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.
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.
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