Direction Cosine Matrix ECEF to NED to Latitude and Longitude

Convert direction cosine matrix to geodetic latitude and longitude


Utilities/Axes Transformations


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)


Combining the two axis transformation matrices defines the following DCM.


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



Action for invalid DCM

Block behavior when direction cosine matrix is invalid (not orthogonal).

  • Warning — Displays warning and indicates that the direction cosine matrix is invalid.

  • Error — Displays error and indicates that the direction cosine matrix is invalid.

  • None — Does not display warning or error (default).

Tolerance for DCM validation

Tolerance of direction cosine matrix validity, specified as a scalar. Default is eps(2). The block considers the direction cosine matrix valid if these conditions are true:

  • The transpose of the direction cosine matrix times itself equals 1 within the specified tolerance (transpose(n)*n == 1±tolerance)

  • The determinant of the direction cosine matrix equals 1 within the specified tolerance (det(n) == 1±tolerance).

Inputs and Outputs

InputDimension TypeDescription


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


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.


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