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.
This study introduces novel algorithms and the underlying mathematics to process pictures of planetary illuminated bodies and use them for navigation purposes. The goal is to accurately estimate the observer-to-body relative position in inertial coordinates. The main motivation is to provide autonomous navigation capabilities to spacecrafts by observing a planet or a moon. This is needed, for example, in humanrated vehicles in order to provide navigation capabilities in a loss of communications scenario. The algorithm is derived for the general case of a triaxial ellipsoid which is observed bounded by an elliptical cone. The orientation of the elliptical cone reference frame is obtained by eigenanalysis and the offset between the elliptical cone axis and the body center direction as well as the equation of the terminator are quantified. The main contribution of this paper is in the image processing approach adopted to derive centroid and distance to the body. This is done by selecting a set of pixels around the body limb and modeling the smooth limb transition using two-dimensional circular and elliptical sigmoid functions. More accurate estimates of centroid and distance are then obtained by iterative nonlinear least-squares using these models. Sensitivity analysis is performed and numerical examples using a real Moon picture taken from Earth is provided to clarify the image processing steps and to validate the proposed theory.
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