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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. INDEX TERMS adaptive control, delay compensation, linear predictor, permanent magnet spherical actuator, robust sliding mode control, trajectory tracking.

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Point-to-point (PTP) motion of the end effector is an essential part for using robot manipulators to perform precision machining task. Traditionally, it is implemented by several motors with complex transmission mechanism, which causes slow responses, low positioning precision and dynamic performance. Therefore, dynamics analysis and modified electrifying method of a permanent magnet spherical motor (PMSM), which can provide up to three degrees-of-freedom independently for PTP motion is presented in this study. First, a new inverse kinematics method for solving PMSM's rotation angle using a crossproduct between two points is proposed. Through dynamics analysis, the control torque is designed by using sinusoidal function to plan the PTP motion. Second, a modified electrifying method by combined coils has been proposed based on the currenttorque equation, which deduced the calculation burden by reducing the dimensions of characteristic matrix. Finally, the simulations and experimental results show that PTP motion trajectory planning based on sinusoidal function and the modified electrifying control method can not only reduce the torque disturbance and increase the electromagnetic torque, but also has the advantages of relatively simple control and better tracking control performance.

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