# Structural-Parametric Synthesis of the RoboMech Class Parallel Mechanism with Two Sliders
## 4 Kinematic Analysis of the PM with Two Sliders
### Position Analysis  
For position analysis of the considered PM, we use the method of conditional generalized coordinates [50]. According to this method, we remove the link 5 by disconnecting the elements of the joints \(D\) and \(E\) and select the slider \(P\) as a conditional input link due to the additional DOF that appears. Then, this PM of the third class is transformed into a mechanism of the second class with the structural formula  
\[I(1)\rightarrow\text{II}(3,4)\gets I(2). \tag{28}\]  
Derive the function  
\[\Delta=l_{DE}-\widetilde{l}_{DE}, \tag{29}\]  
where \(\widetilde{l}_{DE}\) is a variable distance between the centers of the disconnected joints \(D\) and \(E\), which is determined by the expression  
\[\widetilde{l}_{DE}=\Big{[}\big{(}Kh_{D_{i}}-Kh_{E}\big{)}^{2}+\big{(}Y_{D_{i}} -Y_{E}\big{)}^{2}\Big{]}^{\frac{1}{2}}. \tag{30}\]  
Coordinates \(Kh_{D_{i}}\) and \(Y_{D_{i}}\) of the joint \(D\) center in Equation (30) are determined by the equation  
\[\left[\begin{array}{c}X_{D_{i}}\\ Y_{C_{i}}\end{array}\right]=\left[\begin{array}{c}X_{C_{i}}\\ Y_{C_{i}}\end{array}\right]+\left[\begin{array}{cc}\cos\varphi_{4i}&-\text{ sin}\varphi_{4i}\\ \text{ sin}\varphi_{4i}&\cos\varphi_{4i}\end{array}\right]\cdot\left[\begin{array} []{c}x_{D}^{(4)}\\ y_{D}^{(4)}\end{array}\right], \tag{31}\]  
where  
\[\left[\begin{array}{c}X_{C_{i}}\\ Y_{C_{i}}\end{array}\right]=\left[\begin{array}{c}-\Big{(}s_{P_{i}}+x_{C}^{( 2)}\Big{)}\\ y_{C}^{(2)}\end{array}\right]. \tag{32}\]  
To determine the angle \(\varphi_{4i}\) in Equation (31), we derive a vector \(ABC\) loop-closure equation  
\[l_{AB}\varepsilon_{3i}-l_{CDB}\varepsilon_{4i}+l_{(CA)i}\varepsilon_{(CA)i}=0, \tag{33}\]  
\[l_{(CA)_{i}}=[(X_{A_{i}}-X_{C_{i}})^{2}+(Y_{A_{i}}-Y_{C_{i}})^{2}]^{\frac{1}{ 2}}, \tag{34}\]  
\[\varphi_{(CA)_{i}}=\text{tg}^{-1}\frac{Y_{A_{i}}-Y_{C_{i}}}{X_{A_{i}}-X_{C_{i }}}, \tag{35}\]  
\[\left[\begin{array}{c}X_{A_{i}}\\ Y_{A_{i}}\end{array}\right]=\left[\begin{array}{c}-x_{A}^{(1)}\\ s_{Q_{i}}+y_{A}^{(1)}\end{array}\right]. \tag{36}\]  
In Equation (33) \(e\) denotes the unit vector.  
We transfer \(l_{AB}\varepsilon_{3i}\) to the right side of Equation (33) and square both sides. As a result, we obtain  
\[\varphi_{4i}=\varphi_{(CA)_{i}}+\cos^{-1}\frac{l_{CB}^{2}+l_{(CA)_{i}}^{2}-l_{ AB}^{2}}{2l_{CB}l_{(CA)_{i}}}, \tag{37}\]  
Next, we define  
\[\left[\begin{array}{c}X_{B_{i}}\\ Y_{B_{i}}\end{array}\right]=\left[\begin{array}{c}X_{C_{i}}\\ Y_{C_{i}}\end{array}\right]+l_{CB}\left[\begin{array}{c}cos\varphi_{4i}\\ \text{sin}\varphi_{4i}\end{array}\right], \tag{38}\]  
\[\varphi_{3i}=\text{tg}^{-1}\frac{Y_{B_{i}}-Y_{A_{i}}}{X_{B_{i}}-X_{A_{i}}}. \tag{39}\]  
Thus, Equation (29) is a function of one variable: the conditional generalized coordinate \(s_{P_{i}}\), for the given values of the real generalized coordinate \(s_{Q_{i}}\). MinimizingEquation (29) with respect to a variable \(s_{P_{i}}\) by the bisection method [53], we determine its values for given values \(s_{Q_{i}}\). In this case, the angles \(\varphi_{3i}\) and \(\varphi_{4i}\) are simultaneously determined. The angle \(\varphi_{5i}\) is determined by the expression  
\[\varphi_{5i}=\text{tg}^{-1}\frac{Y_{D_{i}}-Y_{E}}{X_{D_{i}}-X_{E}}. \tag{40}\]