Trajectory optimization with risk minimization for military aircraft

Abstract

Methods of time-controlled optimal flight-trajectory generation are developed that include the effects of risk from a threat environment. Lateral and vertical algorithms are developed for military jet aircraft with the intent of near real-time application. Simple analytic functions are used for threat models, as the focus of this paper is on general problem formulation and not detailed solutions for specific threats. A constant altitude, lateral flight-trajectory generation method is developed that optimizes with respect to time, fuel, final position, and risk exposure. Existing vertical plane trajectory generation methods that use standard direct operating costs of time and fuel are modified to include the effect of risks. Singular perturbation methods are used to obtain reduced-order airplane models that allow static rather than dynamic optimization. Pontryagin's Minimum Principle is used with a Fibonacci search method to minimize the cost functional. Formulation and numerical results are presented for both the horizontal and vertical plane problems. N the 1970's, rising fuel prices and improved micropro- cessor capability made consideration of onboard flight- path optimization appealing as a way to reduce operating costs. More recently, increased air traffic and the associated control problems have shifted research emphasis toward four- dimensional trajectory optimization. The four-dimensional problem typically includes a cost of time that allows trajecto- ries to meet required times of destination arrival. The majority of research in this area has been directed toward commercial operations and involves using the energy state approximation for generating vertical flight paths. It has been shown1^ that singular perturbation theory (SPT) may be used successfully to reduce the order of this problem. A reduced-order, often static, optimization technique can then be employed that greatly reduces computational burden. In this paper, methods of trajectory generation are developed that expand the basic SPT techniques to address the needs of military operations. Trajectory generation will be a fundamental requirement for future military aircraft flight management systems. These systems will be required to take advantage of all available information to perform integrated task processing and reduce pilot workload. In addition, the systems should be able to provide updates at any time throughout a mission or at regular time intervals sufficient for threat avoidance. Much of the previous work in trajectory generation for military air- craft57 has concentrated on feasible directions algorithms that use dynamic programming. These methods tend to be computationally intense and, therefore, are not well suited for onboard applications in dynamic threat environments. An ideal flight trajectory for military operations meets the mission requirements within the constraints of the aircraft limitations while minimizing exposure to threats. Several lev- els of information are used in the selection of such a strategic flight path and velocity profile. The trajectory will be a function of mission requirements (time of arrival, point of arrival, etc.), aircraft performance limitations (for example, fuel quantity and thrust limits), and the threat environment. The threat models used in this study do not refer to any specific military scenario. They are simple analytic functions