View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UPCommons. Portal del coneixement obert de la UPC 64rd International Astronautical Congress, Beijing, China. Copyright 2012 by the International Astronautical Federation. All rights reserved. IAC-13. C1.8.10 JET TRANSPORT PROPAGATION OF UNCERTAINTIES FOR ORBITS AROUND THE EARTH Daniel Perez´ ∗, Josep J. Masdemontz, Gerard Gomez´ x Abstract In this paper we present a tool to study the non-linear propagation of uncertainties for orbits around the Earth. The tool we introduce is known as Jet Transport and allows to propagate full neighborhoods of initial states instead of a single initial state by means of usual numerical integrators. The description of the transported neighborhood is ob- tained in a semi-analytical way by means of polynomials in 6 variables. These variables correspond to displacements in the phase space from the reference point selected in an orbit as initial condition. The basis of the procedure is a standard numerical integrator of ordinary differential equations (such as a Runge-Kutta or a Taylor method) where the usual arithmetic is replaced by a polynomial arithmetic. In this way, the solution of the variational equations is also obtained up to high order. This methodology is applied to the propagation of satellite trajectories and to the computation of images of uncertainty ellipsoids including high order nonlinear descriptions. The procedure can be specially adapted to the determination of collision probabilities with catalogued space debris or for the end of life analysis of spacecraft in Medium Earth Orbits. 1 Introduction of a particle that at time t0 is at x0. As it has been stated, our objective is to propagate a full neighborhood U of x0, Any trajectory, from the ones of small debris to big aster- i.e. to compute '(t; t0;U). For this purpose, the initial oids, or artificial satellites, is under uncertainty when its condition x0 is replaced by a simple polynomial that pa- orbit is tried to be determined. As it is well known, small rameterizes its neighbourhood: Pt0;x0 (ξ) = x0 + ξ = T differences in the initial conditions can produce big devi- x0 + (ξ1; : : : ; ξn) . ations from predicted states after some time. Therefore it Then we can select any ordinary differential equation is very important to study how these uncertainties evolve integrator (RK, Taylor, symplectic,. ) and to use it to and how can they be controlled along the time. propagate the polynomial. We note that, since the initial The study of uncertainties is usually performed by condition is a polynomial, all the computations required means of Monte Carlo methods or analyzing the evolu- in the propagation method have to be done using polyno- tion of the covariance matrix. The first option, based mial algebra, i.e. the algebraic operations are done using on massive integrations of initial conditions, has a slow truncated polynomials up to a certain order. rate of convergence towards a meaningful result. On the other hand, the study of the covariance matrix is a faster ξ2 xT P (ξ)] T;x0 2 method but only takes into account a linear approxima- x0 P (ξ)] ξ T;x0 1 tion of the uncertainties, while higher order approxima- 1 tions could play an important role. In this paper we present a new way to study uncer- Figure 1: Schematic idea of the Jet Transport. tainty regions: The Jet Transport. Given an initial state x0, the Jet transport consist in the integration of a full neigh- In this way we obtain the expression of the neighbourhood borhood of states around x0, instead of doing the single U propagated at time t by means of truncated Taylor’s se- x propagation of 0. This procedure is also known as Dif- ries up to order n, Pt;x0 (ξ). These series give us the state ferential Algebra and it was introduced by M. Wertz and of the particle, that initially is at x0 + ξ, at time t, i.e. K. Makino for the study of particle beams and particle ac- Pt;x0 (ξ) = '(t; t0; x0 + ξ). celerators. This kind of techniques have been also applied Using this procedure the time required for Monte in celestial mechanics to work with Kalman Filters and Carlo simulations can be dramatically reduced since now differential Kalman Filters.7 the propagation of an orbit is just a mater of evaluating a To briefly describe the methodology, let us consider polynomial (of course, as a minor drawback, the time to a dynamical system x_ = f(t; x) and the associated flow compute the propagation of the neighbourhood is longer map '(t; t0; x0) = xt which give us the position at time t than the required for a single trajectory). Moreover, this ∗Universitat de Barcelona, Spain, [email protected] zIEEC & Universitat Politecnica` de Catalunya, Spain. [email protected] xIEEC & Universitat de Barcelona, Spain, [email protected] 1 64rd International Astronautical Congress, Beijing, China. Copyright 2012 by the International Astronautical Federation. All rights reserved. procedure also allows the study of the covariance matrix control. Finally other applications of this procedure will up to high order, due to the fact that the different orders of be commented. the final polynomials provide the variational equations up to that order. The polynomial algebra As an interesting application of this jet transport methodology, in Alessi et al1 the authors studied the close As already stated, one of the basic ingredients of the Jet approach of the Near Earth Asteroid (NEO) Apophis. Transport is the polynomial algebra. In this section we More concretely they study the close approach of 2019. will explain some procedures and considerations for its Other Apophis encounters have been studied in.2 implementation. The relevance of the polynomial alge- bra is due to the fact that any algebraic operation in the method will use it. Some of these questions have been 2 Jet Transport studied by Jorba.5 The first question that any polynomial algebra must The objective of Jet Transport techniques for the numeri- deal with is how the polynomials are stored and accessed. cal integration of Ordinary Differential Equations (ODEs) There are several storage procedures in the literature, each is to obtain the solutions of the high order variational one with its own strengths and weaknesses. Some of them equations associated to a certain ODE. have been developed to deal with sparse polynomials, oth- The direct computation of high order variational equa- ers are memory efficient and others are fast accessing to tions requires a large amount of work just to obtain the the data. system of differential equations that must be integrated. In general, the polynomials that appear when the Jet To overcome this difficulty, the basic idea is to propagate, Transport procedure is applied to the usual vector-fields instead of just one initial condition x0, a full neighbour- of Astrodynamics and Celestial Mechanics are not sparse, hood of it, given by a polynomial Pt0;x0 (ξ) = x0 + ξ, therefore methods thought for these kind of polynomials where ξ = (ξ1; :::; ξn) represents the deviation with re- are not the best option. Two other storage systems are: spect to x0. Then, the ODEs can be integrated using any of the usual methods but instead of evaluating the vector- • Nested arrays: this system consist in a nest with a field that defines the differential equation at a certain state number of arrays as the number of variables of the using real arithmetic, polynomial arithmetic must be used system. It is a fast system to write, since the co- to do the integrations. efficients are easily stored using the exponents of The procedure requires two main ingredients: the variables. As main drawback, their structure is difficult to insert in a general implementation. The 1. A polynomial algebra package. The polynomial dimension of the dynamical system determines the algebra package has to contain all the basic func- number of arrays that must be nested and, therefore, tions needed for the implementation of the integra- they cannot be introduced in a general code. tion method. The usual functions that appear in the package are: addition, product, division, composi- • Reverse lexicographical order: in a first step the tion of polynomials as well as the elementary func- monomials are ordered according to their degree. tions (trigonometric, exponential, logarithm, ...). Inside each degree the following rule is applied: the monomial pj precedes the monomial pk if the 2. An integration method. Any standard integration exponent of the first variable of pj is higher than method can be used, for instance a Runge-Kutta, the exponent of the first variable of pk. In case a Taylor method or a symplectic integrator (i.e. that both monomials have the same exponent for leap-frog integrator). The arithmetic of the se- the first (second, ...) variable then the second lected method has to be adapted to the Jet Trans- (third, ...) variables should be used for the com- port, transforming the standard floating point arith- parison. In other words, using the multi-indices metics of the numerical integrator by a polynomial j = fj1; : : : ; jng and k = fk1; : : : ; kng then we arithmetic. say that j < k if the minimum i such that ji 6= ki verifies j < k . In this section we will show the basic characteristics of i i the Jet Transport methods. First we will describe the basic The selection of one of these two systems is done depend- characteristics of the polynomial algebra; later we will see ing on the characteristics of the problem. Usually, if the how we should mix the polynomial algebra with a Taylor number of variables is small the first method is used; when integration method, which is the integrator that has been the number of variables is large, then the second method is selected for the computations that follow.