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 equation. Based on this expression and properties of air-core stator coils, Lorentz force law is employed for the study of the relationship between the rotor torque and coil input currents. By using linear superposition, the expression of the actuator torque in terms of current input to the stator coils can be obtained in a matrix form. The linear expression of the actuator torque will facilitate real-time motion control of the actuator as a servo system. Experimental works are carried out to measure the actual magnetic field distribution of the PM rotor in three-dimensional (3-D) space as well as to measure the actual 3-D motor torque generated by the actuator coils. The measurement results were coincident with analytical study on the rotor magnetic field distribution and actuator torque expressions. The linearity and superposition of the actuator torque were also verified through the experiments

This paper has proposed a 3-DOF spherical actuator consisting of a ball-shaped rotor with a full circle of permanent- magnet (PM) poles and a spherical-shell-like stator with two layers of circumferential air-core coils. One key feature of this design is the parametrization of PM and coil poles. Based on the torque model of the PM spherical actuator, the relationship between poles' parameters and torque output can be demonstrated. As a result, the actuator design aiming at achieving maximum torque output can be carried out from the relationships. Another advantage of this spherical actuator is its singularity-free workspace, which is verified with the actuator torque model and condition numbers.

Although numerous researches on s-curve motion profiles have been carried out, up to date, no systematic investigation on the general model of polynomial s-curve motion profiles is considered. In this paper, the model of polynomial s-curve motion profiles is generalized in a recursive form. Based on that, a general algorithm to design s-curve trajectory with time-optimal consideration is proposed. In addition, a special strategy for planning s-curve motion profiles using a trigonometric model is also presented. The algorithms are implemented on a linear motor system. Experimental results show the effectiveness and promising application ability of the algorithms in s-curve motion profiling.