# Implementation of Fault-Tolerant Control for
## 4 Fault Tolerant Control with Synchronous Sliding Mode Control  
In this section, the fault-tolerant control based on synchronous sliding mode control is proposed. Some definitions are necessary to propose the fault-tolerant control law. Synchronization error [28] is given as  
\[ \begin{array}{l}\varepsilon_{1}=\psi_{1}\varepsilon_{1}-\psi_{2}e_{2}\\ \varepsilon_{2}=\psi_{2}e_{2}-\psi_{3}e_{3}\\ \vdots\\ \varepsilon_{n-1}=\psi_{n-1}e_{n-1}-\psi_{n}e_{n}\\ \varepsilon_{n}=\psi_{n}e_{n}-\psi_{1}e_{1}\end{array}  \tag{18}\]  
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. The synchronization control goal [28] is stated as \(e_{1}=e_{2}=e_{3}=\ldots=e_{n}\) and \(e_{i}\to 0\) at \(t\rightarrow\infty\). In a traditional controller, only the position error converges to zero, but in the synchronization control, the kinematic relationship among the errors as well as the position error converges to zero. For ease of practical implementation, \(\psi_{i}=1\) is chosen. Then, Equation (18) can be written as  
\[ \begin{array}{l}\varepsilon_{1}=e_{1}-e_{2}\\ \varepsilon_{2}=e_{2}-e_{3}\\ \vdots\\ \varepsilon_{n}=\varepsilon_{n}-\varepsilon_{1}\end{array}  \tag{19}\]  
Cross-coupling error [28] is given as  
\[ \begin{array}{l}\xi_{1}=\gamma_{1}\varepsilon_{1}-\gamma_{n}\varepsilon_{n }\\ \xi_{2}=\gamma_{2}\varepsilon_{2}-\gamma_{1}\varepsilon_{1}\\ \vdots\\ \xi_{n}=\gamma_{n}\varepsilon_{n}-\gamma_{n-1}\varepsilon_{n-1}\end{array}  \tag{20}\]  
where \(\gamma_{i}(i=1,2,\ldots,n)\) is the positive gain. In this paper, \(\gamma_{i}=1\) was chosen. Then, Equation (20) can be written as  
\[ \begin{array}{l}\xi_{1}=\varepsilon_{1}-\varepsilon_{n}\\ \xi_{2}=\varepsilon_{2}-\varepsilon_{1}\\ \vdots\\ \xi_{n}=\varepsilon_{n}-\varepsilon_{n-1}\end{array}  \tag{21}\]  
The coupling position error [28], which includes the position and synchronization errors, is defined as  
\[E_{i}=\mu_{i}e_{i}+\eta_{i}\int\xi_{i}dt \tag{22}\]  
where \(\eta_{i}\) and \(\mu_{i}\) are positive gains. The synchronous sliding surface [26] is defined as  
\[S=\dot{E}+cE \tag{23}\]  
where \(S=\left[S_{1},S_{2},\ldots,S_{n}\right]^{T}\in\Re^{n};c\in\Re^{n\times n}\) is the diagonal positive matrix; and \(E=\left[E_{1},E_{2},\ldots,E_{n}\right]^{T}\in\Re^{n}\) with \(E_{i}\) defined in Equation (22). For ease of implementation, the synchronous sliding surface is rewritten as  
\[s=\dot{e}+c\varepsilon+\alpha\xi+\kappa\int\xi dt \tag{24}\]  
where \(s=\left[S_{1}/\mu_{1},S_{2}/\mu_{2},\ldots,S_{n}/\mu_{n}\right]^{T}\in\Re^{n}\), \(\alpha=diag(\eta_{i}/\mu_{i})\in\Re^{n\times n}\) and \(\kappa=diag(c_{i}\eta_{i}/\mu_{i})\in\Re^{n\times n}\)\((i=1,2,\ldots,n)\) are positive gain matrices. Here, the fault-tolerant control law is proposed as  
\[\tau=\tau_{0}+\tau_{SSMC}+\tau_{ob} \tag{25}\]  
where  
\[\tau_{0}=\mathbf{M}(q)\ddot{q}_{d}+\mathbf{H}(q,\dot{q}),\,\tau_{ssmc}=\mathbf{ M}(q)(\dot{c}\dot{e}+\alpha\dot{\xi})+k_{1}sgn(s)+k_{2}s+k_{3}\xi,\,\text{and}\,\tau_{ob}=- \mathbf{M}(q)\widehat{\phi}.\]  
**Theorem 2**. : _The system described in (7), using controller specified in (25) guarantees that \(e\to 0\) and \(\varepsilon\to 0\) as time \(t\to\infty\) under the condition \(k_{3}>\kappa\)._  
**Proof. ** Following [22], the Lyapunov function can be selected as  
\[V=\frac{1}{2}s^{T}s+\frac{1}{2}{\sum\limits_{i=1}^{n}\Lambda_{ \varepsilon}\varepsilon_{i}^{2}}+\frac{1}{2}{\sum\limits_{i=1}^{n}\kappa_{ \varepsilon}\Lambda_{\varepsilon}\left(\int\varepsilon_{i}-\varepsilon_{i-1} \right)^{2}}dt\geq 0 \tag{26}\]  
The time derivative of V is therefore  
\[