## 4 3D FEM Simulation  
To compare the torque output of the original 3-DOF spherical motor and the proposed 3-DOF spherical motor, ANSYS, a commercial software, was utilized to demonstrate their performance. Here, the 3D FEM was used to demonstrate the results of torque output and magnetic flux density distribution because the original 3-DOF spherical motor and the proposed 3-DOF spherical motor are not full symmetry. Figure 7a,b show the 3D FEM models of the original 3-DOF spherical motor and the proposed 3-DOF spherical motor, respectively. Figure 6: Flowchart to actuate proposed 3-DOF spherical motor for rotating around the \(X\)-axis. Figure 7: 3D FEM models of (**a**) original and (**b**) proposed 3-DOF spherical motor, respectively. From Figures 1 and 5, because the structure of the original 3-DOF spherical motor and the proposed 3-DOF spherical motor for rotating around the \(Y\)-axis is the same as that for rotating around the \(X\)-axis, only 3D FEM simulations for rotating around the \(X\)-axis were performed and compared. In the simulations, the materials of the yokes, the coils, and the permanent magnets are selected as steel-1008, copper, and NdFe35, respectively. In the original 3-DOF spherical motor, each coil for rotating around the \(X\)-axis, \(Y\)-axis, and \(Z\)-axis has 900, 900, and 250 turns, respectively. By calculating, the inductance and resistance of each coil for rotating around the \(X\)- and \(Y\)-axes are 7.9 \(\times\) 10\({}^{-7}\) H and 19.2 \(\Omega\), respectively; those of each coil for rotating around the \(Z\)-axis are 2.2 \(\times\) 10\({}^{-7}\) H and 6.5 \(\Omega\), respectively. In the proposed 3-DOF spherical motor, each coil for rotating around the \(X\)-axis, \(Y\)-axis, and \(Z\)-axis have 450, 450, and 250 turns, respectively. By calculating, the inductance and resistance of each coil for rotating around the \(X\)- and \(Y\)-axes are 4.0 \(\times\) 10\({}^{-7}\) H and 9.6 \(\Omega\), respectively; those of each coil for rotating around the \(Z\)-axis are 2.2 \(\times\) 10\({}^{-7}\) H and 6.5 \(\Omega\), respectively. The current 0.25 A was selected to pass each coil of the original and proposed 3-DOF spherical motor. Therefore, the current density of the original and proposed structure is 1.4 \(\times\) 10\({}^{7}\) A/m\({}^{2}\). The diameter of the coils is 0.15 mm for both the original 3-DOF spherical motor and proposed 3-DOF spherical motor. It is noted that for the original 3-DOF spherical motor [25] and proposed 3-DOF spherical motor, the main driving range of both motors is around the origin, and the coils are switched at the same high speed for both motors. Here, the power consumption issue for the driving circuit is ignored. For a fair comparison, the original 3-DOF spherical motor and the proposed 3-DOF spherical motor are assumed to have the same power consumption. For the original 3-DOF spherical motor, in the simulation case for rotating around the \(X\)-axis, the power consumption P\({}_{\rm o}\) can be expressed as:  
\[\rm{P_{o}=2\,I_{o}^{\rm{\rm{  
part of the coils in the left side is very weak, which agrees to the viewpoint in Section 2. Figures 8 and 9 only show the magnetic flux density distribution of the coils and the magnet. Based on [PERSON]'s circuital law, the magnetic field generated from the energized coils must be considered. The vector diagram of the magnetic flux density distribution in the yoke and magnet is shown in Figure 10.  
Under the same input current of 0.25 A, Figure 11 shows the numerical results with a step size of 1\({}^{\circ}\) obtained for the torque change with respect to the different rotation angle of the original and the proposed 3-DOF spherical motor either in a counter-clockwise direction or in a clockwise direction. When the rotor rotates in a counter-clockwise direction, the direction of the current is out of the page, such as in Figure 4. By contrast, when the rotor rotates in a clockwise direction, the direction of the current is into the page.