## 5 Numeric Simulation  
### Simulation Setup  
In order to verify the performance of the proposed PP-NFTSMC-ESO controller, computer simulations were conducted based on a simulation model of the electric oil-filled joint actuator, which is shown in Figure 3. The nominal parameters of the simulation model were set with the actual parameters of the electric oil-filled joint actuator listed in Table 1. The simulations were performed with a fixed sampling time 1 ms, and the initial states of the simulation model were set as \(x=[0,\,0]^{T}\), while that of the ESO was set as \(\hat{x}=[0,\,0,\,0]^{T}\).  
Figure 3: Simulation model of the electric oil-filled joint actuator.  
The oil stirring viscos loss was set as Equation (6) with the working pressure and temperature in two cases, \(P=10\) MPa, \(T=2\)\({}^{\circ}\)C, and \(P=0\) MPa, \(T=25\)\({}^{\circ}\)C, which were used to simulate the deep-sea working condition with 1000 m depth (the temperature of deep sea is generally about \(2\)\({}^{\circ}\)C) and the normal laboratory working condition. The values of \(\mu_{0}\) and \(\alpha\) are listed in Table 2 (the oil used in this work was the ISO VG 22 hydraulic oil), and the geometry parameters of the researched electric oil-filled joint actuator were presented in Table 3.  
The external disturbance torque, output shaft dynamic seal friction, and the core loss are all unknown but deteriorate the performance of the electric oil-filled joint actuator. In this work, in order to simulate their influence, we set the external disturbance torque as time varying sinusoidal signals, and the influence of core loss as parameter variance of the electric oil-filled joint actuator, which are presented as Equations (40) and (41) respectively. Since the output shaft dynamic seal friction is related to joint speed and working pressure, we set the variation regulation of it as Equation (42), of which the amplitude surface is shown in Figure 4 with the maximum and minimum values are 10 Nm (at the deep-sea working condition \(P=10\) MPa and \(T=2\)\({}^{\circ}\)C) and 3 Nm (at the normal laboratory working condition \(P=0\) MPa and \(T=25\)\({}^{\circ}\)C), respectively. Although these regulations are not from their real law of changes, they are able to simulate the influence of the dynamic uncertainties and unknown disturbances on the electric oil-filled joint actuator to some extent.  
\[T_{d}=20(1+0.5\sin(0.5\pi t)) \tag{40}\]  
\[\left\{\begin{array}{l}J=J_{n}(1+0.2\sin(0.5\pi t))\\ k_{t}=k_{tn}(1+0.3\cos(0.3\pi t))\\ k_{b}=k_{bn}(1+0.2\cos(0.3\pi t))\end{array}\right. \tag{41}\]  
\[T_{f}=(1+\exp(0.3591|x_{2}|)+\exp(0.1504P))\text{sign}(x_{2}) \tag{42}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(J_{n}\) **(kg m\({}^{2}\))** & \(k_{tn}\) **(Nm/A)** & \(k_{bn}\) **(Nm/(rad/s))** & \(N\) \\ \hline value & \(3.34\times 10^{-5}\) & \(0.105\) & \(1.95\times 10^{-5}\) & \(100\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The nominal parameters of the electric oil-filled joint actuator.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \(d\) & \(d_{1}\) & \(d_{2}\) & \(\delta\) & \(h\) & \(L\) \\ \hline value & \(37.5\) mm & \(63\) mm & \(37.5\) mm & \(0.3\) mm & \(38\) mm & \(15.5\) mm \\ \hline \hline \end{tabular}
\end{table}
Table 3: The geometry parameters of the researched electric oil-filled joint actuator.