## 2 System Description  
The magnetic levitation actuation and motion control system based on force imbalance consists of a main control unit, power supply modules, electromagnetic actuators, input control modules of current, PWM output modules, differential calculation modules of signals and A/D modules of input signals. The schematic diagram of the magnetic levitation actuation and motion control system based on force imbalance is shown in Figure 1. The electromagnetic actuator adopts dual electromagnets with a symmetrical layout to generate a superimposed magnetic field to attract the control component of a permanent magnet ball. The system forms a feedback closed loop through the position signals and current signals collected by sensors, so that the control component can move according to the required path and accuracy, thereby realizing the operation and control of the permanent magnet ball in a non-liquid environment. During the actuating process, the ball can move with variable speeds and controllable trajectory. It can be seen from Figure 2 that the permanent magnet ball is located on the central axis of the electromagnetic solenoid along the \(Z\) direction, and the electromagnetic force generated by the external magnetic field actuates the ball to move in the direction where the magnetic induction intensity changes most rapidly. The force with cylindrical symmetry in the \(XY\) plane can be ignored because it has a total effect of zero on the permanent magnet \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(N\) & number of coils & 1445 \\ \(R_{1}\) & inner radius & 7.5 (mm) \\ \(R_{2}\) & outer radius & 20 (mm) \\ \(L\) & winding height & 60 (mm) \\ \(R\) & resistance of coil & 9.3 (\(\Omega\)) \\ \(d\) & diameter of copper wire & 0.71 (mm) \\ \(L_{0}\) & overall length of coil & 80,743.13 (mm) \\ \(S\) & coil cross-sectional area & 678.24 (mm\({}^{2}\)) \\ \(A\) & surface area of coil & 10,399.68 (mm\({}^{2}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Related parameters of electromagnetic actuators. Figure 1: Mechanical structure and system control model of magnetic levitation system. (**a**) External magnetic field actuation device; (**b**) System control model. ball. From \((\partial B_{Z})/\partial z\), the electromagnetic force \(F\) exerted by the hollow solenoid on the permanent magnet ball [19; 20; 21] is:  
\[F=(m\bullet\
abla)B=|m|\frac{\Delta B_{z}}{\Delta z}=\frac{3B_{\mathrm{s}}}{2 \mu_{0}}V_{\mathrm{b}}\bullet\frac{\partial B_{z}}{\partial z} \tag{1}\]  
where \(F\) denotes the electromagnetic force on the permanent magnet ball; \(m\) denotes the mass of the permanent magnet ball; \(B\) denotes magnetic field of electromagnetic actuators; \(\
abla\) denotes gradient; \(B_{x}\) denotes magnetic field of electromagnetic actuators in vertical direction; \(B_{\mathrm{s}}\) denotes magnetic field of the permanent magnet ball; \(V_{\mathrm{b}}\) denotes the volume of the permanent magnet ball; \(\mu_{0}\) denotes vacuum permeability. \(F\) on the permanent magnet ball is correlated to the current \(i\) and the levitation gap \(\delta\) of the permanent magnet ball, and \(F\) is proportional to the magnitude of current \(i\) through the coil from (1). Moreover, the electromagnetic force exerted by the dual electromagnetic actuators on the permanent magnet ball is a superposition of both electromagnetic forces. The dimensionless equation of \(F\) is simplified from (1) based on the external magnetic field actuation, and then the electromagnetic force \(F\) exerted by the dual electromagnetic actuators on the permanent magnet ball is superimposed in the \(Z\) axis direction to obtain the following equation.