# Real-Time Tension Distribution Design for Cable-Driven Parallel Robot
## 5 Tracking Control of Cable-Driven Robot Considering Model Uncertainties in Jacobian Matrix
### Fault Detection and Isolation-Based Adaptive Algorithm  
First, as in [19; 20], we can consider the dynamics of the generalized momentum of the auxiliary system in Equation (41), as follows:  
\[p=\dot{q_{e}} \tag{49}\]  
By differentiating Equation (49), we obtain  
\[\begin{split}\dot{p}&=-(K_{d}\dot{q_{e}}+K_{p}q_{e }+K_{i}\!\!\int_{0}^{\mathtt{t}}\!q_{e}\mathtt{dt})+M^{-1}(A^{T}\hat{f}-\tau_{ d})\\ &=\tau_{r}+M^{-1}A^{T}\hat{f}\end{split} \tag{50}\]  
For notational brevity, vector \(\tau_{r}=-(K_{d}q_{e}+K_{p}q_{e}+K_{i}\!\!\int_{0}^{\mathtt{t}}\!q_{e} \mathtt{dt})-M^{-1}\tau_{d}\) is introduced.  
Based on \(\tau_{r}\), we can construct a residual generator as follows:  
\[r(t)=\Gamma[p(t)-\int_{0}^{t}\left(\tau_{r}+r(s)\right)ds] \tag{51}\]  
where \(\Gamma\) denotes a positive scalar and the initial value of \(r\) satisfying \(r(0)=0\). \(r(t)\) is calculable as \(p\) in \(r(t)\), and can be calculated using \(\dot{q}-\dot{q}^{\prime}\), where \(\dot{q}\) can be obtained using Equation (39) and \(\dot{q}^{\prime}\) can be measured by sensors.  
By differentiating Equation (51) with respect to time and considering Equation (50), we can obtain  
\[\dot{r}+\Gamma r=\Gamma M^{-1}A^{\frac{T}{2}}\dot{f}=\Gamma\dot{\gamma}a \tag{52}\]  
From this equation, it can be seen that the state \(r\) of this constructed system is only affected by the term \(\Gamma\dot{\gamma}a\). If we introduce an estimation method using \(\dot{a}\) instead of \(a\), we obtain  
\[\dot{\dot{r}}+\Gamma\dot{r}=\Gamma\dot{\gamma}\dot{a} \tag{53}\]  
where \(\dot{r}\) represents new system states after using \(\dot{a}\). By comparing the error between these two systems (Equations (52) and (53)), we can obtain the following error system:  
\[\dot{\dot{r}}+\Gamma\dot{r}=\Gamma\dot{\gamma}\dot{a} \tag{54}\]  
where \(\dot{\bar{r}}=\dot{r}-r\) and \(\dot{a}=\dot{a}-a\). By appropriately adjusting \(\dot{a}\), \(\ddot{a}\) and \(\bar{r}\) can converge to zero and system Equation (54) becomes stable. The stability of this error system can be analyzed using a Lyapunov theory-based stability analysis, with the Lyapunov candidate selected as follows:  
\[V=\frac{1}{2}\bar{r}^{T}\bar{r}+\frac{1}{2}\dot{a}^{T}\dot{a} \tag{55}\]  
Therefore, the time derivative of \(V\) can be calculated as follows:  
\[\dot{V} =\bar{r}^{T}\dot{r}+\bar{a}^{T}\dot{a} \tag{56}\] \[=-\bar{r}^{T}\Gamma\bar{r}+\bar{r}^{T}\Gamma\dot{\gamma}\dot{a}+ \bar{a}^{T}\dot{a}\]  
Thus, if we designate the time change of \(\dot{a}\) as  
\[\dot{\dot{a}}=-\Gamma Y^{T}\bar{r} \tag{57}\]  
we obtain  
\[\dot{V}=-\bar{r}^{T}\Gamma\bar{r}\leq 0 \tag{58}\]  
which means that the system of Equation (54) is stable. Here, \(\bar{r}\) converges to 0, which denotes that the error term \(Y\dot{a}\) converges to zero. With the dynamics in Equation (48), the tuning law of Equation (42), and the existence of the integral term in the nonlinear compensated PID controller, \(q_{e}\) and \(q_{e}\) converge to zero, and \(q^{\prime}\) finally converges to the desired position \(q_{d}\).  
Here, in Equation (57), \(\bar{r}\) can be calculated using \(r-\hat{r}\), where \(\hat{r}\) can be obtained using Equation (53) and \(r\) can be computed with Equation (51).