# 6DOF ECEF (Quaternion)

Implement quaternion representation of six-degrees-of-freedom equations of motion in Earth-centered Earth-fixed (ECEF) coordinates

• Library:
• Aerospace Blockset / Equations of Motion / 6DOF

## Description

The 6DOF ECEF (Quaternion) block Implement quaternion representation of six-degrees-of-freedom equations of motion in Earth-centered Earth-fixed (ECEF) coordinates. It considers the rotation of a Earth-centered Earth-fixed (ECEF) coordinate frame (XECEF, YECEF, ZECEF) about an Earth-centered inertial (ECI) reference frame (XECI, YECI, ZECI). The origin of the ECEF coordinate frame is the center of the Earth. For more information on the ECEF coordinate frame, see Algorithms.

## Limitations

• This implementation assumes that the applied forces act at the center of gravity of the body, and that the mass and inertia are constant.

• This implementation generates a geodetic latitude that lies between ±90 degrees, and longitude that lies between ±180 degrees. Additionally, the MSL altitude is approximate.

• The Earth is assumed to be ellipsoidal. By setting flattening to 0.0, a spherical planet can be achieved. The Earth's precession, nutation, and polar motion are neglected. The celestial longitude of Greenwich is Greenwich Mean Sidereal Time (GMST) and provides a rough approximation to the sidereal time.

• The implementation of the ECEF coordinate system assumes that the origin is at the center of the planet, the x-axis intersects the Greenwich meridian and the equator, the z-axis is the mean spin axis of the planet, positive to the north, and the y-axis completes the right-handed system.

• The implementation of the ECI coordinate system assumes that the origin is at the center of the planet, the x-axis is the continuation of the line from the center of the Earth toward the vernal equinox, the z-axis points in the direction of the mean equatorial plane's north pole, positive to the north, and the y-axis completes the right-handed system.

## Ports

### Input

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Applied forces, specified as a three-element vector.

Data Types: `double`

Applied moments, specified as a three-element vector.

Data Types: `double`

Greenwich meridian initial celestial longitude angle, specified as a scalar.

#### Dependencies

To enable this port, set Celestial longitude of Greenwich to `External`.

Data Types: `double`

### Output

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Velocity of body with respect to ECEF frame, expressed in ECEF frame, returned as a three-element vector.

Data Types: `double`

Position in ECEF reference frame, returned as a three-element vector.

Data Types: `double`

Position in geodetic latitude, longitude, and altitude, in degrees, returned as a three-element vector or M-by-3 array, in selected units of length, respectively.

Data Types: `double`

Body rotation angles [roll, pitch, yaw], returned as a three-element vector, in radians. Euler rotation angles are those between body and NED coordinate systems.

Data Types: `double`

Coordinate transformation from ECI axes to body-fixed axes, returned as a 3-by-3 matrix.

Data Types: `double`

Coordinate transformation from NED axes to body-fixed axes, returned as a 3-by-3 matrix.

Data Types: `double`

Coordinate transformation from ECEF axes to NED axes, returned as a 3-by-3 matrix.

Data Types: `double`

Velocity of body with respect to ECEF frame, returned as a three-element vector.

Data Types: `double`

Relative angular rates of body with respect to NED frame, expressed in body frame and returned as a three-element vector, in radians per second.

Data Types: `double`

Angular rates of the body with respect to ECI frame, expressed in body frame and returned as a three-element vector, in radians per second.

Data Types: `double`

Angular accelerations of the body with respect to ECI frame, expressed in body frame and returned as a three-element vector, in radians per second squared.

Data Types: `double`

Accelerations of the body with respect to the ECEF coordinate frame, returned as a three-element vector.

Data Types: `double`

Accelerations in body-fixed axes with respect to ECEF frame, returned as a three-element vector.

#### Dependencies

To enable this point, Include inertial acceleration.

Data Types: `double`

## Parameters

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### Main

Input and output units, specified as `Metric (MKS)`, `English (Velocity in ft/s)`, or `English (Velocity in kts)`.

UnitsForcesMomentAccelerationVelocityPositionMassInertia
`Metric (MKS)` NewtonNewton-meterMeters per second squaredMeters per secondMetersKilogramKilogram meter squared
`English (Velocity in ft/s)` PoundFoot-poundFeet per second squaredFeet per secondFeetSlugSlug foot squared
`English (Velocity in kts)` PoundFoot-poundFeet per second squaredKnotsFeetSlugSlug foot squared

#### Programmatic Use

 Block Parameter: `units` Type: character vector Values: `Metric (MKS)` | `English (Velocity in ft/s)` | `English (Velocity in kts)` Default: `Metric (MKS)`

Select the type of mass to use:

Mass TypeDescriptionDefault for
`Fixed`

Mass is constant throughout the simulation.

`Simple Variable`

Mass and inertia vary linearly as a function of mass rate.

`Custom Variable`

Mass and inertia variations are customizable.

The `Fixed` selection conforms to the previously described equations of motion.

#### Programmatic Use

 Block Parameter: `mtype` Type: character vector Values: `Fixed` | `Simple Variable` | `Custom Variable` Default: `'Simple Variable'`

Initial location of the aircraft in the geodetic reference frame, specified as a three-element vector. 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.

#### Programmatic Use

 Block Parameter: `xg_0` Type: character vector Values: `'[0 0 0]'` | three-element vector Default: `'[0 0 0]'`

Initial velocity in body axes, specified as a three-element vector, in the body-fixed coordinate frame.

#### Programmatic Use

 Block Parameter: `Vm_0` Type: character vector Values: `'[0 0 0]'` | three-element vector Default: `'[0 0 0]'`

Initial Euler orientation angles [roll, pitch, yaw], specified as a three-element vector, in radians. Euler rotation angles are those between the body and north-east-down (NED) coordinate systems.

#### Programmatic Use

 Block Parameter: `eul_0` Type: character vector Values: `'[0 0 0]'` | three-element vector Default: `'[0 0 0]'`

Initial body-fixed angular rates with respect to the NED frame, specified as a three-element vector, in radians per second.

#### Programmatic Use

 Block Parameter: `pm_0` Type: character vector Values: `'[0 0 0]'` | three-element vector Default: `'[0 0 0]'`

Initial mass of the rigid body, specified as a double scalar.

#### Programmatic Use

 Block Parameter: `mass_0` Type: character vector Values: `'1.0'` | double scalar Default: `'1.0'`

Inertia of the body, specified as a double scalar.

#### Dependencies

To enable this parameter, set Mass type to `Fixed`.

#### Programmatic Use

 Block Parameter: `inertia` Type: character vector Values: `eye(3)` | double scalar Default: `eye(3)`

Select this check box to add an inertial acceleration port.

#### Dependencies

To enable the Abe port, select this parameter.

#### Programmatic Use

 Block Parameter: `abi_flag` Type: character vector Values: `'off'` | `'on'` Default: `off`

### Planet

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

#### Programmatic Use

 Block Parameter: `ptype` Type: character vector Values: `'Earth (WGS84)'` | `'Custom'` Default: `'Earth (WGS84)'`

Radius of the planet at its equator, specified as a double scalar, in the same units as the desired units for the 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'`

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'`

Rotational rate of the planet, specified as a scalar, in rad/s.

#### Dependencies

To enable this parameter, set Planet model to `Custom`.

#### Programmatic Use

 Block Parameter: `w_E` Type: character vector Values: double scalar Default: `'7292115e-11'`

Source of Greenwich meridian initial celestial longitude, specified as:

 `Internal` Use celestial longitude value from Celestial longitude of Greenwich. `External` Use external input for celestial longitude value.

#### Dependencies

Setting this parameter to `External` enables the LG(0) port.

#### Programmatic Use

 Block Parameter: `angle_in` Type: character vector Values: `'Internal'` | `'External'` Default: `'Internal'`

Initial angle between Greenwich meridian and the x-axis of the ECI frame, specified as a double scalar.

#### Dependencies

To enable this parameter, set Celestial longitude of Greenwich source to `Internal`.

#### Programmatic Use

 Block Parameter: `LG0` Type: character vector Values: double scalar Default: `'0'`

### State Attributes

Assign a unique name to each state. You can use state names instead of block paths during linearization.

• To assign a name to a single state, enter a unique name between quotes, for example, `'velocity'`.

• To assign names to multiple states, enter a comma-separated list surrounded by braces, for example, `{'a', 'b', 'c'}`. Each name must be unique.

• If a parameter is empty (`' '`), no name is assigned.

• The state names apply only to the selected block with the name parameter.

• The number of states must divide evenly among the number of state names.

• You can specify fewer names than states, but you cannot specify more names than states.

For example, you can specify two names in a system with four states. The first name applies to the first two states and the second name to the last two states.

• To assign state names with a variable in the MATLAB® workspace, enter the variable without quotes. A variable can be a character vector, cell array, or structure.

Quaternion vector state names, specified as a comma-separated list surrounded by braces.

#### Programmatic Use

 Block Parameter: `quat_statename` Type: character vector Values: `''` | comma-separated list surrounded by braces Default: `''`

Body rotation rate state names, specified comma-separated list surrounded by braces.

#### Programmatic Use

 Block Parameter: `pm_statename` Type: character vector Values: `''` | comma-separated list surrounded by braces Default: `''`

Velocity state names, specified as comma-separated list surrounded by braces.

#### Programmatic Use

 Block Parameter: `Vm_statename` Type: character vector Values: `''` | comma-separated list surrounded by braces Default: `''`

ECEF position state names, specified as a comma-separated list surrounded by braces.

#### Programmatic Use

 Block Parameter: `posECEF_statename` Type: character vector Values: `''` | comma-separated list surrounded by braces Default: `''`

Inertial position state names, specified as a comma-separated list surrounded by braces.

Default value is `''`.

#### Programmatic Use

 Block Parameter: `posECI_statename` Type: character vector Values: `''` | comma-separated list surrounded by braces Default: `''`

Celestial longitude of Greenwich state name, specified as a character vector.

#### Programmatic Use

 Block Parameter: `LG_statename` Type: character vector Values: `''` | scalar Default: `''`

## Algorithms

The origin of the ECEF coordinate frame is the center of the Earth. In addition, the body of interest is assumed to be rigid, an assumption that eliminates the need to consider the forces acting between individual elements of mass. The representation of the rotation of ECEF frame from ECI frame is simplified to consider only the constant rotation of the ellipsoid Earth (ωe) including an initial celestial longitude (LG(0)). This excellent approximation allows the forces due to the Earth's complex motion relative to the “fixed stars” to be neglected.

The translational motion of the ECEF coordinate frame is given below, where the applied forces [Fx Fy Fz]T are in the body frame and the mass of the body m is assumed constant.

${\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{b}+{\overline{\omega }}_{b}×{\overline{V}}_{b}+DC{M}_{bf}{\overline{\omega }}_{e}×{\overline{V}}_{b}+DC{M}_{bf}\left({\overline{\omega }}_{e}×\left({\overline{\omega }}_{e}×{\overline{X}}_{f}\right)\right)\right)$

where the change of position in ECEF ${\stackrel{˙}{\overline{x}}}_{f}$ is calculated by

`${\stackrel{˙}{\overline{x}}}_{f}=DC{M}_{fb}{\overline{V}}_{b}$`

and the velocity of the body with respect to ECEF frame, expressed in body frame $\left({\overline{V}}_{b}\right)$, angular rates of the body with respect to ECI frame, expressed in body frame $\left({\overline{\omega }}_{b}\right)$. Earth rotation rate $\left({\overline{\omega }}_{e}\right)$, and relative angular rates of the body with respect to north-east-down (NED) frame, expressed in body frame $\left({\overline{\omega }}_{rel}\right)$, are defined as

`$\begin{array}{l}{\overline{V}}_{b}=\left[\begin{array}{c}u\\ v\\ w\end{array}\right],{\overline{\omega }}_{rel}=\left[\begin{array}{c}p\\ q\\ r\end{array}\right],{\overline{\omega }}_{e}=\left[\begin{array}{c}0\\ 0\\ {\omega }_{e}\end{array}\right],{\overline{\omega }}_{b}={\overline{\omega }}_{rel}+DC{M}_{bf}{\overline{\omega }}_{e}+DC{M}_{be}{\overline{\omega }}_{ned}\\ {\overline{\omega }}_{ned}=\left[\begin{array}{c}\stackrel{˙}{l}\mathrm{cos}\mu \\ -\stackrel{˙}{\mu }\\ -\stackrel{˙}{l}\mathrm{sin}\mu \end{array}\right]=\left[\begin{array}{c}{V}_{E}/\left(N+h\right)\\ -{V}_{N}/\left(M+h\right)\\ -{V}_{E}•\mathrm{tan}\mu /\left(N+h\right)\end{array}\right]\end{array}$`

The rotational dynamics of the body defined in body-fixed frame are given below, where the applied moments are [L M N]T, and the inertia tensor I is with respect to the origin O.

`$\begin{array}{l}{A}_{bb}=\left[\begin{array}{c}{\stackrel{˙}{u}}_{b}\\ {\stackrel{˙}{v}}_{b}\\ {\stackrel{˙}{\omega }}_{b}\end{array}\right]=\frac{1}{m}{\overline{F}}_{b}-\left[{\overline{\omega }}_{b}×{\overline{V}}_{b}+DC{M}_{bf}{\overline{\omega }}_{e}×{\overline{V}}_{b}+DC{M}_{bf}\left({\overline{\omega }}_{e}×\left({\overline{\omega }}_{e}×{\overline{X}}_{f}\right)\right)\right]\\ {A}_{b}{\text{​}}_{ecef}=\frac{{F}_{b}}{m}\\ {\overline{M}}_{b}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I{\stackrel{˙}{\overline{\omega }}}_{b}+{\overline{\omega }}_{b}×\left(I{\overline{\omega }}_{b}\right)\\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}$`

The integration of the rate of change of the quaternion vector is given below.

`$\left[\begin{array}{c}{\stackrel{˙}{q}}_{0}\\ {\stackrel{˙}{q}}_{1}\\ {\stackrel{˙}{q}}_{2}\\ {\stackrel{˙}{q}}_{3}\end{array}\right]=-1}{2}\left[\begin{array}{cccc}0& {\omega }_{b}\left(1\right)& {\omega }_{b}\left(2\right)& {\omega }_{b}\left(3\right)\\ -{\omega }_{b}\left(1\right)& 0& -{\omega }_{b}\left(3\right)& {\omega }_{b}\left(2\right)\\ -{\omega }_{b}\left(2\right)& {\omega }_{b}\left(3\right)& 0& -{\omega }_{b}\left(1\right)\\ -{\omega }_{b}\left(3\right)& -{\omega }_{b}\left(2\right)& {\omega }_{b}\left(1\right)& 0\end{array}\right]\left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]$`

Aerospace Blockset™ uses quaternions that are defined using the scalar-first convention.

## References

[1] Stevens, Brian, and Frank Lewis. Aircraft Control and Simulation, 2nd ed. Hoboken, NJ: John Wiley & Sons, 2003.

[2] McFarland, Richard E. "A Standard Kinematic Model for Flight simulation at NASA-Ames." NASA CR-2497.

[3] "Supplement to Department of Defense World Geodetic System 1984 Technical Report: Part I - Methods, Techniques and Data Used in WGS84 Development." DMA TR8350.2-A.

## Version History

Introduced in R2006a