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

Singular controls in optimal collision avoidance for participants with unequal linear speeds

“…The first letter in the strategy notation, R or L, indicates the strategy of the faster participant (located at the origin of Figure 1). As was shown by Merz, Tarnopolskaya et al [4,6,7,8], for given initial conditions, one of these strategies is optimal for the whole duration of the encounter if the linear speeds of the participants are equal.…”

“…We make the same assumptions as in the close proximity encounter models [4,5,6,7,8,9,10,11,12]. Because of a short encounter time, a common assumption in such models is that the linear speeds of the participants are constant.…”

“…The problem of optimal cooperative collision avoidance of two participants in a planar close proximity encounter can be formulated as an optimal control problem characterized by a threedimensional state vector, a two-dimensional control vector, a terminal cost functional, and a free terminal time [4,5,6,7,8,9,10,11,12]. The performance objective is to maximize the distance between the participants at the terminal time T .…”

“…The performance objective is to maximize the distance between the participants at the terminal time T . In this problem, the controls are the non-dimensional angular speeds of the two participants, which are bounded functions of time on a Merz, Tarnopolskaya et al [4,5,6,7,8,9,10,11,12] showed that optimal collision avoidance strategies for participants with equal linear speeds require bang-bang control strategies that remain constant for the whole duration of the encounter. This means that, in order to avoid a collision, each participant should turn with maximum allowed angular speed.…”

Synthesis of Optimal Control for Cooperative Collision Avoidance in a Close Proximity Encounter: Special Cases

“…Such representation is especially useful when establishing and validating the air traffic rules. Recently, Merz's solution has been re-examined (Tarnopolskaya and Fulton, 2009) and extended to the encounter of participants with unequal turn capabilities but equal linear speeds (Tarnopolskaya and Fulton, 2010a, b, c).…”