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    Re: Unistar Principle etc.
    From: Alexandre Eremenko
    Date: 2012 Apr 21, 00:56 -0400

    Dear Kermit,
    
    > Please do explain the Unistar principle to us.
    
    I'll try. I cannot say that I recovered all mathematical details.
    (And even less, the engineering details).
    As I said, there is no satisfactory exlanation in the literature.
    
    > And I bet that your will find that the star which
    > is to bring the highest accuracy update is  ... just the one on top of
    > the target when the missile reaches it.
    
    I afraid your bet is wrong:-) The adjustment from the sight is made
    just before the engine switches off. That is about 6 min after launch.
    So it is the position of the star at the time of launch, at the place
    of launch which is relevant. They point the "sextant" at the star
    before the launch.
    The star you propose will be too low, (and even can be obscured by the
    the Earth, if the target is 6000 miles away at this moment:-)
    
    According to one account, the best star is in front of the of the missile
    at the moment when the engin is stopped.
    My "theory" shows that any star which is reasonably high over the horizon
    and has approximately the same azimuth as the target will work.
    (Will work from the mathematical point of view. There are also engineering
    restrictions on the size, placement and width of the window in
    the missile).
    
    But let me try to explain what I understand. There are 9 parameters
    whose adjustment can be relevant:
    3 coordinates of the missile (say, latitude, longitude and altitude).
    3 components of velocity.
    3 parameters of what they call "attitude" which characterize the
    orientation of the missile with respect to the stars (more precisely,
    the orientation of the gyroscopic platform).
    
    The first 6 parameters determine whether the missle will hit the target or
    not, and their value at the moment when the engine switches off is
    important. (The engine works for about 6 minutes, after which
    you cannot change anything). Shortly
    before this moment, the star is sighted,
    and the last 3 parameters (the attitude) are only
    needed to recompute the
    trajectory and adjust the first six.
    
    The observation of one star gives you two numbers (for example,
    altitude and azimuth). If you measure altitude and azimuth of one
    star exactly, with respect to the frame connected to the Earth,
    then you derive your position exactly.
    The problem is that you measure not with respect to the Earth related
    frame but with respect to your gyroscopic platform frame, which can
    be off by some error.
    
    The problem is how you adjust many (9) parameters from only 2
    measurements. One account says that they do this statisticaly.
    The relative importance of these parameters is not the same.
    They collect the empirical data from the tests and make
    on the base of these data a 9 times 2 "correlation matrix"
    which gives
    adjustment of 9 parameters from 2 measurements. This looks like a very
    naive and non-efficient way to solve the problem,
    and I do not believe that this is really the way it works.
    
    Here is a simple argument which shows that information in two
    parameters from the star sight should be sufficient for exact adjustment.
    Suppose you do not adjust anything. Then you miss,
    your missile hits the ground somewhere near the target.
    Where exactly it hits is described by TWO numbers: miss in distance and
    miss in direction.
    That's why two adjustments must be enough:-)
    According to some authors, it is only the direction of the velocity vector
    that is adjusted. The optimal star for this is in front of the missile.
    
    When I look at this more closely, I find the following.
    First of all, 2 of the 9 parameters cannot be corrected from any number
    of star sights, namely the altitude of the missile over the sea level,
    and its speed (speed is the magnitude of the velocity vector, it is
    just a number, mesasured in miles/sec).
    Because from the two missiles flying at different altitudes
    at different speeds, the sky looks EXACTLY the same.
    Of course, altitude does not really have to be adjusted from the
    star sight, because there are other good ways to measure it.
    Speed is a different matter, but in no way it can be measured
    from a star sight.
    (They make ONLY ONE sight, there is no time for two, and no space
    to set two sighting devices).
    
    I could solve comletely the simplified problem where everything is in
    one (vertical) plane: the center of the Earth, the missile, the target and
    the star. In this case, second star does not help,
    but EXACT adjustment cannot be made. I suppose the same is true
    in the general situation.
    Where is the star in this plane does not matter (privided it is high
    enough, so that refraction has no influence.
    Complete justification of this needs some formulas which I am reluctant to
    write:-)
    
    One account says that the pure mathematical problem caused some
    difficulty to the designer team...
    But much more difficult are the engineering problems, and one of them
    is how to make the window through which the sight is made:-)
    It must be strong enough to stand the launch (the active phase lasts
    about 6 minutes), but perfect optitally, not to distort the sight.
    Another problem is how to make sure that the telescope finds the
    right star. They also pre-compute all possible occultation, to make
    sure that the selected star is not occulted by the Moon or Sun or some
    planet. The telescope is supposed to find the star in the daylight as
    well as at night:-) It helps that the missile is already high
    enough at the time of the sight, so the sky is not blue but dark.
    
    This message is too long, but I can continue...
    
    Alex.
    
    
    
    
    
    
    
    
    
    

       
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