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

Utilities/Axes Transformations

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

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

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 (* R_{N}*)
and the radius of curvature in the meridian (

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

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 (

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

**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 only available with

**Planet model Custom**.**Equatorial radius of planet**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.

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

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

Second | Scalar | Contains the reference height from surface of Earth to flat Earth frame with regard to Earth frame, in same units as flat Earth position. |

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

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

Second | Scalar | Contains the altitude above the input reference altitude, in same units as flat Earth position. |

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.

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.

Direction Cosine Matrix ECEF to NED

Direction Cosine Matrix ECEF to NED to Latitude and Longitude

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