Analytical 3D model of a spherical induction actuator with multi-DoF for industrial applications

Nowadays, advances in the automation of traditional industrial processes demand the massive use of robots. Most applications require multidirectional movements that have to be precise and, possibly, easy to control. Conventionally, multi-DoF actuation systems have been implemented with the connection of several actuators, introducing drawbacks that can be overcome by considering devices with unusual configurations, such as spherical actuators. An example could be a spherical electromagnetic induction actuator with 3 DoF. Studies on such a device can bring significant advantages in robotics, biomedical, and industrial systems, reducing the complexity and increasing the performance of the systems. In general, the availability of an analytical solution for electromagnetic devices provides a point of reference for validating FE models, which are essential for analyzing multi-DoF prototypes. In this particular case, the importance of an analytical model lies in the possibility of understanding which system of supply currents is suitable to obtain specific magnetic field distributions along given directions (e.g., the equatorial circumference). Thus, even though simplified, an analytical model can help determine the required parameters. Given the complexity of the problem, the analyzed system was reduced to a conductive sphere (rotor) and a single excitation coil in which a sinusoidal current flows. The excitation coils are fed to create a traveling wave whose motion direction has both latitudinal and longitudinal components. Analyzing this geometry with spherical coordinates, the separation of variables method alone was not sufficient: we included the Second Order Vector Potential (SOVP) in the analytical formulation so that scalar equations were used to formulate the problem. Spherical Bessel functions and Legendre-associated polynomials express the solutions.