# Robust Multi-Performances Control for Four-Link Manipulator Arm
## Appendix A
\begin{bmatrix}C_{11}&C_{12}&0&0\\ C_{21}&C_{22}&C_{23}&C_{24}\\ C_{31}&C_{32}&C_{33}&C_{34}\\ C_{41}&C_{42}&C_{43}&C_{44}\end{bmatrix}  \tag{131}\]  
where  
\[ \begin{array}{l}C_{11}=\left\{m_{2}r_{2}^{2}\sin(2\theta_{2})-m_{3}l_{2}^{2 }\sin(2\theta_{2})-2m_{3}l_{2}r_{3}\sin(2\theta_{2}+\theta_{3})\right.\\ \left.-m_{3}r_{3}^{2}\sin(2\theta_{2}+2\theta_{3})-m_{4}l_{2}^{2}\sin(2\theta_ {2})-2m_{4}l_{2}l_{3}\sin(2\theta_{2}+\theta_{3})\right.\\ \left.-2m_{4}l_{2}r_{4}\sin(2\theta_{2}+2\theta_{3}+\theta_{4})-m_{4}l_{2}^{2} \sin(2\theta_{2}+2\theta_{3})-2m_{4}l_{3}r_{4}\right.\\ \left.\sin(2\theta_{2}+1\theta_{3}+\theta_{4})-m_{4}r_{4}^{2}\sin(2\theta_{2}+ 2\theta_{3}+2\theta_{4})\right\}\hat{\theta}_{2}\\ \left.+\left\{-2m_{3}l_{2}r_{3}s_{23}c_{2}-2m_{3}r_{3}^{2}s_{23}c_{23}-2m_{4}l _{2}s_{23}c_{2}-2m_{4}l_{2}r_{4}s_{234}c_{2}-2m_{4}l_{3}^{2}s_{23}c_{23}\right. \\ \left.-2m_{4}l_{3}r_{4}s_{234}c_{23}-2m_{4}l_{3}r_{4}s_{23}c_{23}-2m_{4}r_{4}^{2} s_{234}c_{234}\right\}\hat{\theta}_{3}\\ +\left\{-2m_{4}l_{2}r_{4}s_{234}c_{2}-2m_{4}l_{3}r_{4}s_{234}c_{23}-2m_{4}r_{4}^{2 }s_{234}c_{234}\right\}\hat{\theta}_{4}\end{array}  \tag{132}\]  
\[ \begin{array}{l}C_{12}=\left\{m_{4}l_{2}^{2}s_{1}^{2}c_{1}+2m_{4}l_{2}l_{3}s_{ 1}s_{2}s_{23}c_{1}+m_{4}l_{3}^{2}s_{1}^{2}s_{23}^{2}c_{1}\right.\\ \left.+2m_{4}l_{3}r_{4}s_{1}s_{23}s_{234}c_{1}+2m_{4}l_{2}r_{4}s_{1}s_{23}^{2}c _{1}+m_{4}r_{4}^{2}s_{1}s_{234}^{2}c_{1}\right\}\hat{\theta}_{2}\end{array}  \tag{133}\]  
Using (A35) yields the inverse of the matrix \(D\) as follows:  
\[D^{-1}= \begin{bmatrix}I_{11}&0&0&0\\ 0&I_{22}&I_{23}&I_{24}\\ 0&I_{23}&I_{33}&I_{34}\\ 0&I_{24}&I_{34}&I_{44}\end{bmatrix} \] (A63)  
\[I_{11}=\frac{1}{\Delta_{1}}\] (A64)  
\[I_{22}=\frac{D_{33}D_{44}-D_{34}^{2}}{\Delta_{1}}\] (A65)  
\[I_{23}=\frac{-D_{23}D_{44}+D_{24}D_{34}}{\Delta_{1}}\] (A66)  
\[I_{24}=\frac{D_{23}D_{34}-D_{24}D_{33}}{\Delta_{1}}\] (A67)  
\[I_{33}=\frac{D_{22}D_{44}-D_{24}^{2}}{\Delta_{1}}\] (A68)  
\[I_{34}=\frac{-D_{22}D_{34}+D_{23}D_{24}}{\Delta_{1}}\] (A69)  
\[I_{44}=\frac{D_{22}D_{33}-D_{23}^{2}}{\Delta_{1}}\] (A70)  
From (A43) and (A63), we obtain  
\[D^{-1}C= \begin{bmatrix}I_{11}&0&0&0\\ 0&I_{22}&I_{23}&I_{24}\\ 0&I_{23}&I_{33}&I_{34}\\ 0&I_{24}&I_{34}&I_{44}\end{bmatrix}\begin{bmatrix}C_{11}&C_{12}&0&0\\ C_{21}&C_{22}&C_{23}&C_{24}\\ C_{31}&C_{32}&C_{33}&C_{34}\\ C_{41}&C_{42}&C_{43}&C_{44}\end{bmatrix} \equiv \begin{bmatrix}D_{c11}&D_{c12 }&0&0\\ D_{c21}&D_{c22}&D_{c23}&D_{c24}\\ D_{c31}&D_{c32}&D_{c33}&D_{c34}\\ D_{c41}&D_{c42}&D_{c43}&D_{c44}\end{bmatrix} \] (A71)  
where  
\[D_{c11}=I_{11}C_{11}\] (A72)  
\[D_{c21}=I_{22}C_{21}+I_{23}C_{31}+I_{24}C_{41}\] (A73)  
\[D_{c31}=I_{23}C_{21}+I_{33}C_{31}+I_{34}C_{41}\] (A74)  
\[D_{c41}=I_{24}C_{21}+I_{34}C_{31}+I_{44}C_{41}\] (A75)  
\[D_{c12}=I_{11}C_{12}\] (A76)  
Multiplying (A63) with (A19) yields  
\[D^{-1}\overset{\rightharpoonup}{\bar{g}}= \begin{bmatrix}I_{11}&0&0&0\\ 0&I_{22}&I_{23}&I_{24}\\ 0&I_{23}&I_{33}&I_{34}\\ 0&I_{24}&I_{34}&I_{44}\end{bmatrix}\begin{bmatrix}g_{1}\\ g_{2}\\ g_{3}\\ g_{4}\end{bmatrix} = \begin{bmatrix}D_{g1}\\ D_{g2}\\ D_{g3}\\ D_{g4}\end{bmatrix}