The problem of estimating the state vector of a dynamical system from vector measurements, when it is known that the state vector satisfies norm equality constraints is considered. The case of a linear dynamical system with linear measurements subject to a norm equality constraint is discussed with a review of existing solutions. The norm constraint introduces a nonlinearity in the system for which a new estimator structure is derived by minimizing a constrained cost function. It is shown that the constrained estimate is equivalent to the brute force normalization of the unconstrained estimate. The obtained solution is extended to nonlinear measurement models and applied to the spacecraft attitude filtering problem.

Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational burden given a large number of Gaussian components that may not be needed. This work examines an improved method implementing a Gaussian mixture model that is adapted online via splitting of the Gaussian mixture model components triggered by an entropy-based detection of nonlinearity during the probability density function evolution. In doing so, the Gaussian mixture model approximation adaptively includes additional components as nonlinearity is encountered and can therefore be used to more accurately approximate the probability density function. This paper introduces this strategy, called adaptive entropy-based Gaussian-mixture information synthesis. The adaptive entropy-based Gaussian-mixture information synthesis method is demonstrated for its ability to accurately perform inference on two cases of uncertain orbital dynamical systems. The impact of this work for orbital dynamical systems is that the improved representation of the uncertainty of the space object can then be used more consistently for identification and tracking

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Abstract-This paper proposes a modular and flexible approach to adaptive Kalman filtering using the framework of a mixture-of-experts regulated by a gating network. Each expert is a Kalman filter modeled with a different realization of the unknown system parameters such as process and measurement noise. The gating network performs on-line adaptation of the weights given to individual filter estimates based on performance. This scheme compares very favorably with the classical Magill filter bank, which is based on a Bayesian technique, in terms of i) estimation accuracy, ii) quicker response to changing environments, and iii) numerical stability and computational demands. The proposed filter bank is further enhanced by periodically using a search algorithm in a feedback loop. Two search algorithms are considered. The first algorithm uses a recursive quadratic programming approach which extremizes a modified maximum likelihood function to update the parameters of the best performing filter in the bank. This particular approach to parameter adaptation allows a real-time implementation. The second algorithm uses a genetic algorithm to search for the parameter vector and is suited for post-processed data type applications. The workings and power of the overall filter bank and the suggested adaptation schemes are illustrated by a number of examples.