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    Re: Navigational Algorithms - 3 COPs analytical solution
    From: James R. Van Zandt
    Date: 2006 Oct 21, 08:13 -0400

    "Andres Ruiz"  wrote:
    >>   (It does require inverting 3x3 matrices, so it's not well suited to
    >>   hand calculation.)
    >>   If someone remembers seeing something like this, I would appreciate a
    >>   reference.  Otherwise I'll write it up in more detail.
    >   Now I am writing some articles to be published on my web page
    >   under the name Navigational Algorithms.
    That is indeed the first step I had in mind.
    >   I am looking for some information about sight reduction with
    >   matrices, I have find two articles:
    >   Watkins. R. and Janiczek. P. M., Sight Reduction with Matrices,
    >   NAVIGATION, Journal of The Institute of Navigation, Vol. 25,
    >   No. 4, Winter 1978-79, pp. 447-48.
    Wolfgang K�berer has kindly supplied a copy of this article, and their
    least squares method is the second step.  However I note they use
    equally weighted least squares.  I believe the weight should be higher
    for high altitude sights.  That's what the Kalman filter method would
    Paul Hirose  wrote:
    >Have you read Kaplan's "navigation solution" paper at the USNO site?
    >His algorithm may give an error estimate. I can't remember. 
    Yes, it uses an extended Kalman filter, so it does give an error
    estimate but requires an estimated position.  The thing I think is
    really neat about a Kalman filter is that it lets you update a
    multi-dimensional state with a one-dimensional measurement.  I planned
    to update a three-dimensional state (position in cartesian
    coordinates).  Kaplan is using a four-dimensional state (latitude,
    longitude, course, and speed) which I admit has more practical interest.
    >It's used in
    >the Navy's STELLA software, which unfortunately is not available to
    >civilians. If you could create an open-source implementation of Kaplan's
    >algorithm, that would be some accomplishment. If nobody else steps up, I
    >may attempt that someday. I have experience in the related area of
    >orbital element least-squares adjustment. 
    The main technical obstacle is the calculation of all the partial
    derivatives.  I would recommend using an unscented filter [1] which
    eliminates that chore, but is otherwise similar to the Kalman filter.
    >I don't believe that requiring an assumed position for a sight reduction
    >is a significant problem in practice. And I suspect proficient
    >navigators will pay little attention to a computer's error estimate.
    >They will rely on their experience and intuition, based on the observing
    >conditions and the way the LOPs look on the chart.
    Alas, you're probably right.  
                - Jim Van Zandt
    [1] http://citeseer.ist.psu.edu/julier97new.html
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