## 5 Simulation Setup
### Camera Operational Space Development  
As in Figure 16, set the camera lens so that it is always parallel to the face of the marker and the baseline is parallel to the \(\overline{X}_{V}\) axis. In this case, the lens is kept facing downward when detecting the marker on the ground. As discussed in the above section, the detectable space of the camera is modeled as a cone. This cone intersects with the horizontal plane and forms an oval. To have the camera detect the marker, the marker must be contained inside the oval. Figure 16: The detectable elliptic cone in the inertial visual system frame. An inertia coordinate system \(\overline{O}_{V}\overline{X}_{V}\overline{Y}_{V}\overline{Z}_{V}\) is set up with its origin seated at the bottom center of the visual system. Make point C the location of the camera center, and its coordinates in the inertial frame are \((X_{C}^{V},Y_{C}^{V},Z_{C}^{V})\). \(Q\) is the vertex of the detectable elliptic cone, and \(Q^{\prime}\) is the projection of \(Q\) on the horizontal plane. \(M\) is the marker. The elliptic cone of the camera's detectable space always intersects with the horizontal plane. Consider an extreme case when point \(C\) is located at the apex of the visual system workspace, as shown in Figure 16. Then, the \(Z_{C}^{V}\) coordinate of \(C\) at the inertial frame is:  
\[Z_{C}^{V}\{max\}=L_{1}+r_{V}=2.211\ \mathrm{m}\ \leq\ Z_{max} \tag{68}\]  
where \(Z_{max}\), defined above, is the maximum depth of the camera to detect the markers. Therefore, the cone intersects with the horizontal plane when the camera center \(C\) is placed at any location of the workspace's contour. From the expression of Cartesian coordinates in the inertia frame:  
\[a_{Q^{\prime}}=(Z_{C}^{V}-d)tan(\frac{\alpha}{2}),\ b_{Q^{\prime}}=(Z_{C}^{V}-d )tan(\frac{\beta}{2}) \tag{69}\]  
where \(d\) is the same offset as in Equation (62). \(a_{Q^{\prime}}\) and \(b_{Q^{\prime}}\) are the major radius and the minor radius of the oval. Also, the coordinates of points \(Q^{\prime}\) and \(M\) are \((X_{C}^{V},Y_{C}^{V},\ 0)\) and \((L_{M},0,\ 0)\). \(Q^{\prime}\) is the center of the oval. To ensure the marker \(M\) is inside the projected oval, it is required that:  
\[\frac{\left(X_{C}^{V}-L_{M}\right)^{2}}{{a_{Q^{\prime}}}^{2}}+\frac{\left(Y_{C} ^{V}\right)^{2}}{{b_{Q^{\prime}}}^{2}}\leq 1 \tag{70}\]  
Plug Equation (69) into Equation (70):  
\[\frac{\left(X_{C}^{V}-L_{M}\right)^{2}}{tan(\frac{\alpha}{2})^{2}}+\frac{\left( Y_{C}^{V}\right)^{2}}{tan(\frac{\beta}{2})^{2}}\leq\left(Z_{C}^{V}-d \right)^{2} \tag{71}\]  
Inequality (71) is exactly a mathematic expression of all points \(C(X_{C}^{V},Y_{C}^{V},Z_{C}^{V})\) within a cone whose opening is in the positive \(Z\) direction with its vertex at \((L_{M},0,d)\) and the opening parameters as \(\tan(\frac{\alpha}{2})=0.934\) and \(\tan(\frac{\beta}{2})=0.524\). Therefore, this inequality defines the second geometric constraint of the camera center. It forms an elliptical cone with its vertex located in Figure 17 at point E, which is the offset point of the marker \(M\) from the ground. As \(C(X_{C}^{V},Y_{C}^{V},Z_{C}^{V})\) should also be within the sphere of the workspace (abiding by the first geometric constraint), the camera operational space is presented as the overlap between the cone and the sphere (the shaded area in Figure 17). Figure 17: Illustration of camera operational space (the shaded area). From Figure 17, the marker should not be too far from the visual system, otherwise there is no overlap region between the cone and the sphere. The largest value of \(L_{M}\), which is the distance from the marker to the coordinate center, occurs when the cone is tangent to the sphere, as shown in Figure 18. To find the limit of \(L_{M}\), draw a horizontal line across point \(O_{V}\) so that it intersects the surface of the cone at point A and the center line of the cone at point B, as shown in Figure 18. Let \(\overline{O_{V}A}\) and \(\overline{AB}\) be the line segment lengths.