## 2 Problem Formulation
Similarly, the exact positions of the calibration targets are not known to us, and the nominal versions denoted by \(\mathbf{c}_{k}=[x_{c,k},y_{c,k},z_{c,k}]^{\mathrm{T}}\) (\(k=1,2,\ldots,K\)) are given as  
\[\mathbf{c}_{k}=\mathbf{c}_{k}^{\mathrm{o}}+\mathbf{\Delta c}_{k} \tag{9}\]  
where \(\mathbf{\Delta c}_{k}\) is the position error of the \(k\)th calibration target. Collecting (9) for all the \(K\) calibration targets forms a \(3K\)-by-\(1\) calibration target position vector as  
\[\mathbf{c}=\mathbf{c}^{\mathrm{o}}+\mathbf{\Delta c} \tag{10}\]  
where \(\mathbf{c}=[\mathbf{c}_{1}^{\mathrm{T}},\mathbf{c}_{2}^{\mathrm{T}},\ldots, \mathbf{c}_{K}^{\mathrm{T}}]^{\mathrm{T}}\) is the nominal calibration target position vector, \(\mathbf{c}^{\mathrm{o}}=[(\mathbf{c}_{1}^{\mathrm{o}})^{\mathrm{T}},(\mathbf{ c}_{2}^{\mathrm{o}})^{\mathrm{T}},\ldots,(\mathbf{c}_{K}^{\mathrm{T}})]^{ \mathrm{T}}\) is the true calibration target position vector, and \(\mathbf{\Delta c}=[\mathbf{\Delta c}_{1}^{\mathrm{T}},\mathbf{\Delta c}_{2}^{ \mathrm{T}},\ldots,\mathbf{\Delta c}_{K}^{\mathrm{T}}]^{\mathrm{T}}\) is the calibration target position error vector that is usually supposed to obey Gaussian distribution with zero-mean and covariance \(\mathbf{Q}_{\mathbf{c}}\). Herein, it should be pointed out that, the positions of calibration targets are generally considered to be more precise compared with those of the transmitters and receivers, although they are also contaminated by errors. Then, the distance from the \(m\)th transmitter to the \(k\)th calibration target is given by  
\[R_{c,k,m}^{\mathrm{o}}=\|\mathbf{c}_{k}^{\mathrm{o}}-\mathbf{s}_{\mathrm{t},m }^{\mathrm{o}}\| \tag{11}\]  
and the distance from the \(k\)th calibration target to the \(n\)th receiver is given by  
\[R_{c,k,r,n}^{\mathrm{o}}=\|\mathbf{c}_{k}^{\mathrm{o}}-\mathbf{s}_{\mathrm{t}, n}^{\mathrm{o}}\| \tag{12}\]  
Based on this, the BR measurement corresponding to the \(k\)th calibration target, \(m\)th transmitter and \(n\)th receiver can be modeled as  
\[ \begin{array}{rl}r_{\mathrm{c},k,m,n}&=r_{\mathrm{c},k,m,n}^{\mathrm{o}}+ \mathbf{\Delta r}_{\mathrm{c},k,m,n}\\ &=R_{c,k,m}^{\mathrm{o}}+R_{c,k,r,n}^{\mathrm{o}}-R_{\mathrm{t},m,n,n}^{\mathrm{ o}}+\mathbf{\Delta r}_{\mathrm{c},k,m,n}\\ &=\|\mathbf{c}_{k}^{\mathrm{o}}-\mathbf{s}_{\mathrm{t},m}^{\mathrm{o}}\|+\| \mathbf{c}_{k}^{\mathrm{o}}-\mathbf{s}_{\mathrm{t},n}^{\mathrm{o}}\|-\| \mathbf{s}_{\mathrm{t},m}^{\mathrm{o}}-\mathbf{s}_{\mathrm{t},n}^{\mathrm{o}}\|+ \mathbf{\Delta r}_{\mathrm{c},k,m,n}\end{array}  \tag{13}\]  
where \(\mathbf{\Delta r}_{\mathrm{c},k,m,n}\) represents measurement noise in \(r_{\mathrm{c},k,m,n}\), \(r_{\mathrm{c},k,m,n}^{\mathrm{o}}=R_{c,k,t,m}^{\mathrm{o}}+R_{\mathrm{c},k,r,n}^ {\mathrm{o}}-R_{\mathrm{t},m,x,n}^{\mathrm{o}}\) represents the true BR with respect to the \(k\)th calibration target, \(m\)th transmitter and \(n\)th receiver. Collecting (13) for the set of \(K\) calibration targets, \(M\) transmitters and \(N\) receivers, results in a _KMN_-by-\(1\) vector as  
\[\mathbf{r}_{\mathrm{c}}=\mathbf{r}_{\mathrm{c}}^{\mathrm{o}}+\mathbf{\Delta r }_{\mathrm{c}} \tag{14}\]  
where \(\mathbf{r}_{\mathrm{c}}=[\mathbf{r}_{\mathrm{c},1}^{\mathrm{T}},\mathbf{r}_{ \mathrm{c},2}^{\mathrm{T}},\ldots,\mathbf{r}_{\mathrm{c},K}^{\mathrm{T}}]^{ \mathrm{T}}\) with \(\mathbf{r}_{\mathrm{c},k}=[\mathbf{r}_{\mathrm{c},k,1}^{\mathrm{T}},\mathbf{r}_{ \mathrm  
corresponding error vector, which is presumed to be a Gaussian random vector with zero mean and covariance \(\mathbf{Q}_{\text{rc}}\). Now, the purpose of this work is to determine the target position from the noisy BR measurements and the inaccurate transmitter/receiver positions. In particular, the calibration targets with known position and the corresponding BR measurements are also available to reduce the transmitter/receiver position error and improve localization accuracy.