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

Utilities/Axes Transformations

The
LLA to Flat Earth block converts a geodetic latitude $$\left(\overline{\mu}\right)$$, longitude $$\left(\overline{\iota}\right)$$, and altitude (* h*)
into a 3-by-1 vector of Flat Earth position$$\left(\overline{p}\right)$$. Latitude and longitude values
can be any value. However, latitude values of +90 and -90 may return
unexpected values because of singularity at the poles. The flat Earth
coordinate system assumes the

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

To convert geodetic latitude and longitude to the North and
East coordinates, the estimation uses 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 $$f$$ is
the flattening of the planet.

Small changes in the North (dN) and East (dE) positions are approximated from small changes in the North and East positions by

$$\begin{array}{l}dN=\frac{d\mu}{\text{atan}\left(\frac{1}{{R}_{M}}\right)}\\ dE=\frac{d\iota}{\text{atan}\left(\frac{1}{{R}_{N}\mathrm{cos}{\mu}_{0}}\right)}\end{array}$$

With the conversion of the North and East coordinates to the
flat Earth * x* and

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

where

$$\left(\psi \right)$$

is the angle in degrees
clockwise between the * x*-axis and north.

The flat Earth * z*-axis value is the negative
altitude minus the reference height (

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

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

Units

Position

Equatorial Radius

Altitude

`Metric (MKS)`

Meters

Meters

Meters

`English`

Feet

Feet

Feet

This option is available only 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 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 available only 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**Specifies angle for converting flat Earth

and*x*coordinates to North and East coordinates.*y*

Input | 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. |

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

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

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

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 has higher accuracy with smaller variations in latitude.
Additionally, longitude is singular at the poles.

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

Stevens, B. L., and F. L. Lewis. *Aircraft Control
and Simulation*, 2nd ed. New York: John Wiley & Sons,
2003.

Direction Cosine Matrix ECEF to NED

Direction Cosine Matrix ECEF to NED to Latitude and Longitude

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