# Sensorless Posture Detection of Reluctance Spherical Motor Based on Mutual Inductance Voltage
## 2 Basic Principle of Detecting Rotation Angle from Mutual Inductance Voltage
### Numerical Relationship between Mutual Inductance Voltage and Single-DoF Position Angle
As can be seen from Figure 5, the value of the coil mutual inductance voltage reaches the maximum value at the position of zero stator and rotor poles misalignment angle and is symmetrically distributed on both sides of the peak position. According to Equation (6), as the amplitude of the input signal \(u_{A}\) increases, the amplitude of the mutual inductance voltage \(u_{B}\) increases. As can be seen from Figure 5, when the amplitude of the input signal \(u_{A}\) rises from 0 V to 5 V, the mutual inductance voltage at each deviation angle position almost linearly rises accordingly, i.e., the spherical motor iron core is still in unsaturated state. Considering that the higher the input signal, the stronger the driving torque generated by the coil, and the higher the risk of position change, the amplitude of the excitation signal should not be too high. According to Figure 5, when the input signal amplitude is 2 V, the variation range of the mutual inductance voltage is approximately 10-120 mV, which is distinguishable already. Therefore, the input signal for the subsequent content is a sinusoidal signal with 50 Hz and 2 V amplitude. Figure 4: The block diagram of the experimental platform. There are three layers of coils in the stator, while the rotor teeth are only distributed in one single layer. In contrast to the 60\({}^{\circ}\) symmetrical periodicity of the coil mutual inductance generated by the rotor yawing, the mutual inductance voltage generated during the rotor pitch movement is only symmetrical but not periodic. In order to verify this analysis, an experiment is carried out by spinning the rotor for 60\({}^{\circ}\) from the initial position and recording the mutual inductance voltages of the fifth coil group as shown in Figure 6. While the rotor is pitching 3\({}^{\circ}\)-45\({}^{\circ}\) from the initial position, the mutual inductance voltages of the first group of coils are shown in Figure 7. From Figures 6 and 7, it can be seen that limited by the fact that the mutual inductance voltage caused by rotor yawing only has a monotonic variation range of 30\({}^{\circ}\), each group of coils can detect that the angle range between rotor teeth and the coil axis is within \(\pm\)30\({}^{\circ}\). Figure 5: (**a**) Inductance voltages calculated by Finite Element Method (FEM), (**b**) inductance voltages measured, (**c**) comparison. Figure 6: Mutual inductance voltage measurement value of group 5 coil at yaw range of 60\({}^{\circ}\). After the measurement is completed, take the latitude and longitude position of the rotor pole relative to the coil axis on the spherical surface as the coordinate, mark the mutual inductance voltage value on a 2D plane, and draw a contour map of the mutual inductance voltage accordingly. The predicted value of the mutual inductance voltage when the coil is at this assumed position can be obtained by interpolating the contour map with the longitude and latitude of the assumed position point. This predicted value would be used to compare with the measured value to judge the rationality of the hypothetical position point. For the same group of coils, the variation patterns of the mutual inductance voltage caused by rotor yawing and pitching are not completely consistent.