Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

Woollands, Robyn; Read, Julie L.; Probe, Austin; Junkins, John L.

doi:10.1007/s40295-017-0116-6

Cited by 17 publications

(15 citation statements)

References 10 publications

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“…t F P between the runtime for the computation of a k-th order Taylor polynomial t k D A (either map (7) or (11)) and its three for map (11). Similar results are obtained when the dynamics are propagated numerically.…”

Section: A Illustrative Examplessupporting

confidence: 63%

“…Therefore, the proposed approach will be suitable to study the optimal insertion of corrective maneuvers for long-duration rendezvous maneuvers. Besides, the availability of high order expansions of the solution of MRPLP could be profitably used in space situational awareness to study the linkage of radar observations, as already suggested in [11]. Note that the expansion order can be tuned based on accuracy requirements, using as input the estimation of the truncation error.…”

Section: A Illustrative Examplesmentioning

confidence: 99%

“…Remarkably transfers with more than one thousand revolutions were presented, although limited details on how to define the continuation path were provided. Another solver suitable for MRPLP was published recently by Woollands et al [11]. This method combines the MCPI method with the method of particular solutions [12] and has the favorable property of not requiring the computation of the state transition matrix, which is particularly advantageous for high-fidelity dynamical models they considered.…”

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confidence: 99%

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Multiple Revolution Perturbed Lambert Problem Solvers

Armellín

Gondelach

San-Juan

2018

Journal of Guidance, Control, and Dynamics

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I. Introduction The Lambert problem is one of the most extensively studied problems in astrodynamics as its solution is a building block for many problems, including interplanetary transfer optimization, rendezvous missions design, collision avoidance maneuvers design, and orbit determination. Although this problem was solved more than 200 years ago [1], many researchers are still working on devising robust and efficient resolution procedures [2, 3], including analytical approximations [4]. In particular, when the transfer time is long, Lambert's problem becomes a multi-revolution Lambert's problem (MRLP) which has the additional difficulty of admitting many solutions associated with different number of revolutions. More specifically, there exists 2N max + 1 number of solutions to a MRLP, in which N max is the maximum number of revolutions compatible with the time-of-flight of interest [5]. The original formulation of Lambert's problem is based on two-body dynamics. However, when the MRLP solutions are used to design missions around a planetary body or for orbit determination and data association problems, the effect of perturbations cannot be neglected. The perturbations, e.g. J 2 or atmospheric drag, can cause large violations of the terminal constraints, up to the point that classical Lambert's solutions fail to provide a good initial guess for the multi-revolution perturbed Lambert problem (MRPLP). Different methods have been proposed over the years to solve perturbed Lambert's problems. Engles and Junkins [6] proposed a variation-of-parameter approach combined with Kustaanheimo-Stiefel (KS) transformation to algebraically solve the J 2 perturbed problem. Bai and Junkins [7] proposed a modified Chebyshev-Picard iteration (MCPI) method for the solution of two-point boundary values problems, which was later regularized by Woollands et al. [8] using the KS time transformation and applied to high-fidelity dynamics. Der [9] developed a Lambert solver that can be modified to include the effect of J 2 − J 4 via a targeting technique based on Vinti's approximation. However, these approaches are not suitable for cases with many revolutions, due to the fact that the initial Keplerian guess for the velocity is not close to the perturbed one. More recently, Yang et al. [10] developed a homotopic approach suitable for solving the MRPLP, which they applied to the design of rendezvous missions around Mars. This approach employs a homotopy on the residuals. For each of the multiple solutions of the MRLP, a sequence of MRPLP is solved for decreasing values of the homotopic parameter.

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Section: A Illustrative Examplessupporting

confidence: 63%

Section: A Illustrative Examplesmentioning

confidence: 99%

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confidence: 99%

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Multiple Revolution Perturbed Lambert Problem Solvers

Armellín

Gondelach

San-Juan

2018

Journal of Guidance, Control, and Dynamics

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“…Therefore, the proposed approach will be suitable to study the optimal insertion of corrective maneuvers for long-duration rendezvous maneuvers. Besides, the availability of high order expansions of the solution of MRPLP could be profitably used in space situational awareness to study the linkage of radar observations, as already suggested in [10]. Note that the expansion order can be tuned based on accuracy requirements, using as input the estimation of the truncation error.…”

Section: A Illustrative Examplesmentioning

confidence: 99%

“…Remarkably transfers with more than one thousand revolutions were presented, although limited details on how to define the continuation path were provided. Another solver suitable for MRPLP was published recently by Woollands et al [10]. This method combines the MCPI method with the method of particular solution and has the favorable property of not requiring the computation of the state transition matrix.…”

Section: Introductionmentioning

confidence: 99%

Multi-revolution perturbed Lambert problem

Armellín

Gondelach

San-Juan

2018

2018 Space Flight Mechanics Meeting

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The solution of multiple-revolution perturbed Lambert problems is a challenging task due to the high sensitivity of the final state to variations of the initial velocity. In this work two different solvers based on high order Taylor expansions and an analytical solution of the J 2 problem are presented. In addition, an iteration-less procedure is developed to refine the solutions in a dynamical model that includes J2 − J4 perturbations. The properties of the proposed approached are tested against transfers with hundreds of revolutions including those required to solve the Global Trajectory Optimisation Competition 9.

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Unified Lambert Tool for Massively Parallel Applications in Space Situational Awareness

et al. 2017

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This paper introduces a parallel-compiled tool that combines several of our recently developed methods for solving the perturbed Lambert problem using modified Chebyshev-Picard iteration. This tool (unified Lambert tool) consists of four individual algorithms, each of which is unique and better suited for solving a particular type of orbit transfer. The first is a Keplerian Lambert solver, which is used to provide a good initial guess (warm start) for solving the perturbed problem. It is also used to determine the appropriate algorithm to call for solving the perturbed problem. The arc length or true anomaly angle spanned by the transfer trajectory is the parameter that governs the automated selection of the appropriate perturbed algorithm, and is based on the respective algorithm convergence characteristics. The second algorithm solves the perturbed Lambert problem using the modified Chebyshev-Picard iteration two-point boundary value solver. This algorithm does not require a Newtonlike shooting method and is the most efficient of the perturbed solvers presented herein, however the domain of convergence is limited to about a third of an orbit and is dependent on eccentricity. The third algorithm extends the domain of convergence of the modified Chebyshev-Picard iteration two-point boundary value solver to about 90% of an orbit, through regularization with the Kustaanheimo-Stiefel transformation. This is the second most efficient of the perturbed set of algorithms. The fourth algorithm uses the method of particular solutions and the modified ChebyshevPicard iteration initial value solver for solving multiple revolution perturbed transfers. This method does require "shooting" but differs from Newton-like shooting methods in that it does not require propagation of a state transition matrix. The unified Lambert tool makes use of the General Mission Analysis Tool and we use it to compute thousands of perturbed Lambert trajectories in parallel on the Space Situational Awareness computer cluster at the LASR Lab, Texas A&M University. We demonstrate the power of our tool by solving a highly parallel example problem, that is the generation of extremal field maps for optimal spacecraft rendezvous (and eventual orbit debris removal). In addition we demonstrate the need for including perturbative effects in simulations for satellite tracking or data association. The unified Lambert tool is ideal for but not limited to space situational awareness applications.

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Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

Cited by 17 publications

References 10 publications

Multiple Revolution Perturbed Lambert Problem Solvers

Multiple Revolution Perturbed Lambert Problem Solvers

Multi-revolution perturbed Lambert problem

Unified Lambert Tool for Massively Parallel Applications in Space Situational Awareness

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