuators rotor complete spherical anywhere positioning can be achieved. The distribution along the equatorial line of the coil is energized, the rotation movement; coil distribution along the line of longitude electricity, the yaw motion and pitching motion. Therefore, the three degrees of freedom spherical actuators connected to the rotor load to three degrees of freedom movement, due to its own structure and manufacturing process factors, however, permanent magnet spherical actuators the work area is limited within a deflection range.

2 Magnetic field analysis of PM spherical actuator

The magnetic field generated by spherical actuator rotor pole is the main part of the air gap magnetic field, the new structure of spherical actuator has two layers of rotor. The magnetic field can be divided into six parts:

① From the outer edge of outer back iron to external air field, denoted by the area one. The magnetization characteristics

B1=μ0H1

(1)

whereμ0is the air permeability.

② From the outer edge of outer back iron to its inner edge, denoted by the area two. The magnetization characteristics:

B2=μ0μrH2

(2)

whereμris the relative permeability of iron.

③ From the inner edge of outer back iron to outer edge of outer PM denoted by the area three. The magnetization characteristics:

B3=μ0μmH3+μ0M0

(3)

whereμmis the relative permeability of PM. Which is NdFeB magnet, takeμm=1.2;M0=Br/μ0for the remanent magnetization,Brresidual magnetic flux density.

④ Outside the outer rotor air-gap magnetic field, denoted by field area 4. The magnetization characteristics

B4=μ0H4

(4)

⑤ Rotor pole internal magnetic field, is denoted by the site area. The magnetization characteristics

B5=μ0μmH5+μ0M0

(5)

⑥ The rotor core magnetic field is denoted by the field area. The magnetization characteristics

B6=μ0μrH6

(6)

By Eqs.(1)-(6) we have

(7)

By Eq.(7), by the gradient of the scalar magneticΦk

(8)

Available from the above kinds of

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

2.1 Through the solution of the scalar magnetic

(17)

Eqs.(9)-(16) show that solving the problem of the magnetic field generated by the rotor pole is actually solving the Laplace equation under certain boundary conditions. The Laplace equation is solved, the general solution

(18)

where

(19)

(20)

Cnmis a coefficient determined by the surface integral of the following forms:

(21)

(22)

Note that Eq.(21) is only effective in the rotor pole.

Therefore, the residual magnetization vector is limited in the following range:

For the rest of the non-magnetized rotor region equal to zero. By putting Eq.(22) into Eq.(21) we are

(23)

where

(24)

2.2 Boundary conditions

By the above boundary conditions can be scalar magnetic particular solution.

(1) BC (Boundary conditions)-A (r→∞)

B1r|r→∞=0,B1φ|φ→∞=0,B1θ|θ→∞=0

(2) BC-B (B1r|r=Ra=B2r|r=Ra)

(3) BC-C

(H1φ|r=Ra=H2φ|r=Ra,H1θ|r=Ra=H2θ|r=Ra)

(4) BC-D (B2r|r=Rb=B3r|r=Rb)

(5) BC-E

(H2φ|r=Rb=H3φ|r=Rb,H2θ|r=Rb=H3θ|r=Rb)

(6) BC-F (B3r|r=Rc=B4r|r=Rc)

(7) BC-G

(H3φ|r=Rc=H4φ|r=Rc，H3θ|r=Rc=H4θ|r=Rc)

(8) BC-H (B4r|r=Rd=B5r|r=Rd)

(9) BC-I

(H4φ|r=Rd=H5φ|r=Rd,H4θ|r=Rb=H5θ|r=Rd)

(10) BC-J

(H6φ|r=0=H6φ|r=0,H6θ|r=0=H6θ|r=0)

(11) BC-K (B5r|r=Re=B6r|r=Re)

(12) BC-L

(H5φ|r=Re=H6φ|r=Re,H5θ|r=Re=H6θ|r=Re)

Refer to Fig.5, the power generated by each component in the direction of flux density can be determined. The length of the DL on behalf of poor line tangentOpoints on the sphere, only the radial component of the magnetic flux density BRI can generate torque to change the direction of the rotor. We focus on the BRI after the discussion.

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

In this section, a prototype of the spherical actuator and a magnetic field test platform has been developed. Experiments on 3D magnetic field distribution have been conducted to validate the analytical model. Furthermore, the result from analytical model is compared with the numerical result. Although FEM simulation is time-consuming, its precision is relatively high, because it has no model simplification and influence of manufacturing error. Therefore, it is a practical way to validate the analytical model. The material characteristics and design parameters are in accordance with those in experiments. The finite element solutions are obtained in natural boundary condition.

3 Permanent magnet spherical actuators experimental study

A spherical actuator with an outer rotor is developed for experimental investigation on the magnetic field as shown in Fig.3. The specification is listed in Tab.1. The air-core coils are symmetrically distributed about the equatorial plane of the stator. By using nonmetal material PTFE, the eddy current effect can be reduced by a large margin. The rotor mainly consists of alternate-polarity cylindrical PMs for the convenience of manufacturing and cost reduction, which are sintered by NdFeB35 withBrem=1.2 T andμm=1.099 7. Around the PM poles is a holder (back iron) made of ferromagnetic material (steel 1010). A joint bearing that can achieve 3-DOF rotation is assembled at the center to support the rotor.

Fig.3 Spherical actuator with an outer rotor

The main design parameters are shown in Tab.1.

Tab.1 Structure specifications of the spherical actuator

Since the magnetic flux density B is a 3D vector, it is necessary to measure all three components of the flux density vector at the measuring point. A three-axis hall probe is thus employed in the measurement. Fig.4 shows the complete flux density measurement system. The three-axis hall probe is mounted on three-axis translational motion stage. Through translation of the stage, the hall probe can achieve any desired location. The hall probe is also connected to a 3-channel Gauss meter to display the measured flux density. The rotor is fixed on a step motor that can spin along the rotor axis for 360°. The movement of 3-axis stage and the step motor can be controlled simultaneously by the motion driven controller. The Gauss meter and the motion controller are linked to a personal computer through serial ports. Through a Labwindows CVI program, the motion of 3-axis stage and the step motor can be controlled with ease and the three components of magnetic flux density can be stored at the same time.

Fig.4 Testbed for measurement of flux density

The hall probe moves along a path controlled by the algorithm of the program and takes measurements of flux density at sampling points along the path. Firstly, the hall probe arrives at the start point of the measuring arc. The neighboring sampling points keep a constant angle of Δθwith respect to the rotor center. After the probe reaches the finish point of the measuring arc, the step motor rotates to an angle Δφ. Then the probe returns to the start point and repeats the process until the motor reaches the predeterminedφ. Thus, flux density values on a spherical surface have been recorded. Then measurement on the neighboring spherical surface can be conducted by moving the probe with an radial offset Δr. This process is repeated until the flux density is significantly small. The entire process of positioning the probe, taking measurement of the flux density, and turning the rotor can be fully automated through the communication between the motion controller, Gauss meter and PC.

Comparison of analytical model, numerical result and experimental result is shown in Fig.5 by 3D view.

Fig.5 Comparision of analytical and experimental results of B4r (3D view)

Fig.5 shows the distribution ofB4ralong latitudinal (φ) and longitudinal (θ) directions(θ=75°-105°，φ=0°-360°)at a fixed radial distance from the rotor center,r=75. The analytical and experimental results of the flux densityB4rare approximately the same. Because of the model error, the analytical result has a little difference. However, the peak value and the main trend are the same. The peak value of the flux density is at around 0.35-0.38 T.

Fig.6 shows the comparison of the analytical, numerical and experimental results in 2D view, wherer=75, andθ=90. The black line is the analytical result. The hollow blue squares represent the numerical result and the small solid circle shows the experimental result. From the comparison, the maximum differences occur between two rotor poles. The difference is less than 5%. It indicates that the accuracy is relatively high. And the proposed magnetic field is acceptable. With the inclusion of high-order harmonics, the difference may be reduced further.

When the two layers rotor spherical actuator coil (600 laps), given the 3.3 A current, peak torque 2.6 N·m, larger than single layer rotor spherical actuator (1.2 N·m), 2.166 7 times, as shown in Fig.7.

Fig.6 Comparision of analytical, numerical and experimental results of B4r (2D view, θ=90°)

Fig.7 Spherical actuator torque (3D view)

4 Conclusions

A novel PM spherical actuator with an outer rotor is proposed in this paper. The 3D magnetic field distribution is formulated analytically based on Laplace’s equations. With the FEM method, numerical results of magnetic field are obtained. A prototype and a field test platform are developed. Experimental verification on the magnetic field around the rotor has been carried out. Comparisons between analytical, numerical and experimental data have shown the consistency. The errors may come from the modeling error and the omission of the higher order terms of the harmonics.

Hall sensors are mounted on thespherical actuator statorwhich location is difficult to detect in a small magnetic field. And two layers rotor in which the magnetic field increases twice, therefore it is easier to detect the rotor magnetic field.

Because of its structural features, doublelayers rotor at the same input current, the output torque is larger than the single layer rotor spherical actuator(about twice).

[1] Williams F, Laithwaite E, Eastham J F. Development and design of spherical induction actuatorss [J]. Proceedings of the IEEE (S0368-2692), 1959, 106(12): 471-484.

[2] Lee K M, Kwan C K. Design concept development of a spherical stepper for robotic applications[J]. IEEE Transactions on Robotics and Automation (S1042-296X), 1991, 7(1): 175-181．

[3] Lee Kok-Meng, Zhou Debao. A real-time optical sensor for simultaneous measurement of three-DOF motions [J]. IEEE/ASME Transactions on Mechatronics (S1083-4435), 2004, 9(3): 499-507.

[4] Gregory S Chirikjian, David Stein. Kinematic design and commutation of a spherical stepper actuators[J]. IEEE Transactions on Mechatronics (S1083-4435), 1999, 4(4): 342-353.

[5] Wang Qunjing, Li Zheng, Ni Youyuan, et al. Magnetic field computation of a PM spherical stepper actuators using integral equation method[J]. IEEE Transactions on Magnetics (S0018-9464),2006, 42(4): 31-734.

[6] Wang Qunjing, Li Zheng, Chen Lixia, et al. Kinematic analysis and simulation of permanent magnet spherical stepper actuators[J]. Journal of System Simulation (S1004-731X), 2005, 17(9): 2260-2264.

[7] Yan Liang, Chen I-Ming, Yang Guilin, et al. Torque modeling of a spherical actuator based on lorentz force law[C]∥International Conference on Robotics and Automation, Barcelona, Spain, USA: IEEE, 2005: 3646-3651.

[8] David Stein, Edward R S, Gregory S C. Mathematical models of binary spherical-motion encoders[J]. IEEE/ASME Transactions on Mechatronics (S1083-4435), 2003, 8(2): 234-244.

[9] Lee Kok-Meng, Zhou Debao. A real-time optical sensor for simultaneous measurement of 3-DOF motions[C]∥IEEE/ASME International Conference on Advanced Intelligence Mechatronics, Kobe, Japan, 2003: 472-477.

(Edited by Cai Jianying)

10.15918/j.jbit1004-0579.201524.0114

TM 351 Document code: A Article ID: 1004- 0579(2015)01- 0097- 08

Received 2013-12- 20

Supported by the National Key Basic Research and Development Program (973 Program)( 2014CB046405)

E-mail: mengzju@163.com