ECEF Position to LLA

Calculate geodetic latitude, longitude, and altitude above planetary ellipsoid from Earth-centered Earth-fixed (ECEF) position

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Libraries:
Aerospace Blockset / Utilities / Axes Transformations

Description

The ECEF Position to LLA block converts a 3-by-1 vector of ECEF position $(\bar{p})$ into geodetic latitude $(\bar{μ})$ , longitude $(\bar{ι})$ , and altitude $(\bar{h})$ above the planetary ellipsoid. For more information on the ECEF position, see Algorithms.

Limitations

This implementation generates a geodetic latitude that lies between ±90 degrees, and longitude that lies between ±180 degrees. The planet is assumed to be ellipsoidal. By setting the flattening to 0, you model a spherical planet.
The implementation of the ECEF coordinate system assumes that its origin lies at the center of the planet, the x-axis intersects the prime (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-handed system.

Ports

Input

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X_f — Position
3-by-1 vector

Position in ECEF frame, specified as a 3-by-1 vector.

Data Types: double

Output

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μ l — Geodetic latitude and longitude
2-by-1 vector

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

Data Types: double

h — Altitude
scalar

Altitude above the planetary ellipsoid, returned as a scalar, in the same units as the ECEF position.

Data Types: double

Parameters

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Units — Output units
`Metric (MKS)` (default) | `English`

Output units, specified as:

Units	Position	Equatorial Radius	Altitude
`Metric (MKS)`	Meters	Meters	Meters
`English`	Feet	Feet	Feet

Dependencies

To enable this parameter, set Planet model to Earth (WGS84).

Programmatic Use

Block Parameter: units

Type: character vector

Values:

'Metric
									(MKS)'

| 'English'

Default:

'Metric
									(MKS)'

Planet model — Planet model
`Earth (WGS84)` (default) | `Custom`

Planet model to use, Custom or Earth (WGS84).

Programmatic Use

Block Parameter: ptype

Type: character vector

Values: 'Earth (WGS84)' | 'Custom'

Default: 'Earth (WGS84)'

Flattening — Flattening of planet
`1/298.257223563` (default) | scalar

Flattening of the planet, specified as a double scalar.

Dependencies

To enable this parameter, set Planet model to Custom.

Programmatic Use

Block Parameter: F

Type: character vector

Values: double scalar

Default: '1/298.257223563'

Equatorial radius of planet — Radius of planet at equator
`6378137` (default) | scalar

Radius of the planet at its equator, specified as a double scalar, in the same units as the desired units for the ECEF position.

Dependencies

To enable this parameter, set Planet model to Custom.

Programmatic Use

Block Parameter: R

Type: character vector

Values: double scalar

Default: '6378137'

Algorithms

The ECEF position is defined as:

$\bar{p} = [\begin{matrix} {\bar{p}}_{x} \\ {\bar{p}}_{y} \\ {\bar{p}}_{z} \end{matrix}] .$

Longitude is calculated from the ECEF position by

$ι = atan (\frac{p_{y}}{p_{x}}) .$

Geodetic latitude $(\bar{μ})$ is calculated from the ECEF position using Bowring's method, which typically converges after two or three iterations. The method begins with an initial guess for geodetic latitude $(\bar{μ})$ and reduced latitude $(\bar{β})$ . An initial guess takes the form:

$\begin{matrix} \bar{β} = atan (\frac{p_{z}}{(1 - f) s}) \\ \bar{μ} = atan (\frac{p_{z} + \frac{e^{2} (1 - f)}{(1 - e^{2})} R {(\sin β)}^{3}}{s - e^{2} R {(\cos β)}^{3}}) \end{matrix}$

where R is the equatorial radius, f is the flattening of the planet, e² = 1−(1−f)², the square of first eccentricity, and:

$s = \sqrt{p_{x}^{2} + p_{y}^{2}} .$

After the initial guesses are calculated, the reduced latitude $(\bar{β})$ is recalculated using

$β = atan (\frac{(1 - f) \sin μ}{\cos μ})$

and geodetic latitude $(\bar{μ})$ is reevaluated. This last step is repeated until $\bar{μ}$ converges.

The altitude $(\bar{h})$ above the planetary ellipsoid is calculated with

$h = s \cos μ + (p_{z} + e^{2} N \sin μ) \sin μ - N,$

where the radius of curvature in the vertical prime $(\bar{N})$ is given by

$N = \frac{R}{\sqrt{1 - e^{2} {(\sin μ)}^{2}}} .$

References

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

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

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

Extended Capabilities

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

Version History

Introduced before R2006a

ECEF Position to LLA

Description

Limitations

Ports

Input

X_f — Position
3-by-1 vector

Output

μ l — Geodetic latitude and longitude
2-by-1 vector

h — Altitude
scalar

Parameters

Units — Output units
`Metric (MKS)` (default) | `English`

Dependencies

Programmatic Use

Planet model — Planet model
`Earth (WGS84)` (default) | `Custom`

Programmatic Use

Flattening — Flattening of planet
`1/298.257223563` (default) | scalar

Dependencies

Programmatic Use

Equatorial radius of planet — Radius of planet at equator
`6378137` (default) | scalar

Dependencies

Programmatic Use

Algorithms

References

Extended Capabilities

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

Version History

See Also

Topics

ECEF Position to LLA

Description

Limitations

Ports

Input

Xf — Position 3-by-1 vector

Output

μ l — Geodetic latitude and longitude 2-by-1 vector

h — Altitude scalar

Parameters

Units — Output units Metric (MKS) (default) | English

Dependencies

Programmatic Use

Planet model — Planet model Earth (WGS84) (default) | Custom

Programmatic Use

Flattening — Flattening of planet 1/298.257223563 (default) | scalar

Dependencies

Programmatic Use

Equatorial radius of planet — Radius of planet at equator 6378137 (default) | scalar

Dependencies

Programmatic Use

Algorithms

References

Extended Capabilities

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

Version History

See Also

Topics

X_f — Position
3-by-1 vector

μ l — Geodetic latitude and longitude
2-by-1 vector

h — Altitude
scalar

Units — Output units
`Metric (MKS)` (default) | `English`

Planet model — Planet model
`Earth (WGS84)` (default) | `Custom`

Flattening — Flattening of planet
`1/298.257223563` (default) | scalar

Equatorial radius of planet — Radius of planet at equator
`6378137` (default) | scalar

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