Time-Optimal Turn to a Heading: An Improved Analytic Solution

Abstract

Aircraft turning performance in the horizontal plane is addressed. The task is to reach a speci ed heading and speed in minimum time. The objective is to obtain a closed-form feedback expression for the optimal turn rate in the case in which load-factor constraints do not become active. The starting point is the boundary-value problem derived from the necessary conditionsof optimal control. An analyticsolution results for constant maximum thrust and a parabolicdrag polar with constant coef cients. The extension to previous results is twofold: First, the solution is determined for all boundary conditions in the ight envelope. In many cases a composite structure made up of two different control types occurs. Second, the dynamic model includes the drag induced by the vertical lift component, which was neglected in previous work. The feedback expression for the optimal turn rate is utilized to guide an aircraft with realistic dynamics. The desired terminal conditions are reached accurately as long as load-factor constraints do not become active. The agreement of simulated ight paths using this guidance law and optimal turns obtained from open-loop trajectory optimization depends on the particular boundary conditions. Nomenclature c D0 = zero-lift drag coef cient c L = lift coef cient D = drag g = gravitational acceleration k = factor in the drag polar M = Mach number m = aircraft mass q = dynamic pressure S = reference wing area T = thrust u = turn rate P  u s = steady-state turn rate V = speed W = weight, m £ g ¹ = bank angle ½ = air density  = heading angle