# A Study of Friction Nonlinearity and Compensation for Turntable Servo Systems
## 2 Friction Modeling and Identification
### Friction Parameter Identification
#### 2.2.2 Parameter Identification Experiment  
The determination of the maximum static friction torque is presented below:  
Under the control mode of \(i_{d}\)* = 0, a slowly increasing \(q\)-axis voltage was applied, thereby applying a slowly increasing electromagnetic torque. Considering the hysteresis caused by the motor's rotational inertia, the relatively stable maximum value of the \(q\)-axis current shortly before this moment was identified by examining the electrical current data. This current value was then multiplied by the torque coefficient to calculate the maximum static friction torque. The experiment was repeated, and the average values are reported in Table 1.  
The determination of the Coulomb friction coefficient and the viscous friction coefficient is shown below:  
With a given \(q\)-axis voltage signal of \(U_{q}=10\) V, after the motor had operated stably for a period, the power supply was cut off and the motor circuit was opened to allow it to decelerate freely without the influence of back electromotive force (EMF). Disregarding the minimal currents, the motion equation of the motor at this point is:  
\[J_{m}\ddot{\theta}_{m}=-T_{g}-(T_{c}+B_{v}\ddot{\theta}_{m}) \tag{3}\]  
The variables in (3) are explained in Figure 2. In a dual-inertia system, \(T_{c}\) is the Coulomb friction moment, \(B_{v}\) is the viscous friction coefficient, \(J_{m}\) is the motor inertia on the drive side, \(\theta_{m}\) is the motor angle on the drive side, and \(T_{g}\) denotes the axial torque component, a variable that is specific to a dual-inertia system and whose mathematical expression is shown in Figure 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of** & **Current \(Iq\) (A)** & \
\begin{tabular}{c} **Maximum Static** \\ **Friction Torque \(T_{s}\) (Nm)** \\ \end{tabular} &
\begin{tabular}{c} **Average Maximum Static** \\ **Friction Torque \(T_{s}\) (Nm)** \\ \end{tabular} \\ \hline
1 & 0.051 & 0.041 & \\
2 & 0.057 & 0.046 & \\
3 & 0.048 & 0.038 & \\
4 & 0.069 & 0.055 & 0.043 \\
5 & 0.050 & 0.040 & \\
6 & 0.049 & 0.039 & \\
7 & 0.053 & 0.042 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Determination of maximum static friction torque. It is posited that the torque equals the moment of inertia on the load side. In this, experimental setup, the inertia of the motors on both the drive and load sides are equal; therefore, (3) is revised to (4). \[2J_{m}\ddot{\theta}_{m}=-(T_{c}+B_{v}\ddot{\theta}_{m}) \tag{4}\]  
A solution to the differential equation provides the time expression for the motor's rotational speed. \[\dot{\theta}_{m}=-\frac{T_{c}}{B_{v}}+(\frac{T_{c}}{B_{v}}+\theta_{0})e^{- \frac{B_{v}}{2J_{m}}t} \tag{5}\]  
Through two repeated experiments, the relationship between the motor speed and time during the power-off deceleration process was measured, and its curve was fitted by PSO. The results are shown in Figure 3. The coefficients were calculated by bringing the above test results into Equation (5). And these coefficients were deformed to obtain the parameters to be identified. Let the identification coefficients \(T_{c}/B_{v}\), \(\theta_{0}\), and \(B_{v}/J_{m}\) be denoted as \(a\), \(b\), and \(c\), respectively. The formula for solving the friction parameters is given in (6), and the results are presented in Table 2.  
\[\left\{ \begin{aligned} & B_{v}=cJ_{m}\\ T_{c}=acJ_{m}\end{aligned} \right. \tag{6}\]  
At the same time, the above experiments also proved that, when the number of iterations was below 20, the optimal fitness decreased as the number of iterations increased; and after the number of iterations exceeded 20, the optimal fitness tended to be close to 0, indicating that the iterative computation of the PSO accomplished its fitting objective. Therefore, the number of iterations for PSO was set to be 20.