## 2 Single-Camera Trilateration Algorithm
### Error-Prone Regions in Performance Indices  
Figure 4, Figure 5, Figure 6, Figure 8, Figure 9, Figure 10, Figures 12-14 indicate that a sharp high peak consistently appears in \(\mathrm{B}_{\mathrm{p}}\), \(\mathrm{S}_{\mathrm{p}}\), \(\mathrm{B}_{\mathrm{\theta}}\) and \(\mathrm{S}_{\mathrm{\theta}}\) of \(\delta^{\mathcal{M}}\mathbf{f}\), \(\delta^{\mathcal{M}}\mathbf{q}_{i}\) and \(\delta^{\mathcal{M}}\mathbf{q}_{0}\). Each performance index increases to form a high peak as \(\theta\) increases from \(0^{\circ}\), and in general decreases as \(\theta\) further increases towards \(90^{\circ}\). (Though \(\mathrm{B}_{\mathrm{p}}(\delta^{\mathcal{M}}\mathbf{f})\) and \(\mathrm{B}_{\mathrm{p}}(\delta^{\mathcal{M}}\mathbf{q}_{0})\) have a second peak later on, it is not so significant as the first peak). The high peak stands out sharply from its neighborhood. On the other side, the crests and transitions in Figures 3, 7 and 11, which show the effect of landmark position errors on the performance indices, are relatively mild. This high peak is of particular interest. It reflects:  
1. The camera localization algorithm is more sensitive to the errors from the calibration of the camera parameters and segmentation of the landmarks from the image than the errors in positioning the landmarks in the environment. 2. The neighborhood of a high peak represents a range of camera orientations (relative to the base plane) from which the camera localization is highly sensitive to the errors in camera parameters and image segmentation, resulting in high estimation bias and uncertainty. Thus, improving the accuracy of camera calibration and image segmentation will help to improve the accuracy of camera localization. Moreover, knowing the values of \(\theta\) corresponding to the peaks will help to avoid those error-prone regions in practice. The simulation results reveal that:  
1. Given a specific geometric arrangement of the landmarks and a specific observation distance, the peaks of all these performance indices appear at the same \(\theta\) value, as shown in Figures 8-10;
2. The \(\theta\) value at which the peaks appear shifts as the observation distance changes, as shown in Figures 12-14. This means that the \(\theta\) value corresponding to the peaks in the above performance indices is mostly determined by the geometric arrangement of the landmarks and camera. Although a closed-form estimation of the peak \(\theta\) value is not available yet, a close approximation can be found through simulations. Specifically in our example where the three reference points define an equilateral triangle, the simulation results show that the peaks occur when the camera is located above the inscribing circle of the base triangle. That is, the \(\theta\) value corresponding to the peaks is well approximated as:  
\[\theta=\arcsin(\frac{r}{R}), \tag{26}\]  
where \(r\) denotes the base radius. This relationship has been verified by comparing the numerically estimated peak locations in these performance indices with \(\theta\) calculated from Equation (26) (Table 1). Since the \(\theta\) value at which the peaks appear is consistent across B\({}_{\text{p}}\), S\({}_{\text{p}}\), B\({}_{\theta}\) and S\({}_{\theta}\) of \(\delta^{m}\mathbf{f}\), \(\delta^{m}\mathbf{q}_{i}\) and \(\delta^{m}\mathbf{q}_{0}\), only B\({}_{\text{p}}\)(\(\delta^{m}\mathbf{f}\)) is reported here as a representative. Table 1 shows a high consistency between the numerically estimated \(\theta\) and that calculated from Equation (26). Since the high peaks in these performance indices mean high bias and uncertainty in camera pose estimation, tracking \(\theta\) at which the observation is made can alert the localization process the error-prone region in practice. The above discussion shows that the singular cases represented by the sharp high peaks existing in the plots of performance analysis are related to both the spatial relationship between the camera and landmarks and the usage of camera imaging as the input.