# Robust Multi-Performances Control for Four-Link Manipulator Arm
## 2 Complete Mathematical Model for Four-Link Manipulator Arm
\begin{array}{l}\dot{\vec{\eta}}_{non\_k}=\stackrel{{\rightarrow }}{{\rho}}_{non\_k}\\ +\sum\limits_{j=1}^{P}\left(\frac{\partial}{\partial X}\vec{\sigma}_{non\_k}( \stackrel{{\rightarrow}}{{X}})\right)\stackrel{{ *}}{{q}}_{j}^{*}\left(\theta_{un\_j}+\theta_{noise\_j}\right),k=d_{o}+1, \ldots,n\end{array}  \tag{56}\]  
\[y_{oi}=\xi_{lin\_1}^{i},1\leq i\leq 4 \tag{57}\]  
Hence,  
\[\begin{array}{l}\left[ \begin{array}{l}\dot{\xi}_{l\_d_{ci}}^{1}\\ \dot{\vec{\xi}}_{l\_d_{ci}}^{2}\\ \dot{\vec{\xi}}_{l\_d_{ci}}^{3}\\ \dot{\vec{\xi}}_{l\_d_{ci}}^{4}\\ \dot{\vec{\xi}}_{l\_d_{ci}}^{4}\\ \dot{\vec{\xi}}_{l\_1}^{4}\end{array} \right]=\left[ \begin{array}{l}\dot{\xi }_{l\_1}^{1}\\ \dot{\vec{\xi}}_{l\_1}^{1}\\ \dot{\vec{\xi}}_{l\_1}^{1}\\ \dot{\vec{\xi}}_{l\_1}^{2}\end{array} \right]=\left[ \begin{array}{l}u_{c1} \\ u_{c2}\\ u_{c3}\\ u_{c4}\end{array} \right]+\left[ \begin{array}{l}u_{d11}\\ u_{d21}\\ u_{d31}\\ u_{d41}\\ u_{d42}\\ \end{array} \right]+\left[ \begin{array}{l}u_{d11}\\ u_{d22}\\ u_{d23}\\ u_{d33}\\ u_{d44}\end{array} \right]+\left[ \begin{array}{l}u_{d11}\\ u_{d22}\\ u_{d32}\\ u_{d5}\end{array} \right]\\ +\left[\begin{array}{l}\sum\limits_{j=1}^{P}\left(\frac{\partial}{\partial X }L_{\frac{d_{ci}-1}{f}}^{d_{ci}-1}\sigma_{01}\right)\stackrel{{ *}}{{q}}_{j}^{*}\left(\theta_{un\_j}+\theta_{noise\_j}\right)\\ \sum\limits_{j=1}^{P}\left(\frac{\partial}{\partial X}L_{\frac{d_{ci}-1}{f}}^ {d_{ci}-1}\sigma_{02}\right)\stackrel{{*}}{{q}}_{j}^{*}\left( \theta_{un\_j}+\theta_{noise\_j}\right)\\ \sum\limits_{j=1}^{P}\left(\frac{\partial}{\partial X}L_{\frac{d_{ci}-1}{f}}^ {d_{ci}-1}\sigma_{03}\right)\stackrel{{we use the vector  
\[ \begin{split}\overset{\rightharpoonup}{u}_{b}\equiv\left[\begin{array}{ ccc}u_{b1}&u_{b2}&u_{b4}&u_{b4}\end{array}\right]^{T}\equiv\left[\begin{array}{ ccc}L_{f}^{d_{\text{oi}}}\sigma_{\text{o1}}&L_{f}^{d_{\text{oi}}}\sigma_{ \text{o2}}&L_{f}^{d_{\text{oi}}}\sigma_{\text{o3}}&L_{f}^{d_{\text{oi}}}\sigma _{\text{o4}}\end{array}\right]^{T}\\ =\left[\begin{array}{cccc}u_{c1}&u_{c2}&u_{c3}&u_{c4}\end{array}\right]^{T} \equiv\overset{\rightharpoonup}{u}_{c}\end{split}  \tag{62}\]  
and the virtual input [37]  
\[\overset{\rightharpoonup}{u}_{v}\equiv\left[u_{v1}&u_{v2}&u_{v3}&u_{v4} \right]^{T} \tag{63}\]  
Then, we can transform (58) into the following model:  
\[\begin{split}\left[ \begin{array}{c}\overset{\rightharpoonup}{ \xi}_{l,d_{\text{oi}}}^{1}\\ \overset{\rightharpoonup}{\xi}_{l,d_{\text{oi}}}^{1}\\ \overset{\rightharpoonup}{\xi}_{l,d_{\text{oi}}}^{1}\\ \overset{\rightharpoonup}{\xi}_{l,d_{\text{oi}}}^{1}\\ \overset{\rightharpoonup}{\xi}_{l,d_{\text{oi}}}^{1}\\ \overset{\rightharpoonup}{\xi}_{l,d_{\text{oi}}}^{1}\end{array} \right]= \left[ \begin{array}{c}u_{v1}\\ u_{v2}\\ u_{v3}\\ u_{v4}\end{array} \right]+\left[\begin{array}{c}\overset{\rightharpoonup}{ \lambda}\\ \overset{\rightharpoonup}{\lambda}\\ \overset{\rightharpoonup}  
\[A_{c}^{i}\equiv \begin{bmatrix}0&1&0&\dots&0\\ 0&0&1&\dots&0\\ &\vdots&&&\vdots\\ 0&0&0&\dots&1\\ -\alpha_{1}^{i}&-\alpha_{2}^{i}&-\alpha_{3}^{i}&\dots&-\alpha_{d_{di}}^{i}\end{bmatrix}  _{d_{d_{di}}\times d_{di}}=\left[-50\right]_{1\times 1},i=1,\dots,4 \tag{68}\]  
Based on the feedback linearization approach, we then propose that the robust controller possesses the pre-specified tracking signals \(y_{track}^{1}=y_{track}^{2}=y_{track}^{3}=y_{track}^{4}=\pi/6\) as follows:  
\[\overset{\rightarrow}{u}_{fb}=A^{-1}\Big{(}-\overset{\rightarrow}{u}_{b} +\overset{\rightarrow}{u}_{v}\Big{)}=A^{-1}\Big{(}-\Big{[}u_{b1}\quad\dots \quad u_{b4}\Big{]}^{T}+\Big{[}u_{v1}\quad\dots\quad u_{b4}\Big{]}^{T}\Big{)} \tag{69}\]  
\[A^{-1}= \begin{bmatrix}D_{11}&0&0&0\\ 0&D_{22}&D_{23}&D_{24}\\ 0&D_{23}&D_{33}&D_{34}\\ 0&D_{24}&D_{34}&D_{44}\end{bmatrix}  \tag{70}\]  
\[\overset{\rightarrow}{u}_{v}= \begin{bmatrix}-\epsilon^{-1}(50)(x_{1}+x_{2}- \pi/6)\\ -\epsilon^{-1}(50)(x_{3}+x_{4}-\pi/6)\\ -\epsilon^{-1}(50)(x_{5}+x_{6}-\pi/6)\\ -\epsilon^{-1}(50)(x_{7}+x_{8}-\pi/6)\end{bmatrix}  \tag{71}\]  
\[\overset{\rightarrow}{u}_{b}=