## 3 Multi-Objective Optimization of System Construction  
### Characteristics of System Construction and Its Mathematical Description  
In order to optimize the construction of the mobile robot system, the objective function optimized by the construction system should be determined first, which means, the evaluation criteria of the optimal system construction should be determined, and the mathematical model of the optimal system construction method should be established. The omnidirectional mobile platform, mechanical arm, flexible position changing system have positioning errors, transmission errors and clearance errors. The visual measurement system has measurement errors and random errors caused by environmental factors. In the actual system construction process, the above errors will lead to different final accuracies guaranteed by the constructed system. Generally, the higher the cost of equipment or system is, the higher the accuracy is, this is because of the high cost of precision improvement methods and hardware. Different milling processes (different station planning, different precision allocation, etc.) will lead to a different system running time and efficiency. Therefore, the three factors of system accuracy, cost and time consumption will be coordinated and selected according to actual requirements. Based on the above analysis and actual engineering application requirements, the characteristics required for optimal system construction can be defined as follows:  
1. The error accumulation of the constructed mobile robot system is minimal;
2. The total cost of the mobile robot system is the lowest;
3. The milling time of all the large cabin faces of the constructed mobile robot system is the shortest. The above description form cannot be used for quantitative analysis of the construction process of the optimal system of mobile robots by mathematical operations, so it is necessary to use mathematical language to express and optimize the above description. The mathematical model of the optimal system construction can be defined as:  
\[ \begin{array}{l}min\ f_{1}=P_{\rm R}+P_{\rm V}+P_{\rm E}+P_{\rm D}\\ min\ f_{2}=C_{\rm O}+C_{\rm R}+C_{\rm V}+C_{\rm E}+C_{\rm D}\\ min\ f_{3}=T_{\rm O}+T_{\rm R}+T_{\rm V}+T_{\rm E}+T_{\rm D}\end{array}  \tag{1}\]  
In Equation (1), The precision \(P\) here mainly refers to the precision of the product after the mobile robot milling system is moved to the specified position and processed. The precision is affected by the positioning error of the manipulator, the machining error of the milling end actuator, the measurement error of the visual measurement system, and the positioning error of the flexible displacement system. It is the superposition of the above link errors, \(P>0\). The cost \(C\) here mainly refers to the hardware cost of constructing the mobile robot milling system, which is the superposition of the hardware costs of the omni-directional mobile platform, the manipulator, the milling end actuator, the visual measurement system, and the flexible displacement system, \(C>0\). The efficiency \(T\) here mainly refers to the time required for the mobile robot milling system to process from moving to the specified position until the processing is completed. This time includes the moving time of the omni-directional mobile platform, the movement time of the manipulator, the processing time of the milling end actuator, the measurement time of the visual measurement system, and the movement time of the flexible displacement system. The superposition of the above link time is the efficiency, \(T>0\). Since the positioning error of the omnidirectional mobile platform was corrected through the measurement of local target points by the visual measurement system, the positioning error \(P_{\rm O}\) of the omnidirectional mobile platform was subtracted from the final milling error \(f_{1}\). The total cost of the mobile robot system \(f_{2}\) is the sum of the market price of all components and \(f_{3}\) is the sum of the time spent on all milling processes.