TRIAD-Aided EKF for Satellite Attitude Estimation and
Sensor Calibration
MEHMET ASIM GOKCAY
Hezârfen Aeronautics and Astronautics Technologies Institute for Space Sciences
Turkish National Defense University
Yeşilyurt, 34149, Bakırköy, Istanbul
TURKEY
CHINGIZ HAJIYEV
Faculty of Aeronautics and Astronautics
Istanbul Technical University
Ayazağa, 34469, Maslak, Istanbul
TURKEY
Abstract: - Increasing demand for the space operations, space industry turns its face to cost effective solutions.
Small satellites, due to their size and cost, are receiving interest from many organizations. Two of most
common sensors that are being used in nanosatellites are magnetometers and sun sensors. In this work,
magnetometer and sun sensor measurements are fused together with the TRIAD algorithm to produce body
angle estimation. Combining with gyroscopes measurements, an Extended Kalman Filter is used to estimate
body angles, angular velocities, gyroscope and magnetometer biases.
Key-Words: Extended Kalman filter, satellite attitude estimation, magnetometer, sun sensor, gyroscope
Received: November 29, 2021. Revised: November 2, 2022. Accepted: November 29, 2022. Published: December 31, 2022.
1 Introduction
Increasing usage of small satellites led to an
increase in research about their hardware and
software technologies. Attitude determination and
control (ADC) is a key part for any type of mission
[1]. It directly affects the mission success. Due to
their size, sensor variety is considerably lower than
regular size satellites. This fact is a challenge for
ADC researchers. Commonly used sensors are used
for body angle and angular velocity estimations and
Euler angles are used to represent satellite attitude.
Many different methods have been proposed for
body angle estimation [9]. In this work, sun sensor
and magnetometer are fed to the TRIAD
algorithm to produce body angle
estimations [2],[3],[4],[5],[8]. Combining with
gyroscope measurements, these TRIAD outputs
are then fed to an extended Kalman filter as
measurements in order to estimate body angles,
angular velocities, gyroscope and
magnetometer biases. These estimated biases are
fed back to sensor models for calibration.
2. Equations of Motion
To define satellite motion, kinematic and dynamic
equations are derived. In this paper, the DCM is
constructed using 2-1-3 rotation.
213 3 1 3
A ( , , ) A ( )A ( )A ( )
cos sin 0 1 0 0 cos 0 sin
sin cos 0 0 cos sin 0 1 0
0 0 1 0 sin cos sin 0 cos
θ φ ψ = ψ φ θ
ψ ψ θ − θ
= − ψ ψ φ φ
− φ φ θ θ
(2.1)
After matrix multiplication, dcm has been derived.
c c s s s c s c s s c s
c s s c s c c s s c c s
c s s c c
ψ θ + φ ψ θ φ ψ θ φ ψ − ψ θ
ψ φ θ − θ ψ φ ψ ψ θ + ψ θ φ
φ θ − φ φ θ
(2.2)
where c(.) and s(.) are cosine and sine functions
respectively. From 2-1-3 rotation matrix, Euler
angles can be extracted from matrix elements with
equations below.
A(1,2)
atan 2 A(2,2)
ψ =
(2.3)
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A(3,1)
atan
A(3,3)
θ =
(2.4)
)
A(3,2)cos
atan A(2,2)
− ψ
φ =
(2.5)
where atan and atan2 are both arctangent functions.
atan2 is the arc tangent function with two arguments
for complete coverage of four quadrants.
2.1 Kinematic equations
Obtaining a frame from another frame is possible
with rotation matrices. Using the fact that angular
velocities are additive, angle rates can be
determined in terms of these velocities. Considering
2-1-3 rotation, the initial frame rotates about the y
axis by θ. Second rotation is about x′ by ϕ and final
rotation is about w by ψ. Summing angular velocity
vectors for each rotation [1]
y x
w
ω = θ + φ
′ + ψ
ɺ
ɺ
ɺ
(2.6)
Taking the components of ω in
u, v, w
r r r
v
u
w
yu x
yv x
u
v
yw wx
ω = θ + φ
θ + φ
θ +
′
ω = ′
ω = ′ + ψ
φ
ɺ ɺ
ɺ ɺ
ɺ ɺ
ɺ
(2.7)
In order to derive kinematic equations, (2.7) needs
to be determined.
yu,yv
r r r r
and
yw
r r
can be calculated
from DCM.
y
r
indicates that second column of the
DCM.
yu cos sin
= φ ψ
(2.8)
yv cos cos
= φ ψ
(2.9)
yw sin
= − φ
(2.10)
For second rotation, R3(ψ)R1(ϕ) matrix needs to be
calculated. Taking first column of the second
rotation matrix
cx
u os
′ = ψ
(2.11)
sx
v in
′ = − ψ
(2.12)
wx
0
′ =
(2.13)
Hence, angular velocity equations become
u
cos sin cos
ω = θ φ ψ + φ ψ
ɺ ɺ
(2.14)
v
cos cos sin
ω = θ φ ψ − φ ψ
ɺ ɺ
(2.15)
wsin
ω = −θ φ + ψ
ɺ
ɺ
(2.16)
In matrix form,
bo
cos sin cos 0
cos cos sin 0
sin 0 1
φ ψ ψ θ
ω = φ ψ − ψ φ
− φ ψ
ɺ
ɺ
ɺ
(2.17)
Taking inverse of the matrix, Euler angle rates can
be determined in terms of angular velocities in body
frame with respect to reference frame [1]. Rate
equations are given below
u v
cos sin
φ = ω ψ − ω ψ
ɺ
(2.18)
u v
( sin cos )sec
θ = ω ψ + ω ψ φ
ɺ
(2.19)
u v w
tan ( sin cos )
ψ = φ ω ψ + ω ψ + ω
ɺ
(2.20)
Rotations need to be defined with respect to inertial
frame. For this reason, a transformation matrix is
needed. Using the relation below, angular velocities
with respect to the inertial frame can be calculated.
u x
v y 0
w z
w 0
w A
w 0
ω
ω = − −ω
ω
(2.21)
where 0
3
r
µ
ω =
is angular velocity at the altitude r.
µ
is gravitational constant of the Earth and r is the
distance between center of mass of the satellite and
the Earth,
r r
=
.
2.2 Satellite dynamics
Attitude dynamics is related with time
derivative of the angular momentum vector.If the
origin of the body frame is selected as the center of
mass, c, then [1],
c
h Jw
=
(2.22)
where h is angular momentum vector, w is angular
velocity vector in body frame wrt inertial frame and
J is moment of inertia matrix. J is defined as,
x
y
z
J 0 0
J 0 J 0
0 0 J
=
(2.23)
The relation between time derivative of the angular
momentum and the angular velocity is,
c b c
h h w (h )
= + ×
ɺ ɺ
(2.24)
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where
b
h
ɺ
is the time derivative of angular
momentum as seen in a body-fixed frame. Knowing
the time rate of change of the angular momentum is
equal to external torque, Tc and using (2.22)
c b
T h w (Jw)
= + ×
ɺ
(2.25)
Rewriting this equation considering time derivative
of the w is same in both body-fixed and reference
frames,
b
h Jw
=
ɺ
ɺ
(2.26)
( )
1
c
w J T w Jw
−
= − ×
ɺ
(2.27)
The external torque can be defined as the sum of the
gravity gradient torque, aerodynamic torque,
magnetic torque and solar pressure disturbance. In
this thesis, only magnetic torque is considered as
external torque.
c m
T T
=
(2.28)
Finally, rewriting (2.25) for discrete time,
( )
k 1 k
x x z y y z m
x
t
w w w w J J T
J
+
∆
= + − +
(2.29)
( )
k 1 k
y y x z z x m
y
t
w w w w J J T
J
+
∆
= + − +
(2.30)
( )
k 1 k
z z x y x y m
z
t
w w w w J J T
J
+
∆
= + − +
(2.31)
Equations (2.14)-(2.16) and (2.29)-(2.31) describe
the satellite attitude motion
3 Models of Sensor Measurements
3.1 Magnetometer Measurement Model
Magnetometer is the most commonly used sensor
particularly in nanosatellite applications. For
magnetic field vector, dipole model is used. Sensor
model is given below,
x 1
y 2 m m
z 3
B ( , , , t) B (t)
B ( , , , t) A B (t) b
B ( , , , t) B (t)
φθψ
φ θ ψ = + + η
φ θ ψ
(3.1)
where B1(t), B2(t) and B3(t) indicates Earth magnetic
field vector components in orbit frame and given by
[5],
( ) ( )
{
( ) ( )
}
e
1 0 e
3
0 e
M
B (t) cos t cos sin( ) sin( )cos( )cos( t)
r
sin t sin( )sin t
= ω ε ι − ε ι ω
− ω ε ω
(3.2)
( ) ( ) ( ) ( ) ( )
e
2 e
3
M
B (t) cos cos sin sin cos t
r
= − ε ι + ε ι ω
(3.3)
( ) ( ) ( ) ( ) ( ) ( ) ( )
{
( ) ( ) ( )
}
e
3 0 e
3
0 e
2M
B t sin t cos sin sin cos cos t
r
2cos t sin sin t
= ω ε ι − ε ι ω
+ ω ε ω
(3.4)
where
15
e
M 7.943 10
= ×
Wb.m is magnetic dipole
moment of the Earth,
11.7
ε =
o
is the magnetic
dipole tilt,
5
e
7.29 10
−
ω = ×
rad/s is the spin rate of
the Earth,
ι
is the orbit inclination and
14
3.98601 10
µ = ×
m3/s2 is the Earth Gravitational
constant.
Bx(
, , , t
φ θ ψ
), By(
, , , t
φ θ ψ
) and Bz(
, , , t
φ θ ψ
) show
the Earth magnetic field vector components in body
frame as a function of body angles and time. The
magnetometer bias vector is given as
T
m x y z
b b b b
=
. Bias vector is model as rate of
change of the bias vector is constant in time.
m
b 0
=
ɺ
(3.5)
The noise term,
m
,
η
is added linearly to the model
and modeled as zero mean Gaussian white noise
with the characteristic of
T 2
1k 1 j 3x3 m kj
E
η η = σ δ
Ι
(3.6)
where I3x3 is the identity matrix,
m
σ
is the standard
deviation of magnetometer errors and
kj
δ
is the
Kronecker symbol.
3.2 Sun Sensor Measurement Model
In order to construct a sun sensor model, a sun
direction vector is used. Using VSOP87 theory, a
direction cosine matrix is calculated which shows
the sun’s position relative to Earth in ECI frame [6]
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x 1
y 2
3
z
B E
B E s
E
B
s s
s A s
s
s
= + η
(3.7)
Construction of the sun direction vector,
E
s
requires
two assumptions.Comparing the distance between
Sun-Earth, 1 AU, and Earth-satellite, satellite
altitude is negligible. Therefore the satellite’s sun
direction vector is always parallel to Earth’s sun
direction vector. The other assumption is taking the
right ascension node of the Sun’s orbit as zero.
Model is using Julian Date (JD). The reference
epoch of the first order model is the January 1st
2000, 12:00:00 pm. Converting this date to JD
would be 2451545. Satellite’s epoch is selected as
March 16th 2017, 22:46:22. The first step of
calculating the direction vector is to find the mean
anomaly of the Sun. All of the angles that are given
below are in degrees.
UTC
TDB
JD 2451545
T
36525
−
=
(3.8)
Sun TDB
M 357.5277233 35999.05034T
= ° +
(3.9)
Secondly, the ecliptic longitude of the Sun,
ecliptic
λ
is
calculated.
)
)
Sun
ecliptic M Sun Sun
1.914666471sin M 0.019994643sin 2Mλ = λ + +
(3.10)
where
Sun
M
λ
is the mean longitude of the sun. It can
be calculated with the equation below.
Sun
M sat
280.460 36000.770T
λ = +
(3.11)
Initially, Tsat is satellite’s epoch. It will increase by
one second until satellite reach at the end of its first
period. Lastly, the angle between Earth’s orbit and
equator planes, obliquity of the ecliptic needs to be
calculated.
TDB
23.439291 0.0130042T
ε = −
(3.12)
The sun direction vector,
ecliptic
E ecliptic
ecliptic
cos
s sin cos
sin sin
λ
= λ ε
λ ε
(3.13)
The noise term,
s
,
η
is added linearly to the model
and modeled as zero mean Gaussian white noise
with the characteristic of
T 2
1k 1j 3x3 s kj
E
η η = σ δ
Ι
(3.14)
where I3x3 is the identity matrix,
s
σ
is the standard
deviation of sun sensor errors and
kj
δ
is the
Kronecker symbol.
3.3 Gyroscope Measurement Model
The gyro model is constructed with satellite
dynamic equations. A commonly used model for
gyro measurements given by
BI
BI g g
b
ω = ω + + η
(3.15)
where
g
b
is the gyro bias vector and the
g
η
is
modeled as zero mean Gaussian white noise with
the characteristic of
T 2
1k 1j 3x3 g kj
E
η η = σ δ
Ι
(3.16)
where I3x3 is the identity matrix,
g
σ
is the standard
deviation of gyro errors and
kj
δ
is the Kronecker
symbol. Bias vector is given as
g b
b
= η
ɺ
(3.17)
where
b
η
is also the zero mean Gaussian white
noise with the characteristic of
2
3 3
.
T
bk bj x gb kj
E I
η η σ δ
=
(3.18)
Here,
gb
σ
is the standard deviation of gyro biases.
4 Filter Design
In this section, the design methods that are
used for the attitude determination process are
given. Initially, a deterministic attitude
determination process, the algebraic method, also
known as TRIAD method, is given. The outputs of
the TRIAD method are then used in extended
Kalman filters [7].
After each subsection, results are presented. For
simulation, an imaginary satellite is selected. Its
orbital attributes are given in Table 1.
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Table 1 : Orbital Parameters.
RAAN
(deg)
Argument
of
Perigee
(deg)
Inclination
(deg)
Altitude
(km)
310.94 207.4 97.65 400
Using (2.29)-(2.31) and (2.21), satellite’s
mathematical model is constructed. The satellite’s
inertia matrix is given below.
3
3
3
2.1 10 0 0
J 0 2 10 0
0 0 1.9 10
−
−
−
×
= ×
×
(4.1)
4.1 TRIAD Method
A rotation matrix describes the attitude of a
spacecraft with respect to a known reference frame.
It takes at least two measured vectors to determine
the orientation of the vehicle. This directed cosine
matrix has nine elements but only three quantities
are sufficient to build the matrix. Therefore, with
two measurements, four different quantities are
obtained. This leads the problem to be
overdetermined.The TRIAD algorithm discards the
part of the measurements so that a solution can be
found [2].
The algebraic method constructs two triads of
orthonormal vectors. In this thesis, two triads are
expressed by sun sensor and magnetometer unit
vectors in body and reference frames. The
magnetometer measurement vector is denoted by B
and the sun sensor vector is denoted by S. For initial
base vector, a more accurate sensor is selected to be
exact.
u S
=
(4.2)
b b
u S
=
(4.3)
r r
u S
=
(4.4)
b and r subscripts denote the body and reference
frames. For the second base vector, a unit vector
that is perpendicular to the first base vector is
constructed.
v S B
= ×
(4.5)
b b
b
b b
S B
v
S B
×
=×
(4.6)
r r
r
r r
S B
v
S B
×
=×
(4.7)
And the final base vector to complete the triad,
w u v
= ×
(4.8)
b b b
w u v
= ×
(4.9)
r r r
w u v
= ×
(4.10)
Three base vectors form a complete orthogonal
coordinate system. It is important to note that two
vectors, S and B can not be parallel,
S.B 1
<
.Constructing the direction cosine matrix,
[
]
[
]
T
br
b b b r r r
A u v w u v w=
(4.11)
Equation (4.11) results in a 3x3 matrix. Using
equations (2.3)-(2.5), body angles can be obtained.
Final step of the algebraic method is to find
covariance of the algorithm. Calculating with the
formula given below, covariance matrix is
established [4],
2 T 2 T
2 T
m b b s b b
s b b
S S B B
P v v
S B
σ + σ
= + σ
×
(4.12)
Covariance and body angles that are obtained from
the TRIAD algorithm are used as measurement
inputs to the kalman filter. These results are
combined with gyro measurements in order to
achieve better accuracy.
4.2 Extended Kalman Filter
In this section, the traditional EKF, which is based
on the nonlinear measurements, is introduced. The
satellite’s rotational motion about its mass center is
formulated using the discrete-time nonlinear state
space model
(
)
(
)
(
)
1
x k f x k w k
+ = +
(4.13
)
[
]
( ) ( ) ( )
z k h x k v k
= +
(4.14)
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where
[
]
f
⋅
and
[
]
h
⋅
are the nonlinear dynamic and
measurement functions respectively,
( )
x k
is the
n
dimensional state vector at time
k
,
(
)
w k
is the
zero-mean Gaussian noise with covariance
(
)
Q k
,
( )
z k
is the
d
dimensional measurement vector,
(
)
v k
is the zero-mean Gaussian noise with
covariance
(
)
R k
. It is assumed that both noise
vectors
(
)
w k
and
(
)
v k
are linearly additive
Gaussian, temporally uncorrelated with zero
mean,
[
]
[
]
( ) ( ) 0,
E w k E v k k
= = ∀
(4.15)
Filter algorithms based on the described system and
measurements in (4.13)- (4.14) can be given. The
approximations in the prediction and update stages
of the filter can be found based on the EKF. The
estimation value can be found as,
[
]
}
ˆ ˆ ˆ
( 1) ( 1/ ) ( 1) ( 1) ( 1/ )
x k x k k K k z k h x k k
+ = + + + × + − +
(4.16)
The extrapolation value of the dynamic function can
be found as
[
]
ˆ ˆ
( 1 / ) ( )
x k k f x k
+ =
(4.17)
Filter-gain of the EKF is,
[ ]
1
( 1) ( 1 / ) ( 1) ( 1) ( 1 / ) ( 1) ( )
T T
K k P k k H k H k P k k H k R k
−
+ = + + × + + + +
(4.18)
where
ˆ
[ ( 1 / )]
( 1) ˆ
( 1 / )
h x k k
H k
x k k
∂ +
+ = ∂ +
is the partial
derivatives of the measurement function with
respect to the states.
The covariance matrix of the extrapolation error
is,
ˆ ˆ
[ ( )] [ ( )]
( 1/ ) ( / ) ( )
ˆ ˆ
( ) ( )
T
f x k f x k
P k k P k k Q k
x k x k
∂ ∂
+ = × +
∂ ∂
(4.19)
The covariance matrix of the filtering error is,
[
]
( 1 / 1) ( 1) ( 1) ( 1 / )
P k k I K k H k P k k
+ + = − + + +
(4.20)
The filter expressed by the formulas (4.17) - (4.20)
is called the extended Kalman filter.
The state vector considered in this study is,
x y z x y z
T
x y z m m m g g g
x b b b b b b= φ θ ψ ω ω ω
(4.21)
Body angles from the TRIAD algorithm, angular
velocities from gyros and magnetic field
measurements from magnetometers are taken as
measurement vector. 9-state measurement vector is
given below
T
x y z x y z
z B B B
= φ θ ψ ω ω ω
5 Simulations
A 12-state EKF is used for attitude estimation and
magnetometer and gyro bias calibration. Orbit of the
satellite is chosen as circular with an altitude of 550
km. Other orbital parameters are given in the
magnetometer measurement model section.
For magnetometer measurements, sensor noise is
characterized by zero mean Gaussian white noise
with standard deviation of σm = 0.006.
Magnetometers are calibrated in-flight by using
biases estimated by EKF. Sun sensors are calibrated
against sensor biases. Sensor noise is also
characterized by zero means Gaussian white noise
with standard deviation of σs = 0.002. Gyroscope
m
easure
ment
s are also calibrated in-flight.
Gyroscope noise standard deviation is σg = 0.005.
Simulations are conducted for 5910 seconds with
sampling time of 1 sec. Body angles, angular
velocities and sensor biases are estimated. Estimated
biases are fed back to sensor measurements for in-
flight calibration. Hence, calibrated sensor
measurements are used in both TRIAD and EKF. In
Figures 1, Figure 2, Figure 3 and Figure 4,
estimation results are given.
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Figure 1: Body angle estimations
Figure 2: Angular velocity estimations
Figure 3: Magnetometer bias estimations
Figure 4: Gyroscope bias estimations
As can be seen in all of the graphs, there are some
spikes and sudden fluctuations approximately in the
same places. These anomalies are caused by high
body angles. When yaw, pitch or roll angle goes to
their respective boundaries, 180 degrees for yaw and
roll, 90 degrees for pitch angle, The TRIAD
algorithm starts to fail and it leads EKF to the same
behaviour. In roll angle estimation, it can be seen
that when the angle is not close to its limits, both
TRIAD and EKF estimate decently.
6 Conclusion
In this study, magnetometer and sun sensor
measurements are fused together with the TRIAD
algorithm to produce body angle estimation.
Combining with gyroscopes measurements, an
Extended Kalman Filter is used to estimate body
angles, angular velocities, gyroscope and
magnetometer biases.
The TRIAD method, even though it is an aging
algorithm, can estimate satellite altitude well. The
sun sensor and magnetometer are selected for inputs
to the TRIAD algorithm because of their wide usage
in the space industry. Many different filtering
algorithms are presented to this day but proven
algorithms are still getting attention from engineers.
EKF proves itself on many missions. Satellite
dynamic and kinematic equations show that all of
the states are connected. The simulation results
Pitch (deg) Yaw (deg)Roll (deg)
Variance
bxby
bz
Variance
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show that, the proposed TRIAD-aided EKF
algorithm estimates the attitude, attitude rate and
magnetometer and gyroscope biases decently
[1] J. Wertz, Spacecraft Attitude Determination
and Control. Kluwer Academic Publishers,
Dordrecht.
[2] H.D. Black. "A passive system for
determining the attitude of a satellite" AIAA
Journal 7 (1964): 1350-1351
[3] C. Hajiyev and M. Bahar, “Increase of
accuracy of the small satellite attitude
determination using redundancy
techniques,” Acta Astronaut., vol. 50, no.
11, pp. 673–679, 2002.
[4] F.L. Markley and J.L. Crassidis,
Fundamentals of spacecraft attitude
determination and control, Springer New
York, New York, NY, 2014.
[5] D. C. Guler and C. Hajiyev, Gyroless
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.25
Mehmet Asim Gokcay, Chingiz Hajiyev
E-ISSN: 2224-3429
214
Volume 17, 2022
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