**3. FOG Bias instability problem**

The accuracy of satellite attitude estimation depends only on the FOG data during the 30 min of STT-OFF periods; thus, the bias stability of the FOGs is a crucial factor in maintaining the accuracy. With several STTs onboard, such limitation to the accuracy of attitude would not exist. Owing to their small weight and volume, small satellites have little capacity for onboard components, and it is not unusual for a small satellite to have only one sensor or one actuator onboard, whereas most satellites have several sets onboard.

Although the attitude is estimated sufficiently accurately to meet the requirements of the mission during the STT-ON periods, this is not the case during the STT-OFF periods. The cumulative attitude error caused by the bias estimation error increases to a value exceeding the requirements.

There are two sources of bias estimation error: the Kalman filter tuning performance and the bias instability. In REIMEI system, the bias is modelled using Farrenkopf's gyro dynamic model shown in the section 2.4.

Telemetry Data Mining with SVM for Satellite Monitoring 105

The angular difference between the star vectors observed using the STT and the corresponding theoretical star vectors should be monitored during STT-ON periods to verify the accuracy of

Figure 9 shows several examples of time history plots of the angular difference on the same axes. The horizontal axis is the time elapsed from the moment when the STT is switched on, while the vertical axis is the angular difference. We can observe two types of angular difference data on this plot. The data points plotted on or near the vertical axis (t=0) are the angular difference of propagation, and the data points near the center of this plot, with

Figures 10 and 11 show time history plots of these two types of angular difference obtained over the last two years (2007 and 2008). From these plots, we can conclude that (1) the STT does not appear to be developing any signs of problems and (2) the bias estimation error has increased discontinuously several times, i.e., some unknown factors have affected the

Fig. 9. Angular difference plots overlaid aligned for each start from STT-On event to the left

*<sup>u</sup>* , of the FOGs.

σ *<sup>v</sup>* andσ

the STT data. If there is a problem with the STT, this angular difference will increase.

Fig. 8. Cumulative propagation error caused by bias drift of FOGs

times of about 480 s to 1200 s, show the angular difference of the STT.

**3.2 Bias instability observed in telemetry data** 

stability of the characteristic parameters,

σ*v* 2 *T* + 1 3 σ*u* 2 *T*3

σ*n* 2

σθ

2

#### **3.1 Accuracy of bias drift estimation**

Since the parameters σ *<sup>v</sup>* and σ *<sup>u</sup>* were previously carefully tuned using a flight software simulator and actual flight telemetry data, the results of attitude estimation and FOG bias estimation for the REIMEI satellite were sufficiently good to ensure operation without any problems until about 18 months after the launch.

To maintain the estimation performance, it is necessary to continually monitor the temperature of the FOGs (Sakai et al., 2006b). Changes in temperature have a strong effect on the stability of the FOGs. Figure 7 shows an example of changes in bias observed in the telemetry data (Fukushima et al., 2006). Although these FOGs were well calibrated and their thermal environment has been controlled by heaters, some minor problems have arisen in bias estimation since Nov. 2007.

Fig. 7. FOG bias drift caused by change in FOG temperature

The accuracy of the estimated bias can be evaluated using the attitude error. This error can be observed at the moment when the STT is switched on. The attitude error is defined as the angular difference between the observed star vectors and the corresponding theoretical star vectors calculated using the onboard star catalog and the satellite attitude. This angular difference of propagation can be regarded as being equal to the cumulative error caused by the bias estimation error during the STT-OFF periods. Figure 8 is a schematic drawing of the cumulative propagation error, with its standard deviation σθ formulated usingσ *<sup>v</sup>* , σ *<sup>u</sup>* , and the deviation of the STT observation dataσ*<sup>n</sup>* .

$$
\sigma\_{\theta} \,^2 = \sigma\_n \,^2 + \sigma\_v \,^2 T + \frac{1}{3} \sigma\_u \,^2 T^2 \tag{18}
$$

simulator and actual flight telemetry data, the results of attitude estimation and FOG bias estimation for the REIMEI satellite were sufficiently good to ensure operation without any

To maintain the estimation performance, it is necessary to continually monitor the temperature of the FOGs (Sakai et al., 2006b). Changes in temperature have a strong effect on the stability of the FOGs. Figure 7 shows an example of changes in bias observed in the telemetry data (Fukushima et al., 2006). Although these FOGs were well calibrated and their thermal environment has been controlled by heaters, some minor problems have arisen in

FOG T=50

07/02 07/09 07/16 07/23

07/02 07/09 07/16 07/23

07/02 07/09 07/16 07/23

Δ = +3

σθ Δ = -1.7

formulated using

*<sup>u</sup> T* (18)

σ *<sup>v</sup>* , σ*<sup>u</sup>* ,

Δ = +0.5

T=55 T=53

**UT (2006/07/01-07/24)**

The accuracy of the estimated bias can be evaluated using the attitude error. This error can be observed at the moment when the STT is switched on. The attitude error is defined as the angular difference between the observed star vectors and the corresponding theoretical star vectors calculated using the onboard star catalog and the satellite attitude. This angular difference of propagation can be regarded as being equal to the cumulative error caused by the bias estimation error during the STT-OFF periods. Figure 8 is a schematic drawing of the

> σ*<sup>n</sup>* .

2 2 2 2 2

3 1

σ

*<sup>u</sup>* were previously carefully tuned using a flight software

Δ = -0.5

**3.1 Accuracy of bias drift estimation** 

bias estimation since Nov. 2007.



> -2.06 -1.03 0 1.03

Fig. 7. FOG bias drift caused by change in FOG temperature

cumulative propagation error, with its standard deviation

σθ =σ *<sup>n</sup>* +σ*<sup>v</sup> T* +

and the deviation of the STT observation data

**b3 [deg/h]**


**b1 [deg/h]**


σ*<sup>v</sup>* and

problems until about 18 months after the launch.

σ

Since the parameters

#### **3.2 Bias instability observed in telemetry data**

The angular difference between the star vectors observed using the STT and the corresponding theoretical star vectors should be monitored during STT-ON periods to verify the accuracy of the STT data. If there is a problem with the STT, this angular difference will increase.

Figure 9 shows several examples of time history plots of the angular difference on the same axes. The horizontal axis is the time elapsed from the moment when the STT is switched on, while the vertical axis is the angular difference. We can observe two types of angular difference data on this plot. The data points plotted on or near the vertical axis (t=0) are the angular difference of propagation, and the data points near the center of this plot, with times of about 480 s to 1200 s, show the angular difference of the STT.

Figures 10 and 11 show time history plots of these two types of angular difference obtained over the last two years (2007 and 2008). From these plots, we can conclude that (1) the STT does not appear to be developing any signs of problems and (2) the bias estimation error has increased discontinuously several times, i.e., some unknown factors have affected the stability of the characteristic parameters, σ *<sup>v</sup>* andσ*<sup>u</sup>* , of the FOGs.

Fig. 9. Angular difference plots overlaid aligned for each start from STT-On event to the left

Telemetry Data Mining with SVM for Satellite Monitoring 107

We can see that the angular difference of the STT appeared to be larger during some periods in Fig. 11. This difference can be explained using the dilution of precision (DOP) of stars. The DOP of stars is defined in this paper as an index of how stars are aligned in the FOV of

*DOP* = *trace H HT*

where H is the observation matrix used in the Kalman filter of this system. The larger, the value of DOP, the less accurate the attitude estimation, similarly to the DOP used in the accuracy analysis of the Global Positioning System. The DOP indicates how stars are seen in the FOV of the STT, which affects the accuracy of attitude determination. The degradation of accuracy of attitude estimation over time is inevitable for such a system

It has been needed to find any weak sign of event that would link to some system troubles. Such demand called failure detection, a sub category of system monitoring, has been researched for ground systems as well as onboard systems in many fields. Having experience of working for years as an operator of REIMEI, the author also has felt such for failure detection only for system failure but also for detection of degrade of performance: signs of changes in specification that there is no error but something seems to be wrong. In this section, an example of such failure detection is shown using a problem of degrade of

Figure 12 shows time history plots of the angular difference of propagation and the angular difference of the STT over 2 years, which are plotted over different time spans such as one

From the statistics for these time spans, it is easy to observe the level of fitness of the EKF parameters. By taking statistics over a longer time span, randomness can be reduced. For example, there is no sign of parameter mismatch before Oct. 2006, but an event occurred in Oct. 2006; the difference between the median and mode of the distribution of the angular difference of propagation has been increasing since Nov. 2006. The median indicates the accuracy of bias estimation, while the mode indicates the random-walk error

According to these time history plots over 2 years, it can be concluded that the EKF parameters have been mismatched with the actual state of the FOGs since Oct. 2006. This means that the characteristics of the FOG bias changed at that time. The most probable reason for the bias error is that the degradation of fibre transparency due to radiation became sufficiently large to cause bias instability. However, this degradation is still considered to have a limited effect according to the results of radiation tests performed

From the operator's viewpoint, one month is a too long period to take any measures to prevent the degradation of accuracy of attitude determination. Any parameter mismatch

before the launch; there must be another factor causing this degradation.

must be noticed as soon as possible rather from the analysis of monthly statistics.

( )−<sup>1</sup>

*stars* (19)

the STT; its formula is given as follows:

**4. Detection of changes in FOG bias using SVM** 

with only one STT.

FOG bias drifting in REMEI.

**4.1 Signs of changes in bias** 

month, ten days, and one day.

of bias drift.

Fig. 10. Trend of angular difference of propagation

Fig. 11. Trend of angular difference of STT

Fig. 10. Trend of angular difference of propagation

Fig. 11. Trend of angular difference of STT

We can see that the angular difference of the STT appeared to be larger during some periods in Fig. 11. This difference can be explained using the dilution of precision (DOP) of stars. The DOP of stars is defined in this paper as an index of how stars are aligned in the FOV of the STT; its formula is given as follows:

$$DOP\_{\text{stars}} = \sqrt{\text{trace}\left(\boldsymbol{H}^T \boldsymbol{H}\right)^{-1}} \tag{19}$$

where H is the observation matrix used in the Kalman filter of this system. The larger, the value of DOP, the less accurate the attitude estimation, similarly to the DOP used in the accuracy analysis of the Global Positioning System. The DOP indicates how stars are seen in the FOV of the STT, which affects the accuracy of attitude determination. The degradation of accuracy of attitude estimation over time is inevitable for such a system with only one STT.
