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    Re: Navigational Algorithms
    From: Paul Hirose
    Date: 2006 Oct 21, 14:53 -0700

    James R. Van Zandt wrote:
    > The main technical obstacle is the calculation of all the partial
    > derivatives.
    For my satellite orbital element differential adjustment program, I
    obtained the partial derivatives numerically. The program prompted for a
    ∆ to apply to the element being adjusted, say inclination. It then
    displayed the corresponding change in the satellite's predicted position
    (in arc minutes) at each observation, and asked if that was acceptable.
    I aimed to make it about the same order of magnitude as the residuals.
    The program could do a simultaneous least squares adjustment to any
    desired combination of orbital elements while holding the others
    constant. Observations were independently weighted according to the
    estimated arc minutes of accuracy supplied by the user. At the end of
    each adjustment run, residuals for the observation set were displayed,
    as well as an overall figure of merit. It was up to the user to decide
    if another iteration was necessary.
    The number of observations was limited only by available memory. In
    practice, this was no limitation at all. An artificial satellite
    orbiting Earth has very "bumpy road" compared to a planet orbiting the
    Sun. It's affected significantly by atmospheric drag and complex
    irregularities in the gravity field. The orbital models (I used NORAD's
    SGP4 model) allow for these effects in the short term but not over a
    period spanning several years. To generate accurate elements, you must
    use fresh observations.
    My least squares formula came right out of a textbook. Later I read that
    this elementary formula is sensitive to error, and not recommended for
    demanding applications. However, in my program it converged rapidly to a
    stable solution. I never felt the need for anything more sophisticated.
    I used this program to maintain orbital elements for some U.S.
    reconnaisance satellites. The Pentagon won't release the elements for
    those birds, but a number of hobbyists track them visually. My elements 
    for the Lacrosse radar satellites were so accurate, you could set 
    binoculars on a predicted spot against the star background and watch the 
    bird come sailing through on course and within a few seconds of the 
    predicted time.
    All this was done in the C language on an 8-bit computer with 64 k of
    RAM and a 2.5 MHz processor. It was a long time ago -- I no longer have
    a machine able to read my old floppies. But when I read Kaplan's paper I
    recognized many parallels to what I used to do.
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