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

Utilities/Axes Transformations

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:

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

where *R* is the equatorial radius, *f* the
flattening of the planet, *e*^{2} =
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

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

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

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

**Units**Specifies the parameter and output units:

Units

Position

Equatorial Radius

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`

.**Equatorial radius of planet**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`

.

Input | Dimension Type | Description |
---|---|---|

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

Output | Dimension Type | Description |
---|---|---|

First | 2-by-1 vector | Contains the geodetic latitude and longitude, in degrees. |

Second | Scalar | Contains the altitude above the planetary ellipsoid, in the same units as the ECEF position. |

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.

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.

See About Aerospace Coordinate Systems.

Direction Cosine Matrix ECEF to NED

Direction Cosine Matrix ECEF to NED to Latitude and Longitude