## 3 Single Transmitter Station Calibration Method
### Optimization Solution Method and Initial Value Estimates  
Because Equation (13) is nonlinear, the solution to the objective function is considered to be an unconstrained nonlinear optimization problem. In addition, a least-squares equation of these 12 unknown parameters can be solved when \(N\geq 6\). The Levenberg-Marquardt algorithm is considered one of the most effective nonlinear least squares algorithms available owing to its advantages of rapid convergence and high stability. Therefore, this algorithm is employed in the present work to solve Equation (13). Because the Levenberg-Marquardt algorithm is a least-squares fitting method, its calculation process first selects appropriate initial values of the 12 unknown parameters, and then calculates their values step by step according to established iteration rules until their true values are approached to within an established tolerance. Defining the vectors \(\mathbf{J}_{9}=[r_{1}\ r_{2}\ r_{3}\ r_{4}\ r_{5}\ r_{6}\ r_{7}\ r_{8}\ r_{9}\ 1_{1}\ r_{2}\ t_{3}]^{\mathrm{T}}\) and \(\mathbf{J}_{3}=[r_{7}\ r_{8}\ r_{9}]^{\mathrm{T}}\) allows Equation (8) to be expressed as in the following matrix form:  
\[\mathbf{U}\mathbf{J}_{9}+\mathbf{V}\mathbf{J}_{3}+\mathbf{D}=0 \tag{14}\]  
where \(\mathbf{U}\) is a \(2N\times 9\) matrix, \(\mathbf{V}\) is a \(2N\times 3\) matrix, and \(\mathbf{D}\) is a \(2N\times 1\) matrix, which are defined as follows. \[\mathbf{U}=\left[ \begin{array}{cccccccc}a_{11}x_{k}&a_{11}y_{k}&a_{11}z_{k} &b_{11}x_{k}&b_{11}y_{k}&b_{11}z_{k}&a_{11}&b_{11}&c_{11}\\ a_{12}x_{k}&a_{12}y_{k}&a_{12}z_{k}&b_{12}x_{k}&b_{12}y_{k}&b_{12}z_{k}&a_{12} &b_{12}&c_{12}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ a_{N1}x_{k}&a_{N1}y_{k}&a_{N1}z_{k}&b_{N1}x_{k}&b_{N1}y_{k}&b_{N1}z_{k}&a_{N1} &b_{N1}&c_{N1}\\ a_{N2}x_{k}&a_{N2}y_{k}&a_{N2}z_{k}&b_{N2}x_{k}&b_{N2}y_{k}&b_{N2}z_{k}&a_{N2} &b_{N2}&c_{N2}\end{array} \right]\]  
\[\mathbf{V}=\left[ \begin{array}{cccccccc}c_{11}x_{1}&c_{1}y_{1}&c_{1}z_{1} \\ c_{12}x_{1}&c_{2}y_{1}&c_{2}z_{1}\\ \vdots&\vdots&\vdots\\ c_{N1}x_{k}&c_{N1}y_{k}&c_{N1}z_{k}\\ c_{N2}x_{k}&c_{N2}y_{N}&c_{N2}z_{k}\end{array} \right]\]\[\mathbf{D}=[d_{i1},d_{i2},\ldots d_{i1},d_{i2}]^{\mathrm{T}}\]  
Because plane 1 passes through \(O\), we know that \(d_{i1}=0\). In addition, \(d_{i2}\) is relatively small owing to laser module assembly adjustment. Therefore, the initial values of the elements of \(\mathbf{D}\) can be set to 0. This simplifies Equation (14) as  
\[\mathbf{UJ}_{9}+\mathbf{VJ}_{3}=0 \tag{15}\]  
It is also known from constraints (12) that \(r_{7}\), \(r_{8}\), and \(r_{9}\) satisfy the constraint condition \(f_{3}~{}=~{}r_{7}^{2}+r_{8}^{2}+r_{9}^{2}-1=0\), which indicates that  
\[\mathbf{J}_{3}^{\mathrm{T}}\mathbf{J}_{3}-1=0 \tag{16}\]  
Therefore, the initial values of the 12 unknown parameters in Equation (13) can be selected as  
\[\mathbf{F}_{0}=(\mathbf{UJ}_{9}+\mathbf{VJ}_{3})^{\mathrm{T}}(\mathbf{UJ}_{9 }+\mathbf{VJ}_{3})+\lambda\big{(}\mathbf{J}_{3}^{\mathrm{T}}\mathbf{J}_{3}-1 \big{)} \tag{17}\]  
where \(\lambda\) is the Lagrange multiplier. Minimization over successive iterations is determined by obtaining the partial derivatives of \(\mathbf{F}_{0}\) with respect to \(\mathbf{J}_{9}\) and \(\mathbf{J}_{3}\), and setting \(\frac{\partial F_{0}}{\partial J_{9}}=0\) and \(\frac{\partial F_{0}}{\partial J_{3}}=0\). This yields the following set of matrix equations:  
\[(\mathbf{Q}-\lambda\mathbf{I})\mathbf{J}_{3}=0 \tag{18}\]  
where  
\[\mathbf{Q}=\mathbf{V}^{\mathrm{T}}\bigg{(}\mathbf{I}-\mathbf{U}\big{(} \mathbf{U}^{\mathrm{T}}\mathbf{U}\big{)}^{-1}\mathbf{U}^{\mathrm{T}}\bigg{)} \mathbf{V} \tag{19}\]  
and  
\[\mathbf{J}_{9}=-\Big{(}\mathbf{U}^{\mathrm{T}}\mathbf{U}\big{)}^{-1}\mathbf{U }^{\mathrm{T}}\mathbf{VJ}_{3} \tag{20}\]  
It is known from Equation (19) that \(\mathbf{Q}\) is a positive definite matrix.