# Navigation Safety on Shipping Routes during Construction
## 3 Materials and Methods
### Mathematical Model  
During the determination of ship's possibility of passing navigational channel, the fluctuation of ship's trajectory should be considered. This fluctuation may depend on the internal and external factors influencing ships' maneuvering characteristics and the accuracy of ship's position observation. Therefore, it is important to possess available information about ship's position and its coordinates at any time of its movement through navigational channel, as well as assess the possible fluctuations of the ship's trajectory (Figure 4). Ship's trajectory coordinates fluctuation (\(\Delta X^{\prime}\) and \(\Delta Y^{\prime}\)), as shown in Figure 4, can be calculated as follows (Equations (1) and (2)):  
\[\Delta X^{\prime}=\Delta X+P^{\prime}.\sigma_{X}; \tag{1}\]  
\[\Delta Y^{\prime}=\Delta Y+P^{\prime}.\sigma_{Y}, \tag{2}\]  
where \(\Delta X\) is ship's sailing trajectory on \(X\) coordinate fluctuation; \(\Delta Y\) is ship's sailing trajectory on \(Y\) coordinate fluctuation; \(P^{\prime}\) is probability coefficient, in case the coefficient is equal to 1, probability is about 63-68%; in case the coefficient is equal to 2, probability is about 93-95%; in case the coefficient is equal to 3, probability is about 97-99%; \(\sigma_{X}\) and \(\sigma_{Y}\) are accuracies of the ship's position observation in \(X\) and \(Y\) coordinates depending on type of ship's position observation equipment and applied methods; this fluctuation may be between 0.05 m (using Real-Time Kinematic (RTK) system) up to 15-25 m (using ship's or shore radar systems) [63]. Figure 4: Possible ship’s position during maneuvering [own elaboration], where \(l^{\prime}\) is the total distance between calculated trajectory and possible fluctuation from ship’s sailing trajectory (line). The total ship's position fluctuation on the passage may be assessed at any time on the basis of available information about ship's position and its coordinates. This position can be calculated using Equation (3):  
\[l^{\prime}=\sqrt{(\Delta X^{\prime})^{2}+(\Delta Y^{\prime})^{2}}. \tag{3}\]  
This measurement and the obtained values are necessary to plan the path for crossing the channel bands for ships, where the maneuvering accuracy should be estimated in detail (Figure 5). For the purpose of the research, the passage's main parameters were defined as passage minimum width and depth, allowing them to cross the restricted area by ships with possible maximum main parameters (length, width, draft), assuring their maneuverability. Requested passages widths and depths could be calculated or selected based on recommendations provided, for example, by Permanent International Association of Navigation Congresses (PIANC) [11], Recommendation for Design of the Maritime Configuration of Ports, Approach Channels and Harbour Basins (ROM) [12], and others. Ship passage minimum width (\(B_{p}^{\prime}\)), considering the parameters of biggest ships, which can sail via narrow passage, could be calculated based on mentioned recommendations [11; 12], as well as may be achieved from simulators [62] or obtained from the real ships (using Marine traffic system [64] or other platforms).