# Structural-Parametric Synthesis of the RoboMech Class Parallel Mechanism with Two Sliders
## 3 Structural-Parametric Synthesis of the PM with Two Sliders  
The problem of structural-parametric synthesis of the PM with two sliders is to determine the structural scheme and geometric parameters of links, when the first slider Q (the punch of a hydraulic press) takes the lower extreme position with a stroke \(s_{Q_{1}}\), the second slider P takes the left extreme position with a stroke \(s_{P_{1}}\) (Figure 4a) and also, when the first slider Q takes the upper extreme position with a stroke \(s_{P_{N}}\), and the second slider P takes the right extreme position with a stroke \(s_{Q_{N}}\) (Figure 4b). For the convenience of reporting the strokes of the sliders, the absolute coordinate system OXY is located at the point of their intersection. Figure 4: (a) lower extreme and (b) upper extreme positions of the slider \(Q\). As noted above, to form the PM with two sliders, providing their specified positions, we connect the punch Q of the hydraulic press and the slider P using the dyad \(ABC\) with revolute kinematic pairs. The dyad \(ABC\) has zero DOF and it is a passive CKC, which does not impose geometric constraints on the movements of the punch Q and the slider \(P\). Therefore, the passive CKC \(ABC\) allows reproduction of the specified movements of the sliders Q and P. Then we connect the link \(BC\) of the dyad \(ABC\) with a base using a binary link \(DE\) with revolute kinematic pairs, which has one negative DOF, it is a negative CKC. Negative CKC \(DE\) imposes one geometric constraint on the movements of the sliders Q and P, and as a result, we obtain a structural scheme of the PM with structural formula I (0,1) \(\rightarrow\) III (3,4,2,5), where the kinematic chain 3-4-2-5 represents the Assur group of the third class [48]. Figure 5 shows a block structure of the formed PM with two sliders. For parametric synthesis of the PM with two sliders, let us consider its \(i\)-th intermediate position and attach the coordinate systems \(Qx_{1}y_{1}\) and \(Px_{2}y_{2}\) with the sliders (Figure 6), the axes \(Px_{1}\) and \(Qx_{2}\) which are directed parallel to the axis OX of the absolute coordinate system OXY. Then, the movements of the sliders are determined by the parameters \(s_{Q_{i}}\) and \(s_{P_{i}}\) of the coordinate systems \(Px_{2}y_{2}\) and \(Qx_{2}y_{2}\) movements, where \(i\) = 1, 2..., \(N\) (\(N\) is the number of given positions). Parametric synthesis of this PM with two sliders, according to its block structure (Figure 5), consists of a parametric synthesis of the passive CKC \(ABC\) and the negative CKC \(DE\). The synthesis parameters (geometric parameters of the links) of the passive CKC \(ABC\) is a vector \(r_{1}=\left[x_{A}^{(1)},y_{A}^{(1)},x_{C}^{(2)},y_{C}^{(2)},l_{AB},l_{BC} \right]^{T}\), where \(x_{A}^{(1)},y_{A}^{(1)}\) and \(x_{C}^{(2)},y_{C}^{(2)}\) are the coordinates of the joints A and C in the moving coordinate systems \(Qx_{1}y_{1}\) and \(Px_{2}y_{2}\) respectively, \(l_{AB}\) and \(l_{BC}\) are the lengths of the links AB and BC. Since the passive CKC ABC does not impose geometric constraint on the movements of the sliders Q and P, then its synthesis parameters are set, they are varied by the generator of \(LP_{T}\) sequence [49]  
Figure 5: Block structure of the PM with two sliders. Figure 6: Intermediate position of the PM with two sliders. depending on the geometric constraint imposed by the negative CKC DE. Negative CKC DE imposes one geometric constraint on the movements of the links AB and BC of the passive CKC ABC; therefore, its synthesis parameters are determined. Let us consider the parametric synthesis of the negative CKC \(DE\).