## 3 Proposed Solutions  
### Problem Formulation  
The measurement model given in (4) is nonlinear. We shall first formulate an optimization problem related to the unknown UN position, velocity, clock offset, and clock drift. Let us start with (2). Keeping the norm on the right side, moving the rest of the terms to the left, and substituting (3) yield  
\[r_{i}+\gamma_{i}-\gamma^{o}-\ell^{o}t_{i}-\varepsilon_{i}=\|\mathbf{p}^{o}+ \mathbf{v}^{o}t_{i}-\mathbf{s}_{i}-\Delta\mathbf{s}_{i}\|,\quad i=1,\ldots M. \tag{6}\]  
Squaring both sides of (6) above and using the mathematical truth \(\|\mathbf{x}\|^{2}=\mathbf{x}^{T}\mathbf{x}\), we have  
\[\|\mathbf{s}_{i}\|^{2}-\alpha_{i}^{2}-t_{i}^{2}(\ell^{o2}-\| \mathbf{v}^{o}\|^{2})-2\mathbf{s}_{i}^{T}\mathbf{p}^{o}-2t_{i}\mathbf{s}_{i}^{ T}\mathbf{v}^{o}+2t_{i}\gamma^{o}+2t_{i}\alpha_{i}\ell^{o} \tag{7}\] \[-2t_{i}(\gamma^{o}\ell^{o}-\mathbf{p}^{oT}\mathbf{v}^{o})-( \gamma^{o2}-\|\mathbf{p}^{o}\|^{2})=\varrho_{i}+\varepsilon_{i}^{2}-2\Delta \mathbf{s}_{i}^{T}\Delta\mathbf{s}_{i},\quad i=1,\ldots M,\]  
where \(\alpha_{i}=r_{i}+\gamma_{i}\), and the first-order noise and error term \(\varrho_{i}=-2(\alpha_{i}-\gamma^{o}-\ell^{o}t_{i})\varepsilon_{i}+2(\mathbf{ p}^{oT}+t_{i}\mathbf{v}^{oT}-\mathbf{s}_{i}^{oT})\Delta\mathbf{s}_{i}\). To remove the term \(\gamma^{o2}-\|\mathbf{p}^{o}\|^{2}\), we can subtract (7) with that with a fixed \(i\). Without loss of generality, we choose the subtracted equation with subscript \(i=1\) and obtain  
\[\|\mathbf{s}_{1}\|^{2}-\alpha_{1}^{2}-t_{1}^{2}(\ell^{o2}-\| \mathbf{v}^{o}\|^{2})-2\mathbf{s}_{1}^{T}\mathbf{p}^{o}-2t_{1}\mathbf{s}_{1}^ {T}\mathbf{v}^{o}+2\alpha_{1}\gamma^{o}+2t_{1}\alpha_{1}\ell^{o} \tag{8}\] \[-2t_{1}(\gamma^{o}\ell^{o}-\mathbf{p}^{oT}\mathbf{v}^{o})-(\gamma ^{o2}-\|\mathbf{p}^{o}\|^{2})=\varrho_{1}+\varepsilon_{1}^{2}-2\Delta \mathbf{s}_{1}^{T}\Delta\mathbf{s}_{1}.\]  
Subtracting (8) from (7) and ignoring the noise and error terms higher than the first order yield  
\[\|\mathbf{s}_{i}\|^{2}-\|\mathbf{s}_{1}\|^{2}-(\alpha_{i}^{2}- \alpha_{1}^{2})-2(\mathbf{s}_{i}^{T}-\mathbf{s}_{1}^{T})\mathbf{p}^{o}-2(t_{i }\mathbf{s}_{i}^{oT}-t_{1}\mathbf{s}_{1}^{oT})\mathbf{v}^{o}-2(\alpha_{1}- \alpha_{i})\gamma^{o} \tag{9}\] \[-2(t_{1}\alpha_{1}-t_{i}\alpha_{i})\ell^{o}+(t_{1}^{2}-t_{i}^{2} )(\ell^{o2}-\|\mathbf{v}^{o}\|^{2})+2(t_{1}-t_{i})(\gamma^{o}\ell^{o}-\mathbf{ p}^{oT}\mathbf{v}^{o})\] \[=\varrho_{i}-\varrho_{1},\quad i=2,\ldots,M.\]  
The equation above is a pseudolinear equation associated with \(\mathbf{p}^{o}\), \(\mathbf{v}^{o}\), \(\gamma^{o}\), and \(\ell^{o}\). Let the unknown parameters be \(\mathbf{\phi}^{o}=[\mathbf{p}^{oT},\mathbf{v}^{oT},\gamma^{o},\ell^{o2}-\| \mathbf{v}^{o}\|^{2},\gamma^{o}\ell^{o}-\mathbf{p}^{oT}\mathbf{v}^{o}]^{T}\), and (9) can be collected in matrix form. We have  
\[\mathbf{h}-\mathbf{G}\mathbf{\phi}^{o}=\mathbf{B}\mathbf{\varepsilon}+\mathbf{C} \Delta\mathbf{s}, \tag{10}\]  
where  
\[\mathbf{h}= \begin{bmatrix}\|\mathbf{s}_{2}\|^{2}-\|\mathbf{s}_{1}\|^{2}-( \alpha_{2}^{2}-\alpha_{1}^{2})\\ \vdots\\ \|\mathbf{s}_{M}\|^{2}-\|\mathbf{s}_{1}\|^{2}-(\alpha_{M}^{2}-\alpha_{1}^{2} )\end{bmatrix} , \tag{11}\]  
\[\mathbf{G}(:,1:2N+1)= \begin{bmatrix}2(\mathbf{s}_{2}-\mathbf{s}_{1})^{T}&2(t_ {2}\mathbf{s}_{2}-t_{1}\mathbf{s}_{1})^{T}&2(\alpha_{1}-\alpha_{2})\\ \vdots&\vdots&\vdots\\ 2(\mathbf{s}_{M}-\mathbf{s}_{1})^{T}&2(t_{M}\mathbf{s}_{M}-t_{1}\mathbf{s}_{1} )^{T}&2(\alpha_{1}-\alpha_{M})\end{bmatrix} , \tag{12}\]  
\[\mathbf{G}(:,2N+2:2N+4)= \begin{bmatrix}2(t_{1}\alpha_{1}-t_{2}\alpha_{2})&-(t_ {1}^{2}-t_{2}^{2})&-2(t_{1}-t_{2})\\ \vdots&\vdots&\vdots\\ 2(t_{1}\alpha_{1}-t_{M}\alpha_{M})&-(t_{1}^{2}-t_{M}^{2})&-2(t_{1}-t_{M})\end{bmatrix} , \tag{13}\]\[\mathbf{B}= \begin{bmatrix}2(\alpha_{1}-\gamma^{o}-\ell^{o}t_{1})&-2(\alpha_{2}- \gamma^{o}-\ell^{o}t_{2})&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 2(\alpha_{1}-\gamma^{o}-\ell^{o}t_{1})&0&\cdots&-2(\alpha_{M}-\gamma^{o}-\ell^{ o}t_{M})\end{bmatrix} , \tag{14}\]  
and  
\[\mathbf{C}=