Probabilistic data association: the orbit set

Abstract

This paper presents a novel method to obtain the solution to the initial orbit determination problem for optical observations as a continuum of orbits-namely the orbit set-that fits the set of acquired observations within a prescribed accuracy. Differential algebra is exploited to analytically link the uncertainty in the observations to the state of the orbiting body with truncated power series, thus allowing for a compact analytical description of the orbit set. The automatic domain splitting tool controls the truncation error of the polynomial approximation by patching the uncertainty domain with different polynomial expansions, effectively creating a mesh. The algorithm is tested for different observing strategies to understand the working boundaries, thus defining the region for which the admissible region is necessary to extract meaningful information from observations and highlight where the new method can achieve a smaller uncertainty region, effectively showing that for some observing strategies it is possible to extract more information from a tracklet than the attributable. Consequently, the method enables comparison of orbit sets avoiding sampling when looking for correlation of different observations. Linear regression is also implemented to improve the uncertainty estimation and study the influence of the confidence level on the orbit set size. This is shown both for simulated and real observations obtained from the TFRM observatory.

Data association and uncertainty pruning for tracks determined on short arcs

“…In particular, the high-order nonlinear propagation maps only a two-dimensional domain and is thus very efficient. This paper builds on results obtained in Principe et al (2019Principe et al ( , 2017, Pirovano et al (2018).…”

“…Furthermore, propagating in DA may cost as much as two order of magnitude more in time than propagating in double precision. However, as underlined in Pirovano et al (2018), the cost of propagating a state in DA comes with two main advantages: first of all, the possibility to then evaluate every possible initial condition as a function evaluation rather than a new propagation and secondly, the availability of higher-order terms, necessary to (a) Time spent in each part of the code (b) pruning percentage after one day. understand the influence that each variable has on the solution.…”

“…In the case of a geostationary earth orbit (GEO), it translates into observations taken in a 2-3-min span. In Pirovano et al (2018), a detailed analysis of observation conditions for IOD can be found. In the literature, both the tracklet-to-track and the single observation to track association scenarios are referred to as observation-to-track association (OTTA), since the new observations cannot determine the state.…”

Robust initial orbit determination for short-arc Doppler radar observations

“…Armellin and Di Lizia [24] reformulated Lambert's problem in DA sense, providing a mathematical description of the solution uncertainty of the angle-range IOD problem as a function of measurements noise. This result, combined with the use of automatic domain splitting (ADS) [25], was then exploited in [26] to solve the angle-only IOD problem, thus allowing for an analytical description of the so-called orbit set.…”

“…The solution uncertainty quantification is achieved by combining DA and ADS, which provide the Taylor expansion of the state with respect to the observations. The algorithm, here referred to as DAIOD algorithm, represents an extension of its optical [26] and range radar [24] counterparts, and completes the DAIOD estimation framework for the three possible available scenarios: angle-only, angle-range, and angle-Doppler.…”