## 2 Single-Camera Trilateration Algorithm
### Trends of Performance Indices under Representative Input Error Standard Deviations  
To evaluate the impact of \(\delta^{S}\mathbf{p}_{i}\), \(\delta^{m}\mathbf{q}_{i}\), \(\delta^{m}\mathbf{f}\) and \(\delta^{m}\mathbf{q}_{0}\) on \(\delta^{S}\mathbf{p}_{0}\) and \(\delta^{S}\mathbf{\theta}_{c}\), \(\mathrm{B}_{\mathrm{P}}\), \(\mathrm{S}_{\mathrm{P}}\), \(\mathrm{B}_{\theta}\) and \(\mathrm{S}_{\theta}\) are at first estimated with representative input error standard deviations across the data acquisition sphere S with R = 3000. Specifically, \(\mathrm{B}_{\mathrm{P}}(\delta^{S}\mathbf{p}_{i})\), \(\mathrm{S}_{\mathrm{P}}(\delta^{S}\mathbf{p}_{i})\), \(\mathrm{B}_{\theta}(\delta^{S}\mathbf{p}_{i})\) and \(\mathrm{S}_{\theta}(\delta^{S}\mathbf{p}_{i})\) are estimated with \(\sigma_{\mathrm{P}}\) = 25 (Figure 3); \(\mathrm{B}_{\mathrm{P}}(\delta^{m}\mathbf{f})\), \(\mathrm{S}_{\mathrm{P}}(\delta^{m}\mathbf{f})\), \(\mathrm{B}_{\theta}(\delta^{m}\mathbf{f})\) and \(\mathrm{S}_{\theta}(\delta^{m}\mathbf{f})\) are estimated with \(\sigma_{f}\) = 0.4 (Figure 4); \(\mathrm{B}_{\mathrm{P}}(\delta^{m}\mathbf{q}_{0})\), \(\mathrm{S}_{\mathrm{P}}(\delta^{m}\mathbf{q}_{0})\), \(\mathrm{B}_{\theta}(\delta^{m}\mathbf{q}_{0})\) and \(\mathrm{S}_{\theta}(\delta^{m}\mathbf{q}_{0})\) are estimated with \(\sigma_{c}\) = 0.6 (Figure 5); and \(\mathrm{B}_{\mathrm{P}}(\delta^{m}\mathbf{q}_{i})\), \(\mathrm{S}_{\mathrm{P}}(\delta^{m}\mathbf{q}_{i})\), \(\mathrm{B}_{\theta}(\delta^{m}\mathbf{q}_{i})\) and \(\mathrm{S}_{\theta}(\delta^{m}\mathbf{q}_{i})\) are estimated with \(\sigma_{\mathrm{q}}\) = 0.5 (Figure 6). Since S is vertically symmetrical to the base plane (XY plane) and horizontally symmetrical to the vertical planes containing the medians of the equilateral base triangle, we only need to display the variation of \(\mathrm{B}_{\mathrm{P}}\), \(\mathrm{S}_{\mathrm{P}}\), \(\mathrm{B}_{\theta}\) and \(\mathrm{S}_{\theta}\) across a patch constrained by \(\theta\)\(\in\)[0\({}^{\circ}\), 90\({}^{\circ}\)] and \(\phi\)\(\in\)[0\({}^{\circ}\), 60\({}^{\circ}\)], while the values of these performance indices at any point on S outside the patch can be obtained according to the symmetry. Each sub-figure in Figures 3-6 presents the variation of a performance index, under a specific input error standard deviation, with respect to the observation direction (defined by the tilt angle \(\theta\) and pan angle \(\phi\)) at a specific distance. Each sub-figure includes one surface plot and one contour plot, both generated using Matlab. The surface plot presents a 3D surface which visualizes the values of the corresponding performance index as heights and colors above a horizontal grid defined by \(\theta\) and \(\phi\). The general trend of the performance index's variation is reflected by the variation in the height and color of the surface. Meanwhile, the contour plot presents a set of isolines (contour lines) at different value levels of the corresponding performance index which varies across \(\theta\) and \(\phi\). The contour lines are labeled with the associated values of the performance index in order to provide more numerical details of the variation. To match with the aforementioned data acquisition patch on S, the \(\theta\) - \(\phi\) grid in each plot is presented in the form of a 2D sector, where the tilt angle of the observation direction \(\theta\) varies in the radial direction of the sector from 0\({}^{\circ}\) to 90\({}^{\circ}\) and the pan angle \(\phi\) varies along the arc from 0\({}^{\circ}\) to 60\({}^{\circ}\). For the effect of the landmark position error \(\delta^{g}\mathbf{p}_{i}\):  
1. As shown in Figure 3a, \(\mathrm{B}_{\mathrm{P}}(\delta^{g}\mathbf{p}_{i})\) in general decreases as \(\theta\) increases from 0\({}^{\circ}\) towards 90\({}^{\circ}\), although its maximum may not take place exactly at \(\theta\) = 0\({}^{\circ}\). However, it forms a ridge when \(\theta\) approaches 90\({}^{\circ}\), which is lower than the crest near \(\theta\) = 0\({}^{\circ}\).