# Robust Multi-Performances Control for Four-Link Manipulator Arm
## 2 Complete Mathematical Model for Four-Link Manipulator Arm  
The FLMA is a great platform for industrial mechanics as it is a highly nonlinear control system with disturbances. In this section, we apply Euler-Lagrange equation to derive the dynamic equations of the FLMA, as shown in Figure 1.  
The four-link manipulator arm is made up of aluminium. The dynamic model parameters are selected as follows: the length of the link \(l_{1}=l_{2}=l_{3}=l_{4}=1\) m, the distance of the centre of mass \(r_{1}=r_{2}=r_{3}=r_{4}=0.25\) m, the mass of the link \(m_{1}=m_{2}=m_{3}=m_{4}=1\) kg and the inertia moment of the link \(I_{1}=I_{2}=I_{3}=I_{4}=1\) kg \(\cdot\) m\({}^{2}\). In this section, we give complete derivations of the dynamic mathematical equation for the nonlinear FLMA system. We define the state, input and noise variables of the FLMA as the following physical quantities: \(\overset{\rightharpoonup}{\widetilde{x}}\equiv \begin{bmatrix}x_{1}&x_{2}&x_{3 }&x_{4}&x_{5}&x_{6}&x_{7}&x_{8}\end{bmatrix} ^{T}\), \(x_{1}=\theta_{1}\), \(x_{2}=\dot{\theta}_{1}\), \(x_{3}=\theta_{2}\), \(x_{4}=\dot{\theta}_{2}\), \(x_{5}=\theta_{3}\), \(x_{6}=\dot{\theta}_{3}\), \(x_{7}=\theta_{4}\), \(x_{8}=\dot{\theta}_{4}\), \(u_{f\_{b\_1}}=\tau_{1}\), \(u_{f\_{b\_2}}=\tau_{2}\), \(u_{f\_{b\_3}}=\tau_{3}\), \(u_{f\_{b\_4}}=\tau_{4}\) and \(\overset{\rightharpoonup}{\underset{j=1}{\overset{\rightharpoonup}{\sim}}} \overset{\rightharpoonup}{\underset{j}{\sim}}\theta_{noise\_j}\). From Appendix A, the state-space dynamic model of the FLMA with real physical values can be represented as  
\[ \begin{bmatrix}\dot{x}_{1}&\ldots&\dot{x}_{8}\end{bmatrix} ^{T}= \begin{bmatrix} f_{1}(\overset{\rightharpoonup}{X})&\ldots&f_{8}(\overset{\rightharpoonup}{X})\end{bmatrix} ^{T}\] \[+ \begin{bmatrix}\overset{\rightharpoonup}{\underset{g\_{b\_1}} {\rightarrow}}(\overset{\rightharpoonup}{X})&\ldots&\overset{\rightharpoonup}{ \underset{g\_{b\_4}}{\rightarrow}}(\overset{\rightharpoonup}{X})\end{bmatrix} \begin{bmatrix}u_{f\_{b\_1}}(\overset{\rightharpoonup}{X})&\ldots&u_{f\_{b \_4}}(\overset{\rightharpoonup}{X})\end{bmatrix} ^{T}+\overset{\rightharpoonup }{\underset{j=1}{\overset{\rightharpoonup}{\sim}}}\overset{\rightharpoonup}{ \underset{j}{\sim}}\overset{\rightharpoonup}{\underset{j}{\sim}}\theta_{noise\_j} \tag{3}\]  
\[ \begin{bmatrix}y_{\sigma 1}(\overset{\rightharpoonup}{X})&\ldots&y_{\sigma 4}(\overset{\rightharpoonup}{X})\end{bmatrix} ^{T}= \begin{bmatrix}\sigma_{ \sigma 1}(\overset{\rightharpoonup}{X})&\ldots&\sigma_{\sigma 4}(\overset{\rightharpoonup}{X})\end{bmatrix} ^{T} \tag{4}\]  
\[f_{1}\equiv x_{2} \tag{5}\]  
\[f_{2}\equiv-D_{c11}x_{2}-D_{c12}x_{4}-D_{g1} \tag{6}\]  
\[f_{3}\equiv x_{4} \tag{7}\]  
\[f_{4}\equiv-D_{c21}x_{2}-D_{c22}x_{4}-D_{c23}x_{6}-D_{c24}x_{8}-D_{g2} \tag{8}\]  
\[f_{5}\equiv x_{6} \tag{9}\]\[\stackrel{{\rightharpoonup}}{{\delta}}_{\text{{\tiny fb}\_3}} = \begin{bmatrix}0&0&0&I_{23}&0&I_{33}&0&I_{34}\end{bmatrix} ^{T} \tag{15}\] \[\stackrel{{\rightharpoonup}}{{\delta}}_{\text{{\tiny fb }\_4}} = \begin{bmatrix}0&0&0&I_{24}&0&I_{34}&0&I_{44}\end{bmatrix} ^{T}\] (16) \[\stackrel{{\rightharpoonup}}{{q}}_{\text{{\tiny fb }\_1}}^{*}\theta_{\text{{\tiny noise }\_1}} = \begin{bmatrix}\sin t&0&0&0&0&0&0\end{bmatrix} ^{T}\] (17) \[\stackrel{{\rightharpoonup}}{{q}}_{\text{{\tiny 2}}}^{*}\theta_{\text{{\tiny noise }\_2}} = \begin{bmatrix}0&0&\sin t&0&0&0&0&0\end{bmatrix}