peg_example - Ascent Trajectory Optimization Powered...

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Unformatted text preview: Ascent Trajectory Optimization Powered Explicit Guidance (PEG) • Powered Explicit Guidance (PEG) is the iterative guidance algorithm that solves an optimal control problem in order to determine the thrust-steering vector profile for 2 nd stage (after SRB separation) PEG is derived from the Saturn V “Iterative Guidance Mode” (IGM) PEG is currently the baseline guidance for 2 nd-stage of the Crew Launch Vehicle (CLV) PEG: SRB separation to MECO (orbit) PEG: Optimal Control Problem • Determine the thrust-steering direction (unit vector u ) which minimizes J = t f (total flight time) subject to the EOM: • Subject to the four terminal state constraints: g u V V r + = = T a ) ( =- T f r t r ) ( =- T f V t V ) ( =- T f t γ γ ) ( =- T f i t i Reach target radius and velocity (magnitudes) Reach target flight-path angle Reach target inclination (orbital plane) g = Earth gravity a T = T / m = thrust acceleration Hamiltonian and Costate Equations • Form the Hamiltonian: • Costate differential equations: ( 29 ( 29 u u g u λ V λ f λ T T T V T r T a u x H- + + + = = 1 ) , ( σ V T r H λ r g r λ ∂ ∂- = ∂ ∂- = Added constraint to enforce unit- vector length for control term u r V H λ V λ- = ∂ ∂- = “position costate” vector “velocity costate” vector Optimal Steering • Optimality condition: • However, u* must be a unit vector, so “square” both sides (take inner products) 2 =- = ∂ ∂ u λ u σ T V a H Sub for σ in u * equation: T V a λ u σ 2 1 * = 1 4...
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This note was uploaded on 01/12/2011 for the course MAE 4720 taught by Professor Dr.kluever during the Spring '10 term at Missouri (Mizzou).

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peg_example - Ascent Trajectory Optimization Powered...

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