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A while back, I asked Nicolas if he knew how the attitude system that he has
used figured out the local vertical. Reading through the documentation, I'm
relatively convinced that this is done by averaging (a bit more
sophisticated mathematically, but it's fundamentally averaging).

The device is a full-blown IMU (inertial measuring unit) with three fiber
optic gyros and three accelerometers. This IMU could be used for inertial
navigation, and in fact, the specific manufacturer, Ixsea, has one of each:
an IMU for attitude determination and an IMU for inertial navigation. The
former assumes that you already have a GPS for navigation, so it ignores the
navigation solution and uses the data coming from the IMU to determine the
local vertical, true heading, roll, pitch, and yaw, etc. Basically, to do
this, it looks for the direction in space with the largest acceleration over
a period of five minutes or so. That has to be gravity, because any vessel
at sea will not have any significant accelerations (that we can't correct
for) lasting longer than that. The vast majority of accelerations will be
oscillatory and will drop out of the averaging process. In addition, the IMU
observes the change in this vector over time relative to inertial space
caused by the rotation of the Earth. This "delta-g" vector points
horizontally east-west so we get true heading out of it directly.

So what happens when the vessel really is accelerating, steadily picking up
speed or steadily turning for some rather long period of time? Imagine a
vessel accelerating from 0 to 30 knots over the course of sixty seconds,
maybe a high-speed ferry. That would introduce an apparent horizontal
acceleration, pointing toward the stern, which would be indistinguishable
from gravity. If it lasts a minute and our averaging time is five minutes,
that could throw off the vertical by a good fraction of a degree. But we can
deal with that simply by feeding the vessel's indicated speed through the
water back into the device. Though the resulting acceleration is not exactly
equal to the true acceleration (due to currents), it's enough to remove
almost all of the error. The vertical would be accurate within 1 or 2
minutes of arc. And suppose the vessel is making a 90 degree turn to
starboard at high-speed. Here we would have a horizontal acceleration --an
apparent addition to gravity pointing to port. But the IMU can directly
detect the steady change in the vessel's heading so when this is combined
with the (unchanging) speed through the water, once again we can subtract
out the acceleration.

Another problem with a vertical produced by averaging is that it's really
telling us which gravity was pointing a couple of minutes ago. But again,
since the IMU knows the heading and we feed in the speed, we can "get ahead
of" the average vector. That is, if my gravity vector is "stale" by two
minutes, and I'm travelling due north at 30 knots, then I need to rotate it
toward the south by one minute of arc. Short term accelerations relative to
these long-term stable vectors, gravity and true north, then yield roll,
pitch, heave, etc. which are the primary outputs from this device.

Anyway, that's my best guess. Devices like these are "black boxes". The
computer code that does the work is proprietary.

Getting back to a hypothetical question that got us off on this tangent,
could you use the vertical determined by the IMU for celestial navigation? I
would say absolutely, yes. You could measure angles relative to the vertical
coming out of the device, and from those zenith distances yield a
"traditional" celestial fix accurate to about a minute of arc. But so far, I
can't think of any way that this would be useful, apart from amusement. Can
anyone else? An ordinary bubble attachment for a sextant will give the
vertical to +/-6 minutes of arc or so and costs perhaps $25 second-hand.
This IMU can do between 3 and 10 times better but costs thousands of
dollars. Maybe if a system like this could be improved by two orders of
magnitude, then you could get a mixed celestial-inertial fix that would be
competitive with a GPS fix... Just maybe.

 -FER

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