6DoF ECEF (Quaternion) - Implement quaternion representation of six-degrees-of-freedom equations of motion in Earth-centered Earth-fixed (ECEF) coordinates

Library

Equations of Motion/6DoF

Description

The 6DoF ECEF (Quaternion) block considers the rotation of a Earth-centered Earth-fixed (ECEF) coordinate frame (X_ECEF , Y_ECEF , Z_ECEF ) about an Earth-centered inertial (ECI) reference frame (X_ECI , Y_ECI , Z_ECI ). 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 (ω_e ) including an initial celestial longitude (L_G(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 [F_x F_y F_z]^T are in the body frame, and the mass of the body m is assumed constant.

where the change of position in ECEF is calculated by

and the velocity of the body with respect to ECEF frame, expressed in body frame , angular rates of the body with respect to ECI frame, expressed in body frame . Earth rotation rate , and relative angular rates of the body with respect to north-east-down (NED) frame, expressed in body frame are defined as

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.

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

Dialog Box

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.
`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 (see Simple Variable Mass 6DoF (Quaternion)).

The Fixed 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.

Initial velocity in body axes

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.

Initial mass

The mass of the rigid body.

Inertia

The 3-by-3 inertia tensor matrix I, in body-fixed axes.

Planet model

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

Flattening

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

Inputs and Outputs

Input	Dimension Type	Description
First	Vector	Contains the three applied forces in body-fixed axes.
Second	Vector	Contains the three applied moments in body-fixed axes.

Output	Dimension Type	Description
First	Vector	Contains the velocity of the body with respect to ECEF frame, expressed in ECEF frame.
Second	Three-element vector	Contains the position in 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 north-east-down (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 matrix	Applies to the coordinate transformation from ECEF axes to geodetic axes.
Eighth	Three-element vector	Contains the velocity of the body with respect to ECEF frame, expressed in the body frame.
Ninth	Three-element vector	Contains the relative angular rates of the body with respect to NED frame, expressed in the body frame, in radians per second.
Tenth	Three-element vector	Contains the angular rates of the body with respect to the ECI frame, expressed in body frame, in radians per second.
Eleventh	Three-element vector	Contains the angular accelerations of the body with respect to ECI frame, expressed in the body frame, in radians per second.
Twelfth	Three-element vector	Contains the accelerations in body-fixed axes.

Assumptions and Limitations

This implementation assumes that the applied forces are acting 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 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.