Aerospace Blockset

Custom Variable Mass 6DoF (Euler Angles)

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

Library

Equations of Motion/6DoF

Description

The Custom 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 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 Custom 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.

Inputs and Outputs

The first input to the block is a vector containing the three applied forces.

The second input is a vector containing the three applied moments.

The third input is a scalar containing the rate of change of mass.

The fourth input is a scalar containing the mass

The fifth input is a 3-by-3 matrix for the rate of change of inertia tensor matrix.

The sixth input is a 3-by-3 matrix for the inertia tensor matrix.

The first output is a three-element vector containing the velocity in the Earth-fixed reference frame.

The second output is a three-element vector containing the position in the Earth-fixed reference frame.

The third output is a three-element vector containing the Euler rotation angles [roll, pitch, yaw], in radians.

The fourth output is a 3-by-3 matrix for the coordinate transformation from Earth-fixed axes to body-fixed axes.

The fifth output is a three-element vector containing the velocity in the body-fixed frame.

The sixth output is a three-element vector containing the angular rates in body-fixed axes, in radians per second.

The seventh output is a three-element vector containing the angular accelerations in body-fixed axes, in radians per second.

The eighth output is a three-element vector containing the accelerations in body-fixed axes.

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.