# 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
#### 2.2.2 Error Model Caused by Components except the Zero Point Clamping Unit  
In addition to the zero point clamping unit, only the fixture plate and the foundation plate remain in the flexible quick-change process system, whose position error and posture error are shown schematically in Figure 5. Two adjacent mounting holes for the zero point clamping unit and mounting holes for positioning pins are spaced apart by 0.015 mm on the outer rings of the foundation plate and the fixture plate, respectively. The foundation plate and fixture plate have a surface flatness tolerance of 0.015 mm. When both position tolerances exist as \(\pm\)M and \(\pm\)S, the error \(\mathrm{\Delta M}\) will result in the existence of errors on the X- and Y-axes of \(\pm\)MX and \(\pm\)MY for the fixture plate, whereas \(\mathrm{\Delta S}\) will cause the errors between the positioning pins on the X- and Y-axes to exist as \(\pm\)SX and \(\pm\)SY. As the positioning pin mounting holes on the outer ring of the foundation plate and fixture plate, as well as the two neighboring zero point clamping units, are all shifted in the same direction, the remaining mounting holes are shifted in the opposite direction. The rotational angle errors \(\mathrm{\theta}_{\mathrm{MW}}=\mathrm{\Delta M}/\mathrm{L}_{\mathrm{w}}\) and \(\mathrm{\theta}_{\mathrm{SW}}=\mathrm{\Delta S}/\mathrm{L}_{\mathrm{w}}\) around the Z-axis will occur. In the meantime, the three holes of the inner circle of the foundation plate and the fixture plate rotate along the circumference at the same time. The errors of its rotation angle around the Z-axis are \(\mathrm{\theta}_{\mathrm{MN}}=\mathrm{\Delta M}/\mathrm{L}_{\mathrm{N}}\) and \(\mathrm{\theta}_{\mathrm{SN}}=\mathrm{\Delta S}/\mathrm{L}_{\mathrm{N}}\), because \(\mathrm{L}_{\mathrm{N}}>\mathrm{L}_{\mathrm{W}}\), \(\mathrm{L}_{\mathrm{W}}>\mathrm{L}_{\mathrm{N}}\), then \(\mathrm{\theta}_{\mathrm{MW}}>\mathrm{\theta}_{\mathrm{MN}}\) and \(\mathrm{\theta}_{\mathrm{SW}}<\mathrm{\theta}_{\mathrm{SN}}\), which also indicates that the maximum value of rotational error can only be \(\mathrm{\theta}_{\mathrm{MW}}\) and \(\mathrm{\theta}_{\mathrm{SN}}\). Suppose there are \(\mathrm{\Delta R}_{\mathrm{w}}\in\mathrm{R}^{3\times 1}\) and \(\mathrm{\Delta\theta}_{\mathrm{w}}\in\mathrm{R}^{3\times 1}\) deviations in the workpiece position and attitude on the flexible quick-change process system. Then the posture deviation of the workpiece is \(\mathrm{q}_{\mathrm{w}}=[\mathrm{\Delta R}_{\mathrm{w}},\mathrm{\Delta\theta }_{\mathrm{w}}]^{\mathrm{T}}\). Consequently, the error formula of the flexible quick-change process system can be obtained by adding the error model of the zero location element with the error module algebra caused by the foundation plate and fixture plate. \[\mathrm{q}_{\mathrm{w}}=[\mathrm{\Delta R}_{\mathrm{w}},\mathrm{\Delta\theta }_{\mathrm{w}}]^{\mathrm{T}}=\left[(\delta+\mathrm{M}+\mathrm{S})\mathrm{cos} \,\alpha,\ (\delta+\mathrm{M}+\mathrm{S})\mathrm{sin}\,\alpha,\ \frac{\delta}{2}+\mathrm{P}_{ \mathrm{M}}+\mathrm{P}_{Z},\ \frac{\delta}{\mathrm{L}_{\mathrm{w}}},\ \frac{\delta}{ \mathrm{L}_{\mathrm{w}}},\ \frac{\pm\mathrm{M}}{\mathrm{L}_{\mathrm{w}}}+\frac{\pm\mathrm{S}}{ \mathrm{L}_{\mathrm{w}}}\right]^{\mathrm{T}} \tag{10}\]  
where \(\mathrm{q}_{\mathrm{w}}\), \(\mathrm{\Delta R}_{\mathrm{w}}\), and \(\mathrm{\Delta\theta}_{\mathrm{w}}\) indicate the overall error of the flexible quick-change process system workpiece, position error, and attitude error; M indicates the position tolerance of two adjacent positioning pin mounting holes on the outer ring of the fixture plate; S indicates the flatness tolerance of the surface of the base plate and fixture plate; \(\mathrm{P}_{\mathrm{M}}\) and \(\mathrm{P}_{Z}\) indicate the Z-directional error brought about by the flatness tolerance of the surface of the base plate and fixture plate; and \(\mathrm{L}_{\mathrm{w}}\) indicates half of the distance between the zero point positioning pin and the one-way positioning pin of the outer ring of the fixture plate.