Approximate Analytical Solutions for Launch-Vehicle Ascent Trajectory

Abstract

Approximate analytical solutions for the ascent trajectory of a solid-fuel launch vehicle are derived. To the best of the authors' knowledge, the solutions are the ones achieving the highest accuracy in ascent-trajectory prediction under the condition of high angle-of-attack (AOA) maneuvers, where the magnitude of AOA can even be greater than 60 deg at high altitudes. As the first step of deriving the solutions, a reduced-order dynamics model with a normalized mass as the independent variable is put forward for longitudinal-plane flight. To further simplify the model, the sine of AOA is chosen as the key parameter for trajectory control and designed as a polynomial of the normalized mass. Meanwhile, an improved aerodynamic model with the sine of AOA as the independent variable is developed. Due to high AOA, the simplified dynamics model is still highly nonlinear and difficult to solve. To overcome this difficulty, by performing force analysis, several approximate polynomials and first-order Taylor series expansions are created to replace some highly nonlinear but relatively small terms in the dynamical equations, while the errors of the approximate formulae relative to the original nonlinear terms are retained and treated as minor perturbations. The benefit is that the modified dynamics model can be divided into two analytically solvable subsystems using a perturbation method. By solving the subsystems, the approximate analytical solutions for downrange, altitude, velocity and flight-path angle (FPA) of the vehicle are obtained successfully. Simulation results verify that the proposed solutions are much more accurate than the existing solutions in the scenario of high AOA flight.