# Navigation Safety on Shipping Routes during Construction
## 3 Materials and Methods
### Mathematical Model
To receive the final results on width of the passages, approaches, and inside navigational channels provided by simulators, the maximum distribution method could be used, and its application may be expressed as follows (Equation (6)) [2]:  
\[B_{P}=B_{BM}+k_{n}.P^{\prime}.\sigma_{Y}.R_{n}, \tag{6}\]  
where \(B_{P}\) is width of the approach channel; \(B_{BM}\) is maximum width of the biggest ship or ship's sailing band; \(P^{\prime}\) is factor in probabilistic maintenance (in the case of \(\pm 1\), probability is 68.3%, in the case of \(\pm 2\), probability is 95.3% and in the case of \(\pm 3\), probability is 99.7%); \(k_{n}\) is the coefficient depending on the number of measurements; in case the number of measurements is 3, the coefficient is equal to 0.55, in case the number of measurements is 4, the coefficient is equal to 0.47, in case the number of measurements is 5, the coefficient is  
where \(v_{i}\) is ship's speed module (can be taken from ship's navigational equipment or could be calculated) m/s; \(\omega_{i}\) is turning velocity, rad/s; \(\beta_{i}\) is drift angle, rad; \(v_{i}\), \(\omega_{i}\) and \(\beta_{i}\) can be calculated depending on the sailing conditions using methods presented in [2, 63]. In case of acting current and wind drift, as well as sailing using ship propulsion system, the vessel's trajectory coordinates could be determined as follows, applying Equations (9) and (10) [63]:  
\[X_{0i}=\int_{0}^{t}v_{x}dt+\int_{0}^{t}v_{xsr}dt+\int_{0}^{t}v_{xd}dt, \tag{9}\]  
\[Y_{0i}=\int_{0}^{t}v_{y}dt+\int_{0}^{t}v_{ysr}dt+\int_{0}^{t}v_{yd}dt. \tag{10}\]  
Equations (9) and (10), considering the way of conducting calculations presented in Equations (4) and (5), with the evaluation of the shallow water effect, could be expressed as follows (Equations (11) and (12)):  
\[\Delta Y^{\prime}=\Delta Y+P^{\prime}.\sigma_{Y}, \tag{2}\]0  
\[\Delta Y^{\prime}=\Delta Y+P^{\prime}.\sigma_{Y}, \tag{2}\]1  
Accuracy of the ship's trajectory calculated by applying Equations (7) and (8), according to methodology expressed in [2], including tug assistance for the increasing of ships' maneuverability, could be less than 10% of the ship's circulation diameter. Finally, ship's sailing trajectory on the bends during maneuvering could be presented as it was shown in \[\Delta Y^{\prime}=\Delta Y+P^{\prime}.\sigma_{Y}, \tag{2}\]2  
Considering that channels are crossed by various ship types, it is rather difficult to adopt maneuvering characteristics of all ships'; therefore, it is necessary to evaluate the parameters of biggest ships when they are sailing via passages with obstacles. Therefore, according to developed methodology, the navigational conditions for the analyzed sailing area, as well as vessel's possibilities to cross it, should be evaluated and the accuracy of ship's position should be determined. Ship's trajectory on the bends, in case acting of wind and current, as well as shallow water effect influence, could be calculated using Equations (11) and (12). It should be highlighted that presented methodology for the passage parameters calculation can be applied to any navigational area. In the case of acting wind and current on the maneuvering ship, the ship's kinematic parameters and the part of the ship's trajectory could be calculated by applying the methodology presented in Section 3. The appropriate calculation results for the selected conditions (taken as an example to present the possibility of implementing the developed methodology) are shown in Figures 6 and 7. \[\Delta Y^{\prime}=\Delta Y+P^{\prime}.\sigma_{Y}, \tag{2}\]3 Kinematic parameters of ship motion during circulation with current perpendicular to the port side at the beginning of the circulation (\(v_{sr}=2.0\) knots) and wind perpendicular to the port side at the beginning of the circulation (\(v_{a}=15\) m/s) [own elaboration based on [62]].