## 2 Materials and Methods
### Data Preparation  
This section presents the method of preparing data for laboratory and field tests, which in the next stage was used to generate a virtual point cloud for coordinate transformation using machine learning (ANN).  
In laboratory tests, a set of reference data (ideal) for training neural networks was prepared, along with a set of data disturbed by random errors generated according to a normal (Gaussian) distribution. Two datasets were prepared in the field trials. Total Station measurements were taken on a ship that was docked on the water. Based on the measurement data, two different sets of common points generated two different sets of points for transformation.  
The research was carried out in the first phase on the basis of laboratory data. Ideal model data based on 7 points on the sphere were adopted for the calculations. Then, the  
Figure 1: Coordinate transformation algorithm for offshore purposes using ANN.  
set of reference points was transformed according to the relation (1), rotated by (assumed) angles, and shifted by the translation vector (Figure 2). The scale factor was adopted as 1 due to the fact that the nature of geodetic measurements with the use of a Total Station instrument does not change the scale between the individual measurement stations.  
\[\left[\begin{array}{c}X^{\prime}\\ Y^{\prime}\\ Z^{\prime}\end{array}\right]=\lambda\cdot R\cdot\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]+\left[\begin{array}{c}\overline{X}_{0}\\ \overline{Y}_{0}\\ \overline{Z}_{0}\end{array}\right], \tag{1}\]  
where:  
* \([X,\ Y,\ Z]^{T}\)--vector (point) in the original (primary) coordinate system,
* \([X^{\prime},\ Y^{\prime},\ Z^{\prime}]^{T}\)--vector (point) in the secondary coordinate system,
* \([\overline{X}_{0},\ \overline{Y}_{0},\ \overline{Z}_{0}]^{T}\)--translation vector,
* \(\lambda\)--scale factor,
* \(R\)--rotation matrix.  
The sphere was generated in the Rhino 3D program. This is a CAD program used for modeling and creating 3D objects. In this way, common points were obtained in two coordinate systems; input was designated by the authors as local frame (LF), and output was labeled as vessel frame (VF). There can be more than one LF, which must be combined with a VF. A virtual point cloud was generated in two coordinate systems (input and output) according to the algorithm described in Section 2.2. In the second step, an additional set of data was prepared, to which random disturbances generated according to a normal (Gaussian) distribution were added.  
Figure 2: Reference points for generating a virtual point cloud (general view).  
Experimental field tests were carried out on the vessel Navigator XXI, which was in the dock, on the water. Measurements were taken with the use of a Total Station (Trimble (XX10), without its leveling, in the technology of Total Free Station (TFS). Measurements on the ship were made from two measurement stations in random local coordinate systems related to the Total Station. An observation plate with a magnet feature was used for common points (Figure 3). Measurements were made in windless weather.  
As a result of the measurements, a set of measurement data was created. The base points altogether between the stations were used to generate a virtual point cloud.