Probabilistic Optical and Radar Initial Orbit Determination

Abstract

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.

Bounded Relative Orbits in the Zonal Problem via High-Order Poincaré Maps

“…This map is generally valid in the zonal problem when orbits with limited eccentricity are considered, such that the precession of the argument of perigee causes limited deviations from the expansion point (i.e., the fixed point). In Wittig et al [38] and Armellin and Lizia [32] it was illustrated that it was possible to estimate the size (the order of magnitude) of the region in which the truncation error of the Taylor approximation was lower than a prescribed value relying exclusively on the use of the coefficients of the map. However, in this work we evaluate the accuracy of the Taylor representation by numerically validating the results, as this method provides a more precise assessment and it represents a viable approach during the mission design phases.…”

“…This limitation shows up when searching for relative motion with large amplitudes, e.g. when δr = 0.11 is used (note that δr ≈ 1 is the estimated order of magnitude at which the truncation error of the 6-th order expansion of the Poincaré map is ≈ 10 −8 , according to the coefficient-based method presented in Armellin and Lizia [32]). This is clearly shown in the top panels of Fig.…”

“…As in [26], the high-order Taylor polynomials computed in this work are obtained via differential algebra (DA) techniques. These were first introduced in the field of particle accelerator physics by Berz [27] and later applied in a variety of problems in astrodynamics, including asteroid encounter analysis [28], uncertainty propagation [29], orbital conjunctions assessment [30], space debris evolution [31], and initial orbit determination [32]. In this work the key capability of DA to efficiently expand both the flow of ordinary differential equations (ODEs) and the solution of parametric implicit equations (PIEs) is exploited to compute the high-order Taylor expansion of Poincaré maps with respect to both initial conditions and an arbitrary number of relevant parameters.…”

Multiple Revolution Perturbed Lambert Problem Solvers

“…To include deep-space maneuvers, it is thus necessary to study how a perturbation of the position vector when p max > 1 affects the total mission ∆v. Although the study of the inclusion of deep space maneuvers is beyond the scope of this work, in the following we provide a procedure to exploit DA computation to expand the solution of the perturbed Lambert problem with respect to a perturbation in both the initial and final position vectors (building on the algorithm presented in [26]). We start from the solution of the perturbed Lambert problem, i.e.…”

Probabilistic data association: the orbit set

“…This problem is not found in the deterministic approach, where a function may be expressed in another state without any restriction. Different authors follow the probabilistic approach, such as Armellin and Di Lizia (2016), Fujimoto (2013), DeMars and Jah (2013). All three of them use a different approach: the first paper uses differential algebra (DA), the second one maps the AR into Delaunay variables, while the third uses Gaussian mixture methods (GMMs), where a generic PDF over the AR is approximated as the sum of Gaussian PDFs.…”