## 4 Multi-Objective Optimization with Equivalent Static Loads for Parallel Mechanism
### Application in the Lander  
In this section, the lander introduced in Section 3 is optimized with modified ESLs and the desirability approach. Before landing on the ground, the legged lander will adjust the axis \(R_{i}\) (\(i=1\sim 3\)) to fit the landing surface environment. Therefore, the lander has a different configuration to land on the ground. We added the landing configuration as a new dimension in the structural optimization. When hitting on the ground, the footpad suffers severe vibration force. To reduce the mass and have better vibration performance, the leg of the lander should be optimized. The design parameter includes the tube radius in the main strut \(r_{mO},r_{ml}\) and the tube radius in the secondary struts \(r_{sO},r_{sl}\). It also includes mechanism parameters of the main body \(\theta,a,b\) and mechanism parameters of the platform \(a_{p},b_{p}\), as shown in Figure 4b. The main body \(\mathbf{u}_{b}\) is fixed. The rotational DOF (degree of freedom) of the footpad is fixed. The landing force exerted on the footpad \(\mathbf{F}_{j}\) is chosen as the force boundary condition. The parameter value of \(F_{f}\) is listed in Table 2. The lunar gravity is also considered, which is along the opposite direction of the \(z\) axis. \(g\) is the gravitational acceleration. \[\mathbf{u}_{b}=\left[0,0,0,0,0,0\right]^{T}\] \[\mathbf{u}_{f}=\left[u_{fx},u_{fy},u_{fz},0,0,0\right]^{T}\] \[\mathbf{F}_{f}=\left[F_{fx},F_{fy},F_{fz},0,0,0\right]^{T}\] \[F_{fx}=A_{x}e^{-\xi_{x}\omega_{z}t}\sin(\omega_{x}t)\] \[F_{fy}=A_{y}e^{-\xi_{y}\omega_{y}t}\sin(\omega_{y}t)\] \[F_{fz}=A_{z}\big{(}1-e^{-\xi_{z}\omega_{z}t}\sin(\omega_{z}t) \big{)}\] \[\mathbf{g}=\left[0,0,-1.63\right]^{T}\]  
The buffer force magnitude \(f_{1}\) in the main strut is the function of buffer stroke \(x\). Their relationship is shown in Figure 9a. The \(x\) axis is the main strut buffer stroke. It is the relative translation distance between nodes \(\overline{u}_{O1}\) and \(\overline{u}_{I1}\), which are expressed in local coordinates. \[x=\overline{u}_{O1}(1)-\overline{u}_{I1}(1) \tag{33}\]  
The buffer force magnitude in two secondary struts has the same function, \(f_{2m}=f_{3m}\), which is shown in Figure 9b. The \(x\) axis is the secondary strut buffer stroke computed by the relative translation distance between the \(\overline{u}_{O2}\) and \(\overline{u}_{I2}\), which are also expressed in local coordinates. The parameter values in Figure 9 are listed in Table 2.  
\[\mathbf{u}_{b}=\left[0,0,0,0,0,0\right]^{T}\]0 \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \(A_{x}\)(**kN**) & \(A_{y}\)(**\(\mathbf{\delta}\)**N**) & \(A_{z}\)(**k**N**) & \(\xi_{x}\) & \(\xi_{y}\) & \(\xi_{z}\) & \(\omega_{x}\) & \(\omega_{y}\) & \(\omega_{z}\) & \(x_{c1}^{1}\)(**mm**) & \(x_{c1}^{2}\)(**mm**) & \(x_{c2}^{2}\)(**mm**) \\ \hline
5 & 5 & 15 & 0.05 & 0.05 & 0.05 & 600\(\pi\) & 600\(\pi\) & 600\(\pi\) & 0.01 & 0.01 & 0.01 \\ \hline \(f_{\text{max}}(\mathbf{k}\mathbf{N})\) & \(f_{\text{max}}^{\text{T}}(\mathbf{k}\mathbf{N})\) & \(f_{\text{max}}^{\text{T}}(\mathbf{k}\mathbf{N})\) & \(L_{m}(\mathbf{k}\mathbf{g})\) & \(H_{m}(\mathbf{k}\mathbf{g})\) & \(w_{m}\) & \(L_{\omega}(\mathbf{Hz})\) & \(H_{\omega}(\mathbf{Hz})\) & \(\mathbf{w}_{\omega}\) & \(\mathbf{num}_{\epsilon}\) & \(\mathbf{num}_{\epsilon}\) & \(\mathbf{num}_{\epsilon}\) & \(\left[\sigma_{s}\right]\)(**MPa**) \\ \hline
25 & 10 & 15 & 30 & 90 & 2 & 30 & 300 & 1 & 20 & 30 & 200 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter values related to the landing force \(F_{f}\), the buffer force magnitude \(f_{im}\), the desirability approach, and the ESLs. Figure 9: The buffer force magnitude \(f_{im}\) with respect to buffer stroke \(x\) in (**a**) the main strut and (**b**) two secondary struts. The vibration equation can be derived from Equations (19) and (20). The dimension of \(M_{11}\) is 270. The time span was set from 0 to 0.02\(s\). The mass of the leg \(m\) and the average fundamental frequency in the workspace \(\omega\) were chosen as the individual objective functions.