## 2 Single-Camera Trilateration Algorithm
### Camera Orientation Estimation
Since \(\mathbf{B}^{T}{}^{\delta}\mathbf{R}_{c}\mathbf{A}\) is an orthogonal matrix and \(\mathbf{D}\) is a non-negative diagonal matrix, we must have:  
\[\text{max}Tr(\mathbf{D}\mathbf{B}^{T}{}^{\delta}\mathbf{R}_{c}\mathbf{A})=Tr( \mathbf{D}), \tag{17}\]  
and correspondingly  
\[\mathbf{B}^{T}{}^{\delta}\mathbf{R}_{c}\mathbf{A}=\mathbf{I}. \tag{18}\]  
Then, we obtain from Equation (18)  
\[{}^{\delta}\mathbf{R}_{c}={}^{\delta}\Delta\mathbf{P}^{\delta}\Delta\mathbf{P }^{T}({}^{c}\mathbf{P}^{\delta}\mathbf{P}^{T})^{-1}. \tag{14}\]0  
which is guaranteed to be an orthogonal matrix. Based on the above derivation, it is clear that the camera orientation can be determined by first calculating \({}^{c}\mathbf{p}_{i}\) from Equation (12), next conducting the singular value decomposition of the matrix \({}^{c}\mathbf{P}{}^{\delta}\Delta\mathbf{P}^{T}\), and then getting \({}^{\delta}\mathbf{R}_{c}\) from Equation (19). where \(|\mathbf{a}|\)denotes the norm of a vector \(\mathbf{a}\), \(\mathit{Tr}\) (\(\mathbf{M}\)) denotes the trace of a matrix \(\mathbf{M}\), E(.) denotes the mean of a random variable, and var(.) denotes the variance of a random variable. Similar to \({}^{\delta}\mathbf{p}_{0}\), \({}^{\delta}\mathbf{R}_{c}\) is also a function of \(\mathbf{x}\), i.e., \({}^{\delta}\mathbf{R}_{c}={}^{\delta}\mathbf{R}_{c}(\mathbf{x})\). To make the presentation consistent, we discuss the orientation estimation error in the vector form, and correspondingly define an error metric of \({}^{\delta}\mathbf{R}_{c}\) as:  
\[{}^{\delta}\mathbf{R}_{c}={}^{\delta}\Delta\mathbf{P}^{\delta}\Delta\mathbf{P }^{T}({}^{c}\mathbf{P}^{\delta}\mathbf{P}^{T})^{-1}. \tag{14}\]1  
where  
\[{}^{\delta}\mathbf{R}_{c}={}^{\delta}\Delta\mathbf{P}^{\delta}\Delta\mathbf{P }^{T}({}^{c}\mathbf{P}^{\delta}\mathbf{P}^{T})^{-1}. \tag{14}\]2  
in the unit of degrees. Here \(\mathbf{n}_{i}\) denotes the \(i\)th column of \({}^{\delta}\mathbf{R}_{c}\) which corresponds to the x, y or z directional vector of the camera frame with respect to the global frame. As a result, \(\delta^{\delta}\boldsymbol{\theta}_{c}\) shows the angular difference between the actual and erroneous camera orientations. Similar to Equations (20) and (21), we define the corresponding performance indices--the normalized total bias \(\mathrm{B}_{\theta}\) and the normalized total standard deviation error \(\mathrm{S}_{\theta}\) as:  
\[{}^{\delta}\mathbf{R}_{c}={}^{\delta}\Delta\mathbf{P}^{\delta}\Delta\mathbf{P }^{T}({}^{c}\mathbf{P}^{\delta}\mathbf{P}^{T})^{-1}. \tag{14}\]3  
\[{}^{\delta}\mathbf{R}_{c}={}^{\delta}\Delta\mathbf{P}^{\delta}\Delta\mathbf{P }^{T}({}^{c}\mathbf{P}^{\delta}\mathbf{P}^{T})^{-1}. \tag{14}\]4  
where \(\mathrm{B}_{\theta}\) represents the sensitivity of the systematic orientation estimation error to the input error, and \(\mathrm{S}_{\theta}\) represents the sensitivity of the orientation estimation uncertainty to the input error. In the following performance analysis, we will report the variation of \(\mathrm{B}_{p}\), \(\mathrm{S}_{p}\), \(\mathrm{B}_{\theta}\) and \(\mathrm{S}_{\theta}\) across the simulated spaces of \({}^{\delta}\mathbf{p}_{0}\) under the representative \(\sigma_{v}\) values of \(\delta^{\delta}\mathbf{p}_{i}\), \(\delta^{m}\mathbf{q}_{i}\), \(\delta^{m}\mathbf{f}\) and \(\delta^{m}\mathbf{q}_{0}\).