## 3 Multi-Objective Optimization of System Construction
### Characteristics of System Construction and Its Mathematical Description
The meanings of each parameter are shown in Table 1.  
It is assumed that the installation surface of the large cabin body to be processed is square, and the side length is \(l_{\rm K}\), and the installation surfaces are evenly distributed on the outer surface of the cabin body with the spacing \(h\). \(L_{\rm S}\) is the length of the large cabin body, \(D\) is the diameter of the large cabin body (as shown Figure 3), \(l_{\rm R}\) is the length of the working space side of the manipulator (regarded as a cube, and \(l_{\rm R}/l_{\rm K}>1\)). In general, positioning errors of the manipulator are related to the working space, and the larger the working space is, the larger the positioning error will be. Therefore, the manipulator's working space is taken as the constraint, and the large cabin body is divided along the axial and radial directions. The segmentation method is as follows:  
\[Row=\frac{L_{\rm S}}{l_{\rm R}},Col~{}=\frac{D\pi}{l_{\rm R}} \tag{2}\]  
The following parameter calculation methods can be obtained:  
\[ \begin{split} T_{\rm O}&=Col\times Row\times\frac {l_{\rm R}}{v_{\rm O}}\\ T_{\rm R}&=Col\times Row\times((n+2)\times n\times \frac{h}{v_{\rm R}}),\\ T_{\rm V}&=Col\times Row\times M\\ T_{\rm E}&=Col\times Row\times(n+1)^{2}\times\frac {l_{\rm K}}{d_{\rm K}}\times\frac{l_{\rm K}}{v_{\rm E}}\\ T_{\rm D}&=Col\times\frac{l_{\rm R}}{v_{\rm D}} \end{split}  \tag{3}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Meaning** \\ \hline \(P_{\rm R}\) & the positioning error of the manipulator \\ \(P_{\rm V}\) & the measurement error of the visual measurement system \\ \(P_{\rm E}\) & machining error of end effector in milling \\ \(P_{\rm D}\) & the rotation error of the flexible position changing system \\ \(C_{\rm O}\) & the market price (cost) of an omnidirectional mobile platform \\ \(C_{\rm R}\) & the market price of the mechanical arm \\ \(C_{\rm V}\) & the market price of a visual measurement system \\ \(C_{\rm E}\) & the market price of milling end-effector \\ \(C_{\rm D}\) & the market price of the flexible position changing system \\ \(T_{\rm O}\) & the time when the omnidirectional mobile platform participates in the milling process \\ \(T_{\rm R}\) & the time for the manipulator to participate in the milling process \\ \(T_{\rm V}\) & the time when the visual measurement system participates in the milling process \\ \(T_{\rm E}\) & the time that the end effector participates in the milling process \\ \(T_{\rm E}\) & the time that the flexible position changing system participates in the milling process \\ \(T_{\rm D}\) & the time that the flexible position changing system participates in the milling process \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The meaning of each parameter. Figure 3: Parameter meanings. In Equation (3), \(v_{\rm O}\) represents the movement speed of the omnidirectional mobile platform; \(v_{\rm R}\) represents the speed at which the manipulator moves; \(M\) is a constant, representing the time of the target point of visual measurement; \(d_{\rm K}\) stands for milling spindle tool diameter; \(v_{\rm E}\) represents the feed speed of the main shaft; \(D\) is the diameter of the large cabin; \(v_{\rm D}\) represents the moving speed of the flexible positioning system, and \(n\) is calculated by the following pseudo-code, \(n\) = 0, 1, 2,... :  
\[n=0;\] \[Do\{\] \[if(\frac{f_{\rm R}}{n\times M+f_{\rm K}}<1)\ \{\] \[break;\] \[\}\] \[n=n+1;\] \[\}while({\rm true});\]  
Based on the movement speed of the omnidirectional mobile platform, \(T_{\rm O}\) is calculated. In the manipulator working space, the surface milling of multiple mounting surfaces is carried out. As the workspace becomes larger or smaller, the number of mounting surfaces to be milled will increase or decrease. \(T_{\rm R}\) is calculated by combining with the manipulator moving speed.