The rotational motion of a satellite with a magnetic stabilization system is discussed. The motion is described by a nonautonomous differential equation, with the magnetic moment of the satellite as a parameter. The global phase portrait of the problem is investigated in a wide range of magnetic-parameter values, using a numerical realization of the method of Poincaré point maps. New periodic solutions of the problem are found, and an analysis is carried out of the evolution of the phase portrait and the bifurcation of periodic solutions with varying magnetic-parameter values. The values of the magnetic parameter that must be avoided in the design of the satellite magnetic stabilization system are discussed.

High image quality and complex, refractive optical systems, as those used in remote sensing applications, are, in general, very difficult to be manufactured with the required performance. This can be charged to the high sensitivity of such systems to the fabrication tolerances, mainly concerning the relative alignment of the optical components with respect to each other. When the system does not achieve the expected quality, the puzzle is to identify where the problems lies. This is even worsened when the number of optical elements becomes high. Due to these facts, some misalignment characterization and estimation techniques based on Bayesian estimators and wavefront measurements have been proposed in the literature. This paper is the result of a deep study and investigation of these techniques, with emphasis on an application to an intentionally simple system for the sake of illustration that highlights conceptual issues that could be extended to more realistic, complex optical systems. With this purpose, the sensitivity of the wavefront Zernike coefficients to the misalignment parameters, its use in a parameter estimator design that includes nonlinear terms, the study of the system observability, and a statistical analysis of the estimator performance considering the observation noise are addressed in details. Numerical simulation results for the simple system are shown. We also present insights on how to apply the technique to the alignment of a 11-lens optical system used in the Brazilian remote sensing camera MUX, that will fly on-board the upcoming Sino-Brazilian satellites CBERS 3&4.

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ResumoNeste artigo é apresentada a descrição do algoritmo TRIAD para a determinação da atitude tridimensional de um corpo, por meio da observação de dois vetores. Esse algoritmo, apesar de não constituir a solução ótima para o problema, pode ser implementado eficientemente em computadores. Também é apresentada a matriz cartesiana de covariância da atitude para o algoritmo. Os pormenores do cálculo desta matriz são esmiuçados, resultando em uma equação analítica simples em função dos conjuntos de medidas tomados de tríades de dois diferentes sensores no sistema de coordenadas do corpo. As soluções apresentadas podem ser utilizadas para implementar sistemas de determinação da atitude altamente integrados, usando sensores inerciais e sensores de referência com tecnologia de micro-fabricação (Micro-Electro-Mechanical Systems -MEMS) e microcontroladores de baixo custo. Palavras-chave: Determinação da atitude. Triad. Matriz de covariância AbstractThis article presents a description of the TRIAD algorithm, which utilizes two vector observations for determining a body three-dimensional attitude. This algorithm describes a deterministic form to calculate the attitude and it does not constitute the optimal solution for the problem, but can be implemented efficiently on computers. The Cartesian covariance matrix for this algorithm is also presented. This matrix calculation are addressed in details, resulting in a simple analytical formula based on two different sensor sets measurements in the body frame. The solutions presented are suited to implement highly integrated attitude determination systems that employ MEMS (Micro-Electro-Mechanical Systems) technology sensors and low cost microcontrollers.

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