Optimal Cooperative Collision Avoidance Strategy for Coplanar Encounter: Merz’s Solution Revisited

• Published in: Journal of optimization theory and applications Vol. 140; no. 2; pp. 355 - 375
• Main Authors: Tarnopolskaya, T., Fulton, N.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.02.2009; Springer; Springer Nature B.V
• Subjects: Aircraft; Altitude; Applications of Mathematics; Applied sciences; Approximation; Automation; Calculus of Variations and Optimal Control; Optimization; Collisions; Computer science; control theory; systems; Control theory. Systems; Engineering; Exact sciences and technology; Information science; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimal control; Optimization; Optimization techniques; Studies; Theory of Computation; Velocity; Cooperative maneuvers; Collision avoidance; Optimal control; Mayer problem; Partition; Navigation; Modeling; Exact solution; Airplane; Optimal strategy; Variational inequality; Air traffic; Automatic control; Ship; Aircraft
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-008-9452-9

Analytic solutions for optimal collision avoidance strategies are of great importance when setting and validating air traffic rules and as a benchmark when validating automated proximity management and collision avoidance systems. Such a solution for optimal air collision avoidance strategies for a coplanar cooperative encounter between two identical aircraft (or ships) was first presented by Merz (Proc. Joint Automatic Control Conf., Pap. 15-3:449–454, 1973 ; Navigation 20(2):144–152, 1973 ). Unfortunately, Merz provided only a very brief indicative justification for his solution. This paper presents a rigorous analysis of the problem. New results include a characterization of a complete set of extremals, justification for optimal strategies and an analysis of the properties of the regions of different optimal strategies. A simple, practical and sufficiently accurate closed form approximation for dispersal curves that partition the plane of initial positions into the regions of different optimal strategies is also presented.