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Yu. N. Chelnokov - One of the best experts on this subject based on the ideXlab platform.
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Inertial Navigation in Space Using the Regular Quaternion Equations of Astrodynamics
Mechanics of Solids, 2019Co-Authors: Yu. N. ChelnokovAbstract:Quaternion equations are proposed for the ideal operation of spatial inertial navigation systems with an azimuthally stabilized platform and a gyrostabilized platform that keeps its orientation invariant in inertial space, quaternion equations for the ideal operation of strapdown inertial navigation systems in the regular four-dimensional Kustaanheimo-Stiefel variables with consideration of zonal, tesseral, and sectorial harmonics of the Earth's gravitational field. The equations are dynamically similar to the regular equations of perturbed spatial two-body problem in the Kustaanheimo-Stiefel variables, which enables to use the results obtained in regular celestial mechanics and Astrodynamics theory in inertial astronavigation. The development of operational algorithms for these navigation systems using these equations is considered.
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Inertial navigation in space using quaternion regular equations of Astrodynamics
2018 25th Saint Petersburg International Conference on Integrated Navigation Systems (ICINS), 2018Co-Authors: Yu. N. ChelnokovAbstract:The new quaternion equations of perfect functioning of the spatial inertial navigation systems with azimuthally stabilized platform and with gyrostabilized platform which retains its orientation in an inertial reference frame, and the quaternion equations of perfect functioning of the strapdown inertial navigation systems in regular four-dimensional Kustaanheimo-Stiefel variables are proposed. These INS equations have a dynamic analogy with the quaternion regular equations of the perturbed spatial two-body problem, which makes it possible to use the results, obtained in the theory of regular celestial mechanics and Astrodynamics, in space inertial navigation. This paper discusses the development of INS algorithms using the proposed quaternion equations of INS perfect functioning. This paper develops the results, derived in [1, 2].
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Quaternion regularization and trajectory motion control in celestial mechanics and Astrodynamics: II
Cosmic Research, 2014Co-Authors: Yu. N. ChelnokovAbstract:Problems of regularization in celestial mechanics and Astrodynamics are considered, and basic regular quaternion models for celestial mechanics and Astrodynamics are presented. It is shown that the effectiveness of analytical studies and numerical solutions to boundary value problems of controlling the trajectory motion of spacecraft can be improved by using quaternion models of Astrodynamics. In this second part of the paper, specific singularity-type features (division by zero) are considered. They result from using classical equations in angular variables (particularly in Euler variables) in celestial mechanics and Astrodynamics and can be eliminated by using Euler (Rodrigues-Hamilton) parameters and Hamilton quaternions. Basic regular (in the above sense) quaternion models of celestial mechanics and Astrodynamics are considered; these include equations of trajectory motion written in nonholonomic, orbital, and ideal moving trihedrals whose rotational motions are described by Euler parameters and quaternions of turn; and quaternion equations of instantaneous orbit orientation of a celestial body (spacecraft). New quaternion regular equations are derived for the perturbed three-dimensional two-body problem (spacecraft trajectory motion). These equations are constructed using ideal rectangular Hansen coordinates and quaternion variables, and they have additional advantages over those known for regular Kustaanheimo-Stiefel equations.
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quaternion regularization in celestial mechanics Astrodynamics and trajectory motion control iii
Cosmic Research, 2013Co-Authors: Yu. N. ChelnokovAbstract:The present paper1 analyzes the basic problems arising in the solution of problems of the optimum control of spacecraft (SC) trajectory motion (including the Lyapunov instability of solutions of conjugate equations) using the principle of the maximum. The use of quaternion models of Astrodynamics is shown to allow: (1) the elimination of singular points in the differential phase and conjugate equations and in their partial analytical solutions; (2) construction of the first integrals of the new quaternion; (3) a considerable decrease of the dimensions of systems of differential equations of boundary value optimization problems with their simultaneous simplification by using the new quaternion variables related with quaternion constants of motion by rotation transformations; (4) construction of general solutions of differential equations for phase and conjugate variables on the sections of SC passive motion in the simplest and most convenient form, which is important for the solution of optimum pulse SC transfers; (5) the extension of the possibilities of the analytical investigation of differential equations of boundary value problems with the purpose of identifying the basic laws of optimum control and motion of SC; (6) improvement of the computational stability of the solution of boundary value problems; (7) a decrease in the required volume of computation.
Dario Izzo - One of the best experts on this subject based on the ideXlab platform.
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revisiting high order taylor methods for Astrodynamics and celestial mechanics
Monthly Notices of the Royal Astronomical Society, 2021Co-Authors: Francesco Biscani, Dario IzzoAbstract:We present heyoka, a new, modern and general-purpose implementation of Taylor's integration method for the numerical solution of ordinary differential equations. Detailed numerical tests focused on difficult high-precision gravitational problems in Astrodynamics and celestial mechanics show how our general-purpose integrator is competitive with and often superior to state-of-the-art specialised symplectic and non-symplectic integrators in both speed and accuracy. In particular, we show how Taylor methods are capable of satisfying Brouwer's law for the conservation of energy in long-term integrations of planetary systems over billions of dynamical timescales. We also show how close encounters are modelled accurately during simulations of the formation of the Kirkwood gaps and of Apophis' 2029 close encounter with the Earth (where heyoka surpasses the speed and accuracy of domain-specific methods). heyoka can be used from both C++ and Python, and it is publicly available as an open-source project.
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on the Astrodynamics applications of weierstrass elliptic and related functions
arXiv: Earth and Planetary Astrophysics, 2016Co-Authors: Dario Izzo, Francesco BiscaniAbstract:Weierstrass elliptic and related functions have been recently shown to enable analytical explicit solutions to classical problems in Astrodynamics. These include the constant radial acceleration problem, the Stark problem and the two-fixed center (or Euler's) problem. In this paper we review the basic technique that allows for these results and we discuss the limits and merits of the approach. Applications to interplanetary trajectory design are then discussed including low-thrust planetary fly-bys and the motion of an artificial satellite under the influence of an oblate primary including $J_2$ and $J_3$ harmonics.
Edward Belbruno - One of the best experts on this subject based on the ideXlab platform.
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resonant motion ballistic escape and their applications in Astrodynamics
Advances in Space Research, 2008Co-Authors: Francesco Topputo, Edward Belbruno, Marian GideaAbstract:Abstract A special set of solutions governing the motion of a particle, subject to the gravitational attractions of the Earth, the Moon, and, eventually, the Sun, is discussed in this paper. These solutions, called resonant orbits, correspond to a special motion where the particle is in resonance with the Moon. For a restricted set of initial conditions the particle performs a resonance transition in the vicinity of the Moon. In this paper, the nature of the resonance transition is investigated under the perspective of the dynamical system theory and the energy approach. In particular, using a new definition of weak stability boundary, we show that the resonance transition mechanism is strictly related to the concept of weak capture. This is shown through a carefully computed set of Poincare surfaces, at different energy levels, on which both the weak stability boundary and the resonant orbits are represented. It is numerically demonstrated that resonance transitioning orbits pass through the weak stability boundaries. In the second part of the paper the solar perturbation is taken into account, and the motion of the resonant orbits is studied within a four-body dynamics. We show that, for a wide class of initial conditions, the particle escapes from the Earth–Moon system and targets an heliocentric orbit. This is a free ejection called a ballistic escape. Astrodynamical applications are discussed.
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low energy trajectories and chaos applications to Astrodynamics and dynamical astronomy
Annals of the New York Academy of Sciences, 2005Co-Authors: Edward BelbrunoAbstract:: In recent years the application of methods of chaos theory to Astrodynamics has produced new revolutionary types of low energy trajectories in space with important applications. They have uncovered new routes to the Moon demonstrated by various spacecraft—the Japan Hiten in 1991 and the European SMART-1 in 2004. These applications have also played an important role in a new theory on the origin of the Moon.
Francesco Topputo - One of the best experts on this subject based on the ideXlab platform.
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approximate solutions to nonlinear optimal control problems in Astrodynamics
International Scholarly Research Notices, 2013Co-Authors: Francesco Topputo, Franco BernellizazzeraAbstract:A method to solve nonlinear optimal control problems is proposed in this work. The method implements an approximating sequence of time-varying linear quadratic regulators that converge to the solution of the original, nonlinear problem. Each subproblem is solved by manipulating the state transition matrix of the state-costate dynamics. Hard, soft, and mixed boundary conditions are handled. The presented method is a modified version of an algorithm known as “approximating sequence of Riccati equations.” Sample problems in Astrodynamics are treated to show the effectiveness of the method, whose limitations are also discussed.
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resonant motion ballistic escape and their applications in Astrodynamics
Advances in Space Research, 2008Co-Authors: Francesco Topputo, Edward Belbruno, Marian GideaAbstract:Abstract A special set of solutions governing the motion of a particle, subject to the gravitational attractions of the Earth, the Moon, and, eventually, the Sun, is discussed in this paper. These solutions, called resonant orbits, correspond to a special motion where the particle is in resonance with the Moon. For a restricted set of initial conditions the particle performs a resonance transition in the vicinity of the Moon. In this paper, the nature of the resonance transition is investigated under the perspective of the dynamical system theory and the energy approach. In particular, using a new definition of weak stability boundary, we show that the resonance transition mechanism is strictly related to the concept of weak capture. This is shown through a carefully computed set of Poincare surfaces, at different energy levels, on which both the weak stability boundary and the resonant orbits are represented. It is numerically demonstrated that resonance transitioning orbits pass through the weak stability boundaries. In the second part of the paper the solar perturbation is taken into account, and the motion of the resonant orbits is studied within a four-body dynamics. We show that, for a wide class of initial conditions, the particle escapes from the Earth–Moon system and targets an heliocentric orbit. This is a free ejection called a ballistic escape. Astrodynamical applications are discussed.
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a sixth order accurate scheme for solving two point boundary value problems in Astrodynamics
Celestial Mechanics and Dynamical Astronomy, 2006Co-Authors: Roberto Armellin, Francesco TopputoAbstract:A sixth-order accurate scheme is presented for the solution of ODE systems supplemented by two-point boundary conditions. The proposed integration scheme is a linear multi-point method of sixth-order accuracy successfully used in fluid dynamics and implemented for the first time in Astrodynamics applications. A discretization molecule made up of just four grid points attains a O(h 6) accuracy which is beyond the first Dahlquist's stability barrier. Astrodynamics applications concern the computation of libration point halo orbits, in the restricted three- and four-body models, and the design of an optimal control strategy for a low thrust libration point mission.
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an optimal h6 scheme for solving tpbvp in Astrodynamics
4th International Meeting on Celestial Mechanics (CELMEC IV), 2005Co-Authors: Roberto Armellin, Francesco TopputoAbstract:The present paper presents an accurate scheme for the solution of boundary value problems with two-point nonlinear boundary conditions. The proposed scheme is a linear multi-point method of sixth-order accuracy successfully used in uid dynamics and here implemented for the rst time in Astrodynamics applications. It is an optimal scheme since a discretization molecule made up of just four grid points assures an h6 order of accuracy. This kind of discretization allows to attain an accuracy beyond the rst Dahlquist's stability barrier and simultaneously has a simple formulation and numerical e ciency. Astrodynamics applications concern the computation of libration point halo orbits, in the restricted three- and four-body models, and the design of an optimal control strategy for a low thrust libration point mission.
Roberto Armellin - One of the best experts on this subject based on the ideXlab platform.
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differential algebra space toolbox for nonlinear uncertainty propagation in space dynamics
2016Co-Authors: M Rasotto, Roberto Armellin, P. Di Lizia, A Morselli, Mauro Massari, Alexander Wittig, C Valles, G OrtegaAbstract:This paper is aimed at presenting a new tool developed by Dinamica, with the support of ESA, for the efficient non-linear propagation of uncertainties in space dynamics. The newly implemented software is based on Differential Algebra, which provides a method to easily extend the existing linearization techniques and allows the implementation of efficient arbitrary order methods. These theoretical concepts represent the building blocks over which the Differential Algebra Space Toolbox is implemented. The application areas for the tool are plenty. To illustrate the power of the method in general and to give the user a better understanding of the various features, several different examples in the field of Astrodynamics and space engineering are presented.
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algebraic manipulators new perspectives in analytical or semi analytical solutions to Astrodynamics problems
2016Co-Authors: Roberto Armellin, R Lopez, Jf San JuanAbstract:The evolution of the hardware and the capabilities of the general computer algebra system have supplied us with the possibility of developing an environment called MathATESAT embedding in Mathematica, which is not linked to a Poisson series processor. MathATESAT implements all the necessary tools to carry out high accuracy analytical or semi-analytical theories in order to analyze the quantitative and qualitative behavior of a dynamic system.
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application of high order expansions of two point boundary value problems to Astrodynamics
Celestial Mechanics and Dynamical Astronomy, 2008Co-Authors: P. Di Lizia, Roberto Armellin, Michele LavagnaAbstract:Two-point boundary value problems appear frequently in space trajectory design. A remarkable example is represented by the Lambert’s problem, where the conic arc linking two fixed positions in space in a given time is to be characterized in the frame of the two-body problem. Classical methods to numerically solve these problems rely on iterative procedures, which turn out to be computationally intensive in case of lack of good first guesses for the solution. An algorithm to obtain the high order expansion of the solution of a two-point boundary value problem is presented in this paper. The classical iterative procedures are applied to identify a reference solution. Then, differential algebra is used to expand the solution of the problem around the achieved one. Consequently, the computation of new solutions in a relatively large neighborhood of the reference one is reduced to the simple evaluation of polynomials. The performances of the method are assessed by addressing typical applications in the field of spacecraft dynamics, such as the identification of halo orbits and the design of aerocapture maneuvers.
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a sixth order accurate scheme for solving two point boundary value problems in Astrodynamics
Celestial Mechanics and Dynamical Astronomy, 2006Co-Authors: Roberto Armellin, Francesco TopputoAbstract:A sixth-order accurate scheme is presented for the solution of ODE systems supplemented by two-point boundary conditions. The proposed integration scheme is a linear multi-point method of sixth-order accuracy successfully used in fluid dynamics and implemented for the first time in Astrodynamics applications. A discretization molecule made up of just four grid points attains a O(h 6) accuracy which is beyond the first Dahlquist's stability barrier. Astrodynamics applications concern the computation of libration point halo orbits, in the restricted three- and four-body models, and the design of an optimal control strategy for a low thrust libration point mission.
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an optimal h6 scheme for solving tpbvp in Astrodynamics
4th International Meeting on Celestial Mechanics (CELMEC IV), 2005Co-Authors: Roberto Armellin, Francesco TopputoAbstract:The present paper presents an accurate scheme for the solution of boundary value problems with two-point nonlinear boundary conditions. The proposed scheme is a linear multi-point method of sixth-order accuracy successfully used in uid dynamics and here implemented for the rst time in Astrodynamics applications. It is an optimal scheme since a discretization molecule made up of just four grid points assures an h6 order of accuracy. This kind of discretization allows to attain an accuracy beyond the rst Dahlquist's stability barrier and simultaneously has a simple formulation and numerical e ciency. Astrodynamics applications concern the computation of libration point halo orbits, in the restricted three- and four-body models, and the design of an optimal control strategy for a low thrust libration point mission.
