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    Re: FOG's, was Re: automatic celestial navigation
    From: Frank Reed
    Date: 2008 Jan 29, 01:58 -0500

    George, you wrote:
    "Nevertheless, isn't it the case that there is effectively zero long-term
    drift in the orientation sensing of these devices; quite different from the
    behaviour of their mechanical predecessors?"
    No, they drift for many little reasons. But they're a lot cheaper than the
    fine mechanical gyros that they have mostly replaced. Also, they're getting
    better all the time. So in ten or twenty years, it may well be said that
    "for all practical purposes, they have no drift".
    Nicolas sent me a brief message saying he's gotten busy with work so he
    can't get back into this discussion right now. But he did send me some docs
    on the FOG system he uses so I'll see if there's anything of interest in
    It occurs to me that it's worth mentioning that there are two fairly
    distinct ways of doing inertial navigation. There's the "pure" approach
    where we maintain three fixed directions in space. It's the sort of thing
    that's well-suited to a spacecraft. Then there's a "gravity" approach where
    the system tilts the platform to keep it level based on the calculated
    current position. If we travel ten nautical miles along the equator, the
    system turns the platform ten minutes of arc in the same direction (the
    "platform" can be a real table in gimbals or it can be a "virtual platform"
    in software). This has the advantage that a fixed leveling error leads to
    oscillatory errors in position: errors obey a harmonic equation (at the
    84-minute local tidal period** at the surface of the Earth), but to do this
    it sacrifices accuracy in the vertical direction: errors propagate
    anti-harmonically. So altitude above (or below) sea level has to be
    determined (or constrained) in some other way.
    And back to the point, can we use the determined level/vertical for
    celestial navigation as an independent check on position? The answer again
    is 'no', or at best 'not really'. The error in the vertical is the same as
    the error in the position. If my calculated position is oscillating back and
    forth with an amplitude of two nautical miles, then my calculated vertical
    is oscillating back and forth with an amplitude of two minutes of arc. But
    the stars CAN be used as a compass, drastically reducing any error that
    results from gyro drift. They calculate where the stars should be, based on
    the gyros, then they automatically point the star tracker towards that
    location and do a little spiral search pattern for the star (with a fixed
    star tracker, this is all done in software). If it's thirty seconds of arc
    off-center, then the platform is mis-aligned by that amount. So you feed in
    that error and re-do the calculation. That's what these "stellar-inertial"
    systems apparently do (I say apparently because I don't really know what
    they do --some of them, after all, are highly classified). On the other
    hand, the FOG devices that are not serving as inertial navigation systems,
    like the one Nicolas has described, might still be used for a celestial
    navigation vertical. That part I'm still thinking about... It all depends on
    how they do it.
    **Above I mentioned the tidal period. So there's no confusion, I'm not
    talking about ocean tides due to the variation in the gravitational forces
    of the Sun and the Moon over the surface of the Earth, but rather the local
    "tidal" acceleration in the vicinity of any point on Earth due to the
    variation in the gravitational force of the Earth itself. For an example,
    imagine an air table with a disc on it ("air hockey" with a floating puck).
    If I level the table perfectly at its center, and if the table is perfectly
    flat, then the floating disc, if released near one edge, will "fall" towards
    the center because that's the only place where the gravity vector is exactly
    perpendicular to the table --everywhere else, the vector is slightly tilted,
    pointing towards the center of the Earth, so a small component of the
    gravity vector is pointing towards the center of the table. The amount is
    directly proportional to the distance from the center of the table. The disc
    will execute simple harmonic motion [natural frequency
    omega=sqrt(g/R)=sqrt(GM/R^3)], as if it's attached to the center of the
    table by a perfect spring, gliding back and forth across the table with a
    period of 84 minutes (ignoring air resistance). Give it a slight sideways
    motion and the disc will travel in a circular or elliptical path around the
    center of the table with that same 84 minute period (no matter how big the
    circle). That's the "tidal field" of the Earth.
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