^{1}

^{1}

^{2}

^{1}

^{2}

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.

The attitude of a spacecraft is defined by its orientation in space related to some reference system. The importance of determining the attitude is related not only to the performance of attitude control but also to the precise usage of information obtained by payload experiments performed by the satellite. The attitude estimation is the process of calculating the orientation of the spacecraft in relation to a reference system from data supplied by attitude sensors. Chosen the vector of reference, an attitude sensor measures the orientation of these vectors with respect to the satellite reference system [

The attitude of an artificial satellite is directly related to its orientation in space. Through the attitude one can know the spatial orientation of the satellite, since in most cases it can be considered as a rigid body, where the attitude is expressed by the relationship between two coordinate systems, one fixed on the satellite and another associated with a reference system, for example, inertial system [

1st rotation of an angle

2nd rotation of an angle

3rd rotation of an angle

The matrix obtained through the 3-2-1 rotation sequence is given by

By representing the attitude of a satellite with Euler angles, the set of kinematic equations are given by [

In order to estimate the satellite attitude accurately, several types of sensors, including gyros, earth sensors, and solar sensors, are used in the attitude determination system. The equations of these sensors are introduced here.

The advantage of a gyro is that it can provide the angular displacement and/or angular velocity of the satellite directly. However, gyros have an error due to drifting, meaning that their measurement error increases with time. In this work, the rate-integration gyros (

Thus, the measured components of the angular velocity of the satellite are given by

One way to compensate for the drifting errors present in gyros is to use the earth sensors. These sensors are located on the satellite and aligned with their axes of

Since an earth sensor is not able to measure the

The goal of an estimator is to calculate the state vector (attitude) based on a set of observations (sensors) [

The basic premise behind the Unscented Kalman Filter (UKF) is that it is easier to approximate a Gaussian distribution than it is to approximate an arbitrary nonlinear function. Instead of linearizing to first order using Jacobian matrices, the UKF uses a rational deterministic sampling approach to capture the mean and covariance estimates with a minimal set of sample points. The nonlinear function is applied to each point, in turn, to yield a cloud of transformed points. The statistics of the transformed points can then be calculated to form an estimate of the nonlinearly transformed mean and covariance. We present an algorithmic description of the UKF omitting some theoretical considerations, left to [

Consider the system model given by

Performing the necessary simplifications (small Euler angles) in the set of (

Given the state vector at step

Once

With

As the last part of the prediction step, we calculate the

To compute the correction step, we first must transform the columns of

With the mean measurement vector

Finally, the last calculation in the correction step is to compute the

Here, the results and the analysis from the algorithm developed to estimate the attitude are presented. To validate and to analyze the performance of the estimators, real sensors data from the CBERS-2 satellite were used (see [

initial attitude:

initial

initial covariance (^{2} (error related to the attitude), and ^{2} (error related to the drift of gyro);

observation noise covariance (^{2} (sun sensor), are ^{2} (earth sensor);

dynamic noise covariance (^{2} (noise related to the attitude), ^{2}, and ^{2} (noise related to the drifting of gyro).

The real measurements obtained by the attitude sensors (digital sun sensors, infrared Earth sensors, and gyroscopes) are shown in Figure

Real measurements supplied by attitude sensors.

In Figures

Attitude estimated by the Unscented Kalman filter end Extended Kalman filter.

Figures

Attitude errors (Standard Deviation) estimated.

Bias errors (Standard Deviation) estimated.

In Figures

Residuals related to DSS attitude sensors.

Residuals related to IRES attitude sensors.

These results seem consistent because in this case it is not possible to compare the estimated values with true values, since these values are not known.

In this paper, the Unscented Kalman Filter estimator applied in nonlinear systems was presented for use in real-time attitude estimation. The main objective was to estimate the attitude of a CBERS-2 like satellite, using real data provided by sensors that are on board of the satellite. To verify the consistency of the estimator, the attitude was estimated by two different methods. The usage of real data from on-board attitude sensors poses difficulties like mismodelling, mismatch of sizes, misalignments, unforeseen systematic errors, and postlaunch calibration errors. However, it is observed that, although the EKF and UKF have roughly the same accuracy, the UKF leads to a convergence of the state vector faster than the EKF. This fact was expected, since the UKF prevents the linearizations needed for EKF, when the system has some nonlinearity in their equations.