# Structural-Parametric Synthesis of the RoboMech Class Parallel Mechanism with Two Sliders
## 3 Structural-Parametric Synthesis of the PM with Two Sliders
To do this, it is necessary to first determine the angle \(\varphi_{4i}\) by the expression  
\[\varphi_{4_{i}}=\varphi_{(CA)_{i}}+\cos^{-1}\frac{I_{BC}^{2}+l_{(CA)_{i}}^{2}- l_{AB}^{2}}{l_{BC}l_{(CA)_{i}}}, \tag{1}\]  
where  
\[\varphi_{(CA)_{i}}=\mathrm{tg}^{-1}\frac{Y_{A_{i}}-Y_{C_{i}}}{X_{A_{i}}-X_{C_{ i}}}, \tag{2}\]  
\[l_{(CA)_{i}}=\left[(X_{A_{i}}-X_{C_{i}})^{2}+(Y_{A_{i}}-Y_{C_{i}})^{2}\right]^{ \frac{1}{2}}. \tag{3}\]  
Coordinates \(X_{A_{i}},Y_{A_{i}}\) and \(X_{C_{i}},Y_{C_{i}}\) of the joints \(A\) and \(C\) in the absolute coordinate system OXY in Equations (2) and (3) are determined by the expressions  
\[X_{A_{i}}=x_{A}^{(1)},Y_{A_{i}}=s_{Q_{i}}+y_{A}^{(1)},X_{C_{i}}=-s_{P_{i}}+x_{ C}^{(2)},Y_{Ci}=y_{C}^{(2)}. \tag{4}\]  
With the link \(CB\) of the dyad \(ABC\), we attach the coordinate system \(Cx_{4}y_{4}\), the axis \(Cx_{4}\) of which is directed along the link \(CB\). Then, the synthesis parameters of the negative CKC \(DE\) are the vector \(r_{2}=\left[x_{E}^{(4)},y_{E}^{(4)},X_{D},Y_{D},l_{DE}\right]^{T}\), where \(x_{E}^{(4)},y_{E}^{(4)}\) and \(X_{D},Y_{D}\) are the coordinates of the joints \(E\) and \(D\) in the coordinate systems \(Cx_{4}y_{4}\) and OXY, respectively. Let us consider the motion of the plane \(Cx_{4}y_{4}\) in the absolute coordinate system OXY. In this case, the joint \(E\left(x_{E}^{(4)},y_{E}^{(4)}\right)\) moves along an arc of a circle with a center at the joint \(D(X_{D},Y_{D})\) and a radius \(l_{DE}\). Consequently, an equation of the geometric constraint imposed by the negative CKC \(DE\) of the RR type on the motion of the moving plane \(Cx_{4}y_{4}\) is expressed as a weighted difference  
\[\Delta q_{i}=\left(X_{E_{i}}-X_{D}\right)^{2}+\left(Y_{E_{i}}-Y_{D}\right)^{2} -l_{DE}^{2}, \tag{5}\]  
where \(X_{E_{i}}\) and \(Y_{E_{i}}\) are the coordinates of the joint \(E\) in the absolute coordinate system OXY, which are determined by the expression  
\[\left[ \begin{array}{c}X_{E_{i}}\\ Y_{E_{i}}\end{array} \right]=\left[ \begin{array}{c}X_{C_{i}}\\ Y_{C_{i}}\end{array} \right]+\left[ \begin{array}{cc}\cos\varphi_{4i}&-\text{ sin}\varphi_{4i}\\ \text{ sin}\varphi_{4i}&\cos\varphi_{4i}\end{array} \right]\cdot\left[ \begin{array}[ ]{c}x_{E}^{(2)}\\ y_{E}^{(2)}\end{array} \right] \tag{6}\]  
After substituting Equation (6) into Equation (5) and replacing the synthesis parameters of the form  
\[\left[ \begin{array}{c}p_{1}\\ p_{2}\end{array} \right]=\left[ \begin{array}{c}X_{D}\\ Y_{D}\end{array} \right],\ \left[ \begin{array}{c}p_{4}\\ p_{5}\end{array} \right]=\left[ \begin{array}{c}x_{E}^{(4)}\\ y_{E}^{(4)}\end{array} \right],\ p_{3}=\frac{1}{2}\left(X_{D}^{2}+Y_{D}^{2}+{x_{ E}^{(4)^{2}}+y_{E}^{(4)^{2}}-l_{ED}^{2}}\right) \tag{7}\]  
then Equation (5) is expressed linearly in two groups of synthesis parameters \(\mathbf{p}_{2}^{(1)}=\left[p_{1},p_{2},p_{3}\right]^{T}\) and \(\mathbf{p}_{2}^{(2)}=\left[p_{4},p_{5},p_{3}\right]^{T}\) in the form  
\[\Delta q_{i}^{(k)}\left(\mathbf{p}_{2}^{(k)}\right)=2{\left({\mathbf{g}_{i}^{(k)}}^{T} \cdot\mathbf{p}_{2}^{(k)}-g_{oi}^{(k)}\right)},k=1,2, \tag{8}\]where \(\mathbf{g}_{i}^{(k)}\) and \(g_{oi}^{(k)}\) are the coefficients of the vectors \(\mathbf{p}_{2}^{(k)}\) and free terms depending on the remaining synthesis parameters, which have the form  
\[\mathbf{g}_{i}^{(1)}=-\left[ \begin{array}{cc}X_{C_{i}}\\ Y_{C_{i}}\\ 1\end{array} \right]-\left[ \begin{array}{cc}\mathbf{\Gamma}(\mathbf{\varphi}_{4i})&0\\ 0&0\end{array} \right]\cdot\left[ \begin{array}{c}p_{4}\\ p_{5}\\ 0\end{array} \right] \tag{9}\]  
\[\mathbf{g}_{i}^{(2)}=\left[ \begin{array}{cc}\mathbf{\Gamma}(\mathbf{\varphi}_{4i})&0\\ 0&0\end{array} \right]\cdot\left[ \begin{array}{c}X_{C_{i}}\\ Y_{C_{i}}\\ 1\end{array} \right]+\left[ \begin{array}{cc}\mathbf{\Gamma}(\mathbf{\varphi}_{4i})&0\\ 0&0\end{array} \right]\cdot\left[ \begin{array}{c}p_{1}\\ p_{2}\\ 0\end{array} \right] \tag{10}\]  
\[g_{oi}^{(1)}=-\frac{1}{2}\Big{[}X_{C_{i}}^{2}+Y_{C_{i}}^{2}\Big{]}+\big{[}X_{ C_{i}},Y_{C_{i}}\big{]}\cdot\mathbf{\Gamma}(\mathbf{\varphi}_{4i})\bigg{[}