# Initial Structure Design and Optimization of an Automotive Remote Head-Up Display
## 3 Proposed Methods  
### Initial Structure Design Based on the Rodrigues Rotation Formula  
At present, the initial structural design of HUD is basically aimed at the short-distance display system, that is, the optical system with a virtual visual range of between 2 and 2.5 m. And there is no detailed design scheme for the long-distance projection AR-HUD. In this paper, based on the off-axis triple inverse structure, the following AR-HUD initial structure design method is proposed, combined with the HUD installation position and prototype volume constraints. According to the internal structure size of the instrument panel, the maximum volume of the automotive HUD that can be installed is about 15 L. In the initial design, in order to prevent the occlusion of the lens in the later stage as much as possible, the reflector space should be extended as far as possible with a certain installation gap. The initial placement position is roughly shown in Figure 4. The spacing between the components can be measured according to the installation position. When designing the initial structure of the HUD, it usually starts from the reverse, that is, the virtual image plane is regarded as the object plane, which is conducive to the construction of the initial light path. In order to facilitate the initial structural design, the source image, reflectors M\({}_{1}\) and M\({}_{2}\), and windshield are set in the same meridian plane as the main ray intersection. The optical path structure shown in Figure 5 can be obtained by pursuing the main ray. In the figure, the vertical axis always corresponds to the y axis. \begin{table}
\begin{tabular}{c c} \hline \hline
**HUD System Parameter** & **Value** \\ \hline Wavelength (mm) & 486–656 (visible light) \\ Virtual image distance (m) & 7.5 \\ Angle of field (\({}^{\circ}\)) & 10 \(\times\) 5 \\ Eyebox (mm) & 130 \(\times\) 50 \\ Source image size (mm) & 92.08 \(\times\) 42.04 \\ Virtual image space MTF & \[EMAIL_ADDRESS] lp/mm \\ Distortion (\%) & \textless{}2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: AR-HUD system parameter settings. Figure 4: HUD prototype. In Figure 5, \(S_{0},S_{1},S_{2},S_{3},S_{4}\) are intersection points of the main ray with the eyebox, windshield, free-form surfaces M1 and M2, and source image surface, respectively. The tilt angle of the windshield surface can be measured in advance, and the direction vector \(\overset{\rightarrow}{N}\) can be calculated from the surface profile data. The reflection angle \(\alpha\) of the surface can be calculated as follows:  
\[\alpha=\arccos(\frac{\overset{\rightarrow}{S_{1}S_{0}}}{\left|\overset{ \rightarrow}{S_{1}S_{0}}\right|}\cdot\frac{\overset{\rightarrow}{N}}{\left| \overset{\rightarrow}{N}\right|}). \tag{4}\]  
Given that the angle of reflection \(\alpha\) is equal to the angle of incidence, the reflected light \(\overset{\rightarrow}{S_{1}S_{2}}\) can be calculated as  
\[\frac{\overset{\rightarrow}{S_{1}S_{2}}}{\left|\overset{\rightarrow}{S_{1}S_ {2}}\right|}=2\cdot\frac{\overset{\rightarrow}{N}}{\left|\overset{ \rightarrow}{N}\right|}\cdot\cos\alpha-\frac{\overset{\rightarrow}{S_{1}S_{0 }}}{\left|\overset{\rightarrow}{S_{1}S_{0}}\right|}. \tag{5}\]  
According to the HUD installation position measurement before the design, \(\left|\overset{\rightarrow}{S_{1}S_{2}}\right|\) is about 230 mm, and then the vector \(\overset{\rightarrow}{S_{1}S_{2}}\) can be calculated. According to the installation spacing of each surface in Figure 4, the reflection angle and reflected light vector parameter coordinates \(\overset{\rightarrow}{S_{2}S_{3}}\), \(\overset{\rightarrow}{S_{3}S_{4}}\) of the surface \(S_{3},S_{4}\) can be calculated. After calculating the reflection vector of each surface, it is necessary to calculate the rotation angle and tilt angle of each surface, including the three \(x\), \(y\), and \(z\) axis directions, which can help to calculate the rotation angles.