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Direction Cosine Matrix ECEF to NED to Latitude and Longitude

Convert direction cosine matrix to geodetic latitude and longitude

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  • Direction Cosine Matrix ECEF to NED to Latitude and Longitude block

Libraries:
Aerospace Blockset / 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 geodetic latitude and longitude. For more information on the direction cosine matrix, see Algorithms.

Limitations

The DCM matrix performs the coordinate transformation of a vector in ECEF axes, (ox0, oy0, oz0), into geodetic latitude and longitude. The order of the axis rotations required to bring this about is:

  • 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. For more information, see About Aerospace Coordinate Systems.

Ports

Input

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Direction cosine matrix from which to geodetic latitude and longitude, specified as a 3-by-3 matrix.

Data Types: double

Output

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Geodetic latitude and longitude, returned as a 2-by-1 vector in degrees.

Data Types: double

Parameters

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

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

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

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

Programmatic Use

Block Parameter: action
Type: character vector
Values: 'None' | 'Warning' | 'Error'
Default: 'None'

Data Types: char | string

Tolerance of the direction cosine matrix validity, specified as a scalar. 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).

Programmatic Use

Block Parameter: tolerance
Type: character vector
Values: 'eps(2)' | scalar
Default: 'eps(2)'

Data Types: double

Algorithms

The DCM matrix performs the coordinate transformation of a vector in ECEF axes, (ox0, oy0, oz0), into geodetic latitude and longitude. 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]=[−sinμ0cosμ010−cosμ0−sinμ][cosιsinι0−sinιcosι0001][ox0oy0oz0]

Combining the two axis transformation matrices defines the following DCM.

DCMef=[−sinμcosι−sinμsinιcosμ−sinιcosι0−cosμcosι−cosμsinι−sinμ]

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))

References

[1] Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Second Edition. Reston, VA: AIAA Education Series, 2000.

[2] Recommended Practice for Atmospheric and Space Flight Vehicle Coordinate Systems, R-004-1992, ANSI/AIAA, February 1992.

[3] Stevens, B. L., and F. L. Lewis. Aircraft Control and Simulation, Hoboken, NJ: John Wiley & Sons, 1992.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced before R2006a

See Also

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