Reentry Trajectory Optimization: Evolutionary Approach

Abstract

This paper shows the application of Genetic Algorithm to Reentry trajectory optimization problem in the presence of constraints such as dynamic pressure and heat flux. The coupled non linear equations of motion of reentry body are difficult to solve using conventional methods like finding costate equations and then applying standard optimization methods. As an alternate, stochastic way of optimization through genetic algorithm is proposed. The results for a typical Reusable Launch Vehicle – Technology Demonstrator are in good match with that of Space Shuttle. Nomenclature CD = drag coefficient CDo = constant CL = lift Coefficient CLα = CL as a function of α, /rad D = drag force, N FT = thrust force, N go = gravity at earth surface, m/s 2 g = gravity at some altitude, m/s h = altitude, m Isp = specific impulse, N s/kg k = constant L = lift force, N LH = local horizontal plane M = total mass, kg Q = dynamic pressure, Pa q = heat flux, W/cm ro = radius of earth, m r = radial vector, m RN = nose radius, m S = area, m U = control vector V = velocity, m/s α = angle of attack, deg β = constant φ = latitude, deg γ = flight path angle, deg ρo = density at earth surface, kg/m ρ = density at some altitude,kg/m θ = longitude, deg σ = bank angle, deg ψ = heading angle, deg Introduction The concept of Reusable Launch Vehicle (RLV) is evolved in order to minimize cost per unit payload. Nearly 70% of cost of a launch vehicle lies in structure and avionics system, recovering the same and using it for further flights (after refurbishing) will be less expensive. The technologies like TPS material, structure and avionics system for a full fledged RLV are highly challenging. As a cost effective measure, a smaller version of RLV is proposed called as Technology Demonstrator (RLV-TD) as given in Fig.1 which will validate the above mentioned technologies in a step by step manner. Fig.1 RLV – Technology Demonstrator In the reentry trajectory design one has to ensure that with minimum fuel consumption the desired landing site is reached satisfying constraints such as dynamic pressure and heat flux. Traditionally reentry trajectory optimization is carried out by non dimensionalising the equations of motion and then solving it through gradient method using algorithms such as SGRA , Brysons. As the equations of motion of a reentry body are coupled non linear equation, it gives complicated co state equations which are difficult to solve and simulate. As an alternate, evolutionary approach is proposed for this type of problems. Genetic Algorithms(GA) are search algorithms based on the mechanism of natural selection. They lay on one of the most important principles of Darwin : survival of the fittest. Globally we make use of the population, submitted to many transformations. After some generations, the population endues no more; the best individual represents the optimal solution. GAs do not require 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization 4-6 September 2002, Atlanta, G orgia AIAA 2002-5466 Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 2 American Institute of Aeronautics and Astronautics derivative or continuity of a function to be evaluated. They are simple, fast converging and easy to implement and simulate. Equations of motion of a reentry body The equations of motion for a lifting trajectory are given by (Fig.2) ψ γ φ φ ψ γ θ γ γ σ φ ψ γ ψ σ γ γ γ sin cos cos cos cos sin cos sin tan cos cos cos cos sin r V r V V r MV L r V MV L V g r V M F M D g V T = = = + − =   +   − = − − − =