1.1

Why minimizing the low frequency error

In order to obtain a good image localization for Earth observation satellites, an accurate onboard attitude knowledge is necessary, usually given by star trackers. Among the star tracker error classes, the bias can be inflight calibrated, and the noise is usually filtered with gyroscopes data by the AOCS (Attitude and Orbit control System). So, the most penalizing error class for the attitude control of the satellite is the low frequency error. That’s why in the frame of the Pleiades program, an effort was brought to reduce that error class.

1.1

Presentation of SED36 star tracker

The SED36 star tracker was developed by SODERN for Pleiades, the new Earth observation satellite developed by the CNES. It is based on the SED26 (see [1]), which is a space qualified ITAR free sensor (International Traffic in Arms Regulation); the same objective, detector, electronic and software are used. The SED36 has been separated into three parts in order to minimize thermal coupling effects: the optical head, which is mounted directly on the instrument, the electronic box and the baffle. The main features of this star tracker are indicated in Table 1.

Fig. 1.

SED36 star tracker: the optical head (up – left), the electronic box (up – right), and the baffle (down)

Table 1:

Over view of the acquired ASAR data in Huazhaizi site

2.1

An optimized thermo-mechanical design

The first step in order to reduce the attitude low frequency error is to minimize the thermal effects. Indeed, the thermal drift of the optical axis has the same value and direction for all the tracked stars, so it is not averaged among the number of stars. Moreover, approximately one half of this error is due to the contribution of thermal dissipation of the baffle, when it is exposed to sun light. So, SED36 has been separated into three parts, in order to minimize thermal coupling effects. The optical head is directly mounted on the Pleiades instrument in order to achieve the best accuracy and avoid additional frame transformations, whereas the baffle is mounted on the satellite structure.

The other half of the thermal drift error is due to thermal conductive coupling at the optical head mounting plane. This is why the electronic box has been separated from the optical head, to minimize power dissipation at the optical head interface. Moreover, one of the major constraints to achieve the required accuracy is to have a very stable thermal regulation at the satellite interface, about 20°C ± 3°C. Taking into account this thermal stability and the nature of the Pleiades instrument interface (SiC), a judicious choice of materials and of geometrical configuration was performed in order to minimize thermal differential dilatations. The aluminium main structure of SED16/26 was replaced by a titanium/AlSiC combination, very well matching with the main instrument SiC interface, while keeping a very rigid and stable mounting.

Then, software improvements were developed, in order to go one step further in minimizing the low frequency error.

2.2

A more accurate calibration of optical distortion

The second main error item of Table 2 is the residual optical distortion error. On SED16/26 star trackers, a third order polynomial law is used, which gives residual errors after correction of about ±9’’ at 3σ. This law is calibrated on the BCG16 (Banc de Calibration Géométrique) test bench available in SODERN, with simulated stars of equivalent colour temperature of 6000K, and with a mounting plane temperature of +20°C. It has been decided to perform a dedicated study in the frame of a CNES contract, in order to develop a more accurate distortion calibration method.

The first step was to evaluate the test bench accuracy, and the sensibility of the existing correction to the spectral types of the stars, and to a mounting plane temperature variation.

The repeatability of the distortion calibration on BCG16 test bench is measured to about ±1” at 3σ, and the absolute angular precision of the rotation table to ±1” 3σ. Consequently, the calibration limit of the bench over duration of the test is ±2”; this is the precision limit of any distortion calibration method that can be implemented. Nevertheless, it is possible to average a deterministic periodic part of the error of the rotation table, by positioning the calibration stars in judicious positions in the field of view.

Moreover, taking into account the spectral distribution of stars of the Tracking mode catalogue, the effect of chromaticity on the distortion residue appears to be negligible. Indeed, in almost the whole celestial vault, most tracked stars have a colour temperature close to 6000K (simulated colour temperature used for law calibration), and for other stars the influence of chromaticity is averaged on the attitude.

Lastly, when the temperature of the mounting plane varies about ±3°C compared to the calibration temperature of 20°C, a potential degradation of ±0.5” 3σ should be added to the obtained performances.

Then several distortion calibration laws have been compared, in order to evaluate the ratio between the performance that can be reached and the corresponding constraints. The calibration laws studied are:

The comparison of these calibration laws has been performed thanks to measurements done on the BCG16 test bench on 4 SED16 star trackers, with about 1000 points in the field of view (in order to have an accurate estimation). A specific algorithm has been implemented under Matlab for that purpose, in order to calculate the laws and to evaluate the corresponding residues.

The results are the following: high order X/Y polynomials have almost the same efficiency as Zernike polynomials. When the order of the polynomial increases, the value of the residues decrease, with however a limit: indeed, after the 10th order, the calibration law becomes very close to calibration points, but can have high frequency variations between these points, making the correction less efficient.

With the correction using several small areas in the field of view, the correction is more efficient. When the number of areas increases, the value of the residues decrease, and the only limitation is the test bench accuracy.

Fig. 2 compares the 3σ values of the distortion residues for the different types of corrections, in function of the number of needed parameters. The values shown are the averages of the performances obtained among the 4 SED16 star trackers. The red curve refers to the XY polynomials, the blue one to the small areas method, and the black one to Zernike polynomials.

Fig. 2.

Distortion residues in function of the number of parameters.

Since the calibration using small areas in the field of view is the most efficient, its implementation constraints have been evaluated, in order to assess the maximum allowable number of areas (and so the minimum performance achievable):

But when the number of areas increases such as the limit of the test bench accuracy is reached, the calibration measurement lasts less than ten hours, so it is compatible of the maximum duration constraint.

In synthesis, the calibration method which presents the best performances is the method using small areas in the field of view of the star tracker, and its implementation constraints are compatible with SED36 design.

The method using small areas has then been implemented in the star tracker software. The correction process is presented in Fig. 3. In order to have an efficient correction, a third order polynomial law still needs to be applied before the correction by areas; it is then a two step correction. Moreover, in order to avoid a step on the attitude restitution when a tracked star shifts from one calibration area to an adjacent one, an interpolation law is used between the adjacent areas.

Fig. 3.

Flight software distortion correction process.

The test bench software has also been modified in order to perform an automatic calibration measurement process for this specific correction. The calibration measurements are performed on BCG16 test bench, measuring several points per calibration area. The following precautions are taken into account during the process:

After implementing the calibration method using small areas, it has been validated on the EM (Engineering Model) SED36 star tracker, on the BCG16 test bench.

Fig. 4 presents the distortion residues in the field of view using a representation with small arrows, and Fig. 5 presents the error histograms measured along X or Y axes.

Fig. 4.

Distortion residues after correction on EM SED36

Fig. 5.

Histograms of residual errors along X axis (up) and Y axis (down)

The single-star distortion residue measured during this validation is ±2” at 3σ along X or Y axis. This allows reaching a global calibration precision of ±3” at 3σ for the single star distortion error (see Table 2), when taking also into account the influence of the variation of mounting plane temperature, or the test bench accuracy.

2.3

An update of the star catalogue

After having optimized both the contribution of thermal errors and the distortion residues, another low frequency error contributor which can be easily reduced is the star catalogue error. This error is mainly due to the star proper motion, the position measurement error of the reference astronomical catalogue being negligible. The catalogue error equals ±3” at 3σ for a duration of 15 years, the star positions in the catalogue being given at the 01/01/2000.

As the Pleiades mid-mission is scheduled for year 2013, the in-flight SED36 tracking catalogue was updated by propagating stars positions up to 01/01/2013. The same stars were kept, only their positions were corrected with the proper motion information available in the reference astronomical catalogue. Simple validations of the modification have been performed, the original mission catalogue being considered as in-flight validated on board SED16 and SED26. After this update, the star catalogue error is reduced to ±1" at 3σ for the duration of the Pleiades mission.

2.4

An increase of the number of tracked stars

Finally, in order to better average the field of view errors, the number of tracked stars is also increased to 12 instead of 10 nominally on SED16/26.

It has been checked that the SED36 star tracker could still operate with 12 measured stars at the required 8Hz frequency.

It has been also verified that in more than 99% of the celestial vault, there are more than 12 stars in the field of view of the star tracker that are referenced in the star catalogue.

Finally, this improvement allows an additional reduction of the field of view error on the attitude of about 10% (with the hypothesis of a uniform star weighting).

2.5

Attitude performance achieved

Taking into account all the above described improvements, an assessment of performances has been performed with the single-star errors presented in Table 2, in order to evaluate the low frequency error on the attitude of the SED36 star tracker.

The result is a low frequency error of:

It is recalled that this type of improvement can only be achieved in case of stable thermal regulation (±3°C) at the satellite interface of the optical head.