Permanent Magnet Spherical Actuator Research Articles

Spherical Joint Actuator with Backstepping Sliding Mode Control for Robotic Manipulator Rigid-Flexible Coupling System

At present, the output shaft of a permanent magnet spherical actuator (PMSA) is a rigid structure, which has the problems of small load weight ratio and high energy consumption. It is difficult to meet the application requirements of lightweight, high speed, and low energy consumption. In this paper, a PMSA is proposed as the driving motor for the rigid-flexible coupling system (R-FCS) of the robotic manipulator, and a flexible mechanical arm is used to replace the rigid output shaft of the PMSA to connect the hand claw to form a spherical joint. Firstly, the partial differential dynamics model of the robotic manipulator in the coordinate system is established based on the Hamilton method, and the R-FCS is constructed according to the momentum conservation equation combined with the rigid axis dynamics model of the PMSA. Then, a backstepping sliding mode controller (BSMC) is designed to form a closed-loop control system by selecting an appropriate nonlinear gain function to estimate the unknown disturbances of the system, and the stability of the controller is guaranteed by the Lyapunov theory. Finally, the simulation and experimental results verify that the R-FCS can achieve better trajectory tracking and provide an effective driving method for robotic manipulator spherical joint.

Adaptive Sliding‐Mode Control for Permanent Magnet Spherical Actuator Based on Trajectory Re‐Planning

This study proposes an adaptive sliding‐mode control strategy based on trajectory re‐planning for a permanent magnet spherical actuator (PMSA) to address the large overshoot caused by substantial initial error while maintaining tracking precision. The proposed re‐planning technique reconstructs a local trajectory without the preceding knowledge of the desired position in the future, which makes this technique more suitable for reconstructing trajectory given online. First, a re‐planning cool‐down term is added to maximize the capability of controller to improve the precision and speed of tracking. Moreover, a smooth switching technique is also applied to further improve the precision of trajectory tracking and stop redundant re‐planning when the system enters satisfactory state. Finally, the proposed trajectory re‐planning strategy is combined with a robust adaptive sliding‐mode controller (RASC), which effectively reduces friction, delay, and external uncertainty disturbances. The effectiveness of the proposed control design is verified by both simulation and experiment on PMSA, which can provide reference for the further engineering application of multi‐degree‐of‐freedom system. © 2023 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC.

Continuous non‐singular terminal sliding‐mode control of permanent magnet spherical actuator for trajectory tracking based on a modified nonlinear disturbance observer

Multi-degree-of-freedom permanent magnet spherical actuators (PMSAs) are highly nonlinear, coupled, and multivariable systems that are subject to different types of disturbances including unknown payloads, unmodelled dynamics, frictions, and other external disturbances, which adversely affect the trajectory tracking performance of a system. In this paper, a disturbance-observer-based continuous non-singular terminal sliding-mode control (NDOB-CNTSMC) strategy is developed for a PMSA to enhance its trajectory tracking performance. Firstly, a nonlinear disturbance observer (NDOB) with a parameter design method is proposed to suppress uncertainty and reject disturbance in a PMSA trajectory tracking control system. Secondly, a continuous non-singular terminal sliding-mode controller (CNTSMC) is presented to compensate for the effect of the disturbance observation error and unobservable disturbances. The stability of the proposed control strategy is proved by the Lyapunov theorem. Both simulation and experimental results have shown that the control strategy proposed in this research can effectively address the issues of the inaccuracy of modelling and various disturbances and improve the trajectory tracking precision of the PMSA.

Designing Active Disturbance Rejection Control of Permanent Magnetic Spherical Actuator Based on Nonlinear Extended State Observer

Aiming at the problems of low control precision and poor anti-disturbances ability of permanent magnet spherical actuator (PMSA), an active disturbance rejection trajectory tracking control method based on nonlinear extended state observer (NESO) is proposed in this paper. Firstly, the multivariable and strong coupling mathematical model of the PMSA is obtained by rigid body rotation coordinate transformation and the Lagrange dynamic equation. Then, the tracking differentiator is used to reduce the influence caused by the expected trajectory jump. In addition, NESO is designed to observe and calculate coupling of the system, external disturbance, and model error online. Finally, the nonlinear control law is used to compensate for the observed disturbance, and the stability is proved based on the Lyapunov equation. Simulation and experimental results show that the proposed control scheme can achieve decoupling and tracking control of the complex system in the presence of model error, random disturbance, and other uncertain factors, has good control accuracy and response speed, and has strong robustness to uncertain disturbances.

Time Delay Estimation Control of Permanent Magnet Spherical Actuator Based on Gradient Compensation

A multi-degree-of-freedom Permanent Magnet Spherical Actuator (PMSpA) has a special mechanical structure and electromagnetic fields, and is easily affected by nonlinear disturbances such as modeling errors and friction. Therefore, the quality of a PMSpA control system may be deteriorated. In order to keep the PMSpA with good trajectory tracking performance, this paper designs a time delay estimation controller based on gradient compensation. Firstly, the dynamic model of the PMSpA with nonlinear terms is derived. The nonlinear terms in the complex dynamic model can be simplified and estimated by the time delay estimation method. Secondly, for the estimation errors caused by time delay control, a gradient compensator is introduced to further correct and compensate for it. Furthermore, the stability of the designed controller is proved by the Lyapunov equation. Finally, the correctness and effectiveness of the controller are validated by comparison with other controllers through simulation. In addition, experimental results have also shown that the control accuracy of the spherical motor can be effectively improved using the proposed controller.

Adaptive friction compensation robust control for permanent magnet spherical actuator under compound disturbance.

The permanent magnet spherical actuator (PMSA) is a multi-variable featured and strongly coupled nonlinear system, and its motion control will unavoidably be affected by various factors. In order to reduce the influence of model uncertainty, friction, and external disturbance on the actual control during the position tracking of PMSA, this paper proposes a novel adaptive friction compensation robust control method under compound disturbance to improve the motion stability of PMSA. In this method, an adaptive compensator is designed to compensate for the unmodeled friction with unknown but constant parameters, and a robust compensator is used to deal with friction parameter variation and non-parametric interference factors that cannot be modeled. The combination of the two controllers can achieve better friction compensation and interference suppression with smaller feedback gain, thus reducing the trajectory tracking errors. In addition, in order to add the smoothness of the velocity value during the experiment, a first-order filter is introduced into the controller. Finally, the stability of the designed controller for the closed-loop system is proved by the Lyapunov method. Furthermore, a simulation model is established and experiments are conducted on the research prototype, and the feasibility and effectiveness under compound disturbance of the proposed control method are verified by comparing with the simulation and experimental results of other controllers.

Robust Adaptive Sliding-Mode Control of a Permanent Magnetic Spherical Actuator With Delay Compensation

This study proposes an adaptive robust sliding-mode control strategy with time delay compensation to address the issues of the inaccuracy of modeling, friction, uncertain disturbances, and time delay in a permanent magnet spherical actuator trajectory tracking control system. First, an improved linear predictor is designed to compensate for the time delay in position information. Second, a robust sliding mode controller is designed to suppress the influence of uncertain disturbance. Third, the constant parameters of the spherical actuator are estimated using the adaptive law and compensated at the control input. The stability of the adaptive robust sliding-mode controller is proved by the Lyapunov theorem. Simulation and experimental results show that the control strategy proposed in this research has good dynamic and static performance, which can provide reference for the further engineering application of multi-degree of freedom control system.

Robust Adaptive Sliding-Mode Control for Permanent Magnet Spherical Actuator with Uncertainty Using Dynamic Surface Approach

This paper presents a position tracking control method for a three degree-of-freedom permanent magnet spherical actuator (PMSA). The control method is designed based on a dynamic model of the PMSA with uncertainties including modelling errors and external disturbance. Sliding-mode surface are adopted to restrain and eliminate the effect of external disturbances. To compensate modelling errors, an adaptive law is employed to estimate unknown model parameters so that model information can be updated in real time. By the use of dynamic surface approach, three first-order filters are introduced to avoid the explosion of derivative terms caused by traditional adaptive backstepping approach. The stability of the closed-loop system using the proposed controller is confirmed through Lyapunov theorem. The test bench consisted of the prototype of PMSA, the host computer, the current controller and the orientation detection device is established for experiments. Simulation and experimental results are provided to validate effectiveness of the proposed method.

Magnetic field modeling and validation for a spherical actuator with cylindrical permanent magnets

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.

Adaptive fuzzy sliding mode algorithm-based decentralised control for a permanent magnet spherical actuator

ABSTRACTThe dynamic model of multi-degree-of-freedom permanent magnet (PM) spherical actuators is multivariate and nonlinear due to strong inter-axis couplings, which affects the trajectory tracking performance of the system. In this paper, a decentralised control strategy based on adaptive fuzzy sliding mode (AFSM) algorithm is developed for a PM spherical actuator to enhance its trajectory tracking performance. In this algorithm, the coupling terms are separated as subsystems from the entire system. The AFSM algorithm is applied in such a way that the fuzzy logic systems are used to approximate the subsystem with uncertainties. A sliding mode term is introduced to compensate for the effect of coupling terms and fuzzy approximation error. The stability of the proposed method is guaranteed by choosing the appropriate Lyapunov function. Both simulation and experimental results show that the proposed control algorithm can effectively handle various uncertainties and inter-axis couplings, and improve the trajectory tracking precision of the spherical actuator.

Single neural adaptive controller and neural network identifier based on PSO algorithm for spherical actuators with 3D magnet array.

Permanent magnet spherical actuator (PMSA) is a multi-variable featured and inter-axis coupled nonlinear system, which unavoidably compromises its motion control implementation. Uncertainties such as external load and friction torque of ball bearing and manufacturing errors also influence motion performance significantly. Therefore, the objective of this paper is to propose a controller based on a single neural adaptive (SNA) algorithm and a neural network (NN) identifier optimized with a particle swarm optimization (PSO) algorithm to improve the motion stability of PMSA with three-dimensional magnet arrays. The dynamic model and computed torque model are formulated for the spherical actuator, and a dynamic decoupling control algorithm is developed. By utilizing the global-optimization property of the PSO algorithm, the NN identifier is trained to avoid locally optimal solution and achieve high-precision compensations to uncertainties. The employment of the SNA controller helps to reduce the effect of compensation errors and convert the system to a stable one, even if there is difference between the compensations and uncertainties due to external disturbances. A simulation model is established, and experiments are conducted on the research prototype to validate the proposed control algorithm. The amplitude of the parameter perturbation is set to 5%, 10%, and 15%, respectively. The strong robustness of the proposed hybrid algorithm is validated by the abundant simulation data. It shows that the proposed algorithm can effectively compensate the influence of uncertainties and eliminate the effect of inter-axis couplings of the spherical actuator.

Adaptive backstepping sliding mode control for 3-DOF permanent magnet spherical actuator

In this paper, a robust adaptive control system combining backstepping and sliding mode control method has been implemented for the 3-DOF permanent magnet (PM) spherical actuator in order to improve its trajectory tracking performance. Due to the complexity of both mechanical structure and electromagnetic field, the dynamic model of a PM spherical actuator inevitably contains uncertainties, such as disturbances and model errors, which will gravely influence the performance of conventional control system. Therefore, a backstepping sliding mode approach is firstly applied, where the backstepping design is to synthesize the controller and the sliding mode term is to compensate disturbances. Then, an adaptive law is presented to estimate the model errors during the control process, where the parameters of the model are initially assumed to be accuracy. Correspondingly, the stability can be guaranteed by choosing the appropriate Lyapunov function. Both simulations and experiments are designed to demonstrate the effectiveness of the proposed control scheme.

Robust dynamic decoupling control for permanent magnet spherical actuators based on extended state observer

This study presents a robust dynamic decoupling control strategy to solve the trajectory tracking problem for the permanent magnet spherical actuator (PMSA). The dynamic model of PMSA obtained by the Lagrange–Euler formalism is obviously a multi-variable non-linear system with strong cross-couplings. Furthermore, uncertainties such as model errors and external disturbances will also affect the precision of the control system. In the active disturbance rejection control (ADRC) framework, the decoupling problem can be reformulated as disturbance rejection by merging the cross channel interference into the lumped disturbance, which consists of internal dynamics and external disturbances. The lumped disturbance is then estimated using extended state observer (ESO) and canceled out in the control law. Herein, the linear active disturbance rejection control is selected for PMSAs, as the tuning process can be greatly simplified by making all the parameters of ESO or controller a function of bandwidth. Simulations and experiments are presented to corroborate the effectiveness and robustness of the proposed strategy, showing that the proposed control algorithm can decouple and linearise the system in the presence of model errors as well as the load and random disturbances. Meanwhile, the modified system has better static and dynamic performances with strong robustness to uncertainties.

A Robust Adaptive Iterative Learning Control for Trajectory Tracking of Permanent-Magnet Spherical Actuator

This paper presents a robust adaptive iterative learning control (ILC) algorithm for 3-DOF permanent magnet (PM) spherical actuators to improve their trajectory tracking performance. The dynamic model of a PM spherical actuator is a multivariable nonlinear system with interaxis coupling terms. Uncertainties such as modeling errors, loads, and external disturbances exist in the model unavoidably, which will affect the performance, including the precision of the control system. Hence, to compensate for these uncertainties, a new hybrid control scheme that consists of a proportional–derivative (PD) feedback control with varying gains, a PD-type ILC with adjustable gains, and a robust term is developed. The new control law combines the advantages of simplicity and easy design of the PD control, the effectiveness of the ILC to handle model uncertainties and repetitive disturbances, and the robustness of the robust term to random disturbances. In addition, to expedite the convergence rate, the gains in the PD feedback control and the PD-type ILC are adaptively adjusted according to the iteration times. It is shown that the system tracking error approaches zero as the number of iterations increases. To demonstrate the proposed algorithm and verify the established theoretical results, both simulations and experiments are conducted. The results have shown that the proposed control algorithm can effectively compensate for various uncertainties and can thus improve the trajectory tracking performance of spherical actuators.

Closed-loop magnetic bearing and angular velocity control of a reaction sphere actuator

This article presents the first closed-loop magnetic bearing and angular velocity experimental results of a reaction sphere actuator for satellite attitude control. The proposed reaction sphere is a permanent magnet spherical actuator whose rotor is supported by magnetic bearing and can be torqued electronically about any desired axis. The spherical actuator is composed of an 8-pole permanent magnet spherical rotor and of a 20-pole stator with electromagnets. The electromechanical model of the reaction sphere is summarized together with procedures to estimate the rotor magnetic state, the back-EMF voltage, and the rotor angular velocity, which are all fundamental ingredients for controller design. Dynamic controllers are developed to levitate the rotor inside the stator (magnetic bearing) and to control the rotor angular velocity. The magnetic bearing is based on a state-space controller with reduced-order displacement velocity estimator whereas the angular velocity controller is a simple proportional controller with a dedicated angular velocity estimator. The developed control algorithms are experimentally validated using the developed laboratory prototype showing the ability of simultaneously levitating the rotor while rotating it about a given arbitrary axis.

An orientation measurement method based on Hall-effect sensors for permanent magnet spherical actuators with 3D magnet array.

An orientation measurement method based on Hall-effect sensors is proposed for permanent magnet (PM) spherical actuators with three-dimensional (3D) magnet array. As there is no contact between the measurement system and the rotor, this method could effectively avoid friction torque and additional inertial moment existing in conventional approaches. Curved surface fitting method based on exponential approximation is proposed to formulate the magnetic field distribution in 3D space. The comparison with conventional modeling method shows that it helps to improve the model accuracy. The Hall-effect sensors are distributed around the rotor with PM poles to detect the flux density at different points, and thus the rotor orientation can be computed from the measured results and analytical models. Experiments have been conducted on the developed research prototype of the spherical actuator to validate the accuracy of the analytical equations relating the rotor orientation and the value of magnetic flux density. The experimental results show that the proposed method can measure the rotor orientation precisely, and the measurement accuracy could be improved by the novel 3D magnet array. The study result could be used for real-time motion control of PM spherical actuators.

Accuracy enhancement of the spherical actuator with a two-level geometric calibration method

This paper presents a two-level geometric calibration method for the permanent magnet (PM) spherical actuator to improve its motion control accuracy. The proposed actuator is composed of a stator with circumferential coils and a rotor with multiple PM poles. Due to the assembly and fabrication errors, the real geometric parameters of the actuator will deviate from their design values. Hence, the identification of such errors is critical for the motion control tasks. A two-level geometric calibration approach is proposed to identify such errors. In the first level, the calibration model is formulated based on the differential form of the kinematic equation, which is to identify the geometric errors in the spherical joint. In the second level, the calibration model is formulated based on the differential form of torque formula, which is to calibrate the geometric parameters of the magnetization axes of PM poles and coils axes. To demonstrate the robustness and availability of the calibration algorithm, simulations are conducted. The results have shown that the proposed two-level calibration method can effectively compensate the geometric parameter errors and improve the positioning accuracy of the spherical actuator.

Dynamic modelling and parameter identification of a three-degree-of-freedom spherical actuator

This paper addresses both dynamic modelling and dynamic parameter identification of a permanent magnet spherical actuator, which is capable of performing three-degree-of-freedom (DOF) motion in one single joint. The dynamic model of the spherical actuator is derived from Lagrange’s equations, but the parameters, called dynamic parameters, in the model are usually uncertain. Then the dynamic model is represented in a form that is linear in these parameters. A new identification method based on the output error (OE) method and recursive least square (LS) estimation is proposed to identify the parameters. This method only requires the current measurement of the stator coils in the identification procedure, which greatly simplifies the experimental process and improves the identification accuracy. Lastly, simulation and experimental results illustrate the effectiveness of the proposed method and its robustness to external disturbances. The proposed method can be also applied to other electromagnetic driving spherical actuators.

Modeling and Iron-Effect Analysis on Magnetic Field and Torque Output of Electromagnetic Spherical Actuators With Iron Stator

This paper presents a three-degree-of-freedom permanent magnet (PM) spherical actuator with an iron stator. The major contribution of this paper is to study the effect of iron stator on the magnetic field and torque output of the electromagnetic spherical actuator quantitatively and qualitatively. It could be helpful for actuator design optimization. Based on the poles' arrangement and the iron boundary, the magnetic field of the PM-pole rotor and the actuator torque are formulated analytically. The effect of iron stator on the magnetic field and torque output is analyzed with respect to structural parameters. The result shows that the iron stator can increase the radial component of the flux density and thus the actuator torque output significantly.

Design and control of a three degree-of-freedom permanent magnet spherical actuator

This paper presents a permanent magnet (PM) spherical actuator embedded with a novel three-dimensional (3D) orientation measurement system. The study covers actuator design, torque modeling, and motion control. The spherical actuator consists of a stator with 24 coils and a rotor with 8 PM poles. A high accuracy and resolution, non-blinding 3D measurement mechanism is designed for the rotor orientation detection. The torque output is formulated from finite element (FE) computation and curve fitting method. Due to the nonlinear property of dynamic model, computed torque control law is utilized for the three degree-of-freedom (3-DOF) system motion control. The space geometry method is employed to study the redundancy feature of current inputs, which offers the opportunity for control optimization to improve the power efficiency and the fault tolerance capability of the spherical actuator. Simulations and experiments are then conducted on the developed prototype to validate the design concept, mathematic models and control strategy.