## 2 Single-Camera Trilateration Algorithm
### Trends of Performance Indices under Representative Input Error Standard Deviations
(**a**) Surface plot and contour plot of \(\mathrm{B_{p}}(\delta^{m}\mathbf{q}_{0})\); (**b**) Surface plot and contour plot of \(\mathrm{S_{p}}(\delta^{m}\mathbf{q}_{0})\); (**c**) Surface plot and contour plot of \(\mathrm{B_{\theta}}(\delta^{m}\mathbf{q}_{0})\); (**d**) Surface plot and contour plot of \(\mathrm{S_{\theta}}(\delta^{m}\mathbf{q}_{0})\). Figure 6: \(\mathrm{B_{P}(\delta^{m}\mathbf{q}_{j})}\), \(\mathrm{S_{P}(\delta^{m}\mathbf{q}_{j})}\), \(\mathrm{B_{\theta}(\delta^{m}\mathbf{q}_{j})}\) and \(\mathrm{S_{\theta}(\delta^{m}\mathbf{q}_{j})}\) with \(\sigma_{c}=0.5\) and \(\mathrm{R}=3000\). (**a**) Surface plot and contour plot of \(\mathrm{B_{P}(\delta^{m}\mathbf{q}_{j})}\); (**b**) Surface plot and contour plot of \(\mathrm{S_{P}(\delta^{m}\mathbf{q}_{j})}\); (**c**) Surface plot and contour plot of \(\mathrm{B_{\theta}(\delta^{m}\mathbf{q}_{j})}\); (**d**) Surface plot and contour plot of \(\mathrm{S_{\theta}(\delta^{m}\mathbf{q}_{j})}\). Figure 9: Comparing \(\mathrm{B_{P}}(\delta^{m}\mathbf{q}_{0})\), \(\mathrm{S_{P}}(\delta^{m}\mathbf{q}_{0})\), \(\mathrm{B_{\theta}}(\delta^{m}\mathbf{q}_{0})\) and \(\mathrm{S_{\theta}}(\delta^{m}\mathbf{q}_{0})\) with \(\alpha_{c}=\) [0.6,0.8,1.0] and \(\mathrm{R}=3000\). (a) Variation of \(\mathrm{B_{P}}(\delta^{m}\mathbf{q}_{0})\); (b) Variation of \(\mathrm{S_{P}}(\delta^{m}\mathbf{q}_{0})\); (c) Variation of \(\mathrm{B_{\theta}}(\delta^{m}\mathbf{q}_{0})\); (d) Variation of \(\mathrm{S_{\theta}}(\delta^{m}\mathbf{q}_{0})\). Figure 12: Comparing \(\mathrm{B_{p}}(\delta^{m}\mathbf{f})\), \(\mathrm{S_{p}}(\delta^{m}\mathbf{f})\), \(\mathrm{B_{\theta}}(\delta^{m}\mathbf{f})\) and \(\mathrm{S_{\theta}}(\delta^{m}\mathbf{f})\) with \(\sigma_{\mathrm{f}}=0.4\) and \(\mathrm{R}=\{3000,4000,5000\}\). **(a)** Variation of \(\mathrm{B_{p}}(\delta^{m}\mathbf{f})\); **(b)** Variation of \(\mathrm{S_{p}}(\delta^{m}\mathbf{f})\); **(c)** Variation of \(\mathrm{B_{\theta}}(\delta^{m}\mathbf{f})\); **(d)** Variation of \(\mathrm{S_{\theta}}(\delta^{m}\mathbf{f})\). Figure 13: Comparing \(\mathrm{B}_{\mathrm{P}}(\delta^{m}\mathbf{q}_{0})\), \(\mathrm{S}_{\mathrm{P}}(\delta^{m}\mathbf{q}_{0})\), \(\mathrm{B}_{\mathrm{\theta}}(\delta^{m}\mathbf{q}_{0})\) and \(\mathrm{S}_{\mathrm{\theta}}(\delta^{m}\mathbf{q}_{0})\) with \(\sigma_{c}=0.6\) and \(\mathrm{R}=\{3000,4000,\)\(5000\}\). (**a**) Variation of \(\mathrm{B}_{\mathrm{P}}(\delta^{m}\mathbf{q}_{0})\); (**b**) Variation of \(\mathrm{S}_{\mathrm{P}}(\delta^{m}\mathbf{q}_{0})\); (**c**) Variation of \(\mathrm{B}_{\mathrm{\theta}}(\delta^{m}\mathbf{q}_{0})\); (**d**) Variation of \(\mathrm{S}_{\mathrm{\theta}}(\delta^{m}\mathbf{q}_{0})\). For the effect of \(\delta^{\mathcal{S}}\mathbf{p}_{i}\), the simulation results show that, as R increases,  
1. \(\mathrm{B}_{\mathrm{p}}(\delta^{\mathcal{S}}\mathbf{p}_{i})\) and \(\mathrm{S}_{\mathrm{p}}(\delta^{\mathcal{S}}\mathbf{p}_{i})\) increases across S (Figure 11a,b);
2. \(\mathrm{B}_{\mathrm{p}}(\delta^{\mathcal{S}}\mathbf{p}_{i})\) and \(\mathrm{S}_{\mathrm{\theta}}(\delta^{\mathcal{S}}\mathbf{p}_{i})\) have very small changes and can be considered being essentially independent of the variation of R (Figure 11c,d). It means that a longer observation distance tends to increase the sensitivity of the positioning bias and uncertainty to the landmark position error, while it has little impact on the corresponding estimation bias and uncertainty of the camera orientation. For the effect of \(\delta^{\mathcal{M}}\mathbf{q}_{i}\), \(\delta^{\mathcal{M}}\mathbf{f}\) and \(\delta^{\mathcal{M}}\mathbf{q}_{0}\), the simulation results show that, as R increases,  
1.