Attitude Determination of a Nanosatellite Using a Magnetometer
Hyunsam Myung, Hyochoong Bang, Choongsuk Oh, Min-Jea Tahk Korea Advanced Institute of Science and Technology (KAIST)
373-1, Guseong, Yuseong, Deajeon, Korea
E-mail: {hsmyung, csoh, hcbang, mjtahk} ＠fdcl.kaist.ac.kr
Abstract
The objective of this paper is to simulate attitude determination of a nanosatellite using a magnetometer data assuming that the satellite is at the designed orbital motion.
From the satellite being assumed to be in its orbit, the assumed measurement data of the three-axis magnetometer is obtained. They contain the reference measurement of the Earth’s magnetic field which can be translated reference attitude information of the satellite and that of measurement uncertainties of the magnetometer. The reference measurement data is made from the assumed attitude angles during its operation. Obtained measurement data are analyzed using TRIAD and QUEST (QUaternion ESTimator) algorithm, and then the satellite attitude information in the forma of Euler angles is estimated. In order to produce the measurement data, the Earth’s magnetic field is simulated earlier. Simulations and the performance examination based upon the proposed algorithm are conducted.
1 Introduction
The attitude estimation by measuring two (or more) vectors fixed in the inertial coordinate is a frequently used method in attitude determination. The measuring properties could be the Sun vectors by a sun sensor, the star orientation by a star sensor or the Earth’s magnetic field vectors by a magnetometer. The last case of these observations is considered as being quite attractive if the system to be applied is a nano-satellite with less than 10 kg weight, so all the subsystem should be even lighter. In spite of the relatively low accuracy in measuring the Earth’s magnetic field, the low cost and small weight of a magnetometer make it possible to implement an attitude determination system for the nano-satellite [4].
There are representative attitude determination algorithms such as TRIAD and QUEST using vector observations for the Earth’s magnetic field. Theoretically, only two vectors can be used in the TRIAD algorithm to produce deterministic solution. That is, it is not optimal, but the algebraic calculation is very simple. On the other hand, the QUEST algorithm is based on minimizing a least-square loss function proposed by Wahba [3]. The attitude information is obtained from the orthogonal matrix minimizing the cost function. In this algorithm, the multiple numbers of measurement vectors can be
considered to enhance accuracy of the attitude estimations.
Many researches have been in quaternion attitude estimation [2,3], this study is limited to using Euler angles.
2 The Earth’s Magnetic Field
The Earth’s magnetic field, B, is assumed as a spherical harmonic model [6]. B is represented as a gradient of a scalar potential function V.
= ∇V
B (1) The potential function V is expressed by harmonic functions as [6]
( )
1
1 0
( , , ) cos sin ( )
k n n
m m
n n n
n m
V r a a g m h m P
r
θ φ + φ m
= =
=
∑
∑
+ φ θ(2) where a is the Earth’s equatorial radius, and gnm and are Gaussian coefficients, which are determined empirically. The radius a is 6371.2km and Gaussian coefficients are given from the International Geomagnetic Reference Field (IGRF) for Epoch 2000. Furthermore, r,
m
hn
θ, and φ are geocentric distance, coelevation, and East longitude from Greenwich, respectively. Schmidt functions
m(
Pn θ) can be represented in terms of Gauss functions. [6]
, , nm n m
P =S Pn m (3) Since the factors Sn m, are not functions of r, θ , and φ , they can be calculated with Gaussian coefficients such as
, ,
, ,
n m m
n m n
n m m
n m n
g S g
h S h
=
= (4) And then, the factors and Gauss functions are written in recursive form as follows. [6]
,
Sn m
0,0
,0 1,0
1
, , 1
1
2 1 1
( 1)( 1) 1
n n
n m n m m
S
S S n n
n
S S n m m
n m δ
−
−
=
−
= ≥
− + +
= ≥
+
(5) The 4th Asian Control Conference
September 25-27, 2002
Singapore
FA4-2
0,0
, 1, 1
, 1, ,
1 sin
cos
n n n n
n m n m n m n m
P
P P
P P K P
θ θ
− −
− −
=
=
= − 2,
j
(6)
where Kronecker delta, if and otherwise, and [6]
j 1
δi = i= 0
2 2
,
,
( 1) 1
(2 1)(2 3) 0
n m
n m
n m
K n n
K
− −
≡ >
− −
≡ =1
n n
(7)
The partial derivatives of Pn m, ( )θ satisfy [6]
0,0
, 1, 1
1, 1
, 1,
1, ,
0
sin cos 1
cos sin
n n n n
n n
n m n m n m
n m n m
P
P P P n
P P P K P
θ
θ θ
θ θ
θ θ
θ θ
− −
− −
− −
∂ =
∂
∂ = ∂ +
∂ ∂
∂ = ∂ − − ∂
∂ ∂
2,
θ
−
≥
∂
(8)
The Earth’s magnetic field in spherical coordinates is represented as [6]
( )
( )
(
2
, ,
1 0
2 ,
, ,
1 0
2
, , ,
1 0
( 1) cos sin (
cos sin ( )
sin cos ( )
k n n
n m n m n m
r n m
n n m
k n
n m n m
n m
k n n
n m n m n m
n m
B a n g m h m P
r
a P
B g m h m
r
B a m g m h m P
r
θ
φ
)
, )
φ φ θ
φ φ θ
θ
φ φ
+
= =
+
= =
+
= =
= + +
∂
= − + ∂
= − +
∑ ∑
∑ ∑
∑ ∑
θ(9) 3 Satellite Orbit
In this paper, the system under study is a nano-satellite which is assumed to be operational in 600 km altitude polar orbit. Table 1 shows the key orbital parameters of the satellite. The epoch is March 21st 2002 00:00:00.
Table 1. Satellite orbit elements
Eccentricity 0.001129 RAAN 23.273 deg Semi-major
axis 6978.14 km Argument of
perigee 86.329 deg True
by using the given orbit elements.
4 Attitude Determination Algorithms The representative method in the attitude determination is to use vector observation. Let’s assume the sensor is aligned with the body axes. Then a three-axis magnetometer will provide the Earth’s magnetic field vectors in the spacecraft body frame. There are algorithms relatively simple but still useful algorithms such as TRIAD and QUEST applicable to the nano-satellite,. The former is direct algebraic method creating orthogonal unit vectors using the cross product of the two measurement vectors, and the latter estimates the optimized quaternion from the multiple observations vectors. These two algorithms are investigated in the attitude determination of the satellite model.
4. 1 TRIAD algorithm
First, review on TRIAD algorithm is presented in this section. Two vectors, W and are observed from the magnetic field in the body frame. If the position information is obtained, the two reference unit vectors, V and V , can be found. From these vectors, two orthogonal coordinate frames are calculated as [5]
1 W2
1 2
1 1 1 1
2 1 2 1 2 2 1 2 1 2
3 1 2 3 1 2
/ /
s W r V
s s W s W r r V r V
s s s r r r
= =
= × × = × ×
= × = ×
(10)
The orthogonal unit vectors obtained in Eq. (10) formulate the orthonormal matrices presented in each coordinate frame [5].
1 2 3
Mobs= s s s , Mref = r1 r2 r3 (11) The two matrices have the following relationship with the direction cosine matrix A [5].
ref obs
A M =M (12) or
obs refT
A M= M (13) The Euler angles between the reference frame and the body frame can be obtained in a straightforward manner from A.
4. 2 QUEST algorithm
where a is the weighting parameter for i-th measurement vector. The goal of the QUEST algorithm is to find the optimal A minimizing the cost function in Eq. (14).
i
The direction cosine matrix A is parameterized with the quaternion q=[Q q4]T as
42 4
( ) ( ) 2 T 2
A q = q − ⋅Q Q I+ QQ − q ×Q (15) Then, Eq. (14) is rewritten as [1]
1
( ) 1 ( ) / 1 2
k i i
J q J q a
=
′ = − ∑ (16) That is, J q′( ) is maximized for the optimally estimated q.
Rearranging the problem, Eq. (16) is expressed by [1] Fig. 1. The Earth's magnetic field in the orbit
max ( ) T 1
q J q′ =q Kq under q qT = (17) where
( )
1 1 1
1
1 1
, ,
, 1 ,
k k k
k i i iT i i i i
i ki ki
T k
i i i T
ki
m a a W V B a W V
m m
S I Z
S B B Z a W V K
m Z
σ
σ
T
σ
= = =
=
= = =
−
= + = × =
∑ ∑ ∑
∑
(18)
The Earth’s magnetic field is simulated using Eqns. from (4) to (9). The total simulation duration is three times of the orbit period, about 5 hours. Fig. 1 shows the Earth’s magnetic field in the orbit with magnified images.
The variation of the magnetic field in accordance with simulation time in spherical coordinates is presented in Figs. 2. and 3, respectively.
In Eq. (17), the optimal quaternion is the eigenvector q maximizing . In other words, q is the eigenvector corresponding to the maximum eigenvalue of K [1].
qopt T opt
q K q
K qopt=λmaxqopt (19) Introducing the Gibbs vector as
/ ˆtan( / Y=Q q n= θ 2)
S Z
(20) one can extend Eq. (19) into
1
[( max ) ]
Y= λ +σ I− − (21) Consequently qopt can be expressed in terms of Y [1].
Fig. 2. B field in the orbit in spherical coordinates.
2
1 1 1
opt
q Y
Y
=
+ (22)
Fig. 3 shows the magnitude of the Earth’s magnetic field corresponding to the orbit position, geocentric distance, coelevation and longitude. The magnitude is maximized near the north and the south poles. The maximum value of the magnetic field in the orbit is about 5 10 nT× 4 or 0.5G. 5 Simulation Studies
5. 1 The Earth’s magnetic field simulations
Table 2. RMSE of Euler angle for each case RMSE of
Euler angle (rad)
RMSE of Euler angle
(rad) TRIAD 2.5446 10× −2 QUEST4 7.7398 10× −3 QUEST2 2.5440 10× −2 QUSET5 5.5632 10× −3 QUSET3 1.2452 10× −2
Simulation results of TRIAD, QUEST2 and QUEST5 are presented in Fig. 4 over the specified simulation time.
QUEST5 shows relatively very good performance. The overall accuracy of the attitude determination by using the magnetometer data with certain level of uncertainty introduced seems to be satisfactory. However, it is a fairly well accepted fact that the magnetometer provides about few degrees in the attitude error.
Fig. 3. Magnitude of the B field corresponding to the orbit position.
5. 2 Simulations of the attitude determination
It is assumed that the sensor frame aligned with the magnetometer is identical to the body fame. Therefore the magnetometer data are represented in the Cartesian coordinates of the body frame. The reference attitude orientation of the body frame is assumed constant throughout the simulation time as
3 6 4
T π π π T
Φ Θ Ψ =
(rad) (23) These Euler angles are those for 3-2-1 transformation from the reference Earth Centered Inertial (ECI) frame to the body frame. The measurement data in the body frame are perturbed with random noise, which is normally distributed with zero mean, and whose standard deviation is 0.1% of the reference B field data.
Two algorithms, TRIAD and QUEST, are tested through simulation, and the number of the measurement vectors for QUEST algorithm is varied from 2 to 5. Corresponding simulation result is denoted as TRIAD, QUEST2, QUEST3, and so on. The weighting parameters for the measurement vectors in the QUEST algorithm are all assumed unity.
Simulation results for the attitude determination of the body frame are presented in Table 2. The values are the mean values of the estimated Euler angles from 5 random simulations. It can be shown that the more the measurement vectors, the higher the accuracy results.
Fig. 4. Attitude determination using TRIAD, QUEST2, and QUEST5.
As mentioned earlier, the magnetometer is an attractive choice as a three-axis attitude determination sensor for nano-satellite missions. Therefore, the results in this paper imply the practical merit derived from implementing the magnetometer in conjunction with associated algorithm for
measurement vectors is also investigated. As a further study, the attitude determination for slowly time varying reference and the selection of weighting factors need to be considered. Also, the algorithm tested in this study may be implemented in the actual nano-satellite mission after additional enhancement and modification. The inherent reliability and cost-effectiveness of the magnetometer- based three-axis attitude determination provides plenty of room for continuing research and applications for the future nano-satellite missions.
This work was supported by grant No. 2000-1-30500- 002-3 from the Basic Research Program of the Korea Science and Engineering Foundation.
References
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[2] L. Y. Bar-Itzhack, and Y. Oshman, “Attitude determination from vector observations: Quaternion estimation”, IEEE Transactions on Aerospace and Electronic Systems, 21 (1), pp.128-135, 1985.
[3] L. F. Markley, and D. Mortari, "Quaternion Attitude estimation using vector observations", The Journal of the Astronautical Sciences, 48 (2/3), pp.359-380, 2000.
[4] M. L. Psiaki, F. Martel, and P. K. Pal, “Three-axis attitude determination via Kalman filtering of magnetometer data”, Journal of Guidance, 13 (3), pp.506-514, 1990.
[5] M. D. Shuster and S. D. Oh, “Three-axis attitude determination from vector observation”, Journal of Guidance and Control, 4 (1), pp.70-77, 1981.
[6] J. R. Wertz (Ed.), “Spacecraft attitude determination and control”, Reidel, Boston, 1978.
