Three Station Tracking - Altitude Only Method

In the "Handbook of Model Rocketry" By H. Stine there is mention of a 3 station tracking method to determine the altitude of your rocket. It
makes use of only the measured angle of the rocket at peak altitude (with respect to the ground) and a baseline that is a specific length that has an observer at each end and one observer exactly in the middle. In the appendix there are tables that you plug your measured angles into and, like magic, you get the altitude of the rocket. There is no attempt to show the derivation of these tables or even the method for deriving them.
My question is, has anyone done the math to derive the equations used to generate the tables in the book? I was trying to do this and got stuck at a point with 3 equations and 4 unknowns. I'm not sure what I'm missing but I'm sure there is some trigonometric identity or some characteristic of triangles that I am overlooking. If anyone can offer some guidance or a pointer to where the equations are it would be greatly appreciated.
Thank you, Michael Nycz
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I can't really comment on GHS's works. I could probably derive similar stuff, but gosh, what if your flight is not directly over your baseline?
I have published n-station Elevation only tracking equations, but I did not publish the solution. To be roughly equivalent to two station AZ-IL tracking, you need four independent EL only measurements. There are three unknowns to solve for, the estimated rocket position, e.g. x, y, and altitude. You need at least three measurements to solve, but you also need at least one more measurement as a check on accuracy or closure. You solve a ll east squares problem for x, y, and altitude, that minimizes the sum of the squares of the distance between x,y, and altitude, and the n line of sight vectors.

Sorry, I'm unwilling to lay it all out for you here in ASCII. There are some NARTS pubs that you may want to look at.
Alan
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Alan Jones wrote:

Stine didn't derive the 3-station elevation-only tracking method, he just reported it in his book. J. Talley Guill developed the method in 1972.
The method doesn't depend on the apogee being right over the baseline.
Two-station elevation-only doesn't give a unique altitude, of course, but it does produce an arc as it's "solution". A third station elevation is enough to pinpoint the apogee on that arc.
I don't have the derivation of the tables, sorry Michael. It's possible Guill published the method in an NAR magazine (or maybe an R&D report) in '72 or '73 or thereabouts.
Steve Humphrey
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wrote:

Doh, That should be the n cones defined by the elevation measurements, not LOS vectors.

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Had this lying around:
Let T1, T2, and T3 be the trackers from left to right.
Let T2 be situated at the origin. Let A be the
distance along the ground from T1 to the point below
the rocket, R. Let B be the same distance from T2,
and C be that distance from tracker 3. Then,
A*Tan(Theta1) = Z
B * Tan(Theta2) = Z
C * Tan(Theta3) = Z
where the thetas are the elevation angles, and Z is
the rocket's altitude.
From simple trigonometry,
A == SQRT{(BASELINE + X)^2 + Y^2}
B == SQRT{X^2 + Y^2}
C == SQRT{(BASELINE - X)^2 + Y^2}
For convenience, define the following
ALPHA == 1/Tan^2(Theta1)
BETA == 1/Tan^2(Theta2)
GAMMA == 1/Tan^2(Theta3)
Substituting in the above and rearranging:
1) BASELINE^2 + 2*X*BASELINE + X^2 + Y^2 = Z^2* ALPHA
2) X^2 + Y^2 = Z^2 * BETA
3) BASELINE^2 - 2*X*BASELINE + X^2 + y^2 = Z^2 * GAMMA
The second equation can be substituted into the first and third, yielding
4) BASELINE^2 + 2*X*BASELINE = Z^2* (ALPHA - BETA)
5) BASELINE^2 - 2*X*BASELINE = Z^2 * (GAMMA - BETA)
Adding equations 4) and 5) and solving for Z, we get
Z = BASELINE * SQRT{2/(ALPHA + GAMMA - 2 * BETA)}
Subtracting the equation 5) from 4), we get
X = Z^2 * (ALPHA - GAMMA) / (4 * BASELINE)
where Z is taken from the equation above.
Using equation 2), we can solve for Y as
Y = SQRT{Z^2 * BETA - X^2}
where Z and X are already known.
The answers become extremely sensitive to angular error as the three angles get closer to each other. Equivalently, the answers can become very poor when the rocket is far from the baseline. Best answers occur when the rocket is somewhere over the head of the middle tracker, and where the distance between trackers is great.
Regards, -Larry Curcio

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Thank you Larry! I now see the trick I was missing. By making the triangle lying flat on
the ground into a right triangle by adding and subtracting the value X makes everything work out. That was the part I was missing... How to incorporate the baseline into the equations. Thanks again for your help.
Michael Nycz
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Larry Curcio wrote:

The three angles won't be close except with extreme situations, i.e. when the apogee is extremely high or the rocket travels horizontally extremely far from the baseline. Other typical model rocket tracking methods suffer the same problem.
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Alas, this assessment is at variance with my experience. Unless the experiment is well engineered, three-station elevation-only tracking is extraordinarily sensitive to errors in angular measurement. To test for sensitivity in data, suggest that each angle in turn be increased and decreased by 1 degree. If the resulting altitude changes a lot, then you have problems.
Would also suggest that the tracking line be parallel with prevailing wind.
Luck and Regards, -Larry Curcio
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wrote:

Nice work. I'd be interested in knowing what your experience was. Why did you use co linear three station elevation only tracking? Was this done just for R&D, or was there some particular reason?

I've never given much consideration to elevation only tracking, other than just be prepared in case you get stuck with a class of students armed with Altitracks and no theodolites. Conventional wisdom is to lay out the baseline for convenience to the terrain and range operations, with some consideration for sun tracking angles. My inclination would be to locate the trackers (as many as practical) spaced out radially around the launch site, but then I'd also crunch the numbers differently.
ALan

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Hi, Alan.
Years ago (1993) I was doing a project that used ascent time data instead of drag coefficient as an input to simulations. The idea was to adjust the drag coefficient until the simulation yielded the observed ascent time. The result was not only a flight summary Cd, but also an estimate of altitude. The nice thing about altitude is that it is observable, so that the method can be verified or refuted.
At the time, I was a born again grad student, working for a professor several years my junior. The professor, as it happened, had a son who was interested in rockets, and his son had friends. I had little money for equipment, but we had enough people to do three-station elevation-only tracking.
We tried the technique, but we didn't want the kids too near the rocket when it went off. We displaced the launch pad from the line quite a bit. At the time, I thought the only concern was to have a large distance between trackers. This we had. It didn't help. The data were too sensitive to be of use. Reviewed the math and discussed the problem on RMR and CompuServe. (I still have the data, but they are on big diskettes.)
Never actually repeated the experiment with three station tracking. Never, IOW, did a properly engineered experiment. I assume it can be done, but in fact I have no direct experience. (Eventually resorted to single station tracking with long baselines on perfectly calm evenings.)
More trackers and different layouts might indeed help. Never did as much of the math as you did. You are certainly right in implying that the biggest practical consideration is usually the real estate you have.
Altitracks, BTW, have a nasty habit of shifting a couple of degrees when the trigger is released. Found we had to hold the cursor in place with the free hand before releasing the trigger.
Regards, -Larry Curcio

wind.
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NOTE: Having come com trouble. Apologies in the event that this post appears twice.
Hi, Alan.
Years ago (1993) I was doing a project that used ascent time data instead of drag coefficient as an input to simulations. The idea was to adjust the drag coefficient until the simulation yielded the observed ascent time. The result was not only a flight summary Cd, but also an estimate of altitude. The nice thing about altitude is that it is observable, so that the method can be verified or refuted.
At the time, I was a born again grad student, working for a professor several years my junior. The professor, as it happened, had a son who was interested in rockets, and his son had friends. I had little money for equipment, but we had enough people to do three-station elevation-only tracking.
We tried the technique, but we didn't want the kids too near the rocket when it went off. We displaced the launch pad from the line quite a bit. At the time, I thought the only concern was to have a large distance between trackers. This we had. It didn't help. The data were too sensitive to be of use. Reviewed the math and discussed the problem on RMR and CompuServe. (I still have the data, but they are on big diskettes.)
Never actually repeated the experiment with three station tracking. Never, IOW, did a properly engineered experiment. I assume it can be done, but in fact I have no direct experience. (Eventually resorted to single station tracking with long baselines on perfectly calm evenings.)
More trackers and different layouts might indeed help. Never did as much of the math as you did. You are certainly right in implying that the biggest practical consideration is usually the real estate you have.
Altitracks, BTW, have a nasty habit of shifting a couple of degrees when the trigger is released. Found we had to hold the cursor in place with the free hand before releasing the trigger.
Regards, -Larry Curcio

wind.
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wrote:

I did see the first post. I found it to be complete, with nothing further to say except thanks, and that I can relate to your grad school experience. :) So thanks again, and I apologize for thanking you twice. ;)
Alan
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