# ECEF Position to LLA

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

## Library

Utilities/Axes Transformations

## Description

The ECEF Position to LLA block converts a 3-by-1 vector of ECEF position $\left(\overline{p}\right)$ into geodetic latitude $\left(\overline{\mu }\right)$, longitude $\left(\overline{\iota }\right)$, and altitude $\left(\overline{h}\right)$ above the planetary ellipsoid.

The ECEF position is defined as

$\overline{p}=\left[\begin{array}{c}{\overline{p}}_{x}\\ {\overline{p}}_{y}\\ {\overline{p}}_{z}\end{array}\right]$

Longitude is calculated from the ECEF position by

$\iota =\text{atan}\left(\frac{{p}_{y}}{{p}_{x}}\right)$

Geodetic latitude $\left(\overline{\mu }\right)$ 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 $\left(\overline{\mu }\right)$ and reduced latitude $\left(\overline{\beta }\right)$. An initial guess takes the form:

$\begin{array}{c}\overline{\beta }=\text{atan}\left(\frac{{p}_{z}}{\left(1-f\right)s}\right)\\ \\ \overline{\mu }=\text{atan}\left(\frac{{p}_{z}+\frac{{e}^{2}\left(1-f\right)}{\left(1-{e}^{2}\right)}R{\left(\mathrm{sin}\beta \right)}^{3}}{s-{e}^{2}R{\left(\mathrm{cos}\beta \right)}^{3}}\right)\end{array}$

where R is the equatorial radius, f 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 $\left(\overline{\beta }\right)$ is recalculated using

$\beta =\text{atan}\left(\frac{\left(1-f\right)\mathrm{sin}\mu }{\mathrm{cos}\mu }\right)$

and geodetic latitude $\left(\overline{\mu }\right)$ is reevaluated. This last step is repeated until $\overline{\mu }$ converges.

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

$h=s\mathrm{cos}\mu +\left({p}_{z}+{e}^{2}N\mathrm{sin}\mu \right)\mathrm{sin}\mu -N$

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

$N=\frac{R}{\sqrt{1-{e}^{2}{\left(\mathrm{sin}\mu \right)}^{2}}}$

## Dialog Box

Units

Specifies the parameter and output units:

Units

Position

Altitude

`Metric (MKS)`

Meters

Meters

Meters

`English`

Feet

Feet

Feet

This option is only available when Planet model is set to `Earth (WGS84)`.

Planet model

Specifies the planet model to use, `Custom` or ```Earth (WGS84)```.

Flattening

Specifies the flattening of the planet.

This option is available only with Planet model set to `Custom`.

Specifies the radius of the planet at its equator. The equatorial radius units should be the same as the desired units for ECEF position.

This option is available only with Planet model set to `Custom`.

## Inputs and Outputs

InputDimension TypeDescription

First

3-by-1 vector Contains the position in ECEF frame.

OutputDimension TypeDescription

First

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

Second

ScalarContains the altitude above the planetary ellipsoid, in the same units as the ECEF position.

## Assumptions and 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.

## 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.