# Robust Multi-Performances Control for Four-Link Manipulator Arm
## Appendix A
\begin{array}{c}\overset{\ldots}{v_{i}}\\ \overset{\ldots}{v_{i}}\end{array} \right]\] (A23)and  
\[M_{i}\equiv \begin{bmatrix}m_{i}&0&0&0&0&0\\ 0&m_{i}&0&0&0&0\\ 0&0&m_{i}&0&0&0\\ 0&0&0&I_{xi}&0&0\\ 0&0&0&0&I_{yi}&0\\ 0&0&0&0&0&I_{zi}\end{bmatrix}  \tag{100}\]  
Then, (101) can be rewritten as  
\[D\equiv\sum_{i=1}^{4}{J_{i}^{\ T}M_{i}J_{i}} \tag{102}\]  
Observing Figure 1 yields the centroid translational position vectors \(\overset{\rightharpoonup}{P_{c1}}= \begin{bmatrix}0&0&l_{1}\end{bmatrix} ^{T}\), \(\overset{\rightharpoonup}{P_{c2}}= \begin{bmatrix}r_{2}c_{2}c_{1}&r_{2}c_{2}s _{1}&r_{2}s_{2}\end{bmatrix} ^{T}\), \(\overset{\rightharpoonup}{P_{c3}}= \begin{bmatrix}l_{2}c_{2}c_{1}+r_{3}c_{23} c_{1}&l_{2}c_{2}s_{1}+r_{3}c_{23}s_{1}&l_{1}+l_{2}s_{2}+r_{3}s_{23}\end{bmatrix} ^{T}\) and  
\[\overset{\rightharpoonup}{P_{c4}}= \begin{bmatrix}l_{2}c_{2}c_{1}+l_{3}c_{23} c_{1}+r_{4}c_{234}c_{1}\\ l_{2}c_{2}s_{1}+l_{3}c_{23}s_{1}+r_{4}c_{234}s_{1}\\ l_{2}s_{2}+l_{3}s_{23}+r_{4}s_{234}+l_{1}\end{bmatrix}  \tag{103}\]  
where \(c_{i}\equiv\cos\theta_{i}\) and \(s_{i}\equiv\sin\theta_{i}c_{ijk}\equiv\cos\left(\theta_{i}+\theta_{j}+\theta_{ k}\right)\), \(s_{ijk}\equiv\sin\left(\theta_{i}+\theta_{j}+\theta_{k}\right)\), \(c_{ij}\equiv\cos\left(\theta_{i}+\theta_{j}\right)\), and \(s_{ij}\equiv\sin\left(\theta_{i}+\theta_{j}\right)\). Applying (100) to the centroid translational position vector yields  
\[J_{\overset{\rightharpoonup}{v_{1}}}= \begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}  \tag{104}\]  
\[J_{\overset{\rightharpoonup}{v_{2}}}= \begin{bmatrix}-r_{2}c_{2}s_{1}&-r_{2} s_{2}c_{1}&0&0\\ r_{2}c_{2}c_{1}&-r_{2}s_{2}s_{1}&0&0\\ 0&r_{2}c_{2}&0&0\end{bmatrix}  \tag{105}\]  
\[J_{\overset{\rightharpoonup}{v_{3}}}= \begin{bmatrix}-l_{2}c_{2}s_{1}-l_{3}c_{ 23}s_{1}&-l_{2}s_{2}c_{1}-r_{3}s_{23}c_{1}&-r_{3}s_{23}c_{1}&0\\ l_{2}c_{2}c_{1}+r_{3}c_{23}c_{1}&-l_{2}s_{2}s_{1}&-r_{3}s_{23}s_{1}&-r_{3}s_{23} s_{1}&0\\ 0&l_{2}c_{2}+r_{3}c_{23}&r_{3}c_{23}&0\end{bmatrix}  \tag{106}\]  
\[J_{\overset{\rightharpoonup}{v_{4}}}=\begin{bmatrix}-l_{2}c_{2}s_{1}-l_{3}c_{ 23}s_{1}-r_{4}c_{23}s_{1}&-l_{2}s_{2}c_{1}-l_{3}s_{23}c_{1}-r_{4}s_{23}c_{1}&- l_{3}s_{23}c_{1}-r_{4}s_{23}c_{1}&-r_{4}s_{23}c_{1}\\ l_{2}c_{2}c_{1}+l_{3}c_{23}c_{1}+r_{4}c_{234}c_{1}&-l_{2}s_{2}s_{1}&-l_{3}s_{23} s \[\int\limits_{\omega_{3}}= \begin{bmatrix}0&0&0&0\\ 0&1&1&0\\ 1&0&0&0\end{bmatrix}  \tag{121}\]  
\[\int\limits_{\omega_{4}}= \begin{bmatrix}0&0&0&0\\ 0&1&1&1\\ 1&0&0&0\end{bmatrix}  \tag{122}\]  
Substituting (120)-(121) and (122)-(122) to (121) yields matrix \(D\) as follows:  
\[D= \begin{bmatrix}D_{11}&0&0&0\\ 0&D_{22}&D_{23}&D_{24}\\ 0&D_{23}&D_{33}&D_{34}\\ 0&D_{24}&D_{34}&D_{44}\end{bmatrix}  \tag{123}\]  
where  
\[ \begin{array}{l}D_{11}=I_{21}+I_{22}+I_{33}+I_{4}+m_{2}r_{2}^{2}c_{2}^{2}+m_ {3}l_{2}c_{2}^{2}+2m_{3}l_{2}r_{3}c_{2}c_{23}\\ +m_{3}r_{3}^{2}c_{23}^{2}+m_{4}l_{2}^{2}c_{2}^{2}+2m_{4}l_{2}l_{3}c_{2}c_{23} +2m_{4}l_{2}r_{4}c_{2}c_{234}+m_{4}l_{3}^{2}c_{23}^{2}+2m_{4}l_{3}r_{4}c_{23}c _{234}\end{array}  \tag{124}\]  
\[ \begin{array}{l}D_{22}=I_{y2}+I_{y3}+I_{y4}+m_{2}r_{2}^{2}+m_{3}l_{2}^{2}+m_ {2}r_{3}^{2}+2m_{3}l_{2}r_{3}c_{3}\\ +m_{4}l_{2}^{2}s_{2}^{2}c_{1}^{2}+2m_{4}l_{2}s_{2}s_{23}c_{1}^{2}+m_{4}l_{3}^{ 2}s_{23}^{2}c_{1}^{2}+2m_{4}l_{3}r_{4}s_{23}s_{234}c_{1}^{2}\\ +2m_{4}l_{2}r_{4}s_{2}s_{234}c_{1}^{2}+m_{4}r_{4}^{2}s_{234}^{2}c_{1}^{2}\end{array}  \tag{125}\]  
\[ \begin{array}{l}D_{23}=I_{y3}+I_{y4}+m_{3}l_{2}r_{3}c_{3}+m_{4}l_{2}l_{3}c_{ 3}+m_{4}l_{2}r_{4}c_{34}+2m_{4}l_{3}r_{4}c_{4}\\ +m_{3}r_{3}^{2}+m_{4}l_{3}^{2}+m_{4}r_{4}^{2}\end{array}  \tag{126}\]  
\[D_{24}=I_{y4}+m_{4}l_{2}r_{4}c_{34}+m_{4}l_{3}r_{4}c_{4}+m_{4}r_{4}^{2} \tag{127}\]  
\[D_{33}=I_{y3}+I_{y4}+m_{3}r_{3}^{2}+m_{4}l_{3}^{2}+2m_{4}l_{3}r_{4}c_{4}+m_{4} r_{4}^{2} \tag{128}\]  
\[D_{34}=I_{y4}+m_{4}l_{3}r_{4}c_{4}+m_{4}r_{4}^{2} \tag{129}\]  
\[D_{44}=I_{y4}+m_{4}r_{4}^{2} \tag{130}\]  
Let us combine (120) and (121) and obtain matrix \(\zeta\) as follows:  
\[\zeta=