Nonlinear Mapping of Uncertainties in Celestial Mechanics

Zanetti

2016

Celest Mech Dyn Astr

A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map the observation uncertainties from the observation space to the state space. When a minimum set of observations is available, DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available, high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology.

Section: Discussionmentioning

confidence: 99%

Section: Nonlinear Mapping Of the Estimate Statisticsmentioning

confidence: 99%

Section: Da-based Iod Updatementioning

confidence: 99%

See 1 more Smart Citation

Dealing with uncertainties in angles-only initial orbit determination

Zanetti

2016

Celest Mech Dyn Astr

“…The O2S map can be profitably used to map analytically samples from the observation space into the RSO orbital parameters space, thus enabling for example efficient Monte Carlo (MC) simulations. In addition, for given statistical properties of the measurements, the availability of a Taylor approximation of the O2S allows one to obtain the statistical moments of the states by computing the expectation of its monomials, thus avoiding the need of samples [26,27]. More importantly, the O2S map (and the Taylor representation of its inverse, the S2O map) can be manipulated to analytically represent the probability density function (pdf) of the IOD solution for any arbitrary pdf of the measurements.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Optical and Radar Initial Orbit Determination

Journal of Guidance, Control, and Dynamics

2018

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.

“…Differential algebra has already proven its efficiency in the nonlinear propagation of uncertainties within different dynamical models, including two-body dynamics (Valli et al 2012), (n + 1)-body dynamics , and geocentric models (including Earth's gravitational harmonics, solar radiation pressure, shadows, and third body perturbations) (Morselli et al 2010). Nonetheless, the accuracy of the method tends to decrease drastically when the uncertainty domain becomes too stretched in one or more directions.…”

Section: Introductionmentioning

confidence: 99%

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

Wittig

et al. 2015

Celest Mech Dyn Astr

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