Gradient-based approach to formalizing the distance at the closest point of approach for ship collision avoidance problem

This study develops a gradient-based formalization of the closest point of approach (CPA) distance of ships, intended as a foundation for analyzing navigational situations and planning collision avoidance maneuvers. The relevance of the study stems from the necessity for a rigorous, clear, and visually interpretable description of the CPA distance. The research investigates the process of ship collision avoidance and examines the properties of the scalar field of the CPA distance within the space of controlled ship speed components along both the meridian and the parallel. The objective of the work is to obtain a gradient-based representation of the CPA distance and to interpret the resulting data from the perspective of practical navigation. The methods employed comprise established mathematical approaches in navigation, which describe navigational parameters through the magnitude and direction of their gradients, as well as the geometric interpretation of both the scalar field of the CPA distance and the vector field of its gradient. Both the magnitude and direction of the CPA gradient are explicitly determined, and their relationship with the time to CPA and the geometry of CPA isolines is systematically analyzed. The study demonstrates that the orthogonality of the gradient relative to the corresponding isolines provides a natural and effective basis for selecting the most appropriate collision avoidance maneuver. Moreover, the research substantiates the use of gradient-based formalization in collision risk assessment and the design of algorithms for ship collision avoidance. The main conclusions confirm that the CPA gradient possesses independent navigational significance and should be regarded as a fundamental characteristic of the ship collision avoidance process, providing both analytical insight and practical guidance for the safe and efficient maneuvering of controlled vessels within their navigational domain.