# Optimized Design of Flexible Quick-Change System Based on Genetic Algorithm and Monte Carlo Method
## 3 Results and Discussion
### Mechanical Properties of the Different Zero Point Positioning Unit Layouts
\begin{bmatrix}(-0.02649,0.02559)\mathrm{mm} \\ (-0.02741,0.02479)\mathrm{mm}\\ (-0.03237,0.03225)\mathrm{mm}\\ (-1.334\times 10^{-5},1.533\times 10^{-5})\mathrm{rad}\\ (-1.293\times 10^{-5},1.485\times 10^{-5})\mathrm{rad}\\ (-5.489\times 10^{-5},5.381\times 10^{-5})\mathrm{rad}\end{bmatrix} . \tag{11}\]  
The displacement deviations \(\Delta\mathrm{x}\), \(\Delta\mathrm{y}\), and \(\Delta\mathrm{z}\) in three directions and the rotation angle deviations \(\Delta\mathrm{x}\), \(\Delta\mathrm{y}\), and \(\Delta\mathrm{z}\) around three axes are obtained individually by using the Monte Carlo model to carry out random sampling of the error source parameter variables and substituting them into the MATLAB R 2019 b error model for calculation, as the error distribution is shown in Figure 8.  
From the figure, we can observe that the posture deviation distribution of the six components of the workpiece is extremely similar, all of which obey normal distribution in a certain range, whereas the influence degree of each error source on the posture deviation of the workpiece is different. In the deviation of workpiece posture, the rotation errors \(\Delta\alpha\) and \(\Delta\beta\) of the workpiece around the X- and Y-axes are relatively close, with a smaller error range, while the rotation angle error \(\Delta\gamma\) around the Z-axis is more widely distributed, meaning that the limit error values that occur are greater, but all meet the technical requirements. In the displacement deviation, the distribution range of \(\Delta\mathrm{x}\) and \(\Delta\mathrm{y}\) errors is similar, while the distribution range of \(\Delta\mathrm{z}\) errors is larger because the dowel pins limit the degrees of freedom of motion in the X- and Y-directions. By analyzing the reasons, we can see that the distance between the installation holes of the inner ring and the outer ring of the fixture plate and the foundation plate is basically the same in the X- and Y-directions, while the positioning block will have a height difference on the Z-axis, so the error distribution on the Z-axis is relatively dispersed and the error distribution is large. In addition, it also shows that each error source has different degrees  
Figure 7: Optimized layout by genetic algorithm. of influence on the displacement deviation of the workpiece in the three directions, and the Z-axis error is more sensitive. According to the technical requirements of the flexible quick-change process system, all the error values should not be greater than 0.01 mm; hence, the axial error of the Z-axis failed to meet the technical demands. For that reason, it is necessary to conduct further optimization design for the foundation plate and fixture plate. Generally, the foundation plate is fixed to the machine tool; it does not need to be removed. The fixture plate must regularly be lifted, moved, and subjected to other operations while being directly attached to the fixture and the workpiece, which would reduce its accuracy. As a result, allow the Z-axis axial error of the flexible quick-change process system to be resampled with Monte Carlo sampling while allowing the design tolerance of the flatness of the fixture plate to be decreased to 0.01 mm. The outcomes are displayed in Figure 9.  
After the flatness tolerance design of the fixture plate is optimized, the axial error of the Z-axis is [\(-\)0.02638 mm, 0.02453 mm]. At this point, the error value meets the technical requirements. Therefore, when designing the positioning element of the fixture, the influence of the deviation of each positioning element on the position error should be considered, and the sensitivity of each error source to the overall position and pose error of the workpiece should be analyzed. For the error source with higher sensitivity, the  
Figure 8: Error distribution diagram. (**a**) X-direction displacement error. (**b**) Y-direction displacement error. (**c**) Z-direction displacement error; (**d**) X-axis rotational error. (**e**) Y-axis rotational error. (**f**) Z-axis rotational error.