Optimal Aircraft Routing in General Wind Fields

Abstract

A IRCRAFT navigation involves routing and guidance of airplanes as a part of flight planning. In this Note, optimal aircraft routing refers to route determination for an airplane traveling horizontally between two given points so that the transit time is minimized, assuming that the meteorological conditions (in particular, the wind field) are fully known beforehand for the complete passage. Aircraft guidance refers to guiding the aircraft along this route. The determination of the minimal-time path for an airplane was first accomplished by Zermelo [1] in a lecture in Prague, which gave rise to a number of publications in this field, including those in which the analogy between optical and minimum-flight problems was pointed out [2]. The actual interest in the problem came after the Second World War, when regular long-distance flights came into operation. After some less successful attempts, a useful manual method for the calculation of minimal-flight paths was developed by introducing time fronts analogous to wave fronts in geometrical optics [3]. The introduction of the computer made this manual method obsolete. The obvious next step was to program appropriate methods such as applications of the calculus of variations and graph theory for computer application. Although the minimal-flight problem can be formulated very simply as a problem of the calculus of variations, the solutionmust generally be obtained by iteration so that apart from the fact that convergence problems may occur, the iteration process could converge to a relative minimum instead of converging to an absolute minimum. On the other hand, the graph method always yields an absoluteminimum, but at the cost of large computation time and memory space. As an ultimate application of the calculus of variations, an extension of the technique of neighboring optimal control was introduced to compute near-optimal trajectories in general wind fields, starting from nominal solutions to the Zermelo problem obtained with simple analytical or zero wind fields [4]. In [5], it was claimed that the excellent performance of this approach in practice is achieved because winds typically vary in a smooth manner and do not contain many sharp nonlinearities or discontinuities. However, the wind does not generally vary in a smooth manner. Consequently, the choice of a nominal solution is not always obvious and neither is the convergence to a near-optimal trajectory, as we will see in the next section. Inefficient routingmay result in excess fuel burn and, accordingly, in excess emissions. Therefore, combining the approaches of the calculus of variations and graph theory, we propose a method that always yields an optimal solution with moderate computational effort and memory space. The minimal-flight problem is a simple example of the control problem of Bolza [6] from the calculus of variations. Integration of the model equations and varying the initial heading yields a one-parameter family of solutions (extremals) emanating from the point of departure and, under certain conditions, continuous in their dependence on this parameter. New extremals can be inserted at a specific time if the distance between two adjacent extremals becomes too large at that time. If this distance is chosen to be sufficiently small, the starting values of the new extremal can be obtained by linear interpolation between the corresponding values of its neighbors. In this way, a network is built up containing points that can be reached along minimal-time tracks after a certain number of time steps. The ultimate minimal-time track from origin to destination is obtained by selecting that extremal, which ends closest to the destination. This method was tested extensively during the 1970s at the Royal Netherlands Meteorological Institute in many practical situations related to minimal-time ship routing [7]. In the following section, the problem of Bolza is discussed in relation to minimal-time aircraft routing for constant airspeed. Next, this discussion is generalized for airspeeds, which may depend on position, time, and heading, and the relation with minimal-time ship routing is elucidated. Conclusions are presented in the last section.