Maria Cecília Zanardi

Mathematical methods applied to the celestial mechanics of artificial satellites

1 INPE-DMC, Avenida dos Astronautas 1758, 12227-010 Sao Jose dos Campos, SP, Brazil 2 IEEC and Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain 3 Universidade Estadual Paulista, UNESP, CEP 12516-410, Campus de Guaratingueta, Guaratingueta, SP, Brazil 4Universidade Estadual Paulista, UNESP, Caixa Postal 178, CEP 13500-970, Campus de Rio Claro, Rio Claro, SP, Brazil

Unscented Kalman Filter Applied to the Spacecraft Attitude Estimation with Euler Angles

The aim of this work is to test an algorithm to estimate, in real time, the attitude of an artificial satellite using real data supplied by attitude sensors that are on board of the CBERS-2 satellite (China Brazil Earth Resources Satellite). The real-time estimator used in this work for attitude determination is the Unscented Kalman Filter. This filter is a new alternative to the extended Kalman filter usually applied to the estimation and control problems of attitude and orbit. This algorithm is capable of carrying out estimation of the states of nonlinear systems, without the necessity of linearization of the nonlinear functions present in the model. This estimation is possible due to a transformation that generates a set of vectors that, suffering a nonlinear transformation, preserves the same mean and covariance of the random variables before the transformation. The performance will be evaluated and analyzed through the comparison between the Unscented Kalman filter and the extended Kalman filter results, by using real onboard data.

Study of Stability of Rotational Motion of Spacecraft with Canonical Variables

This work aims to analyze the stability of the rotational motion of artificial satellites in circular orbit with the influence of gravity gradient torque, using the Andoyer variables. The used method in this paper to analyze stability is the Kovalev-Savchenko theorem. This method requires the reduction of the Hamiltonian in its normal form up to fourth order by means of canonical transformations around equilibrium points. The coefficients of the normal Hamiltonian are indispensable in the study of nonlinear stability of its equilibrium points according to the three established conditions in the theorem. Some physical and orbital data of real satellites were used in the numerical simulations. In comparison with previous work, the results show a greater number of equilibrium points and an optimization in the algorithm to determine the normal form and stability analysis. The results of this paper can directly contribute in maintaining the attitude of artificial satellites.

Spin-Stabilized Spacecrafts: Analytical Attitude Propagation Using Magnetic Torques

An analytical approach for spin-stabilized satellites attitude propagation is presented, considering the influence of the residual magnetic torque and eddy currents torque. It is assumed two approaches to examine the influence of external torques acting during the motion of the satellite, with the Earths magnetic field described by the quadripole model. In the first approach is included only the residual magnetic torque in the motion equations, with the satellites in circular or elliptical orbit. In the second approach only the eddy currents torque is analyzed, with the satellite in circular orbit. The inclusion of these torques on the dynamic equations of spin stabilized satellites yields the conditions to derive an analytical solution. The solutions show that residual torque does not affect the spin velocity magnitude, contributing only for the precession and the drift of the spacecrafts spin axis and the eddy currents torque causes an exponential decay of the angular velocity magnitude. Numerical simulations performed with data of the Brazilian Satellites (SCD1 and SCD2) show the period that analytical solution can be used to the attitude propagation, within the dispersion range of the attitude determination system performance of Satellite Control Center of Brazil National Research Institute.

Movimento rotacional com torque de gradiente de gravidade nas regiões de libração e circulação

Neste Trabalho e analisado o comportamento temporal das variaveis que representam o movimento rotacional de um satelite de medio porte nao simetrico, sob a acao do torque de gradiente de gravidade, nas regioes de libracao e circulacao ao redor dor pontos de equilibrio do sistema. A Hamiltoniana que descreve o movimento e expressa em termos das variaveis canonicas de Andoyer [3], que sao importantes em analises envolvendo metodos analiticos de pertubacao, e estao indicadas na Figura 1. Neste caso, as aplicacoes sao feitas com o auxilio dos softwares MATLAB e MATHEMATICA [...]

O Filtro H estendido de segunda ordem para estimação de atitude e bias de giros

Este trabalho descreve a determinacao de atitude e a estimacao de bias de giros usando do Filtro H Estendido e Segunda Ordem para sistemas nao lineares. Tal filtro usa a serie de Taylor para aproximar as nao linearidade da dinâmica conhecida e assume que os ruidos tem propriedades estatisticas conhecidas. A aplicacao utiliza dados de medidas de um satelite real CBERS-2 (China Brazil Earth Resources Satellite 2). O modelo cinematico da atitude e descrito por equacoes nao lineares envolvendo os ângulos de Euler. Os sensores de atitude disponiveis sao dois DSS (Digital Sun Sensors), dois IRES (Infra-Red Earth Sensor) e um triedro de giros mecânicos. De acordo com a teoria, em comparacao com o filtragem Kalman, a filtragem H tem algumas vantagens na estimacao de estados. No Filtro H, a natureza e considerada perversa e procura ativamente degradar a estimacao de estados tanto quando possivel, enquanto isso no Filtro de Kalman, a natureza e considerada indiferente. Assim, o Filtro H Estendido e simplesmente uma versao robusta dos Filtro de Kalman Estendido pois adiciona tolerâncias a ruidos e dinâmica nao modelados. Ao usar o Filtro H Estendido de Segunda Ordem, a meta e destacar e ampliar as propriedades do Filtro H em termos de suas caracteristica favoraveis. Os resultados neste trabalho mostram que se pode melhorar a precisao na determinacao de atitude com os requerimentos prescritos, alem de fornecer a estimativa dos bias dos giros que pode ser usada para realcar o modelo de erro dos giros. Sabe-se que giros apresentam algumas fontes de erros tal com os bias que e o mais problematico, pois com o tempo, a acumulacao de erros pioram a precisao no processo de estimacao, alem disso os bias devem ser levados em conta no processo de determinacao de atitude para garantir o sucesso da missao.

Nonlinear Least Squares Method for Gyros Bias and Attitude Estimation Using Satellite Attitude and Orbit Toolbox for Matlab

The knowledge of the attitude determination is essential to the safety and control of the satellite and payload, and this involves approaches of nonlinear estimation techniques. Here one focuses on determining the attitude and the gyros drift of a real satellite CBERS-2 (China Brazil Earth Resources Satellite) using simulated measurements provided by propagator PROPAT Satellite Attitude and Orbit Toolbox for Matlab. The method used for the estimation was the Nonlinear Least Squares Estimation (NLSE). The attitude dynamical model is described by nonlinear equations involving the Euler angles. The attitude sensors available are two DSS (Digital Sun Sensor), two IRES (Infra-Red Earth Sensor), and one triad of mechanical gyros. The two IRES give direct measurements of roll and pitch angles with a certain level of error. The two DSS are nonlinear functions of roll, pitch, and yaw attitude angles. Gyros are very important sensors, as they provide direct incremental angles or angular velocities. However gyros present several sources of error of which the drift is the most troublesome. Results show that one can reach accuracies in attitude determination within the prescribed requirements, besides providing estimates of the gyro drifts which can be further used to enhance the gyro error model.