S. SERIANI, Y.L. BRAMA, P. GALLINA, G. MANZONI
27 JANUARY 2016
In-Orbit Offline Estimation of the Residual Magnetic Dipole Biases
of the POPSAT-HIP1 Nanosatellite a
S. Seriani
Lead and Corresponding author
Dept. of Architecture and Engineering, University of Trieste, via A. Valerio 10, 34127, Trieste, Italy.
e-mail: stefano.seriani@phd.units.it
phone: +39 040 5582540
Y.L. Brama
Microspace Rapid Pte Ltd
e-mail: yesie.l.brama@micro-space.org
P. Gallina
Dept. of Architecture and Engineering, University of Trieste, via A. Valerio 10, 34127, Trieste, Italy.
e-mail: pgallina@units.it
G. Manzoni
Microspace Rapid Pte Ltd
e-mail: giulio.manzoni@micro-space.org
ABSTRACT — The nanosatellite POPSAT-HIP1 is a Cubesat-class spacecraft launched on the 19th of June 2014 to test cold-gas based micro-thrusters; it
is, as of April 2015, in a low Earth orbit at around 600km of altitude and is equipped, notably, with a magnetometer. In order to increment the
performance of the attitude control of nanosatellites like POPSAT, it is extremely useful to determine the main biases that act on the magnetometer
while in orbit, for example those generated by the residual magnetic moment of the satellite itself and those originating from the transmitter.
Thus, we present a methodology to perform an in-orbit offline estimation of the magnetometer bias caused by the residual magnetic moment of the
satellite (we refer to this as the residual magnetic dipole bias, or RMDB). The method is based on a genetic algorithm coupled with a simplex
algorithm, and provides the bias RMDB vector as output, requiring solely the magnetometer readings. This is exploited to compute the transmitter
magnetic dipole bias (TMDB), by comparing the computed RMDB with the transmitter operating and idling.
An experimental investigation is carried out by acquiring the magnetometer outputs in different phases of the spacecraft life (stabilized, maneuvering,
free tumble). Results show remarkable accuracy with an RMDB orientation error between 3.6° and 6.2°, and a module error around 7%. TMDB values
show similar coherence values.
Finally, we note some drawbacks of the methodologies, as well as some possible improvements, e.g. precise transmitter operations logging. In general,
however, the methodology proves to be quite effective even with sparse and noisy data, and promises to be incisive in the improvement of attitude
control systems.
1
Introduction
The field of miniaturized satellites has been growing exponentially since the 2000s; lightweight structures and payloads,
possible through the most recent miniaturization technologies, have allowed for more and more compact and light
spacecraft designs. Because of this, in the last couple of years, sending a nano- or micro-satellite has become affordable
for many small companies and research institutes.
As with all small satellites, complex and redundant attitude instrumentation is often impractical due to its weight. Cheap
miniaturized sensors (e.g. MEMS or systems-on-a-chip), on the other hand, offer generally sub-optimal results;
furthermore, non-redundancy contributes to generally poorly precise measurements of the satellite attitude. This is
especially important when designing the attitude control system [1]. In particular, the magnetometer readings are often
disrupted because of the magnetic field produced by the spacecraft subsystems; this leads to a deviation from the nominal
value of the magnetic field. On top of this, for Low Earth Orbit (LEO) objects, the magnetic field influence dominates
a
Abbreviations:
MEMS – Micro Electro-Mechanical Systems
KF – Kalman Filter
UKF – Unscented Kalman Filter
EKF – Extended Kalman Filter
RMDB – Residual Magnetic Dipole Bias
TMDB – Transmitter Magnetic Dipole Bias
DE – Differential Evolution
ECI – Earth-Centered Inertial
LCI – Local-Centered Inertial
TLE – Two-Line Elements
IGRF – International Geomagnetic Reference Field
IN-ORBIT OFFLINE ESTIMATION OF THE RESIDUAL AND TRANSMITTER MAGNETIC DIPOLE BIASES OF THE POPSAT-HIP1 NANOSATELLITE
ACTA ASTRONAUTICA
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S. SERIANI, Y.L. BRAMA, P. GALLINA, G. MANZONI
27 JANUARY 2016
over other environmental effects like atmospheric drag or solar radiation pressure. Generally, this is addressed using
dynamics filtering techniques, like the Kalman Filter (KF); Farahanifar et al. [2] develop an attitude estimation technique
using an Unscented KF (UKF) coupled to a dynamics model of the spacecraft to determine and thus compensate the
magnetometer bias and scale factors. Along the same lines, an Extended KF (EKF) is described by Inamori et al. in [3],
for use in nano-satellite missions. Gebre-Egziabher et al. describe, in [4], an algorithm for the calibration of
magnetometers based on ellipsoid fitting; Wu et al. present a methodology [5] based on particle swarm optimization to
the same objective. Similarly, Springmann et al. in [6] use a least squares regression method to estimate the residual
magnetic dipole moment of the Radio Aurora Explorer.
POPSAT-HIP1 is an experimental nanosatellite of the Cubesat class that has been developed by Microspace Rapid Pte
Ltd to test in space a micro-propulsion system based on cold-gas micro thrusters [7]. The satellite has been launched on
the 19th of June 2014 and has been active in Low Earth Orbit (LEO) since then, at an altitude of approximately 600km.
The basic attitude control is performed by means of magnetorquers while the attitude is measured by using a
magnetometer and six Sun sensors placed on the satellite surfaces as well as by three single axis gyroscopes. Attitude
data can be stored in sequences of 60 points with intervals variable from few seconds to a maximum of 4 minutes and are
downloaded during the satellite passes. Since the attitude control is mainly based on the magnetometer output, being able
to determine its biases can substantially improve the performance of the control system. For this purpose, in this paper we
present an estimation of the Residual Magnetic Dipole Bias RMDB for its possible consideration in the correction of the
attitude control algorithm. Additionally, we note that the RMDB appears to be greatly altered during the activation of the
transmitter. This is supported by the evidence of a strong bias in the attitude data downloaded from the satellite during
transmission events. The origin of this effect may be traced down to the high currents needed by this sub-system, which
is one of the most power-demanding of the entire satellite. By comparing the RMDB value when the transmitter is active
to that when it is non-active, we extrapolate the effect of the transmitter dipole alone. Hence, we will refer to the
generated dipole with the term transmitter magnetic dipole bias (TMDB).
In order to determine the effect on the magnetometer readings of the RMDB and thus of the TMDB on POPSAT, we
develop an algorithm that takes advantage of a differential evolution (DE) algorithm coupled with a direct search method;
specifically a genetic algorithm and the simplex method. Contrary to many of the cited, our approach does not take into
account the dynamics of the satellite, thus not needing attitude computation.
The determination of the RMDB and TMDB allows compensating the disturbances in the magnetometer readings, thus
leading to a better magnetic model of the spacecraft, which can then be coupled to one of the well-established KF
technique, in a feed-forward loop. Especially in the case of transmission events, the attitude control system could easily
take advantage of the TMDB value to counteract, in real-time, the moment caused by the active transmitter sub-system,
thus reducing the need for the control-loop to stabilize. This reasoning can be extended to virtually every sub-system in
the spacecraft.
In Section 2 the methodology is outlined, covering the model, the parameters used in the analysis and providing the
validation. In Section 3 the experimental results are reported, and the actual analysis is performed and discussed. Finally
in the Conclusion we summarize the work, stress the strong points of the method and provide insight in future practical
improvements and applications.
2
Methodology
Let us define the local 3D frame of reference 𝑅𝐿𝑂𝐶 = [𝒙𝐿𝑂𝐶 , 𝒚𝐿𝑂𝐶 , 𝒛𝐿𝑂𝐶 ], with 𝒙𝐿𝑂𝐶 , 𝒚𝐿𝑂𝐶 , 𝒛𝐿𝑂𝐶 ∈ ℝ3 , as fixed to the
frame of the satellite, and centred in the magnetometer location; please consider Fig. 1 as reference. Furthermore, let us
define the Earth Centred Inertial (𝑅𝐸𝐶𝐼 ) 3D frame of reference; this is centred in the Earth barycentre, with its x-axis lying
in the direction of the vernal equinox and the z-axis oriented along the rotation axis. We then define another frame of
reference, the Local Centred Inertial (𝑅𝐿𝐶𝐼 ), which is the translation of 𝑅𝐸𝐶𝐼 in the origin of 𝑅𝐿𝑂𝐶 .
The rotational transformation between the frames of reference 𝑅𝐿𝑂𝐶 and 𝑅𝐿𝐶𝐼 is given by the attitude quaternion 𝒒𝑎𝑡𝑡 , as
described in [8,9]. If a general vector 𝑼𝐿𝑂𝐶 ∈ ℝ𝟑 is defined in 𝑅𝐿𝑂𝐶 , then the same vector in the 𝐿𝐶𝐼 frame of reference is
given by the following,
∗
𝑼𝐿𝐶𝐼 = 𝒒𝑎𝑡𝑡
𝑼𝐿𝑂𝐶 𝒒𝑎𝑡𝑡 ,
(1)
∗
where 𝒒𝑎𝑡𝑡
is the conjugate of 𝒒𝑎𝑡𝑡 .
The inverse operation is not so straightforward; finding the attitude quaternion can prove difficult, since, given the two
vectors 𝑼𝐿𝐶𝐼 and 𝑼𝐿𝑂𝐶 , one can find an infinite number of quaternions 𝒒𝑎𝑡𝑡 that hold Eq. 1 as true. In order to eliminate
this uncertainty, another vector 𝑽𝐿𝑂𝐶 ∈ ℝ𝟑 should be considered, along with its counterpart 𝑽𝐿𝐶𝐼 ; the following ensues,
IN-ORBIT OFFLINE ESTIMATION OF THE RESIDUAL AND TRANSMITTER MAGNETIC DIPOLE BIASES OF THE POPSAT-HIP1 NANOSATELLITE
ACTA ASTRONAUTICA
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S. SERIANI, Y.L. BRAMA, P. GALLINA, G. MANZONI
𝑼𝐿𝐶𝐼
𝑽𝐿𝐶𝐼
{
𝑼𝐿𝑂𝐶 × 𝑽𝐿𝑂𝐶
𝑼𝐿𝐶𝐼 × 𝑽𝐿𝐶𝐼
27 JANUARY 2016
= 𝒒∗𝑎𝑡𝑡 𝑼𝐿𝑂𝐶 𝒒𝑎𝑡𝑡
= 𝒒∗𝑎𝑡𝑡 𝑽𝐿𝑂𝐶 𝒒𝑎𝑡𝑡
,
≠0
≠0
(2)
that is, for each and every pair of vectors 𝑼 and 𝑽 which are not co-linear, only one quaternion 𝒒𝑎𝑡𝑡 exists which holds to
Eq. 2. The determination of this quaternion is carried out by Singular Value Decomposition (SVD), as described in [10].
This leads to the following,
𝑼𝐿𝑂𝐶 , 𝑼𝐿𝐶𝐼 , 𝑽𝐿𝑂𝐶 , 𝑽𝐿𝐶𝐼 →
𝑆𝑉𝐷
𝒒𝑎𝑡𝑡 ,
(3)
which is univocal when,
𝑼𝐿𝑂𝐶 ∙ 𝑽𝐿𝑂𝐶 = 𝑼𝐿𝐶𝐼 ∙ 𝑽𝐿𝐶𝐼 .
(4)
Fig. 1. Main definitions of the frame of references and vectors. In a) the 𝑅𝐸𝐶𝐼 and 𝑅𝐿𝐶𝐼 frames of reference are shown, along with the origin 𝑶𝐸𝐶𝐼 , and
the position 𝑷𝐸𝐶𝐼 of the satellite. In b) the 𝑅𝐿𝑂𝐶 and 𝑅𝐿𝐶𝐼 frames of reference are shown, along with the magnetic field vectors as measured in the
POPSAT local frame of reference. The origin of the axis coincides with the magnetometer.
These concepts can be applied to the problem of attitude determination of a satellite, when at least a pair of vectors can
be defined in 𝑅𝐿𝑂𝐶 and 𝑅𝐿𝐶𝐼 . In our case, as stated in Section 1, two 3-axial sensors are available, a magnetometer, and a
solar-incidence sensor; these provide two vectors, relative to 𝑅𝐿𝑂𝐶 , that is, the raw – or measured – magnetic field vector
𝑩′𝐿𝑂𝐶 (𝑡), and the solar incidence vector 𝑺𝐿𝑂𝐶 (𝑡). The dependence from time 𝑡 is due to the fact that the satellite moves
and tumbles following a path 𝑃 (which is called its orbit) thus continually changing both its orientation and position
relative to the Earth’s magnetic field and the Sun. Let us indicate with 𝑩𝑑,𝐿𝑂𝐶 (𝑡) a variable disturbance vector which also
contains the RMDB; in fact, in our work, we assume that the time-invariant portion of this disturbance is indeed
coincident to the RMDB. We can then define the pure magnetic field reading 𝑩𝐿𝑂𝐶 as follows,
𝑩𝐿𝑂𝐶 (𝑡) = 𝑩′𝐿𝑂𝐶 (𝑡) − 𝑩𝑑,𝐿𝑂𝐶 (𝑡).
(5)
In order to define the correspondent vectors in the 𝑅𝐿𝐶𝐼 frame of reference, the position of the satellite must be
determined; the SGP4 orbital propagation algorithm [11] is used; the necessary two-line elements (TLE) relative to the
POPSAT nanosatellite are retrieved through the database of the STK software from AGI Inc. (http://www.agi.com) built
from data provided by the NORAD. The relation follows,
𝑡, 𝑇𝐿𝐸𝑃𝑂𝑃𝑆𝐴𝑇 →
𝑆𝐺𝑃4
𝑷𝐸𝐶𝐼 (𝑡),
(6)
The magnetic field vector 𝑩𝐿𝐶𝐼 (𝑡, 𝑷𝐸𝐶𝐼 (𝑡)) is time-dependent, the same being true for the solar incidence vector
𝑺𝐿𝐶𝐼 (𝑡, 𝑷𝐸𝐶𝐼 (𝑡)) = 𝑷𝑆𝑢𝑛,𝐸𝐶𝐼 (𝑡) − 𝑷𝐸𝐶𝐼 (𝑡). The value of the magnetic field 𝑩𝐿𝐶𝐼 is computed using the International
Geomagnetic Reference Field (IGRF) model [12], whereas the geocentric sun vector 𝑷𝑆𝑢𝑛,𝐸𝐶𝐼 is determined from the
precise solar ephemeris as given by Bretagnon [13].
Let us apply the SVD method, described by Eq. 3, in the real world, as follows,
̂𝐿𝑂𝐶 , 𝑺
̂𝐿𝐶𝐼 →
̂ 𝐿𝑂𝐶 , 𝑩
̂ 𝐿𝐶𝐼 , 𝑺
𝑩
𝑆𝑉𝐷
𝒒𝑎𝑡𝑡 ,
IN-ORBIT OFFLINE ESTIMATION OF THE RESIDUAL AND TRANSMITTER MAGNETIC DIPOLE BIASES OF THE POPSAT-HIP1 NANOSATELLITE
ACTA ASTRONAUTICA
(7)
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S. SERIANI, Y.L. BRAMA, P. GALLINA, G. MANZONI
27 JANUARY 2016
where the vectors are all normalized. In this case the condition in Eq. 4 is generally not valid, due to a certain deviation
that exists between the two pairs of vectors in the different frames of reference. This is indeed considered by Eq. 5. The
SVD algorithm copes well with uncertainties of this type.
Suppose, at this point, that an estimated disturbance 𝑩∗𝑑,𝐿𝑂𝐶 (𝑡) is computed and subtracted from 𝑩′𝐿𝑂𝐶 (𝑡), so as to have
𝑩∗𝐿𝑂𝐶 (𝑡) = 𝑩′𝐿𝑂𝐶 (𝑡) − 𝑩∗𝑑,𝐿𝑂𝐶 (𝑡).
(8)
By considering Eq. 5, one can realize that the following is true,
𝑩∗𝑑,𝐿𝑂𝐶 (𝑡) ≅ 𝑩𝑑,𝐿𝑂𝐶 (𝑡) ⟹ 𝑩∗𝐿𝑂𝐶 (𝑡) ≅ 𝑩𝐿𝑂𝐶 (𝑡).
(9)
An extension is then made, in form of the following general assumption,
̃ 𝑑,𝐿𝑂𝐶 = 𝑐𝑜𝑛𝑠𝑡
𝑩𝑑,𝐿𝑂𝐶 (𝑡) ≜ 𝑩
.
∗
̃ ∗𝑑,𝐿𝑂𝐶 = 𝑐𝑜𝑛𝑠𝑡
𝑩𝑑,𝐿𝑂𝐶 (𝑡) ≜ 𝑩
(10)
This is made to account for a disturbance that is predicted to be constant over-time, as the RMDB (or at least a substantial
̃ ∗𝑑,𝐿𝑂𝐶 is considered.
part of it) generally is. A correspondent constant estimate 𝑩
If Eq. 10 holds, then a set 𝕋 can be built, with 𝑡𝑖 = 𝑡1 , … , 𝑡𝑛 ∈ 𝕋 for which the model described in the Equations 5-10,
holds. 𝕋 generates a set of different configurations ℚ of the satellite, which are defined by the attitude quaternions
𝒒𝑎𝑡𝑡, (𝑡𝑖 ).
̃ ∗𝑑,𝐿𝑂𝐶 , common to all the elements of ℚ, can be considered as a minimization problem. In order to
The determination of 𝑩
solve it, a numerical minimization technique can be used, such as a DE algorithm.
Note that for the solution to converge, the configurations ℚ must be well distributed in the rotational sense; indeed, this
allows for a good definition of the topology of the vectors.
Finally, let us consider two distinct sets, ℚ𝑡𝑥𝑂𝐹𝐹 and ℚ𝑡𝑥𝑂𝑁 , the first with an idling transmitter, and the second with
continuous transmission. Since the transmitter is a major source of electromagnetic disturbance, its influence in the
̃ ∗𝑑,𝐿𝑂𝐶 value for the two sets. If
magnetometer readings is non-negligible. This results in the determination of a different 𝑩
we indicate this with the appropriate subscripts, we have that the TMDB is,
̃ ∗𝑡𝑥,𝐿𝑂𝐶 = 𝑩
̃ ∗𝑑,𝐿𝑂𝐶,𝑡𝑥𝑂𝑁 − 𝑩
̃ ∗𝑑,𝐿𝑂𝐶,𝑡𝑥𝑂𝐹𝐹 .
𝑩
2.1
(11)
Genetic Algorithm
̃ 𝑑,𝐿𝑂𝐶 is constant over time (Eq. 10) then the solution to the
If we work under the hypothesis that the disturbance 𝑩
∗
problem of estimating the 𝑩𝑑,𝐿𝑂𝐶
is a minimization problem over a certain time interval 𝑡 = 0, … , 𝑇. The time domain
can be discretized in 𝑡𝑖 = 𝑡1 , … 𝑡𝑛 with 𝑡1 = 0 and 𝑡𝑛 = 𝑇. Following the right-hand term of Eq. 9, the objective function
can be defined in general terms as,
𝜀
𝑇
= ∫0 ‖𝑩∗𝐿𝑂𝐶 (𝑡) − 𝑩𝐿𝑂𝐶 (𝑡)‖ 𝑑𝑡
𝑇
̃ 𝑑,𝐿𝑂𝐶 − 𝑩𝐿𝑂𝐶 (𝑡)‖ 𝑑𝑡
.
= ∫0 ‖𝑩′𝐿𝑂𝐶 (𝑡) − 𝑩
𝑛
∗
′
̃ 𝑑,𝐿𝑂𝐶 − 𝑩𝐿𝑂𝐶 (𝑡𝑖 )‖⁄𝑛)
≅ ∑𝑖 (‖𝑩𝐿𝑂𝐶 (𝑡𝑖 ) − 𝑩
(12)
However, since the term 𝑩𝐿𝑂𝐶 (𝑡) is unknown, and since its computation would require solving from 𝑩𝐿𝐶𝐼 an implicit
problem given by Eq. 5 and 7, an alternative, explicit formulation of the objective function, is as follows,
𝜀̃
𝑇
∗ (𝑡)‖
= ∫0 (‖𝑩𝐿𝑂𝐶
− ‖𝑩𝐿𝐶𝐼 (𝑡)‖) 𝑑𝑡
𝑇
̃ 𝑑,𝐿𝑂𝐶 ‖ − ‖𝑩𝐿𝐶𝐼 (𝑡)‖) 𝑑𝑡
= ∫0 (‖𝑩′𝐿𝑂𝐶 (𝑡) − 𝑩
.
(13)
̃ ∗𝑑,𝐿𝑂𝐶 ‖ − ‖𝑩𝐿𝐶𝐼 (𝑡𝑖 )‖)/𝑛 )
≅ ∑𝑛𝑖 ((‖𝑩′𝐿𝑂𝐶 (𝑡𝑖 ) − 𝑩
This, contrary to Eq. 13, considers the scalar difference between the nominal vector 𝑩𝐿𝐶𝐼 (given by IGRF) and the
̃ ∗𝑑,𝐿𝑂𝐶 . The remarkable advantage of this
magnetometer readings 𝑩′𝐿𝑂𝐶 , purified from the estimated disturbance vector 𝑩
approach is the fact that no attitude computation is necessary, since all the terms in Eq. 12 are readily available or require
only minor computations, in particular Eq. 6 and the IGRF model. Furthermore, because 𝒒𝑎𝑡𝑡 is unnecessary, from Eq. 7
IN-ORBIT OFFLINE ESTIMATION OF THE RESIDUAL AND TRANSMITTER MAGNETIC DIPOLE BIASES OF THE POPSAT-HIP1 NANOSATELLITE
ACTA ASTRONAUTICA
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S. SERIANI, Y.L. BRAMA, P. GALLINA, G. MANZONI
27 JANUARY 2016
one can appreciate that this methodology does not require the knowledge of the solar vectors 𝑺𝐿𝐶𝐼 and 𝑺𝐿𝑂𝐶 , allowing this
to work with the magnetometer-derived data alone.
Finally, the minimization problem statement can be written as,
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
̃ ∗𝑑,𝐿𝑂𝐶
𝑩
̃ ∗𝑑,𝐿𝑂𝐶 , 𝑩𝐿𝐶𝐼 )
𝜀̃(𝑩′𝐿𝑂𝐶 , 𝑩
.
(14)
In order to solve Eq. 14, we employ the mono-objective Genetic Algorithm (GA) described in [14-16], which we couple
with a Simplex algorithm [17] to refine the solution. The parameters used are summarized in Table I.
TABLE I
GENETIC ALGORITHM PARAMETERS
Parameter or method
Value or type
Selection method
Crossover method
Initial population creation method
Mutation function
Crossover fraction
Elites
Fitness relative improvement limit
Population size
Population type
Stochastic uniform
Heuristic
Uniform distribution
Gaussian
0.8
2
10−6 over 500 generations
50
Double
The Simplex algorithm convergence criterion is a relative improvement limit of 10−6 .
2.2
Validation Method
In order to validate the methodology presented in Sec. 2 and Sub-sec. 2.1 we construct an ad-hoc situation for which all
the data relative to the magnetometer readings 𝑩′𝐿𝑂𝐶 , the IGRF reference value 𝑩𝐿𝐶𝐼 and, more importantly, to the
disturbance vector 𝑩𝑑,𝐿𝑂𝐶 (𝑡) are known. Additionally, we superimpose on the readings a random noise error, to account
for magnetic pollution. Operatively, the first step is the definition of a realistic random orbital path 𝑃𝑑𝑒𝑓 and a
corresponding attitude sequence ℚ𝑑𝑒𝑓 defined by the attitude quaternions 𝒒𝑎𝑡𝑡,𝑑𝑒𝑓 . A disturbance vector is then defined
as follows,
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 + 𝑹𝑑𝑒𝑓 (𝑡),
𝑩𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 (𝑡) = 𝑩
(15)
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 corresponds to 𝑩
̃ 𝑑,𝐿𝑂𝐶 , defined in Eq. 10, and 𝑹𝑑𝑒𝑓 (𝑡) = (𝑟𝑥 (𝑡), 𝑟𝑦 (𝑡), 𝑟𝑧 (𝑡)) is a random noise, where
Where, 𝑩
its components 𝑟𝑥,𝑦,𝑧 are random scalar values for which the following is true,
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 ‖, +𝜌‖𝑩
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 ‖] ,
𝑟𝑥,𝑦,𝑧 ∈ [−𝜌‖𝑩
(16)
where 𝜌 ∈ ℝ+ is a time-invariant coefficient that gives control over the average magnitude of the imposed noise. Note
that 𝑹𝑑𝑒𝑓 (𝑡) is not time-invariant, since the components 𝑟𝑥,𝑦,𝑧 are generated at each time-step 𝑡.
Finally, the 𝑩′𝐿𝑂𝐶 value is computed as follows,
𝑩′𝐿𝑂𝐶 = 𝒒∗𝑎𝑡𝑡 𝑩𝐿𝐶𝐼 𝒒𝑎𝑡𝑡 + 𝑩𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 .
(17)
Having defined all the variables that come into play, we can exploit the GA described in Sec. 2.1 to solve the
̃ ∗𝑑,𝐿𝑂𝐶 .
minimization problem formalized in Eq. 14, and compute the value of 𝑩
̃ ∗𝑑,𝐿𝑂𝐶 and the a priori defined value 𝑩
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 .
The validation consists in the comparison between the estimated value 𝑩
The methodology can be considered accurate when the difference between the two vectors is comparably small; to this
purpose the following definitions for this discrepancy are given,
𝜉
Δ%
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 ∙ 𝑩
̃ ∗𝑑,𝐿𝑂𝐶 /(‖𝑩
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 ‖‖𝑩
̃ ∗𝑑,𝐿𝑂𝐶 ‖))
= acos (𝑩
=
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 ‖−‖𝑩
̃ ∗𝑑,𝐿𝑂𝐶 ‖|
100|‖𝑩
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 ‖
‖𝑩
,
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27 JANUARY 2016
where 𝜉 is the angle between the two vectors, and Δ is the difference (defined positive) of the norms.
2.3
Validation Results
A set of runs of the method described in Sec. 2.2 were carried out in order to provide an insight on the accuracy of the
methodology described in Sec. 2 and Sub-sec. 2.1.
The validation was organized as illustrated in Table II.
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 components
𝑩
[𝑛𝑇]
x
y
z
+0.2
+1.0
-0.6
+0.9
-1.0
+0.0
-0.1
+0.9
+1.0
-0.4
-0.2
+0.3
+0.3
+0.4
-0.1
𝜌
[−]
0.1
+0.2
+0.9
-0.1
-0.4
+0.3
+1.0
-1.0
+0.9
-0.2
+0.4
-0.6
+0.0
+0.1
+0.3
-0.1
0.5
+0.2
+0.9
-0.1
-0.4
+0.3
+1.0
-1.0
+0.9
-0.2
+0.4
-0.6
+0.0
+0.1
+0.3
-0.1
2.0
TABLE II
VALIDATION RESULTS
̃ ∗𝑑,𝐿𝑂𝐶 components
𝑩
min(𝜀̃)
[𝑛𝑇]
[𝑛𝑇]
x
y
z
1591.61
+0.201
+1.001
-0.607
2011.02
+0.880
-0.999
+0.062
2120.26
-0.110
+0.894
+1.019
844.86
-0.402
-0.200
+0.315
686.86
+0.299
+0.397
-0.097
Average:
6611.20
+0.176
+1.056
-0.424
8098.04
+0.906
-0.911
+0.163
7396.17
-0.117
+0.951
+1.164
3069.32
-0.443
-0.173
+0.394
3027.35
+0.282
+0.410
-0.048
Average:
36368.55
+0.179
+1.058
-0.582
42207.47
+0.909
-0.865
+0.073
41712.39
-0.214
+0.877
+0.870
11927.65
-0.320
+0.022
+0.503
12046.23
+0.292
+0.518
-0.007
Average:
𝜉
Δ%
[°]
[𝑛𝑇]
0.281
2.753
0.830
1.163
0.224
1.050
8.992
7.775
2.747
7.182
6.270
6.593
2.482
5.538
6.444
31.288
12.966
11.744
0.489
1.244
1.111
0.984
0.400
0.846
3.189
5.019
15.880
7.953
0.986
6.605
3.718
8.903
9.522
5.755
8.438
7.267
̃ 𝑑,𝐿𝑂𝐶,𝑑𝑒𝑓 were defined; for each, 3 levels of noise 𝜌 were applied. The 𝜉 and Δ% value were
A set of 5 different 𝑩
calculated for each experimental set.
It is immediately apparent that the methodology performs well for a noise coefficient value of 𝜌 = 0.1, with an error of
alignment with a mean value of 1.05°, and an average magnitude error of 0.85%. The performance at higher values of
noise (𝜌 = 0.5 and 𝜌 = 2) tends to reach a plateau for what concerns the magnitude error, with a mean magnitude error
around 6.6% and 7.3%, respectively. Regarding the deviation, we have values of 6.6° and 11.7°, respectively, which
seems appropriate, given the large uncertainty given by the higher 𝜌.
To summarize the analysis given above, it is apparent that the methodology is consistent and accurate, when white noisepolluted data is fed to the algorithm. This is to be expected since, in general, the genetic algorithm works on mean values,
as Eq. 13 clearly shows; white noise tends thus to be eliminated.
3
Results and Discussion
The methodology presented in Section 2 was applied to the real-world case scenario represented by the artificial nanosatellite POPSAT-HIP1, which is described in the Introduction. Our team was able to acquire, from the currently orbiting
spacecraft telemetry, a total of 18 datasets for the sun incidence sensor and magnetometer, each set being relative to a
specific epoch during the lifetime of the craft. A summary of these is visible in TABLE III. The data ranges from the 24th
November 2014 to the 23rd February 2015.
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TABLE III
DATASETS SUMMARY
Number of data points
Transmitter Transmitter
y
m
d
Total M+S
OFF
ON
1
2014
11
24
Tumb.
61
61
55
6
2
2014
11
25
Tumb.
66
42
66
—
3
2014
11
26
Prop.
57
57
55
2
4
2014
12
3
Tumb.
64
42
64
—
5
2014
12
7
Tumb.
65
43
65
—
6
2014
12
8
Prop.
57
57
53
4
7
2014
12
16
Prop.
48
48
44
4
8
2015
1
6
Prop.
60
60
55
5
9
2015
1
19
Stab.
59
59
59
—
10
2015
1
20
Tumb.
60
—
60
—
11
2015
2
8
Tumb.
60
60
51
9
12
2015
2
17
Prop.
51
51
46
5
13
2015
2
18
Prop.
60
60
60
—
14
2015
2
19
Prop.
60
60
59
1
15
2015
2
20
Prop.
61
61
54
7
16
2015
2
21
Prop.
44
44
44
—
17
2015
2
22
Prop.
69
69
69
—
18
2015
2
23
Prop.
49
49
49
—
Datasets for the experiment on POPSAT-HIP1. A dataset identifier is provided in the first column; the “status” column specifies whether the satellite is
in free tumbling mode (Tumb.), is performing a maneuver with its thrusters (Prop.) or is stabilized by means of its magnetorquers (Stab.). The last four
columns are relative to the datasets size; specifically, the “M+S” column lists, for each dataset, the number of points that provide information both on
the magnetometer and the sun incidence sensor. The last two columns regard the number of points collected with the transmitter in idle, and active,
respectively.
Dataset
ID
Date
Status
For this experiment, as illustrated in detail in Section 2.1, only the data from the magnetometer is used; the sun incidence
sensor data is discarded. From the Table we can see that, though the number of points with the transmitter idle is quite
high (58 points on average), the active transmitter datasets are small (4.8 on average) if present at all.
̃ ∗𝑑,𝐿𝑂𝐶 computation results are illustrated and discussed. Emphasis is given to the analysis of stray
In the following, the 𝑩
̃ ∗𝑑,𝐿𝑂𝐶 vectors which present a very different value with respect to the general trend. Finally, the
computed vectors, i.e. 𝑩
∗
𝑩𝑡𝑥,𝐿𝑂𝐶 is computed and discussed.
3.1
̃ ∗𝒅,𝑳𝑶𝑪 and TMDB Vectors
Computed 𝑩
̃ ∗𝑑,𝐿𝑂𝐶 , the data-points relative to the actively transmitting
In order to have a clean dataset for the determination of the 𝑩
̃ ∗𝑑,𝐿𝑂𝐶 vectors are
spacecraft, are pruned; a thorough analysis of the data shows that, with specific datasets, stray 𝑩
computed. This is clearly visible in Fig. 2, where these vectors are present mainly in the active transmitter vectors (blue).
̃ ∗𝑑,𝐿𝑂𝐶 and TMDB vectors. The 𝑩
̃ ∗𝑑,𝐿𝑂𝐶 vectors as computed using the methodology described in Section 2 are indicated by the
Fig. 2. Complete set of 𝑩
“TxN” expression (red), which stands for the transmitter idle mode; the vectors relative to the transmitter dipole being active are indicated with “TxY”
(blue). All numeric values are in nT.
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On the other hand, a pair of clusters of vectors can be discerned with some clarity in the same figure, one for the idle
transmitter and one for the active one. The TMDBs, which are shown in Fig. 2, present a rather large number of stray
vectors, namely ID#1, 3, 7, 8, 14. However, these are partly due to the poor definition of the relevant datasets, as can be
seen from TABLE III.
The vectors from the idle transmitter datasets have strong coherence, with the exception of ID#1, as is immediately
visible from Fig. 2, despite its dataset being extensive (55 data-points); this can be explained by looking at Fig. 3a and b.
In general, Fig. 3 illustrates the measured magnetic field vector 𝑩′𝐿𝑂𝐶 during the orbital path and tumbling motion of the
satellite. Dataset ID#1, which is in fact shown in Fig. 3a, displays a “path” of limited extent (roughly ⅛ of the sphere
surface), in the rotational sense (see Section 2). This, combined with the naturally present background measurement̃ ∗𝑑,𝐿𝑂𝐶
noise, leads naturally to a certain degree of indetermination for the problem, which, in turn, translates in a stray 𝑩
vector; one can compare this to the vector path in Fig. 3b, pertaining to ID#13, which presents a path which covers a
large part of the sphere. Along the same line, the TMDB vectors ID#1, 3, 4, 7, 14 have this problem as well, which
corroborates the general thesis; please refer to Fig. 3c and d, relative to ID#7 and ID#15, this last given as a comparison.
Fig. 3. Measured magnetic field vector shown in the LOC frame of reference for a subset of datasets. All values are normalized on the IGRF value,
shown in the figure as a dark blue sphere.
Having defined a selection rationale, the vector sets are pruned of the incorrect entries, which are the following: ID#1, 3,
4, 7, 8, 14 for the active transmitter, and ID#1 for the idle one. By intersecting the sets and using Eq. 11, the TMDB
̃ ∗𝑡𝑥,𝐿𝑂𝐶 is computed. This is illustrated in Fig. 4b, where the transmitter dipole vectors for datasets ID#6, 11, 12 (the only
𝑩
ones that fulfil all the requirements) are shown. Numerical results are shown in TABLE IV.
̃ ∗𝑑,𝐿𝑂𝐶 and TMDB. In a) the 𝑩
̃ ∗𝑑,𝐿𝑂𝐶 vector clusters can be seen for the idle and transmitting transmitter; in b) the resulting TMDB can
Fig. 4. Results for 𝑩
be seen. All numeric values are in nT.
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TABLE IV
RESULTS
Vector components (Cartesian)
Orientation error
Vector
Transmitter
x
y
z
Transmitter
Deviation [°]
Mean
0.1346
0.0307
-0.0832
OFF
Mean
3.6314
STD
0.0174
0.0079
0.0081
OFF
̃ ∗𝑑,𝐿𝑂𝐶
𝑩
Mean
0.0955
0.0574
-0.0546
STD
2.5731
ON
STD
0.0120
0.0154
0.0142
Mean
6.1558
ON
Mean
-0.0365
0.0225
0.0289
̃ ∗𝑡𝑥,𝐿𝑂𝐶 —
𝑩
STD
0.0149
0.0032
0.0046
STD
4.5152
Vector components (polar)
Module error
Transmitter
azimuth elevation
module
Transmitter
Error [%]
Mean
0.2233 -0.5457
0.1616
OFF
Mean
6.79%
STD
0.0481
0.0678
0.0166
OFF
̃ ∗𝑑,𝐿𝑂𝐶
𝑩
Mean
0.5390 -0.4485
0.1250
STD
7.51%
ON
STD
0.1431
0.0826
0.0155
Mean
8.34%
ON
Mean
2.5542
0.6161
0.0528
̃ ∗𝑡𝑥,𝐿𝑂𝐶 —
𝑩
STD
0.1602
0.2396
0.0089
STD
7.09%
Final results are illustrated. On the left side, the mean vectors are reported in Cartesian and polar coordinates, along with the standard deviation (STD)
of its component. On the right side, the mean errors are shown, both for the orientation and the module. The orientation error is defined as the angle in
degrees between the mean vector and the considered vector, whereas the module error is the relative error of their modules.
3.2
Discussion
̃ ∗𝑑,𝐿𝑂𝐶 and TMDB 𝑩
̃ ∗𝑎𝑛𝑡,𝐿𝑂𝐶 vectors were presented in Section 3.1. Following a
The results of the determination of the 𝑩
systematic approach, a pruning is made to discard the datasets which do not meet the necessary vectorial topology
requirements; this leads to a highly consistent group of vectors, representing the RMDB and TMDB.
̃ ∗𝑑,𝐿𝑂𝐶 , by looking at Fig. 4a and by referring to TABLE IV, one can see that there indeed exists a pair of
Regarding the 𝑩
distinct vector clusters, one for each transmitter status. Amongst each cluster, the vectors show remarkable coherence,
with an orientation error between 3.6° and 6.2°, and a module error around 7%. Furthermore, as reported in the table, the
STD of the vector components (both Cartesian and polar) is comparably low. All these values are similar to those found
in the validation phase, described in Section 2.3, which is a clear indication that supports the results. In accordance to the
̃ ∗𝑑,𝐿𝑂𝐶 is indeed
previously stated assumption that the RMDB tends to be constant, we can infer that the computed 𝑩
correspondent.
̃ ∗𝑑,𝐿𝑂𝐶,𝑡𝑥𝑂𝑁 and 𝑩
̃ ∗𝑑,𝐿𝑂𝐶,𝑡𝑥𝑂𝐹𝐹 , which can be computed to be
Let us consider the angle 𝛾 between the mean vectors 𝑩
approximately 𝛾 = 16.7°. Given that the STD of the orientation error is 2.6° and 4.5° respectively for the idle and active
transmitter vectors, which are substantially lower values than the angle 𝛾, it can be concluded, with adequate accuracy,
that there is a non-negligible and consistent statistical difference between the vector clusters. This can be attributed with
̃ ∗𝑡𝑥,𝐿𝑂𝐶 . We can see that, both from Fig. 4b and TABLE IV, the variation,
sufficient certainty to the TMDB vector 𝑩
indicated by the STD value, is relatively low for this vector, a fact which strengthens our thesis.
In general, it is fair to note that both the breadth of the datasets we acquired, as well as the accuracy of each
measurement, cannot be considered on par with the state-of-the-art. This has practical reasons related to the small buffer
in the POPSAT-HIP1 satellite (ca. 60 points), as well as in the transmitter status tracking, which was not originally
provided and had to be extrapolated. Despite this, the general trend, as well as the comparable low dispersion of the
results, seems to strongly support the methodology we have here outlined.
All things considered, in order to take full advantage of the methodology, a large number of data-points should be
acquired for the active transmitter, to match the number of points taken for the idle status. Furthermore, precise tracking
and determination of the transmitter system status is advisable. Furthermore, a necessary condition is that, during the
measurements of the magnetic field, the satellite is made to cover as many orientations as possible, covering all the
rotational sphere.
Conclusion
In this paper we present a methodology based on a differential evolution (DE) algorithm to perform an in-orbit estimate
of the residual magnetic dipole bias (RMDB) of the artificial satellite POPSAT-HIP1 from the magnetometer readings
alone. The rationale for this investigation comes from the need to provide efficient attitude control to nanosatellites;
accurately determining the principal constant disturbances (or biases) can greatly increase the effectiveness of the control
system.
Since one of the largest disturbances to these readings is likely given by the transmitter and its sub-systems, we then
apply the methodology to extrapolate the time-invariant part of this disturbance, which we refer to as transmitter
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magnetic dipole bias (TMDB). The methodology is validated using ad-hoc data. The formalism associated to the
algorithm is defined with the intention of being as general as possible; this translates in the strong point of being able to
work regardless of the nature or origin of the bias.
An experimental investigation is carried out by acquiring a total of 18 datasets during various phases of the spacecraft life
(stabilization, maneuvering, free tumbling), each dataset consisting of magnetometer and sun incidence sensor readings.
The sun incidence sensor is used for the determination of the satellite attitude, which is necessary for validation.
The experimental results for the RMDB are shown to be coherent with the expected accuracy, with an orientation error
between 3.6° and 6.2°, and a module error around 7%, while the TMDB vector cluster shows a general common trend,
though the quantity of computed vectors is too small to provide a statistical value of significance.
This being said, while the methodology in general is proven to be sound, some steps were defined to better the overall
accuracy, namely a precise temporal tracking of the transmitter sub-systems, and acquisition of datasets with sufficient
rotational covering. Moreover, a complete laboratory-based experimental campaign to measure the actual bias is planned
for the next iteration of the nanosatellite. This would allow for a final validation of the algorithm.
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