# Fusion of Land-Based and Satellite-Based Localization Using Constrained Weighted Least Squares
## 3 Pseudo-Linear Equation
### TDOA Pseudo-Linear Equation for Satellite  
Let \(g_{i}^{o}=r_{i1}^{o}-||\mathbf{s}_{N_{1}+i}^{o}-\mathbf{t}||_{2}+||\mathbf{s}_{N_{1}+1}^{o }-\mathbf{t}||_{2}\). Shifting the terms and squaring both sides simultaneously of (10) yields  
\[\Big{(}g_{i}^{o}+||\mathbf{u}-\mathbf{s}_{N_{1}+1}^{o}||_{2}\Big{)}^{2}=||\mathbf{u}-\mathbf{s }_{N_{1}+i}^{o}||_{2}^{2},2\leq i\leq N_{2} \tag{30}\]Expanding and simplifying (30) yields  
\[2\Big{(}\mathbf{s}^{o}_{N_{1}+i}-\mathbf{s}^{o}_{N_{1}+1}\Big{)}^{\text{T}}\mathbf{u}+2g^{o}_{ i}||\mathbf{u}-\mathbf{s}^{o}_{N_{1}+1}||_{2}=\Big{(}\mathbf{s}^{o}_{N_{1}+i}\Big{)}^{\text{T}} \mathbf{s}^{o}_{N_{1}+i}-\Big{(}\mathbf{s}^{o}_{N_{1}+1}\Big{)}^{\text{T}}\mathbf{s}^{o}_{N_ {1}+1}-\big{(}\mathbf{s}^{o}_{i}\big{)}^{2} \tag{31}\]  
Therefore, the pseudo-linear equation for the satellite can be given as  
\[\mathbf{G}^{o}_{t}\mathbf{\eta}^{o}_{t}=\mathbf{h}^{o}_{t} \tag{32}\]  
where  
\[\mathbf{G}^{o}_{t}=\begin{bmatrix}2\Big{(}\mathbf{s}^{o}_{N_{1}+2}-\mathbf{s}^{o}_{N_{1}+1 }\Big{)}^{\text{T}}&2g^{o}_{2}\\ 2\Big{(}\mathbf{s}^{o}_{N_{1}+3}-\mathbf{s}^{o}_{N_{1}+1}\Big{)}^{\text{T}}&2g^{o}_{3} \\ \vdots&\vdots\\ 2\Big{(}\mathbf{s}^{o}_{N_{1}+N_{2}}-\mathbf{s}^{o}_{N_{1}+1}\Big{)}^{\text{T}}&2g^{o} _{N_{2}}\end{bmatrix}\in\mathbb{R}^{(N_{2}-1)\times 4} \tag{33}\]  
\[\mathbf{h}^{o}_{t}=\begin{bmatrix}\mathbf{s}^{o\text{T}}_{N_{1}+2}\mathbf{s}^{o}_{N_{1}+2} -\mathbf{s}^{o\text{T}}_{N_{1}+1}\mathbf{s}^{o\text{T}}_{N_{1}+1}-\big{(}\mathbf{s}^{o}_{2 }\big{)}^{2}\\ \mathbf{s}^{o\text{T}}_{N_{1}+3}\mathbf{s}^{o}_{N_{1}+3}-\mathbf{s}^{o\text{T}}_{N_{1}+1} \mathbf{s}^{o\text{T}}_{N_{1}+1}-\big{(}\mathbf{s}^{o}_{3}\big{)}^{2}\\ \vdots\\ \mathbf{s}^{o\text{T}}_{N_{1}+N_{2}}\mathbf{s}^{o}_{N_{1}+N_{2}}-\mathbf{s}^{o\text{T}}_{N _{1}+1}\mathbf{s}^{o}_{N_{1}+1}-\Big{(}\mathbf{s}^{o}_{N_{2}}\Big{)}^{2}\end{bmatrix} \in\mathbb{R}^{(N_{2}-1)\times 1} \tag{34}\]  
\[\mathbf{\eta}^{o}_{t}=\begin{bmatrix}\mathbf{u}\\ ||\mathbf{u}-\mathbf{s}^{o}_{N_{1}+1}||_{2}\end{bmatrix}\in\mathbb{R}^{4\times 1} \tag{35}\]  
In (35), \(||\mathbf{u}-\mathbf{s}^{o}_{N_{1}+1}||_{2}\) is similarly an introduced auxiliary variable. And, \(\mathbf{\eta}^{o}_{t}\) satisfies  
\[(\mathbf{\eta}^{o}_{t})^{\text{T}}\mathbf{\Lambda}_{3}\mathbf{\eta}^{o}_{t}+2\Big{[}-\Big{(} \mathbf{s}^{o}_{N_{1}+1}\Big{)}^{\text{T}},0\Big{]}\mathbf{\eta}^{o}_{t}=-\Big{(}\mathbf{s }^{o}_{N_{1}+1}\Big{)}^{\text{T}}\mathbf{s}^{o}_{N_{1}+1} \tag{36}\]  
where \(\mathbf{\tilde{\Lambda}}_{3}=diag\{1,1,1,-1\}\).  
**Remark 1**.: _Any two equations having the form (27) and (32) can be combined into one equation by expanding the matrix dimension. However, this is a simple stacking of matrices and vectors and does not take advantage of the connection between the two. Since the auxiliary variables utilized in (27) and (32) are different, the unknown variables in the two equations are also different. Thus, it is not possible to directly combine (27) and (32) into one equation. Although it is possible to combine (27) and (32) to obtain an equation, the combination of the two equations is a simple matrix combination that does not take advantage of the connection between the two equations._