# Cuckoo Coupled Improved Grey Wolf Algorithm for PID Parameter Tuning
## 4 Numerical Data Experiment and Simulation Analysis  
### Mathematical Model of a Servo Motor System  
In recent years, with the increasing industrialization levels in China, servo control systems have been widely used in all walks of life, from industrial production of mechanical arms to computer numerical control (CNC) machine tools, military weapon servo systems, and even aerospace satellites. Servo control systems are ubiquitous in use. The motor control determines the working performance of the entire system as the main component of the servo system. Therefore, it is necessary to study the drive control part of the servo motor in depth [27, 28].  
The experiment in this section is conducted on a self-developed small CNC polishing machine. This equipment has three linear axes (X, Y, and Z) and two rotational axes (A and C). To validate the practicality of the optimisation algorithm proposed in this paper, an experimental motion control of the Y-axis, whose model is Somotics s-1k7(SIEMENS, Munich, Germany), was performed on this machine. The physical illustration is depicted in Figure 5.  
The servomotor model is shown in Figure 6; it includes the motor, coupling, roller lead screw, guide rail, and workbench. Driven by the motor, the ball screw and motor are coaxially rotated through the coupling, so that the rotating motion of the rotating motor is transformed into the linear motion of the workbench. The symbols in the figure are listed in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Physical Meaning** & **Parameter** \\ \hline \(J_{a}\) & Moment of inertia of the drive shaft & 0.1 kg \(\cdot\) m\({}^{2}\) \\ \(T_{m}\) & Motor output torque & 50 N \(\cdot\) m \\ \(K_{p}\) & Torsional stiffness coefficient of a driven shaft & 899 Nm/rad \\ \(T_{b}\) & Output torque of a ball screw driven by a driven shaft & 50 N \(\cdot\) m \\ \(M_{0}\) & Moving platform mass & 38 kg \\ \(X_{0}\) & Moving platform displacement & 0–130 cm \\ \(T_{p}\) & Coupling transmission output torque & 50 N \(\cdot\) m \\ \(f_{p}\) & Moment of inertia of a driven shaft & 0.15 kg\(\cdot\)m\({}^{2}\) \\ \(\theta_{m}\) & Motor output angle & 0–360\({}^{\circ}\) \\ \(D_{p}\) & Driven shaft damping coefficient & 3.83 Nm/rad \\ \(\theta_{b}\) & Output angle of a ball screw driven by a driven shaft & 0–360\({}^{\circ}\) \\ \(D_{0}\) & Damping coefficient of a moving platform & 10 Ns/m \\ \(f_{0}\) & Load resistance of a moving platform & 44 N \(\cdot\) m \\ \(\theta_{p}\) & Coupling transmission output angle & 0–360\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Symbols used for the servo motor model.  
Figure 5: The physical illustration of the Y-axis.  
According to the servo motor model shown in Figure 5, the dynamic differential equation can be obtained as follows:  
\[T_{b}=\frac{{{{I_{0}}}{{\ddot{\theta}}_{b}}+{D_{0}}{{\dot{\theta}}_{b}}+{f_{0}}{R _{b}}}}{{{R_{b}^{2}}}}, \tag{14}\]  
\[T_{m}={{J_{a}}}{{\ddot{\theta}}_{m}}+T_{p}, \tag{15}\]  
\[T_{p}={{K_{p}}}\big{(}{{\theta_{p}}-{\theta_{b}}}\big{)}+{D_{p}}\Big{(}{{\dot{ \theta}}_{p}}-{{\dot{\theta}}_{b}}\Big{)}={{J_{p}}}{{\ddot{\theta}}_{b}}+T_{b}. \tag{16}\]  
The above dynamic differential equation is further simplified into a transfer function form by the Laplace transformation:  
\[{{\theta_{b}}(s)}\approx\frac{{{R_{b}^{2}}\big{(}{{k_{p}}+{D_{p}}s}\big{)}{{ \theta_{p}}(s)}}}{{\big{(}{{J_{p}}{R_{b}^{2}}+{M_{0}}}\big{)}{{s^{2}}}+\big{(} {{D_{0}}+{R_{b}^{2}}{D_{p}}}\big{)}{{s}}+{R_{b}^{2}}{k_{p}}}}. \tag{17}\]