## 5 Conclusions and Future Work  
This paper has introduced an effective single-camera trillateration scheme for regular forward-looking cameras. Based on the classical pin-hole model and principles of perspective geometry, the position and orientation of such a camera can be calculated successively from a single image of a few landmarks at known positions. The proposed scheme is an advantageous addition to the category of trillateration techniques:  
1. The camera position is trillaterated from multiple simultaneously captured landmarks, which naturally resolves the common concern of the synchronization among multiple distance measurements in ranging-based trillateration systems. 2. The estimation of the camera orientation becomes an integrated part of the proposed trillateration scheme, taking advantage of the geometrical information contained in a single image of multiple landmarks. 3. The instantaneous camera pose is estimated from only one single image, which naturally resolves the issue of mismatching among a pair of images in the stereovision-based trillateration. The proposed single-camera trillateration scheme provides a convenient tool for the self-localization of mobile robots and other vehicles which are equipped with cameras. Depending on the identification of the landmarks involved, the proposed algorithm targets environments with identifiable landmarks, such as indoor and outdoor environments with artificial landmarks, metropolitan environments, and natural environments with distinct landmarks. The performance of the proposed algorithm has been analyzed in this work based on extensive simulations with a representative set of geometric arrangements among the landmarks and camera and a representative set of input errors. In practice, simulations with the settings of a targeted environment and the input errors of a targeted camera will provide an effective way to predict the algorithm performance in the specific environment and system. The simulation results will provide valuable guidance for the implementation of the proposed algorithm, and facilitate mobile robots to achieve accurate self-localization by avoiding sensitive regions of the performance indices. Nevertheless, a closed-form solution will further reduce the time complexity of the proposed algorithm, and a closed-form estimation of the performance indices will enable more efficient performance analysis and prediction of the proposed algorithm. As pointed out, existing closed-form \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{6}{c}{**Errors of Camera Pose**} & \multicolumn{3}{c}{**Errors of Single-Camera Trillateration**} \\ \multicolumn{6}{c}{**Calibration**} \\ \hline \multirow{2}{*}{\(\mathbf{R_{h}}\) (mm)} & **STD(\(\delta\)S\({}_{\text{P0}}\))** & **STD(\(\delta\)S\(\theta\)\({}_{\mathcal{O}}\))** & **|E(A\({}^{S}\)P\({}_{\text{p0}}\))** & **STD(A\({}^{S}\)P\({}_{\text{p0}}\))** & **|E(A\({}^{S}\)\(\theta\)\({}_{\mathcal{O}}\))** & **STD(A\({}^{S}\)\(\theta\)\({}_{\mathcal{O}}\))** \\  & **(mm)** & **(deg)** & **(mm)** & **(mm)** & **(deg)** & **(deg)** \\ \hline
2000 & 4.5627 & 0.1806 & 1.3539 & 10.9633 & 0.0838 & 0.3564 \\
2500 & 5.5080 & 0.1448 & 1.4588 & 12.4060 & 0.2302 & 0.2878 \\
3000 & 6.3335 & 0.1348 & 5.0328 & 17.5257 & 0.2874 & 0.3189 \\
3500 & 7.2251 & 0.1273 & 4.4541 & 29.3335 & 0.1970 & 0.3084 \\
4000 & 8.2313 & 0.1228 & 6.1605 & 31.9332 & 0.2548 & 0.4927 \\ \hline \hline \end{tabular}  
* In this table, \(\mathbf{R_{h}}\) denotes the horizontal distance between the camera and the \(Z\) axis of the global frame which passes through the center of the base triangle, E(.) denotes mean, STD(.) denotes standard deviation, and \(|.|\) denotes the norm of a vector.  
\end{table} 
Table 2: Camera pose estimation error—the proposed algorithm versus camera calibration 1.  
solutions can be adopted into the steps (2) and (3) of the proposed framework (Algorithm 1) to solve the 3-landmark case.