Measurement Model Nonlinearity in Estimation of Dynamical Systems
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asurement Model Nonlinearity in Estimation of Dynamical Systems1 Manoranjan Majji2, J. L. Junkins3, and J. D. Turner4
Abstract The role of nonlinearity of the measurement model and its interactions with the uncertainty of measurements and geometry of the problem is studied in this paper. An examination of the transformations of the probability density function in various coordinate systems is presented for several astrodynamics applications. Smooth and analytic nonlinear functions are considered for the studies on the exact transformation of uncertainty. Special emphasis is given to understanding the role of change of variables in the calculus of random variables. The transformation of probability density functions through mappings is shown to provide insight in to understanding the evolution of uncertainty in nonlinear systems. Examples are presented to highlight salient aspects of the discussion. A sequential orbit determination problem is analyzed, where the transformation formula provides useful insights for making the choice of coordinates for estimation of dynamic systems.
Introduction Parameter estimation from algebraic models is a key topic of continued research in science and engineering. Contributions of Gauss [1] and others in parameter estimation seek the best estimate for the true parameters of interest, while incorporating the uncertainty associated with individual measurements in the decision process. This paper builds on the second author’s tutorial lecture [2] “How Nonlinear is it?” To this end, we explore the fundamental role of nonlinear algebraic transformations in estimation theory. Junkins and Singla propose a nonlinearity index as a quantitative measure for nonlinearity of functions and dynamical systems. This 1
Dedicated to Professor Kyle T. Alfriend for his contributions in Astronautics. Research Associate, Aerospace Engineering Department, Texas A&M University, College Station, TX, 77840. 3 Distinguished Professor, Regents Professor, Royce Wisenbaker Chair in Engineering, Aerospace Engineering Department, Texas A&M University, College Station, TX, 77843-3141, Fellow, AAS, Fellow, AIAA. E-mail: [email protected]. 4 Research Professor, Aerospace Engineering Department, Texas A&M University, College Station, TX, 77843-3141, Associate Fellow, AAS. E-mail: [email protected]. 2
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metric is used to investigate the nonlinearity of orbital and attitude dynamics models. Recent work [3,4,5] in the area of uncertainty propagation in dynamical systems made significant progress in understanding the evolution of the probability density function governing the state uncertainty. Progress has also been made to incorporate the propagated state and measurement uncertainty in nonlinear estimators that make use of analytic, semi-analytic and numerical methods (cf. Park and Scheeres [6], Majji et al. [7, 8], Julier and Uhlmann [9]). In astronautics, Junkins et al. [10, 11] show that uncertainty propagation through nonlinear dynamical systems is an important problem in astronauti
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