## 1 Introduction
Nevertheless, this method is relatively complex, as exhibited in the high dimensionality of the matrix that is suitable for the calculation of the stiffness of spatial parallel mechanisms [20; 21; 22; 23].The principle of the strain energy approach is to establish the function relationship between force and displacement based on strain energy and [PERSON]'s second theorem [24; 25; 26; 27]. Notwithstanding, the applied wrench analysis of over-constrained parallel mechanisms involves the statically indeterminate problem, whose solution requires first stiffness models. Figure 1: Aircraft model support system in the wind tunnel. The Jacobian matrix of the mechanism is integrated by utilizing screw theory, which is widely used for the stiffness analysis of parallel mechanisms at present [28; 29; 30; 31; 32]. Moreover, a \(6\times 6\) stiffness matrix is established while considering the constraint stiffness. This method can separate active stiffness from constraint stiffness so that the overall Jacobian can provide a more intuitive parameter relationship for the stiffness optimization. Regardless of what method is used to calculate the stiffness of a mechanism, a stiffness matrix will always be obtained. In the design stage, the stiffness matrix should be refined into a stiffness index to evaluate the stiffness performance of the mechanism quantitatively; this approach provides an intuitive reference for the optimal design of the mechanism. The application of the eigenvalues of the stiffness matrix [33; 34; 35] and stiffness ellipsoid [36; 37] is the most common method for evaluating stiffness performance, which reflects the isotropy of the mechanism. However, this approach has less effect on assessing the stiffness characteristic of an asymmetric planar mechanism. The elastic behavior of the mechanism can provide a physical interpretation by using the eigen-screw decomposition [29; 38; 39]. However, the fully coupled mechanism does not have an elastic center. Some researchers evaluated the deformation under unit torque or unit force through the principle of virtual work and strain energy theory [26; 27; 28]. Nevertheless, the index is changed with varying external loads. The linear and angular stiffness are separated from a stiffness matrix to evaluate the explicit physical significance of stiffness [40; 41]. Notably, the variation of a component parameter may affect linear and angular stiffness, and only some given directional stiffness changes. For instance, the parallel planar part has movable ability in X, Y, and yaw directions, but it lacks the movable ability in Z, roll, and pitch directions. We define X, Y, and yaw as movable directions and Z, roll, and pitch as non-movable directions. Therefore, it is reasonable to expect that some of the component parameters may only affect movable (active) stiffness or non-movable (constant) stiffness, or both of them. The existing stiffness indexes have mainly indicated the isotropy of mechanism configuration or the ability to resist comprehensive deformations. However, revealing the influence of component parameters on the specific directional stiffness performance is challenging. This study aims to achieve a considerable spatial stiffness of the 3-DoF parallel part of the support mechanism. Furthermore, exploring the influence of a component parameter on the active and constraint stiffness is purposeful for optimizing the structure. The main contributions of this study are summarized as follows:  
1. Two types of 3-DoF planar parallel parts for a model support mechanism are proposed and the stiffness models are established while considering constraint stiffness. 2. Two novel stiffness indexes are proposed to conduct a quantitative evaluation of comprehensive stiffness characteristics in the movable and non-movable directions. 3. The effects of the variation of the actuator parameter on stiffness characteristics are investigated on the basis of the proposed indexes. The research steps are presented as follows.