Launch Vehicle Trajectories with a Dynamic Pressure Constraint

Abstract

The problem of three-dimensional launch trajectoriesfortheadvanced launch system to minimize fuel orequivalently maximize payload isconsidered. Adynamicpressureinequality constraintis involved dueto itssignie cance on the resulting trajectories and structure of the launch vehicle. Highly accurate solutions and exact switching structures of the optimal trajectories are presented by using a multiple shooting method. The multiple shooting structure is modie ed to satisfy the internal boundary conditions and discontinuities in variables at entry and exit points of the boundary arc. A methodology is also developed to visualize the optimal controls in a geometric sense using hodograph analysis. Nomenclature Ab = cross-sectional area Ae = exit area of a engine a = speed of sound CD = drag coefe cient CL = lift coefe cient Cm = pitching moment coefe cient Cmcg = pitching moment coefe cient about the center of gravity c = number of engines operating D = drag e = eccentricity of orbit f = dynamic equations G = function of states, controls, and time g = Earth’ s gravitational force gs = Earth’ s gravitational force at sea level H = Hamiltonian function h = altitude above mean sea level ha = apogee altitude h f = altitude at tf h p = perigee altitude h1 = density scale height h2 = pressure scale height Isp = specie c impulse J = performance index L = lift l = total core vehicle length lT = distance from the center of mass to the exit plane of engine M = Mach number Maero = aerodynamic pitching moment MT = pitching moment due to thrust m = mass mref = reference mass, sum of the masses of the payload, payload margin, and payload fairing p = atmospheric pressure q = dynamic pressure r = distance from the center of the Earth to the vehicle center of gravity, rs C h ra = radius at apogee from the center of the Earth rp = radius at perigee from the center of the Earth rs = radius of the Earth at sea level S = state inequality constraint Sp = pth time derivative of S