## 6 Static Analysis and Stiffness Matrix Generation  
Assuming that the MP is loaded from an external wrench \(\mathbf{F}_{\text{W}}=\left[ \begin{array}{cc}\mathbf{f}^{\text{T}}&\mathbf{w} ^{\text{T}}\end{array} \right]^{\text{T}}\), and it is expressed in the Plucker ray coordinate, \(\mathbf{f}=\left[ \begin{array}{cc}f_{x}&f_{y}&f_{z}\end{array} \right]^{ \text{T}}\) denotes the force vector, \(\mathbf{w}=\left[ \begin{array}{cc}w_{x}&w_{y}&w_{z}\end{array} \right]^{ \text{T}}\) is a wrench vector. Let \(\mathbf{f}_{\text{a}}\) and \(\mathbf{f}_{\text{c}}\) represent the reaction forces/torques of the actuators and constraints:  
\[\mathbf{F}_{\text{W}}=\mathbf{J}_{\text{a}}^{\text{T}}\mathbf{f}_{\text{a}}+ \mathbf{J}_{\text{c}}^{\text{T}}\mathbf{f}_{\text{c}}, \tag{31}\]  
\[\mathbf{f}_{\text{a}}=\mathbf{k}_{\text{a}}\Delta\mathbf{x}_{\text{a}}, \tag{32}\]  
\[\mathbf{f}_{\text{c}}=\mathbf{k}_{\text{c}}\Delta\mathbf{x}_{\text{c}}, \tag{33}\]  
where \(\mathbf{k}_{\text{a}}\) and \(\mathbf{k}_{\text{c}}\) represent active and constraint stiffness, respectively; \(\Delta\mathbf{x}_{\text{a}}\) and \(\Delta\mathbf{x}_{\text{c}}\) are the displacements caused by the driving and constraint reactions of the limb, respectively. According to the principle of virtual work, we can achieve Equation (34), \(\Delta\mathbf{x}_{\text{a}}=\left[\Delta\chi\;\Delta y\;\Delta z\right]\) denotes the linear displacement, \(\Delta\mathbf{x}_{\text{c}}=\left[\Delta\theta_{\text{x}}\;\Delta\theta_{ \text{y}}\;\Delta\theta_{\text{z}}\right]\) denotes the angular displacement. \[\mathbf{F}_{\text{W}}^{\text{T}}\Delta\mathbf{x}=\mathbf{f}_{\text{a}}^{\text {T}}\Delta\mathbf{x}_{\text{a}}+\mathbf{f}_{\text{c}}^{\text{T}}\Delta \mathbf{x}_{\text{c}}. \tag{34}\]  
Gravity should be considered in external forces. \(\mathbf{F}_{\text{E}}\) denotes the applied external force and \(\mathbf{F}_{\text{G}}\) denotes the gravity of components:  
\[\mathbf{F}_{\text{W}}=\mathbf{F}_{\text{E}}+\mathbf{F}_{\text{G}}, \tag{35}\]  
\[\mathbf{F}_{\text{G}}^{\text{I}}=\mathbf{F}_{\text{G,S}}+\mathbf{F}_{\text{G, L}}+\mathbf{F}_{\text{G,M}},\mathbf{F}_{\text{G}}^{\text{Z}}=\mathbf{F}_{\text{G,H}}+ \mathbf{F}_{\text{G,M}}. \tag{36}\]  
The applied external force \(\mathbf{F}_{\text{E}}=\left[ \begin{array}{cc}\mathbf{F}_{0}^{\text{T}}& \mathbf{\tau}_{0}^{\text{T}}\end{array} \right]^{\text{T}}\) takes the position of the center of gravity as the action point, \(\mathbf{F}_{\text{G,L}}\) is the gravity of the leg, \(\mathbf{F}_{\text{G,S}}\) is the gravity of the slider, \(\mathbf{F}_{\text{G,H}}\) is the gravity of the hydraulic cylinder, and \(\mathbf{F}_{\text{G,M}}\) is the gravity of the MP. \[\mathbf{F}_{\text{G,L}}=\sum\limits_{i=1}^{8}\text{m}_{\text{L}}\otimes \mathbf{g}\left[ \begin{array}{c}\mathbf{s}_{1}\\ (\mathbf{R}\mathbf{a}_{i}-\frac{1}{2}\mathbf{b}_{i})\times\mathbf{s}_{1}\end{array}  \right]\mathbf{F}_{\text{G,S}}=\sum\limits_{j=1}^{4}\text{m}_{\text{S}}\otimes \mathbf{s}_{1}\end{array}\right]\mathbf{F}_{\text{G,H}}=\sum\limits_{i=1}^{8} \text{m}_{\text{H}}\otimes\left[ \begin{array}{c}\mathbf{s}_{1}\\ (\mathbf{R}\mathbf{a}_{i}-\frac{1}{2}\mathbf{q}_{\text{L}}\mathbf{i}_{\text{ }})\times\mathbf{s}_{1}\end{array} \right]\]  
where \(\text{m}_{\text{L}}\) denotes the mass of the leg, \(\text{m}_{\text{S}}\) denotes the mass of the slider, \(\text{m}_{\text{H}}\) denotes the mass of the hydraulic cylinder, \(\text{m}_{\text{M}}\) denotes the mass of the manipulator, and \(\mathbf{g}\) denotes the acceleration of gravity; the parameters of mechanism are presented in Table 15 in Appendix C.  
According to [PERSON]'s law [34], \(\Delta\mathbf{X}\) is the deformation in six directions, \(\mathbf{K}\) is a \(6\times 6\) stiffness matrix.