## 4 Simulation Analyses  
To demonstrate the effectiveness of the proposed control strategy, the strategy was applied to a pathway tracking control for the first three joints of a PUMA560 manipulator, and its tracking performance was compared with those of a classical SMC [16, 17] and NFTSMC [49, 50]. The dynamic model with the crucial parameters found in a 3-DOF PUMA560 robot manipulator was explained by [PERSON] et al. [51]. We utilized the MATLAB/Simulink environment for all simulation analysis with the sampling rate set to \(10^{-3}\) s. In this work, only the first three joints of a robot manipulator were investigated (the last three joints were blocked). The simulations were implemented to compare the controllers in terms of their positional accuracy, response speed, and the resulting chattering phenomenon in their control inputs. To ascertain the robustness of all control methods, we evaluated the system performance in three operation stages, where disturbances and uncertainties were modeled as follows:  
\[F_{r}\big{(}\dot{q}\big{)}+\tau_{d}(t)=\left[ \begin{array}{c}0.9\dot{q}_{1}+1.0\sin(3q_{1})+1.7\sin(\dot{q}_{1})\\ 1.8\dot{q}_{2}+1.85\sin(2q_{2})+1.65\sin(\dot{q}_{2})\\ -2.1\dot{q}_{3}+2.5\sin(2q_{3})+0.57\sin(\dot{q}_{3})\end{array} \right]. \tag{55}\]  
**Stage 1**: Robot system was assumed to run under normal operation from time 0 s to 15 s.  
**Stage 2**: Robot system was assumed to run under operation condition, but there was an external disturbance impacting the first joint between 15 s and 50 s. This external disturbance had a value defined as \(\big{(}15\sin(q_{1}q_{2})+1.5\cos(\dot{q}_{1}q_{2})+5.5\cos(\dot{q}_{1}\dot{q} _{2})\big{)}\). **Stage 3**: Robot system was assumed to run under operation condition, but there was a partial loss (75%) of control input effectiveness at the second joint between 25 s and 50 s.  
The desired joint pathways for the position tracking were  
\[q_{r}=\Big{[} \begin{array}{cc}\cos(\frac{t}{5\pi})-1,&\sin(\frac{t}{5\pi}+ \frac{\pi}{2}),&\sin(\frac{t}{5\pi}+\frac{\pi}{2})-1\end{array} \Big{]}^{T}. \tag{56}\]  
The RBFNN architecture consisted of seven nodes, the initial weight matrix of the network was selected as 0, the width and center of the Gaussian function was set as \(\delta=0.2\), and the center of the Gaussian function \(\mu\) was selected in range (\(-1.5\div 1.5\)) with \(\mu_{l}=0.5\). The matrix used in an adaptive law of RBFNN was selected as \(\omega=15l_{7}\), and the NN input was selected as \(\upsilon=\Big{[} \begin{array}{cccc}e&\dot{e}&q_{r}&\dot{q}_{r}&\ddot{q}_{r}& \end{array} \Big{]}\). The SMC control input was set as  
\[\tau(t)=-B^{-1}(q)\big{(}\Xi\big{(}q,\dot{q}\big{)}+\eta\big{(}\dot{q}-\dot{q} _{r}\big{)}-\ddot{q}_{r}+(\Phi_{2}+\rho_{2})sign(s)\big{)}. \tag{57}\]  
Here, \(\eta,\Phi_{2},\rho_{2}\) are positive constants, \(s\) is a linear sliding function, and \(q_{r}\) is defined as a desired trajectory value. The NFTSMC control input was set as  
\[\tau(t)=-B^{-1}(q)\bigg{(}\Xi\big{(}q,\dot{q}\big{)}-\ddot{q}_{r}+\beta\frac{ h}{d}\big{(}\dot{e}\big{)}^{2-\frac{d}{h}}+(\Phi_{3}+\rho_{3})\frac{s}{\|s\|+ \upsilon}\bigg{)}. \tag{58}\]  
Here, \(\beta,\Phi_{3},\rho_{3}\) are positive constants, \(s\) is a nonlinear sliding function, \(\upsilon\) is a small positive scalar, \(q_{r}\) is defined as a desired trajectory value, and \(d,\ h\) are positive odd integers satisfying the condition \(1<d/h<2\). The control parameter selection for the varying control strategies, including classical SMC, NTSMC, and the proposed control strategy is shown in Table 1.  
The averaged tracking errors were calculated according to the following equation \(E_{i}^{q_{r}}=\sqrt{\frac{1}{n}\sum\limits_{k=1}^{n}\Big{(}\|e_{i}\|^{2}\Big{)} }\)\(i=1,\ 2,\ 3\) in which \(n\) is the number of simulation steps. The trajectory tracking performances, including tracking positions and tracking errors at each of the first three joints with three controllers, are illustrated in Figures 3 and 4.