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Direction Cosine Matrix ECEF to NED to Latitude and Longitude

Convert direction cosine matrix to geodetic latitude and longitude

Library

Utilities/Axes Transformations

Description

The Direction Cosine Matrix ECEF to NED to Latitude and Longitude block converts a 3-by-3 direction cosine matrix (DCM) into geodetic latitude and longitude. The DCM matrix performs the coordinate transformation of a vector in Earth-centered Earth-fixed (ECEF) axes (ox0, oy0, oz0) into a vector in north-east-down (NED) axes (ox2, oy2, oz2). The order of the axis rotations required to bring this about is:

  1. A rotation about oz0 through the longitude (ι) to axes (ox1, oy1, oz1)

  2. A rotation about oy1 through the geodetic latitude (μ) to axes (ox2, oy2, oz2)

[ox2oy2oz2]=DCMef[ox0oy0oz0][ox2oy2oz2]=[sinμ0cosμ010cosμ0sinμ][cosιsinι0sinιcosι0001][ox0oy0oz0]

Combining the two axis transformation matrices defines the following DCM.

DCMef=[sinμcosιsinμsinιcosμsinιcosι0cosμcosιcosμsinιsinμ]

To determine geodetic latitude and longitude from the DCM, the following equations are used:

μ=asin(DCM(3,3))ι=atan(DCM(2,1)DCM(2,2))

Inputs and Outputs

InputDimension TypeDescription

First

3-by-3 direction cosine matrixTransforms ECEF vectors to NED vectors.

OutputDimension TypeDescription

First

2-by-1 vectorContains the geodetic latitude and longitude, in degrees.

Assumptions and Limitations

This implementation generates a geodetic latitude that lies between ±90 degrees, and longitude that lies between ±180 degrees.

The implementation of the ECEF coordinate system assumes that the origin is at the center of the planet, the x-axis intersects the Greenwich meridian and the equator, the z-axis is the mean spin axis of the planet, positive to the north, and the y-axis completes the right-hand system.

References

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.

Zipfel, P. H., Modeling and Simulation of Aerospace Vehicle Dynamics, AIAA Education Series, Reston, Virginia, 2000.

"Atmospheric and Space Flight Vehicle Coordinate Systems," ANSI/AIAA R-004-1992.

Introduced before R2006a

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