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.

Human-robot collaboration (HRC) [1], [2] is a vastly developing field in diverse industries such as health care [3], industrial assembly [4], search and rescue, home service [5], and construction [6], [7]. Many researchers believe that robots will enhance human workers, not replace them, as they do not have the same capability to evaluate and correct their work in real time. Lee [8] introduces case studies on glazing robot technology for installing glass panels on construction sites. A human-robot dialogue system [9] has been developed in joint-action science and technology to solve a construction task collaboratively with a human. The use of immersive virtual environments is also reported in [10] to evaluate human trust and perceived safety in response to robot actions during a collaborative construction. Deploying robots in collaboration with humans is seen as an enabler of major changes in construction productivity for various tasks, such as interior finishing.