I'm finding an interesting quandary in using an inertial system for
calculating rocket altitude:
An accelerometer alone is unsuitable to determine changes in altitude over
distances where the vertical decrease of gravity is significant, such as
for aircraft and rockets. In the presence of a gravitational gradient, the
calibration and data reduction process is numerically unstable.
I wonder how this is resolved with launch vehicles?
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Ed Hahn writes on the sci.aeronautics.airliners newsgroup:
The IRS (inertial reference system) is unable to give an accurate position
OR speed in the *vertical* dimension due to an effect called the
"Gravitational Gradient".
If you take the accelerometer's reading and integrate it once, you get
the velocity.
If you integrate that velocity once more, you get the position. Of course, you
have to worry about constants - but the alignment procedure takes care of that.
Unfortunately, while this works great for the two accelerometers in the
horizontal plane, the vertical axis has a problem: in steady level flight, the
vertical accelerometer will still experience gravitational accelertion (~1g).
However, again thanks to Newton, we know that the acceleration due to
gravity falls off as the aircraft climbs higher (f = Gmm'/r^2). There
is, in other
words, a *gradient* to the gravitational acceleration with altitude.
So, if a small bias (error) exists in the vertical accelerometer output, the IRS
cannot tell whether this is due to the aircraft changing altitude
within the gravitational gradient, or whether this is just a problem
in the signal. This leads to divergent behavior of the output.
For example:
1) Say the bias makes the accelerometer read slightly high (i.e. indicating a
slightly greater vertical acceleration from gravity than is really present).
2) Since the IRS cannot detect this, it assumes the aircraft is accelerating
upwards slightly (i.e. climbing).
3) Because the aircraft is moving away from the centre of the earth, the IRS
then *must* assume that the acceleration due to gravity should also decrease.
(The effect of the gravitational gradient.)
4) But the accelerometer output hasn't changed because it really hasn't moved
vertically.
5) Since the output is the same, the IRS must assume that there is an additional
real acceleration in the aircraft upwards.
This cycle (2-5) continues until the velocity and position solutions blow-up
divergently. Note that if the bias was in the other direction, the solutions
would still diverge, but in a downward direction. Since you can't build a
perfect accelerometer, you're out of luck.
This effect is why barometric altitude rate is always the source of
the source of
vertical speed.
