A person standing in Sydney ( = 151, = 34 ) is trying to determine the exact rel

Question

A person standing in Sydney ( = 151, = 34 ) is trying to determine the exact relative orientation of a person standing on the summit of Chomolungma ( = 86 55 30” E, = 27 59 16.08”N ).

(a) Using the longitude and latitude coordinates of each location, compute the direction cosine matrix (coordinate transformation) CCS that relates the two local Earth surface frames FES (Sydney) and FEC (Chomolungma).

(b) Verify your answer by performing a second unique set of coordinate transformation rotations.

(c) Describe the relative orientation of the FEC frame up vector with respect to the surface frame in Sydney (FES) in terms of NED components by transforming the location vector of the peak in the Chomolungma frame {0, 0, 8488}m, into the Sydney frame.

Explanation / Answer

Answer:

1) location coordinate, Sydney = 151°, = -34° and chomolunga = 86.925 °, = 27.9878 where , are longitude & latitude.

When rotation from one geodetic point to another point happens through the

longitude rotation (l) = 86.925 – 151 = -64.075

latitude rotation (k) = 27.9878- (-34) = 61.9878

directional cosine matrix =       

- sinkcosl -sinksinl cosk

    -sin l        cos l           0

-coskcosl    -cosksinl –sin k

= -0.386   0.794 0.4696

0.8994   0.4372 -0.8828

-0.2053   0.4223 -0.8828

2) apply for any two coordinates,

m1= [0,0,0] and m2= [45, 45, 0]

DCM= 0.49   -0.49 0.7

            -0.49 0.7   0

        -0.49    -0.49   -0.7

3)

For conversion from FEC to NED system, position vector p,

P= R * (P1-P2) where P2 reference frame taken-point.

Here, p1-P2 = {0,0,0} – {0,0,-8488} = {0,0,8488}

R= sin cos    -sin       cos

       -sin            cos          0

-cos cos       -cos sin   -sin

Then, p=   cos    *8488

                     0

                 -sin * 8488

For Sydney, p= [7036.552 0   4744.792 ]       for chomolungma p = [7494.904   0   -3983.4184]

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