## 3 Theoretical Basis of the Method of Least Squares with Conditional Equations  
In geodetic surveys, there are alignment tasks in which the parameters of the system of observation equations must satisfy additional conditions [41]. In geodetic networks, these conditions may concern the coordinates of certain points being part of a particular network. Such a case occurs when the distance between points is determined with high accuracy, or the conditions between the coordinates result from the geometric structure of the network. Such a case is found in the research problem under consideration, where the position of antennas on the surveying platform are determined from satellite observations as well as from tacheometric surveys conducted prior to the measurement session using satellite measurement techniques. The distances between GNSS receiver antennas were precisely measured prior to commencing dynamic surveys. The geodetic inventory of GNSS antennas was carried out using an electronic total station and the values obtained were adopted as observations determined with high accuracy, which justifies the assumption that they are relatively error-free. Therefore, it follows that the parameters of the observation system (coordinates of GNSS antennas' measuring points on the surveying platform) must satisfy additional conditions, i.e., the distances between GNSS antennas on the MSP. For the cumulative processing of the obtained observations, the Method of Least Squares with conditional equations (LSce) can then be applied. The theory behind the method was discussed in detail inter alia in [42; 43]. Its further developments and applications to prepare geodetic observations are discussed inter alia in [41], [44; 45]. It should also be noted that conditional equations have also been used in other estimation methods, e.g., in M\({}_{\text{split}}\) estimation [46]. The initial tests of the application of LSce to estimate the GNSS receiver antennas' coordinates on surveying platforms were carried out over the test section of tram tracks in the urban agglomeration of Gdansk [47]. Below are the main assumptions and the ways to apply LSce in the context of the surveying platform and the associated observation systems considered in this study. Figure 1: Mobile surveying platform. Let us assume that the coordinates of antennas \((x_{i},y_{i})\), \(i=1,2,\cdots,s\), on the surveying platform, are contained in the vector \(\mathbf{X}=\left[x_{1},y_{1},\ldots,x_{s},y_{s}\right]^{T}=\left[X_{1},X_{2}, \ldots,X_{r}\right]^{T}\) (s--the number of antennas, \(r\)--the number of unknown coordinates). Following the previous assumptions about the adopted measurement structure, these quantities must satisfy the following system of conditional equations:  
\[ \begin{array}{l}\left.\begin{array}{l}\Psi_{1}(X_{1},X_{2},\ldots,X_{r})=0 \\ \Psi_{2}(X_{1},X_{2},\ldots,X_{r})=0\\ \vdots\\ \Psi_{w}(X_{1},X_{2},\ldots,X_{r})=0\end{array}\right\}\Leftrightarrow\Psi( \mathbf{X})=0\end{array}  \tag{1}\]  
For example, if we assume that \((x_{i},y_{i})\)\((x_{j},y_{j})\) are the coordinates of two antennas between which the distance \(d_{ij}\) was measured with high accuracy, then \(\left[(x_{j}-x_{i})^{2}+(y_{j}-y_{i})^{2}\right]^{1/2}-d_{ij}=0\). Moreover, in accordance with the general principles of determining geodetic network coordinates, the vector \(\mathbf{X}\) is determined based on GNSS satellite observations which enable the formulation of the following system of observation equations:  
\[\left. \begin{array}{l}y_{1}+v_{1}=F_{1}(X_{1},X_{2},\ldots,X_{r})\\ y_{2}+v_{2}=F_{2}(X_{1},X_{2},\ldots,X_{r})\\ \vdots\\ y_{n}+v_{n}=F_{n}(X_{1},X_{2},\ldots,X_{r})\end{array} \right\}\Leftrightarrow \mathbf{y}+\mathbf{v}=\mathbf{F}(\mathbf{X}) \tag{2}\]  
where \(\mathbf{y}=[y_{1},y_{2}\cdots,y_{n}]^{T}\) is the observation vector.