Exact Analytical Solution for Three-Dimensional Interception of a Maneuvering Target

Abstract

The three-dimensional guidance problem has been extensively discussed in the open literature for several decades. However, because of the nonlinearity of the equations of relative motion, the analytical solution is generally expressed in the form of nonintegrable quadratures. A partial closed-form solution for a guidance problem, in which the relative commanded acceleration is applied in the direction normal to the line-of-sight, was obtained by previous researchers. In this paper, we generalize that guidance law to include a component of the relative commanded acceleration along the line-of-sight. In this more general case, in the plane of relative motion which is always parallel to a fixed plane, the relative trajectory is obtained explicitly in closed form. Furthermore, while the previous analysis considered only special integer values for the sole navigation constant, the present analysis obtains the analytical solution for a more general guidance law, and arbitrary values of the two control parameters involved. The relative range, the line-of-sight swept angle, and the flight time are analytically expressed in terms of each other. An application to the interception of a satellite in circular orbit is presented.