**2.2 Attitude control requirements**

96 Modern Telemetry

and mechanism of estimation of characteristic parameters of hardware as health indicators of satellite systems throughout operations. To dealing with change in value of such parameters, a complete set of analytical process of attitude determination is shown. The

The Institute of Space and Astronautical Science, Japan Exploration Agency (ISAS/JAXA) has launched a series of scientific satellites including planetary spacecraft as well as astronomical observation satellites. Although the missions have achieved fruitful scientific results, these satellites, including our own M-V launch vehicle, have cost nearly 160 million dollars and taken over eight years to develop. As a result, the launch frequency of scientific satellites has decreased significantly in the last decade. In addition to these large and expensive missions with a long development time, an inexpensive mission with a short development time involving a small piggyback satellite has been planned. This satellite should be an effective tool for demonstrating new technology and performing scientific

A small satellite named "REIMEI" was developed from 2000 and was launched to a height of 610 Km by the Dnepr Launching Vehicle from Baikonur Cosmodrome launch site in Kazakhstan on August 24, 2005 (Saito et al., 2005). Since then it has followed a near Sun synchronous orbit. The Japanese word "REIMEI" means dawn, and the satellite name was chosen to celebrate the new era of high-performance small satellites developed in Japan. REIMEI's mission objective is to observe dynamic aurora phenomena using the three spectrum imagers (MAC) and two particle analysers (E/ISA) installed onboard. Observations are carried out with the aim of studying the small-scale dynamics of terrestrial aurora, namely, their spatial distributions and time variations, and their correspondence to the spectral properties and spatial distributions of charged particles. Figure 1 shows the flight configuration of REIMEI. There are two solar-concentrated deployable paddles on the top surface that can generate a power of 150 W, three camera lens holes for MAC in the black-kapton-covered front surface, and E/ISA are installed covered by the silver-Teflon-

telemetry data used in this chapter was obtained in an actual satellite operation.

**2.1 Aurora science small satellite REIMEI** 

observations.

lined right surface.

Fig. 1. Flight configuration of REIMEI

REIMEI has a bias-momentum three-axis attitude control subsystem (ACS) to meet the requirements: the attitude control accuracy should be less than 0.5 degree, and the attitude determination accuracy should be less than 0.05 degree. These requirements are specified for one of the most important observation modes, the image/particle simultaneous observation mode, in which E/ISA detect particles to count their number as well as to measure their energy, while MAC captures aurora phenomena emerging at a magnetic line foot-point, where the aurora is energized by the particles observed.

ACS inputs data from sensors such as the spin type sun sensor (SSAS), the two-dimensional sun sensor (NSAS), the star tracker (STT), the three-axis geomagnetic field aspect sensor (GAS), and three-axis fibre-optic gyroscopes (FOGs). On the other hand, ACS outputs data to actuators such as the small momentum wheel (WHL), which provides the satellite with bias momentum (0.5 Nms), and the three-axis magnetic torquers (MTQ). Figure 2 shows the flight models of FOG of REIMEI.

ACS can be divided into two function blocks: the attitude control and the attitude determination blocks. Since several papers have been published (Sakai et al., 2006a; Fukushima et al., 2006) dealing with the REIMEI attitude control block that refer to its algorithm, formulation, and flight results, here we will focus on the attitude determination block illustrated in Fig.3. The inputs for the attitude determination block are the angular increment angle data measured by FOG and the five star vectors measured by STT. The outputs are the attitude and the attitude rate of REIMEI, which are expressed in an inertial coordinate system. Attitude is expressed using quaternions in this paper and is propagated by the integration of an initial or current attitude quaternion using the attitude rate with respect to time. The attitude rate is also called the angular velocity. Note that the attitude rate is calculated from both FOG outputs and the FOG bias. The FOG bias is residual output when FOG is motionless in the inertial coordinate system.

Fig. 2. Flight models of three FOGs assembled on one aluminium plate.

An error correction procedure must be installed into the attitude determination block since there will be a modeling error or observation error in the a priori initial attitude, the attitude

Telemetry Data Mining with SVM for Satellite Monitoring 99

If *q* is considered as reference, the state equations can be linearized around *q* . In other words, *q* can be regarded as an ideal determined attitude under the condition that there is

First, it is necessary to define estimation variables and formulate state equations expressed in terms of these variables. In this paper, the attitude estimation error vector and FOG bias estimation error vector are used for this purpose. The attitude estimation error vector

δδδδ

δ

() ( ) ( ) <sup>2</sup> *En tn vv v* τ

() ( ) ( ) <sup>2</sup> *En tn uu u* τ

characteristics, respectively. They are usually estimated through experiments or given by

Since we did not measure the precise alignment between FOG and STT, there is probably some misalignment that may cause the attitude rate of one axis to have a non zero projection onto the other axes. In addition, there are unobservable parameters of FOG in principle;

Figure 4 shows the update sequence in EKF. There are two types of update: the time update with an 8 Hz cycle and the observation update with a 1 Hz cycle. Since the state variables are updated in these cycles, their accuracy will vary during estimation. In the time update steps, the attitude is propagated by the integration of *q* using the small Euler angle

> φφφφ[ <sup>123</sup> ] . Δ

sampled during the STT update time interval.

by the propagation time interval. In the observation update steps, the estimation is

*b* from the calculated

According to Farrenkopf's paper (Farrenkopf, 1978), the following equations hold.

<sup>3</sup> represents the difference between the estimated attitude and the actual

represents small rotation angles with respect to the X, Y, and Z axes. The

( ) *<sup>v</sup>*

*u*

 σδ

 σδ *T*

 τ

 τ

*<sup>u</sup>* indicate the level of FOG bias drift and the random walk

φ

ω

θ

θ

is the incremental angle data

*FOG* is obtained by dividing

and *b* is obtained from the

is recalculated by integrating

= −*t* (7)

= −*t* (8)

*bbbb* = represents the difference between

=−− + *b n* (5)

*b n* <sup>=</sup> (6)

no noise or uncertainties when Eqn. 1 is integrated with respect to time.

the estimated bias and the actual bias with respect to the X, Y, and Z axes.

δθ ω θ

**2.4 Estimation variables** 

[ ]

θ

FOG bias estimation error vector [ ] <sup>123</sup>

where *nv* and *nu* have the following statistics characteristics.

σ

**2.5 Formulation of attitude determination** 

approximation for the three axes Δ =Δ Δ Δ

including both the bias and the misalignment uncertainty.

δθ and δ

latest observation update using Eqns. 2 and 3. After that,

thus, *b* should be regarded as the net equivalent bias vector.

attitude, while

The constants

Δφ

the manufacturers.

performed by subtracting

φ

the sum of the Δ

σ*<sup>v</sup>* and

δθ = δθ1 δθ 2 δθ

rate observed by FOG, and the star vectors obtained by STT. For example, if the FOG bias is not suitably taken into account in the attitude propagation, the attitude rate error will accumulate to a value that will corrupt the attitude propagation. In addition, the FOG bias fluctuates with respect to time and temperature of FOG. Thus, REIMEI has a Kalman filter programmed in the attitude determination block to simultaneously estimate the attitude estimation error and the FOG bias estimation error.

Fig. 3. Diagram of attitude control of REIMEI

#### **2.3 Attitude kinematics and attitude determination**

An extended Kalman filter (EKF) theory was employed to determine the attitude and attitude rate simultaneously. This filter algorithm is based on an attitude uncertainty model in rotating body coordinates that is discretized with intervals equal to the attitude update time interval. The algorithm estimates both the attitude estimation error angle theta and the attitude rate bias b. Bias b is also called FOG bias.

The satellite rotation equation and the relation between the attitude rate and the FOG bias are given by

$$
\dot{q} = \frac{1}{2} \Omega(\omega) q \tag{1}
$$

$$a \bullet = a \bullet\_{\text{FOG}} - b \tag{2}$$

*b* = 0 **,** (3)

where *q* is a quaternion vector, ω is an attitude rate vector, *b* is an rate bias vector, and ω*FOG* is an attitude rate vector measured by FOG. <sup>Ω</sup> ( ) ω is the following well-known 4x4 skew matrix

$$
\Omega(\alpha) = \begin{pmatrix}
0 & \alpha\_3 & -\alpha\_2 & \alpha\_1 \\
\alpha\_2 & -\alpha\_1 & 0 & \alpha\_3 \\
\end{pmatrix} \tag{4}
$$

If *q* is considered as reference, the state equations can be linearized around *q* . In other words, *q* can be regarded as an ideal determined attitude under the condition that there is no noise or uncertainties when Eqn. 1 is integrated with respect to time.

#### **2.4 Estimation variables**

98 Modern Telemetry

rate observed by FOG, and the star vectors obtained by STT. For example, if the FOG bias is not suitably taken into account in the attitude propagation, the attitude rate error will accumulate to a value that will corrupt the attitude propagation. In addition, the FOG bias fluctuates with respect to time and temperature of FOG. Thus, REIMEI has a Kalman filter programmed in the attitude determination block to simultaneously estimate the attitude

An extended Kalman filter (EKF) theory was employed to determine the attitude and attitude rate simultaneously. This filter algorithm is based on an attitude uncertainty model in rotating body coordinates that is discretized with intervals equal to the attitude update time interval. The algorithm estimates both the attitude estimation error angle theta and the

The satellite rotation equation and the relation between the attitude rate and the FOG bias

( ) <sup>1</sup> <sup>2</sup> *q q* = Ω ω

*FOG*

0

<sup>−</sup>

−−−

ωωω

ω

= − *b* (2)

*b* = 0 **,** (3)

is the following well-known 4x4

is an attitude rate vector, *b* is an rate bias vector, and

ω

0

 ω

3 21 3 12 21 3 123

 ωω

 ωω

0

ω ω

0

ω

<sup>−</sup> Ω = <sup>−</sup>

ωω

ω

( )

ω

*FOG* is an attitude rate vector measured by FOG. <sup>Ω</sup> ( )

(1)

(4)

estimation error and the FOG bias estimation error.

Fig. 3. Diagram of attitude control of REIMEI

**2.3 Attitude kinematics and attitude determination** 

attitude rate bias b. Bias b is also called FOG bias.

where *q* is a quaternion vector,

are given by

ω

skew matrix

First, it is necessary to define estimation variables and formulate state equations expressed in terms of these variables. In this paper, the attitude estimation error vector and FOG bias estimation error vector are used for this purpose. The attitude estimation error vector [ ] δθ = δθ1 δθ 2 δθ<sup>3</sup> represents the difference between the estimated attitude and the actual attitude, while θ represents small rotation angles with respect to the X, Y, and Z axes. The FOG bias estimation error vector [ ] <sup>123</sup> *T* δδδδ *bbbb* = represents the difference between the estimated bias and the actual bias with respect to the X, Y, and Z axes.

According to Farrenkopf's paper (Farrenkopf, 1978), the following equations hold.

$$
\delta\dot{\theta} = \omega - \dot{\theta} - \left(b + n\_v\right) \tag{5}
$$

$$
\delta \dot{b} = n\_u \tag{6}
$$

where *nv* and *nu* have the following statistics characteristics.

$$E\left[n\_v\left(t\right)n\_v\left(\tau\right)\right] = \sigma\_v^{-2}\delta\left(t-\tau\right)\tag{7}$$

$$E\left[n\_u\left(t\right)n\_u\left(\tau\right)\right] = \sigma\_u^{-2}\delta\left(t-\tau\right)\tag{8}$$

The constants σ *<sup>v</sup>* and σ *<sup>u</sup>* indicate the level of FOG bias drift and the random walk characteristics, respectively. They are usually estimated through experiments or given by the manufacturers.

Since we did not measure the precise alignment between FOG and STT, there is probably some misalignment that may cause the attitude rate of one axis to have a non zero projection onto the other axes. In addition, there are unobservable parameters of FOG in principle; thus, *b* should be regarded as the net equivalent bias vector.

#### **2.5 Formulation of attitude determination**

Figure 4 shows the update sequence in EKF. There are two types of update: the time update with an 8 Hz cycle and the observation update with a 1 Hz cycle. Since the state variables are updated in these cycles, their accuracy will vary during estimation. In the time update steps, the attitude is propagated by the integration of *q* using the small Euler angle approximation for the three axes Δ =Δ Δ Δ φφφφ [ <sup>123</sup> ] . Δφ is the incremental angle data including both the bias and the misalignment uncertainty. ω*FOG* is obtained by dividing Δφ by the propagation time interval. In the observation update steps, the estimation is performed by subtracting δθ and δ*b* from the calculated θ and *b* is obtained from the latest observation update using Eqns. 2 and 3. After that, θ is recalculated by integrating the sum of the Δφsampled during the STT update time interval.

Telemetry Data Mining with SVM for Satellite Monitoring 101

Since we apply the reset-type Kalman filter (Ninomiya et al, 1994), x is the zero vector; in

<sup>Δ</sup> <sup>Δ</sup> <sup>−</sup>

*<sup>u</sup> <sup>u</sup>*

 

*<sup>u</sup> <sup>u</sup> <sup>v</sup>*

2 2

2

<sup>Δ</sup> <sup>Δ</sup> <sup>−</sup>

2 2 2 2

3 2

*<sup>I</sup> <sup>t</sup> tI <sup>t</sup>*

2

σ

σ

*<sup>I</sup> tI <sup>t</sup>*

 

*r*

σ

( ) ( ) ( ) <sup>−</sup><sup>1</sup> *K* = *P t H H P t H* + *R <sup>T</sup>*

The observation update shown in Fig.4 is performed up to i so that the inverse of a matrix

We have now defined all the matrices required for EKF. Even if the EKF process must be turned off for some reason, the time update process should be continued using Eqns.2 and 3

The attitude determination system of REIMEI mainly depends on the STT, which has 3 arcmin accuracy. If the STT is not available, the FOGs take the role of the principal sensor to acquire the current satellite attitude by propagation. Two periods in which the STT is available and not available follow each other cyclically as illustrated in Fig. 5. In science operations, the earth enters the field of view (FOV) of the STT when the satellite attitude is controlled at a fixed value and pointing in a particular direction with respect to an inertial coordinate system. The durations of the periods when the STT is unavailable (STT-OFF period) and available (STT-ON period) are 67 min and 30 min,

This EKF has been operating as expected for more than five years (from Sep. 2005 to Mar. 2011) and no serious failures have occurred. The observed stars were scattered inside STT-FOV, in other words, they were not gathered within a small area of STT-FOV, resulting in attitude error correction being performed efficiently. Figure 6 shows a sample of EKF telemetry data including the determined quaternion vector *q* , the observed error angle

δθ

, and the attitude rate bias vector *b* . REIMEI performed Z-axis maneuvers at 1:30

data for the identified stars viewed by STT. Note

*i i*

0

 =

σ

0

*<sup>r</sup> R*

*T*

 

(14)

[ ] ( ) <sup>2</sup> <sup>2</sup> 0 *H* = *CSc* <sup>×</sup> (15)

*<sup>i</sup> <sup>i</sup>* (17)

(16)

other words, x will be reset to zero after every observation update.

 

=

where suffix i is the sequence number of the identified stars.

*G*

 

σ

 Δ +

σ

σ

2

G is the process error matrix written as follows:

H is the observation matrix written as follows:

R is the following observation noise matrix:

K is the following Kalman gain matrix:

larger than 3x3 does not appear in Eqn.17.

**2.6 Flight results of attitude determination** 

every 125 ms.

respectively.

vector

δθ

and 2:40 on Aug 5, 2006. There are four

Repeat as the number of identified stars

#### Fig. 4. EKF block diagram showing data-flow

Observations are made by obtaining a residual vector *y* expressed as

$$y = S\_{\rm obs} - \left(\delta\theta CS\_c + n\_r\right) \tag{9}$$

$$\mathbf{S} = \mathbf{S}\_{obs} + \left(\mathbf{C} \mathbf{S}\_{\mathbf{C}}\right) \delta \boldsymbol{\theta} - \mathbf{n}\_r \tag{10}$$

$$
\delta\theta = \begin{bmatrix}
0 & \delta\theta\_3 & -\delta\theta\_2 \\
\delta\theta\_2 & -\delta\theta\_1 & 0
\end{bmatrix} \tag{11}
$$

where "~" indicates the tilde operator used to form a 3x3 skew matrix from a 3x1 vector, *Sobs* is the observed star vector, *Sc* is the corresponding catalog star vector, and *C* is the direction cosine matrix of the satellite composed using *q* . The observation noise vector *nr* can be expressed as follows using the delta function:

$$\mathbb{E}[n\_r(t)n\_r(\tau)] = \sigma\_r^{-2}\delta(t-\tau) \tag{12}$$

Figure 11 shows a block diagram of EKF (Farrenkopf, 1978). The symbols G, R, and K are described as follows. x is a state vector composed from δθ andδ*b* . The state transition matrix of x can be written as

$$\mathbf{x}(t+\Delta t) = \Phi \mathbf{x}(t) = \begin{bmatrix} \Delta \phi & -\Delta t I \\ 0 & I \end{bmatrix} \begin{bmatrix} \delta \theta(t) \\ \delta \theta(t) \end{bmatrix} \tag{13}$$

σr σu σv Δt Δφ Pt

constant memory

Pt+1 = F Pt F + GGT T Pt = (I - ΣKiHi)Pt <sup>i</sup>

x = ΣKiyi i

x

output

(9)

(10)

(11)

<sup>2</sup> (12)

<sup>0</sup> (13)

*b* . The state transition

Pt+1

G Φ

R

Ki

Repeat as the number of identified stars

can be expressed as follows using the delta function:

matrix of x can be written as

described as follows. x is a state vector composed from

*x t t x t*

Observations are made by obtaining a residual vector *y* expressed as

δθ

( ) *obs c r y* =− + *S CS n* δθ

( ) *S CS n obs C r* =+ −

0

 δθ

*E*[ ] *n* () ( ) *t n* τ =σ δ ( ) *t* −τ

δθ

2 1

where "~" indicates the tilde operator used to form a 3x3 skew matrix from a 3x1 vector, *Sobs* is the observed star vector, *Sc* is the corresponding catalog star vector, and *C* is the direction cosine matrix of the satellite composed using *q* . The observation noise vector *nr*

*r r r*

Figure 11 shows a block diagram of EKF (Farrenkopf, 1978). The symbols G, R, and K are

( ) () ( )

*I tI*

Δ − Δ <sup>+</sup> <sup>Δ</sup> <sup>=</sup> <sup>Φ</sup> <sup>=</sup> *<sup>b</sup> <sup>t</sup>*

φ

 δθ

<sup>−</sup> = − <sup>−</sup>

δθ

3 1

0

δθ

3 2

 δθ

 δθ

0

δθ andδ

( ) 

δ

δθ

*t*

Fig. 4. EKF block diagram showing data-flow

Hi

q Sc

yi

Sobsi

input

Since we apply the reset-type Kalman filter (Ninomiya et al, 1994), x is the zero vector; in other words, x will be reset to zero after every observation update.

G is the process error matrix written as follows:

$$\begin{aligned} \prescript{}{G}{G} = \begin{bmatrix} \left(\sigma\_v^2 + \frac{\sigma\_u^2 \Delta t^2}{3}\right) \Delta t I & -\frac{\sigma\_u^2 \Delta t^2}{2} I \\ -\frac{\sigma\_u^2 \Delta t^2}{2} I & \sigma\_u^2 \Delta t I \end{bmatrix} \end{aligned} \tag{14}$$

H is the observation matrix written as follows:

$$H = \begin{bmatrix} \text{(CS}\_{\mathcal{L}} \text{)} & \text{0}\_{2 \times 2} \end{bmatrix} \tag{15}$$

R is the following observation noise matrix:

$$R = \begin{bmatrix} \sigma\_r & 0\\ 0 & \sigma\_r \end{bmatrix} \tag{16}$$

K is the following Kalman gain matrix:

$$\mathcal{K}\_i = \overline{P}(t) H\_i^{\,T} \left( H\_i \overline{P}(t) H\_i^{\,T} + R \right)^{\perp 1} \tag{17}$$

where suffix i is the sequence number of the identified stars.

The observation update shown in Fig.4 is performed up to i so that the inverse of a matrix larger than 3x3 does not appear in Eqn.17.

We have now defined all the matrices required for EKF. Even if the EKF process must be turned off for some reason, the time update process should be continued using Eqns.2 and 3 every 125 ms.

## **2.6 Flight results of attitude determination**

The attitude determination system of REIMEI mainly depends on the STT, which has 3 arcmin accuracy. If the STT is not available, the FOGs take the role of the principal sensor to acquire the current satellite attitude by propagation. Two periods in which the STT is available and not available follow each other cyclically as illustrated in Fig. 5. In science operations, the earth enters the field of view (FOV) of the STT when the satellite attitude is controlled at a fixed value and pointing in a particular direction with respect to an inertial coordinate system. The durations of the periods when the STT is unavailable (STT-OFF period) and available (STT-ON period) are 67 min and 30 min, respectively.

This EKF has been operating as expected for more than five years (from Sep. 2005 to Mar. 2011) and no serious failures have occurred. The observed stars were scattered inside STT-FOV, in other words, they were not gathered within a small area of STT-FOV, resulting in attitude error correction being performed efficiently. Figure 6 shows a sample of EKF telemetry data including the determined quaternion vector *q* , the observed error angle vector δθ , and the attitude rate bias vector *b* . REIMEI performed Z-axis maneuvers at 1:30 and 2:40 on Aug 5, 2006. There are four δθdata for the identified stars viewed by STT. Note

Telemetry Data Mining with SVM for Satellite Monitoring 103

STT-ON STT-ON STT-ON STT-ON

Z axis -110 [deg] rotaion observation in pointing control

01:00 01:30 02:00 02:30 03:00 03:30 04:00 04:30 05:00 05:30 06:00

01:00 01:30 02:00 02:30 03:00 03:30 04:00 04:30 05:00 05:30 06:00 <sup>0</sup>

01:00 01:30 02:00 02:30 03:00 03:30 04:00 04:30 05:00 05:30 06:00 0.494

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

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

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

Fig. 6. Attitude and bias estimation results from actual telemetry data

one actuator onboard, whereas most satellites have several sets onboard.

**UT (2006/08/05)**


q1

q2

q3

q4


> -1 0 1

0.7 0.8 0.9

0 0.05 0.1

0 0.05 0.1

0 0.05 0.1

0.05 0.1


b1

b2

b3

[deg/h]

[deg/h]

[deg/h]


0.536 0.578

the requirements.

model shown in the section 2.4.

**3. FOG Bias instability problem** 

that the δθ in Fig. 6 were calculated from y shown in Fig. 4. The observed error angle includes the effects of *nr* .

The resultant accuracy drived from obtained telemetry are attitude determination is 0.04 deg ±0.003( ) σ and accuracy of rate bias is 0.1 deg/h ±0.08( ) σ , respectively. The accuracy of *b* can be evaluated using the first several δθ data obtained shortly after beginning the STT-ON period. δθ for the first several data are equivalent to the angle given by the sum of the STT output errors, which can be modelled by a Gaussian, and the cumulative error angle of δ*b* Δ*t* . In other words, if δ*b* is sufficiently larger than the STT noise, then we can regard the first δθ as δ*b* Δ*t* . Note that the accuracy may vary depending on the operation maneuverer plan, for example, how long STT has been turned off due to observation attitude requirements, or how rapidly the attitude is changing. However, the accuracy of estimation is typical for the most frequent observation operations using the telemetry data.

Fig. 5. STT-ON and STT-OFF operation: since the attitude of the satellite is fixed in an inertial coordinate system during the observation periods, STT must be turn on and off cyclically to prevent the Earth's albedo (reflection of the sunlight) from coming inside STT-FOV. This is one of a specific limitation of the use of STT in REIMEI system.

The resultant accuracy drived from obtained telemetry are attitude determination is 0.04

STT output errors, which can be modelled by a Gaussian, and the cumulative error angle of

plan, for example, how long STT has been turned off due to observation attitude requirements, or how rapidly the attitude is changing. However, the accuracy of estimation

Earth

S

Sun syncronous orbit

Southen Sky

Fig. 5. STT-ON and STT-OFF operation: since the attitude of the satellite is fixed in an inertial coordinate system during the observation periods, STT must be turn on and off cyclically to prevent the Earth's albedo (reflection of the sunlight) from coming inside STT-FOV. This is one of a specific limitation of the use of STT in REIMEI

N

δθ

and accuracy of rate bias is 0.1 deg/h ±0.08( )

is typical for the most frequent observation operations using the telemetry data.

STT

Camera

in Fig. 6 were calculated from y shown in Fig. 4. The observed error angle

for the first several data are equivalent to the angle given by the sum of the

*b* Δ*t* . Note that the accuracy may vary depending on the operation maneuverer

σ

*b* is sufficiently larger than the STT noise, then we can regard the

STT-ON Period

S

aurora

, respectively. The accuracy of

data obtained shortly after beginning the STT-

Sunlight

nli

Sunlight

that the

deg ±0.003( )

ON period.

δ

first δθ as δ

system.

δθ

includes the effects of *nr* .

δθ

*b* Δ*t* . In other words, if

*b* can be evaluated using the first several

δ

Northen Sky

STT-OFF Period

σ

Fig. 6. Attitude and bias estimation results from actual telemetry data
