Nonlinear Transformations and Filtering Theory for Space Operations



Journal Title

Journal ISSN

Volume Title



Decisions for asset allocation and protection are predicated upon accurate knowledge of the current operating environment as well as correctly characterizing the evolution of the environment over time. The desired kinematic and kinetic states of objects in question cannot be measured directly in most cases and instead are inferred or estimated from available measurements using a filtering process. Often, nonlinear transformations between the measurement domain and desired state domain distort the state domain probability density function yielding a form which does not necessarily resemble the form assumed in the filtering algorithm. The distortion effect must be understood in greater detail and appropriately accounted for so that even if sensors, state estimation algorithms, and state propagation algorithms operate in different domains, they can all be effectively utilized without any information loss due to domain transformations.

This research presents an analytical investigation into understanding how non-linear transformations of stochastic, but characterizable, processes affect state and uncertainty estimation with direct application to space object surveillance and space- craft attitude determination. Analysis is performed with attention to construction of the state domain probability density function since state uncertainty and correlation are derived from the statistical moments of the probability density function. Analytical characterization of the effect nonlinear transformations impart on the structure of state probability density functions has direct application to conventional non- linear filtering and propagation algorithms in three areas: (1) understanding how smoothing algorithms used to estimate indirectly observed states impact state uncertainty, (2) justification or refutation of assumed state uncertainty distribution for more realistic uncertainty quantification, and (3) analytic automation of initial state estimate and covariance in lieu of user tuning.

A nonlinear filtering algorithm based upon Bayes? Theorem is presented to ac- count for the impact nonlinear domain transformations impart on probability density functions during the measurement update and propagation phases. The algorithm is able to accommodate different combinations of sensors for state estimation which can also be used to hypothesize system parameters or unknown states from available measurements because information is able to appropriately accounted for.