## 2 Materials and Methods
### Kinematic Relations
Projecting Equations (1)-(3) onto the \(x,y\) axes, we obtain six scalar equations with eight unknowns \(\varphi_{i}\), \(i=2,3,4,5,6,7\) for a given \(\varphi_{1}\):  
(a) for a four-link (0,1,2,3)  
\[\left\{\overset{l_{1}cos(\varphi_{1})+\overset{l}{l}_{2}^{\prime}cos(\varphi _{2}^{\prime})=-b+l_{3}cos(\varphi_{3})}{l_{1}sin(\varphi_{1})+\overset{l}{l}_{ 2}sin(\varphi_{2}^{\prime})=l_{3}sin(\varphi_{3})}\right., \tag{4}\]  
(b) for the group (0,2,3,4,5,7)  
\[\left\{\overset{-b+l_{3}cos(\varphi_{3})-\overset{l}{l}_{2}cos(\varphi_{2}^ {\prime}+\overset{\rightarrow}{\alpha}_{2})+\overset{l}{l}_{4}cos(\varphi_{ 4})=-(a+b)+\overset{l}{l}_{7}cos(-\varphi_{7}^{\prime}+\overset{\rightarrow} {\alpha}_{7})+\overset{l}{l}_{5}cos(\varphi_{5})}{l_{3}sin(\varphi_{3})-\overset{ l}{l}_{2}sin(\varphi_{2}^{\prime}+\overset{\rightarrow}{\alpha}_{2})+\overset{l}{l}_{4}sin( \varphi_{4})=\overset{l}{l}_{7}sin(-\varphi_{7}^{\prime}+\overset{\rightarrow} {\alpha}_{7})+\overset{l}{l}_{5}sin(\varphi_{5})}\right. \tag{5}\]  
(c) for the group (4,5,6,7)  
\[\left\{\overset{l_{7}cos(-\varphi_{7}^{\prime}+\overset{\rightarrow}{ \alpha}_{7})+\overset{l}{l}_{5}cos(\varphi_{5})+\overset{l}{l}_{4}cos(- \varphi_{4}^{\prime}+\overset{\rightarrow}{\alpha}_{4})=\overset{l}{l}_{7} cos(\varphi_{7}^{\prime})+\overset{l}{l}_{6}cos(\varphi_{6})}{l_{7}sin(- \varphi_{7}^{\prime}+\overset{\rightarrow}{\alpha}_{7})+\overset{l}{l}_{5} sin(\varphi_{5})+\overset{l}{l}_{4}sin(-\varphi_{4}^{\prime}+\overset{\rightarrow} {\alpha}_{4})=\overset{l}{l}_{7}sin(\varphi_{7}^{\prime})+\overset{\rightarrow} {l}_{6}sin(\varphi_{6})}\right., \tag{6}\]  
To find the angular velocities \(\omega_{2}\)_and_\(\omega_{3}\) of links 2 and 3, we differentiate the system of Equation (4). \[\left\{\overset{\omega_{1}l_{1}sin(\varphi_{1})+\omega_{2}l_{2}^{\prime}sin( \varphi_{2}^{\prime})=\omega_{3}l_{3}sin(\varphi_{3})}{\omega_{1}l_{1}cos( \varphi_{1})+\omega_{2}l_{2}^{\prime}cos(\varphi_{2}^{\prime})=\omega_{3}l_{3} cos(\varphi_{3})}\right., \tag{7}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \(DC\)=\(l_{1}\) & \(BC\)=\(l_{2}\) & \(BP\)=\(l_{2}^{\prime}\) & \(AB\)=\(l_{3}\) & \(PD_{1}\)=\(l_{4}\) & \(PA_{1}\)=\(l_{4}^{\prime}\) \\ \hline
1178.8 mm & 358.49 mm & 1532.72 mm & 704.628 mm & 1500.0 mm & 648.3 mm \\ \(C_{1}D_{1}=l_{6}\) & \(A_{1}B_{1}=l_{5}\) & \(P_{1}B_{1}=l_{7}\) & \(P_{1}C_{1}=l_{7}^{\prime}\) & \(P_{1}A=a\) & \(AD=b\) \\
1178.8 mm & 704.628 mm & 1532.7 mm & 1171.5 mm & 648.3 mm & 851.7 mm \\ \hline \end{tabular}
\end{table} 
Table 1: The constant geometric dimensions of the links of the lifting lever mechanism. Resolving Equation (7) we find the angular velocities \(\omega_{2}\)_and_ \(\omega_{3}\). To find the links of the angular velocities \(\omega_{4},\ \omega_{5},\omega_{6}\)_and_ \(\omega_{7}\), we differentiate the system of Equations (5) and (6). \[\left\{ \begin{array}{l}-\omega_{3}l_{3}sin(\varphi_{3})+\omega_{2}l_{2}^{ \prime}sin(\varphi_{4}^{\prime}+\alpha_{4})-\omega_{4}l_{4}sin(\varphi_{4})= \omega_{7}l_{7}sin(-\varphi_{7}^{\prime}+\alpha_{7})-\omega_{5}l_{5}sin( \varphi_{5})\\ \omega_{3}l_{3}cos(\varphi_{3})-\omega_{2}l_{2}^{\prime}cos(\varphi_{4}^{ \prime}+\alpha_{4})+\omega_{4}l_{4}cos(\varphi_{4})=\omega_{7}l_{7}cos(- \varphi_{7}^{\prime}+\alpha_{7})+\omega_{5}l_{5}cos(\varphi_{5})\\ \omega_{7}l_{7}sin(-\varphi_{7}^{\prime}+\alpha_{7})-\omega_{5}l_{5}sin( \varphi_{5})+\omega_{4}l_{4}^{\prime}sin(-\varphi_{4}^{\prime}+\alpha_{4})= \omega_{7}l_{7}^{\prime}sin(\varphi_{7}^{\prime})-\omega_{6}l_{6}sin(\varphi _{6})\\ \omega_{7}l_{7}cos(-\varphi_{7}^{\prime}+\alpha_{7})+\omega_{5}l_{5}cos( \varphi_{5})+\omega_{4}l_{4}^{\prime}cos(-\varphi_{4}^{\prime}+\alpha_{4})= \omega_{7}l_{7}^{\prime}cos(\varphi_{7}^{\prime})+\omega_{6}l_{6}cos(\varphi _{6})\end{array} \right. \tag{8}\]  
Resolving Equation (9), we find the angular velocities \(\omega_{4},\ \omega_{5},\omega_{6}\)_and_ \(\omega_{7}\). To find the angular accelerations \(\varepsilon_{2}\)_and_ \(\varepsilon_{3}\) of links 2 and 3, we differentiate the system of Equation (7). \[\left\{