## 5 Conclusions  
This contribution investigated a case of code-based differential GNSS for two lunar orbiters as it has never been conducted before. In particular, it assessed the potential of ISR estimation both through theoretical simulations and HWil tests using the soon-to-be-flying NaviMoon Galileo-GPS receiver. Doing so, it has shown that commonly used terrestrial algorithms for kinematic code-based differential GNSS ranging techniques are not applicable for a scenario of GNSS receivers in lunar orbits, despite their individual ability to navigate using GNSS signals from terrestrial constellations. Indeed, when using the standard code-based DGNSS algorithm, a significant bias is introduced, whose magnitude increases with the angular separation between the two users with respect to the GNSS constellation. This geometrical bias is introduced by the parallelism assumption between steering vectors from the two users involved to each visible GNSS satellite, which has been taken for granted in ground applications. This contribution hence analyzed the potential to improve the DGNSS ranging algorithm in space with respect to the violation of the parallelism assumption while keeping the algorithm low complexity and linear for lunar scenarios. It showed that potential updates of the steering vectors in the design matrix of the SD and DD LS estimation solutions can significantly reduce or even cancel this bias. In particular, a technique called \"ideal\" in the present contribution and found in  
Figure 14: ISR estimation from simulations vs from NaviMoon. the literature in the context of Doppler-based LEO positioning based on SoP, has proven to show added value. It is based on defining a specific steering vector at each epoch for which the projection of the baseline on this given direction exactly equals the difference in geometrical ranges to the GNSS satellite. Hence, this technique reduces the bias that the terrestrial assumption was introducing in this linear estimator to zero. The variance of the presented estimators is consistent with the DOP calculated based on the GNSS satellites commonly visible to both users at the same time. This paper further analyzed the presented algorithms by assessing their potential to cancel errors correlated between the involved users despite the large-scale geometry of such an environment. In the presence of a worst-case scenario of 400 m of GNSS orbital errors, the presented differential algorithms proved to overtake the APD technique in case of a large baseline. Another point this work looked into is the sensitivity of the algorithms to the quality of the receiver's clock bias estimation. This is particularly important for a scenario of orbiters given the high relative velocities and the large ISR variation rate over the timestamp offset between the set of GNSS measurements of the two users. Given the large Doppler shifts characteristic of this kind of GNSS scenarios, an error in the receiver's clock bias estimation can significantly decrease the ranging performances. Conceptualization, A.D., A.M. and F.D. ; methodology, A.D. and A.M.; software, A.D.; validation, A.D., A.M. and F.D. ; formal analysis, A.D.; investigation, A.D.; resources, A.M. and F.D. ; data curation, A.D.; writing--original draft preparation, A.D.; writing--review and editing, A.D., A.M. and F.D. ; visualization, A.D.; supervision, A.M. and F.D. ; project administration, F.D. ; funding acquisition, A.M. and F.D. All authors have read and agreed to the published version of the manuscript. This research was co-funded by European Space Agency Contract No. 4000134565/21/NL/GLC/my. [PERSON] acknowledges funding from the research contract no. 32-G-13427-5 DM 1062/2021 funded within the Programme Operativo Nazionale (PON) Ricerca ed Innovazione of the Italian Ministry of University and Research (MUR). The content of the present article reflects solely the authors' view and by no means represents the official view of the European Space Agency (ESA).