Aerospace Blockset™

Simple Variable Mass 6DoF ECEF (Quaternion)

Simple Variable Mass 6DoF Wind (Wind Angles)

Simple Variable Mass 6DoF Wind (Quaternion)

Implement quaternion representation of six-degrees-of-freedom equations of motion of simple variable mass with respect to wind axes

Library

Equations of Motion/6DoF

Description

The Simple Variable Mass 6DoF Wind (Quaternion) block considers the rotation of a wind-fixed coordinate frame about an Earth-fixed reference frame . The origin of the wind-fixed coordinate frame is the center of gravity of the body, and the body is assumed to be rigid, an assumption that eliminates the need to consider the forces acting between individual elements of mass. The Earth-fixed reference frame is considered inertial, an excellent approximation that allows the forces due to the Earth's motion relative to the "fixed stars" to be neglected.

The translational motion of the wind-fixed coordinate frame is given below, where the applied forces [F_x F_yF_z]^Tare in the wind-fixed frame.

The rotational dynamics of the body-fixed frame are given below, where the applied moments are [L MN]^T, and the inertia tensor is with respect to the origin O. Inertia tensor I is much easier to define in body-fixed frame.

The inertia tensor is determined using a table lookup which linearly interpolates between I_full and I_empty based on mass (m). While the rate of change of the inertia tensor is estimated by the following equation.

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.
`Custom Variable`	Mass and inertia variations are customizable.

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

Representation

Select the representation to use:

`Wind Angles`	Use wind angles within equations of motion.
`Quaternion`	Use quaternions within equations of motion.

The Quaternion selection conforms to the previously described equations of motion.

Initial position in inertial axes

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

Initial airspeed, sideslip angle, and angle of attack

The three-element vector containing the initial airspeed, initial sideslip angle and initial angle of attack.

Initial wind orientation

The three-element vector containing the initial wind angles [bank, flight path, and heading], in radians.

Initial body rotation rates

The three-element vector for the initial body-fixed angular rates, in radians per second.

Initial mass

The initial mass of the rigid body.

Empty mass

A scalar value for the empty mass of the body.

Full mass

A scalar value for the full mass of the body.

Empty inertia matrix

A 3-by-3 inertia tensor matrix for the empty inertia of the body, in body-fixed axes.

Full inertia matrix

A 3-by-3 inertia tensor matrix for the full inertia of the body, in body-fixed axes.

Inputs and Outputs

Input	Dimension Type	Description
First	Vector	Contains the three applied forces in wind-fixed axes.
Second	Vector	Contains the three applied moments in body-fixed axes.
Third	Scalar	Contains the rate of change of mass.

Output	Dimension Type	Description
First	Three-element vector	Contains the velocity in the Earth-fixed reference frame.
Second	Three-element vector	Contains the position in the Earth-fixed reference frame.
Third	Three-element vector	Contains the wind rotation angles [bank, flight path, heading], in radians.
Fourth	3-by-3 matrix	Applies to the coordinate transformation from Earth-fixed axes to wind-fixed axes.
Fifth	Three-element vector	Contains the velocity in the wind-fixed frame.
Sixth	Two-element vector	Contains the angle of attack and sideslip angle, in radians.
Seventh	Two-element vector	Contains the rate of change of angle of attack and rate of change of sideslip angle, in radians per second.
Eighth	Three-element vector	Contains the angular rates in body-fixed axes, in radians per second.
Ninth	Three-element vector	Contains the angular accelerations in body-fixed axes, in radians per second.
Tenth	Three-element vector	Contains the accelerations in body-fixed axes.
Eleventh	Scalar element	Contains a flag for fuel tank status: 1 indicates that the tank is full. 0 indicates that the integral is neither full nor empty. -1 indicates that the tank is empty.

Assumptions and Limitations

The block assumes that the applied forces are acting at the center of gravity of the body.

References

Mangiacasale, L., Flight Mechanics of a μ-Airplane with a MATLAB Simulink Helper, Edizioni Libreria CLUP, Milan, 1998.

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.