## 5 Simulation Study  
In this section, Matlab simulations of a two-degrees-of-freedom FJR are reported to verify the effectiveness of the identification and control algorithm proposed in this study. The schematic diagram of the \(i\)th link of the FJR is depicted in Figure 1. The matrices \(\mathbf{M}(q)\), \(\mathbf{C}(q,\dot{q})\), \(\mathbf{G}(q)\) of the FJR model (1) are given by  
\[M(q)= \begin{bmatrix}M_{11}&M_{12}\\ M_{21}&M_{22}\end{bmatrix} ,\;C(q,\dot{q})= \begin{bmatrix}C_{1}\\ C_{2}\end{bmatrix} ,\;G(q)= \begin{bmatrix}G_{1}\\ G_{2}\end{bmatrix} ,\] \[M_{11}= m_{1}l_{c1}^{2}+l_{1}+l_{2}+m_{2}(l_{1}^{2}+l_{c2}^{2}+2l_{1}l_{c2 }\cos q_{2}),\] \[M_{12}= m_{2}l_{c2}^{2}+l_{2}+m_{2}l_{2}l_{c2}\cos q_{2}\] \[M_{21}= m_{2}l_{c2}^{2}+l_{2}+m_{2}l_{2}l_{c2}\cos q_{2},\;M_{22}=m_{2}l_{c2 }^{2}+l_{2}\] \[C_{1}= -m_{2}l_{1}l_{c2}\sin q_{2}q_{2}-m_{2}l_{1}l_{c2}\sin q_{2}(q_{1} +q_{2}),\;C_{2}=m_{2}l_{1}l_{c2}\sin q_{2}\vartheta_{1}\] \[G_{1}= m_{2}gl_{1}\cos(q_{1}+q_{2})+(m_{1}+m_{2})gl_{1}\cos q_{1},\;G_{2}= m_{2}gl_{2}\cos(q_{1}+q_{2})\]  
and the nominal system parameters used in the simulation are listed in Table 1. For simplicity, no friction is considered in the simulation; thus, \(D=[0,\;0]^{T}\). To verify the effectiveness of the proposed methods for systems with model uncertainties, the real system parameters \(m_{1}^{*}\), \(m_{2}^{*}\), \(l_{1}^{*}\), \(l_{2}^{*}\), \(l_{1}^{*}\), \(l_{2}^{*}\), \(l_{c1}^{*}\), \(l_{c2}^{*}\) are set to be 1.2 times of their nominal values. The reference signal is \(q_{d}=[\sin(1.6\pi t),\sin(1.6\pi t+\pi/3)]^{T}\). The initial values for \(x_{1}\) and \(x_{2}\) are \(x_{1}(0)=0.8\) and \(x_{2}(0)=0\). To transform the original system model (1) into a singular perturbation model (2), \(K_{0}=\text{diag}\{0.01,0.01\}\) is used. The prescribed performance continuous terminal sliding mode controller based on the singular perturbation technique and the composite-learning optimal bounded ellipsoid algorithm (SOBE) is used to control this system, and the following parameters are chosen for the controller: \(K_{1}^{\text{s}}=5\), \(K_{2}^{\text{s}}=1\), \(K_{3}^{\text{s}}=1\), \(\epsilon_{s}=0.1\), \(\epsilon_{s}=0.9\), \(K_{1}^{f}=250\), \(K_{2}^{f}=1\), \(K_{3}^{f}=1\), \(\epsilon_{f}=0.1\), \(\varrho_{f}=0.9\). To compensate the uncertainties, an NN with 30 neurons is employed, and the identification algorithm parameters are chosen as: \(\gamma_{s}=0.02\), \(\gamma_{f}=0.02\), \(L_{i0}=0.2\), \(F_{il}=201\), \(F_{iu}=200\), \(\lambda_{i}=5\), \(\overline{\zeta}_{fi}=0.001\). For the purpose of comparison, a modified OBE algorithm (MOBE)-based adaptive controller [13] is also tested in the simulation. The choice of this particular control scheme is motivated by the fact that our identification methodology builds upon the MOBE framework. While MOBE exclusively utilizes the identification error for updating NN weights, our proposed method incorporates both identification and tracking errors in the NN training process, thereby extending the MOBE approach. Additionally, whereas [13] employs a conventional sliding mode controller, our study develops a terminal sliding mode PPC. Consequently, [13] constitutes an apt and informative reference point against which to assess the performance of our novel controller. For the MOBE method, all the shared parameters are the same as those used for SOBE. It should be pointed out that the control \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(m_{1},m_{2}\) & 0.1 & kg \\ \(l_{1},l_{2}\) & 0.4 & m \\ \(l_{1},l_{2}\) & 0.02 & kg\(\cdot\)m\({}^{2}\) \\ \(l_{c1},l_{c2}\) & 0.2 & m \\ \(l_{1},l_{2}\) & 0.01 & kg\(\cdot\)m\({}^{2}\) \\ K & 9 & N\(\cdot\)m\(/\)rad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nominal parameters of the manipulator. method proposed in [13] is only used to control the reduced slow subsystem. The controller for the reduced fast subsystem is the same as the one used in SOBE. The simulation outcomes are presented in Figures 2-17.