Aerospace Blockset™

Simple Variable Mass 6DoF (Euler Angles)

Implement Euler angle representation of six-degrees-of-freedom equations of motion of simple variable mass

Library

Equations of Motion/6DoF

Description

The Simple Variable Mass 6DoF (Euler Angles) block considers the rotation of a body-fixed coordinate frame about an Earth-fixed reference frame . The origin of the body-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 body-fixed coordinate frame is given below, where the applied forces [F_x F_yF_z]^Tare in the body-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.

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 relationship between the body-fixed angular velocity vector, [p q r]^T, and the rate of change of the Euler angles, [ ]^T, can be determined by resolving the Euler rates into the body-fixed coordinate frame.

Inverting then gives the required relationship to determine the Euler rate vector.

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:

`Euler Angles`	Use Euler angles within equations of motion.
`Quaternion`	Use quaternions within equations of motion.

The Euler Angles 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 velocity in body axes

The three-element vector for the initial velocity in the body-fixed coordinate frame.

Initial Euler rotation

The three-element vector for the initial Euler rotation angles [roll, pitch, yaw], 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.

Full inertia matrix

A 3-by-3 inertia tensor matrix for the full inertia of the body.

Inputs and Outputs

Input	Dimension Type	Description
First	Vector	Contains the three applied forces.
Second	Vector	Contains the three applied moments.
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 Euler rotation angles [roll, pitch, yaw], in radians.
Fourth	3-by-3 matrix	Applies to the coordinate transformation from Earth-fixed axes to body-fixed axes.
Fifth	Three-element vector	Contains the velocity in the body-fixed frame.
Sixth	Three-element vector	Contains the angular rates in body-fixed axes, in radians per second.
Seventh	Three-element vector	Contains the angular accelerations in body-fixed axes, in radians per second.
Eight	Three-element vector	Contains the accelerations in body-fixed axes.
Ninth	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.

Reference

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