## 5 Simulation Setup
### Mathematic Expression for Node Coordinates within the Camera Operational Space  
From Equation (73), the closer the marker's distance \(L_{M}\) to its lower bound, the larger the operational space. In general, a larger operational space is preferred because the camera has a larger space for searching the optimal location. Set \(L_{M}\) to equal its lower bound:  
\[L_{M}=2.83\text{ m} \tag{74}\]  
Get the cross-section area at the \(\overline{X}_{V}-\overline{Z}_{V}\) plane (Figure 19). Figure 19: The camera operational space cross-section at \(\overline{X}_{V}-\overline{Z}_{V}\) plane. The function of the semi-circle and the line EFG is:  
\[\left(\overline{Z}_{V}-L_{1}\right)^{2}+\left(\overline{X}_{V}-a_{1}\right)^{2}=r _{V}{}^{2} \tag{75}\]  
\[\overline{X}_{V}=-tan(\frac{\alpha}{2})(\overline{Z}_{V}-d)+L_{M} \tag{76}\]  
Combine Equations (75) and (76) and solve:  
\[ \begin{array}{c}(1+tan(\frac{\alpha}{2})^{2})\overline{Z}_{V}{}^{2}-2(tan( \frac{\alpha}{2})^{2}d+tan(\frac{\alpha}{2})(L_{M}-a_{1})+L_{1})\overline{Z}_{ V}+\tan(\frac{\alpha}{2})^{2}d^{2}+\\ 2\tan(\frac{\alpha}{2})(L_{M}-a_{1})d+(L_{M}-a_{1})^{2}-r_{V}{}^{2}+L_{1}{}^{2} =0\end{array}  \tag{77}\]  
Solve Equation (77) by plugging numbers:  
\[1.872\ \overline{Z}_{V}{}^{2}-6.061\overline{Z}_{V}+4.670=0 \tag{78}\]  
\[\text{Solve:}\ \ \overline{Z}_{V}{}_{max},\overline{Z}_{V}{}_{min}=1.972\ \text{m},1.265\ \text{m} \tag{79}\]  
\(\overline{Z}_{V}{}_{max}\) and \(\overline{Z}_{V}{}_{min}\) are the largest and smallest \(\overline{Z}_{V}\) coordinates of the camera center \(C\) that is within the camera operation space. Take a value of \(Z_{C}^{V}\) (\(\overline{Z}_{V}\) coordinates of the camera center \(C\)) in the range (\(\overline{Z}_{V}{}_{min}\), \(\overline{Z}_{V}{}_{max}\)) = (1.265 m, 1.972 m). Set a plane \(\overline{Z}_{V}=Z_{C}^{V}\), and that plane intersects the camera operational space and forms a shade area, which is the overlap between the circle and the oval, as shown in Figure 14. The inequality governing points \(C\) within the sphere at a specific \(Z_{C}^{V}\) is:  
\[\left(\overline{X}_{V}-a_{1}\right)^{2}+\overline{Y}_{V}{}^{2}\leq r_{V}{}^{2 }-\left(Z_{C}^{V}-L_{1}\right)^{2} \tag{80}\]  
The inequality governing points \(C\) within the cone at \(Z_{C}^{V}\) is:  
\[\frac{\left(\overline{X}_{V}-L_{M}\right)^{2}}{\tan(\frac{\alpha}{2})^{2}}+ \frac{\left(\overline{Y}_{V}\right)^{2}}{\tan(\frac{\beta}{2})^{2}}\leq\left( Z_{C}^{V}-\text{d}\right)^{2} \tag{81}\]  
By plotting those inequalities in Figure 20, the shaded area is where \(X_{C}^{V}\) and \(Y_{C}^{V}\) (\(\overline{X}_{V}\) and \(\overline{Y}_{V}\) coordinates of the camera center \(C\)) should be located within. The intersection of the two curves occurs when equality holds for (80) and (81):  
\[\left(\overline{X}_{V}-a_{1}\right)^{2}+\overline{Y}_{V}{}^{2}=r_{V}{}^{2}- \left(Z_{C}^{V}-L_{1}\right)^{2} \tag{82}\]  
\[\frac{\left(\overline{X}_{V}-L_{M}\right)^{2}}{\tan(\frac{\alpha}{2})^{2}}+ \frac{\left(\overline{Y}_{V}\right)^{2}}{\tan(\frac{\beta}{2})^{2}}=\left(Z_{C }^{V}-d\right)^{2} \tag{83}\]  
Figure 20: The camera operational space cross-section at \(\overline{X}_{V}-\overline{Y}_{V}\) plane. Solve by plugging numbers:  
\[(\overline{Y}_{Vmax},\ \overline{Y}_{Vmin})=\pm 0.5*\sqrt{3.44*Z_{\rm C}^{V2}-2.0241*Z_{ \rm C}^{V}-29.802+2.451*\sqrt{-29.475*Z_{\rm C}^{V2}+23.707*Z_{\rm C}^{V}+137.28}} \tag{84}\]  
\(\overline{Y}_{Vmax}\) and \(\overline{Y}_{Vmin}\) are the largest and the smallest \(\overline{Y}_{V}\) coordinates of the camera center \(C\) in the camera operation space with specific \(Z_{\rm C}^{V}\).