Analytical Gradients for Gravity Assist Trajectories Using Constant Specific Impulse Engines

Abstract

A procedure for calculating the analytical derivatives required to optimize long duration constant specific impulse finite burns and multiple gravity assist trajectories is presented. The analytical derivatives are calculated using the state transition matrix associated with the complete set of the Euler-Lagrange equations of the optimal control problem on each trajectory segment. Another transition matrix maps perturbations across any discontinuities in the state due to a zero sphere of influence patched conic flyby or discontinuities in the equations of motion that occur when the engine turns on or off. As applications, the method is used to find optimal Earth to Saturn trajectories. The state transition matrix derivatives are shown to find optimal trajectories from sets of initial conditions where finite difference derivatives fail to converge. NomenclatureC = constraint function c = specific impulse e = unit vector for adjoint control transformation g = gravitational acceleration acting on spacecraft H = Hamiltonian function h = orbital angular momentum of spacecraft I = identity matrix J = cost function m = spacecraft mass q = number of gravity assists R = transformation matrix for adjoint control transformation R fb = transformation matrix used to express the flyby v in an inertial frame r = spacecraft position r m = periapse on hyperbolic flyby trajectory S = switching function T = thrust magnitude t = time u = thrust direction unit vector v = velocity v ∞ = velocity of spacecraft relative to flyby planet X = augmented spacecraft state (position, velocity, mass, and costates) x = perturbation of true state from nominal state y = position α = adjoint control parameter α = adjoint control parameter β = flyby angle γ , γ = adjoint control parameter δ = time-free variation δ fb = flyby turn anglẽ δ = time-fixed variation . Member AIAA.ζ = relaxation parameter κ = parameter that affects spacecraft state λ m , λ r , λ v = spacecraft mass, position, and velocity costate λ ⊥ v fb− = component of velocity costate at t fb− that is perpendicular to v ∞i λ ⊥ v fb+ = component of velocity costate at t fb+ that is perpendicular to v ∞o λ v fb− = component of velocity costate at t fb− that is parallel to v ∞i λ v fb+ = component of velocity costate at t fb+ that is parallel to v ∞o µ fb = gravitational parameter of gravity assist planet ρ, υ = Lagrange multiplier Φ = state transition matrix for spacecraft state described by X Subscripts f = final fb = flyby i = inbound max = maximum min = minimum o = outbound p = flyby planet s = switching t = target planet 0 = initial − = immediately before + = immediately after Superscripts * = nominal − = perturbed backward + = perturbed forward & = partial derivative mapped to a common time

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“…However, in this paper we limit our discussion and application to low-thrust, constant specific impulse (I sp ) trajectories. For a sampling of other recent calculus of variations applications to low-thrust trajectory design, see [14][15][16][17][18][19][20].…”

Analytic Gradient Computation for Bounded-Impulse Trajectory Models Using Two-Sided Shooting

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Reducing Orbit Covariance for Continuous Thrust Spacecraft Transfers

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“…Using the state transition matrix to compute derivatives instead of finite differences provides a faster, more robust method to 'walk' from solutions that ignore observability to ones that incorporate observability. 20,21 SECTION IV.…”