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3-DOF motion of the spherical actuator. (a) First tilting motion. (b) Second tilting motion. (c) Spinning motion.
Source publication
This paper presents the torque model of a ball-joint-like three-degree-of-freedom (3-DOF) permanent magnet (PM) spherical actuator. This actuator features a ball-shaped rotor with multiple PM poles and a spherical stator with circumferential air-core coils. An analytical expression of the magnetic field of the rotor is obtained based on Laplace's e...
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... working principle of the spherical actuator is illustrated in Fig. 1. The rare-earth PMs (NdFeB) mounted along the rotor equator can produce high flux density. The air-core coils are assembled on the stator, which can simplify the torque model of the spherical actuator in a linear fashion. By activating pairs of coils in two longitudinal directions, the rotor can tilt in two orthogonal directions as ...
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... 1. The rare-earth PMs (NdFeB) mounted along the rotor equator can produce high flux density. The air-core coils are assembled on the stator, which can simplify the torque model of the spherical actuator in a linear fashion. By activating pairs of coils in two longitudinal directions, the rotor can tilt in two orthogonal directions as shown in Fig. 1(a) and (b). Energizing all circumferential coils, the rotor can spin about its own axis [ Fig. 1(c)]. Therefore, by varying the input currents of the coils, any desirable 3-DOF spherical motion within the workspace can be ...
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... The air-core coils are assembled on the stator, which can simplify the torque model of the spherical actuator in a linear fashion. By activating pairs of coils in two longitudinal directions, the rotor can tilt in two orthogonal directions as shown in Fig. 1(a) and (b). Energizing all circumferential coils, the rotor can spin about its own axis [ Fig. 1(c)]. Therefore, by varying the input currents of the coils, any desirable 3-DOF spherical motion within the workspace can be ...
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... objective of this experiment is to verify the torque generated by a single coil as described in (29). A testbed is developed for the measurement of the force/torque generated between the PM rotor and a single coil as shown in Fig. 10. The coil is supplied by a dc power (Topward 3303D). A six-axis force/torque sensor (ATI Nano43) is mounted on the shaft of the rotor which in turn is connected to a guide shaft. This guide shaft can slide along the slot of the arc guide so that a tilting motion of the rotor is able to be achieved. In addition, the guide shaft can ...
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... 110 • , which cover the range of coils on the stator. The torque is a 3-D vector with components T x , T y , and T z defined on the rotor frame. The torque vector is pose (θ, φ) dependant, where θ and φ specify the coil-axis orientation in the rotor frame. The variation of torque components can be visualized as a surface with respect to (θ, φ). Fig. 12 presents the torque variation of experimental and analytical results. In the experiment, 3-A current is supplied to the coil. The change of T y is the same as T x with 30 • shift in the φ-direction. It can be seen that the analytical model fits the experimental result well with maximum difference of 8% of the experimental result. ...
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... Torque Generated by Multiple Coils (Two Coils): Ac- cording to (30), the total torque of the spherical actuator is the Fig. 11. Force/torque measurement on two coils. superposition of torques generated by individual coils. The ob- jective of this experiment is to verify the superposition principle of the spherical actuator torque. The testbed of this experiment is the same as that in Fig. 10 except that the stator is used to replace the single coil. To verify ...
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... Coils): Ac- cording to (30), the total torque of the spherical actuator is the Fig. 11. Force/torque measurement on two coils. superposition of torques generated by individual coils. The ob- jective of this experiment is to verify the superposition principle of the spherical actuator torque. The testbed of this experiment is the same as that in Fig. 10 except that the stator is used to replace the single coil. To verify the superposition principle, force/torque measurement is conducted on two arbitrarily cho- sen coils indicated in Fig. 11. The initial positions of these coils in the rotor frame are (θ = 79 • , φ = 0) and (θ = 79 • , φ = 30 • ). In this experiment, the rotor shaft ...
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... jective of this experiment is to verify the superposition principle of the spherical actuator torque. The testbed of this experiment is the same as that in Fig. 10 except that the stator is used to replace the single coil. To verify the superposition principle, force/torque measurement is conducted on two arbitrarily cho- sen coils indicated in Fig. 11. The initial positions of these coils in the rotor frame are (θ = 79 • , φ = 0) and (θ = 79 • , φ = 30 • ). In this experiment, the rotor shaft changes the orientation by sliding along the arc guide with an angle θ r . For any specified rotor orientations, three sets of data are captured: torque pro- duced by energizing Coil 1 alone, ...
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... an angle θ r . For any specified rotor orientations, three sets of data are captured: torque pro- duced by energizing Coil 1 alone, by Coil 2 alone, and by Coil 1 and 2 simultaneously. By comparing the vector sum of the first two sets of torque values with the last set, the superposition of actuator torque can be evaluated. It can be seen from Fig. 13(a) -(c) that the superposition of torque generated by two individual coils fits well with the torque generated by energizing two coils simultaneously. Hence, the linear model of (30) can be used for the spherical actuator. 3) Linearity of Torque Model: According to (29) and (30), the torque output is proportional to the current input. ...
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This paper presents an analytical study of the air-gap magnetic field of a Surface Permanent Magnet (SPM) linear machine under no load. By means of the method of images, the complex expression of the magnetic field of a conductor, inside the air gap between two smooth iron surfaces, is retrieved. Then, integrating the conductor expression, the form...
Citations
... These mechanical torque variables are in turn associated with the currents of stator's EMs through an algebraic matrix relation (Yan et al., 2014;Rossini et al., 2013;Yan et al., 2010). Thus, the EMs currents are the final control inputs of the spherical motor (Gan et al., 2020;Yan et al., 2006;Wang et al., 2016). The motion control problem of spherical motors is nontrivial because of the complicated nonlinear and multivariable dynamics of these electric machines (Guo et al., 2020;Liu et al., 2018), (Kumagui et al., 2013). ...
Purpose
Permanent magnet synchronous spherical motors can have wide use in robotics and industrial automation. They enable three-DOF omnidirectional motion of their rotor. They are suitable for several applications, such as actuation in robotics, traction in electric vehicles and use in several automation systems. Unlike conventional synchronous motors, permanent magnet synchronous spherical motors consist of a fixed inner shell, which is the stator, and a rotating outer shell, which is the rotor. Their dynamic model is multivariable and strongly nonlinear. The treatment of the associated control problem is important.
Design/methodology/approach
In this paper, the multivariable dynamic model of permanent magnet synchronous spherical motors is analysed, and a nonlinear optimal (H-infinity) control method is developed for it. Differential flatness properties are proven for the spherical motors’ state-space model. Next, the motors’ state-space description undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a time-varying operating point, which is defined by the present value of the motors’ state vector and by the last sampled value of the control input vector. For the approximately linearized model of the permanent magnet synchronous spherical motors, a stabilizing H-infinity feedback controller is designed. To compute the controller’s gains, an algebraic Riccati equation has to be repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. Finally, the performance of the nonlinear optimal control method is compared against a flatness-based control approach implemented in successive loops.
Findings
Due to the nonlinear and multivariable structure of the state-space model of spherical motors, the solution of the associated nonlinear control problem is a nontrivial task. In this paper, a novel nonlinear optimal (H-infinity) control approach is proposed for the dynamic model of permanent magnet synchronous spherical motors. The method is based on approximate linearization of the motor’s state-space model with the use of first-order Taylor series expansion and the computation of the associated Jacobian matrices. Furthermore, the paper has introduced a different solution to the nonlinear control problem of the permanent magnet synchronous spherical motor, which is based on flatness-based control implemented in successive loops.
Research limitations/implications
The presented control approaches do not exhibit any limitations, but on the contrary, they have specific advantages. In comparison to global linearization-based control schemes (such as Lie-algebra-based control), they do not make use of complicated changes of state variables (diffeomorphisms) and transformations of the system's state-space description. The computed control inputs are applied directly to the initial nonlinear state-space model of the permanent magnet spherical motor without the intervention of inverse transformations and thus without coming against the risk of singularities.
Practical implications
The motion control problem of spherical motors is nontrivial because of the complicated nonlinear and multivariable dynamics of these electric machines. So far, there have been several attempts to apply nonlinear feedback control to permanent magnet-synchronous spherical motors. However, due to the model’s complexity, few results exist about the associated nonlinear optimal control problem. The proposed nonlinear control methods for permanent magnet synchronous spherical motors make more efficient, precise and reliable the use of such motors in robotics, electric traction and several automation systems.
Social implications
The treated research topic is central for robotic and industrial automation. Permanent magnet synchronous spherical motors are suitable for several applications, such as actuation in robotics, traction in electric vehicles and use in several automation systems. The solution of the control problem for the nonlinear dynamic model of permanent magnet synchronous spherical motors has many industrial applications and therefore contributes to economic growth and development.
Originality/value
The proposed nonlinear optimal control method is novel compared to past attempts to solve the optimal control problem for nonlinear dynamical systems. Unlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation which is used for computing the feedback gains of the controller is new, and so is the global stability proof for this control method. Compared to nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (H-infinity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of state dependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems which can be transformed into the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes, which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions. Furthermore, the second control method proposed in this paper, which is flatness-based control in successive loops, is also novel and demonstrates substantial contribution to nonlinear control for robotics and industrial automation.
... The stator structure is shown in Figure 1a, with layered structure and nested structure. The layered stator coil has two layers [3][4][5] or three layers [6][7][8] along the axial direction. For the nested structure [9,10], two small inclined coils are vertically arranged inside the rotating coil of the stator. ...
A double‐stator swing rotating permanent magnet spherical motor (SR‐PMSM) is designed, which can realize rotation, swing and spiral motion, and is used in the multi‐degree‐of‐freedom motion fields of industrial robot joints, torque gyroscopes, panoramic photography tables and so on. The inner surface of the rotor of the motor is attached with cylindrical PM poles, and the outer surface is attached with trapezoidal PM poles. Cylindrical and trapezoidal PM poles are simple to manufacture, but the magnetic field model of a spherical motor with two kinds of PM poles is complex. Therefore, a magnetic field modelling method combining the geometric equivalence principle and analytical method is proposed. Using the method of shape approximation, according to the principle of equivalence, the key parameters of cylindrical PM poles and trapezoidal PM poles are replaced by the parameters of approximate dihedral PM poles in a spherical coordinate. Select the most suitable dihedral PM poles, establish the analytical model of the magnetic field in the spherical coordinate system and verify the analytical results by FEM method. A SR‐PMSM motor with 6/4 pole (swing) and 12/10 pole (rotation) structure was assembled, and the experimental device was set up. The experimental results show that the experimental curve, analytical curve and FEM curve are in good agreement, which proves the accuracy of the equivalent model and is suitable for the magnetic field model of cylindrical and trapezoidal PM poles of the spherical motor. The research results of magnetic field lay a foundation for further analysis of the stator–rotor composite magnetic field considering cogging effect and edge effect, and analysis of load characteristics of the motor.
... It improves the motion mode of motor and develops from simple onedimensional circular rotation to multi-degree-of-freedom motion, which meets the application requirements of industrial field [1][2][3]. Among them, the permanent magnet spherical motor has become a research hotspot because of its high power density and high efficiency [4,5]. ...
A new type of double‐stator permanent magnet spherical motor (DSSRPMSM) is proposed. Compared with most existing spherical motors, this motor uses a large number of stator iron cores, which can generate larger magnetic flux density and significantly improve torque capacity. The proposed DSSRPMSM can continuously rotate, swing and spiral, and can be used in the multi‐degree‐of‐freedom motion fields of industrial robot joints, moment gyroscopes, panoramic photography tables and so on. Because the motor has the dual characteristics of rotary motor and arc linear motor, the comprehensive effects of stator slotting and stator iron core breaking should be considered. Cylindrical and trapezoidal PM poles are equivalent to dihedral PM poles, and stator coils are equivalent to PM pole models. Analytical models of PM pole magnetic field and stator magnetic field in a spherical coordinate system are established, and the air gap magnetic flux density under unequal magnetic potential boundary conditions is calculated by conformal transformation. The analytical calculation method proposed is consistent with the results of finite element method (FEM) simulation, and the experimental test further verifies the effectiveness and accuracy of this method.
... Spherical actuators have several advantages, such as a simple structure and a simple control method compared with conventional multi-degree-of-freedom (DOF) actuating systems that consist of several single-DOF motors [1]. Therefore, they have been actively developed for applications in robotics [2,3] and industrial machineries [4]. ...
Multi‐degree‐of‐freedom (multi‐DOF) spherical actuators have been developed for the fields of robotics and industrial machineries. We have proposed an outer rotor type three‐DOF spherical actuator that can realize a high torque density. Each coil input current is calculated using a torque generating equation based on the torque constant matrix. The permanent magnet type actuators have a problem with generating unexpected cogging torque due to various manufacturing errors. Manufacturing errors mainly mean differences between the ideal dimensions at the motor design stage and the actual dimensions in mass production. In this case, the actuator would exceed the limitations of classical proportional‐integral‐differential (PID) controllers. Therefore, we propose a current compensator using reinforcement learning by introducing a deep neural network that is expected to improve the robustness of spherical actuators. This current compensator was applied to uncertainty problems such as manufacturing fluctuations of cogging torque. We examined the reward, which is the main parameter of deep reinforcement learning, and reduced the control error compared to classical PID controller and simple neural network (NN) controller. © 2022 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.
... Song Spherical motors are electric devices capable of achieving multiple DOF rotary motions in a single joint [19][20][21]. Spherical motors have several advantages, such as a compact structure, a flexible motion mode, no accumulation of motion bias, and a simple control method compared with conventional multi-DOF actuating systems that consist of several single-DOF motors [1,22,23]. Therefore, they have been actively developed and have a wide application prospect in joints and eyeballs of the humanoid, unmanned aerial vehicles, autonomous cars, walking robots, and other fields. For these autonomous systems, environment and space recognition is a crucial function. ...
In keeping with consumers’ preferences for electromagnetic motors of ever smaller power consumption, it is necessary to improve the power efficiency of the electromagnetic motors used in unmanned aerial vehicles and robots without sacrificing their performance. Three-degree-of-freedom (3-DOF) spherical motors have been developed for these applications. Accordingly, this study modifies the 3-DOF spherical motor proposed by Hirata’s group in a previous study (Heya, A.; Hirata, K.; Niguchi, N., Dynamic modeling and control of three-degree-of-freedom electromagnetic actuator for image stabilization, IEEE Transactions on Magnetics 2018, 54, 8207905.) to accomplish a 3-DOF spherical motor for camera module with higher torque output in the large rotation angle. The main contribution of this study is to improve the static torque in the X- and Y-axes with an improved electromagnetic structure and a particular controlling strategy. In the structural design, eight symmetrical coils with specific coil combination are used instead of conventional four symmetrical coils. In this study, the development of the proposed 3-DOF spherical motor was constructed and verified by using a 3D finite-element method (3D FEM). The simulation results show that the proposed 3-DOF spherical motor has higher torque output in the large rotation angle when compared to the original 3-DOF spherical motor.
... However, due to the specific set of magnets used, the movable angle along the Z-axis was only ±5 • . Chu et al., Fusayasu et al., and Yang et al. also proposed other spherical VCMs of this type [28][29][30]. ...
Multi-DOF motion is realized in the eyes and joints of robots mostly through the combination of multiple one-degree-of-freedom (1-DOF) motors. However, this results in reduced efficiency, a large volume, reduced response speed, and inaccurate positioning. To solve these problems, this study proposes a novel 3-DOF spherical voice coil motor (VCM). In this VCM, 16 coils and a radially oriented ring magnet are used to generate a Lorentz force to achieve motion. In particular, coils for Z-axis rotation are sandwiched between the coils for X- and Y-axis rotation. Furthermore, the proposed VCM can achieve 360° rotation about the Z-axis. The commercial software ANSYS was used to design and verify the performance of the proposed VCM. Simulation results indicate that this VCM affords improved power efficiency because only a suitable combination of coils, rather than all coils, needs to be powered on. The results demonstrate the feasibility of the proposed 3-DOF spherical VCM.
... In the work of [5,6], a torque measurement system is designed with a pressure sensor mounted in a torsion cantilever bracket structure. In [7,8], a torque measurement system with a torque sensor installed in the circular and curved-guide rail is designed for different PMSMs. Moreover, for an inductive spherical motor, an output torque measurement system consists of a brushless motor and a six-axis force sensor installed on the rotor ball, is developed in the papers [9,10]. ...
The output torque is an important performance indicator of a motor. Due to the special structure of a permanent magnet spherical motor (PMSM), it is difficult to measure its torque. This paper proposes a novel method to measure the output torque of the PMSM. The proposed method uses the microelectromechanical system (MEMS) gyroscope to measure the rotor motion acceleration, which is then used to calculate the output torque based on the rotor dynamics equation. In this paper, we firstly simulate and analyze the output torque of the PMSM. Secondly, we design a torque-measuring device to measure the output torque. Thirdly, we compare and discuss the experimental and simulation results. The comparison results show that the proposed method is feasible.
... Analytical magnetic modeling has been investigated adopting methods including spherical harmonics, magnetic equivalent circuit, distributed multipole and charge models. Spherical harmonics (SH) is a method for analyzing the magnetic field distribution of permanent magnet arrays in 3D spherical coordinate [21][22][23]. Magnetic equivalent circuit (MEC) is employed by meshing the geometry into elements [24]. ...
... (21)-(23). After that, we calculate the magnetic field distribution of point P in l th local cylindrical coordinate system ...
This paper presents modeling of an integrated design of an electromagnetic driven spherical motion generator with multi-layer structure. An analytical method to calculate the complex magnetic field distribution of the rotor array by combining the equivalent charge model and transformation method is developed, upon which torque model is further established analytically by using the moment principle. Both the analytical magnetic model and torque model are validated with the numerical finite element method by Ansoft Maxwell and experimental measurements. With the model, two different designs of the electromagnetic driven SMGs are analyzed to illustrate the application of the developed model.
... Spherical actuators have several advantages such as a simple structure, a low moment of inertia, and a simple control method compared with conventional multi-degree-of-freedom (DOF) actuating systems that are composed of several single-DOF motors [1]. Therefore, they are expected to be applied in the fields of robotics [2][3][4][5], industrial machinery [6,7] and aerospace [8,9], and have been actively developed. ...
... In particular, a permanent magnet synchronous spherical actuator is popular because its torque can be controlled using a torque generating equation around an arbitrary axis. For this reason, a variety of studies have focused on the permanent magnet synchronous spherical motor [1,[3][4][5][6]8,9]. ...
... where u 1 1. The interpolation of these two currents v 1 and v 2 under copper loss limiting is given as follows. ...
Multi-Degree-of-Freedom (Multi-DOF) actuating systems are composed of several single-DOF motors, which results in large, heavy and complicated structures. In order to solve these problems, various multi-DOF spherical actuators have been actively studied. Among them, a spherical actuator can generate torques in arbitrary positions and directions. However due to this, it is difficult to evaluate the torque characteristics at any position all over the movable range. In order to evaluate the spherical actuators, we proposed a torque evaluation method under copper loss limiting. However, in the proposed method, only copper losses can be considered. In this paper, we propose a new torque evaluation method considering a maximum current limiting. Finally, the torque characteristics of our multi-DOF actuators are evaluated using the proposed method and the effectiveness of the proposed method is verified.
... In this method, the permanent magnets are replaced with magnetic dipoles, and then the magnetic scalar potential at any point can be obtained by the superposition of all the magnetic dipoles. Spherical harmonics is an analytical method for the design and analysis of spherical actuators with dihedral permanent magnets [19][20][21]. Spherical harmonics method can be also used to analyze approximately the magnetic field distribution of cylindrical permanent magnets [22]. In [23], the analytical torque model of spherical actuators with cylindrical permanent magnets is obtained by adjusting the analytical terms with the measured experimental results of the actual magnetic field. ...
... where k ζ is not zero only when = m 1, and = ·k k . Substituting Eq. (19) into Eq. (21), we obtain ...
A permanent magnet spherical actuator is an electromagnetic device characterized by achieving multi-degree-of-freedom rotations in one driving unit. Analytical magnetic field modeling of the rotor array is needed for the design, performance analysis and optimization of the spherical actuator. This paper presents an analytical method to formulate the complex magnetic field distribution of rotor array with cylindrical permanent magnets in 3D spherical space. This method combines the current model by toroidal functions and transformation method, thus the magnetic field can be calculated efficiently with reasonable accuracy in order for further design optimization and real-time control. The analytical magnetic model of the permanent magnet rotor array is validated with the numerical finite element method by Ansoft Maxwell and experimental measurement.
