## 3 Problem Research
### Ship Control Technology Problem
#### 3.2.2 Trajectory Control in Use
According to the proposed algorithm, the dimensions may change (\(u_{max}\) and \(v_{max}\)) as the vessel moves if the vessel deviates from the path leading to the next point in the trajectory (e.g., due to external disturbances), as the angle \(\rho\) will then change. However, it was assumed that the dimensions of both axes cannot be completely arbitrary (e.g., zero), but can take values from the ranges:  
- For the long axis: a range of (8-20) m,  
- For the short axis: a range of (2-5) m.  
The limiting values of the angles \(\rho^{I}\) and \(\rho^{II}\) are derived from the assumed range of velocity components u and v and are: \(\rho^{I}\) = arc kg (0.08/0.2) and \(\rho^{II}\) = arc kg (0.08/0.1). These values are derived from the maximum permissible velocities \(u_{max}\) and \(v_{max}\). It was also assumed that, in zone I (outside the ellipse), the ship moves with maximum velocities, resulting from the instantaneous value of the angle \(\rho\). Inside the ellipse, both velocities decrease in proportion to the distance of the vessel from the trajectory point. Figure 12: The method of determining the value of \(r\) depending on the magnitude of the course deviation \(\psi\), with \(\psi_{k}\) denoting the course set on the previous section of the trajectory, and \(\psi_{k+1}\) denoting the value set on the current section [24]. The position of the vessel relative to the next point of the trajectory (Figure 13). The current trajectory of a vessel outside the current trajectory section (dotted line) determines the angle \(\rho\), and this, in turn, determines the maximum values of \(u_{max}\) and \(v_{max}\). The waveforms of both velocities during the approach to the turning point here are included in the time diagrams: parallel to the trajectory section--the velocity \(u\)--whereas perpendicular--velocity \(v\). It can be noted that the boundaries of the ellipse and circle change the calculation of these velocities. It was also assumed that, in zone I (outside the ellipse), the ship moves with maximum velocities, resulting from the instantaneous value of the angle \(\rho\). Inside the ellipse, both velocities decrease in proportion to the distance of the vessel from the trajectory point. If the vessel finds itself in zone III, the further behavior of the trajectory controller depends on whether the point reached is the last one (\(x_{N}\), \(y_{N}\)) or just the next one (\(x_{k}\), \(y_{k}\)). In the first situation, both velocities are set to zero. Otherwise, the algorithm is switched to the next trinity of numbers (\(x_{k+1}\), \(y_{k+1}\), \(\bar{y}_{k+1}\)), determining the next point of the trajectory. Under such circumstances, the ship will be in zone I relative to the new point and the control begins again. \"Hitting\" zone III (circle) is not an easy task, especially when accompanied by stronger external disturbances. Atmospheric conditions in this case refer to the strength of the wind and the waves that occur on Lake Silm. A mathematical model of the wave module of Lake Silm is currently being developed. A number of wind measurements were made using an ultrasonic anemometer made by GILL Instruments located on the training ship model. Wave measurements were made using a capacitive sensor described in the work [52]. Additional wave measurements were taken on the lake using an experimental system designed and built by Dr. inz. [PERSON], and Dr. inz. [PERSON]. This team of researchers is currently working on a paper about this. Currently, the authors of this paper, in cooperation with Prof. [PERSON] of the University of Gdansk, are analyzing these measurements in order to create a mathematical model of the waves of Lake Silm. Due to the complexity of the problem, this will be the subject of another article. However, it should be emphasized that atmospheric conditions significantly affect the operation of a ship, which is difficult to verify with computer simulations. Therefore, it is important to test the system performance in real conditions.