The purpose of this work is to analyze the performance of the Cubature Kalman Filter, Unscented Kalman Filter and Extended Kalman Filter estimators in the attitude estimation problem when submitted to real attitude sensors data.The Extended Kalman Filter (EKF) is the most used nonlinear filtering algorithm for the attitude estimation in real time. The EKF is the nonlinear version of the Kalman Filter which linearizes about an estimate of the current mean and covariance. However, when the filter is subjected to poor conditions, the linearization of the system may not be efficient and lead to an estimation of low accuracy and divergence of the filter. The Unscented Kalman Filter (UKF) is an algorithm that was developed in order to avoid the linearizations required by the EKF. Basically, the UKF uses a set of points chosen deterministically, called "sigmapoints", to capture the probability distribution and generalizes to nonlinear system without the burdensome analytic derivation as in the EKF. More recently, the Cubature Kalman Filter (CKF) was proposed as an alternative estimation algorithm for general nonlinear systems. The CKF, which builds on the numerical-integration perspective of Gaussian filters, employs a third-degree spherical-radical cubature rule to compute Gaussian-weighted integrals, derivative-free nonlinear filtering algorithm with improved performance over the UKF in terms of estimation accuracy, numerical stability and computational costs.In this work, the application uses the real measurement data for orbit and attitude of the CBERS-2 (China Brazil Earth Resources Satellite) satellite. The attitude dynamical model is described by nonlinear equations involving the Euler angles.The attitude sensors available are two DSS (Digital Sun Sensors), two IRES (Infra-Red Earth Sensor), and one triad of mechanical gyros. The analyzes are based on the robustness of the filter, in relation to the precision, computational cost and convergence speed in attitude estimation. As the use of real data makes it impossible to compare the estimated results with the real attitude of the satellite, then the results obtained via EKF are taken as reference for comparison with the UKF and CKF. The results in this work Feedback/Corrections? (/feedback/correctabstract?bibcode=2018cosp...42E3523V)show that, for the case studied in this article, the filters are very competitive and present advantages and disadvantages that should be evaluated according to the need of each problem.
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 Earth's 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 spacecraft's 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.
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.
One problem that afflicts teachers, particularly those who teach Differential and Integral Calculus, is the difficulty of students in mathematics content and the consequent lack of understanding of what is presented in this discipline. Numerous studies have been carried out with the aim of creating and improving new teaching-learning methodologies that make students active elements of the learning process. This work proposes the application of the Problem Based Learning methodology in the definite Integral topic of the Calculus discipline, in order to guide the teacher and contribute to the student having contact with the discipline in problems that represent situations closer to their reality. In this proposal the student will also make use of technologies that, although they are known, are often underutilized by them.
his work is applied to the dynamics of rotational motion of artificial satellites, that is, itsorientation (attitude) with respect to an inertial reference system. The attitude determination involvesapproaches of nonlinear estimation techniques, which knowledge is essential to the safety and controlof the satellite and payload. Here one focuses on determining the attitude of a real satellite: CBERS-2(China Brazil Earth Resources Satellite). This satellite was launched in 2003 and were controlled andoperated in turns by China (Xi’an Control Center) and Brazil (Satellite Control Center). Its orbit isnear polar sun-synchronous with an altitude of 778km, crossing Equator at 10:30am in descendingdirection, frozen perigee at 90 degrees, and providing global coverage of the world every 26 days.The attitude dynamical model is described by nonlinear equations involving the Euler angles. Theattitude sensors available are two DSS (Digital Sun Sensor), two IRES (Infra-Red Earth Sensor), andone triad of mechanical gyros. The two IRES give direct measurements of roll and pitch angles with acertain level of error. The two DSS are nonlinear functions of roll, pitch, and yaw attitude angles. Thegyros furnish the angular measurements in the body frame reference system. Gyros are very importantsensors, as they provide direct incremental angles or angular velocities. They can sense instantaneousvariations of nominal velocities. An important feature is that it allows the replacement of complexmodels (different torques acting on the space environment) by using their measurements to turn thedynamical equations into simple kinematic equations. However gyros present several sources of errorof which the drift is the most troublesome. Such drifts yield along time an accumulation of errorswhich must be accounted for in the attitude determination process. Herein one proposes to estimatethe attitude and the drift of the gyros using the Least SquaresMethod. Results show that one can reachaccuracies in attitude determination within the prescribed requirements, besides providing estimatesof the gyro drifts which can be further used to enhance the gyro error model.
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