## 5 Numeric Simulation
### Simulation Study  
Considering the actual working conditions, computer simulation studies were performed on the electric oil-filled joint actuator for trajectory tracking control in the presence of unmeasured system states, dynamic uncertainties, and unknown external disturbances. Case 1: This case is to verify the effectiveness of the proposed PP-NFTSMC-ESO controller for the trajectory tracking with different working conditions. In this case, the desired trajectories were set with two kinds: slow motion trajectory \(y_{d}=\frac{2}{9}\pi[0.5-\cos(\pi t)][1-\exp(-t)]\) rad, and fast motion trajectory \(y_{d}=\frac{2}{9}\pi[0.5-\cos(4\pi t)][1-\exp(-t)]\) rad. The physical parameters of the joint were set as Equation (41) with the nominal values given in Table 1 (i.e., there exist dynamic uncertainties), and the design parameters of the proposed controller and the ESO were set as the values shown in Table 4. Suppose unknown external disturbances, including the oil stirring loss \(T_{v}\), the external disturbance torque \(T_{d}\), and output shaft dynamic seal friction \(T_{f}\), are also acted on the joint actuator, of which the expressions are expressed in Equations (6), (40) and (42). The position-tracking performance of the slow motion and fast motion are illustrated in Figures 5 and 6 and Table 5, in which the subscripts 'n' and's' of the figure legends indicate the normal laboratory working condition and deep-sea working condition with 1000 m depth, respectively. As seen, the proposed PP-NFTSMC-ESO controller has excellent position-tracking performance in both slow motion and fast motion with different working conditions. Figures 5a and 6a demonstrate that the actual position response almost overlaps with the desired trajectories. Although the two working conditions impose different influences on the joint actuator and dynamic uncertainties and unknown disturbances vary with time and system states, the position tracking errors were always kept in a very small value (presented in Table 5, with a maximum steady state tracking error about 4.89 \(\times\) 10\({}^{-4}\) rad for slow motion and 6.96 \(\times\) 10\({}^{-4}\) rad for fast motion) and maintained in the predefined performance boundaries, which are shown in Figures 5b and 6b. As far as position-tracking performance is concerned, the effectiveness of the proposed PP-NFTSMC-ESO controller was clearly shown. The reason for such a good tracking performance may be explained by the fact that, with the help of the ESO, the unmeasured velocity and lumped disturbances (dynamic uncertainties and unknown external disturbances) of the joint actuator can be precisely estimated, which are depicted in Figure 5c,d and Figure 6c,d, and thus the proposed PP-NFTSMC-ESO controller is able to accomplish by employing both the system \begin{table}
\begin{tabular}{c c} \hline \hline
**Controller/Observer** & **Parameter** \\ \hline PI & \(k_{p}=500\), \(k_{i}=300\) \\ SMC-ESO & \(k_{1}=20,000\), \(k_{2}=10,000\), \(\eta=100\) \\ NFTSMC-ESO & \(k_{1}=1000\), \(k_{2}=20\), \(k_{3}=200\), \(\alpha_{1}=1.1\), \(\alpha_{2}=1.3\), \(\eta=0.1\) \\ \multirow{2}{*}{PP-NFTSMC-ESO} & \(k_{0}=100\), \(k_{1}=1000\), \(k_{2}=20\), \(k_{3}=200\), \(\alpha_{1}=1.1\), \(\alpha_{2}=1.3\), \\  & \(\eta=0.1\), \(\tau=0.06\), \(\rho_{0}=0.25\), \(\rho_{00}=0.05\), \(m=0.5\), \(l=1\), \(h=1\) \\ ESO & \(l_{1}=15\), \(l_{2}=75\), \(l_{3}=125\), \(\omega_{0}=60\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The design parameters of the controllers and observers. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Normal Laboratory Working** & **Deep-Sea Working** \\  & **Condition** & **Condition** \\ \hline slow motion & \(2.55\times 10^{-4}\) rad & \(4.89\times 10^{-4}\) rad \\ fast motion & \(4.36\times 10^{-4}\) rad & \(6.96\times 10^{-4}\) rad \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The maximum steady-state tracking error of the PP-NFTSMC-ESO controllers under different working conditions.