## 7 Discussion  
This paper has calculated the spatial stiffness of the planar parallel mechanism and presented two novel stiffness indexes based on overall Jacobian. Compared with the previous research methods, it has the following advantages:  
* Most studies only consider the stiffness in plane movement when analyzing the stiffness of the planar parallel mechanism. For example, the work [14] researched performance of three planar parallel manipulators, and finally a \(3\times 3\) stiffness matrix was achieved. However, the mechanism bears the force in six directions in the actual work. We build a stiffness model considering the constraint stiffness, which can reflect the actual stiffness characteristic of the mechanism. * Most of the existing stiffness indexes are focused on evaluating the isotropic characteristics of the mechanism and the ability to resist deformations. The total linear stiffness index (TLSI) and the total angular stiffness index (TASI) are separated to evaluate the ability of the mechanism to resist deformation in [40; 41]. As shown in Equation (44), \(\mathbf{c_{tr}}\) is the linear compliance element matrix, \(\mathbf{c_{rr}}\) is the angular compliance element matrix. It can be found from Figure 12 that, with the increase of the hydraulic cylinder range, TLSI and TASI of the two types of mechanisms have the same decreasing trend. However, it is difficult to analyze the phenomenon from the theoretical formulas, and it brings obstacles for subsequent optimization of the component size. The stiffness index KSI was proposed in [34], which is a traditional mechanism stiffness evaluation method. As shown in Equation (45), \(\lambda_{\max}\), \(\lambda_{\min}\) are the maximum and minimum eigenvalues of the stiffness matrix \(\mathbf{K_{p}}\), respectively This index reflects the isotropy of the mechanism geometry. From index values in Figure 13, we can find the hydraulic cylinder range has less effect on the KSI of the two mechanisms. This method lacks effective output parameter support for evaluating the dimensional characteristics of planar asymmetric mechanisms. When investigating the effect of the parameters on the stiffness in given directions, the proposed indexes are more clear explanations. \[\mathbf{C}_{p}=\left[ \begin{array}{cc}\mathbf{c_{tt}}&\mathbf{c_{rr}}\\ \mathbf{c_{tr}^{T}}&\mathbf{c_{rr}}\end{array} \right],\]  
\[\text{TLSI}=\frac{1}{\sqrt[3]{\det(\mathbf{c_{tt}})}},\text{TASI}=\frac{1}{ \sqrt[3]{\det(\mathbf{c_{rr}})}}. \tag{44}\]  
\[\text{KSI}=\sqrt{\frac{\lambda_{\min}}{\lambda_{\max}}} \tag{45}\]  
Figure 12: TLSI and TASI variation caused by a range of hydraulic cylinder. (**a**) TLSI index variation of two type mechanisms; (**b**) TASI index variation of two type mechanisms; The blue solid line stands for index of typeI mechanism, the red dashed line stands for index of type II mechanism. The application to two types of parallel planar part for a hybrid mechanism and some interesting results have been illustrated. We can find the type I has better stiffness characteristics according to deformation distributions of the sub-workspace. In the constraint directions, the reason is that in the establishment of the constraint stiffness model, the guideway of type I provides a part of the constrained stiffness. It can also be concluded from the ASI and CSI that type has better stiffness characteristics. It is worth noting that the influence of the hydraulic cylinder range on the active stiffness of type II is more significant, and there exists a turning point when the length of the rod that sticks out of the hydraulic cylinder is 0.485 m. This may be a noteworthy research direction for stiffness optimization in future work. In future work, the proposed indexes will be applied to the spatial parallel mechanism, and multi-objective optimization may be researched based on ASI and CSI.