196,279 research outputs found
Long term nonlinear propagation of uncertainties in perturbed geocentric dynamics using automatic domain splitting
No full textCurrent approaches to uncertainty propagation in astrodynamics mainly refer tolinearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationallyintensive. Differential algebra has already proven to be an efficient compromiseby replacing thousands of pointwise integrations of Monte Carlo runs with thefast 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 in highly nonlinear dynamics or long term propagation. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial of the current state is split in two polynomials when its accuracy reaches a given threshold. The resulting polynomials accurately track uncertainties, even in highly nonlinear dynamics and long term propagations. Furthermore, valuable additional information about the dynamical system is available from the pattern in which those automatic splits occur. From this pattern it is immediately visible where the system behaves chaotically and where its evolution is smooth. Furthermore, it is possible to deduce the behavior of the system for each region, yielding further insight into the dynamics. In this work, the method is applied to the analysis of an end-of-life disposal trajectory of the INTEGRAL spacecraft
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Remarks on the differential algebraic approach to particle beam optics by M. Berz
No full textThe underlying mathematical structure of the differential algebraic approach of M. Berz to particle beam optics is isomorphic to the familiar truncated polynomial algebra. Concrete examples of derivations in this algebra, consistent with the truncation operation, are given
High order optimal feedback control of lunar landing and rendezvous trajectories
No full textOptimal feedback control is classically based on linear approximations, whose accuracy drops off rapidly in highly nonlinear dynamics. A high order optimal control strategy is proposed in this work, based on the use of differential algebraic techniques. In the frame of orbital mechanics, differential algebra allows the dependency of the spacecraft state on initial conditions and environmental parameters to be represented by high order Taylor polynomials. The resulting polynomials can be manipulated to obtain the high order expansion of the solution of two-point boundary value problems. Based on the reduction of the optimal control problem to an equivalent two-point boundary value problem, differential algebra is used in this work to compute the high order expansion of the solution of the optimal control problem about a reference trajectory. New optimal control laws for displaced initial states are then obtained by the mere evaluation of polynomials
Rigorous Global Optimization of Impulsive Planet-to-Planet Transfers
No full textAbstract. The rigorous solution of a generic impulsive planet-to-planet transfer by means of a Taylor Model based global optimizer is presented. Although a planet-to-planet transfer represents the simplest case of in- terplanetary transfer, its formulation and solution is a challenging task as far as the rigorous global optimum is sought. A customized ephemeris function is derived from JPL DE405 to allow the Taylor Model evalu- ation of planets’ positions and velocities. Furthermore, the validated solution of Lambert’s problem is addressed for the rigorous computa- tion of transfer fuel consumption. The optimization problem, which consists in finding the optimal launch and transfer time to minimize the required fuel mass, is complex due to the abundance of local minima and relatively high search space dimension. Its rigorous solution by means of COSY-GO is presented considering Earth–Mars and Earth–Venus transfers as test cases
Nonlinear mapping of uncertainties: a differential algebraic approach
No full textA method for the nonlinear propagation of uncertainties in celestial mechanics based on differential algebra is presented. The arbitrary order Taylor expansion of the flow of ordinary differential equations with respect to the initial condition delivered by differential algebra is exploited to implement an accurate and computationally efficient Monte Carlo algorithm, in which thousands of pointwise integrations are substituted by polynomial evaluations. The algorithm is applied to study the close encounter of asteroid Apophis with our planet in 2029. To this aim, we first compute the high order Taylor expansion of Apophis’ close encounter distance from the Earth by means of map inversion and composition; then we run the proposed Monte Carlo algorithm to perform the statistical analysis
Rigorous and accurate enclosure of invariant manifolds on surfaces
No full textKnowledge about stable and unstable manifolds of hyperbolic fixed points of certain maps is desirable in many fields of research, both in pure mathematics as well as in applications, ranging from forced oscillations to celestial mechanics and space mission design. We present a technique to find highly accurate polynomial approximations of local invariant manifolds for sufficiently smooth planar maps and rigorously enclose them with sharp interval remainder bounds using Taylor model techniques. Iteratively, significant portions of the global manifold tangle can be enclosed with high accuracy. Numerical examples are provided.</p
High-order robust guidance of interplanetary trajectories based on differential algebra
No full textSpace trajectory design always requires the solution of an optimal control problem in order
to maximize the payload launch-mass ratio while achieving the primary mission goals. A certain level
of approximation always characterizes the dynamical models adopted to perform the design process. Furthermore the state identification is usually affected by navigation errors. Thus, after the nominal optimal
solution is computed, a control strategy that assures the execution of mission goals in the real scenario
must be implemented. In this frame differential algebraic techniques are here proposed as an effective alternative tool to design the guidance law. By using differential algebra the final state dependency on initial
conditions, environmental and control parameters is represented by high order Taylor series expansions.
The mission constraints can then be solved to high order using a so-called high order partial inversion of
the polynomial relationship for every admissible uncertainty. The control strategy is eventually reduced to
a simple function evaluation. The performances of the proposed methods are assessed by two examples of
space mission trajectory design: a continuous propelled Earth-Mars transfer and an aerocapture maneuver
at Mars
High Order Integration and Sensitivity Analyisis of Multibody Systems Using Differential Algebra
No full textTaylor models and floating-point arithmetic: proof that arithmetic operations are validated in COSY
No full textThe goal of this paper is to prove that the implementation of Taylor models in COSY, based on floating-point arithmetic, computes results satisfyin- g the «containment property», i.e. guaranteed results. First, Taylor models are defined and their implementation in the COSY software by Makino and Berz is detailed. Afterwards IEEE-754 floating-point arithmetic is introduced. Then the core of this paper is given: the algorithms implemented in COSY for multiplying a Taylor model by a scalar, for adding or multiplying two Taylor models are given and are proven to return Taylor models satisfying the containment property.L'objectif de ce travail est de démontrer que l'implantation des modèles de Taylor, telle qu'elle est réalisée dans le logiciel COSY, calcule des résultats qui sont garantis, c'est à dire qu''ils satisfont la propriété d'inclusion.Tout d'abord, les modèles de Taylor sont définis et leur implantation par Makino et Berz dans le logiciel COSY est détaillée. Ensuite l'arithmétique flottante, telle qu'elle est spécifiée par la norme IEEE-754, est présentée. Enfin on arrive au cœur du sujet : les algorithmes implantés dans COSY pour la multiplication d'un modèle de Taylor par un scalaire et pour la somme et le produit de deux modèles de Taylor sont donnés; il est démontré que ces algorithmes retournent de s modèles de Taylor qui satisfont la propriété d'inclusion
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