Asteroid close encounters characterization using differential algebra: the case of Apophis

Armellín, Roberto; Lizia, P. Di; Bernelli‐Zazzera, Franco; Berz, Martin

doi:10.1007/s10569-010-9283-5

Cited by 71 publications

(43 citation statements)

References 19 publications

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“…One of the key points is that, evaluating a polynomial, the image under the flow of points inside U can be computed very fast. This technique, introduced by M. Berz and K. Makino to study particle accelerators, has been used in several fields, for instance: to study the close approaches of Near Earth Asteroids [3,2] or to compute a Gaussian particle filter for spacecraft navigation [21].…”

Section: Introductionmentioning

confidence: 99%

Tools to detect structures in dynamical systems using Jet Transport

Pérez-Palau

Masdemont

Gómez

2015

Celest Mech Dyn Astr

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This paper is devoted to the development of some dynamical indicators that allow the determination of regions and structures that separate different dynamic regimes in autonomous and non-autonomous dynamical systems. The underlying idea is closely related to the Lagrangian Coherent Structures concept introduced by Haller. In the present paper, instead of using the Cauchy-Green tensor, that determines the domains where the flow associated to a differential equation is expanding in the normal direction, the Jet Transport methodology is used. This is a semi-numerical tool, that has as basic ingredients a polynomial algebra package and a numerical integration method, allowing, at each integration step, the propagation under a flow of a neighbourhood U instead of a single initial condition. The output of the procedure is a polynomial in several variables that represents the image of U up to a selected order, containing high order terms of the variational equations. Using these high order representation, the places where the normal direction expands can be easily detected, in a similar manner as the procedures for calculating the Lagrangian Coherent Structures do. In order to illustrate the methodology, first the results obtained in the determination of the separatrices of the simple and the periodically perturbed pendulum are given. Later, the applications to the circular restricted three body problem are considered, where the aim is the detection of invariant manifolds of libration point orbits, as well as in the non-autonomous vector field defined by the elliptic restricted three body problem.

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Section: Introductionmentioning

confidence: 99%

Tools to detect structures in dynamical systems using Jet Transport

Pérez-Palau

Masdemont

Gómez

2015

Celest Mech Dyn Astr

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“…It is worth noting that, in the DA-based approach, the Taylor expansion T f shall be computed only once, since the same polynomial can be used in case new statistics need to be mapped. This is an additional advantage with respect to classical MC, as already pointed out in [42].…”

Section: Da-based Monte Carlomentioning

confidence: 80%

Probabilistic Optical and Radar Initial Orbit Determination

Armellín

Lizia

2018

Journal of Guidance, Control, and Dynamics

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Future space surveillance requires dealing with uncertainties directly in the initial orbit determination phase. We propose an approach based on Taylor differential algebra to both solve the initial orbit determination (IOD) problem and to map uncertainties from the observables space into the orbital elements space. This is achieved by approximating in Taylor series the general formula for probability density function (pdf ) mapping through nonlinear transformations. In this way the mapping is obtained in an elegant and general fashion. The proposed approach is applied to both angles-only and two position vectors IOD for objects in LEO and GEO.

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“…The differential algebra used for the computations in this work was implemented in the software COSY INFINITY (Berz and Makino 2006). An important application of DA is the automatic high order expansion of the solution of an ODE in terms of the initial conditions (Berz 1999a;Armellin et al 2010). This can be achieved by replacing the operations in a classical numerical integration scheme, including evaluation of the right hand side, by the corresponding DA operations.…”

Section: Differential Algebra and High Order Flow Expansionmentioning

confidence: 99%

“…It was shown that a good agreement with Monte Carlo simulations can be achieved, however the derivation of the dynamics of the high order tensor can be complex from the computational standpoint, especially for high fidelity dynamics. These difficulties were addressed by the authors in past works, in which a technique called differential algebra (DA) was used to automatically expand the flow of the dynamics up to an arbitrary order (Di Lizia et al 2008;Armellin et al 2010). Nonlinear uncertainty propagation can take advantage of the high order expansions of the flow.…”

Section: Introductionmentioning

confidence: 99%

Propagation of large uncertainty sets in orbital dynamics by automatic domain splitting

Wittig

Lizia

Armellín

et al. 2015

Celest Mech Dyn Astr

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Current approaches to uncertainty propagation in astrodynamics mainly refer to linearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationally intensive. Differential algebra has already proven to be an efficient compromise by replacing thousands of pointwise integrations of Monte Carlo runs with the fast evaluation of the arbitrary order Taylor expansion of the flow of the dynamics. However, the current implementation of the DA-based high-order uncertainty propagator fails when the non-linearities of the dynamics prohibit good convergence of the Taylor expansion in one or more directions. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial expansion of the current state is split into two polynomials whenever its truncation error reaches a predefined threshold. The resulting set of polynomials accurately tracks uncertainties, even B Alexander Wittig

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Asteroid close encounters characterization using differential algebra: the case of Apophis

Cited by 71 publications

References 19 publications

Tools to detect structures in dynamical systems using Jet Transport

Tools to detect structures in dynamical systems using Jet Transport

Probabilistic Optical and Radar Initial Orbit Determination

Propagation of large uncertainty sets in orbital dynamics by automatic domain splitting

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