Closed-Loop Endoatmospheric Ascent Guidance

Abstract

This paper will present a complete formulation of the optimal control problem for atmospheric ascent of rocket powered launch vehicles subject to usual load constraints and final conditionconstraints. We shall demonstrate that the classical finite difference method for two-point-boundaxy-value-problems (TPBVP) is suited for solving the ascent trajectory optimization problem in real time, therefore closed-loop optimal endoatmospheric ascent guidance becomes feasible. Numerical simulations with a the vehicle data of a reusable launch vehicle will be provided. AscentGuidance Problem FormulationThe equations of motion of the RLV in an inertial coordinate system can be expressed aswhere r and V are inertial position and velocity vectors; g the gravitational acceleration;T the thrust magnitude; A and N are aerodynamic axial and normal forces, respectively; lb the unit vector defining the RLV body longitudinal axis; m is the mass of the RLV. The magnitudes of the aerodynamic forces and thrust are modeled bywhere p is the atmospheric density, and Vr is the magnitude of the Earth-relative velocity V, = V -_E X r with _s being the Earth angular rotation rate vector.

Galerkin Optimal Control

“…Of the direct methods, specifically the global orthogonal collocation methods (a.k.a. pseudospectral methods) have proven to solve difficult problems with great ac-curacy [4,8,34,35] after becoming an actively researched topic in the 1990s by Elnagar et al [1] and Fahroo et al [2].…”

“…The nodal interpolation of the function x(t) can be accomplished by the N -th order expansion 34) wherex N j = x N (t j ), for j = 0, 1, . .…”

A Unified Pseudospectral Framework for Nonlinear Controller and Observer Design

“…Therefore, an optimal solution always exists. More importantly, (19) shows the existence of a feasible discrete solution around any neighborhood of the continuous trajectory. The convergence of PS methods for nonlinear optimal control problems has been proved in [12], [13] in a way similar in spirit to Polak's theory of consistent approximations [24].…”

“…This is the approach used in DIDO [31], a MATLAB application package for solving dynamic optimization problems. In recent years, PS methods have been successfully applied to solve a wide variety of optimal control problems, [6], [26], [12], [32], [34], [15], [19], [23], [16], [27]. As a result of its success, PS methods are now part of OTIS [22], NASA's software package for solving trajectory optimization problems.…”

Costate Computation by a Chebyshev Pseudospectral Method

“…As exemplified by its experimental and flight applications in national programs [6][7][8][9][10], it is not surprising that the Legendre PS method has become the method of choice [11][12][13][14][15][16][17][18][19] in both industry and academia for solving optimal control problems. Efforts to improve the Legendre PS methods by using either other polynomials [20][21][22] or point distributions [23,24] have not yet resulted in any rigorous framework for convergence of these approximations [24,25].…”