Papers by Vladimir Martinusi
AIAA Guidance, Navigation, and Control Conference, 2009
ABSTRACT The paper offers a vectorial approach to the J2–perturbed relative orbital motion. The m... more ABSTRACT The paper offers a vectorial approach to the J2–perturbed relative orbital motion. The model uses mean orbital elements in order to derive closed form expressions for the relative position and relative velocity with respect to the LVLH frame attached on a main satellite. The advantage of the present approach is that only the motion of one satellite must be known, together with the initial conditions with respect to LVLH of the other satellites. As in the previous works of the authors, a tensorial instrument is used in order to transfer the motion from inertial to non-inertial reference frames and back. In this way, a concise instrument for the study of the satellites formations under the influence of the oblateness factor J2 is offered.
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ABSTRACT The paper introduces a unified methodology for the study of bounded motion in central fo... more ABSTRACT The paper introduces a unified methodology for the study of bounded motion in central force fields, which serves both for qualitative insights, as well as for the derivation of closed-form vectorial solutions for the equations of motion. The paper offers a full regularization of the equations of motion in a central force field, starting from the polar equations in the plane of motion. A time transformation, as well as a coordinate transformation, are performed together, and the motion is (i) reduced to a Kepler motion in a rotating frame, with respect to a new time variable (ii) regularized further to a harmonic oscillator. In addition, some new results pertaining to the existence of a Laplace-Runge-Lenz vector in a central force motion are presented.
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Celestial Mechanics and Dynamical Astronomy, 2013
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AIAA Guidance, Navigation, and Control Conference, 2011
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Celestial Mechanics and Dynamical Astronomy, 2011
... Solutions and periodicity of satellite relative motion under even zonal harmonics perturbatio... more ... Solutions and periodicity of satellite relative motion under even zonal harmonics perturbationsVladimir Martinusi · Pini Gurfil Received: 26 January 2011 / Revised: 1 July 2011 / Accepted: 16 August 2011 © Springer Science+Business Media BV 2011 ...
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Journal of Guidance, Control, and Dynamics, 2014
ABSTRACT Keeping a cluster of satellites within bounded relative distances requires active contro... more ABSTRACT Keeping a cluster of satellites within bounded relative distances requires active control, because natural perturbations (the most significant of which is the J2 term in the geopotential) tend to move the satellites apart. Whereas there is abundant literature on controlling the relative drift using multiple impulsive maneuvers and approximate astrodynamical models involving mean orbital elements, the works that attempt to minimize the number of impulses while using the inertial position and velocity vectors of the satellites are scarce. In this paper, single-impulse distance-keeping maneuvers are derived, without approximating the J2-perturbed dynamics. An analytical derivation of minimum-fuel impulsive maneuvers between equatorial orbits is provided, while relying on radial period and orbital angle matching conditions. Then, a continuation procedure is used to obtain single-impulse relative distance control between inclined orbits. The development of the impulsive maneuvers relies on the inertial position and velocity vectors of the satellites and does not involve mean elements. In each step, necessary and sufficient conditions for the existence of a single-impulse maneuver are provided. The results are illustrated using a number of realistic scenarios, such as a simulation that includes a 21×21 gravitational model, drag, and lunisolar attraction.
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ABSTRACT The paper introduces a unified methodology for the study of bounded motion in central fo... more ABSTRACT The paper introduces a unified methodology for the study of bounded motion in central force fields, which serves both for qualitative insights, as well as for the derivation of closed-form vectorial solutions for the equations of motion. The paper offers a full regularization of the equations of motion in a central force field, starting from the polar equations in the plane of motion. A time transformation, as well as a coordinate transformation, are performed together, and the motion is (i) reduced to a Kepler motion in a rotating frame, with respect to a new time variable (ii) regularized further to a harmonic oscillator. In addition, some new results pertaining to the existence of a Laplace-Runge-Lenz vector in a central force motion are presented.
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Celestial Mechanics and Dynamical Astronomy, 2015
ABSTRACT In the present paper, an averaging perturbation technique leads to the determination of ... more ABSTRACT In the present paper, an averaging perturbation technique leads to the determination of a time-explicit analytic approximate solution for the motion of a low-Earth-orbiting satellite . The two dominant perturbations are taken into account: the Earth oblateness and the atmospheric drag. The proposed orbit propagation algorithm comprises the Brouwer–Lyddane transformation (direct and inverse), coupled with the analytic solution of the averaged equations of motion. This solution, based on equinoctial elements, is singularity-free, and therefore it stands for low inclinations and small eccentricities as well. The simplifying assumption of a constant atmospheric density is made, which is reasonable for near-circular orbits and short-time orbit propagation. Two sets of time-explicit equations are provided, for moderate and small eccentricities (O(e^4) = 0 and O(e^2) = 0, respectively), and they are obtained by performing (1) a regularization of the original averaged differential equations of motion for the vectorial orbital elements, and (2) Taylor series expansions of the aforementioned equations with respect to the eccentricity. The numerical simulations show that the errors due to the use of the proposed analytic model in the presence of drag are almost the same as the errors of the Brouwer first-order approximation in the absence of drag.
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The paper presents an approach to the motion with respect to a rotating reference frame from the ... more The paper presents an approach to the motion with respect to a rotating reference frame from the point of view of the Analytical Mechanics. Surprisingly, this subject is barely considered in classical textbooks, especially the motion in a rotating frame from the Hamiltonian perspective. We offer here a way of determining the Hamiltonian in a rotating frame and we offer an interesting parallel between this type of motion and the motion in an inertial frame in the presence of a generalized potential. The applications which are presented here are in relative orbital dynamics, with a special attention given to the relative motion of satellites.
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"The paper offers a general method to study the motion of a mass particle with respect to a non-i... more "The paper offers a general method to study the motion of a mass particle with respect to a non-inertial reference frame. By using an adequate tensor instrument, we obtain a simplified form of the initial value problem that models the non-inertial motion. The study of the motion in a non-inertial reference frame in a central positional force field is then reduced to the study of the classic motion in a central force field. The applications to this method are in solving Kepler’s problem in non-inertial reference frames, solving the relative orbital motion problem and deriving the explicit solution to a classic Theoretical Mechanics problem: The Foucault Pendulum.
The advantage of using this method in the study of the motion in a non-inertial reference frames is that is significantly reduces the amount of computations and it offers, in some important particular cases, closed form analytical solutions."
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The paper offers a comprehensive study of the two-body problem in an arbitrary rotating reference... more The paper offers a comprehensive study of the two-body problem in an arbitrary rotating reference frame. By using a tensorial symbolic procedure, the problem is reduced to two uni-particle problems with respect to a rotating reference frame. Unprecedented prime integrals are deduced that generalize those from the classic inertial two body problem. The results apply in astrodynamics problems (orbital relative dynamics, space rendezvous, satellite formation flight), as well as in theoretical mechanics (Kepler’s problem in rotating reference frames, the Foucault Pendulum problem).
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Advances in the Astronautical Sciences
The closed-form solution to the problem of the relative motion in gravitational field is expressed... more The closed-form solution to the problem of the relative motion in gravitational field is expressed with the help of the universal functions. The full nonlinear model of the relative motion is considered, and no linearization or approximation assumptions are made. The results are free of singularities and they hold for any reference and targeted orbits, as well as for any time scale. A unified way to propagate the relative orbit with respect to time is introduced, by making use of the solution to a generalized Kepler equation.
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Presented at the 2nd IAA Conference on Dynamics and Control of Space Systems, Rome, 24-26 March 2014
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Papers by Vladimir Martinusi
The advantage of using this method in the study of the motion in a non-inertial reference frames is that is significantly reduces the amount of computations and it offers, in some important particular cases, closed form analytical solutions."
The advantage of using this method in the study of the motion in a non-inertial reference frames is that is significantly reduces the amount of computations and it offers, in some important particular cases, closed form analytical solutions."