Flat Earth to LLA

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

Libraries:
Aerospace Blockset / Utilities / Axes Transformations

Description

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

Examples

Lightweight Airplane Design

Address the technical and process challenges of aircraft design using the design of a lightweight aircraft.

Open Live Script

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 when close to the reference geodetic latitude and longitude and nearer to the equator than farther away. The longitude will have higher accuracy when there are smaller the variations in latitude. Additionally, longitude is singular at the poles.

Ports

Input

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X_e — Position in flat Earth frame
3-by-1 vector

Position in flat Earth frame, specified as a 3-by-1 vector.

Data Types: double

h_ref — Reference height
scalar

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

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

This port is enabled if the Input reference position and orientation check box is selected.

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

This port is enabled if the Input reference position and orientation check box is selected.

Data Types: double

Output

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μ l — Geodetic latitude and longitude
2-by-1 vector

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

Data Types: double

h — Altitude
scalar

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

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

Dependencies

To enable this parameter, set Planet model to Earth (WGS84).

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, specified as either Custom or Earth (WGS84).

Dependencies

Selecting the Custom option 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 planet
`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, specified as a double scalar, in the same units as the Units parameter.

Dependencies

This parameter is enabled when Planet model is set 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`

Select this check box to enable ports for reference position and angle to convert flat Earth. Select this check box if you want to specify the reference positions and angle as dynamic values.

Dependencies

Selecting this check box replaces these parameters:

Reference geodetic latitude and longitude [deg]
Direction of flat Earth x-axis (degrees clockwise from north)

with these input ports:

μ_ref l_ref
ψ_ref input ports.

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 the Input reference position and orientation check box.

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) — Angle
`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

This parameter is disabled if the Input reference position and orientation check box is selected.

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 transforming the flat Earth x and y coordinates to North and East coordinates. The transformation has the form of:

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

where $(\bar{ψ})$ 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 (R_M) are used.

(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 $(\bar{f})$ is the flattening of the planet.

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

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

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

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

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

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

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 before R2006a

Flat Earth to LLA

Description

Examples

Lightweight Airplane Design

Limitations

Ports

Input

Xe — Position in flat Earth frame 3-by-1 vector

href — Reference height scalar

μref lref — Reference location 2-by-1 vector

Dependencies

ψref — Direction of flat Earth x-axis scalar

Dependencies

Output

μ l — Geodetic latitude and longitude 2-by-1 vector

h — Altitude scalar

Parameters

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

Dependencies

Programmatic Use

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

Dependencies

Programmatic Use

Flattening — Flattening of planet 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

Dependencies

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) — Angle 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

X_e — Position in flat Earth frame
3-by-1 vector

h_ref — Reference height
scalar

μ_ref l_ref — Reference location
2-by-1 vector

ψ_ref — Direction of flat Earth x-axis
scalar

μ l — Geodetic latitude and longitude
2-by-1 vector

h — Altitude
scalar

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

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

Flattening — Flattening of planet
`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) — Angle
`0` (default) | scalar

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