VIEWS: 1 PAGES: 12 POSTED ON: 3/27/2011
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 to ADFF, because the smallest slice that can be made in Emerson is about 1/16in, which 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; %From ADCS Final Design 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'); ylabel('Steady State Angle (degrees)'); xlabel('Length of tape');