## 3 Adaptive Robust Control Design
### Deign of Adaptive Robust Control for Delta Robot
Considering the adaptive law in (25), we have  
\[\dot{V}=(\dot{r}+Ar)^{T}M(\ddot{r}+A\dot{r})+\frac{1}{2}(\dot{r}+Ar)^{T}\dot{M}( \dot{r}+Ar)+r^{T}(B+AD)\dot{r}+(\dot{\rho}-\rho)^{T}(v_{1})^{-1}\dot{\rho}. \tag{32}\]3 \[\dot{V}=(\dot{r}+Ar)^{T}M(\ddot{r}+A\dot{r})+\frac{1}{2}(\dot{r}+Ar)^{T}\dot{M}( \dot{r}+Ar)+r^{T}(B+AD)\dot{r}+(\dot{\rho}-\rho)^{T}(v_{1})^{-1}\dot{\rho}. \tag{32}\]4 \[\dot{V}=(\dot{r}+Ar)^{T}M(\ddot{r}+A\dot{r})+\frac{1}{2}(\dot{r}+Ar)^{T}\dot{M}( \dot{r}+Ar)+r^{T}(B+AD)\dot{r}+(\dot{\rho}-\rho)^{T}(v_{1})^{-1}\dot{\rho}. \tag{32}\]5 \[\dot{V}=(\dot{r}+Ar)^{T}M(\ddot{r}+A\dot{r})+\frac{1}{2}(\dot{r}+Ar)^{T}\dot{M}( \dot{r}+Ar)+r^{T}(B+AD)\dot{r}+(\dot{\rho}-\rho)^{T}(v_{1})^{-1}\dot{\rho}. \tag{32}\]6 \[\dot{V}=(\dot{r}+Ar)^{T}M(\ddot{r}+A\dot{r})+\frac{1}{2}(\dot{r}+Ar)^{T}\dot{M}( \dot{r}+Ar)+r^{T}(B+AD)\dot{r}+(\dot{\rho}-\rho)^{T}(v_{1})^{-1}\dot{\rho}. \tag{32}\]7 \[\dot{V}=(\dot{r}+Ar)^{T}M(\ddot{r}+A\dot{r})+\frac{1}{2}(\dot{r}+Ar)^{T}\dot{M}( \dot{r}+Ar)+r^{T}(B+AD)\dot{r}+(\dot{\rho}-\rho)^{T}(v_{1})^{-1}\dot{\rho}. \tag{32}\]8 \[\dot{V}=(\dot{r}+Ar)^{T}M(\ddot{r}+A\dot{r})+\frac{1}{2}(\dot{r}+Ar)^{T}\dot{M}( \dot{r}+Ar)+r^{T}(B+AD)\dot{r}+(\dot{\rho}-\rho)^{T}(v_{1})^{-1}\dot{\rho}. \tag{32}\]9 \[\left(\dot{r}+Ar\right)^{T}M\left(\ddot{r}+A\dot{r}\right)\]0 \[\left(\dot{r}+Ar\right)^{T}M\left(\ddot{r}+A\dot{r}\right)\]1 \[\left(\dot{r}+Ar\right)^{T}M\left(\ddot{r}+A\dot{r}\right)\]2  
where \(\varpi_{2}=\epsilon\). From the above results in (43) and (44), \(\dot{V}\) can be formulated as  
\[\left(\dot{r}+Ar\right)^{T}M\left(\ddot{r}+A\dot{r}\right)\]3  
Thus, \(\dot{V}\) is negative definite for all \(\|\varrho\|\) such that  
\[\left(\dot{r}+Ar\right)^{T}M\left(\ddot{r}+A\dot{r}\right)\]4  
Upon invoking the standard arguments as in [36], we can conclude the uniform boundedness and the uniform ultimate boundedness with \[d(y) = \begin{cases}\sqrt{\frac{\Psi}{\Psi}}Y,&\text{if}\quad y\leq Y;\\ \sqrt{\frac{\Psi}{\Psi}}y,&\text{if}\quad y>Y,\end{cases}  \tag{47}\] \[Y = \frac{1}{2\sigma_{1}}\left(\varpi_{2}+\sqrt{\varpi_{2}^{2}+4 \varpi_{1}\delta_{3}}\right),\] (48) \[\underline{d} = \sqrt{\frac{\Psi}{\Psi}}Y,\] (49) \[T(\overline{d},y) = \begin{cases}0,&\text{if}\quad y\leq\overline{d}\sqrt{\frac{\Psi }{\Psi}};\\ \frac{\Psi y^{2}-(\Psi^{2}/\Psi)\overline{d}^{2}}{\varpi_{1}\overline{d}^{2} (\Delta/\bar{\Psi})-\varpi_{2}\overline{d}(\Delta/\bar{\Psi})-\varpi_{3}},& \text{otherwise.}\end{cases}  \tag{50}\]  
\(\square\)  
**Remark 7**. : _Each portion of the control scheme (20) possesses different impacts for Delta robot. The first portion \(C_{1}\) is designed for the nominal system. The second portion \(C_{2}\) is proposed in the absence of uncertainty to handle the possible deviation of initial position. This means \(\tau=C_{1}+C_{2}\), \(\dot{V}<0\). The third portion \(C_{3}\), which is constructed as an adaptive parameter-dependent function, is designed to compensate the uncertainty. Then the deterministic performance of Delta robot is guaranteed based on the control scheme \(\tau=C_{1}+C_{2}+C_{3}\), that is, \(\|\varrho\|\leq\overline{d},t\to\infty\)._  
**Remark 8**. : _By (47)-(50), we conclude that the control gain \(\gamma\) is closely connected with the size of ultimate boundedness \(\overline{d}\). The size represents the tendency of decreasing with the increase of \(\gamma\), which leads to the greater control cost. In practice, therefore, the designer needs to make a trade-off between the performance and control cost to find an optimal design for the control gain, which will be shown in the next section._