Satellite attitude determination based on star sensors

Three algorithms for satellite three-axis attitude determination based on star sensors: q-Method, improved TRIAD algorithm, and QUEST algorithm.

For the improved TRIAD dual vector pose determination algorithm, the implementation process and information flow of the algorithm are provided, and the limitations of the algorithm are described.

For the QUEST pose determination algorithm, the measurement model, implementation process, and information flow of the algorithm are provided. At the same time, the selection principle of attitude determination algorithm is provided, which is to select and output the final attitude calculation results based on the star identification given to the navigation star during the selection process and the advantages and disadvantages of the minimum cost function values calculated by different attitude determination algorithms. Afterwards, the improved TRIAD and QUEST pose determination results of three measured star maps near Cassiopeia were provided, and according to the results of this chapter

The principle of posture optimization described is used to output the camera’s three axis pose along the line of sight.

Finally, based on the installation matrix of the star sensor on the satellite, a method for obtaining the three-axis attitude of the satellite is provided.

For a three-axis stationary meteorological satellite, within a day, the satellite orbits the Earth once. In order for the scanner to maintain ground pointing, its attitude must continuously rotate. Due to the fact that one side of the three-axis stable satellite is constantly heated by solar radiation, while the other side is constantly facing cold space for heat dissipation, causing serious thermal deformation and affecting the optical axis of the instrument, resulting in inaccurate image positioning. In order to achieve accurate image registration and positioning, the constantly changing attitude of a three-axis stabilized satellite must be accurately measured. The task of a star sensor mounted on a three-axis stabilized satellite body is to continuously monitor the satellite’s attitude for real-time determination of remote sensing instrument pointing. The method for determining the attitude of a three-axis stabilized satellite based on star sensors is to use the vector information of the observed star measured by the star sensor. After the star map recognition process, the navigation star vector information of the corresponding celestial coordinate system is identified. Using the corresponding attitude determination algorithm, the attitude of the star sensor’s line of sight relative to the celestial reference coordinate system is obtained. Then, based on the installation position of the star sensor on the satellite, determine the orientation of the star sensor’s line of sight or three-axis attitude; Then, based on the coordinate transformation relationship, the three-axis attitude of the satellite relative to the celestial reference coordinate system is obtained.

Attitude determination algorithm based on star sensors

(1) Wahba problem

Almost all spacecraft attitude estimation based on vector observation is based on the constrained least squares estimation method proposed by Grace Wahba in 1965 for attitude determination through vector observation – the Wahba problem. Its core is to find the optimal orthogonal direction cosine matrix A with determinant 1, so as to minimize the value of the cost function J (A).

(2) Q Method

In 1968, Paul Davenport first provided a solution to the Wahba problem for determining spacecraft attitude: q-Method, which parameterized the attitude matrix using a unit quaternion.

(3) TRIAD algorithm

TRIAD (Tri Axial Attitude Determination) is an algebraic pose determination algorithm proposed by Harold D. Black in 1964, which uses two absolute vectors in two different coordinate systems to obtain the attitude transformation matrix. However, due to the high dependence of the TRIAD algorithm on the order of vectors, the attitude matrix varies depending on the first vector.

Itzhack Y.Bar and Richard R. Harman proposed an improved TRIAD algorithm, which selects two vectors as the first vector to obtain two attitude conversion matrices. Then, the optimal attitude conversion matrix is obtained by measuring the weighted average factor of the error.

The improved TRIAD algorithm is described as follows:

1) Check if the observation vector is collinear, if so, exit;

2) Calculate the standard deviation of the observation vector to generate a weighting factor for the attitude conversion matrix;

3) Construct an orthogonal coordinate system R with 1rG as the first axis in the reference coordinate system, construct an orthogonal coordinate system B with b1K as the first axis in the star sensor coordinate system, and obtain the attitude conversion matrix A1;

4) Construct an orthogonal coordinate system R with 2 rG as the first axis in the reference coordinate system, construct an orthogonal coordinate system B with b2K as the first axis in the star sensor coordinate system, and obtain the attitude conversion matrix A2;

5) Calculate the attitude conversion matrix A based on the weighting factor of attitude conversion ˆ ， For A ˆ  Perform orthogonalization to obtain the optimal attitude conversion matrix A;

6) Convert the attitude matrix into an Euler angle equation and output the three-axis attitude of the star sensor’s visual axis.

Due to the fact that dual vector attitude determination only uses two observation stars and their corresponding reference stars to determine the satellite’s attitude, this limits the accuracy of attitude estimation. When more than two observation stars are identified, if the least squares method is used to solve the optimal attitude conversion matrix and obtain the minimum cost function (Wahba problem) to estimate the satellite’s attitude, the calculation speed will be very slow.

The information flow of the improved TRIAD algorithm is shown in Figure 6.1.

(4) QUEST algorithm

1. D. Shuster’s QUEST (Quaternion ESTimator) algorithm is a deterministic method that utilizes a point by point solution. It calculates the Davenport feature roots through a Newton-Raphson iteration of the optimal initial value. This method solves the rotation problem in the attitude calculation process and introduces the attitude error covariance matrix between the ontology coordinate system and the reference coordinate system, This weakens the attitude error covariance matrix of the inertial reference frame in terms of Euler angles.

The QUEST algorithm was first applied to the MAGSAT task in 1979 and is currently the most commonly used algorithm to solve the Wahba problem.

The QUEST algorithm is described as follows:

1) Determine whether the output result of star map recognition is more than 3, and if it is less than 3, exit;

2) Check if the observation vector is collinear, if so, exit;

3) Calculate the standard deviation of the observation vector and use the reciprocal of variance as the weight coefficient of the minimum cost function;

4) Calculate the covariance matrix of quaternions;

5) If the number of iterations set is equal to 0, the feature root is λ 0 can be obtained by formula (6.33);

6) If the set number of iterations is greater than 0, a new feature root is calculated through n (n is the set number of iterations) Newton Raphson iterations, with an initial value of λ 0

7) Calculate the attitude quaternion based on the feature roots, where the quaternion is non orthogonal;

8) Check the calculation accuracy of quaternions, and if the predetermined accuracy requirements are met, orthogonalize the non orthogonal quaternion;

9) If the accuracy requirements are not met, rotate the reference vector 180 degrees around the X, Y, or Z axis, solve for the quaternion, select the optimal value, and orthogonalize the non orthogonal quaternion;

10) Convert the orthogonalized attitude quaternions into Euler angles and output the three-axis attitude of the star sensor’s visual axis.

The information flow of the QUEST algorithm is shown in Figure 6.2.

The improved TRIAD pose determination results are superior to the QUEST pose determination results due to the lack of calibration of the camera’s focal length and image plane center point coordinates during the star map preprocessing process, as well as the failure to consider errors introduced by factors such as lens distortion, atmospheric refraction, and Earth rotation. Therefore, the more stars participate in attitude calculation, the more error sources are introduced.

After determining the axis of view direction of the star sensor (star camera), the three-axis attitude of the satellite can be obtained through the installation matrix of the star sensor on the star.