attitude determination method in eclipse for qsat (version 2)
TRANSCRIPT
19 September 2007
1
Attitude Determination Method in Eclipse for QSAT
(Version 2)
Tomohiro Narumi
Space Systems Dynamics Laboratory, Kyushu University
744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
19 September 2007
2
Contents
1. Introduction……………………………………………………………………….4
2. Coordinate Systems………………………………………………………………5
2.1 Geocentric Equatorial Coordinate System, IJK………………………………..5
2.2 Orbit Coordinate System……………………………………………………….5
2.3 Body Frame…………………………………………………………………….6
2.4 Relation of Quaternions and Tait-Bryan Angles……………………………….7
3. Attitude Determination Method ……….………………………………………..8
3.1 Concept…………………………………………………………………………8
3.2 Construction of the Kalman Filter……………………………………………..8
3.3 Sequence of the Estimation…………………………………………………...12
4. Simulation…………..……………………………………………………………13
4.1 Simulator………………………………………………………………………13
4.2 Results…………………………………………………………………………14
5. Discussion, Conclusion and Future Works…………………………………….18
5.1 Discussion……………………………………………………………………..18
5.2 Conclusion…………………………………………………………………….18
5.3 Future Works………………………………………………………………….18
Appendix A. Calculus of Quaternions…………………………………………..19
A.1 Definition…………………………………………………………………...…19
A.2 Notation………………………………………………………………………..20
A.3 Direction Cosine Matrix………………………………………………………..20
A.4 Multiplication…………………………………………………………………..20
A.5 Coordinate transformation matrix using quaternion……………………………22
Appendix B. Kinematic Equation of Quaternion………………………………22
B.1 General Equation……………………………………………………………….22
B.2 Kinematic Equation for Multiplication of Quaternion………………………….23
B.3 Kinematic Equation for Small Quaternion…………………………………..…24
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Appendix C. Observability………………………………………………………26
C.1 Linear System…………………………………………………………………..26
C.2 Nonlinear System……………………………………………………………….26
Appendix D. Discretization of the System………………………………………28
D.1 Discrete-Time Case……………………………………………………………..29
D.2 Discretization of Covariance Matrices………………………………………….30
Appendix E. Geomagnetic Field………………………………………………...33
E.1 Geomagnetic Flux………………………………………………………………33
E.2 Recursive Calculation of Legendre Function…………………………………..34
Appendix F. External Torques and Noise Generation…………………………36
F.1 Gravity Gradient Torques……………………………………………………….36
F.2 Solar Radiation Torques………………………………………………………...37
F.3 Atmospheric Drag Torques……………………………………………………..37
F.4 Magnetic Torques……………………………………………………………….37
F.5 Sensor Outputs………………………………………………………………….38
Bibliography………………………………………………….………………………41
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1. Introduction
In northern Kyushu-island in Japan, a small satellite for observing polar plasma environment and
geomagnetic field named QSAT has been being developed. To achieve the mission, the satellite has
to control the attitude, and the requirement of the error is about 5 degrees. QSAT has three types of
sensors, two 3-axis Sun sensors, 3-axis magnetometer, and 3-axis rate gyros. In sunlight, the
satellite can determine the attitude using three kinds of sensors. However the Sun sensors do not
work when the satellite is in eclipse naturally. Accordingly, attitude determination algorithms using
only magnetometer and rate gyros have to be developed. To achieve it, the Extend Kalman Filter
which can estimate the state variables including attitude is introduced. In order to circumvent
problems of singular matrix, some techniques are adopted. In this paper, construction and
performance of the EKF and the some results are described.
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2. Coordinate Systems
We mainly use three frames, inertial frame (geocentric equatorial coordinate system), orbit frame
(orbit coordinate system), and body frame. One axis of orbit frame is oriented to radial coordinate,
and another axis points in the direction of the velocity vector and is perpendicular to the radius
vector. Quaternion (Euler-parameters) and Tait-Bryan angles (Euler angles) are defined as
transformation of axes from orbit frame to body frame.
2.1 Geocentric Equatorial Coordinate System, IJK
This system has its origin at the center of Earth. The fundamental plane is the Earth’s equator, as
shown in Fig. 2.1. The I axis points towards the vernal equinox; the K axis extends through the north
pole; the J axis is defined as IKJ ˆˆˆ ×= .
2.2 Orbit Coordinate System
The R axis always points from the Earth’s center along the radius vector toward the spaced object
as it moves in its orbit. The W axis is normal to the orbital plane. The S axis is defined as the cross
product between the W axis and R axis. The S axis is usually not aligned the direction of the velocity
except for the perigee and apogee. Fig. 2.2 shows the geometry of this coordinate system.
J
I
Earth
Equatorial Plane
K
Fig. 2.1 Geocentric Equatorial Coordinate System, IJK
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2.3 Body Frame
The frame is fixed to the satellite body as shown in the Fig. 2.3.
x axis: The normal direction to the plane which the Sun sensor unit I is attached.
y axis: The normal direction to the plane which the Sun sensor unit II is attached.
z axis: The negative direction to the direction of the boom extending.
Fig. 2.2 Orbit Coordinate System, RSW
Ω i
Satellite
Vernal
Equinox
I
AN: Ascending Node
R S
W
J
K
u
ˆ x
ˆ y
ˆ z
Fig. 2.3 Definition of Body frame of QSAT
Sun sensor unit I
Sun sensor unit II
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2.4 Relation of Quaternions and Tait-Bryan Angles
The direction cosine matrix (DCM) of quaternions is [1]
2 2
1 4 1 2 3 4 1 3 2 4
2 2
1 2 3 4 2 4 2 3 1 4
2 2
1 3 2 4 2 3 1 4 3 4
2( ) 1 2( ) 2( )
2( ) 2( ) 1 2( )
2( ) 2( ) 2( ) 1
q q q q q q q q q q
q q q q q q q q q q
q q q q q q q q q q
+ − + −
− + − + + − + −
(2.1)
and DCM of 3-2-1 Tait-Bryan angles (Euler angles) [φ, θ, ψ] (yaw, roll, pitch) is
cos cos cos sin sin
sin sin cos cos sin sin sin sin cos cos sin cos
cos sin cos sin sin cos sin sin sin cos cos cos
θ ψ θ ψ θ
φ θ ψ φ ψ φ θ ψ φ ψ φ θ
φ θ ψ φ ψ φ θ ψ φ ψ φ θ
− − + + −
(2.2)
NOTE! : The Tait-Bryan angles rotation [φ, θ, ψφ, θ, ψφ, θ, ψφ, θ, ψ] is defined by successive rotations by angles
φ, θ,φ, θ,φ, θ,φ, θ, and ψ,ψ,ψ,ψ, respectively, about coordinate axes x, y, z. because 3-2-1 rotation.
Comparing the DCMs,
( )( )
2 2
2 3 1 4 3 4
1 3 2 4
2 2
1 2 3 4 1 4
arctan 2 2( ),2( ) 1
arcsin 2( )
arctan 2(2( ),2( ) 1)
q q q q q q
q q q q
q q q q q q
φ
θ
ψ
= + + −
= − −
= + + −
(2.3)
where, arctan2(y,x) is the function that calculates the arctangent of the two variables x and y in C or
FORTRAN languages. It is similar to calculating the arc tangent of y/x, except that the signs of both
arguments are used to determine the quadrant of the result.
And,
1
2
3
4
sin cos cos cos sin sin2 2 2 2 2 2
cos sin cos sin cos sin2 2 2 2 2 2
cos cos sin sin2 2 2 2
q
q
q
q
φ θ ψ φ θ ψ
φ θ ψ φ θ ψ
φ θ ψ φ
−
+
=
−
sin cos2 2
cos cos cos sin sin sin2 2 2 2 2 2
θ ψ
φ θ ψ φ θ ψ
+
(2.4)
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3. Attitude Determination Method
3.1 Construction of the State Model
QSAT has three types of sensors, two 3-axis Sun sensors, 3-axis magnetometer, and 3-axis rate
gyros. However the Sun sensors do not work when the satellite is in eclipse. Accordingly attitude
determination algorithms using only magnetometer and rate gyros have to be developed.
The quaternions which express the attitude of the satellite are defined as the coordinate
transformation from the reference orbital frame to the body frame. The innovation of the quaternion
is calculated as follows
2 2
1 4 1 2 3 4 1 3 2 4
2 2
1 2 3 4 2 4 2 3 1 4
2 2
1 3 2 4 2 3 1 4 3 4
( ) ( )
2( ) 1 2( ) 2( )
2( ) 2( ) 1 2( )
2( ) 2( ) 2( ) 1
m m o
m o
x x
m o
y y
m o
z z
m o
x x
m o
y y
m o
z z
m m
m mq q q q q q q q q q
m mq q q q q q q q q q
s sq q q q q q q q q q
s s
s s
= − = −
+ − + −
= − − + − + + − + −
y y h q y A q y
(3.1)
where ym is the measured sun vector [sx
m, sy
m, sz
m]
T and magnetic field vector [mx
m, my
m, mz
m]
T, y
o is
the reference sun vector [sxo, sy
o, sz
o]
T and magnetic field vector [mx
o, my
o, mz
o]
T, A is the direction
cosine matrix from the orbit frame to the body frame, and q is the estimated quaternion from the
reference orbit frame to the body frame. This innovation is used as the measurement with the outputs
of the rate gyros in the measurement equation.
ref
∆ = −ω g ω (3.2)
where g is the outputs from the rate gyros, ωωωωref is the angular rates of the orbit frame relative to the
inertial frame (see Appendix B).
3.2 Construction of the State Model
The state variable is preferred to choose the quaternion because of its simple expression of the
state equation and the direction cosine matrix which do not use any transcendental functions
compared to the expression of the Euler angles. The state equation of the Kalman filtering consists
of kinematic equation and dynamic equation, and the accuracy of the attitude estimation depends
on the precision of the state model. However it is very difficult because there are many
unpredictable disturbances. For example, environmental external forces, internal forces, flexible
structure, or mobile objects. Fortunately QSAT has the 3-axis rate gyros. Therefore the satellite can
obtain its motion without consideration of any disturbance. Substituting the kinematic equation
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using the outputs of the rate gyros for dynamic equation, it becomes the independent state equation
of the equation of motion. However the rate gyros have biases because of temperature and so on.
Therefore it is estimated simultaneously as the state variables.
If you use the quaternion, there is a problem that its norm is one and they are not independent.
Hence the error covariance matrix of the quaternion has singularity. Lefferts et al.[2]
suggest three
algorithms to circumvent the singularity. First method is to degenerate the error covariance matrix
from 7x7 to 6x6 when you propagate it. Second method is to delete one of quaternions. And third
method is to estimate small quaternions because one of the small quaternions is almost 1. The third
method is the most suitable for QSAT because it is earth-oriented satellite.
Therefore the state variables are small values of quaternion and biases of the rate gyros. The state
equation becomes (Appendix B)
v∆
= ∆
qx
b (3.3)
= +x Fx Gvɺ (3.4)
1k t t t+ = +x Φ x Γv (3.5)
2
10 0 0
2
10 0 0
2
10 0 0
2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
z y
z x
x
ω ω
ω ω
ω ω
−
= −
F (3.6)
3 3
3 3
1
2
− =
I 0G
0 I
(3.7)
1exp[ ]t tt −= ∆Φ F Φ (3.8)
0exp[ ( )]
t
t t dτ τ = − ∫Γ F G (3.9)
where F is the system matrix, u is the input vector, v is the system noise in continuous system. ΦΦΦΦt, ut,
and vt are the discretized matrix or vectors. ∆qv is small quaternion vector except for q4 (defined in
Appendix B), and ∆b is the bias vector of the rate gyros.
The measurement matrix H is introduced as follows because the measurement equation is not
linear. The extended Kalman Filter (EKF) has to be adopted.
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2 2 3 1 1 4 1 4
2 4 1 2 3 2 4
3 4 3 4 1 2 3
2 2 3 1 1 4
( )
2 2 4 2 2 2 2 0 0 0
2 2 2 4 2 2 2 0 0 0
2 2 2 2 2 2 4 0 0 0
2 2 4 2 2 2
t
o o o o o o o
y x y z z y
o o o o o o o
x z x y z z x
o o o o o o o
x y y x x y z
o o o o o
y x y z z
m q m q m q m q m q m q m q
m q m q m q m q m q m q m q
m q m q m q m q m q m q m q
s q s q s q s q s q s
∂ = ∂
+ + − +
+ + + −
− + + +=
+ + −
h qH
x
1 4
2 4 1 2 3 2 4
3 4 3 4 1 2 3
2 0 0 0
2 2 2 4 2 2 2 0 0 0
2 2 2 2 2 2 4 0 0 0
o o
y
o o o o o o o
x z x y z z x
o o o o o o o
x y y x x y z
q s q
s q s q s q s q s q s q s q
s q s q s q s q s q s q s q
+ + + + −
− + + +
(3.10)
( )1
T
t t t t t t t
−
= +K P H H P H R (3.11)
ˆt t t t t= −P P K H P (3.12)
,
ˆ
ˆ
v t
t t
t
∆ =
∆
qK y
b (3.13)
,
,
ˆˆ
ˆ1
v t
t t
v t
∆ =
− ∆
qq q
q (3.14)
ˆ ˆt t t
= + ∆b b b (3.15)
ˆˆt t t
= −ω g b (3.16)
where the superscript (^) means the estimate, the superscript (
-) means the a-priori value, K is the
Kalman gain, Qt , Pt and Rt are covariance matrices defined in Appendix D.
Update equations become as follows.
1ˆ
t t t t t+ = +P Φ PΦ Q (3.17)
1ˆ
t t t+ =x Φ x (3.18)
and according to Appendix D, Qt becomes
2 3
0
TT T
t t t c t t t t td t t tτ ′ ′′ ′′′= = ∆ + ∆ + ∆∫Q Φ B Q B Φ Q Q Q (3.19)
where,
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1
2
3
4
5
6
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
c
n
n
n
n
n
n
=
Q (3.20)
1
2
3
4
5
6
0 0 0 0 04
0 0 0 0 04
0 0 0 0 04
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
t
n
n
n
n
n
n
′ =
Q (3.21)
4
2 1 3 1 3 2
5
2 1 3 3 2 1
6
3 1 2 3 2 3
4
5
6
1 10 ( ) ( ) 0 0
8 8 4
1 1( ) 0 ( ) 0 0
8 8 4
1 1( ) ( ) 0 0 0
8 8 4
0 0 0 0 04
0 0 0 0 04
0 0 0 0 04
t
nn n n n
nn n n n
nn n n n
n
n
n
ω ω
ω ω
ω ω
− −
− − − −
′′ =
Q
(3.22)
2 2
4 3 2 3 2
2 2
3 5 3 1 1
2 2
2 1 6 2 1
1 1 1( ) 0 0 0
12 12 12
1 1 1( ) 0 0 0
12 12 12
1 1 1( ) 0 0 0
12 12 12
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
y z x y x z
x y x z y z
x z y z x y
n n n n n
n n n n n
n n n n n
ω ω ω ω ω ω
ω ω ω ω ω ω
ω ω ω ω ω ω
+ + − −
− + + −
′′′ = − − + +
Q
(3.23)
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3.3 Sequence of the Estimation
Measurement
( )m= −y y h q
( )1
T
t t t t t t t
−
= +K P H H P H R
ˆt t t t t= −P P K H P
,ˆ
ˆ
v t
t t
t
∆ =
∆
qK y
b
,
,
ˆˆ
ˆ1
v t
t t
v t
∆ =
− ∆
qq q
q
ˆ ˆt t t= + ∆b b b
ˆˆt t t= −ω g b
Update
1ˆ
t t t t t+ = +P Φ PΦ Q
1ˆ
t t t+ =x Φ x
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4. Simulation
4.1 Simulator
The attitude simulator for QSAT is build using Microsoft Visual C++. The simulator can calculate
orbit, attitude, perturbing torques, effect of actuator, power consumption, and can also product sensor
outputs. The reason why C++ was chosen is that the computation time is short and it is easy to build
encapsulated components using object oriented programming.
Lining up the kinematic and dynamic equations, the equation of motion is as follows,
2 3 41
1 3 42
1 2 43
1 2 34
( ) / 2
( ) / 2
( ) / 2
( ) / 2
( ( ) ) /
( ( ) ) /
( ( ) ) /
z y x
z x y
y x z
x y z
x y z y z xx
y z x z x yy
z x y x y zz
q q qq
q q qq
q q qqd
q q qqdt
I I I
I I I
I I I
ω ω ω
ω ω ω
ω ω ω
ω ω ω
τ ω ωω
τ ω ωω
τ ω ωω
− + − + + − +
− − −= + −
+ − + −
(4.1)
where, ωx, ωy, and ωz are angular rate about the respective axes in body frame. Ix, Iy, and Iz are
moment of inertia. τx, τy, and τz are the sum of torques by the actuators (magnetic torquers), gravity
gradient, solar radiation pressure, atmospheric drag.
The specification of the simulator is as follows.
Table 5.1 Specification of the Simulator
Object Method
Numerical Integrator
Orbit Propagator
Sun vector
Geomagnetic Field
Atmospheric Density
Random Number Generator
Dormand-Prince, 8th
-order
SGP8
VSOP87
IGRF Model, 10th
-order
NRLMSISE-00
L’Ecuyer & Box-Muller Method
SGP: Simplified General Perturbations
VSOP: Variations Seculaires des Orbites Planetaires
IGRF: International Geomagnetic Reference Field
MSISE: Mass Spectrometer, Incoherent Scatter Radar Extended Model
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4.2 Results
I investigated above-mentioned method under following condition (Table 5.1). Kalman gain is
always calculated per sampling rate. In eclipse, the outputs of sun sensors are set to zero because
they do not work.
Table 5.1 Test Case
Orbit Ideal Circular Orbit
Attitude Perturbation Gravity Gradient Only
Initial Euler Angles (Yaw, Roll, Pitch) [deg] (3.0, 5.0, 10.0) in Orbit Frame
Initial Angular Rates (ωx, ωy, ωz) [deg/sec] (0.0, 0.0, 0.0)
Moment of Inertia (Ix, Iy, Iz) [kgm2] (0.6205, 1.178, 1.178)
Sensor Noise Nothing
Orbit Determination Error Nothing
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0 20 40 60 80 100-10
-5
0
5
10
15
Time [min]
Pitch
[deg]
Attitude Estimation (EKF)
True
Estimation
Figure 4.1 Estimated Pitch Angle
0 20 40 60 80 100-2
-1
0
1
2
3
4
5
6
Time [min]
Pitch
[deg]
Attitude Estimation (EKF)
Error
Figure 4.2 Error of Pitch Angle
Eclipse
Eclipse
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0 20 40 60 80 100
-5
0
5
10
15
20
Time [min]
Roll [deg]
Attitude Estimation (EKF)
True
Estimation
Figure 4.3 Estimated Roll Angle
0 20 40 60 80 100-5
0
5
10
15
20
Time [min]
Roll [deg]
Attitude Estimation (EKF)
Error
Figure 4.4 Error of Roll Angle
Eclipse
Eclipse
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0 20 40 60 80 100-2
-1
0
1
2
3
4
5
6
Time [min]
Yaw [deg]
Attitude Estimation (EKF)
True
Estimation
Figure 4.5 Estimated Yaw Angle
0 20 40 60 80 100-1
-0.5
0
0.5
1
1.5
2
2.5
Time [min]
Yaw [deg]
Attitude Estimation (EKF)
Error
Figure 4.6 Error of Yaw Angle
Eclipse
Eclipse
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5. Discussion, Conclusion and Future Works
5.1 Discussion
Fig. 4.1-4.6 show the result of the attitude determination method. The errors of pitch angle and
roll angle are small, but the error of yaw angle becomes about 2 degrees although the noise and
position error are 0. If filter tuning (like noise covariance) is more appropriate, the performance of
the Kalman filter will be probably better. Nevertheless, the error angles are smaller than the
requirement from the mission instruments (5 deg.).
5.2 Conclusion
The attitude determination algorithm which can be used in eclipse was constructed and
investigated, and this method which estimates the small quaternion seems to be good for
determining the attitude angles.
5.3 Future Works
Adding noises, bias and quantization error of the sensors.
Adding position error including the effect of delay time.
Speeding up the algorithms in C++ language.
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Appendix A. Calculus of Quaternion
Quaternion expresses a transformation of orthogonal coordinates using four parameters, and is
very useful to deal with kinematics of satellites. Quaternion is also called Euler parameters,
q-parameters, or Euler symmetric parameters. The definition, notation, transformation matrix, and
multiplication of quaternions are shown in this section.
A.1 Definition
Quaternion q is defined as follows.
1 1
2 2
3 3
4
sin( / 2)
sin( / 2)
sin( / 2)
cos( / 2)
q e
q e
q e
q
Φ Φ ≡ ≡ Φ
Φ
q (A.1)
where 1 2 3[ ]Te e e=e is the unit vector in the reference frame called Euler axis, Φ is the
rotation angle about Euler axis. Since the norm of Euler axis is 1, following equation is derived.
2 2 2 2
1 2 3 4 1q q q q+ + + = (A.2)
The components of Euler axis in the two frames (before and after the rotation) are same.
Accordingly vector e is the eigenvector of the coordinate transformation matrix (direction cosine
matrix) expressed by q. The inverse or conjugate of q is defined as the rotation which the angle is
–Φ and the axis is e, and is expressed by *q .
A.2 Notation
Quaternion vector is defined to express as follows.
1
2
3 4
4
v
q
q
q q
q
≡ ≡
qq (A.3)
Meanwhile, in the case of expressing the quaternion as the hyperimaginary number, it is written as
follows.
1 2 3 4q q q q≡ + + +q i j k (A.4)
where i, j, and k are units of hyperimaginary number requiring following conditions.
2 2 2 1+ + = − = − = = − = = − =i j k ij ji k jk kj i ki ik j (A.5)
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Besides, *q is also expressed as follows.
*
1 2 3 4q q q q= − − − +q i j k (A.6)
A.3 Direction Cosine Matrix
The direction cosine matrix (DCM) using e and Φ is expressed as follows.
2
1 1 2 3 1 3 2
2
1 2 3 2 2 3 1
2
1 3 2 2 3 1 3
cos (1 cos ) (1 cos ) sin (1 cos ) sin
(1 cos ) sin cos (1 cos ) (1 cos ) sin
(1 cos ) sin (1 cos ) sin cos (1 cos )
e e e e e e e
e e e e e e e
e e e e e e e
Φ + − Φ − Φ + Φ − Φ − Φ
= − Φ − Φ Φ + − Φ − Φ + Φ − Φ + Φ − Φ − Φ Φ + − Φ
T
(A.7)
Substituting Eq. (A.1), DCM expressed by q becomes
2 2 2 2
1 2 3 4 1 2 3 4 1 3 2 4
2 2 2 2
1 2 3 4 1 2 3 4 2 3 1 4
2 2 2 2
1 3 2 4 2 3 1 4 1 2 3 4
2( ) 2( )
( ) 2( ) 2( )
2( ) 2( )
q q q q q q q q q q q q
q q q q q q q q q q q q
q q q q q q q q q q q q
− − + + −
= − − + − + + + − − − + +
T q (A.8)
The relation between inverse and the conjugate are following.
1 *( ) ( )− =T q T q (A.9)
T(q) has the orthogonality as follows.
1( ) ( ) ( ) ( ) , ( ) ( )T T T−= = =T q T q T q T q I T q T q (A.10)
A.4 Multiplication
Multiplication of a quaternion as a operator is introduced here. Calculation of a quaternion and a
vector can be also defined if you use the hyperimaginary expression. The definitions can simplify
calculations, which become easier than vector expression.
Multiplication of quaternions is defined as follows.
1 2 3 4 1 2 3 4
1 4 2 3 3 2 4 1
1 3 2 4 3 1 4 2
1 2 2 1 3 4 4 3
1 1 2 2 3 3 4 4
( )( )
( )
( )
( )
( )
a b a a a a b b b b
a b a b a b a b
a b a b a b a b
a b a b a b a b
a b a b a b a b
q q q q q q q q
q q q q q q q q
q q q q q q q q
q q q q q q q q
q q q q q q q q
= + + + + + +
= + − +
+ − + + +
+ − + +
+ − − − +
q q i j k i j k
i
i
i
(A.11)
Therefore, multiplication of quaternion q and its conjugate *q becomes
* * 1= =qq q q (A.12)
This means that absolute value of the quaternion is 1.
If you use the vector expression, it becomes
19 September 2007
21
2 1 1
3 4 1 2 2
2 1 4 3 3
1 2 3 4 4
2 1 1
3 4 1 2 2
2 1 4 3 3
1 2 3 4 4
( )
( )
b b b b a
b b b b a
a b b b a
b b b b a
b b b b a
a b a a b
a a a a b
a a b
a a a a b
a a a a b
q q q q q
q q q q q
q q q q q
q q q q q
q q q q q
q q q q q
q q q q q
q q q q q
− − = = − − − −
− − = = − − − −
4 3
4 3
q q M q q
M q q
(A.13)
The above equations are also expressed by
4 3 3
4
4 3 3
4
( )( )
( )( )
v v
a T
v
v v
b T
v
q
q
q
q
×
×
+ ≡ −
− + ≡ −
γ q I qM q
q
γ q I qM q
q
(A.14)
where ( )vγ q is the matrix defined as follows
3 2
3 1
2 1
0
( ) 0 ( )
0
v v
q q
q q
q q
− ≡ − = × −
γ q q (A.15)
This means the cross-product operator of the vector. Therefore, the following relation is found for a
arbitrary vector v.
( )v v= ×γ q v v q (A.16)
The relation of the multiplied conjugate quaternions is as follows.
* * *( )a b b a=q q q q (A.17)
The hyperimaginary expression of a vector in the Euclidean space is defined as
1 2 3u u u≡ + +u i j k (A.18)
Multiplication of a quaternion and a vector is calculated as follows.
19 September 2007
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3 2 2 3 1 4 3 1 1 3 2 4
2 1 1 2 3 4 1 1 2 2 3 3
3 2 1 1
3 1 2 2
2 1 3 3
1 2 3 4
4 3 2
3 4 1
2 1 4
1 2 3
( ) ( )
( ) ( )
0
0( )
0
0
( )
u q u q u q u q u q u q
u q u q u q u q u q u q
u u u q
u u u q
u u u q
u u u q
q q q
q q q
q q q
q q q
= − + + − + +
+ − + + − − −
− − = ≡ − − − −
− − = ≡−− − −
qu i j
k
Ω u q
Ξ q u
1
2
3
u
u
u
(A.19)
where
4 3 3
( )( )
0
( )( )
T
v
T
v
q ×
≡ −
− + ≡ −
γ u uΩ u
u
γ q IΞ q
q
(A.20)
A.5 Coordinate transformation matrix using quaternion
The coordinate transformation matrix expressed by q is given in Eq. (A.8). Another expression
using vq and 4q becomes
2
4 3 3 4( ) ( ) 2 2 ( )T T
v v v v vq q×= − + +T q q q I q q γ q (A.21)
And the following equation is derived using Eq. (13) and Eq. (21).
3 1*
1 3
( )( ) ( )
1b a
×
×
=
T q 0M q M q
0 (A.22)
Besides, transformation ( )′ =u T q u can be also expressed as follows.
3 1
1 3
( )
10 0
×
×
′ =
T q 0u u
0 (A.23)
Accordingly, ′u can be expressed by
* *( ) ( ) ( )b a b
′ = = =u M q M q u M q uq q uq (A.24)
Namely, a coordinate transformation using a quaternion is written as above simply.
Besides, according to Eq. (A.24), following equations are derived when a b=q q q .
* ( ) ( ) , ( ) ( ) ( )b a a b b a a b b a
′ = = =u q q uq q T q T q u T q q T q T q (A.25)
19 September 2007
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Appendix B. Kinematic Equation of Quaternion
In this section, the kinematic equation expressed by quaternion is introduced. Firstly we derive the
most general equation as the result of differentiation of quaternion. Finally, we derive the kinematic
equation of variables which is divided into reference and small quaternion in order to apply to
Kalman filtering.
B.1 General Equation
Consider q(t) that expresses an attitude in a reference frame at time t, and variation of quaternion
∆q(t) from the time t to t+∆t. The reference frame of ∆q(t) is the frame of q(t). Then the attitude in
the reference frame at the time ∆t becomes,
4 3 2 1
3 4 1 2
2 1 4 3
1 2 3 4
( ) ( ) ( ) ( ( )) ( ) ( )a
q q q q
q q q qt t t t t t t
q q q q
q q q q
∆ ∆ −∆ ∆ −∆ ∆ ∆ ∆ + ∆ = ∆ = ∆ = ∆ −∆ ∆ ∆ −∆ −∆ −∆ ∆
q q q M q q q (B.1)
Then, the derivative of quaternion at time t is derived as follows using angular rate
1 2 3( ) [ ]Tt ω ω ω=ω in the satellite’s body frame.
0 0
3 2 1
3 1 2
2 1 3
1 2 3
( ( )) ( ) ( )( ) ( )( ) lim lim
0
01( )
02
0
1( ( )) ( )
2
1( ) ( )
2
a
t t
t t tt t tt
t t
t
t t
t t
ω ω ω
ω ω ω
ω ω ω
ω ω ω
∆ → ∆ →
∆ −+ ∆ −= =
∆ ∆
− − = − − − −
= Ω
=
M q q qq qq
q
ω q
q ω
ɺ
(B.2)
B.2 Kinematic Equation for Multiplication of Quaternion
The kinematic equation which expresses continuous two coordinate transformations is derived
here. This equation expressed by quaternion is nonlinear in the case of setting the angular rates as the
state variables. This is the reason why extended Kalman filter is applied to attitude calculation
generally. Here, a quaternion q is divided into a reference value refq and a small value ∆q, and the
kinematic equation is derived.
Firstly, quaternion q is expressed as follows.
19 September 2007
24
ref= ∆q q q (B.3)
where ∆q is based on the frame of refq which expresses the satellite attitude. The kinematic
equation of the reference quaternion refq becomes following from Eq. (B.2)
1
2ref ref ref
=q q ωɺ (B.4)
Therefore,
* * * *1 1
2 2ref ref ref ref ref
= = −q ω q ω qɺ (B.5)
where, refω is the angular velocity vector in the reference frame, and is expressed in
hyperimaginary number.
B.3 Kinematic Equation for Small Quaternion
∆q is derived from Eq. (B.3),
*
ref∆ =q q q (B.6)
and differentiate Eq. (B.6) and substitute Eq. (B.5),
* *
* *1 1
2 2
1 1
2 2
ref ref
ref ref ref
ref
∆ = +
= − +
= − ∆ + ∆
q q q q q
ω q q q qω
ω q qω
ɺ ɺ ɺ
(B.7)
where,
ref= + ∆ω ω ω (B.8)
and substitute Eq. (B.8) into Eq. (B.7), following equation are derived.
( )
( )
( )
4 3 3 4 3 3
1 1 1
2 2 2
( ) ( )1 1 1
2 2 2
( ) 1
0 2
( ) 1
0 2
ref ref
v v
ref refT T
v v
v
ref
ref v
ref
q × ×
∆ = − ∆ + ∆ + ∆ ∆
∆ + ∆ − ∆ + ∆ = − + + ∆ ∆ − −
− ∆ = + ∆ ∆
− ∆ = + ∆ ∆
q ω q qω q ω
γ q I γ q q Iω ω Ξ q ω
q q
γ qω Ξ q ω
γ ω qω Ξ q ω
ɺ
(B.9)
However, ω is expressed in the frame of q, and refω is in the frame of
refq actually. That is,
19 September 2007
25
*
ref= ∆ ∆ + ∆ω q ω q ω (B.10)
where ∆ωωωω is the difference vector between the angular rate vector in the frame of q and the reference
frame. Substituting Eq. (B.10) into (B.7),
*1 1 1
2 2 2
1
2
ref ref∆ = − ∆ + ∆ ∆ ∆ + ∆ ∆
= ∆ ∆
q ω q q q ω q q ω
q ω
ɺ
(B.11)
This means the kinematic equation is independent of the other frame. However actual angular rate
vector is measured in body frame. Therefore ωωωω and ωωωωref should be dealt as the variables in body
frame, and Eq. (B.9) is appropriate for satellites’ attitude determination. That is
( )
( )
( ) 1
0 2
1
0 2
ref v
ref
ref v
ref
− ∆ ∆ = + ∆ ∆
− × ∆ = + ∆ ∆
γ ω qq ω Ξ q ω
ω qω Ξ q ω
ɺ
(B.12)
if ∆q is small value,
1
2v∆ ≈ ∆q θ (B.13)
where ∆θθθθ is small Euler angle (Tait-Bryan angle) vector corresponding to the small quaternion ∆q.
Substituting this into Eq. (B.12), ∆θθθθ becomes
ref∆ = − × ∆ + ∆θ ω θ ωɺ (B.14)
19 September 2007
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Appendix C. Observability
When you use the Kalman filter, you need to confirm the observability of the system equations. If
there is no observability, it is impossible to estimate the state variables logically, and is also
definitely impossible to apply the Kalman filter. Firstly, the most general determination method for a
linear system is introduced in this section. Secondly, the method for a nonlinear system is introduced.
You can confirm the observability of your system using these methods.
C.1 Linear System
As a linear system, following continuous system is introduced.
( ) ( ) ( )
( )
t t t
t
= +
=
x Ax Bu
y Cx (C.1)
where ( ) , ( )n mt t∈ ℜ ∈ ℜx y . The necessary and sufficient condition of observability is that the
rank of following equation is n. That is,
2
1n
rank n
−
=
C
CA
CA
CA
⋮
(C.2)
If this equation is n, the system is observable as long as matrices A and B are constant.
C.2 Nonlinear System
As a nonlinear system, following system is introduced.
( )
( )
( ) ( ), ( )
( ) ( )
t t t
t t
=
=
X f X u
Y h X
ɺ
(C.3)
where ( ) , ( )n mt t∈ ℜ ∈ ℜX Y . The necessary and sufficient condition of observability is that
the rank of following equation is n. That is,
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27
( )
( )
( )
( )
(0)
(1)
(2)
( )
h
h
h
i
h
L
L
rank nL
L
∂ ∂
∂ ∂
=∂ ∂ ∂ ∂
fX
fX
fX
fX
⋮
(C.4)
where ( )i
hL f is called Lie derivative, and is defined as follows.
( )
( )
( )
(0)
(1) (0)
(2) (1)
( ) ( 1)
( ( ))h
h h
h h
i i
h h
L t
L L
L L
L L−
=
∂=
∂
∂=
∂
∂=
∂
f h X
f f fX
f f fX
f f fX
⋮
(C.5)
19 September 2007
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Appendix D. Discretization of the System[7]
A linear time-invariant m-input p-output N-dimensional continuous-time system can be
represented by the state model
( ) ( ) ( )t t t= +x Ax Bw (D.1)
( ) ( )t t=y Cx (D.2)
where Eq. (D.1) is the state eqution and Eq. (D.2) is the output equation. A is the N x N system
matrix, B is the N x m input matrix, C is the p x N output matrix, w(t) is the m-dimensional input
vector, x(t) is the N-dimensional state vector, and y(t) is the p-dimensional output vector. If the
system has a single output so that p=1, we shall sometimes use lower case “c” for the output matrix,
in which case Eq. (A.2) becomes ( ) ( )y t cx t= .
The complete solution to the state equation Eq. (A.1) staring with initial state x(t0) at initial time t0
is
[ ] [ ]0
0 0 0( ) exp ( ) ( ) exp ( ) ( ) ,t
tt t t t t d t tτ τ τ= − + − >∫x A x A Bw (D.3)
The system is said to be asymptotically stable if the system state x(t) converges to 0 as t → ∞ for
any initial state x(t0) at any initial time t0 and with w(t)=0 for all t>t0. It follows from Eq. (D.3) that
the system is asymptotically stable if and only if all the eigenvalues of A have real parts that are
strictly less than zero (i.e., the eigenvalues of A lie in the open left half plane).
Using Eq. (A.2) and Eq. (A.3), we have that the complete output response of the system is given
by
[ ] [ ]0
0 0 0( ) exp ( ) ( ) exp ( ) ( ) ,t
tt t t t t d t tτ τ τ= − + − >∫y C A x C A Bw (D.4)
If the initial state x(t0) is zero and the initial time t0 is taken to be −∞ , Eq. (D.4) reduces to the
input/output representation
( ) ( ) ( )t
t t dτ τ τ−∞
= −∫y H w (D.5)
where H(t) is the p x m impulse response function matrix given by
0, 0
( )exp[ ], 0
tt
t t
<=
≥H
C A (D.6)
Taking the Laplace transform of both sides of Eq. (D.5) results in the tranfrer function representation
( ) ( ) ( )s s s=Y H W (D.7)
where H(s) is the transfer function matrix equal to the transform of H(t). The Laplace transform of
the matrix exponential exp[At], 0t ≥ is equal to 1( )s −−I A where I is the N x N identity matrix.
Thus, from Eq. (D.6) it is seen that the transfer function matrix H(s) is given by
1( ) ( )s s −= −H C I A B (D.8)
19 September 2007
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D.1 Discrete-Time Case
We can generate a discrete-time state model by discretizing the continuous-time state model. To
accomplish this, let T be the sampling interval, and set t0=nT and t=nT+T in Eq. (D.3), where n is
the discrete-time index. This yields
[ ] [ ]( ) exp ( ) exp ( )nT T
nTnT T T nT nT T dτ τ τ
+
+ = + + −∫x A Bw (D.9)
Note that exp[At] can be determined by inverse transforming 1( )s −−I A and setting t=T in the
result. If w(τ) is approximately constant over each interval nT nT Tτ≤ ≤ + , then
( ) ( )nTτ ≈w w for nT nT Tτ≤ ≤ + , and Eq. (D.9) can be rewritten as
[ ] [ ]( ) exp ( ) exp ( ) ( )nT T
nTnT T T nT nT T d nTτ τ
+ + = + + − ∫x A x A B w (D.10)
Defining
[ ]exp ( )nT T
nTnT T dτ τ
+
= + −∫Γ A B (D.11)
via a change of variables it follows that ΓΓΓΓ can be expressed in the form
[ ]0
expT
dτ τ= ∫Γ A B (D.12)
Finally, defining exp[ ]T=Φ A , from Eq. (D.10) we obtain the following discrete-time output
equation:
( ) ( ) ( )nT T nT nT+ = +x Φ Γw (D.13)
and setting t=nT in both sides of Eq. (D.2), we obtain the following discrete-time output equation:
( ) ( )nT nT=y Cx (D.14)
We shall drop the notation “T” for the sampling interval in Eq. (D.13) and Eq. (D.14), in which case
the general form of the linear time-invariant finite-dimensional state model is
( 1) ( ) ( )n n n+ = +x Φx Γw (D.15)
( ) ( )n n=y Cx (D.16)
In some cases of interest, the output y(n) at time n may depend on the input w(n) at time n, in which
case Eq. (D.16) becomes
( ) ( ) ( )n n n= +y Cx Dw (D.17)
where D is a p x p matrix, called the direct-feed matrix.
The complete solution to the state equation Eq. (D.15) starting with initial state x(n0) at initial time
n0 is
0
0
11
0 0( ) ( ) ( ),n
n n n i
i n
n n n n n−
− − −
=
= + >∑x Φ x Φ Γw (D.18)
The system is said to be asymptotically stable if x(n) converges to 0 as n → ∞ for any initial state
at any initial time and with w(n)=0 for all 0n n≥ . It follows from Eq. (D.18) that the system is
19 September 2007
30
asymptotically stable if and only if all the eigenvalues of ΦΦΦΦ have magnitudes that are strictly less
than one (i.e., all eigenvalues of ΦΦΦΦ lie within the unit circle of the complex plane).
Using Eq. (D.17) and Eq. (D.18) we have that the complete output response is
0
0
11
0 0( ) ( ) ( ) ( ),n
n n n i
i n
n n n n n n−
− − −
=
= + + >∑y CΦ x CΦ Γw Dw (D.19)
If the initial state x(n0) is zero and the initial time n0 is taken to be −∞ , Eq. (D.19) reduces to the
input/output representation
( ) ( ) ( )n
i
n n i w i=−∞
= −∑y H (D.20)
where H(n) is the p x m unit-pulse response function matrix given by
0, 0
( ) , 0
1n
n
n n
n
<
= = ≥
H D
CA B
(D.21)
Taking the z-transform of both sides of Eq. (D.21) results in the transfer function representation
( ) ( ) ( )z z z=Y H W (D.22)
The z-transform of An, 1n ≥ , is equal to
1( )z −−I A . It follows from Eq. (D.21) that the transfer
function matrix H(z) is given by
1( ) ( )z z −= − +H C I A B D (D.23)
D.2 Discretization of Covariance Matrices
For the Kalman filter, the discrete-time signal/measurement model is often a discretized
representation of a continuous-time state model of the form:
( ) ( ) ( )
( ) ( ) ( )
c c c
c c c
t t t
t t t
= +
= +
x Ax Bw
y Cx v (D.24)
and initial state 0 0( )c c
t =x x , ( )c
tw and ( )c
tv are uncorrelated, white-noise, continuous-time
random signals with covariances
[ ][ ]
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
c c c c
c c c c
E t t t t
E t t t t
τ δ τ
τ δ τ
− = −
− = −
w w Q
v v R (D.25)
If ( )c
tw and ( )c
tv are stationary, as if often the case, ( )c c
t =Q Q and ( )c c
t =R R for all t.
( )c
tδ is the continuous-time impulse function:
( ) 0, 0, ( ) 1,c c
t t and dε
εδ δ τ τ
−= ≠ =∫ for any 0ε > (D.26)
Then the discretized signal/measurement model with sampling period T has the form of
19 September 2007
31
( 1) ( ) ( )
( 1) ( ) ( )
n n n
n n n
+ = +
+ = +
x Φx Γw
y Cx v (D.27)
where ( )exp T=Φ A , ( )nw and ( )nv are uncorrelated, white-noise, discrete-time random
signals with covariance matrices.
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
T
T
E i j i i j
E i j i i j
δ
δ
= −
= −
w w Q
v v R (D.28)
In the case of stationary, ( )n =Q Q and ( )n =R R for all n.
It is therefore necessary to relate the discrete-time covariance matrices Q(n) and R(n) to their
continuous-time counterparts Qc(t) and Rc(t). First consider descretization of Rc(t). It may appear
that
( ) ( ) ( ) ( )c c
n n nT nTδ δ=R R (D.29)
However, he impulses δ(n) and δc(t) are not equivalent. The discrete-time impulse δ(n) remains finite
for all n, but the continuous-time impulse δc(t) is unbounded at t=0.
To remedy this incompatibility, Lewis[15]
suggests the following method. Define the
continuous-time unit rectangle by
1 11,
( ) 2 2
0,
tb t
otehrwise
− ≤ ≤
=
(D.30)
Then δc(t) can be written as
0
1( ) lim ( / )
cT
t b t TT
δ→
= (D.31)
and the continuous-time covariance becomes
0
1( ) ( ) lim ( )
c c t nT Tt nT
tt t n T b
T Tδ
= →=
= ×
R R (D.32)
The extra factor of T ensures that the area of the rectangle remains unity. In order for the right-hand
side of this equation to equal the left-hand side, it follows that
( ) ( ) /c
n nT T=R R (D.33)
This expression gives the proper relationship between R(n) and Rc(t).
We may discretize Bwc in two ways. First, we may assume a zero-order hold, i.e., we have
w(n)=wc(nT). Note that if wc(t) is a p-vector, then so is w(n). ΓΓΓΓ is given by Eq. (D.12), and by the
same argument that produced Eq. (D.33), Q(n) is given by
( ) ( ) /c
n nT T=Q Q
(D.34)
Second, it is possible to incorporate the system dynamics from time t=nT to time n=nT+T. In this
19 September 2007
32
case, w(n) is given by
[ ]( ) exp ( ) ( )nT T
cnT
n nT T dτ τ τ+
= + −∫w A Bw (D.35)
The intervals of integration for w(n) and w(n+1) do not overlap, so w(n) is white noise. As a result
of Eq. (D.35), w(n) is a vector that is the same length as x(n). To find the covariance associated with
w(n), write
[ ]
[ ]
( ) ( ) ( )
exp ( ) ( )
( ) exp ( )
exp ( ) ( ) ( )
exp ( )
T
nT T
cnT
nT TT T T
cnT
nT T nT TT
c cnT nT
T T
n E n n
E nT T d
nT T d
nT T E
nT T d d
τ τ τ
ρ ρ ρ
τ τ ρ
ρ τ ρ
+
+
+ +
=
= + −
× + −
= + − ×
× + −
∫
∫
∫ ∫
Q w w
A Bw
w B A
A B w w
B A
(D.36)
Now ( ) ( ) ( ) ( )T
c c c cE τ ρ τ δ τ ρ = − w w Q , so
[ ]( ) exp ( ) ( ) exp ( )nT T
T T
cnT
n nT T nT T dτ τ τ τ+
= + − + − ∫Q A BQ B A (D.37)
By a change of variables, this equation becomes
[ ]( ) exp ( ) expT
T T
co
n dτ τ τ τ = ∫Q A BQ B A (D.38)
Because the dynamics of [ ]exp t BA are directly incorporated into w(n) and Q(n) in Eq. (D.38),
this method used
=Γ I (D.39)
where I is the N x N identity matrix.
19 September 2007
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Appendix E. Geomagnetic Field
E.1 Formulation
International Geomagnetic Reference Field (IGRF) model is given as gradient of geomagnetic
potential.
( , , )
1
1
sin
V r
VX
r
VY
r
VZ
r
θ λ
θ
θ λ
= −∇
∂=
∂
∂= −
∂
∂=
∂
B
(E.1)
where, r is the geocentric radius, θ is the colatitude, and λ is the longitude.
Fig. D.1 IGRF Coordinates
The transformation matrix from IGRF coordinates to inertial frame (IJK) is following,
3 2
cos cos sin cos sin
( ) ( ) cos sin cos sin sin
sin 0 cos
LST LST LST
LST LST LST LST
IJKR R
IGRF
θ λ λ λ θ
λ π θ θ λ λ θ λ
θ θ
− − − = − − = − − −
(E.2)
where LSTλ is local sidereal time which is calculated by use of following equations
1
1
1
2 6 3
1 1
2,451,545.0
36,525
67,310.54841 (876,600 8,640,184.812866 )
0.093104 6.2 10
UT
UT
s h s
GMST UT
UT UT
LST GMST
JDT
T
T T
λ
λ λ λ
−
−=
= + +
+ − ×
= +
(E.3)
19 September 2007
34
where 1UTT is the Coordinated Universal Time, and GMSTλ is the Greenwich Mean Sidereal Time.
Geomagnetic potential is given as follows
( ) ( )1
1 0
( , , ) cos sin cos
nk nm m m
n n n
n m
aV r a g m h m P
rθ φ λ λ θ
+
= =
= +
∑ ∑ (E.4)
where (cos )m
nP θ is the Schmidt seminormalized associated Legendre function, and it is defined as
1/ 2
2 2( )!1( ) (1 ) ( 1)
( )!2 !
m nm m nm
n n m n
n m dP
n mn d
εµ µ µ
µ
+
+
− = − − +
(E.5)
where m
ε is the constant value that becomes 1 when 0m = and becomes 2 when 0m ≠ .
Distributed g and h are coefficients that can calculate geomagnetic field from dipole model to 10th
order model. Hence,
( ) ( )
( ) ( )
( ) ( )
2
1 0
2
1 0
2
1 0
cos sin cos
1sin cos cos
sin
( 1) cos sin cos
nk nm m m
n n n
n m
nk nm m m
n n n
n m
nk nm m m
n n n
n m
aX g m h m P
r
aY m g m h m P
r
aZ n g h m P
r
λ λ θθ
λ λ θθ
λ λ θ
+
= =
+
= =
+
= =
∂ = +
∂
= −
= − + +
∑ ∑
∑ ∑
∑ ∑
(E.6)
E.2 Recursive Equations
Legendre function and its derivative are calculated using recurrence equation based on Chapman
and Bartels method[16]
in order to compute the model quickly.
1/ 2
,
( )!(cos ) (cos )
( )!
m m
n n m
n mP P
n m
εθ θ
− = +
(E.7)
where , (cos )n mP θ is non-normalized associated Legendre function. And following values are
introduced, then
1/ 2
1/ 2
, ,
( )!
( )!
( )!
( )!
(cos ) (cos )
m m m
n n
m m m
n n
n m n m
n mG g
n m
n mH h
n m
dQ P
d
ε
ε
θ θθ
− ≡
+
− ≡ +
≡
(E.8)
Therefore Eq. (E.6) becomes
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( ) ( )
( ) ( )
( ) ( )
2
,
1 0
2
,
1 0
2
,
1 0
cos sin cos
1sin cos cos
sin
( 1) cos sin cos
nk nm m
n n n m
n m
nk nm m
n n n m
n m
nk nm m
n n n m
n m
aX G m H m Q
r
aY m G m H m P
r
aZ n G H m P
r
λ λ θ
λ λ θθ
λ λ θ
+
= =
+
= =
+
= =
= +
= −
= − + +
∑ ∑
∑ ∑
∑ ∑
(E.9)
,n mP and ,n mQ can be calculated as follows based on Chapman and Bartels method.
( ),0 1,0 2,0
1(2 1) cos ( 1)n n nP n P n P
nθ − −= − − − (E.10)
( ),0 ,0 1,0cossin
n n n
nQ P Pθ
θ−= − (E.11)
( ), 1, 1 , 1
1( 1) ( 1)cos
sinn m n m n mP n m P n m Pθ
θ− − −= + − − − + (E.12)
, , ,
cos( )( 1)
sinn m n m n m
mQ n m n m P P
θ
θ= + − + − (E.13)
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Appendix F. External Torques and Noise Generation
F.1 Gravity Gradient Torques
Gravity gradient torque is calculated by following equation [17]
2 2
2 2
3
2 2
( ) ( )3
( ) ( )
( ) ( )
z y xy xz yz
E
gg x z xy xy xz
y x xy yz xy
mn I I nlI lmI n m I
nl I I lmI mnI l n IR
lm I I mnI nlI m l I
µ − + − + −
= − + − + − − + − + −
T (F.1)
where, R=R(l,m,n)T is the orbital radius vector pointing from the center of the Earth to the center of
mass of the satellite in the Body Centered Reference Frame (BCRF).
Fig. F.1 shows the stability condition in terms of inertia moment. It is required that y x zI I I> >
or x z yI I I> > to stabilize the satellite.
Fig. F.1 Stability Condition of Inertia Moments
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F.2 Solar Radiation Torques
Solar radiation pressure torque can be calculated by following
( )
(1 )
srp srp srp m
s
srp s sun
FA q
c
= × −
= +
T F c c
F r (F.2)
where csrp is the center of solar radiation pressure, cm is the center of mass, Fs is the solar constant, As
is the area, q is the reflectivity i is the incident angle, c is the velocity of light, and rsun is the unit sun
vector. The vector of the sun is calculated using the ephemeris which is developed by several
researchers. In QSAT simulator, VSOP87 (Variations Séculaires des Orbites Planétaires) method[20]
is built which guarantees for Mercury, Venus, Earth-Moon barycenter and Mars a precision of 1" for
4,000 years before and after the 2,000 epoch. The same precision is ensured for Jupiter and Saturn
over 2,000 years and for Uranus and Neptune over 6,000 years before and after J2000.
F.3 Atmospheric Drag Torques
Atmospheric drag torque is calculated by following equation
( )
1
2
a a a m
a dC AVρ
= × −
=
T F c c
F V (F.3)
where ca is the center of aerodynamic force, Cd is the drag coefficient, ρ is the atmospheric density,
and V is the velocity.
As the atmospheric density model, NRLMSIS00E-00 is used in the simulator. NRLMSISE-00 is an
empirical, global model of the Earth's atmosphere from ground to space developed by Picone, Hedin,
and Drob is based on the earlier models MSIS-86 and MSISE-90, and updated with actual satellite
drag data. It models the temperatures and densities of the atmosphere's components.
Note!: The Lever Arm of Atmospheric Drag Torque & Solar Radiation Pressure Torque
We assume csrp and ca as 5[cm] which are considered as the worst cases.
F.4 Magnetic Torques
Magnetic torques is expressed by following equation.
mag
= ×T M B (F.4)
where M is the magnetic moment vector, and B is the magnetic flux density vector. The Geomatnetic
field is calculated by IGRF model (see Appendix E).
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F.5 Sensor Outputs
For the sake of simulation of attitude determination and control, sensor outputs must be modeled.
After finishing calibration of sensors, I will build sensor model into the simulator.
[ ]( )output true= + +s A s d n (F.5)
soutput is the simulated sensor outputs, strue is the real values, A is the transformation matrix, d is the
drifts, n is the Gaussian white noise. Gaussian white noise is generated using L’Ecuyer’s method [14]
and Box-Muller method [14]
.
We should consider which sun sensor can see the sunlight. The angle between two vectors (sun
vector and perpendicular vector to the position sensing device (PSD)) has to be calculated because
the sun sensor has 55[deg] view angle.
arccos 55[deg]sun sensor
sun sensor
V V
V Vθ
⋅= ≤
(F.6)
Fig. F.2 and F.3 show the output of magnetometer. Noise (standard deviation: 500[nT]) and bias:
(200[nT] in this test) are added to the true value. Fig. F.4 and F.5 show the output of sun sensor. If
the angle between sun vector and perpendicular vector to PSD exceed 55[deg] or the satellite is in
eclipse, output becomes zero. I add noise and bias to normalized sun vector measured by PSD
because it is difficult to determine the current passing through the PSD. We are using temporary
values of drifts and noise currently. We have to get accurate values by means of calibration
tests.
Fig. F.2 Example of Magnetometer Output with Noise and Bias (a)
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Fig. F.3 Example of Magnetometer Output with Noise and Bias (b)
Fig. F.4 Example of Sun Sensor Output with Noise and Bias (a)
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Fig. F.5 Example of Sun Sensor Output with Noise and Bias (b)
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41
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