# Structural-Parametric Synthesis of the RoboMech Class Parallel Mechanism with Two Sliders
## 4 Kinematic Analysis of the PM with Two Sliders
### Analogues of Velocities and Accelerations  
To solve the problems of analogues of velocities and accelerations of the PM with two sliders, we select its independent vector contours, the number of which is equal to half the number of links of the Assur group, i.e., it is equal to two. As independent vector contours, we choose the contours \(OQ^{\prime}ABC^{\prime}O\) and \(OQ^{\prime}ABDEO\), the vector loop-closure equations of which have the forms  
\[\left. \begin{array}{c}l_{OQ^{\prime}}\mathbf{e}_{OQ^{\prime}}+l_{Q^{\prime}A} \mathbf{e}_{Q^{\prime}A}+l_{AB}\mathbf{e}_{3i}-l_{CB}\mathbf{e}_{4i}+l_{CC^{\prime}}\mathbf{e} _{CC^{\prime}}-l_{(OC^{\prime})_{i}}\mathbf{e}_{OC^{\prime}}=0\\ l_{OQ^{\prime}}\mathbf{e}_{OQ^{\prime}}+l_{Q^{\prime}A}\mathbf{e}_{Q^{\prime}A}+l_{AB} \mathbf{e}_{3i}-l_{DB}\mathbf{e}_{(DB)_{i}}-l_{ED}\mathbf{e}_{Si}-l_{DE}\mathbf{e}_{OE}=0 \end{array} \right\} \tag{41}\]  
Project the system of Equation (41) on the axes \(OX\) and \(OY\) of the absolute coordinate system OXY  
\[\left. \begin{array}{c}l_{Q^{\prime}A}+l_{AB}\text{cos}\varphi_{3i}-l_{CB} \text{cos}\varphi_{4i}-l_{(OC^{\prime})_{i}}=0\\ l_{OQ^{\prime}}+l_{AB}\text{sin}\varphi_{3i}-l_{CB}\text{sin}\varphi_{4i}+l_{CC^ {\prime}}=0\\ l_{Q^{\prime}A}+l_{AB}\text{cos}\varphi_{3i}-l_{DB}\text{cos}(\varphi_{4i}- \alpha_{4})-l_{ED}\text{cos}\varphi_{Si}-l_{OE}\text{cos}\varphi_{OE}=0\\ l_{OQ^{\prime}}+l_{AB}\text{sin}\varphi_{3i}-l_{DB}\text{sin}(\varphi_{4i}- \alpha_{4})-l_{ED}\text{sin}\varphi_{Si}-l_{OE}\text{sin}\varphi_{OE}=0\end{array} \right\} \tag{42}\]  
Differentiate the system of Equation (42) with respect to the generalized coordinate \(s_{Q_{i}}\)  
\[\left. \begin{array}{c}-l_{AB}\text{sin}\varphi_{3i}\cdot\varphi_{3i}^{\prime }-l_{CB}\text{sin}\varphi_{4i}\cdot\varphi_{4i}^{\prime}-u_{P_{i}}=0\\ 1+l_{AB}\text{cos}\varphi_{3i}\cdot\varphi_{3i}^{\prime}-l_{CB}\text{cos} \varphi_{4i}\cdot\varphi_{4i}^{\prime}=0\\ -l_{AB}\text{sin}\varphi_{3i}\cdot\varphi_{3i}^{\prime}+l_{DB}\text{sin}( \varphi_{4i}-\alpha_{4})\cdot\varphi_{4i}^{\prime}+l_{ED}\text{sin}\varphi_{5 i}\cdot\varphi_{5i}^{\prime}=0\\ 1+l_{AB}\text{cos}\varphi_{3i}\cdot\varphi_{3i}^{\prime}-l_{DB}\text{cos}( \varphi_{4i}-\alpha_{4})\cdot\varphi_{4i}^{\prime}-l_{ED}\text{cos}\varphi_{ Si}\cdot\varphi_{5i}^{\prime}=0\end{array} \right\} \tag{43}\]  
From the system of Equation (43) we determine the analogues of velocities  
\[\mathbf{u}=\mathbf{A}^{-1}\ \cdot\mathbf{b}, \tag{44}\]  
where  
\[\mathbf{A}=\left[ \begin{array}{cccc}Y_{A_{i}}-Y_{B_{i}}&Y_{B_{i}}-Y_{C_{i}}&0& -1\\ X_{B_{i}}-X_{A_{i}}&X_{C_{i}}-X_{B_{i}}&0&0\\ Y_{A_{i}}-Y_{B_{i}}&Y_{B_{i}}-Y_{D_{i}}&Y_{D_{i}}-Y_{E}&0\\ X_{B_{i}}-X_{A_{i}}&X_{D_{i}}-X_{B_{i}}&X_{E}-X_{D_{i}}&0\end{array} \right] \tag{45}\]  
\[\mathbf{u}=\left[ \begin{array}{c}\varphi_{3i}^{\prime}\\ \varphi_{4i}^{\prime}\\ \varphi_{5i}^{\prime}\\ u_{P_{i}}\end{array} \right],\mathbf{b}=\left[ \begin{array}{c}0\\ -1\\ 0\\ 0\end{array} \right]. \tag{46}\]  
Differentiate the system of Equation (43) with respect to the generalized coordinate \(s_{Q_{i}}\)  
\[\left.\begin{array}{c}-l_{AB}\text{cos}\varphi_{3i}\cdot\varphi^{\prime 2} {}_{3i}-l_{AB}\text{sin}\varphi_{3i}\cdot\varphi_{3i}^{\prime\prime}+l_{CB} \text{cos}\varphi_{4i}\cdot\varphi_{4i}^{\prime\prime}-u_{P_{i}}^{\prime \prime}=0\\ -l_{AB}\text{sin}\varphi_{3i}\cdot\varphi_{3i}^{\prime\prime}+l_{AB}\text{cos} \varphi_{3i}\cdot\varphi_{3i}^{\prime\prime}+l_{CB}\text{sin}\varphi_{4i}\cdot \varphi_{4i}^{\prime\prime}+l_{CB}\text{cos}\varphi_{4i}\cdot\varphi_{4i}^{ \prime\prime}=0\\ -l_{AB}\text{cos}\varphi_{3i}\cdot\varphi^{\prime\prime}{}_{3i}-l_{AB}\text{ sin}\varphi_{3i}\cdot\varphi_{3i}^{\prime\prime}+l_{DB}  
Then we obtain the analogues of accelerations  
\[\mathbf{w}=\mathbf{A}^{-1}\ \cdot\mathbf{c}, \tag{48}\]  
where  
\[\mathbf{w}=\left[ \begin{array}{c}\varphi_{3_{2}}^{\prime\prime}\\ \varphi_{4_{4}}^{\prime}\\ \varphi_{5_{5}}^{\prime}\\ w_{P_{i}}\end{array} \right],\ \mathbf{c}=\left[