![Simplified BH curve of PermeNorm 5000H2 hysteresis material Behaviour of these materials has been investigated using a Helmholtz cage, which is custom constructed in the clean room at the Faculty of Aerospace Engineering of Delft University of Technology [8]. For evaluating the torque created by the magnetic material, the magnetic field strength caused by Earth magnetic field is calculated [9]. This parameter is governed by: v (4) H v = B ext μ 0](https://www.researchgate.net/profile/Btc-Zandbergen/publication/228414521/figure/fig3/AS:301833668055048@1448974162867/Simplified-BH-curve-of-PermeNorm-5000H2-hysteresis-material-Behaviour-of-these-materials_Q320.jpg)
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1
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CEAS European Air and Space Conference, Berlin, Germany, 10-13 September 2007
1
SIMULATION OF THE ATTITUDE BEHAVIOUR AND AVAILABLE POWER PROFILE OF
THE DELFI-C
3
SPACECRAFT WITH APPLICATION OF THE OPSIM PLATFORM
F. te Hennepe, B.T.C. Zandbergen, R.J. Hamann
Chair of Space Systems Engineering, Faculty of Aerospace Engineering, Delft University of Technology
Kluyverweg 1, 2629 HS, Delft
The Netherlands
f.tehennepe@delfic3.nl, b.t.c.zandbergen@tudelft.nl, r.j.hamann@tudelft.nl
ABSTRACT
Delfi-C
3
is a nanosatellite being developed by the Faculties of
Aerospace Engineering and Electrical Engineering of Delft
University of Technology. Its primary mission is to serve as a
test platform for various technology payloads. Delfi-C
3
is
scheduled for a piggyback launch into a near-circular sun-
synchronous orbit in September 2007.
To simulate the attitude behaviour and the available electrical
power of the spacecraft, use is made of OpSim. This is a
simulation platform based on the EuroSim platform developed
by Dutch Space. It allows for accurate simulation of spacecraft
orbits by a powerful numerical integration subroutine and
detailed physical modelling. Therefore, individual algorithms
modelling the attitude dynamics and the generated power were
added for determining the attitude behaviour and electrical
power. This makes OpSim an even more powerful tool for
general spacecraft simulation.
Delfi-C
3
uses passive attitude control by applying rods of
magnetic material. Delfi-C
3
is subjected to various disturbance
torques. By using the OpSim platform the magnitudes of the
disturbance torques and the magnetic control torque have been
determined.
In addition, OpSim is capable of determining the amount of
solar and albedo power, which is incident to the solar arrays of
the spacecraft. Therefore, the amount of power received by the
spacecraft in its orbital path has been determined.
In this paper, the processes of modelling the disturbance
torques acting on Delfi-C
3
and its available electrical power are
presented. Furthermore, various results of the operational
simulations in OpSim are shown. These results are compared to
the design requirements of Delfi-C
3
with respect to minimum
required power and maximum allowed acceleration rate. A
conclusion is drawn whether Delfi-C
3
will be successful in
performing its mission in these aspects.
KEYWORDS
Delfi-C
3
, operational simulation, attitude control, disturbance
torques, power evaluation
1 INTRODUCTION
Simulation is an important design tool in a spacecraft
development process. It allows for preliminary evaluation of the
spacecraft performance without the necessity of performing time
and money consuming verification tests. An all-embracing
simulation is the so-called operational simulation, in which the
behaviour of the spacecraft is modelled as if it were in orbit
around the Earth.
Operational simulations have been performed for the Delfi-C
3
nanosatellite. These are accomplished on the OpSim simulation
platform, which is a powerful numerical integration tool used
for modelling spacecraft orbits. It was originally developed for
developed for Delfi-1, and later extended to Delfi-C
3
[1][2][3].
Quantities of interest are the attitude behaviour of the spacecraft
and the generated array power. To model these quantities,
dedicated subroutines had to be developed.
In this paper, the modelling of the various relevant parameters
in the OpSim simulator is explained. In section 2 , the Delfi-C
3
spacecraft is shortly introduced. In section 3, the body fixed
reference frame and the set of equations used for attitude
dynamics are introduced. A description of the control torque is
given in section 4, while the disturbance torques are treated in
section 5. The algorithm used for evaluating available power is
illustrated in section 6. Several simulation results are shown in
section 7.
2 DELFI-C
3
SPACECRAFT
Delfi-C
3
, see figure 1, is a nanosatellite developed by students of
the Faculties of Aerospace Engineering and Electrical
Engineering of Delft University of Technology under
supervision of faculty staff. As such, it is an educational
opportunity for students to obtain experience in a spacecraft
development process [4].
FIG 1: Delfi-C
3
spacecraft with body fixed reference frame
General characteristics of the satellite are given in the table 1.
TAB 1: General characteristics of the Delfi-C
3
spacecraft
Parameter Characteristic/value
Length 326.5 mm
Width 100.0 mm
Height 100.0 mm
Mass 2.2 kg
Communication Omnidirectional
Attitude Coarse attitude control
Power consumption in sunlight 2.5 W
Power consumption in eclipse 0 W
Delfi-C
3
’s primary mission is to serve as a testbed platform
for a number of innovative technologies:
• An innovative type of thin-film solar cells developed by
Dutch Space;
• An autonomous wireless sun sensor developed by TNO
Science & Industry;
• A Radio Amateur Platform transponder designed by the
Faculty of Electrical Engineering of Delft University of
Technology and the Technische Hogeschool Rijswijk.
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For electrical power generation, Delfi-C
3
uses four solar
arrays, which, as a minimum, should provide for 3 W of array
power in sunlit conditions. Power conditioning is achieved by a
custom designed shunt switching regulator [5].
Attitude control is passive by means of hysteresis rods and a
single permanent magnet. The main purpose of the hysteresis
rods is to enforce upper and lower boundaries on the rotational
velocity following orbit insertion. Additionally, coarse attitude
control is achieved, thereby allowing for all experiments to
receive sunlight, although not necessarily at the same time and
continuously [6].
Delfi-C
3
is scheduled for a piggyback launch into Low Earth
Orbit with an Indian PSLV (Polar Satellite Launch Vehicle)
launcher from Sriharikota in September 2007. Table 2 lists the
characteristics of the Delfi-C
3
orbit.
TAB 2: Orbit characteristics of the Delfi-C
3
spacecraft
Parameter Value
Orbital altitude 635 km
Inclination 97.91 °
Eccentricity ~0
Local time ascending node 10.30 AM
3 ATTITUDE DYNAMICS
3.1 Reference frame
For the simulations, use is made of the so-called body fixed
reference frame. This coordinate system is shown in figure 1.
The z-axis of the body fixed reference frame runs parallel to
the satellite’s long dimension. The x-axis is perpendicular to the
z-axis. Its positive direction is going in such a way that the
quadrant constituted by the positive z- and x-axes contains a
solar panel hinge. The y-axis completes the right handed
orthogonal coordinate system. Its origin is situated at the
geometrical center of the spacecraft.
3.2 Dynamics and kinematics
Attitude dynamics of the Delfi-C
3
spacecraft are modelled in the
OpSim simulator by a complete set of kinematic and dynamic
equations [7]. By using these equations, the influence of the
torques on the attitude behaviour of the spacecraft is
implemented.
For calculating the rotational acceleration of the spacecraft,
the following expression is applied.
(1)
ωωω
v
v
&
v
v
JJM ˆˆ ×+=
where
M
v
is the sum of the control and the disturbance torque
vector,
J
ˆ is the spacecraft’s inertia tensor, and
ω
v
is the
rotational rate of the spacecraft.
This expression relates the torques acting on the spacecraft to
a rotation in the body fixed reference frame. These rotations are
transformed to time derivatives of the Euler angles by using the
following transformation expression:
(2)
θω
&
v
vC=
where
θ
&
v
is the time derivative vector of the Euler angles, and
C
is the transformation matrix relating body fixed reference frame
to orbit fixed reference frame.
In OpSim, a so-called 312-rotation scheme is implemented. In
this case, the transformation matrix
C
is written as:
(3)
−−
−
−−−
=
213213232132
13131
213213232132
ccccsssscscs
sccsc
sccssscssscc
C
where
ii
c
θ
cos
≡
, and
ii
s
θ
sin
≡
with
i
θ
being the Euler angle
with respect to axis
i
.
4 CONTROL TORQUE
The attitude of the Delfi-C
3
spacecraft is largely determined by
the various torques acting on it. The most important parameter is
the control torque.
Delfi-C
3
makes use of passive attitude control. Two types of
magnetic material are flown. It is common practice to depict the
performance of magnetic material in a so-called BH curve,
which relates the flux density in the material to the magnetic
field strength.
In the longitudinal direction of the spacecraft, i.e. along the
z
-
dimension, a permanent magnet is placed. In both lateral
directions, i.e. along the
x
- and
y
-dimensions, rods of magnetic
hysteresis material are placed. This configuration is shown in
figure 2.
FIG 2: Geometrical configuration of magnetic material
For the permanent magnet, AlNiCo 5 cast magnetic material
is chosen. In figure 3, the BH curve corresponding to this
material is shown.
FIG 3: BH curve of AlNiCo 5 cast magnetic material
PermeNorm 5000H2 hysteresis material is applied for the
hysteresis rods. This material will dissipate energy when placed
in a rotating magnetic field. This is caused because magnetizing
the material requires more energy than is released by
demagnetizing. In figure 4, the simplified BH curve for
PermeNorm 5000H2 hysteresis material is shown.
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FIG 4: Simplified BH curve of PermeNorm 5000H2 hysteresis
material
Behaviour of these materials has been investigated using a
Helmholtz cage, which is custom constructed in the clean room
at the Faculty of Aerospace Engineering of Delft University of
Technology [8].
For evaluating the torque created by the magnetic material,
the magnetic field strength caused by Earth magnetic field is
calculated [9]. This parameter is governed by:
(4)
0
µ
ext
B
H
v
v=
where
ext
B
v
is the magnetic flux density of the Earth magnetic
field, and
0
µ
is the magnetic permeability in vacuum.
For evaluating the magnetic flux density of the Earth, several
models are available. In the OpSim simulator, a simple magnetic
dipole model is used [10][11].
With the magnetic field strength known, the magnetic flux
density is determined running through the magnetic material
using the corresponding BH curves. Then, the so-called
magnetization of the material is calculated using:
(5)
H
B
Mv
v
v−=
0
µ
The magnetic dipole created by a piece of magnetic material
is expressed by:
(6)
VMm
v
v
=
where
V
is the volume of the magnetic material.
As the magnetic dipole represented by the piece of magnetic
material has the tendency to align itself with the external
magnetic field, a torque will be generated. This torque is
expressed by:
(7)
ext
BmT
v
v
v
×=
After substituting design values for the volume of the
magnetic material, all necessary calculations for determining the
control torque can be performed. This yields a magnitude of
10
-5
Nm for the control torque of the Delfi-C
3
spacecraft.
5 DISTURBANCE TORQUES
In addition, the attitude behaviour of Delfi-C
3
is determined by
various disturbance torques acting on the spacecraft along its
orbital path. These include the following [12]:
•
Gravity gradient torque;
•
Aerodynamic torque;
•
Solar radiation torque;
•
Magnetic disturbance torque.
In the following sections, the magnitudes of the disturbance
torques are calculated.
5.1 Gravity gradient torque
Because the moments of inertia about the spacecraft principal
axes are not equal, a gravity gradient torque will exist. Deriving
an expression for the gravity gradient torque leads to the
following result:
(8)
33
2
3aJanM
gg
v
)
v
v
⋅×⋅=
where n is the rotational rate of the spacecraft, and
c
c
R
R
a
v
v
≡
3
,
with
c
R
v
being the position vector of the spacecraft in an Earth
fixed reference frame.
As this expression shows the gravity gradient torque in an
Earth fixed reference frame, a transformation has to be applied
for obtaining the torque in the body fixed reference frame. This
transformation yields the following expression:
(9)
(
)
( )
( )
−
−
−
−=
θϕ
θϕ
θϕ
sin2sin
2sincos
cos2sin
2
3
22
xxyy
zzxx
zzyy
gg
z
y
x
JJ
JJ
JJ
n
M
M
M
where
φ
is the Euler roll angle, and
θ
is the Euler pitch angle.
5.2 Aerodynamic torque
Aerodynamic torque is created by collision of the spacecraft
surface with the resident atmospheric particles in the
spacecraft’s orbit. In general, an aerodynamic drag force is
expressed by:
(10)
(
)
nnVAVCF
DA
v
v
v
v
⋅⋅⋅=
2
2
1
ρ
where C
D
is the drag coefficient of the spacecraft surface,
ρ
is
the mass density of the surrounding atmosphere,
V
v
is the
relative velocity vector of the atmosphere with respect to the
spacecraft, n
v
is the outward normal vector of the exposed
surface, and A is the exposed surface area.
As the aerodynamic drag vector is not likely to go through the
center of mass of the spacecraft, a torque will be generated. For
a specific surface, the corresponding torque is given as:
(11)
(
)
A
CoPCoMAA
xxFT
v
v
v
v
−×=
where
CoM
x
v
is the position of the center of mass of the complete
spacecraft, and
A
CoP
x
v
is the position of the center of
aerodynamic pressure of the exposed surface.
Adding the contributions of all surfaces leads to the resultant
aerodynamic torque acting on the spacecraft.
As the aerodynamic torque depends on the size of the
exposed surface area, a dedicated algorithm is developed for
taking into account shadowing phenomena. After all, in
deployed configuration the solar panels will shadow part of the
main spacecraft body. This leads to alteration of the exposed
surface area and of the position of the center of pressure.
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CEAS European Air and Space Conference, Berlin, Germany, 10-13 September 2007
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5.3 Solar radiation torque
Solar radiation torque is generated due to momentum transfer of
photons in incident sunlight to the spacecraft. The pressure
exerted by the incident photons in case of complete absorption is
equal to:
(12)
c
S
p
=
where S is the solar flux, and c is the speed of light.
Because the spacecraft surface possesses an absorption factor
smaller than unity, the expression for the solar radiation pressure
force has to take into account the optical properties of the
spacecraft.
Assuming no transmission of the incident photons by the
spacecraft structure results in the following generated force:
(13)
( )
nA
c
nS
RF
S
v
v
v
v⋅
⋅
+= 1
where S
v
is the solar flux vector in the body fixed reference
frame, n
v
is the outward normal vector of the exposed surface, R
is the reflectivity factor of the spacecraft surface, c is the speed
of light, and A is the area of the exposed surface.
In a similar fashion as for the aerodynamic torque, a solar
radiation torque will be produced in case the solar radiation
pressure vector is not going through the spacecraft’s center of
mass. Therefore, the corresponding torque can be expressed by:
(14)
(
)
S
CoPCoMSS
xxFT
v
v
v
v
−×=
where
CoM
x
v
is the position of the center of mass of the complete
spacecraft, and
S
CoP
x
v
is the position of the center of solar
radiation pressure of the exposed surface.
By adding all contributions generated by the individual
surfaces, the solar radiation torque acting on the spacecraft is
obtained.
Similarly as for the aerodynamic torque, shadowing effects of
the solar panels are taken into account in assessing the
magnitude of the solar radiation torque.
5.4 Magnetic disturbance torque
Magnetic disturbance torques are produced due to interaction of
electrical currents with the Earth magnetic field.
One of the forces involved in the generation of magnetic
disturbance torques is the Lorentz force, which is generated if an
electrical current is flowing in a magnetic field. It is expressed
by:
(15)
(
)
∑
×=
i
m
lIBF
v
v
v
where
B
v
is the external magnetic field, I is the electrical current
flowing through current carrying wire i, and
l
v
is the length of
current carrying wire i.
As the resultant vector will not likely go through the center of
mass of the spacecraft, a torque is produced.
Another source of disturbance torque is caused by magnetic
induction. As the rotational rate of the spacecraft will lead to a
change in magnetic flux, voltages are induced in electrical
conducting material. These voltages lead to current flows, which
consequently result in a generation of torque.
6 POWER EVALUATION
To determine the maximum useful electrical load of the Delfi-C
3
spacecraft, the minimum amount of array power in orbit is
calculated. For this purpose a custom algorithm has been
developed.
Solar flux at the orbital position of the Delfi-C
3
spacecraft is
determined by evaluating the position of the sun with respect to
the Earth. As the orbital position of Delfi-C
3
is known, the
sunlight vector can be written as:
(16)
DEESDS
rrr
///
v
v
v
+
=
where
ES
r
/
v
is the positional vector of the Earth in the
heliocentric reference frame, and
DE
r
/
v
is the position of Delfi-
C
3
in the geocentric reference frame.
In addition to solar flux, albedo flux is taken into account by
dividing the hemisphere of the Earth directed to the spacecraft
into 180 parts. For every part, the corresponding albedo flux is
determined. The complete albedo flux is obtained by summing
all individual contributions.
The incident flux is evaluated by introducing the normal
vector of the involved solar array. This parameter is governed
by:
(17)
(
)
nrSanrSQ
j
ajs
v
v
v
v
⋅
⋅
+
⋅
⋅
=
where S is the solar flux, n
v
is the outward normal vector of the
involved solar panel,
s
r
v
is the unit direction vector from the
solar panel to the Sun, a
j
is the albedo factor of part j of the
Earth globe, and
(
)
j
a
r
v
is the unit direction vector from the solar
panel to part j of the Earth globe.
In the process of determining the array power, it is assumed
that the solar cells are operating in their maximum power point.
The power generated on a solar array is computed by:
(18)
QP
η
=
where
η
is the efficiency of the solar cells.
The value of efficiency of the solar cells depends on several
factors. An important factor is the cell temperature. In general,
the performance of a solar cell decreases with increasing
temperature. The influence of temperature on the performance is
governed by [12]:
(19)
(
)
refref
TT
−
+
=
λ
η
η
'
where
η
ref
is the efficiency of the photovoltaic cells at the
reference temperature,
λ
is the temperature coefficient of the
array power, and T
ref
is the reference temperature.
Another influence on the array power is the mission duration.
After all, exposure of the cells to UV radiation will lead to a
decrease in their performance. Therefore, the efficiency of the
photovoltaic cells is expressed by:
(20)
(
)
t
µηη
−⋅=
1'
where
µ
is the annual degradation of the photovoltaic cells, and
t
is the mission duration in years.
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The actual power available from the solar arrays depends on
the operation of the electrical power system. As the system does
not always operate in the maximum power point a correction
factor is introduced. This leads to the following expression:
(21)
QkP
η
=
where
k
is the power correction factor set to 0.8.
7 SIMULATION RESULTS
Various simulations have been performed to verify the
compliance of the spacecraft design to the requirements. Three
different test scenarios have been investigated. The scenarios are
listed in the table 3.
TAB 3: Characteristics of the simulation scenarios
Scenario Initial rotational
rate Panel deploy-
ment angles
Nominal 0 rad/s 4 × 35
º
Fast rotating 0.175 rad/s about
x
-axis 4 × 35
º
Single solar panel
deployment failure 0 rad/s 3 × 35
º,
1 × 0
º
For all scenarios, both the stowed and deployed
configurations have been investigated.
7.1 Spacecraft requirements
The following requirements are of relevance for the simulations:
•
Minimum available array power in sunlit conditions and all
panels deployed shall exceed 3 W;
•
Components of the satellite rotational rate shall be kept
below 0.175 rad/s (10 deg/s) for all body axes;
•
Average absolute spacecraft rotational rate with respect to
sun vector shall be in excess of 0.0087 rad/s (0.5 deg/s);
•
Field of view of autonomous wireless sun sensor (AWSS)
shall contain Sun for a significant part of the orbit;
•
In sunlit conditions and stowed configuration, array power
shall equal at minimum 2.5 W in periods of at least 30 s.
7.2 Simulation conditions
The following conditions hold for every performed simulation.
With respect to the temperature of the solar cells, no real-time
thermal model has been implemented in OpSim. Instead, the
temperature of the cells is taken constant at the worst case
temperature of 50 °C. This temperature has been determined
using a thermal model of the spacecraft [13].
The spacecraft is in stowed configuration during launch and
during initial operations. At 1800 seconds after separation, the
appendages are deployed. In the simulations, the deployment
procedure is assumed to occur instantaneously. In reality, the
panels are deployed sequentially, each taking less than 1 minute
to deploy.
Table 5 summarizes the Delfi-C
3
geometrical characteristics
and mass properties used in the simulations.
TAB 4: Relevant geometrical characteristics of the Delfi-C
3
spacecraft
Parameter Value
Moment of inertia
x
-axis 3.494
·
10
-2
kg
·
m
-2
Moment of inertia
y
-axis 3.494
·
10
-2
kg
·
m
-2
Moment of inertia
z
-axis 1.505
·
10
-2
kg
·
m
-2
z
-position of center of mass
w.r.t. geometrical center -1.1 mm
The electrical characteristics of the applied photovoltaic cells
required for simulating the available array power are listed in
table 5 [5].
TAB 5: Relevant electrical characteristics of the Delfi-C
3
spacecraft
Parameter Value
Open circuit voltage @ 28 °C 2.47 V
Short circuit current @ 28 °C 0.389 A
Max power point voltage @ 28 °C 2.13 V
Max power point current @ 28 °C 0.366 A
Open circuit voltage temperature
coefficient -5.5 mV
·
K
-1
Short circuit current temperature
coefficient 0.146 mA
·
K
-1
Max power point voltage temperature
coefficient -5.1 mV
·
K
-1
Max power point current temperature
coefficient 0.141 mA
·
K
-1
Number of cells in single string 5
Number of strings in single array 1
7.3 Array power
In this section, the available array power of the Delfi-C
3
spacecraft is examined for the three simulation scenarios.
In figure 5, the power profiles of Delfi-C
3
for the different
scenarios in stowed configuration including deployment
activities are shown.
a.
b.
FIG 5: Available array power profile including panel deployment
a. nominal scenario; b. fast rotating scenario
In the nominal scenario, the array power shows a large
variation in magnitude before the deployment of the
appendages. This is due to blind spots, in which no solar flux is
captured by the arrays. On the other hand, the maximum
available power is very large reaching values of 4.5 W.
The array power shows a large variation in the fast rotating
scenario as well. Due to the high initial rotational rate, a fast
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oscillation is present. This leads to a variation of the array power
between 0 W and 5 W.
This fast oscillation could lead to problems in powering the
spacecraft deployment systems. However, a slow oscillation is
noticed in the magnitude of minimum power. Hence, during part
of the orbit sufficient power (>2.5 W) is present for deploying
the appendages.
In the panel deployment failure scenario, no changes are
found with respect to the nominal situation until deployment.
This comes as no surprise, because in stowed configuration the
spacecraft is geometrically exactly the same.
The Delfi-C
3
power profiles for the deployed configuration
are shown in figure 6.
a.
b.
c.
FIG 6: Available array power in deployed configuration
a. nominal scenario; b. fast rotating scenario; c. panel deployment
failure scenario
Compared to the stowed configuration, the variation in array
power is significantly decreased. A maximum value of 4W in
available power is noticed, while the minimum power drops
slightly below 3 W.
In the fast rotating scenario, the fast oscillation noticed in
stowed conditions is still present in deployed configuration. The
minimum power value increases to slightly below 3W.
Furthermore, the maximum power value decreases to
approximately 4 W.
In the panel deployment failure scenario, the minimum and
maximum power show significant changes compared to the
stowed configuration. The maximum power shows a slight
increase to 5.2 W, which is explained by the existence of an
attitude with good incidence angles for multiple solar panels.
The decrease in minimum power is more serious. As the
worst case situation yields bad incidence angles for the solar
panels, the minimum power reaches only 1.7 W. These periods
can occur for a reasonably long time limiting the spacecraft
operation.
In case of a single deployment failure, the requirements are
not met. The minimum available power is smaller than the
required power, which will lead to temporary black-outs of the
spacecraft. Although is not a complete failure of the spacecraft’s
mission, the useful mission time is reduced.
In the nominal and fast rotating scenario, the design
requirements are met.
7.4 Rotational velocity
In this section, the rotational velocity of the Delfi-C
3
spacecraft
in the three scenarios is investigated.
In figure 7, the rotational velocity of Delfi-C
3
during the
initial three orbits is shown.
a.
b.
FIG 7: Rotational velocity during Delfi-C
3
initial operations in
nominal scenario
a. orbit 1; b. orbit 2
The figure shows that although no initial rotational velocity is
present, the spacecraft spins up. Cross-coupling leads to the
appearance of oscillations in the rotational velocity about the
body axes. By analyzing this plot, this spin-up reaches a
rotational speed of 0.0087 rad/s (0.5 deg/s) in steady-state.
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a.
b.
FIG 8: Rotational velocity during Delfi-C
3
initial operations in fast
rotating scenario
a. orbit 1; b. orbit 2
Figure 8 shows that the rotational velocity starts out at
0.175 rad/s (10 deg/s). A rapid decrease is noticed, which is the
result of the hysteresis rods. It is derived that the rotational
speed reaches a steady state value of 0.026 rad/s (1.5 deg/s) after
4000 s. This settling time is slightly longer than a single orbital
period. Hence, the hysteresis rods are extremely successful at
damping the rotational rate of the spacecraft.
In the panel deployment failure scenario, no significant
differences in the rotational velocity could be distinguished with
respect to the nominal scenario. This is explained by the
dominating torque produced by the magnetic material, which is
significantly larger than the disturbance torques.
In all three scenarios, the design requirements are met. The
rotational velocity remains below the upper boundary of
0.175 rad/s and above the lower boundary of 0.0087 rad/s.
7.5 Spacecraft attitude
In the following plots, the spacecraft attitude in the period
between 2000 s and 3800 s is shown. This period illustrates a
representative orbit of the Delfi-C
3
spacecraft.
In figure 9, the projection of the positive
z
-axis in the body
reference frame on the celestial sphere is shown. Two different
plots are shown, which correspond with the two hemispheres.
One shows the projection on the hemisphere encompassing the
Earth, while the other gives the projection on the hemisphere
pointing away from the Earth.
In these plots, the positive
x
-axis corresponds with the
velocity vector of the spacecraft. Furthermore, the
y
-axis
denotes the cross track direction. The origin of the first plot is
the vector to the center of Earth (i.e. nadir), while the origin of
the other plot corresponds with the vector away from the center
of Earth (i.e. zenith).
a.
b.
FIG 9: Projection of the positive z-axis on the celestial sphere
Figure 9 shows that at the beginning of the time period
(t = 2000 s), the spacecraft is pointing in the direction of flight.
With respect to nadir, it is at an angle of approximately 60°.
It continues to oscillate about a fixed point. However, after
some time the spacecraft attitude changes. The positive
z
-axis
has the tendency to make a 180° turn. This is done in an
oscillatory motion.
At the end of the period (t = 3800 s), the spacecraft is
pointing with its back towards the direction of motion. The
positive
z
-axis is at an angle of approximately 60° with respect
to zenith.
An explanation for the shift in attitude is found in the change
of the Earth magnetic field. As the magnetic material in the
spacecraft is aligning the spacecraft with the Earth magnetic
field, a pass of the spacecraft over the magnetic poles leads to a
180° turn.
In figure 10, the position of the Sun in relation to the
spacecraft body is shown. Hence, the axes in the figures
correspond to the body fixed reference frame.
1
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CEAS European Air and Space Conference, Berlin, Germany, 10-13 September 2007
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a.
b.
FIG 10: Projection of the Sun position on the celestial sphere
In this figure the fields of view of both autonomous wireless
sun sensors (AWSS) are shown. These sensors are positioned in
the square front and back panels.
At the beginning of the time period, the Sun is positioned in
front of the spacecraft. However, it dwindles via the side of the
spacecraft to the back. At the end, the position oscillates
slightly, which seems to indicate that the spacecraft remains in a
more-or-less fixed attitude with respect to the Sun. Recalling
Delfi-C
3
’s orbit, this resembles a pass over a geographic pole.
It can be concluded that the requirement with respect to the
field of view of the AWSS is fulfilled. For a large part of the
orbit, the Sun is observed by the AWSS. In the shown period,
the position of the Sun even travels through all four quadrants of
the back AWSS field of view.
In figure 11, the projection of the magnetic field vector in the
body fixed reference frame is shown. This projection is plotted
in similar fashion as in figure 10, by using a sphere
encompassing the spacecraft with infinite radius.
a.
b.
FIG 11: Projection of the magnetic field vector on the celestial
sphere
The figure shows that the magnetic field projection oscillates
about a fixed point. Throughout the entire time period, no large
deviations with respect to this point are present. Therefore, it
can be concluded that the spacecraft is in magnetic lock, which
means that the magnetic torque is sufficiently large to make the
spacecraft follow any change in the direction of the magnetic
field. This conception is reinforced by the magnetic torque being
a factor 100 larger in magnitude than the largest disturbance
torque (10
-5
Nm versus 10
-7
Nm).
The exact position of the epicenter of the oscillation coincides
with the resultant dipole vector of the magnetic material in
Delfi-C
3
. This position is at an off-set angle of 70° with respect
to the spacecraft’s
z
-axis with an amplitude of approximately
30°.
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9
7.6 Simulator limitations
A remark has to be placed at the application of the OpSim
simulator for Delfi-C
3
. It has been found that the simulation
algorithms are extremely sensitive to the choice of time step.
Choosing too large time steps leads to numerically instable
results.
This restriction results in a maximum simulation duration of
several days. The complete mission duration of Delfi-C
3
could
not be simulated. It is expected that the overall minimum values
will not deviate from the corresponding values found in the
simulations of the initial orbits.
8 CONCLUSIONS
For simulation of the performance of the Delfi-C
3
spacecraft,
use is made of the OpSim operational simulator. Successful
simulation was accomplished by introduction of specific
algorithms for control and disturbance torques, and available
array power.
Three different scenarios were simulated and the results were
evaluated using the design requirements for available array
power and rotational velocity.
In stowed configuration, the array power shows large
variations. Conditions yielding zero power are possible, which
will lead to a reboot of the spacecraft.
In deployed configuration, the minimum required power of
3W is always generated. Rotational rate has no influence on this.
In case a single solar panel fails to deploy, large dips in array
power are experienced. During these situations, the solar arrays
are not capable of supplying the complete spacecraft with
electrical power.
When the spacecraft possesses no initial rotational velocity, a
spin-up is accomplished to approximately 0.5 °/s. No influence
is found by introducing asymmetry by failing to deploy a solar
panel.
A high initial rotational velocity results in a decrease in
rotational rate due to the hysteresis rods. Steady state rotation
equals approximately 1.5 °/s.
Investigation of the attitude profile shows that the spacecraft
is in a coarse magnetic lock. Its positive
z
-axis does not deviate
from the magnetic field vector by more than 30°.
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