## 2 Background
### Characteristics of a Space DGNSS Scenario
#### 2.2.2 Parallelism Assumption Used for Terrestrial Applications
This technique will be named SD \(\texttt{H}_{\texttt{Aided},\texttt{corrected}}\) in the results presented in Section 4. **Redefinition of the unit steering vectors for the space scenario**  
Another proposed solution, that takes into account the invalidity of the parallelism assumption for space scenarios, while keeping the algorithm linear for its low-complexity advantage, is to redefine the unit vectors considered in the design matrix \(\mathbf{H}\) from Equation (10). Instead of using either \(\mathbf{h}^{s}_{i,k}\) or \(\mathbf{h}^{s}_{j,k^{\prime}}\), a combination of both based on their normalized sum can be used:  
\[\mathbf{h}^{s}_{i+j,k}=\frac{\mathbf{h}^{s}_{i,k}+\mathbf{h}^{s}_{j,k}}{|| \mathbf{h}^{s}_{i,k}+\mathbf{h}^{s}_{j,k}||}. \tag{17}\]  
Consequently, the difference in true ranges changes with respect to Equation (13) and it becomes:  
\[\Delta r^{s}_{ij,k}\approx-\Big{[}\mathbf{h}^{s}_{i+j,k}\Big{]}^{T}\mathbf{d} _{ij,k}. \tag{18}\]This technique will be named SD H\({}_{\text{num}}\), or DD H\({}_{\text{num}}\), if the unit vectors Equation (17) are replaced in Equation (12), in the results presented in Section 4. Finally, another solution presented in the literature in the context of differential Doppler positioning in [61], consists in finding the unit steering vector \(\mathbf{h}_{ideal,k}^{s}\), which at each epoch \(k\) will satisfy exactly Equation (13), such that:  
\[\Delta r_{ij,k}^{s}=-\left[\mathbf{h}_{ideal,k}^{s}\right]^{T}\mathbf{d}_{ij,k}. \tag{19}\]  
The angle \(\alpha_{k}\) between \(\mathbf{h}_{ideal,k}^{s}\) and \(\mathbf{d}_{ij,k}\) is given by:  
\[\Delta r^{s}_{ijk}\approx-\Big{[}\mathbf{h}^{s}_{*,k}\Big{]}^{T}\mathbf{d}_{ij,k} \tag{13}\]0  
where \(\mathbf{h}_{ideal,k}^{s}\) satisfying this condition form a conical surface. A solution consists of defining the vector \(\mathbf{h}_{\perp,k}^{s}\) normal to both \(\mathbf{d}_{ij,k}\) and \(\mathbf{h}_{i,k}^{s}\):  
\[\Delta r^{s}_{ijk}\approx-\Big{[}\mathbf{h}^{s}_{*,k}\Big{]}^{T}\mathbf{d}_{ij,k} \tag{13}\]1  
and then to apply Rodrigues' rotation formula to obtain \(\mathbf{h}_{ideal,k}^{s}\) in the plan that contains both \(\mathbf{d}_{ij,k}\) and \(\mathbf{h}_{i,k}^{s}\) by rotating the baseline unit vector \(\mathbf{h}_{ij,k}^{s}\) around \(\mathbf{h}_{\perp,k}^{s}\) by the angle \(\alpha_{k}\) according to the right-hand rule:  
\[ \begin{split}\mathbf{h}_{ideal,k}^{s}&=\mathbf{h}_{ ij,k}^{s}\cos(\alpha_{k})+(\mathbf{h}_{\perp,k}^{s}\times\mathbf{h}_{ij,k}^{s}) \sin(\alpha_{k})+\\ &\quad\mathbf{h}_{\perp,k}^{s}(\mathbf{h}_{\perp,k}^{s}\cdot \mathbf{h}_{ij,k}^{s})(1-\cos(\alpha_{k}))\\ &=\mathbf{h}_{ij,k}^{s}\cos(\alpha_{k})+(\mathbf{h}_{\perp,k}^{ s}\times\mathbf{h}_{ij,k}^{s})\sin(\alpha_{k}).\end{split}  \tag{22}\]  
Consequently, these newly defined unit vector \(\mathbf{h}_{ideal,k}^{s}\) for the differential LS problem can replace \(\mathbf{h}_{i,k}^{s}\) in the \(\mathbf{H}\) design matrix of Equations (10) and (12). These two techniques will be named SD H\({}_{\text{ideal}}\), or DD H\({}_{\text{ideal}}\) in the results presented in Section 4. All the above-described models will be compared to identify the best solution for DGNSS ranging in space.