# Optimized Design of Flexible Quick-Change System Based on Genetic Algorithm and Monte Carlo Method
## 2 Material and Methods  
### Layout Structure Design Optimization of a Flexible Quick-Change Process System  
Finding the ideal quantity, placement, and direction of zero point clamping units for each component in the assembly is the goal of layout optimization for flexible quick-change systems. In order to reach higher quality standards, accuracy must be the primary goal while also meeting the processing needs of many processes and multiple varieties. The layout of the zero point clamping unit will help reduce the impact of fixture system deformation. The forces on thin-walled cylindrical structural parts during turning and milling were examined in order to ensure that the flexible quick-change process system could be used with workpieces subject to various machining techniques. The maximum cutting force of 4200 N under extreme working conditions was chosen as the mechanical design basis for the flexible quick-change process system. In order to make all the workpieces applicable to the unified production line, as depicted in Figure 1, four kinds of initial layout are designed first for the distribution of zero point clamping units in the flexible quick-change process system, considering the geometric size range of the series of workpieces (from 280 mm to 1200 mm) and its structural characteristics. On the basis of determining the preliminary lattice layout structure (4 positioning pins on the outer ring and 3 positioning pins on the inner ring) according to mechanical characteristics analysis, the layout position of the zero point positioning unit is optimized using a genetic algorithm. Based on the experimental positioning accuracy results of two zero point positioning units with 400 mm spacing (repetitive positioning accuracy of 9 \(\upmu\)m), a multi-objective constrained optimization model for positioning and clamping layout optimization is established. The layout positions of each zero point clamping unit in the polar coordinate system are shown in Figure 2.  
Set the overall layout of the system to be symmetrical to reduce uneven deformation. The position of the positioning pins in the system layout is taken as the design variable, the minimum sum of the spacing between the positioning pins is taken as the objective function, and the maximum number of the distance between the internal points and the  
Figure 1: Initializing the layout. **(a)** Layout I (six outside and four inside). **(b)** Layout II (four outside and four inside). **(c)** Layout III (four outside and three inside). **(d)** Layout IV (three outside and three inside). Figure 2: Layout polar coordinates. external points is ensured to be less than 400 mm. The mathematical model of layout optimization can be expressed as:  
\[ \begin{cases}Find:\ x=[x_{1},x_{2},\cdots x_{n}]\\ \ r=[r_{1},r_{2},\cdots r_{n}]\\ \min S=\sum\limits_{i=A}^{C}\left(\sum\limits_{j=D}^{C}S_{ij}\right)\end{cases}  \tag{1}\]  
where x and i represent the position coordinates of the outer ring positioning pins in the polar coordinate system, and n represents the number of positioning pins. S represents the sum of the spacing between positioning pins (A to C and D to G represent the inner and outer ring positioning pins of the flexible quick-change system, respectively). In the polar coordinate system, the coordinates of each point are: A: \(\left(\frac{250\sqrt{3}}{3},\mathrm{x}\right)\); B: \(\left(\frac{250\sqrt{3}}{3},\mathrm{x}+\frac{2}{3}\pi\right)\); C: \(\left(\frac{250\sqrt{3}}{3},\mathrm{x}+\frac{4}{3}\pi\right)\); D: \(\left(400\sqrt{2},\frac{\pi}{4}\right)\); E: \(\left(400\sqrt{2},\frac{3\pi}{4}\right)\); F: \(\left(400\sqrt{2},\frac{5\pi}{4}\right)\); and G: \(\left(400\sqrt{2},\frac{7\pi}{4}\right)\). The formula for the distance between two points in the polar coordinate system is:  
\[\mathrm{S}=\sqrt{\mathrm{r_{1}}^{2}-2\mathrm{r_{1}}\mathrm{r_{2}}\cos(\theta _{1}-\theta_{2})+\mathrm{r_{2}}^{2}}. \tag{2}\]  
The sum of the distances between the points:  
\[