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    Navigational Algorithms - 3 COPs analytical solution
    From: Andrés Ruiz
    Date: 2006 Oct 18, 12:02 +0200

    [NavList 1437] Re: Navigator's Newsletter, and other forums.
    From: jrvz---net
    Date: 17 Oct 2006 22:45

    (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.

    The draft of one about circles of position is:

     

    3 COPs analytical solution

     

    Input data: 3 observations

     

    K

    Body

    Ho

    GHA

    Dec

    1

    Body1

    Ho1

    GHA1

    Dec1

    2

    Body2

    Ho2

    GHA2

    Dec2

    3

    Body3

    Ho3

    GHA3

    Dec3

     

     

    Fix: BF LF

     

     

    Manual Solution

     

     

     

     

    M3 = x1*y2*z3+y1*z2*x3+z1*x2*y3-(z1*y2*x3+x1*z2*y3+y1*x2*z3)

     

    Mx = y2*z3+y1*z2+y3*z1-(y2*z1+y1*z3+y3*z2)

    My = z2*x3+z1*x2+z3*x1-(z2*x1+z1*x3+z3*x2)

    Mz = x2*y3+x1*y2+x3*y1-(x2*y1+x1*y3+x3*y2)

     


    Example by hand

     

    K

    Body

    Ho

    GHA

    Dec

    1

    Arcturus

    22.2

    166.829

    19.335

    2

    Vega

    70.15

    115.277

    38.758

    3

    Dubhe

    20.35

    226.98

    61.909

     

     

    | -2.432  0.569  0.876 | | 0.006|   | 1 |

    | -0.354  0.750  0.666 | | 0.731| = | 1 |

    | -0.924 -0.990  2.537 | | 0.682|   | 1 |

     

    B [0-90][+N/-S]

    L [0-360][+ W->E]

     

    Fix(42.982085093299339, 89.566484629127913)

     

     

    This method work only for one point of intersection. Otherwise a least squares method must be used to find the most probable celestial position.  

     

    Help:

    I am looking for some information about sight reduction with matrices, I have find two articles:

     

    Dewit, C., “Some Remarks on Sight Reduction with Matrices,” Navigation: J. Inst. Naviga., Vol. 26, No. 3, pp. 252-253 (1979).

    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.

     

    if someone could send me some copies or additional information by e-mail, I,ll be very thankful.

     

    Andrés

    http://www.geocities.com/andresruizgonzalez

     


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