Computation of analytical solutions of the relative motion about a Keplerian elliptic orbit

Ren, Yuan; Masdemont, Josep J.; Marcote, M.; Gómez, Gérard

doi:10.1016/j.actaastro.2012.07.026

Cited by 16 publications

(6 citation statements)

References 18 publications

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“…It must be noted that if the Lorentz force f L vanishes, then (1) become the usual HCW equations, which have been extensively studied in the literature (see, for instance, [8,15], and [33]). In this case, the dynamics of the problem is much simpler, since the HCW equations describe the relative dynamics of two…”

Section: (E )mentioning

confidence: 99%

Analysis of the relative dynamics of a charged spacecraft moving under the influence of a magnetic field

Cheng

Gómez

Masdemont

et al. 2018

Communications in Nonlinear Science and Numerical Simulation

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We analyse a dynamical scenario where a constantly charged spacecraft (follower) moves in the vicinity of another one (leader) that follows a circular Keplerian orbit around the Earth and generates a rotating magnetic dipole. The mass of the follower is assumed to be negligible when compared with the one of the leader and they are supposed to be in a high-Earth orbit, so the Lorentz force on the follower due to the geomagnetic field is ignored. With these as

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Section: (E )mentioning

confidence: 99%

Analysis of the relative dynamics of a charged spacecraft moving under the influence of a magnetic field

Cheng

Gómez

Masdemont

et al. 2018

Communications in Nonlinear Science and Numerical Simulation

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“…x unp ( f ) = According to the following initial conditions [11] e = 0.1 , x 0 = 0.1 , x 0 = 0 , y 0 = 0 , y 0 = −21 110 , z 0 = 0.08 and z 0 = 0, a = 7000km, we can get the relations between the chief true anomaly with the x, y and z components of the deputy relative position in both perturbed and unperturbed cases as shown in figures 2, 3 and 4 respectively. Also in figure 5 these three components are parametrized with the chief true anomaly.…”

Section: Numerical Examplementioning

confidence: 99%

Analytical solutions of the relative orbital motion in unperturbed and in J ₂ - perturbed elliptic orbits

Elshaboury

Ammar

Yousef

2017

Applied Mathematics and Nonlinear Sciences

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This paper introduces a different approach to obtain the exact solution of the relative equations of motion of a deputy (follower) satellite with respect to a chief (leader) satellite that both rotate about central body (Earth) in elliptic orbits by using Laplace transformation. Moreover, the paper will take the perturbation due to the oblateness of the Earth into consideration and simulate this problem with numerical example showing the effect of the perturbation on the Keplerian motion. The solution of such equations in this work is represented in terms of the eccentricity of the chief orbit and its true anomaly as the independent variable.

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“…Higher-order analytical solutions for periodic relative orbits have also been established in the literature. Sengupta et al [33] incorporate second-order nonlinearities via a straightforward expansion perturbation method [34], and Ren et al [35] obtain a third-order solution via the Lindstedt-Poincaré algorithm. Exact analytical solutions to the unperturbed, full, nonlinear model exist as well, with recent work by Condurache and Martinuşi establishing two closedform solutions [36,37].…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the TH solution is selected as the only solution for comparison because, for eccentric chiefs, the TH equations have received the greatest attention over the history of the field. Furthermore, the TH equations would benefit from additional accuracy assessments, given that existing ones have been limited in number and depth [23,24,33,35]. To set the full, nonlinear solution of the new parameterization in context relative to the literature, it is illustrative to see under what circumstances the TH equations are valid and when only a nonlinear solution is available or useful.…”

Section: Introductionmentioning

confidence: 99%

Periodic Orbits in the Elliptical Relative Motion Problem with Space Surveillance Applications

Biria

Russell

2015

Journal of Guidance, Control, and Dynamics

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Periodic orbits with respect to an object in an eccentric Keplerian reference orbit can be found in a variety of ways, including the use of Tschauner-Hempel equations and orbital element differences, which both admit linearized solutions, as well as through direct analyses of two orbits. An alternative parameterization of the last approach is proposed in the current study, using an inertial frame and simple geometrical constructs inherent to the relative motion problem. In this manner, an intuitive and straightforward characterization of periodic orbits is established that retains the nonlinear dynamics. The resulting multidimensional space that defines the periodic orbits is surveyed, archiving 336 orbits and their characteristics, and direct comparisons are made with the Tschauner-Hempel equations to assess the linear region of validity. Applications focus on resident space object (RSO) surveillance and circumnavigation orbits. An orbit's effectiveness is analyzed in terms of object coverage using coverage figures of merit, including the concept of "RSO tracks", the analogy of ground tracks on the bodyfixed surface of an RSO. Nomenclature a = semimajor axis, LU C = chief body-fixed frame e = chief eccentricity vector e = chief eccentricity e D = deputy eccentricity H = Hill frame or local-vertical-local-horizontal frame M = number of grid points distributed over the chief's surface N = number of points evaluated along a trajectory n = chief mean motion, rad · TU −1 P = unit vector directed along the chief eccentricity vector P = percentage of chief surface covered by the deputy P = perifocal framê Q =Ŵ ×P B Q A = rotation matrix from the A frame to the B frame B q A = quaternion describing the orientation of the A frame with respect to the B framê R = unit vector directed along the chief position vector R = Earth-centered rotating frame r = inertial position vector, LU r = magnitude of inertial position vector, LÛ S =Ŵ ×R T = orbital period, TU v = inertial velocity vector, LU · TU −1 v = magnitude of inertial velocity vector, LU · TU −1 W = unit vector normal to the plane of motion of the chief orbit, along the angular momentum vector x, y, z = deputy coordinates in the Hill frame, LU α = deputy position in-plane colatitudelike angle relative to the chief, rad β = deputy position out-of-plane longitudelike angle relative to the chief, rad δ = precedes a deputy quantity defined relative to the chief λ = longitude, rad μ = gravitational parameter of primary body, LU 2 · TU −3 ν = chief true anomaly, rad σ = standard deviation, LU ϕ = latitude, rad B ω A = angular velocity of the A frame relative to the B frame, rad · TU −1 B ω A = magnitude of B ω A , rad · TU −1 Subscripts C = quantity associated with the chief D = quantity associated with the deputy ds = inertially dual-axis spin attitude mode NL = quantity computed using the full, nonlinear equations of motion for Keplerian orbits np = inertially nadir-pointing attitude mode ns = inertially nonspinning attitude mode s = angular velocity spin component about t...

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Computation of analytical solutions of the relative motion about a Keplerian elliptic orbit

Cited by 16 publications

References 18 publications

Analysis of the relative dynamics of a charged spacecraft moving under the influence of a magnetic field

Analysis of the relative dynamics of a charged spacecraft moving under the influence of a magnetic field

Analytical solutions of the relative orbital motion in unperturbed and in J ₂ - perturbed elliptic orbits

Periodic Orbits in the Elliptical Relative Motion Problem with Space Surveillance Applications

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Computation of analytical solutions of the relative motion about a Keplerian elliptic orbit

Cited by 16 publications

References 18 publications

Analysis of the relative dynamics of a charged spacecraft moving under the influence of a magnetic field

Analysis of the relative dynamics of a charged spacecraft moving under the influence of a magnetic field

Analytical solutions of the relative orbital motion in unperturbed and in J 2 - perturbed elliptic orbits

Periodic Orbits in the Elliptical Relative Motion Problem with Space Surveillance Applications

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Analytical solutions of the relative orbital motion in unperturbed and in J ₂ - perturbed elliptic orbits