SATELLITE MAGNETISM: TORQUE RODS FOR EYASSAT

Transcription

(Preprint) AAS 15-777
SATELLITE MAGNETISM: TORQUE RODS FOR EYASSAT3
ECATTITUDE CONTROL
David J. Richie,* Maxime Smets,†Jean-Christophe Le Roy,‡
Michael Hychko,§ and Jean-Remy Rizoud**
Often considered only for satellite reaction wheel desaturation, when employed
correctly, torque rods are an effective, independent means of satellite pointing
control: both on orbit and in the classroom. In fact, the US Air Force Academy
has recently developed a CubeSat classroom demonstrator known as EyasSat3,
complete with reaction wheels, light detecting photo-resistors, a magnetometer,
and three-axis magnetic torque rods as well as several other attitude control sensor and actuator systems. Previous papers have investigated these EyasSat3 systems, but none, including the contractor through its provided documentation,
have focused on the EyasSat3 predicted and demonstrated torque rod performance with and without the one-axis Helmholtz cage, an effective method to
control the background magnetic field in laboratory (thus classroom) conditions.
In this work, spacecraft attitude dynamics, magnetic field dynamics, and magnetic actuation fundamental principles, torque rod and Helmholtz cage hardware
sizing, and the resulting EyasSat3 performance are presented. The benefits are
wide reaching as this simple, yet effective demonstration technique gives tomorrow’s leaders, including Academy cadets, a hands-on learning experience that
will shape their mastery of key attitude control principles.
INTRODUCTION
During the 2003/2004 academic year, the United States Air Force Academy (USAFA) Department of Astronautics (Astro Department) created a classroom satellite in order to teach cadets
about satellite subsystem design and operation through hands-on, experiential learning.1 Dubbed
“EyasSAT” as it prepared third year cadets for success in the Astro Department’s wildy
successful “FalconSAT” capstone program as fourth year cadets and an “eyas” is a baby falcon,
this program extended its influence beyond USAFA. Not only did the Air Force use this program
to show key concepts to intermediate space professionals, it was also used by several universities
to demonstrate subsystem concepts in the classroom at low cost. Over the years since its
inception, a growing need amongst attitude control experts became evident: demonstrating key
*
Associate Professor, Senior Military Faculty, and Deputy for Research, Lieutenant Colonel, Department of Astronautics, 2354 Fairchild Drive, USAF Academy, CO 80840, Member AAS, and Senior Member AIAA.
†
Bachelor of Science, USAF Academy Summer French Cadet Researcher, L’ecole de l’air, France.
‡
Bachelor of Science, USAF Academy Summer French Cadet Researcher, L’ecole de l’air, France..
§
Bachelor of Science in Astronautical Engineering, United States Air Force Academy, 2 nd Lieutenant, Goodfellow Air
Force Base, TX.
**
Master of Science, USAF Academy Engineering and Science Exchange Program, L’ecole de l’air, France.
This Document is for information only. No U.S. Government commitment to sell, loan, lease, codevelop, or co-produce defense articles or provide defense services is implied or intended.
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attitude control concepts to undergraduates in a small classroom test apparatus size. This, then,
led to the recent creation of the EyasSat3 classroom CubeSat program to teach undergraduate cadets satellite Attitude Determination and Control Subsystem (ADCS) concepts.234 A key technology for contemporary ADCS implementation is that of magnetic sensing and actuation wherein
components sense satellite orientation relative to the local magnetic field and actuate motion by
changing the satellite’s dipole which, in turn, seeks to align itself with this local field.
Figure 1. ES3 Overview
Figure 2. ES3 ADCS Board and Actuators Block
Several previous works have investigated the key principles governing magnetic torque generation through current-driven magnetic dipole moment production as well as background magnetic
field manipulation through a single-axis, or in some cases, a three-axis Helmholtz Cage ground
test device. In fact, one should refer to the works of Wie, Hughes, Wertz, and others.5 6 7 8 Although many have studied this issue, few have implemented a portable, CubeSat-based system for
undergraduate education, especially for an initial course satellite rotational dynamics. This effort
does not seek to present theoretically revolutionary concepts, but instead seeks to implement
these concepts simply, in a way an undergraduate student can understand early on, and it also
seeks to report on the progress of this effort at the USAF Academy. More specifically, this research focused on measuring and manipulating the magnetic field in the laboratory with the ulti-
2
mate purpose classroom demonstration. More specifically, a Helmholtz cage was analyzed, calibrated, and controlled to couple with the ambient field and create a uniform magnetic field in order to test the satellite’s on-board magnetic devices. The magnetic field uniformity was proved
and the Helmholtz cage is now ready to use by faculty and cadets for valuable ADCS demonstrations.
Furthermore, an on-board CubeSat magnetometer was tested and calibrated for accurate angular position measurements. Although previous Astro Department work calibrated the on-board
magnetometer, the associated calibration factors were insufficient for all CubeSat uses, including
those measured inside the Helmholtz Cage with the magnetic field increased. More accurately,
through extensive comparison of the magnetometer’s data sheet with laboratory measurements, it
was found that the sensor’s default sensitivity range settings saturates for some of the Helmholtz
cage usable region. This is critically important since the Helmholtz Cage must generate a strong
magnetic field in order for the on-board torque rods to slew the satellite in the laboratory, but this
stronger field lies outside the default magnetometer sensitivity region. However, once the sensitivity settings are increased as per the data sheet, these values do NOT saturate. Interestingly, the
initial magnetometer implementation software only used the factory default region, as the current
effort (in conjunction with a parallel National Technological University of Singapore research
project) clearly shows, and if left unchecked, would have led to errant three-axis attitude position
estimates.
Another research effort was to assess on-board torque rod performance as compared to modelled theory as well as demonstrate single-axis satellite rotation using torque rods in the laboratory. Both of these goals were achieved as the CubeSat, hung on a string and dangled into the
Helmholtz cage, was rotated by applying a well-known B-dot control law presented in the literature.
Finally, this very fruitful research effort has spawned several new ideas for follow on work,
including improving the Helmholtz cage and the EyasSat3 magnetic ADCS components in order
to use the entire system inside a spherical-air-bearing system hamster ball test structure. This,
then, would greatly help demonstrate the key principles underlying three-axis attitude determination and control to cadets for generations to come.
THEORY
Before one can assess performance of ES3’s magnetic field focused hardware, it is critical that
one understand the underlying theory for spacecraft dynamics including bang-off-bang maneuvers, magnetic field and torque production for a single loop coil of differing cross-sectional
shape, magnetic dipole generation of solenoid, and magnetic field generation via a two-solenoid,
single-axis Helmholtz Cage. The next portion of this paper will investigate these key areas and
highlight several relevant variable/parameter relationships and governing equations that provide
the fundamentals framework upon which the predicted and experimental results are based.
Bang-off-Bang Attitude Maneuver Requirements
In this section, we present the fundamental rotation dynamics behind spacecraft motion and
then translate these dynamics into three driving requirements: i) desired slew rate to ensure an
efficient, in-classroom learning experience, ii) desired time from rest to the desired slew rate and
the resulting desired spacecraft angular acceleration in order to meet that time requirement, and
iii) required torque about one axis (for now, the body’s “+ Z” axis, to which we’ll also refer to as
) in order to achieve i) and ii). This last requirement or if/how much magnetic field we need
from the Helmholtz cage in order to meet these requirements.
3
Discussions on torque rod, Helmholtz cage, and torque rod plus Helmholtz cage performance
will occur much later in this paper, but the underlying theory will be investigated in subsequent
sections. For now, the discussion centers on the spacecraft slew rate, slew acceleration, and single-axis torque requirements just mentioned (items i), ii), and iii)). The well-known governing
equation, Euler’s Moment Equation, using Hugh’s Vectrix notation with the Astro 445 course
note development, is, in vector form:
(1)
In scalar form, when expressed with respect to
, is:
the Body Reference Frame,
(2)
In space, typically:
(3)
For now, without loss of generality, we’ll assume the magnetic torque is the only torque acting
on the spacecraft as it is our focus for this paper, so
, and we’ll also let
which for a single axis maneuver about the “+Z” (
axis, this
, we’ll assume that
substitution becomes
for requirements generation purposes,
,
is a near principal frame,
, and
. Thus,
(4)
From Eq. (9), we see that the required (“r” subscript) magnetic torque can be given as:
(5)
Now, previous laboratory work has shown that
and our classroom rotation
rate requirement,
, was established to be 6 RPM in order for the satellite to slew fast enough
to demonstrate key principles to undergraduates efficiently. In addition, there’s a need to accelerate the spacecraft from rest to this desired speed in 12 seconds. These requirements essentially
define a bang-bang maneuver requirement9. In reality, the maneuver would be more effective as
a “bang-off-bang” maneuver. In that case, we also assume a dead band (half the maneuver time)
in which the satellite will coast at the given slew rate. This yields
for the
bang-bang case and
0.21
angular acceleration, and 0.63
for the bang-off-bang case, with required torque
. In short, our maneuver requires 1.04 m Nm torque,
angular velocity.
Now that we’ve defined the key requirements for satellite performance when slewed around
the “+Z” (
axis, we’ll next address theory that will help us analyze the resulting rotational motion during our laboratory tests. More specifically, the restorative string (i.e. spring) torque,
resistive damping torque, and driving magnetic torque will impact this motion. Thus, for predict-
4
ing results in one axis, we’ll make the following substitutions:
Then, our single axis torque equation reduces to:
,
, and
.
(6)
where b is string damping and k the string spring constant. Noting that the magnetic dipole is measured in the body frame, the magnetic field components
and
are
measured in the inertial frame, and the body frame moves away from the inertial frame
an angle about the “+Z”(
axis, then
and Eq.
(14) becomes
(7)
Applying the small angle approximation
(b = 0), and rearranging, we find
, assuming the damping term is negligible
(8)
(9)
(10)
(11)
So, the period (in seconds),
, we can compute is
(12)
Interestingly, if there are negligible magnetic field components, e.g.
and
both are near 0
T, then we use
and the measured
to find k. Likewise, if one knows the
and
magnetic
field components when they are not 0, he or she can apply , and
and predict the oscillatory
period, .
Magnetic Torque and Magnetic Field Generation Theory for a Single Current Loop
In this section, we present the fundamental magnetic field and associated torque theory for a
single current loop. As shown on in Griffiths10, the resulting torque on a current loop that produces a steady magnetic field,
, is driven by the magnetic dipole moment,
, on
that loop. The underlying relationship can be found from a few key principles. First, given a circular loop with current I, radius, R, and that encircles a point O, as shown in Figure 3,
5
Figure 3. Circular Current Loop Encircle Point O
will generate a magnetic field, , normal to the loop’s cross sectional area, as shown in Figure
3,
Figure 4. Current Loop Generates Magnetic Field
Next, per Reference 1, Lorentz’s Law can be used to determine the ensuing force,
different points along the current loop (e.g. A, B, C, and D from Figure 3). Noting that:
, at
(13)
If a point charge of interest, Q, is such that
, and the differential charge
,
where is a line charge and is the length of a current carrying wire, then
and also,
the loop current vector, , can be expressed into terms of line charge and charge velocity as
. Now, Lorentz’s law can be expressed more specifically as:
(14)
Next, if and
lie in the same direction and the current, I, is constant, then
(15)
For our circular loop, this means that travelling in the - plane, produces steady magnetic
field if I is constant, and yields outward force around the loop (in order to prove it to yourself,
calculate
at a few points along the loop), as shown in Figure 4,
6
(a) Top View
Figure 5. Force
(b) Side View
distribution around a Circular, constant-Current Loop
As Figure 4 clearly shows,
, and based on the illustrated geometry,
since the forces about the loop act through point O. Similarly, per Reference 1, if we change the
cross-sectional shape of the loop to a rectangular loop, with side lengths a and b, as shown in
Figure 5,
, and based on the illustrated geometry,
unless we make one
minor change to the scenario,
(a) Top View
Figure 6. Force
(b) Side View
distribution around a Rectangular, constant-Current Loop
However, per Reference 1, if we rotate the loop and assume the background (“net”) magnetic
field remains aligned in the direction, we find a torque that seeks to “restore” the loop’s local
field with the underlying field. Figure 6 and the ensuing discussion shows how this works.
(a) Rotated, Rectangular Current Loop
(b) Simplified View
7
Figure 6. Rectangular, constant-Current Loop rotated about the
axis
Figure 7. Rectangular, constant-Current Loop Free Body Diagram
Since the current is constant, one can divide the rectangular up into sides, apply Eq. (15) in
order to calculate the resulting force magnitude for a side after integrating the points along that
side, and then inserting these forces in the standard torque equation,
. Beginning
with point A in Figure 6 and summing (i.e. integrating the result along its side), we see
(16)
Similarly,
Next, the distribution on the side with point B per Figure 7 is
(18)
and
. Next, moment arms from point O to each of the sides, labelling each
side by the midpoint, are
,
,
,
and
. Now, we combine the force magnitudes and position
vectors for each side, then sum these values to yield:
(19)
As described by Griffiths10, this result can be made more general provided we’re using a constant current (closed) loop, in that any cross-sectional loop (e.g. circular, rectangular) shape meeting this and the other assumptions we’ve made so far, will yield:
(20)
where A is the cross-sectional loop area. Next, one can summarize how a current loop actuator
tries to align itself with the underlying magnetic field as described by Griffiths and several other
sources, 10,3,6
8
(21)
Where
and
Note that, on the ground, we can use a laboratory magnetometer to measure
and
compare it to interpolated/extrapolated IGRF values (i.e.
) in order to back out
. In
contrast, estimating
is often an “art”. Sometimes, we can also work to control it,
either by using magnetic shielding materials around key components or by installing small “neutralizing” coils (current loops) to actively make the residual magnetic dipole reach 0 A m2.
Given these factors, the theory for the remaining of the paper will concentrate the remaining
two terms,
and
. Nevertheless, continuing the torque
discussion, we apply Eq. (39) to Eq. (40), and noting that is perpendicular to the loop crosssectional area from page Griffiths10, we see that
(22)
Building on Eq. (41), one can see that,
(23)
Then, clearly,
(24)
Now that we’ve characterized the induced magnetic torque due to magnetic actuators and the
underlying magnetic field, we next delve deeper into computing (at a displacement, away from
it) then into controlling the underlying magnetic field for a single current loop system. Before so
doing, however, it’s important to note the key topics covered after this subsection. More specifically, in the next subsection, we’ll apply Eq. (24) to a solenoid actuator, where in we’ll produce a
stronger magnetic dipole moment by multiplying the current loops. In actuality, we’ll be using a
single current wire with several loop turns in order to increase this dipole. The theory we’ll develop, then, will govern our sizing/performance predictions and experimental test results. Then,
in subsection thereafter, we investigate the underlying theory governing the Helmholtz cage performance. We’ll look at how we produce, analyze
. Note that this last subsection is an ideal
place to discuss the background magnetic field, but this topic was addressed below Eq. (20).
Continuing on, then, we examine how to calculate the magnetic field vector, , as a function
of displacement away from the loop and more specifically, away from the loop’s center. We’ll
call this resulting function,
. Per the Biot-Savart Law, listed on page 215 of Griffiths10, the
“magnetic field of a steady line” is given by:
(25)
where
is the permittivity of free space,
from this result using Reference as a guide,
. Next, let’s examine the geometry
9
Figure 8. Computing the Magnetic Field at a displacement
(distance OQ)
So point P travels around the loop in
increments as governed by angle . In this case,
, , , and
are constant, thus
is constant per the Pythagorean theorem:
. Stated differently:
Also,
is constant with respect to the
So, expressing
and
in the
,
,
,
,
reference frame, but not the
,
, one.
reference frame:
(26)
(27)
So,
(28)
Next, we express the
,
,
reference frame unit vectors in terms of
,
,
(29)
(30)
(31)
Next, we substitute Eq. (47) and
in Eq. (25) to yield
(32)
Inserting Eqs. (29), (30), and (31) into Eq. (32), integrating
from 0 to
, and simplifying:
(33)
10
Thus, Eq. (33) summarizes the magnetic field strength magnitude and direction for points lying along the axis in relation to a single current loop. This result, and the Biot-Savart Law, will
govern operation of the Helmholtz cage as we describe in the subsection after the next one.
Producing Magnetic Torque with a Single Torque-Rod Solenoid
In this section, we present the underlying solenoid theory behind producing torque with a
torque rod. Essentially, this section builds on the previous subsection, but involves multiple current loops adjoined into a single loop with multiple loop turns.
Recall from Eq. (24) that, for one current loop,
(34)
As is somewhat more clear in Figure 9, the torque rods are solenoids so this becomes:
(35)
Figure 9. Torque Rod Solenoid is really
stacked current loops
As we’ll see, the torque rods can also have coils overlaps, thus the shaded region in Figure 10
below shows where wire exits in overlapping fashion:
Figure 10. Overlapping Coils form Shaded (Pink) Region
One can see in Figures 9 and 10 that the rod may have an inner flux cross sectional area, ,
and an outer flux cross sectional area, , so, the mean cross-sectional area is given in Eq. (36):
(36)
The magnetic dipole moment magnitude for such a rod, then, is:
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(37)
This is the basic equation governing performance for each of the
, , and axis rods.
As alluded to earlier, in a later section of this work, we’ll estimate the number of as built coil
loop turns by counting wiring for a sample region using caliper values and extrapolate the total
from the total rod length and average thickness. From this, we’ll estimate the torque performance
based on Eqs. (36, 37) and applying ohm’s law,
, through measuring loop voltage and resistance with a voltmeter.
Modifying the Background Magnetic Field with the Helmholtz Cage
In this section, we present the fundamental concepts between predicting and measuring the
background magnetic field as well as controlling this background magnetic field via a Helmholtz
cage. The equations governing Helmholtz cage performance, then, are addressed as in a controlled environment such as a laboratory or classroom, one can use current input to control the
total local magnetic field.
Figure 11. Helmholtz Cage Contains Two Solenoids, HC1 and HC2
In deriving Eq. (33), we investigated the magnetic field in relation to a single current loop.
For the Helmholtz Cage show in Figure 11, there are two solenoids with multiple turns, similar to
the solenoid-based torque rods shown in Figures 9 and 10. So, if we examine the bottom solenoid, HC2, and examined the B field a distance z along its axis, we have essentially the same
geometry as in Eq. (33), albeit with several more loop turns. Likewise, we can also examine the
top solenoid, HC1, giving it the same treatment. This helps us generate an interesting modification for Eq. (33), essentially the ratio of
(38)
where
represents the number of coil turns in the top solenoid and
the number of
bottom solenoid turns. Next, if one wants to examine the magnetic field at the center (i.e.
)
of one of the solenoids, e.g. HC2, then Eq. (38) becomes:
(39)
So, one can measured the magnetic field with a laboratory magnetic field at this point and for
the amount current sent into the wire, he or she solve for the number of loop turns,
We will
do this for both solenoids and for different current values (
for the background magnetic
field effect) as an effective coil turn number estimation technique. One can also plot how the
magnetic field strength decays along the axis by adapting Eqs. (33) and (39). In other words,
12
(40)
A comparison plot for doing just that is shown in Figure 12
Figure 12. Magnetic Field Decay with Respect to One Helmholtz Cage Solenoids
Next, we determine the how to generate a uniform magnetic field between the two solenoids
and where uniformity is relevant. Before so doing, it’s interesting to note one issue that arose
during laboratory testing to find such uniformity. In short, if one wires the two solenoids together
in parallel instead of in series, then magnetic field is not uniform unless both solenoids have exactly the same number of turns. This is difficult to do in manufacturing, although both solenoids,
as will be shown later, have nearly the same number of turns. However, in order to ensure one
gets the same magnetic field for each coil, it’s important one have the same input current, thus
wiring them together in series.
Nevertheless, the discussion continues with examining uniformity once wired in parallel. Figure 13 shows a comparison of differing distances between coils,
(a)
Destructive Interference: D’s Too Far Away
(b) Constructive Confluence
Figure 13. Destructive Interference and Constructive Confluence
13
Mathematically, one can examine the “best” separation distance by modifying Eq. (39), with
for solenoid HC1 and
for solenoid HC2. As one can see from Figure 13, this occurs
when D = R and the solenoid currents flow in the same direction. Letting D = 0 and examining
the magnetic field solenoid midpoint, i.e. Z = 0, Eq. (39) becomes
(41)
EXPERIMENTAL CONFIGURATION, DIMENSIONS, AND DESIGN PARAMETERS
In order to get the most out of the results presented in this paper, one needs to understand how
the hardware, both primary and test support hardware, is configured.
Figure 14. Key EyasSat3 Circuit Boards
First, Figures 1and 2 give a great overview of the ES3’s major components as well as the
spherical air-bearing hamster ball. Then, Figure 14 presents the primary ES3 circuit boards, and
Figure 15 shows several testing devices: the power supply to drive the Helmholtz Cage, the
ground magnetometer, the circuit muti-meter, the on-board three-axis magnetometer chip, and the
ground magnetometer in action aboard the Helmholtz cage. Meanwhile, Figure 16 shows more of
the Helmholtz cage in use, especially in the fundamental configuration (Figure 16(b)) for satellite
Z-axis, string-hang rotation.
Thus, the basic testing idea is to measure the local magnetic field with the ground magnetometer, validate it with the on-board magnetometer chip, and then wire the Helmholtz Cage to the
power supply. The current is then set on the power supply and fed to the cage, energizing the
cage. The field is then measurable by the ground magnetometer for operations under 1.0A of input current. If more is needed, then the on-board magnetometer is used to detect this operation.
Note that the magnetometer itself has sensitivity settings that must be modified if the generated
magnetic field exceeds +/- 1.3 Gauss default sensitivity range. Note that the on-board magnetometer can be placed in a mode to sense up to +/- 8 Gauss, far above the safe limit (we’re using
4A) for the Helmholtz Cage.
14
From top left: i) power supply for HC, ii) FVM-400 laboratory magnetometer, iii) a multi-meter
measuring HC current, iv) SparkFun on-board magnetometer, and v) FVM-400 in action
Figure 15. Essentially Test Hardware and Equipment
From top left: i) HC mag field direction, ii) One-axis HC, ES3 Torque Rod Testing, and iii) Required ES3 Torque Rod Test Placement
Figure 16. Helmholtz Cage Operations
SIZING AND PREDICTED RESULTS
Investigating the equipment above, it’s important one grasp the key parameters of these system that impact performance as well as gain insight into the predicted performance for these systems. The experimental results in the ensuing subsection will be compared to these predicted estimates and evaluated in the
next major (“DISCUSSION”) section that follows. Nevertheless, predicted performance for the torque
rods, Helmholtz cage, resulting spacecraft motion, and simulated spacecraft motion will be addressed next.
Preliminary torque rod sizing was conducted through simple analysis of the a-built torque rods’ physical dimensions. From inspection, both the X- and Z- axis torque rods appeared to be of approximately
15
equal shape and size, while the Y-axis rod was longer and more narrow. This is of importance, as torque
rods of varying physical characteristics may yield different performance specifications. A sizing of the Xaxis rod could not be reliably taken due to its cluttered location, so this analysis pertains only to the remaining two torque rods.
The following dimensions were taken from the Y- and Z-axis torque rods: length, outer and inner diameter, visible number of turns per 10 mm sample n, and visible number of turns b in a cross-sectional slice.
The number of turns was thrice sampled at random for both torque rods, then averaged to generate an approximate value of n per 10 mm length. Figure 1 displays this information below:
Figure 2: Physical Dimensions of Torque Rods
From this information, a cross-sectional area estimate was determined for both an outer and inner diameter measurements, and an estimate for the total number of turns was then approximated by linearly extending the 10 mm measurement to the previously collected length of the rod and accounting for the approximate number of turns per single cross-sectional slice b. Or rather,
From research, it was chosen that the estimated flux area A was to be determined using the average of
the outer and inner diameter.
The estimated torque produced, then, was generated from the following equation:
Using a preliminary estimate of B from Ayesha Hein’s research on the EyasSat’s magnetometers, the
findings showed that the Y- and Z-axis torque rods were capable of producing a similar amount of torque –
6.684% difference (reference ‘Torque Rod Dimensions’ on SVN). Using this sizing approach, it was found
that the X-rod should produce 116.22 m Am2 and the Y- and Z-axis torque rods should produce 124.54 m
Am2 magnetic dipole moment.
Besides the torque rods, it is also important to understand key Helmholtz cage parameters. In essence,
the radius, R, of the Helmholtz cage is a critical parameter, but the other items are negligible. R is 9.6 in
(0.244 m), whereas the number of Helmholtz solenoid turns are presented in the Experimental Results section. Otherwise, the Experimental Results tables are the best means for illustrating key parameters of the
system, even in terms of measured versus predicted.
In short, the present section’s aim is to capture key component values or concepts as it relates to the asbuilt hardware that helps one understand predictions listed in the subsequent section. Note that a hands-on
Simulink model was also used to confirm results before and after simulation to ensure realistic results were
achieved (and they were).
EXPERIMENTAL RESULTS
The following Tables summarize the key predicted and experimental results for EyasSat3
torque rod/Helmholtz cage operation.
16
Table 1. Calculating Earth’s Magnetic Field (T) at the Test Point
Measured via
FVM-400
Measured via
FVM-400
Measured via
FVM-400
Measured via
FVM-400
Mean
Field
(nT)
(nT)
(nT)
(nT)
(nT)
Bx
7
3
7
0
4.25
By
4
6
40
1
12.75
Bz
1
-1
10
2
3.00
Earth Field
Component
Table 2. Helmholtz Cage Solenoid 1: Current
(IHC1) vs. Induced Field (BHC1)
Trial
IHC1
BHC1
No
(A)
(nT)
n_HC1
(# of
turns)
1
0
0
N/A
2
0.1
12978
50.365
3
0.2
26000
50.451
4
0.3
39015
50.470
5
0.4
52034
50.484
6
0.5
65055
50.494
7
0.6
78063
50.492
8
0.7
91062
50.485
9
0.8
104040
50.470
10
0.9
117019
50.459
11
1.0
129973
50.440
Mean
Table 3. Helmholtz Cage Solenoid 2: Current
(IHC2) vs. Induced Field (BHC2)
Trial
IHC2
No
(A)
1
0
0
N/A
2
0.1
12962
50.303
3
0.2
25914
50.284
4
0.3
38861
50.271
5
0.4
51808
50.265
6
0.5
64727
50.239
7
0.6
77667
50.235
Mean
BHC2
(nT)
n_HC2
(# of turns)
50.266
50.461
17
Table 4. Satellite Magnetic Torque/Field Induced Dynamic Motion
Tr
HC
IHC
No
Use?
(A)
Predicted
Predicted
Mx
My
2
(mA m )
2
(mA m )
Meas
ured
Measured
Predicted
Measured
Mx
My
T0
T0
(μT)
(μT)
(s)
(s)
% T0
Err
(%)
1
No
0.0
0
124.57
0
47.0
97.47
55
77
2
Yes
0.5
0
124.57
0
93.1
86.27
32
170
3
Yes
1.0
0
124.57
0
186.2
16
350
4
Yes
1.5
0
124.57
0
279.3
71.96
12
425
5
Yes
2.0
0
124.57
0
372.4
56.74
10
467
Notes: 1: Izz = 0.005 kg m2 ; 2. k =1.49 x 10-5 N m
DISCUSSION
Having shown the experimental results through Tables 1, 2, 3, and 4, it is important one examine these results for demonstrated performance error as well as identify potential sources of error,
mitigation plans, and continued research/future work thereafter. Table 1 essentially shows, from
multiple measurements, that the residual magnetic field is near 0 T, thus negligible. Seen another
way, this result is also relevant to the problem of generating magnetic torque from the product of
a local (torque rod provided) magnetic field and the actuator itself. On the other hand, Tables 2
and 3 clearly demonstrate solenoid loop turn count estimation. For both coil1 and coil2, there are
a little more than 50 wire turns. More specifically, the top coil has an estimated 50.46 turns,
whereas the bottom coil has 50.27 turns. These values demonstrate the variability in two contractor provided parts (solenoid 1 and solenoid 2), but also how closely tied they are. These values,
then show why there’s significant error if the Helmholtz Cage wires are connected in parallel vice
connecting in series. Next, Table 4 hits at the heart of the present research. In these results, various input currents are sent to the Helmholtz Cage, thereby producing different magnetic field
magnitudes along predefined axes. Nevertheless, the dynamic equations illustrated back near the
start of the section major paper section have merit and help drive this requirements validation activity. Notice at this juncture the experimental error performance is poor. This has many potential error sources, including operator error in how one of the co-authors saved/logged data.
Other potential error sources for these tests include too many magnetically strong background
systems are generating residual fields that impact the result. Another is the timing process to
evaluate oscillation period is difficult to make precise. Nevertheless, very valuable results have
been shown and bringing these to bear appropriately in the laboratory is crucial. Next, there may
be error in letting the ground-based magnetometer work.
Follow on work for present effort will seek to improve upon on-board torque rod and magnetometer performance as well as Helmholtz caged demonstration/effectiveness and it will also focus on three key ideas: refining the specific performance results for the existing hardware, developing new hardware/software components, and extending the hardware to other tasks (e.g. build-
18
ing a three-axis Helmholtz Cage). Ongoing efforts to interpret/refine magnetometer will have
wide-reaching impact as the envisioned three-axis attitude determination algorithms rely on accurate magnetometer data, this data is relevant to driving in classroom Helmholtz coil operation, it
impacts torque rod command magnitudes/duty cycling, and helps monitor overall ADCS system
operation.
CONCLUSION
This work has shown promising results for employing magnetic sensors, actuators, and ground
test equipment in terms of both demonstrating magnetic field initiated device performance, but
also in imparting knowledge to the next graduating class of lieutenants. As one can see from the
data, the methods demonstrated are effective means for estimating as-built magnetic sensor/actuator sizing, for estimating overall spacecraft operation, and for capturing the key concepts
advanced physics textbook seek to highlight in relation to magnetic system operations. It is
shown here that averaging Helmholtz Cage solenoid coil estimates is quite effective and a great
way to access adequate information quickly.
ACKNOWLEDGMENTS
The authors acknowledge several key contributions from some significant individual. First and
foremost, the partnership between L’ecole de l’air and the USAFA Department of Astronautics
led by Col Martin France was crucial to success of this work. Two co-authors completed several
of these items during an 8-week program at USAFA during its summer process. Of significant
note here is that the resulting, internal research report produced by the co-authors was a foundational contributor to the many aspects of this paper. Although considered “internal production”
quality, this effort was fundamental to the results. Also of significant contribution was from Cadet Ayesha Hein, who through self sacrifice led a long effort to characterize and calibrate the onboard magnetometer. Another key contributor is Lt Col David Barnhart, USAF-retired. He provide significant funding, personnel, and opportunities in order to make the project great. A couple more people that helped this effort include Sara and Noel, undergraduates from the Nanyang
Technical University of Singapore. Their work studying magnetometer operation when attached
to a Parallax board as well as an Arduino board, was foundational in our understanding.
Finally, Hughes’ vectrix notation as well as the USAFA Academy “Spacecraft Attitude Dynamics and Control” (Astro 445) course note materials were heavily used and referred to. However, since these course notes are unpublished, one has to derive these equations from first principles using the developments in such textbooks. For this reason, it was more appropriate to include this source in this section rather than with the other listed references.
NOTATION
L
length, mm
D
diameter, mm
B
magnetic field strength, mN*m
I
current, A
A
flux area, mm2
19
n
number of turns per 10 mm sample
N
number of turns per length L
b
number of coils per cross-sectional area slice
REFERENCES
1
Ritchey, O., Barnhart, D., Sellers, J., White, J., White, T., and Clark, J., “EyasSAT:
Workforce—Today,” in 2nd Responsive Space Conference, Los Angeles, CA, 2004.
Creating a Progressive
2
Sellers, J and Barnhart, D., “EyasSat3: A Low-Cost On-ramp for Nanosatellite Programs,” in AIAA Space 2013
Conference and Exposition, San Diego, CA, 10-12 September 2013.
3
Hein, A., Richie, D., Sobers, D.M., Esselstein, R., Jones, D., and Vergez, P, “USAFA’s EyasSat3 and Hamster Ball:
Innovative Tools for Practical Hands-on Attitude Determination and Control Education,” in AAS Guidance, Navigation,
and Control Conference Proceedings, Breckenridge, CO, 30 January – 4 February 2015.
4
Richie, D., Sobers, D., Neo, C., and Tan, C., “Photocells for Small Satellite, Single-Axis Attitude Determination,”
Journal of Small Satellites, vol. 1, no. 4, pp. 285-299, June 2015.
5
Wie, B. Space Vehicle Dynamics and Control, American Institute of Aeronautics and Astronautics, 2nd ed., 2008.
6
Hughes, P. Spacecraft Attitude Dynamics. Mineola, New York: Dover Publications, Inc., 2004.
7
Wertz, J. Spacecraft Attitude Determination and Control, Boston, Massachusetts: Kluwer Academic Publishers, 3 rd
ed., 1999.
8
Larson, W. and Wertz, J. Space Mission Design and Analysis, Boston, Massachusetts: Kluwer Academic Publishers,
3rd ed., 1999.
9
Richie, D., Lappas, V., and Palmer, P. “Sizing/Optimization of a Small Satellite Energy Storage and Attitude Control
System,” AIAA Journal of Spacecraft and Rockets, vol. 44, no. 4, pp. 941-952, July/August 2007.
10
Griffths, D. Introduction to Electrodynamics, Upper Saddle River, New Jersey: Prentice Hall, 3 rd ed., 1999.
20

SATELLITE MAGNETISM: TORQUE RODS FOR EYASSAT

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