## 5 Experiments
### Parameter Selection and Identification  
Based on the mechanical system finite element analysis results and the device parameters, the model parameters in the system are given in Table 3.  
First, to identify the friction model parameters, a low-frequency sine wave (0.005 Hz, 2 V) is used as the control voltage in the open-loop system to obtain the nonlinear disturbance for the model. The curve for the control voltage and the relative angular velocity are shown in Figure 6.  
It is worth noting that when the control voltage approaches zero, the relative angular velocity exhibits a pronounced dead time phenomenon. \(F_{m}(t)\) is similar to the theoretical friction model, and it is proven that the friction nonlinearity dominates the nonlinear disturbance. The friction feedforward method is designed to compensate its negative influence. If the NDOB is used in this system directly, there will be limitations to performance improvements. This is because the system may have a larger \(\mathbf{d}(t)\) or \(\mathbf{\hat{d}}(t)\) at some time, which is inconsistent with **Assumption 1**. Friction torque \(F_{f}(t)\) is fitted by the model (42). The relationship between the estimated disturbance torque and the relative angular velocity is shown in Figure 7. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Device** & **Model** & **Key Item** \\ \hline Gyroscope & STIM210 Multi-Axis Gyro Module & 262 Hz (Bandwidth) \\ Torque motor & RH-14D-3002-E100 AL & 2–14 \({}^{\circ}\)/s (Resolution) \\ Incremental Encoder & - & 1000 (Pulses/revolution) \\ Harmonic drive & - & 100 (Reduction Ratio) \\ Motor Driver & MAXON ESCON Module 50/5 & 5 A (Constant Current) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main devices used in the experiment. Figure 6: Control voltage and response in the open-loop system. In the experiments, the least squares method is used to fit the friction disturbance, and the friction nonlinear torque fitting parameters are obtained as shown in Table 4.  
Second, to verify the model fitting accuracy and the effectiveness of the friction feedforward compensation method, the residual disturbance is estimated again. The control value and the residual disturbance curve are displayed in Figure 8.  
Compared with Figure 7, the disturbance has been reduced from 3 V (peak to peak value) to less than 1 V. It shows the accuracy of the friction fitting value and the effectiveness of the friction feedforward method. From Figure 8, the residual disturbance and its first derivative are small constants like **Assumption 1**. The system performance will improve by increasing the NDOB gain, and the larger the gain, the stronger the ability to compensate for the disturbance. However, it will be considerably sensitive to sensor noise. Therefore, this parameter is selected in the trade-off between the disturbance suppression ability and noise sensitivity. The NDOB gain is 200 rad/s in the experiment. Finally, the PI parameter can be adjusted by the sine or step response. The feedback controller parameters \(u_{c}^{*}(t)\) are properly tuned to achieve optimum performance in practice, along with \(K_{p}=1\) and \(K_{i}=1\). The modified feedback controller is constructed in the form of Equation (29) with the following matrices. \[\mathbf{A}= \begin{bmatrix}0&1\\ -30276&-139.2\end{bmatrix}  \tag{71}\]  
\[\mathbf{B}= \begin{bmatrix}0&1\end{bmatrix} ^{T} \tag{72}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(A_{c1},A_{c2}\) & Coulomb friction magnitudes in the positive and negative direction & 1.8, 2.0 \\ \(\gamma\) & a large constant & 100 \\ \(K_{v1},K_{v2}\) & Viscous friction magnitudes in the positive and negative direction & 0.005, 0.005 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Nonlinear friction fitting parameters. Figure 8: Residual disturbance curve. Figure 7: Disturbance torque fitting and the estimation values. \[\mathbf{C}= \begin{bmatrix}0&-114.84\end{bmatrix}