## 3 Main Results
### The Designed Control Methodology
According to [8], \(s_{i}\) and \(\dot{s}_{i}\) are equal to zero in finite-time (\(t_{ri}=2L_{2}^{1/2}(t=0)/v\)). Therefore, \(s\) and \(\dot{s}\) equal to zero in finite time (\(T_{r}=\text{max}_{i=1,\ldots,n}[t_{ri}]\)) and both \(x_{ei}\) and \(x_{dei}\) also stabilize to equilibrium in finite time (\(T\leq T_{r}+T_{s}\)) under the control action in Equation (20)-(22). **4. Numerical Simulation Studies**  
In this numerical example, a 3-DOF PUMA560 robot manipulator (with the first three joints and the last three joints blocked) is adopted. The MATLAB/SIMULINK software (2019a MATLAB Versionof The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760 USA) was used for all computation, the sampling time was set to \(10^{-3}\) s, and the solver ode3 was used. The kinematic description for the robot system is displayed in Figure 1. The design parameters and dynamic models of the robot system are referenced from the document [39]. There are many essential parameters of a robot that need to be presented. Therefore, to present briefly, the design parameters and dynamic models of the robot system are reported in [39]. To explore the potential of our designed approach, the robot is controlled to follow the designated trajectory configuration at first. Later, its control performance is then evaluated and compared with the performance of different control algorithms, including SMC and NFFSMC. These control methods for comparison are briefly explained as follows:  
The normal SMC [14] has the following control torque:  
\[u = -\Phi(x,t)^{-1}\Big{[}\Pi(x,t)+c\Big{(}x_{2}-\dot{x}_{d}\Big{)}- \ddot{x}_{d}+(\Sigma+\xi)\text{sgn}(s)\Big{]} \tag{32}\]  
where \(s=x_{de}+cx_{e}\) is the linear sliding manifold, \(c\) is a positive constant. Further, the NFFSMC [40] has the following control torque:  
\[u(t) = -\Phi(x,t)^{-1}\Big{[}\Pi(x,t)+\alpha\frac{q}{l}\dot{x}_{de}^{2-l/q }-\ddot{x}_{d}+(\Sigma+\xi)\text{sgn}(s)\Big{]} \tag{33}\]  
where \(s=x_{e}+\alpha^{-1}x_{de}^{l/q}\) is a nonlinear sliding manifold. The designed parameters of three control methodologies are given in Table 1. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Control Schemes** & **Parameters** & **Values** \\ \hline SMC & \(c\) & \(2\) \\  & \(\xi,\Sigma\) & \(0.01,20\) \\ NFFSMC & \(l,q,\omega\) & \(5,3,2\) \\  & \(\xi,\Sigma\) & \(0.01,20\) \\ Proposed FTSMC & \(\gamma_{1},\gamma_{2},\mu_{1},\mu_{2}\) & \(1,1,1.2,1.4\) \\  & \(\phi,\alpha,\Sigma_{1},\Sigma_{2}\) & \(1,0.6,40,50\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values for three different control methodologies. Figure 1: The kinematic description of 3-Degrees of Freedom (DOF) PUMA560 robot manipulator. The designated trajectory configuration for position tracking when the robot manipulator operates:  
\[\theta_{d}=\left\{ \begin{array}{l}\theta_{d1}=0.6+\cos\left(\frac{t}{6\pi} \right)-1\\ \theta_{d2}=-0.6\sin\left(\frac{t}{6\pi}+0.5\pi\right)-1\\ \theta_{d3}=0.6+\sin\left(0.2\frac{t}{\pi}+0.5\pi\right)-1\end{array} \right. \tag{34}\]  
Friction and disturbance models are hypothesized to analyze the strong capability of the designed FTSMC. It is not amenable to accurately calculate these friction and disturbance terms; therefore, the physical values of frictions and disturbances are not measured. Therefore, the following friction forces and disturbances were modeled, respectively:  
\[f_{r}\Big{(}\dot{\theta}\Big{)}=\left\{ \begin{array}{l}f_{r1}=2.3\dot{ \theta}_{1}+2.14\text{sgn}\Big{(}3\dot{\theta}_{1}\Big{)}\\ f_{r2}=4.5\dot{\theta}_{2}+2.35\text{sgn}\Big{(}2\dot{\theta}_{2}\Big{)}\\ f_{r3}=1.7\dot{\theta}_{3}+1.24\text{sgn}\Big{(}2\dot{\theta}_{3}\Big{)}\end{array} \right. \tag{35}\]  
\[\delta_{d}(t)=\left\{ \begin{array}{l}\delta_{d1}(t)=7.6\sin\Big{(}\dot{ \theta}_{1}\Big{)}\\ \delta_{d2}(t)=6.6\sin\Big{(}\dot{\theta}_{2}\Big{)}\\ \delta_{d3}(t)=4.23\sin\Big{(}\dot{\theta}_{3}\Big{)}\end{array} \right.