## 3 Kinematics Analysis
### Platform Configuration Parameters  
The mobile platform is connected to the base platform through the legs and the joints at both ends of the legs, where these legs have the degree of freedom of rotation. The joint adopted is an offset joint, in which the rotation center lines of the two rotation pairs lie perpendicular to each other but do not intersect, and they also have a specific offset that allows them to form an RR joint, as shown in Figure 6a,b. The lower joint is connected to the sliding block, which moves in a straight line along double guide rails that are arranged at an angle of 30\({}^{\circ}\) with respect to the horizontal plane. Figure 6c shows the composition of the P-RR-R-RR leg drive chain of the parallel platform.  
Figure 5: Effects of each component on the stiffness of the entire mechanism.  
Figure 6: \(P\)-RR-R-RR kinematic chain: (**a**) Schematic diagram of offset RR-joint; (**b**) 3D model of offset RR-joint; (**c**) \(P\)-RR-R-RR kinematic chain composition.  
To describe the motion of the mobile platform, a global coordinate system denoted by \(O_{B}-X_{B}Y_{B}Z_{B}\) is established at the center \(O_{B}\) of the base of the base platform. A local coordinate system denoted by \(O_{P}-X_{P}Y_{P}Z_{P}\) is established at the center \(O_{P}\) of the upper surface of the mobile platform. The six joint points \(P_{i}\) (\(i=1,\,\ldots,6\)) of the mobile platform form a \(120^{\circ}\) symmetrical hexagon, and the circle formed by \(P_{i}\) has a center denoted by \(O_{P}\)\({}^{\prime}\) and a radius \(R_{P}\), and the center angle \(\theta_{P}\) of \(P_{i}\) is distributed. When the mobile platform is at the zero position, the six lower joint points denoted by \(B_{i}\) (\(i=1,\,\ldots,6\)) also form a symmetrical hexagon, and the circle formed by \(B_{i}\) has a center denoted by \(O_{B}\)\({}^{\prime}\), the radius is \(R_{B}\), and the center angle \(\theta_{B}\) of \(B_{i}\) is distributed. The distance between the center \(O_{P}\)\({}^{\prime}\) and the center \(O_{P}\) of the top surface of the mobile platform is denoted by \(H_{P}\). The distance between the center \(O_{B}\)\({}^{\prime}\) and the center \(O_{B}\) of the bottom of the base platform is denoted by \(H_{B}\). When the mobile platform is at the zero position, the distance between the circle center \(O_{P}\)\({}^{\prime}\) and the circle center \(O_{B}\)\({}^{\prime}\) is \(H\). The offset amounts of the upper and lower joints are expressed as \(U_{P}\) and \(U_{B}\), respectively. The above coordinate systems and parameters are shown in Figures 7 and 8. The parallel platform configuration parameters are presented in Table 1.  
Figure 7: Definition of body coordinate systems for the platform and the joints of the \(i_{th}\) leg.  
The spatial pose of the mobile platform in the global coordinate system \(O_{B}-X_{B}Y_{B}Z_{B}\) is determined using the six-dimensional vector \([X,\ Y,\ Z,\ \alpha,\ \beta,\ \gamma]\). Among these dimensions, \([X,\ Y,\ Z]\) and \([\alpha,\ \beta,\ \gamma]\) indicate the position and the orientation of the mobile platform, respectively. The transfer matrix of the moving coordinate system \(O_{P}-X_{P}Y_{P}Z_{P}\), when transformed into the global coordinate system \(O_{B}-X_{B}Y_{B}Z_{B}\) according to the RPY rules, can be expressed as follows:  
\[O_{B}\mathbf{T}_{O_{P}}=\left[\begin{array}{ccc}c\beta c\gamma&-c\beta\varsigma \gamma&s\beta&X\\ cas\gamma+s\alpha s\beta c\gamma&cac\gamma-s\alpha s\beta s\gamma&-sac\beta &Y\\ s\alpha s\gamma-c\alpha s\beta c\gamma&sac\gamma+c\alpha s\beta s\gamma&cac \beta&Z\\ 0&0&0&1\end{array}\right] \tag{1}\]  
where c ( ) = cos ( ) and s ( ) = sin ( ).