## 6 Static Analysis and Stiffness Matrix Generation
\[\mathbf{F}_{\text{W}}=\mathbf{K}\Delta\mathbf{X}. \tag{37}\]  
Taking Equation (34) into Equation (37) to achieve Equation (38), where \(\overline{\mathbf{k}}\)\(=\text{diag}(\mathbf{k}_{\text{a}},\mathbf{k}_{\text{c}})\)  
\[\mathbf{K}=\mathbf{J}^{\text{T}}\overline{\mathbf{k}}\mathbf{J}, \tag{38}\]  
It should be noted that \(\mathbf{K}\) is built in the moving frame, thereby transforming stiffness into the fixed frame to analyze the stiffness performance in the workspace. \[\mathbf{K}_{\text{p}}=\mathbf{T}^{\text{T}}\mathbf{K}\mathbf{T}\]  
\[\mathbf{T}=\left[ \begin{array}{cc}\mathbf{R}&-\mathbf{l}_{\text{op}}\times \\ \mathbf{0}&\mathbf{R}\end{array} \right]\]  
\([\mathbf{l}_{\text{op}}\times]\) is the skew symmetrical matrix relating to vector \(\mathbf{l}_{\text{op}}\) from point P to point O. During the establishment of the stiffness matrix, the following assumptions are made:1. No friction is observed between two contact joints. 2. No preload is observed on the MP. 3. The MP and the fixed platform are set as rigid bodies, and no deformation is observed. 4. Regardless of the antagonistic stiffness, the overall Jacobian matrix does not change caused by external force. From Equation (38), we can obtain \(\overline{\mathbf{k}}\) which can be expressed in a matrix form as 32 \(\times\) 32 diagonal matrix. \[ \begin{array}{l}\overline{\mathbf{k}}=\text{diag}(\text{k}_{\text{act},1}, \text{k}_{\text{act},2},\text{k}_{\text{act},3},\text{k}_{\text{act},4},\text {k}_{\text{act},5},\text{k}_{\text{act},6},\text{k}_{\text{act},7},\text{k}_{ \text{act},8},\\ \text{k}_{\text{con}1,1},\text{k}_{\text{con}2,1},\text{k}_{\text{con}3,1}, \text{k}_{\text{con}1,2},\text{k}_{\text{con}2,2},\text{k}_{\text{con}3,2}, \text{k}_{\text{con}1,3},\text{k}_{\text{con}2,3},\text{k}_{\text{con}3,3}, \text{k}_{\text{con}1,A},\text{k}_{\text{con}2,4},\text{k}_{\text{con}3,4},\\ \text{k}_{\text{con}1,5},\text{k}_{\text{con}2,5},\text{k}_{\text{con}3,5}, \text{k}_{\text{con}1,6},\text{k}_{\text{con}2,6},\text{k}_{\text{con}3,6}, \text{k}_{\text{con}1,7},\text{k}_{\text{con}2,7},\text{k}_{\text{con}3,7}, \text{k}_{\text{con}1,8},\text{k}_{\text{con}2,8},\text{k}_{\text{con}3,8}). \end{array} \]  
When the stiffness matrix of the parallel mechanism is achieved, the parallel mechanism should be evaluated in a virtual way. The reachable workspace of the two mechanisms is large enough compared with the actual work required. Thus, evaluating the stiffness along a specific trajectory is practical. According to the air flow direction of the experiment, the mechanism is affected by the force in the x-direction more strongly than in the other directions. Thus, the stiffness in the x-direction should be prioritized. The given trajectory in the x-direction ought to go throughout two workspaces and the trajectory equation is presented as follows:  
\[\left\{ \begin{array}{l}\text{x}(\text{t})=-0.15+0.15\text{t}\\ \text{y}(\text{t})=1.338\text{sin}(0.3957\,\text{t}-1.9897)\\ \text{\theta}(\text{t})=0\hskip 56.905512 pt\text{t}\in[0,2]\end{array} \right. \tag{39}\]  
To obtain the stiffness in all directions, the external \(\mathbf{F}_{\text{E}}\) is applied to point P, and \(\mathbf{F}_{\text{E}}\) is the unit force [1N, 1N, 1N, 1N\(\cdot\)m, 1N\(\cdot\)m] T. The deformation in all directions can be achieved from \(\Delta\mathbf{X}_{p}=\mathbf{K}_{p}^{-1}\mathbf{F}_{W}=\left[\Delta_{\rho x}, \Delta_{\rho y},\Delta_{\rho z},\Delta_{xx},\Delta_{xy},\Delta_{xz}\right]^{T}\). In order to evaluate two types of mechanisms in the actual workspace, we use trajectory Equation (39) as the sub-workspace. We used \(\Delta_{\rho x}\), \(\Delta_{\rho y}\), and \(\Delta_{\rho z}\) to denote the linear deformation along the x, y, and z directions in frame O-xyz, respectively, and employed \(\Delta_{xx}\), \(\Delta_{xy}\), and \(\Delta_{az}\) to represent the angular deformations around the x, y, and z directions in frame O-xyz, respectively. Figure 9 illustrates all directional deformation distributions of type I in frame O-xyz.