## 3 Problem Research
### Ship Control Technology Problem
#### 3.2.2 Trajectory Control in Use  
The research was conducted using a simple master system. It converts the desired trajectory and the set course at each section into the corresponding signals preset linear velocities \(u,v\) and angular velocity \(r\). A detailed description of the workings of the system is recorded in the paper [24]. The set values in the experiment in question are represented by a set of numbers: the coordinate values of the next point of the trajectory and the ship's course on the section to that point: \([x_{1}\),\(y_{1}\), \(\psi_{1}]\), \([x_{n}\),\(y_{n}\), \(\psi_{n}]\), \([x_{n+1}\),\(y_{n+1}\), \(\psi_{n+1}]\). In the system that determines the set values of the linear components of velocity, two angles can be defined: \(\kappa\) as the angle of the current section of the trajectory, which is calculated relative to north, and \(\rho\) as the angle between the longitudinal axis of symmetry of the ship and the line from the ship to the end of the current section of the trajectory, as shown in Figure 8.  
The angle \(\rho\) determines the maximum instantaneous values of the set velocity \(u\) and \(v\). The values are determined as shown in Figure 9.  
Figure 8: The Reprinted/adapted with permission from Ref. [24] Copyright year, copyright owner’s [PERSON] angles \(\kappa\) and \(\rho\) determining the position of the ship on the current segment of the trajectory between the points \((n)\)\(i\)\((n\)+1) [24]. The points, located on its sides, determine the values of \(u_{max}\) and \(v_{max}\) from the ranges given in the point (Figure 10). The instantaneous setpoint \(r\) at a given section \(\kappa\) of the trajectory (Figure 11) and at a given course \(\psi_{1}\) is determined from the relation. \[r = \left\{ \begin{array}{ll}+0.3&\mathrm{dla}\ \ \psi_{k}-\psi>+40[deg] \\ -0.3&\mathrm{dla}\ \ \psi_{k}-\psi<-40[deg]\\ \end{array} \right.\] \[r = \left\{ \begin{array}{ll}+0.3\frac{\psi_{k}-\psi}{40}&\mathrm{dla} \ \ +2[deg]\leqslant\psi_{k}-\psi\leqslant+40[deg]\\ -0.3\frac{\psi_{k}-\psi}{40}&\mathrm{dla}\ \ -2[deg]\geqslant\psi_{k}-\psi >-40[deg]\\ \end{array} \right.\] \[r = \left\{ \begin{array}{ll}+0.1\frac{\psi_{k}-\psi}{40}&\mathrm{dla} \ \ \psi_{k}-\psi<+2[deg]\\ -0.1\frac{\psi_{k}-\psi}{8}&\mathrm{dla}\ \ \psi_{k}-\psi>-2[deg]\\ \end{array} \right.\]  
Where \(\psi\) is the instantaneous value of the exchange rate. The quantities \(\psi_{min}\) = 2 deg and \(\psi_{max}\) = 40 deg are the limiting values of the course deviation, determining the change (switch) in the formula for calculating \(r\). They were determined on the basis of the analysis of the course of the training ship during turns, so that course changes are made as far as possible without overshooting. The concept of determining the set value of angular velocity is shown in Figure 12.  
Figure 11: Graphical presentation of the relationship between velocity \(r\) and \(\psi\)[24]. Figure 9: Rectangle Reprinted/adapted with permission from Ref. [24] Copyright year, copyright owner’s [PERSON] of maximum allowable set values of linear components of speed. Numbers 1–4 indicate parts of the sides of the rectangle, on which there are different rules for calculating \(u_{max}\) and \(v_{max}\) max. The angles \(\rho^{I}\) and \(\rho^{II}\), as well as the horizontal axis, mark the limits of application of each rule \(\psi\) and \(\rho\)[24]. For the calculation of the current values of the u and v components, it was assumed that the ship, while moving between one point of the trajectory and the next, could be in one of three zones--see also Figure 13.  
- A circle with a fixed radius of 1 m around each trajectory point (Zone III),  
- An approximation ellipse with the center at the trajectory point and a long axis coinciding with the current trajectory segment (Zone II),  
- The space outside the ellipse (Zone I). The dimensions of the ellipse are not fixed. The lengths of the two axes depend proportionally on the maximum values of \(u_{max}\) and \(v_{max}\).