# Initial Structure Design and Optimization of an Automotive Remote Head-Up Display
## 4 Simulation Design and Results  
### Design Example  
In the design process, due to the irregularity of the windshield surface, the aberration introduced cannot be ignored, so it is necessary to fit the windshield surface in advance. The free-form surface expression XY extended polynomial in Zemax is used to fit the front windshield of the automobile. The XY extended polynomial is as follows:  
\[z=\frac{\rho r^{2}}{1+\sqrt{1-(1+k)\rho^{2}r^{2}}}+\sum\limits_{i}A_{i}x^{m}y^{ n} \tag{13}\]  
where \(r^{2}=x^{2}+y^{2}\), \(i=\frac{(m+n)^{2}+m+3n}{2+1}+1\), and \(z\) represents the vector height difference in the direction of the \(z\) axis, \(\rho\) represents the curvature, \(k\) represents the conical surface constant, \(A_{i}\) represents the polynomial coefficient, and \(m\) and \(n\) represent the corresponding \(x\) and \(y\) order, respectively. Figure 7: Constraints of the occluded structure. The windshield point cloud data of a certain vehicle is extracted by LiDAR, and then the obtained data are fitted in Matlab using Formula (13). The windshield data after fitting is shown in Table 2.  
After calculating the expression of the windshield, the initial structure is reverse-built according to the method proposed in Section 3.1. Considering the difficulty and processing requirements of the later processing of the free-form surface, the XY extended polynomial with stronger processability is used to design the two free-form mirrors. In the design process, in order to control aberration, only the even terms of the folded mirror M\({}_{1}\) are set as variables to participate in optimization to reduce the complexity of the surface. The maximum degree of the main mirror M\({}_{2}\) and the folded mirror M\({}_{1}\) is set to 14 terms, that is, the maximum degree of the polynomial \(x+y\) is 4. After the system is built, the M\({}_{1}\) and M\({}_{2}\) mirror curvature radius, cone coefficient, free-form surface type, and so on are optimized as variables. If the position and structure of the surface shape are not controlled during optimization, the structural occlusion problem, as shown in Figure 8a, may occur because the surface is very flexible. In Zemax, the occlusion problem of the structure is not easily discoverable because the occlusion problem of light is not considered. By using the optimization function to limit the edge position of the structure, the occlusion control effect, as shown in Figure 8b, can be achieved. In the process of optimization, it is necessary to always pay attention to the optical trace diagram and optical path structure of each surface. When it is found that the light deviation is too large, it is necessary to fine-tune the deflection angle and eccentricity of the free-form surface and correct the deviation caused by calculation errors and other factors in the initial structural design. However, the adjustment amount should not be too large; otherwise, the structure may be invalid. In the optimization process, the number of variables is gradually increased, or a part of the variables is fixed, and the result is gradually optimized until the structure and image quality meet the design requirements. After simple optimization, the partial structure diagram of the AR-HUD is shown in Figure 9 below. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Radius** **(mm)** & **Conic** & **(X\({}_{1}\)Y\({}_{0}\))** & **(X\({}_{0}\)Y\({}_{1}\))** & **(X\({}_{2}\)Y\({}_{0}\))** & **(X\({}_{1}\)Y\({}_{1}\))** & **(X\({}_{0}\)Y\({}_{2}\))** \\ \hline \(-7526.66\) & \(-2.789\) & \(2.308\) & \(-3.569\) & \(-7.601\) & \(2.430\) & \(-3.564\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Windshield fitting results. Figure 8: Differences between before and after the addition of structural constraints: (**a**) optimization without restrictions; (**b**) optimization with restrictions. According to the system data, the visual distance of the virtual image of the optical path map is 7.5 m, and the imaging size of the virtual image is 1312.67 mm \(\times\) 654.87 mm.