## 4 Finite-Time Fault-Tolerant Control Using Synchronous Terminal Sliding Mode Control  
In this section, two finite-time fault-tolerant controls based on a synchronous terminal sliding mode control are proposed. Some definition will be useful in the rest of the paper. Definition 1: We define \(\left[x\right]^{\Lambda}=\left[\left|x_{1}\right|\right|_{\lambda_{1}}sign(x _{1}),\left|x_{2}\right|_{\lambda_{2}}sign(x_{2}),\ldots\left|x_{n}\right| _{\lambda_{n}}sign(x_{n})\right]^{T}\in\Re^{n}\), where \(\lambda_{i}(i=1,2,\ldots,n)>0\) and \(\Lambda=diag(\lambda_{i})\). \(x=[x_{1},x_{2},\ldots,x_{n}]^{T}\in\Re^{n}\) and \(y=[y_{1},y_{2},\ldots,y_{n}]^{T}\in\Re^{n}\). Definition 2: We define \(x\cdot y=[x_{1}y_{1},x_{2}y_{2},\ldots,x_  
**Definition 3**. : _The time derivative of \(\lceil x\rceil^{\Lambda}\) is \(\frac{d}{dt}\lceil x\rceil^{\Lambda}=\Lambda|x|^{\Lambda-1}\cdot\dot{x}=\left[ \lambda\big{|}x_{1}\big{|}_{\lambda-1}\dot{x}_{1},\lambda\big{|}x_{2}\big{|}_{ \lambda-1}\dot{x}_{2},\ldots,\lambda\big{|}x_{n}\big{|}_{\lambda-1}\dot{x}_{n} \right]^{T}\), and \(x^{\Lambda-I}=\left[x_{1}^{\lambda_{1}-1},x_{2}^{\lambda_{2}-1},\ldots,x_{n}^{ \lambda_{n}-1}\right]\in\Re^{n}\), where \(I=diag(1)\in\Re^{n\times n}\)._  
The novel synchronization error is defined as:  
\[ \begin{array}{c}\varepsilon_{1}=(1+\psi_{1}\psi_{n})e_{1}-\psi_{1}e_{2}-\psi _{1}e_{n}\\ \varepsilon_{2}=(1+\psi_{2}\psi_{1})e_{2}-\psi_{2}e_{3}-\psi_{2}e_{1}\\ \vdots\\ \varepsilon_{n}=(1+\psi_{n}\psi_{n-1})e_{n}-\psi_{n}e_{2}-\psi_{n}e_{1}\end{array}  \tag{11}\]  
where \(e_{i}(i=1,2,\ldots,n)\) is the error at each joint, and \(\psi_{i}(i=1,2,\ldots,n)\) is the corresponding positive gain. In matrix form,  
\[\varepsilon=Te \tag{12}\]  
where \(\varepsilon=[\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{n}]^{T}\in \Re^{n}\), \(e=[e_{1},e_{2},\ldots,e_{n}]^{T}\in\Re^{n}\), \(T\in\Re^{n\times n}\), and  
\[T=\left[ \begin{array}{cccc}(1+\psi_{1}\psi_{n})&-\psi_{1}&0&\cdots&-\psi_{1} \\ -\psi_{2}&(1+\psi_{2}\psi_{1})&-\psi_{2}&\cdots&0\\ 0&-\psi_{3}&(1+\psi_{3}\psi_{2})&\cdots&0\\ \vdots\\ -\psi_{n}&-\psi_{n}&0&\cdots&(1+\psi_{n}\psi_{n-1})\end{array} \right] \tag{13}\]  
The Proposed Active Fault-Tolerant Control with Synchronous Terminal Sliding Mode Control 1 (AFTC S-TSMC1)  
The novel finite-time coupling position error is defined as:  
\[E=\alpha e+\beta\int\lceil\varepsilon\rfloor^{\Lambda}dt \tag{14}\]  
where \(E=[E_{1},E_{2},\ldots,E_{n}]^{T}\in\Re^{n}\), \(\alpha=diag(\alpha_{i})\in\Re^{n\times n}\), and \(\beta=diag(\beta_{i})\in\Re^{n\times n}\) are coupling parameters and positive \(\Lambda=diag(\lambda_{i})\,0<\lambda_{i}<1\). The synchronous terminal sliding surface is defined as:  
\[S=\dot{E}+\Gamma\lceil E\rfloor^{\Lambda} \tag{15}\]  
where \(S=[S_{1},S_{2},\ldots,S_{n}]^{T}\in\Re^{n}\), \(\dot{E}=\left[\dot{E}_{1},\dot{E}_{2},\ldots,\dot{E}_{n}\right]^{T}\in\Re^{n}\), \(\Gamma=diag(\gamma_{i})\in\Re^{n\times n}\), and \(\gamma_{i}>0\). The proposed finite-time active fault-tolerant control is given as:  
\[\tau=\tau_{\alpha q}+\tau_{0}+\tau_{ob} \tag{16}\]  
where \(\tau_{\alpha q}=\mathbf{M}(q)\Big{(}\ddot{q}_{d}+\alpha^{-1}\beta\Lambda| \varepsilon|^{\Lambda-I}\cdot\dot{\varepsilon}+\alpha^{-1}\Gamma\Lambda||^{ \Lambda-I}\cdot\Big{)}+\mathbf{H}(q,\dot{q})\), \(\tau_{0}=\mathbf{M}(q)K_{1}sign(S)\), \(\tau_{\alpha b}=-\mathbf{M}(q)\dot{\phi}\), where \(K_{1}=diag(k_{1i})\in\Re^{n\times n}\). **Theorem 1**. : _The system described in Equation (5), using the controller specified in Equation (16) guarantees that \(e\to 0\) as finite-time._  
**Proof of Theorem 1. ** The Lyapunov function can be selected as:  
\[V=\frac{1}{2}S^{T}S \tag{17}\]The time derivative of V in Equation (17) is:  
\[