A Solution to the Full State Formulated Navigation State Vector Estimation Problem

Charles Shapiro

Abstract:	This paper introduces a new estimation methodology and its solution to the Full Sate formulated navigation state vector estimation problem. Described is a single cycle version of an estimation methodology which, unlike Kalman filter theory, was developed for systems whose real-world dynamics obey the laws of Newtonian physics and therefore the Fundamental Theorem of Calculus. This assumption, valid for real-world rigid body motion but not valid for stochastic processes, leads to an estimator that is able to utilize a first order Taylor series expansion of a high fidelity navigation IMU dynamic model. This represents a major change from the standard Error-Model formulation which uses ad-hoc linear error models in place of the linear Taylor series term. Presented is an explanation why Extended Kalman Filter’s standard linearization of the dynamic model has not been successfully applied to navigation state vector estimation problems. Also presented is a first principles derivation of an expression for the state vector’s probability density function. The ability to derive this expression and then optimize its log-likelihood function leads to an understanding that propagating the state vector’s probability density function is unnecessary, thus removing the difficulties that are associated with the propagation of the covariance matrix. A derivation of the iterative processing steps that maximize the probability density function expression is presented along with a proof that the processing steps for the first iteration are mathematically equivalent to the Extended Kalman filter. Simulations have demonstrated that it is the numerical method for evaluating the mathematical equivalent to the propagated covariance matrix, the transformed covariance matrix, which made it possible to develop a Full State Formulated 33 state vector estimator that, according to Monte Carlo simulation studies, satisfied the accepted theoretical lower limit on the tabulated RMS estimation errors (Crammer Rao Lower Bound matrix).
Published in:	Proceedings of the 2009 International Technical Meeting of The Institute of Navigation January 26 - 28, 2009 Disney's Paradise Pier Hotel Anaheim, CA
Pages:	687 - 697
Cite this article:	Shapiro, Charles, "A Solution to the Full State Formulated Navigation State Vector Estimation Problem," Proceedings of the 2009 International Technical Meeting of The Institute of Navigation, Anaheim, CA, January 2009, pp. 687-697.
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