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-diamensional (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.