A New Method for Optimizing Multiple-Flyby Trajectories

D’Amario, Louis A.; Byrnes, Dennis V.; Stanford, Richard H.

doi:10.2514/3.56115

Cited by 32 publications

(13 citation statements)

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“…If such a formulation is to be solved using gradient-based methods, the solver will require partial derivatives of solutions to the Kepler initial value problem (not the Lambert boundary value problem). A less common, alternative optimization formulation [31] embeds the Lambert problem solution inside each major iteration to implicitly enforce the boundary value constraints without the need for a shooting method. If such a formulation is to be solved using gradient-based methods, the solver will require partial derivatives of solutions to the Lambert boundary value problem.…”

mentioning

confidence: 99%

“…The results serve as validation for the first approach and represent a standalone contribution. A similar method for a custom problem has been previously proposed [31] and was used to optimize Galileo's trajectory [13]. It is noted that D'Amario et al [31] gave explicit formulas, without derivation, for the partials of the solution to the perturbed Lambert problem, provided a state transition matrix was available.…”

mentioning

confidence: 99%

“…A similar method for a custom problem has been previously proposed [31] and was used to optimize Galileo's trajectory [13]. It is noted that D'Amario et al [31] gave explicit formulas, without derivation, for the partials of the solution to the perturbed Lambert problem, provided a state transition matrix was available. In the current study, new equivalent formulas are carefully derived in terms of the partitions of the fundamental perturbation matrices, first introduced by Battin, as stated in his seminal book ( [4] p. 467).…”

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confidence: 99%

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Partial Derivatives of the Solution to the Lambert Boundary Value Problem

Arora

Russell

Strange

et al. 2015

Journal of Guidance, Control, and Dynamics

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Two methods for deriving first-order partial derivatives of the outputs with respect to the inputs of the Lambert boundary value problem are presented. The first method assumes the Lambert problem is solved via the universal vercosine formulation. Taking advantage of inherent symmetries and intermediate variables, the derivatives are expressed in a computationally efficient form. The typical added cost of computing these partials is found to be ∼15 to 35% of the Lambert computed cost. A second set of the same partial derivatives is derived from the fundamental perturbation matrix, also known as the state transition matrix of the Keplerian initial value problem. The equations are formulated in terms of Battin's partitions of the state transition matrix and its adjoint. This alternative approach works with any Lambert formulation, including one that solves a perturbed Lambert problem, subject to the availability of the associated state transition matrix. The analytic partial derivatives enable fast trajectory optimization formulations that implicitly enforce continuity constraints via embedded Lambert problems. Nomenclature a, A = notation for vector (column) and matrix, respectively a b = ∂a∕∂b, where a and b are general variables d = transfer angle parameter (1 for 0 < θ < π, −1 for π < θ < 2π) E = eccentric anomaly F = hyperbolic eccentric anomaly f = Lagrange function (part of f and g functions) g = Lagrange function (part of f and g functions), TU k = universal variable for solving Lambert equation using vercosine formulation k = value of k corresponding to T , TU LU = length unit N rev = number of revolutions q = generic variable name representing a scalar function r 1 ; r 2 = initial and final position vectors, respectively (input to Lambert problem), LU S = geometry parameter appearing in Lambert equation, TU T p = parabolic time of flight, TU T = target time of flight (input to Lambert problem), TU t 1 ; t 2 = initial and final times, respectively, TU TU = time unit v 1 ; v 2 = initial and final velocity vectors, respectively (output to Lambert problem), LU/TU W = auxiliary function appearing in Lambert equation δ; d = variation and differential operators, respectively ϵ = small numerical value (e.g., 0.02) θ = transfer angle μ = gravitational parameter of the primary, LU 3 ∕TU 2 τ = geometry parameter appearing in Lambert equation Φt 2 ; t 1 = state transition matrix from time t 1 to t 2 , with first quadrant units of TU and third quadrant units of TU −1 Subscript LT = expression subjected to the Lambert targeting constraints

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Partial Derivatives of the Solution to the Lambert Boundary Value Problem

Arora

Russell

Strange

et al. 2015

Journal of Guidance, Control, and Dynamics

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“…Some researchers have used the state transition matrix to calculate derivatives in a limited class of problems (impulsive maneuvers). D'Amario et al 1 and Sauer 2 use the state transition matrix to calculate partial derivatives that are used to optimize an impulsive v multiple flyby trajectory. Mirfakhraie and Conway 3 use the state transition matrix to calculate the necessary derivatives for a cooperative time-fixed impulsive rendezvous.…”

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confidence: 99%

Analytical Gradients for Gravity Assist Trajectories Using Constant Specific Impulse Engines

Zimmer

Ocampo

2005

Journal of Guidance, Control, and Dynamics

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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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“…Ganyrnede encounter of the nominal and converged post-G1 to G2 finite burn search trajectory.In general, the initial conditions are obtained in the close neighborhood of the converged spacecraft trajectory from the trajectory design software tools such as the Multiple Orbit Satellite Encounter Software (MOSES, Ref 17). which uses the multi-conic technique to propagate the trajectory, not a numerical integration technique.…”

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Survey of optimization techniques for nonlinear spacecraft trajectory searches

Wang

Stanford

Sunseri

et al. 1988

Astrodynamics Conference

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Mathematical analysis of the optimal search of a nonlinear spacecraft trajectory to arrive at a set of desired targets is presented. A high precision integrated trajectory program and several optimization software libraries are used to search for a converged nonlinear spacecraft trajectory. Several examples for the Galileo Jupiter Orbiter and the Ocean Topography Experiment (TOPEX) are presented that illustrate a variety of the optimization methods used in nonlinear spacecraft trajectory searches.

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A New Method for Optimizing Multiple-Flyby Trajectories

Cited by 32 publications

References 5 publications

Partial Derivatives of the Solution to the Lambert Boundary Value Problem

Partial Derivatives of the Solution to the Lambert Boundary Value Problem

Analytical Gradients for Gravity Assist Trajectories Using Constant Specific Impulse Engines

Survey of optimization techniques for nonlinear spacecraft trajectory searches

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