6DOF (Quaternion)

Implement quaternion representation of six-degrees-of-freedom equations of motion with respect to body axes

Library

Equations of Motion/6DOF

Description

For a description of the coordinate system and the translational dynamics, see the block description for the 6DOF (Euler Angles) block.

The integration of the rate of change of the quaternion vector is given below. The gain K drives the norm of the quaternion state vector to 1.0 should $$\epsilon$$become nonzero. You must choose the value of this gain with care, because a large value improves the decay rate of the error in the norm, but also slows the simulation because fast dynamics are introduced. An error in the magnitude in one element of the quaternion vector is spread equally among all the elements, potentially increasing the error in the state vector.

$$\begin{array}{l}\left[\begin{array}{c}{\dot{q}}_0\\ {\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left[\begin{array}{cccc}0& -p& -q& -r\\ p& 0& r& -q\\ q& -r& 0& p\\ r& q& -p& 0\end{array}\right]\left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]+K\epsilon \left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]\\ \epsilon =1-\left(q_0^2+q_1^2+q_2^2+q_3^2\right)\end{array}$$

Parameters

Main

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 Fixed 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 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 flat Earth 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 mass of the rigid body.

Inertia matrix

The 3-by-3 inertia tensor matrix I.

Gain for quaternion normalization

The gain to maintain the norm of the quaternion vector equal to 1.0.

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.

State Attributes

Assign unique name to each state. You can use state names instead of block paths during linearization.

• To assign a name to a single state, enter a unique name between quotes, for example, 'velocity'.

• To assign names to multiple states, enter a comma-delimited list surrounded by braces, for example, {'a', 'b', 'c'}. Each name must be unique.

• If a parameter is empty (' '), no name assignment occurs.

• The state names apply only to the selected block with the name parameter.

• The number of states must divide evenly among the number of state names.

• You can specify fewer names than states, but you cannot specify more names than states.

  For example, you can specify two names in a system with four states. The first name applies to the first two states and the second name to the last two states.

• To assign state names with a variable in the MATLAB^® workspace, enter the variable without quotes. A variable can be a character vector, cell array, or structure.

Position: e.g., {'Xe', 'Ye', 'Ze'}

Specify position state names.

Default value is ''.

Velocity: e.g., {'U', 'v', 'w'}

Specify velocity state names.

Default value is ''.

Quaternion vector: e.g., {'qr', 'qi', 'qj', 'qk'}

Specify quaternion vector state names.

Default value is ''.

Body rotation rates: e.g., {'p', 'q', 'r'}

Specify body rotation rate state names.

Default value is ''.

Inputs and Outputs

Input	Dimension Type	Description
First	Vector	Contains the three applied forces.
Second	Vector	Contains the three applied moments.

Output	Dimension Type	Description
First	Three-element vector	Contains the velocity in the flat Earth reference frame.
Second	Three-element vector	Contains the position in the flat Earth reference frame.
Third	Three-element vector	Contains the Euler rotation angles [roll, pitch, yaw], in radians.
Fourth	3-by-3 matrix	Contains the coordinate transformation from flat Earth 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 squared.
Eight	Three-element vector	Contains the accelerations in body-fixed axes with respect to body frame.
Ninth (Optional)	Three-element vector	Contains the accelerations in body-fixed axes with respect to inertial frame (flat Earth). You typically connect this signal to the accelerometer.

Assumptions and Limitations

The block assumes that the applied forces are acting at the center of gravity of the body, and that the mass and inertia are constant.

Reference

Stevens, Brian, and Frank Lewis, Aircraft Control and Simulation, Second Edition, John Wiley & Sons, 2003.

Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Second Edition, AIAA Education Series, 2007.