# The original ADCS design was to use three Aero Fins attached to

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```					       The original ADCS design was to use three Aero Fins attached to the back of the
satellite, and use them to stabilize the satellite much like a badminton birdie does. As the
satellite would tilt or turn, the fins would create a drag force in the opposite direction of
C D *  *V 2 * AP ˆ
the satellite’s velocity, expressed as: FD                      V . This drag force acts at
2

the satellite’s center of pressure, found by using the equation:                  . Since the
center of pressure is at a different location than the center of gravity, the satellite
experiences a torque, that returns it to the desired position, expressed as:
TAERO  rcp / cg  FD . This was the chosen design for the satellite’s original orbit of 500km

because at this altitude, the atmospheric density is high enough that the drag force
produces the dominating force.
However, as it goes with every other mission, if you can find a launch window,
you take it. The launch window we found was for 700km, a height at which the drag
force is no longer the dominating one. Although this altitude doesn’t cause negligible
drag forces, these forces aren’t large enough to actually stabilize the satellite. We must
therefore rely on torques from gravity gradients.          A gravity gradient utilizes two
connected masses, modeled as point masses. Each of these masses has its own center of
mass, as does the satellite as a whole.
Unless the satellite is oriented perfectly tangent to the earth, one of the two masses will
be slightly closer to the earth than the other. This difference may only be a meter, as
compared to the satellite’s orbit of 700km, however this small difference is enough to
create a small torque, which will cause the satellite to orient itself perpendicular to the
earth. This is precisely what we want. We need the bottom of the satellite to point
directly at the center of the earth, so by hanging a mass via a spring steel boom from the
bottom of the satellite, we get our gravity gradient.
While the idea of a gravity gradient sounded promising, we were still unsure as to
whether or not it would actually work. In our case, we will have torques from both the
gravity gradient, and the drag forces, but how big will each one be. The torque due to
drag is going to rotate the satellite in one direction, and the torque due to the gravity
gradient is going to rotate the satellite in the other, and at some unknown angle, the
satellite will reach an equilibrium. In other words, at some angle of rotation, the two
torques will be equal and opposite to each other. Since the satellite needs to meet the
pointing requirement of 30°, depending on how big the torque due to drag is, compared to
the torque due to the gravity gradient, the satellite may stabilize at an angle that doesn’t
meet specifications. Furthermore, both torques vary on the boom’s length, so not only
did we need to see if a gravity gradient would work, but for what boom lengths would it
work. I therefore had to do some approximate calculations and simulations.
To do these, I assumed that the satellite, the boom, and the tip mass were all point
masses. I found the approximate center of gravity of the satellite, and dimensions of the
tip mass in the ADCS document. I assumed that the satellite was a 10cm cube, and that
the tape measure and tip mass were symmetric about all three axes, making each center of
gravity located in the geometric center of the object. To use various lengths for the tape
measure, I used a loop in MATLAB, going from 0m to 3m, at increments of 1cm. In
each iteration of the loop, I found the entire satellite’s center of gravity from the top of

the satellite using the center of gravity equation:                   . Once I found the
center of gravity, I redefined the coordinate axes to a radial/tangential system with the
center of gravity as the origin. Doing this made future calculations much simpler.
Then I needed to find the center of pressure as the satellite. Since I wasn’t sure
exactly where the satellite would stabilize, I found the center of pressure as the satellite
would sweep from -45° to 45° by using another loop. Using the previously stated
equation for center of pressure, I calculated the various facing surface areas as functions
of theta. Once I had the center of pressure, I could find the torque due to drag for each of
the iterations of the loop for the sweep of the satellite. Using a coefficient drag value of
2, a circular velocity of 7504 m/s (from Larson&Wertz), an atmospheric density of
1.47x10-13 kg/m3 (from Larson&Wertz), and my previously calculated total surface area, I
found a matrix of torques due to drag.
Then I found a matrix of torques due to the gravity gradient. The equation for this

torque is:                                    .   Iz was taken directly from the ADCS
document, but since the three point masses would change the satellite’s moment of
inertia, the moments due to those point masses needed to be added to the Ix value from
the ADCS document to get a total Ix. I found these three moments using the equation
I=ML2. Taking the μ value from Hale as 3.986x1014 m3/s2, and an R of 7078km being
the radius of the earth plus the altitude of the orbit, I can then get a matrix of torques due
to the gravity gradient as the satellite would sweep from -45° to 45°.
Since we are trying to find the angle where the satellite reaches equilibrium, being
the angle where torques are equal and opposite, I summed the two matrices. Since this
new matrix is basically the difference between the two torques, the minimum value in
this matrix would be where the two torques are equal and opposite. Ideally we’d want
the minimum value to be zero, but this isn’t likely to happen b/c of the step size of the
angles that I used. Furthermore, since we want the very smallest difference between the
two torques, we actually want the minimum value of the absolute values in the difference
matrix. Once the minimum value was found, I was then able to work backwards to figure
out which angle created the equilibrium stabilization, and stored the value in a matrix.
After doing this for each increment of boom length, I was able to come up with the
following plot:
This graph gave us exactly what we were hoping we would see. Starting at a boom
length of about 1m, the satellite reaches an equilibrium of about 17°. Although the plot
shows that at small boom lengths, the satellite wouldn’t meet specifications, the model
doesn’t quite apply to these lengths anyways. Therefore, since 17° is completely within
our pointing requirement, we chose to go with a boom length of 1m.
The next big issue is the possibility of the satellite stabilizing in an inverted
position. The communications antenna is on the bottom of the satellite (the side with the
gravity gradient), while the GPS antenna is on the top of the antenna. Therefore, when
the satellite does reach equilibrium, we want it to stabilize with the communications
antenna pointing towards the Earth, as demonstrated in the following picture:
However, it is also possible for the satellite to stabilize with the communications antenna
pointing away from the Earth, as demonstrated in the following picture:

At first, this was thought to be an unstable equilibrium, similar to the unstable
equilibrium of an inverted pendulum. However, after further analysis, it was determined
that not only is this a stable equilibrium, but that the satellite is just as likely to stabilize
in the inverted position as the desired one. Slightly modifying my original code to
account for the inversion, I ran it to make sure that the code would work for the inverted
position. Not only did it work, but it also told us that the satellite’s equilibrium angles
are phase shifted by 180 degrees when it’s inverted, which was exactly what I expected.
Now that we know that the satellite could wind up inverted, we need to further
explore figuring out how to check to see if we are inverted, and if we are, how to flip
ourselves. This is where the magnetometer and torque coils come into play. Although
the complete control law has yet to be written and tested, the intention is to use the
magnetometer to see if the earth’s magnetic field is pointing in the right direction or not.
If we’re inverted, the magnetic field will seem reversed, and we’ll know we have to flip.
As for the actual flipping process, there are three main possibilities, all of which involve
the torque coils. The first is to produce enough torque so that it rotates around its side:
The other two possibilities would be to produce torques meant to either oppose the
gravity gradient’s torque, or the aerodynamic torque. If we produced enough torque so as
to basically eliminate one of these two, the other would be able to dominate, and
therefore flip the satellite. For all of the possibilities, we needed to make sure that the
torque coils could produce enough torque to actually make the flip. While the exact
amount of torque needed for each method hasn’t been determined yet, the worst case
scenario would be that we’d need to produce a torque exactly opposite and equal to either
the torque created by the gravity gradient, or the torque due to drag. Looking at the
values at the equilibrium position, we find values equal to 4.7206e-008 N and 7.2948e-
008 N for the aerodynamic and gravity gradient torques respectively. Looking at table
8.5.2 in the ADCS document, the torque coil configuration we are using can create a
maximum torque of 4.2889e-006 N in any direction. Since this torque is greater than
either of the two worst-case scenarios, we know that we can in fact flip the satellite.
Since using the torque coils uses precious power, we will now need to figure out exactly
how much torque will be needed for each possibility to work, so that we can figure out
which method the least amount of power. We also need to figure out which of these will
be the most efficient in terms of control law programming and execution. Once we
weigh all of the issues, we can determine which method to use, and therefore establish
more specific details about how we will actually flip the satellite.
Now that we knew the gravity gradient would seemingly give us our desired
stabilization, it was time to either design or redesign various components in order to get a
deployable prototype of the gravity gradient. The first component I looked at was the tip
mass. The tip mass I was provided with was a rectangular piece weighing close to 40g.
This had a number of problems. First of all, the fact that the rectangular piece has
corners means that it is more likely to catch on the housing or satellite back panel, which
would therefore mean we’d also need to create some form of spring loaded release
system. Secondly, we want to maximize our tip mass weight, and although we certainly
have limits to how heavy it can be because of the mass budget, we should still go heavier.
Finally, we’d be storing this mass inside of a coiled length of tape measure, so it would
be an inefficient use of space to put a square inside of a circle. I therefore made the
obvious choice, being to redesign the tip mass so that it’s a cylinder with a thickness
equal to the width of the tape measure. From the ADCS document, I found that the outer
diameter of the tape measure in its coiled stored position would be .0254m. Knowing
that the thickness of the tape measure is .0001m, and that we’d need a total length of 1m,
I was able to integrate to find the inner diameter of this coil to be .0226m. This would be
the maximum possible diameter of our tip mass. To allow for a little leeway, and for
convenience, I decided to slightly reduce the diameter to .875in. In order to actually
machine this piece, you must first cut down the metal until it is close to .875in on each
side. Then, use a center drill to put a small hole in the center of the piece. Since the
piece is so thin, you can’t actually put it in the chuck of a lathe. Instead, you can place
the piece against the chuck of a lathe, and using the centering tool, apply as much
pressure to the piece as possible. Once you have locked the centering tool in place, if you
turn on the lathe, the piece will spin as though it were in the chuck itself, and then it’s just
a matter of turning it down until it’s a circle with a diameter of .875in.
Once the tip mass was machined, there was the issue of how we’d actually attach
it to the tape measure. After looking at a few ideas, I finally settled on a combination of
my two favorite possibilities. We could insert the tape measure into very thin slice cut in
the tip mass, and then secure it by putting a screw through the top, and then through the

tape measure.                      . In order to make the thin slice, I actually needed to go
is much wider than the tape measure, whereas ADFF could make a slice using a saw with
a thickness of .012in. This slice would go all the way to the middle of the tip mass, and
would go through the entire thickness to accommodate for the insertion of the tape
measure. The screw hole, however, was doable by me in Emerson. First, I needed to
mill a small .25in flat on top of the tip mass, and about half way to the center. Once the
flat has been milled, it’s just a matter of picking any screw size you wish, looking at the
chart to see which drill bits and taps you’ll need to use, and then drilling and threading
the hole. This hole should go deeper than the slice, but the exact depth doesn’t really
matter as long as there’s enough clearance to get the screw all the way through the tape
measure. When threading the hole, remember to account for that fact that the bottom of
the hole will be a cone, and not a flat surface, so you will need to make sure that you go

deep enough to account for this loss in length                    . The weight of my fully
machined tip mass is about 76g.
After testing this attachment method, I found a fairly large problem. When we
fold the tape measure so that it tightly wraps around the tip mass, we cause the tape
measure to make a very abrupt right angle where it comes out of the slit. This actually
causes the tape measure to permanently deform. Upon further testing, I found that the
tape measure had a very big problem with this particular bend. It only took 3 complete
cycles (bending the tape measure completely one way, and then completely the other) for
it to break along the entire width, and only 2.5 cycles for it to start to tear. This basically
means that we have to either redesign the tip mass, or change the way we attach the tape
measure to the tip mass. For the first choice, we would basically machine a little curve

into the edge of the slit to make for a less abrupt fold                . The other choice is
to just entirely ignore the slit, wrap the tape measure around the tip mass once, screw it
into place, and then finish wrapping it around. The first method seems like it might be
more secure, but the second is much easier to do. Both of these, as well as possible other
methods should be further explored.
Another big change to the gravity gradient design was its housing. All I was
given were aero-fin housings, all of which have holes for three coiled tape measures.
However the gravity gradient only needs one hole. I therefore needed to redesign the
housing, using the same outer dimensions as before, but oriented towards the single coil
of the gravity gradient. The following is a drawing of the current dimensions to the new
housing (all units are in inches).
I couldn’t find the exact locations for the mounting holes, or how far down they need to
be recessed in order to attach the housing to the back panel, so this still needs to be done.
Furthermore, for this housing, we decided it was better to go with aluminum than printed
plastic. After machining the housing, it is weighs approximately 60g. Since we do have
a mass budget, this weight should be reduced. This can be done by adding “lightening
holes.” These holes go through the side of the piece, have no other purpose than to
lighten the piece, and essentially make the piece look like Swiss cheese. These holes do
need to be out of the way of the screws, but other than that, they can basically go
anywhere as long as they aren’t so big that they’ll compromise the housing’s strength.
After putting these lightening holes in the housing, it should weigh about 30g. As a side
note, a new back panel will need to be machined to accommodate for the single hole in
the housing, and for its new location.
The general idea for the deployment is that the tape measure will be secured
within the center of the satellite, run through a small slit in the bottom of the antenna,
where it will then be coiled, wrapped around the satellite, stored in the housing, and
secured by a burnwire.
There are therefore a couple of things to point out. First of all, there needs to be some
sort of thin wall within the satellite that we can attach the end of the boom to. This can
be as simple as thin strip of aluminum with a small hole in the center suited for a screw or
bolt. Another thing is that we shouldn’t use normal washers or bolts to secure the tape
measure. The curvature of the tape measure is very important to maintaining the stiffness
of the spring steel. If we were to use normal washers, and tightly press them against the
tape measure, this curvature would be lost. Therefore, a special set of washers will need
to be constructed, each with a curvature having a diameter of about 1.2375 inches

. I had time to make a quick prototype of the left washer, but never made the
right one, so this piece still needs to be made.
Another big thing is that we need to put a small slit in the center of the antenna.
This slit doesn’t need to be any bigger than 1/16in x 1/2in, but it can be if necessary. The
important thing about putting a hole in the antenna is to avoid putting this hole on, or near
the plated areas. This really shouldn’t be a problem since the antenna was designed to
allow for a hole in the center of it. However, if for whatever reason the hole has to be
moved, we have to make sure to leave at a very minimum .05in clearance around the
plated areas.
Finally, there is an issue with the deployment itself. When the burnwire melts,
and the coil is released, it will very quickly unwind with a lot of momentum. With no
gravity, and little aerodynamic resistance, the boom will to some degree continue to wrap
around the satellite. This could cause a number of problems. The first is that I don’t
know how much of it will wrap around, and if it does indeed wrap around, will it be able
to unwrap itself. If it doesn’t, something will need to be done in order to not only realize
that the boom is wrapped around the satellite, but to then somehow, unwrap it. The next
concern is that if it is going to wrap around the satellite, we need to figure out how much
it will manage to do so. When it stops winding itself around the satellite, the tip mass
will likely smack into one of the sides. If it hits one of the solar panels, it could cause the
solar panel to shatter, which would be a major problem. However, the antenna is a very
sturdy piece, and even with a little crack, its performance wouldn’t be severely affected.
It would therefore be a wise idea to modify the length of the boom, if needed, so that
when the tip mass does smack into the satellite, it hits the antenna. This winding process
could wind up being a very big problem, so it should definitely be looked into. Also, the
solar panel flap should be deployed after the gravity gradient, because if it were to be
extended before the tip mass winds around the satellite, it would likely be hit at some
point, and therefore shatter. For a more complete explanation of this particular problem,
talk with Mike Hammer.

The following is my MATLAB code:

%THIS IS THE MODIFIED COORDINATE VERSION

clear

SSTheta=zeros(301,3);
for Lgth=0:1:300
L=Lgth/100;
%This value is the Mass of the satellite w/o the fin elements + the
tip mouse housing
MassA=.85354+.0080;
%This value is 7 grams per meter
MassB=.0070*L;
%This value is the 40 gram tip mass
MassC=.0400;

CgA=.05+.012951;
%Length of satellite + 1/2 length of tape
CgB=.1+L/2;
%Length of satellite + length of tape + 1/2 height of tip mass
CgC=.1+L+.004375;
%This is the distance from the center of gravity to the top
Cg=(MassA*CgA+MassB*CgB+MassC*CgC)/(MassA+MassB+MassC);

%Moments of inertia from the three masses
MomentA=MassA*(Cg-CgA)^2;
MomentB=MassB*(Cg-CgB)^2;
MomentC=MassC*(Cg-CgC)^2;

%For Radial Coordinate   System Where Cg Is Origin
%Dist from Center of A   to Cg Along Axis of Sym.
DistA=.05-Cg;
%Dist from Center of B   to Cg Along Axis of Sym.
DistB=.1+L/2-Cg;
%Dist from Center of C   to Cg Along Axis of Sym.
DistC=.1+L+.004375-Cg;

for i=0:1:720

%We start at -45 degrees and go through 45 degrees, searching for
SS Angle
theta=-pi/4+i*pi/1440;

%The surface areas of the three parts
SaA=.1*.1*cos(theta)+.1*.1*sin(abs(theta));
SaB=.0127*L*cos(theta);
SaC=.0169*.00875*cos(theta)+.0169*.0169*sin(abs(theta));

%The radial-component of the distance between Cp and the center
of the satellite
%Using the Dist values, we are thinking as though Cg is now 0
Cp=(SaA*DistA+SaB*DistB+SaC*DistC)/(SaA+SaB+SaC);
%Calculation of TD
Cd=2;
V=7504;
SA=SaA+SaB+SaC;
rho=1.47*10^-13;
FD=-Cd*rho*V^2*SA/2;
TD(i+1)=FD*Cp*cos(theta);

%Calculation of TGG
Ix=0.0020996+MomentA+MomentB+MomentC;
Iz=0.0019676;
TGG(i+1)=3*3.986*10^14/2/(7078000^3)*abs(Iz-Ix)*sin(2*theta);
end
difference=abs(TD+TGG);
[Torque,index]=min(difference);
SSTheta(Lgth+1,1)=Lgth/100;
SSTheta(Lgth+1,2)=Torque;
SSTheta(Lgth+1,3)=(-pi/4+pi/1440*(index-1))*180/pi;

end
plot(SSTheta(:,1),SSTheta(:,3));
title('Steady State Angles vs. Length of tape');