on September 25, 2015. This satellite is equipped with one APS star sensor and one CCD star sensor developed by Shanghai Aerospace Control Technology Research Institute, making it the first model of Shanghai Aerospace Technology Research Institute to use all domestic star sensors as the main component of attitude measurement.
The article first outlines the error of star sensors and explains the accuracy analysis methods. Then, based on the in orbit data of different types of star sensors provided by the Pujiang-1 satellite, a comprehensive analysis method of star sensor measurement accuracy using epoch difference method, sliding window method, and optical axis angle method is proposed. Finally, a conclusion is given.
The errors of star sensors are usually divided into three categories: bias error (BE), low spatial frequency error (LSFE), and noise equivalent angle (NEA). The noise equivalent angle can also be divided into high spatial frequency error (HSFE) and random error (TE), and the causes of each error are summarized as follows.
(1) Bias error: The error between the measurement coordinate system of the star sensor and the mechanical coordinate system, usually caused by gravity release or vibration in the active section. The constant value error of in orbit measurement is usually evaluated based on two star sensors or the load of the star sensor and the Earth observation;
(2) Space low-frequency error: divided into two categories: field of view periodic low-frequency error and orbit periodic low-frequency error;
1) Low frequency error of field of view period: As the satellite moves, stars move within the field of view of the star sensor. Due to optical system distortion, star sensor calibration residuals, navigation catalog errors, etc., they will be reflected in the recognition and orientation of stars within the field of view. This error is a system error, with fluctuation periods ranging from tens of seconds to tens of minutes;
2) Orbital low-frequency error: When a satellite is in orbit, the thermal environment in which the star sensor is located will vary with the orbital period. The resulting thermal deformation of the mounting bracket and the star sensor itself will cause the measured values of the star sensor to fluctuate with the orbital period, which is the main component of the low-frequency error of the orbital period; In addition, the optical aberration also belongs to the low-frequency error of the orbital period, but it is easy to model and correct, so it is not a research focus on low-frequency error suppression of star sensors.
(3) Noise equivalent angle: composed of spatial high-frequency error and random error;
The classification of noise equivalent angles is as follows:
1) Spatial high-frequency error: As the satellite moves, the image spot of the star moves between the image spots of the photodetector. Due to the non-uniformity of pixel response and the pixel filling of some photodetectors, it cannot reach 100%, resulting in periodic fluctuations in the measurement value of the center of mass of the star point, resulting in spatial high-frequency error in attitude measurement;
2) Random error: The error caused by electronic system noise such as pixel readout noise and A/D conversion noise, which is not related to time and space, can be suppressed by improving the signal-to-noise ratio of photoelectric detection imaging. The total error of star sensor attitude measurement is synthesized by formula (1), and the bias error can be corrected by modifying the installation matrix, so it is not included in the total error
The star sensor measurement data transmitted by the Pujiang 1 satellite includes the following combinations:
(1) The quaternion output from a single star sensor, corresponding timestamp and data validity flag, with a download period of 2 seconds;
(2) In the satellite orbit coordinate system, the roll, pitch, and yaw angles converted from quaternions output by the star sensor are transmitted for a period of 2 seconds;
(3) The quaternion output from two star sensors simultaneously collected by the onboard computer, with a download period of 16 seconds;
(4) Auxiliary information such as grayscale mean, number of detected stars, number of navigation stars, number of fixed attitude stars, and installation surface temperature of two star sensors in the digital star map of the star sensor.
The principle of using the epoch difference method to calculate the equivalent angle of noise is to calculate the three-axis attitude angle increment of the star sensor at adjacent times through the difference of adjacent quaternions. The direct current part of this physical quantity reflects the motion of the satellite, and the remaining part reflects the attitude measurement noise of the star sensor. The calculation steps are described as follows:
(1) Take a quaternion sequence of measured attitude for a sufficient duration in orbit (usually taking a complete orbital period of measured attitude quaternion);
(2) Due to errors in the sampling time interval of in orbit attitude data, a cubic spline interpolation algorithm is used to calculate the attitude quaternions at equal intervals from the start point to the end point of time;
(3) Using the epoch difference method, calculate the change in quaternion qi+1 of the quaternion star sensor at time i to time i+1, as shown in formula (2), where represents quaternion multiplication
(4) Order Δ Q=[q0, q1, q2, q3] ^ T (all quaternions used in the text are scalar with q0, the same below), and the results obtained in step (3) Δ Q is converted into three axis Euler angles x, y, z, as shown in formulas (3), (4), (5), and (6).
(5) Based on the results obtained in step (4), its DC component reflects the satellite motion information, which is the DC component of the signal. After removing the DC amount, the noise equivalent angle of the three-axis Euler angle is calculated.
The key points to note in evaluating the equivalent angle of noise using the epoch difference method are as follows:1) Step (2) is necessary. If equal time interval preprocessing is not performed, the resulting noise should be significantly greater than the actual value;2) The sampling interval of attitude quaternion has a significant impact on the analysis results. When the sampling interval of quaternion is similar to the exposure period of star sensor, such as the exposure period of star sensor being 100 ms, sampling quaternion at 10 Hz results in TE. As the sampling interval increases, the obtained result will reflect the influence of HSFE. When the sampling interval is more than 10 times larger than the exposure period, this method is not applicable.
The principle of calculating the total error based on the sliding window method is to fit the reference quaternion based on the measured quaternion 7-order polynomial within the preset width of the sliding window, and use it as a benchmark to calculate the error. When the window width matches the time when the star passes through the field of view, the error obtained is the attitude measurement error except for the low-frequency error of the orbital period.
Calculation steps of sliding window method:
(1) Take a quaternion sequence of measured attitude for a sufficient duration in orbit (usually taking a complete orbital period of measured attitude quaternion);
(2) Set the width of the sliding window. When a typical sun synchronous orbit satellite flies horizontally, the angular rate of the optical axis of the star sensor passing through the celestial sphere is about 0.06 (°)/s, and the field of view of the star sensor is 20 °. Therefore, the maximum time for the star to pass through the field of view of the star sensor is 333 seconds, and the average time for the star to pass through the field of view of the star sensor is about 160 seconds. Currently, the data transmission period of the star sensor measurement is 2 seconds, so the width of the sliding window is set to 160/2=80;
(3) In the sliding window, set the measured quaternion sample points as Qm={q1, q2,…, q80}, and the corresponding timestamp sequence as t={t1, t2,…, t80}. Fit the “time quaternion” curve with a 7-order polynomial to obtain the reference quaternion curve Qr, as shown in formula (7).
(4) Calculate the variation between the measured quaternion and the reference quaternion according to formula (2), and convert it into three axis attitude angles x, y, and z according to formulas (3) to (6) to obtain the attitude error curve;
(5) Calculate the three-axis attitude error.
It should be noted that using this method requires consideration of the impact of satellite platform stability on the calculation results. When observing stars on the ground, the Earth can be regarded as a platform with a three-axis stability better than 0.1 “, so its impact can be ignored. However, for three-axis stable Earth satellites, this factor cannot be ignored, so the error calculation results are larger than those of ground observation. However, in the case where satellites cannot provide higher precision attitude standards, this method is still the most effective method for evaluating the total error of star sensors.
The star sensors configured for the Pujiang 1 satellite are fixedly connected to the satellite body, and the angle between the optical axes of the two star sensors is theoretically fixed. The theoretical value of the angle between the optical axes of the two star sensors on the Pujiang 1 satellite is 25.91 °. The statistical value obtained by the included angle method is the composite value of the total measurement error of the two star sensors.
Assuming the measured quaternion of APS star sensor is qAPS=[q’0, q’1, q’2, q3 ‘] ^ T, and the measured quaternion of CCD star sensor is qCCD=[q’0, q’1, q’2, q3’] ^ T, AAPS and ACCD are calculated according to formula (3) to obtain the APS star sensor optical axis pointing vector vAPS and CCD star sensor optical axis pointing vector vCCD, respectively, as shown in formulas (8) and (9). The angle between the optical axes of two star sensors is calculated according to formula (10):
When using this method, it should be noted that the included angle is constant and does not change with satellite attitude changes. It is only related to star sensors, supports, or star temperature environmental factors; This method requires strict time synchronization or data synchronization in subsequent data processing, otherwise it may introduce significant errors. When using this method, it should be noted that the included angle is constant and does not change with satellite attitude changes. It is only related to star sensors, supports, or star temperature environmental factors; This method requires strict time synchronization or data synchronization in subsequent data processing, otherwise it may introduce significant errors.
Based on the data provided by the Pujiang-1 satellite, the article analyzed the in orbit measurement accuracy of the two star sensors configured using epoch difference method, sliding window method, and angle method, respectively. The following conclusions can be drawn:
(1) It is effective to use the epoch difference method to evaluate the noise equivalent angle based on the attitude measurement data of the star sensor, which is rapidly transmitted (with a period of 2 seconds);
(2) It is effective to evaluate the total error (including noise equivalent angle and field of view periodic low-frequency error, excluding channel periodic low-frequency error) based on the rapid downlink (period of 2 seconds) of star sensor attitude measurement data and the sliding window method;
(3) Based on the attitude measurement data of star sensors transmitted slowly (with a period of 16 seconds), it is effective to use the angle method to evaluate the synthesis error of the two star sensors, taking into account the time misalignment and temperature stability of the two sensors.
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