LLA to Flat Earth

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

Libraries:
Aerospace Blockset / Utilities / Axes Transformations

Description

The LLA to Flat Earth block converts a geodetic latitude $(\bar{μ})$ , longitude $(\bar{ι})$ , and altitude (h) into a 3-by-1 vector of flat Earth position $(\bar{p})$ . 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. For more information on the flat Earth coordinate system, see Algorithms.

Limitations

This estimation method assumes that 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.

Ports

Input

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μl — Geodetic latitude and longitude
2-by-1 vector

Geodetic latitude and longitude, specified as a 2-by-1 vector, in degrees.

Data Types: double

h — Altitude
scalar

Altitude above the input reference altitude, specified as a scalar, in the same units as the flat Earth position.

Data Types: double

h_ref — Reference height
scalar

Reference height from the surface of the Earth to the flat Earth frame, specified as a scalar, in the same units as the flat Earth position. The reference height is estimated with regard to Earth frame.

Data Types: double

μ_ref l_ref — Reference location
2-by-1 vector

Reference location, specified as a 2-by-1 vector, in degrees of latitude and longitude, for the origin of the estimation and the origin of the flat Earth coordinate system. Use this port if you want to specify the reference location as a dynamic value.

Dependencies

To enable this port, select Input reference position and orientation.

Data Types: double

ψ_ref — Direction of flat Earth x-axis
scalar

Angle, specified as a scalar, for converting flat Earth x and y coordinates to North and East coordinates. Use this port if you want to specify the angle as a dynamic value.

Dependencies

To enable this port, select Input reference position and orientation.

Data Types: double

Output

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X_e — Position
3-by-1 vector

Position in flat Earth frame, returned as a vector.

Data Types: double

Parameters

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Units — Units
`Metric (MKS)` (default) | `English`

Parameter and output units:

Units	Position	Equatorial Radius	Altitude
`Metric (MKS)`	Meters	Meters	Meters
`English`	Feet	Feet	Feet

Programmatic Use

Block Parameter: units

Type: character vector

Values:

'Metric
									(MKS)'

| 'English'

Default:

'Metric
									(MKS)'

Planet model — Planet model
`Earth (WGS84)` (default) | `Custom`

Planet model to use, Custom or Earth (WGS84).

Dependencies

Selecting the Custom option disables the Units parameter and enables these parameters:

Flattening
Equatorial radius of planet

Programmatic Use

Block Parameter: ptype

Type: character vector

Values:

'Earth
									(WGS84)'

| 'Custom'

Default:

'Earth
									(WGS84)'

Flattening — Flattening of Earth
`1/298.257223563` (default) | scalar

Flattening of the planet, specified as a double scalar.

Dependencies

To enable this parameter, set Planet model to Custom.

Programmatic Use

Block Parameter: F

Type: character vector

Values: double scalar

Default: 1/298.257223563

Equatorial radius of planet — Radius of planet at equator
`6378137` (default) | scalar

Radius of the planet at its equator, in the same units as the desired units for ECEF position.

Dependencies

To enable this parameter, set Planet model to Custom.

Programmatic Use

Block Parameter: R

Type: character vector

Values: double scalar

Default: 6378137

Input reference position and orientation — Input reference position and orientation as ports
`off` (default) | `on`

To enable input ports for reference position and angle to convert flat Earth, select this check box.
To specify the reference positions and angle as static values, clear this check box.

Select this check box if you want

Programmatic Use

Block Parameter: refPosPort

Type: character vector

Values: 'off' | 'on'

Default: 'off'

Reference geodetic latitude and longitude [deg] — Initial geodetic latitude and longitude
`[0 10]` (default) | 2-by-1 vector

Reference location in latitude and longitude, specified as 2-by-1 vector, in degrees.

Dependencies

To enable this parameter, clear Input reference position and orientation.

Programmatic Use

Block Parameter: LL0

Type: character vector

Values: 2-by-1 vector

Default: [0 10]

Direction of flat Earth x-axis (degrees clockwise from north) — Flat Earth x and y coordinates
`0` (default) | scalar

Angle to convert flat Earth x and y coordinates to North and East coordinates, specified as a scalar double, in degrees.

Dependencies

To enable this parameter, clear Input reference position and orientation.

Programmatic Use

Block Parameter: psi

Type: character vector

Values: double scalar

Default: 0

Algorithms

The flat Earth coordinate system assumes the z-axis is downward positive. The estimation begins by finding the small changes in latitude and longitude from the output latitude and longitude minus the initial latitude and longitude.

$\begin{array}{l} d μ = μ - μ_{0} \\ d ι = ι - ι_{0} \end{array}$

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 (R_M). R_N and R_M are defined by the following relationships:

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

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 latitude and longitude by

$\begin{array}{l} d N = R_{M} d μ \\ d E = R_{N} \cos μ_{0} d ι . \end{array}$

With the conversion of the North and East coordinates to the flat Earth x and y coordinates, the transformation has the form of

$[\begin{matrix} p_{x} \\ p_{y} \end{matrix}] = [\begin{matrix} \cos ψ & \sin ψ \\ - \sin ψ & \cos ψ \end{matrix}] [\begin{matrix} N \\ E \end{matrix}],$

where ψ 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 (h_ref):

$p_{z} = - h - h_{r e f} .$

References

[1] Stevens, B. L., and F. L. Lewis. Aircraft Control and Simulation, Hoboken, NJ: John Wiley & Sons, 2003.

[2] Etkin, B. Dynamics of Atmospheric Flight Hoboken, NJ: John Wiley & Sons, 1972.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2011a

LLA to Flat Earth

Description

Limitations

Ports

Input

μl — Geodetic latitude and longitude 2-by-1 vector

h — Altitude scalar

href — Reference height scalar

μref lref — Reference location 2-by-1 vector

Dependencies

ψref — Direction of flat Earth x-axis scalar

Dependencies

Output

Xe — Position 3-by-1 vector

Parameters

Units — Units Metric (MKS) (default) | English

Programmatic Use

Planet model — Planet model Earth (WGS84) (default) | Custom

Dependencies

Programmatic Use

Flattening — Flattening of Earth 1/298.257223563 (default) | scalar

Dependencies

Programmatic Use

Equatorial radius of planet — Radius of planet at equator 6378137 (default) | scalar

Dependencies

Programmatic Use

Input reference position and orientation — Input reference position and orientation as ports off (default) | on

Programmatic Use

Reference geodetic latitude and longitude [deg] — Initial geodetic latitude and longitude [0 10] (default) | 2-by-1 vector

Dependencies

Programmatic Use

Direction of flat Earth x-axis (degrees clockwise from north) — Flat Earth x and y coordinates 0 (default) | scalar

Dependencies

Programmatic Use

Algorithms

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

Version History

See Also

μl — Geodetic latitude and longitude
2-by-1 vector

h — Altitude
scalar

h_ref — Reference height
scalar

μ_ref l_ref — Reference location
2-by-1 vector

ψ_ref — Direction of flat Earth x-axis
scalar

X_e — Position
3-by-1 vector

Units — Units
`Metric (MKS)` (default) | `English`

Planet model — Planet model
`Earth (WGS84)` (default) | `Custom`

Flattening — Flattening of Earth
`1/298.257223563` (default) | scalar

Equatorial radius of planet — Radius of planet at equator
`6378137` (default) | scalar

Input reference position and orientation — Input reference position and orientation as ports
`off` (default) | `on`

Reference geodetic latitude and longitude [deg] — Initial geodetic latitude and longitude
`[0 10]` (default) | 2-by-1 vector

Direction of flat Earth x-axis (degrees clockwise from north) — Flat Earth x and y coordinates
`0` (default) | scalar

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.