## 2 Issue Description  
Consider the dynamic equation of robot manipulators without the loss of generality:  
\[M(\theta)\ddot{\theta}+Q\Big{(}\theta,\dot{\theta}\Big{)}\dot{ \theta}+G(\theta)+f_{\!f}\Big{(}\dot{\theta}\Big{)}+\delta_{d}(t)=\tau \tag{1}\]  
where \(\theta=\left[\begin{array}{cccc}\theta_{1},&\ldots,&\theta_{n}\end{array} \right]^{T}\in R^{n\times 1}\), \(\dot{\theta}=\left[\begin{array}{cccc}\dot{\theta}_{1},&\ldots,&\dot{\theta} _{n}\end{array}\right]^{T}\in R^{n\times 1}\), and \(\ddot{\theta}=\left[\begin{array}{cccc}\ddot{\theta}_{1},&\ldots,&\ddot{ \theta}_{n}\end{array}\right]^{T}\in R^{n\times 1}\) declare the position angle vector, the velocity angle vector, and the acceleration angle vector, respectively. \(M(\theta)=\dot{M}(\theta)+\Delta M(\theta)\in R^{n\times n}\) declares the real inertia matrix, \(Q\Big{(}\theta,\dot{\theta}\Big{)}=\dot{Q}\Big{(}\theta,\dot{\theta}\Big{)}+ \Delta Q\Big{(}\theta,\dot{\theta}\Big{)}\in R^{n\times n}\) declares the real Coriolis and centrifugal force matrix, and \(G(\theta)=\hat{G}(\theta)+\Delta G(\theta)\in R^{n\times n}\) represents the real gravitational matrix. \(\hat{M}(\theta)\in R^{n\times n}\) declares the estimated inertia matrix, \(\hat{Q}\Big{(}\theta,\dot{\theta}\Big{)}\in R^{n\times n}\) represents the estimated Coriolis and centrifugal force matrix, and \(\hat{G}(\theta)\in R^{n\times n}\) declares the estimated gravitational matrix. \(f_{\!f}\Big{(}\dot{\theta}\Big{)}\in R^{n\times 1}\) and \(\delta_{d}(t)\in R^{n\times 1}\) declare the friction vector and disturbance vector, respectively. \(\Delta M(\theta)\in R^{n\times n}\) and \(\Delta Q\big{(}\theta,\dot{\theta}\big{)}\in R^{n\times n}\) are the errors of the real dynamic model.  
Consequently, the real dynamic equation of robot manipulators is achieved as:  
\[\hat{M}(\theta)\ddot{\theta}+\hat{Q}\big{(}\theta,\dot{\theta}\big{)}\dot{ \theta}+\hat{G}(\theta)+\Delta U=\tau \tag{2}\]  
The lumped uncertain component \(\Delta u\) in Equation (2) is given as:  
\[\Delta u=\Delta M(\theta)\ddot{\theta}+\Delta Q\Big{(}\theta,\dot{\theta} \big{)}\dot{\theta}+\Delta G(\theta)+f_{r}+\delta_{d}(t) \tag{3}\]  
Accordingly, the robotic dynamic in Equation (1) is reorganized as:  
\[\ddot{\theta}=\hat{M}^{-1}(\theta)\tau-\hat{M}^{-1}(\theta)\big{[}\hat{Q} \big{(}\theta,\dot{\theta}\big{)}\dot{\theta}+\hat{G}(\theta)\big{]}-\hat{M}^ {-1}(\theta)\Delta u \tag{4}\]  
Then, the dynamic Equation (4) can be transferred into the following state-space form as:  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\Pi(x,t)+\Phi(x,t)u+D(\theta,\Delta u)\end{array}\right. \tag{5}\]  
where, we set \(u=\tau\) as the control input and \(x=\left[x_{1},x_{2}\right]^{T}\) as the state vector in which \(x_{1},\ x_{2}\) correspond to \(\theta,\ \dot{\theta}\in R^{n\times 1}\). \(\Pi(x,t)=-\hat{M}^{-1}(\theta)\big{[}\hat{Q}\big{(}\theta,\dot{\theta}\big{)} \dot{\theta}+\hat{G}(\theta)\big{]}\), \(\Phi(x,t)=\hat{M}^{-1}(\theta)\), and \(D(\theta,\Delta U)=-\hat{M}^{-1}(\theta)\Delta u\).  
The control target of the system is to further increase the response speed and accuracy of the trajectory tracking control for robot manipulators, even if the effects of uncertain dynamics and external perturbations are valid. First, to enhance the response, fast convergence time against uncertainties, and accuracy of the tracking position, the novel FTSMM is developed. Then, STCL is applied to combat the unknown nonlinear functions in the control system. By using this combined technique, the exterior disturbances and uncertain dynamics will be compensated more rapidly and more correctly with the smooth control torque. Finally, the designed controller is launched from the proposed sliding mode manifold and the STCL to obtain the control efficiency.