## 2 Kinematic Analysis
\begin{array}{c}\mathbf{l_{g}}\mathbf{\cos}\mathbf{\beta}=\mathbf{e}-\mathbf{l_{CP}}\mathbf{\cos} \mathbf{\theta}-\mathbf{l_{1}}\mathbf{\cos}\mathbf{\varphi_{1}},\ \mathbf{\sin}\mathbf{\beta}=\mathbf{h}-\mathbf{l_{CP}}\mathbf{\sin}\mathbf{\theta}-\mathbf{l_{1}}\mathbf{\sin} \mathbf{\varphi_{1}}\\ \mathbf{c}=\frac{1}{2}\mathbf{b}+\mathbf{l_{CP}}\mathbf{\cos}\mathbf{\theta}=\mathbf{const},\ \mathbf{d}=-\mathbf{h}+\mathbf{l_{CP}}\mathbf{\sin}\mathbf{\theta}=\mathbf{const} \end{array}  \tag{4}\]  
In the mechanism under consideration, the variable kinematic parameters are \(\mathbf{\varphi_{1}},\mathbf{\varphi_{2}},\mathbf{\varphi_{3}},\mathbf{\psi}\). It is assumed that the engine will be connected to link 1 and mounted on the frame \(\mathbf{O}\). Hence, the angular coordinate of the input link is taken as the generalized coordinate 1--\(\mathbf{\varphi_{1}}\). According to [PERSON] [6], the mechanism belongs to the fourth class since the driven kinematic chain contains a four-sided closed loop and the structural group (2,3,4,5,6,7) does not break up into groups of a smaller order. At the same time, the analysis of the positions of the mechanism for a given \(\mathbf{\varphi_{1}}\), i.e., system (3) has no analytical solutions. From the analysis of system (3), it can be seen that the first three equations of system (3) relative to \(\mathbf{\varphi_{2}}\), \(\mathbf{\varphi_{3}}\), \(\mathbf{\psi}\) represent a system of nonlinear trigonometric equations, which should be solved by approximate numerical methods. Then, from the last equation of system (3), the coordinate of the slider is determined 7--S.  
To construct an effective numerical algorithm for the kinematic analysis of the mechanism under consideration, we apply the following approach. Based on system (3), we write the kinematics equations of the mechanism in small displacements [2] with respect to \(\varphi_{2}\), \(\psi\), \(\varphi_{3}\) in the form of a matrix equation (\(3\times 3\)):  
\[A\overrightarrow{x}=\overrightarrow{b} \tag{5}\]  
where \(\overrightarrow{x}= \begin{bmatrix}\Delta\varphi_{2}&\Delta\psi&\Delta\varphi_ {3}\end{bmatrix} ^{T}\), \(\overrightarrow{b}= \begin{bmatrix}-I_{1}\Delta\varphi_{1}\text{sin}\varphi_{1 }&-I_{1}\Delta\varphi_{1}\text{cos}\varphi_{1}&\mathbf{0}\end{bmatrix} ^{T}\) and  
\[A= \begin{bmatrix}I_{2}\text{sin}\varphi_{2}&I_{BC}\text{sin}(\psi+\alpha_{6})& -I_{3}\text{sin}\varphi_{3}\\ I_{2}\text{cos}\varphi_{2}&I_{BC}\text{cos}(\psi+\alpha_{6})&-I_{3}\text{cos} \varphi_{3}\\ 0&I_{6}\text{sin}\psi&-I_{3}\text{sin}\varphi_{3}\end{bmatrix}  \tag{6}\]  
The solution to system (5) has the following form:  
\[ \begin{array}{c}\Delta\varphi_{2}=\frac{\Delta\varphi_{1}}{|A|_{det}}I_{3} I_{1}[I_{BC}\text{sin}\varphi_{3}\text{sin}(\psi+\alpha_{6}-\varphi_{1})+I_{6} \text{sin}\psi\text{sin}(\varphi_{3}-\varphi_{1})]\\ \Delta\psi=\frac{\Delta\varphi_{1}}{|A|_{det}}I_{3}I_{2}I_{1}\text{sin}\varphi _{3}\text{sin}(\varphi_{2}-\varphi_{1})\\ \Delta\varphi_{3}=\frac{\Delta\varphi_{1}}{|A|_{det}}I_{6}I_{2}I_{1}\text{sin} \psi\text{sin}(\varphi_{2}-\varphi_{1})\end{array}  \tag{7}\]  
where  
\[ \begin{array}{c}\lceil A\rceil_{det}=I_{2}I_{3}[I_{BC}\text{sin}\varphi_{3} \text{sin}(\psi+\alpha_{6}-\varphi_{2})-I_{6}\text{sin}\psi\text{sin}(\varphi _{3}-\varphi_{2})]\
eq\mathbf{0}\end{array}  \tag{8}\]