## 3 Instantaneous Multi-Constellation Relative Positioning: Methodology and Performance
### Loosely Combined Relative Positioning Model with Parameterised Ionospheric Delays  
Nowadays, there are several functional models for combined processing of multi-constellation signals in a relative mode with double-differences (DD) (see [26]). Basically, we can distinguish between two general approaches: intra-system and inter-system combination. The former uses separate pivot satellite for each constellation when forming double-differenced observables. Thus, the observational model is free from the DD observables formed across constellations. This methodology can be considered a classical approach, applied commonly to constellations with no overlapping frequencies. The latter approach, termed tightly integrated or tightly coupled, uses a single common pivot satellite for all constellations. As a result, in this positioning model there are both intra-system and inter-system DD observables (mixed). Such an approach can be especially beneficial in urban canyons with limited satellite visibility, due to a higher number of DD observables in comparison to the intra-system combination. When using the tight integration we do not lose one satellite as pivot per constellation. The prerequisite of application of the tight integration is the presence of common frequencies in combined constellation, such as L1/E1 and L5/E5a in GPS and Galileo [49]. This model can strengthen the solution; however, it requires handling of inter-system biases. Specifically, in the relative mode these biases are differential values (differential ISBs). As studies show, the phase and code differential ISBs may be considered stable [37]. This creates an opportunity for priori calibration of the biases and further correction of observations. As a result, inter-system phase ambiguities for overlapping frequencies may be fixed to their integer values. At this point we should also highlight the most recent contributions considering an inter-system model with no overlapping frequencies. A tightly combined GPS + BDS RTK model with real-time estimation of differential ISB was proposed in [50]. Nevertheless, as studies show, both tight and loose coupling may provide comparative performance, especially in the case of a high number of tracked satellites [26; 50]. Thus, considering the above, in this study we apply an intra-system model for assessing the performance of the relative kinematic positioning with the use of Galileo E14 & E18 satellites. Over the last decades several approaches were recognised for modelling of the ionospheric delay in relative GNSS positioning. Their application mainly depends on the baseline length, state of the ionosphere, positioning mode or session length. Among others, one highly effective approach is the ionosphere weighted model, commonly applied for medium- and long-range RTK [38; 39; 51; 52; 53; 54] as well as instantaneous positioning [40; 55]. The ionosphere weighted model can be considered a generalised approach to ionosphere delay parameterisation, taking into account information on both a priori values of ionosphere delays and its stochastic properties. Thus, the estimated DD ionospheric delays are weighted in the model estimation process by the application of the pseudo-observables, which allowsfor constraining parameters using a priori variance factors. Thanks to the above, it is possible to tune these parameters to the expected values of the factual double differenced ionospheric delays. If we, however, do not introduce a priori information of the ionosphere delays, the model can be labelled the ionosphere float. On the other hand, positioning supported with ionospheric corrections considered as errorless is commonly termed the ionosphere fixed model [56]. The latter approach is, however, not applied in this study, since we take advantage of ionosphere float model. The relative positioning model based on geometry with parameterised ionospheric delays is given in Equation (5).