## 6 Static Analysis and Stiffness Matrix Generation
By carefully observing the cloud figure: (1) the maximum deformation occurred at the boundary position of the desired trajectory, and the minimum deformation occurred nearby x = 0 m, in which the linear deformation value has symmetrical distribution at approximately x = 0 m; (2) the variations of \(\Delta_{\rho y}\) and \(\Delta_{\rho z}\), are associated with the y-axis, and the variations of \(\Delta_{ax}\), \(\Delta_{xy}\), and \(\Delta_{az}\) mainly have relationship with the \(x\)-axis. Figure 10 demonstrates all directional deformation distributions of type II in frame O-xyz. Evidently, the symmetrical characteristic of the linear deformation distributions are similar to type I. Notably, the increased levels of \(\Delta_{\rho y}\), \(\Delta_{\rho z}\), \(\Delta_{xy}\), and \(\Delta_{az}\) have been associated with the decrease in the y-direction. Generally, through the comparison of deformation value, we can find the stiffness of type I is higher than that of type II, except in the y-direction. \(\Delta_{\rho y}\) numerical value of the two types is relatively close. The reason has two aspects: (1) as for type I, the active stiffness consists of axial stiffness of the leg and hydraulic; actually, the stiffness of the leg is an order of magnitude larger than the hydraulic, according to Equations (16) and (17), \(\text{k}_{\text{act},i}^{1}\) is close to \(\text{k}_{\text{act},i}^{2}\) (2) slider-guideway offers constraint stiffness for the driving chain instead of the hydraulic cylinder. As for type II, the constraint stiffness depended on the hydraulic cylinder. From the deformation distribution values in each direction, we can find that type I has a better stiffness characteristic. The cloud figures can reflect the stiffness in six directions. From the overall Jacobian, we can find that the stiffness in the active directions are highly coupled, and the stiffness in the constraint directions are highly coupled, too. However, the active and constraint stiffnesses are fully decoupled. Therefore, investigating the influence of a certain parameter on active and constraint stiffness in the process of parameter optimization is beneficial. Figure 9: Deformation distributions of the mechanism in the sub-workspace of type I. (**a**) Linear deformation distribution in x direction of type I; (**b**) Linear deformation distribution in y direction of type I; (**c**) Linear deformation distribution in z direction of type I; (**d**) Angular deformation distribution in x direction of type I; (**e**) Angular deformation distribution in y direction of type I; (**f**) Angular deformation distribution in z direction of type I. Extracting the active and constraint stiffness from the overall stiffness is the essence of proposing the suitable indexes. For the convenience of stiffness expression, we use a compliance matrix instead of the stiffness matrix, in which \(\mathbf{K}_{\text{p}}^{-1}=\mathbf{C}_{\text{p}}\), \(\mathbf{\Gamma}\) and \(\mathbf{\Phi}\) denote the selection matrix. \[\mathbf{C}_{\text{p}}=\left[ \begin{array}{cccccc}c_{11}&c_{12}&0&0&0&c_{16} \\ c_{21}&c_{22}&0&0&0&c_{26}\\ 0&0&c_{33}&c_{34}&c_{35}&0\\ 0&0&c_{43}&c_{44}&c_{45}&0\\ 0&0&c_{53}&c_{54}&c_{55}&0\\ c_{61}&c_{62}&0&0&0&c_{66}\end{array} \right], \tag{40}\]  
Figure 10: Deformation distributions of the mechanism in the sub-workspace of type II. (**a**) Linear deformation distribution in x direction of type II; (**b**) Linear deformation distribution in y direction of type II; (**c**) Linear deformation distribution in z direction of type II; (**d**) Angular deformation distribution in x direction of type II; (**e**) Angular deformation distribution in y direction of type II; (**f**) Angular deformation distribution in z direction of type II. \[\mathbf{\Gamma}=\left[ \begin{array}{cccccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&0&0&0&1\end{array} \right],\mathbf{\Phi}=\left[ \begin{array}{cccccc}0&0&1&0&0 &0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\end{array}