# Flat Earth to LLA

Estimate geodetic latitude, longitude, and altitude from flat Earth position

## Library

Utilities/Axes Transformations

## Description

The Flat Earth to LLA block converts a 3-by-1 vector of Flat Earth position$\left(\overline{p}\right)$ into geodetic latitude $\left(\overline{\mu }\right)$, longitude $\left(\overline{\iota }\right)$, and altitude (h). The flat Earth coordinate system assumes the z-axis is downward positive. The estimation begins by transforming the flat Earth x and y coordinates to North and East coordinates. The transformation has the form of

`$\left[\begin{array}{c}N\\ E\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\psi & -\mathrm{sin}\psi \\ \mathrm{sin}\psi & \mathrm{cos}\psi \end{array}\right]\left[\begin{array}{c}{p}_{x}\\ {p}_{y}\end{array}\right]$`

where$\left(\overline{\psi }\right)$ is the angle in degrees clockwise between the x-axis and north.

To convert the North and East coordinates to geodetic latitude and longitude, the radius of curvature in the prime vertical (RN) and the radius of curvature in the meridian (RM) are used. (RN) and (RM) are defined by the following relationships:

`$\begin{array}{l}{R}_{N}=\frac{R}{\sqrt{1-\left(2f-{f}^{2}\right){\mathrm{sin}}^{2}{\mu }_{0}}}\\ {R}_{M}={R}_{N}\frac{1-\left(2f-{f}^{2}\right)}{1-\left(2f-{f}^{2}\right){\mathrm{sin}}^{2}{\mu }_{0}}\end{array}$`

where (R) is the equatorial radius of the planet and$\left(\overline{f}\right)$ is the flattening of the planet.

Small changes in the in latitude and longitude are approximated from small changes in the North and East positions by

`$\begin{array}{l}d\mu =\text{atan}\left(\frac{1}{{R}_{M}}\right)dN\\ d\iota =\text{atan}\left(\frac{1}{{R}_{N}\mathrm{cos}\mu }\right)dE\end{array}$`

The output latitude and longitude are simply the initial latitude and longitude plus the small changes in latitude and longitude.

`$\begin{array}{l}\mu ={\mu }_{0}+d\mu \\ \iota ={\iota }_{0}+d\iota \end{array}$`

The altitude is the negative flat Earth z-axis value minus the reference height (href).

`$h=-{p}_{z}-{h}_{ref}$`

## Parameters

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 only available with Planet model Custom.

Specifies the radius of the planet at its equator. The units of the equatorial radius parameter should be the same as the units for flat Earth position. This option is only available with Planet model Custom.

Initial geodetic latitude and longitude

Specifies the reference location, in degrees of latitude and longitude, for the origin of the estimation and the origin of the flat Earth coordinate system.

Direction of flat Earth x-axis (degrees clockwise from north)

Specifies angle used for converting flat Earth x and y coordinates to North and East coordinates.

## Inputs and Outputs

InputDimension TypeDescription

First

3-by-1 vectorContains the position in flat Earth frame.

Second

ScalarContains the reference height from surface of Earth to flat Earth frame with regard to Earth frame, in same units as flat Earth position.
OutputDimension TypeDescription

First

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

Second

ScalarContains the altitude above the input reference altitude, in same units as flat Earth position.

## Assumptions and Limitations

This estimation method assumes the flight path and bank angle are zero.

This estimation method assumes the flat Earth z-axis is normal to the Earth at the initial geodetic latitude and longitude only. This method has higher accuracy over small distances from the initial geodetic latitude and longitude, and nearer to the equator. The longitude will have higher accuracy the smaller the variations in latitude. Additionally, longitude is singular at the poles.

## References

Etkin, B., Dynamics of Atmospheric Flight, John Wiley & Sons, New York, 1972.

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, Second Edition, John Wiley & Sons, New York, 2003.