Explicit guidance along an optimal space curve

Abstract

An explicit guidance theory is developed for maneuvering to a prescribed destination with terminal constraints on velocity vector direction. Motion is constrained to an optimal, three-dimensional space curve by constraint forces perpendicular to velocity (lift). Lift components are derived by twice differentiating functions specifying radial distance and geocentric latitude as functions of longitude. An optimal space curve is determined by solving a two-point boundary-value problem in the calculus of variations. Necessary conditions for an extremum are 1) a set of coupled, fourth-order, Euler-Lagrange differential equations for the space curve functions; 2) a single, first-order differential equation for the adjoint variable; and 3) boundary conditions specified at two ends of the trajectory. Although energy is not conserved because of drag, motion along the space curve is integrable because lift-induced drag is determined by trajectory curvature. Velocity along the space curve may be expressed by a quadrature evaluated by the method of successive approximation to refine the accuracy of the compressibility drag slowdown. Background F UTURE maneuvering vehicles, such as the space plane, will require advanced midcourse guidance algorithms to optimize performance and arrive at a prescribed destination with terminal constraints on flight path. During flight in the atmosphere, vehicle orientation relative to the velocity vector (angle of attack) is controlled to generate the required acceleration perpendicular to velocity (lift). Many explicit guidance algorithms have been developed for lift-controlled entry vehicles. l~21 The guided trajectory problem is not generally integrable, except in certain cases (discussed shortly). Integrable cases for lifting trajectories include constant lift-to-drag L/D ratio, constant-bank angle, and equilibrium glide at constant flight-path angle. Hodograph space solutions express velocity magnitude by a function of turning angle, and the configuration space trajectory is determined t>y a quadrature.48 These approximate solutions are useful for preliminary design of midcourse trajectories satisfying mission performance objectives within vehicle aerothermodynamic limitations (trim, loads, and heating). Direct methods have been used extensively to develop explicit, optimal guidance algorithms. For a vehicle with bounded lift control, optimal range extension maneuvers consist of maximum and minimum L/D subarcs connected by intermediate cruise segments.911 Numerical solutions may be ill-conditioned because switching points must be determined to satisfy the boundary conditions. Green's Theorem may be applied to determine the sequence of maximum and minimum L/D subarcs (without cruise segments) that optimize performance while satisfying end conditions on altitude and flight path.12-13 Optimal, modulated-lift trajectories admit approximate analytic solutions characterized by a change of independent variable from time to an appropriate trajectory variable. For example, proportional navigation guidance minimizes control effort or time-integrat ed lift acceleration magnitude, and closed-form trajectory solutions are obtained when the new independent variable is line-of-sight angle.1416 For maximum velocity turns to a specified heading, approximate optimal control histories were derived from an integrable system of equations using flight-path angle17 or range18 as new indepen