# Custom Variable Mass 6DOF ECEF (Quaternion)

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

## Library

Equations of Motion/6DOF

## Description

The Custom Variable Mass 6DOF ECEF (Quaternion) block 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, additionally 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. Vreb is the relative velocity in the wind axes at which the mass flow ($\stackrel{˙}{m}$) is ejected or added to the body in body-fixed axes.

`$\begin{array}{c}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{b}+{\overline{\omega }}_{b}×{\overline{V}}_{b}+DC{M}_{bf}{\overline{\omega }}_{e}×{\overline{V}}_{b}+DC{M}_{bf}\left({\overline{\omega }}_{e}×\left({\overline{\omega }}_{e}×{\overline{X}}_{f}\right)\right)\right)\\ +\stackrel{˙}{m}\left(\overline{V}r{e}_{b}+DC{M}_{bf}\left({\overline{\omega }}_{e}×{\overline{X}}_{f}\right)\right)\\ \\ {A}_{bb}=\left[\begin{array}{c}{\stackrel{˙}{u}}_{b}\\ {\stackrel{˙}{v}}_{b}\\ {\stackrel{˙}{w}}_{b}\end{array}\right]=\frac{{\overline{F}}_{b}-\stackrel{˙}{m}\left({\overline{V}}_{r{e}_{b}}+DC{M}_{bf}\left({w}_{e}×{X}_{f}\right)\right)}{m}\\ -\left[{\overline{\omega }}_{{}_{b}}×{\overline{V}}_{b}+DCM{\overline{\omega }}_{e}×{\overline{V}}_{b}+DC{M}_{bf}\left({\overline{\omega }}_{e}\left({\overline{\omega }}_{e}×{X}_{f}\right)\right)\right]\\ \\ {A}_{b}{\text{​}}_{ecef}=\frac{{\overline{F}}_{b}-\stackrel{˙}{m}\left({\overline{V}}_{r{e}_{b}}+DC{M}_{bf}\left({\omega }_{e}×{X}_{f}\right)\right)}{m}\end{array}$`

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.

`$\begin{array}{l}{\overline{M}}_{b}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=\overline{I}{\stackrel{˙}{\overline{\omega }}}_{b}+{\overline{\omega }}_{b}×\left(\overline{I}{\overline{\omega }}_{b}\right)+\stackrel{˙}{I}{\overline{\omega }}_{b}\\ \\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}$`

The rate of change of the inertia tensor is defined by the following equation.

`$\stackrel{˙}{I}=\left[\begin{array}{ccc}{\stackrel{˙}{I}}_{xx}& -{\stackrel{˙}{I}}_{xy}& -{\stackrel{˙}{I}}_{xz}\\ -{\stackrel{˙}{I}}_{yx}& {\stackrel{˙}{I}}_{yy}& -{\stackrel{˙}{I}}_{yz}\\ -{\stackrel{˙}{I}}_{zx}& -{\stackrel{˙}{I}}_{zy}& {\stackrel{˙}{I}}_{zz}\end{array}\right]$`

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

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

## Parameters

Units

Specifies the input and output units:

Units

Forces

Moment

Acceleration

Velocity

Position

Mass

Inertia

`Metric (MKS)`

Newton

Newton meter

Meters per second squared

Meters per second

Meters

Kilogram

Kilogram meter squared

`English (Velocity in ft/s)`

Pound

Foot pound

Feet per second squared

Feet per second

Feet

Slug

Slug foot squared

`English (Velocity in kts)`

Pound

Foot pound

Feet per second squared

Knots

Feet

Slug

Slug foot squared

Mass type

Select the type of mass to use:

 `Fixed` Mass is constant throughout the simulation (see 6DOF ECEF (Quaternion)). `Simple Variable` Mass and inertia vary linearly as a function of mass rate (see Simple Variable Mass 6DOF ECEF (Quaternion)). `Custom Variable` Mass and inertia variations are customizable.

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

Initial position in geodetic latitude, longitude and altitude

The three-element vector for the initial location of the body in the geodetic reference frame. 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.

Initial velocity in body-axis

The three-element vector containing the initial velocity of the body with respect to ECEF frame, expressed in body frame.

Initial Euler orientation

The three-element vector containing the initial Euler rotation angles [roll, pitch, yaw], in radians. Euler rotation angles are those between the body and north-east-down (NED) coordinate systems.

Initial body rotation rates

The three-element vector for the initial angular rates of the body with respect to NED frame, expressed in body frame, in radians per second.

Include mass flow relative velocity

Select this check box to add a mass flow relative velocity port. This is the relative velocity at which the mass is accreted or ablated.

Planet model

Specifies the planet model to use, `Custom` or ```Earth (WGS84)```.

Specifies the radius of the planet at its equator. The units of the equatorial radius parameter should be the same as the units for ECEF position. This option is only available when Planet model is set to `Custom`.

Flattening

Specifies the flattening of the planet. This option is only available when Planet model is set to `Custom`.

Rotational rate

Specifies the scalar rotational rate of the planet in rad/s. This option is only available when Planet model is set to `Custom`.

Celestial longitude of Greenwich source

Specifies the source of Greenwich meridian's initial celestial longitude:

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

The initial angle between Greenwich meridian and the x-axis of the ECI frame.

Include inertial acceleration

Select this check box to enable an additional output port for the accelerations in body-fixed axes with respect to the inertial frame. You typically connect this signal to the accelerometer.

## Inputs and Outputs

InputDimension TypeDescription

First

VectorContains the three applied forces in body-fixed axes.

Second

VectorContains the three applied moments in body-fixed axes.

Third (Optional)

VectorContains one or more rates of change of mass (positive if accreted, negative if ablated).

Fourth

ScalarContains the mass.

Sixth

3-by-3 matrixApplies to the inertia tensor matrix.

Seventh (Optional)

1-by-1-by-m arrayContains one or more relative velocities at which the mass is accreted to or ablated from the body in body-fixed axes. m is three times the size of the third input vector.
OutputDimension TypeDescription

First

Three-element vectorContains the velocity of the body with respect to ECEF frame, expressed in ECEF frame.

Second

Three-element vector Contains the position in the ECEF reference frame.

Third

Three-element vector Contains the position in geodetic latitude, longitude and altitude, in degrees, degrees and selected units of length respectively.

Fourth

Three-element vector Contains the body rotation angles [roll, pitch, yaw], in radians. Euler rotation angles are those between the body and NED coordinate systems.

Fifth

3-by-3 matrix Applies to the coordinate transformation from ECI axes to body-fixed axes.

Sixth

3-by-3 matrix Applies to the coordinate transformation from NED axes to body-fixed axes.

Seventh

3-by-3 matrixApplies to the coordinate transformation from ECEF axes to NED axes.

Eighth

Three-element vector Contains the velocity of the body with respect to ECEF frame, expressed in body frame.

Ninth

Three-element vector Contains the relative angular rates of the body with respect to NED frame, expressed in body frame, in radians per second.

Tenth

Three-element vector Contains the angular rates of the body with respect to ECI frame, expressed in body frame, in radians per second.

Eleventh

Three-element vectorContains the angular accelerations of the body with respect to ECI frame, expressed in body frame, in radians per second squared.

Twelfth

Three-element vector Contains the accelerations in body-fixed axes with respect to body frame.

Thirteenth (Optional)

Three-element vector Contains the accelerations in body-fixed axes with respect to ECEF frame.

## Assumptions and Limitations

This implementation assumes that the applied forces are acting at the center of gravity of the body.

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 through the center of the Sun 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.

## References

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, Second Edition, John Wiley & Sons, New York, 2003.

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

“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.

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