## 3 Design Procedure for a Control Strategy
### Design an Adaptive Neural Non-Singular Fast-Terminal Sliding-Mode Control for Robotic Manipulators  
To achieve the desired control performance for the system in Equation (3), the control method is performed as follow:  
Substituting Equation (8) into Equation (7) provides  
\[s=\dot{e}+\Gamma_{1}e^{[2I_{n}-\theta]}+\Gamma_{2}e+\Gamma_{3}e^{[\theta]}+h_{1 }sign[\varsigma]+h_{2}\varsigma^{[\alpha]}, \tag{24}\]  
where \(s=\left[ \begin{array}{cccc}s_{1},&\ldots,&s_{n}\end{array} \right]^{T}\), \(I_{n}\) is the unit matrix, \(\vartheta=diag(\phi_{1},\ \cdots,\ \phi_{n})\), \(\alpha=diag(\alpha_{1},\cdots,\alpha_{n})\), \(\Gamma_{1}=diag(\Gamma_{11},\cdots,\Gamma_{1n})\), \(\Gamma_{2}=diag(\Gamma_{21},\cdots,\Gamma_{2n})\), \(\Gamma_{3}=diag(\Gamma_{31}\), \(\Gamma_{32}\), \(\cdots,\ \Gamma_{3n})\), \(h_{1}=diag(h_{11},\cdots,h_{3n})\), \(h_{2}=diag(h_{21},\cdots,h_{2n})\), \(sign[\varsigma]=\left[ \begin{array}{cccc}sign[\varsigma_{1}],&\ldots,&sign[ \varsigma_{n}]\end{array} \right]^{T}\), \(e=[\varsigma_{1},\ldots,\varsigma_{n}]^{T}\). \(e^{[2I_{n}-\theta]}\), \(e^{[\theta]}\), and \(\varsigma^{[\alpha]}\) are vectors defined as  
\[e^{[\theta]}=diag(sign[e])\cdot|e|^{\vartheta}=\left[ \begin{array}{cccc}e_{1 }^{[\theta_{1}]},&e_{2}^{[\theta_{2}]},&\cdots,&e_{n}^{[\theta_{n}]}\end{array}  \right]^{T}. \tag{25}\]  
To simplify the analysis, the following notion is applied  
\[\frac{de^{[\theta]}}{dt}=\vartheta diag\Big{(}|e|^{\vartheta-I_{n}}\Big{)} \cdot\dot{e}. \tag{26}\]  
Using Equation (26), the time derivative of Equation (24) is derived as  
\[\dot{s}=\ddot{e}+\Gamma_{1}(2I_{n}-\vartheta)diag\Big{(}|e|^{I_{n}-\vartheta} \Big{)}\dot{e}+\Gamma_{2}\dot{e}+\Gamma_{3}\vartheta diag\Big{(}|e|^{\vartheta -I_{n}}\Big{)}\dot{e}+h_{2}adiag\Big{(}|\varsigma|^{\alpha-I_{n}}\Big{)}\dot{ \varsigma}. \tag{27}\]  
From Equation (4), \(\ddot{e}\) is presented as  
\[ \begin{array}{ll}\ddot{e}&=\ddot{q}-\ddot{q}_{d}\\ &=\Xi\big{(}q,\dot{q}\big{)}+B(q)\tau(t)+\Delta_{u}\big{(}q,\dot{q},t\big{)}- \ddot{q}_{d}\end{array} . \tag{28}\]  
Substituting Equation (28) into Equation (27) gives  
\[\dot{s}=\Xi\big{(}q,\dot{q}\big{)}+B(q)\tau(t)+\Delta_{u}\big{(}q,\dot{q},t \big{)}-\ddot{q}_{d}+\Pi(e,\varsigma), \tag{29}\]  
where \(\Pi(e,\varsigma)=\Gamma_{1}(2I_{n}-\vartheta)diag\Big{(}|e|^{I_{n}-\vartheta} \Big{)}\dot{e}+\Gamma_{2}\dot{e}+\Gamma_{3}\vartheta diag\Big{(}|e|^{\vartheta -I_{n}}\Big{)}\dot{e}+h_{2}adiag\Big{(}|\varsigma|^{\alpha-I_{n}}\Big{)}\dot{\varsigma}\). To obtain the desired performance, the proposed control algorithm is designed for system (3) as  
\[\tau(t)=B^{+}(q)\big{(}\tau_{eq}(t)+\tau_{\varsigma}(t)\big{)}, \tag{30}\]  
where \(B^{+}(q)=B^{T}(q)\big{[}B(q)B^{T}(q)\big{]}^{-1}\), the equivalent control law is constructed as  
\[\tau_{eq}(t)=-\big{(}\Xi\big{(}q,\dot{q}\big{)}+\Pi(e,\varsigma)-\ddot{q}_{d} \big{)}, \tag{31}\]  
and the switching control term is designed as  
\[\tau_{s}=-(\Omega+\rho_{1})sign(s) \tag{32}\]  
in which \(\Omega\) and \(\rho_{1}\) are positive constants. Substituting control laws (30)-(32) into Equation (29) provides  
\[\dot{s}=-(\Omega+\rho_{1})sign(s)+\Delta_{u}\big{(}q,\dot{q},t\big{)}. \tag{33}\]  
The positive-definite Lyapunov functional is selected as\[V_{3}=\frac{1}{2}s^{T}s. \tag{34}\]  
With Equation (33), the time derivative of Equation (34) is derived as  
\[ \begin{array}{ll}\dot{V}_{3}&=s^{T}\dot{s}\\ &=s^{T}\big{(}-(\Omega+\rho_{1})sign(s)+\Delta_{u}\big{(}q,\dot{q},t\big{)} \big{)}\\ &=-\Omega|s|-\rho_{1}|s|+\Delta_{u}\big{(}q,\dot{q},t\big{)}s\leq-\rho_{1}|s| \end{array} . \tag{35}\]  
Accordingly, based on the Lyapunov criterion [47], it can be verified that the stability of the tracking error is secured under control laws (30)-(32) despite the presence of external disturbances and system uncertainties. Unfortunately, robot manipulators have complicated dynamic models with many parametric uncertainties (e.g., friction, sensor noise, payload, perturbations).