Implement Euler angle representation of six-degrees-of-freedom equations of motion (2024)

Table of Contents

Description Limitations Ports Input Fxyz(N) — Applied forcesthree-element vector Mxyz(N-m) — Applied moments three-element vector Output Ve — Velocity in flat Earth reference framethree-element vector Xe — Position in flat Earth reference frame three-element vector φ θ ψ (rad) — Euler rotation angles three-element vector DCMbe — Coordinate transformation 3-by-3 matrix Vb — Velocity in the body-fixed frame three-element vector ωb (rad/s) — Angular rates in body-fixed axes three-element vector dωb/dt — Angular accelerations three-element vector Abb — Accelerations in body-fixed axesthree-element vector Abe — Accelerations with respect to inertial frame three-element vector Parameters Main Units — Input and output unitsMetric (MKS) (default) | English (Velocity in ft/s) | English (Velocity in kts) Mass Type — Mass type Fixed (default) | Simple Variable | Custom Variable Representation — Equations of motion representation Euler Angles (default) | Quaternion Initial position in inertial axes [Xe,Ye,Ze] — Position in inertial axes [0 0 0] (default) | three-element vector Initial velocity in body axes [U,v,w] — Velocity in body axes [0 0 0] (default) | three-element vector Initial Euler orientation [roll, pitch, yaw] — Initial Euler orientation [0 0 0] (default) | three-element vector Initial body rotation rates [p,q,r] — Initial body rotation [0 0 0] (default) | three-element vector Initial mass — Initial mass 1.0 (default) | scalar Inertia — Inertia eye(3) (default) | scalar Include inertial acceleration — Include inertial acceleration port off (default) | on State Attributes Position: e.g., {'Xe', 'Ye', 'Ze'} — Position state name '' (default) | comma-separated list surrounded by braces Velocity: e.g., {'U', 'v', 'w'} — Velocity state name '' (default) | comma-separated list surrounded by braces Euler rotation angles: e.g., {'phi', 'theta', 'psi'} — Euler rotation state name '' (default) | comma-separated list surrounded by braces Body rotation rates: e.g., {'p', 'q', 'r'} — Body rotation state names '' (default) | comma-separated list surrounded by braces Algorithms References Extended Capabilities C/C++ Code Generation Generate C and C++ code using Simulink® Coder™. Version History See Also Topics MATLAB Command Americas Europe Asia Pacific FAQs References

Implement Euler angle representation of six-degrees-of-freedom equations of motion

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Libraries:
Aerospace Blockset / Equations of Motion / 6DOF

Description

The 6DOF (Euler Angles) block implements the Euler angle representation of six-degrees-of-freedom equations of motion, taking into consideration the rotation of a body-fixed coordinate frame (X_b, Y_b, Z_b) about a flat Earth reference frame (X_e, Y_e, Z_e). For more information about these reference points, see Algorithms.

Limitations

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

Ports

Input

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F_xyz(N) — Applied forces
three-element vector

Applied forces, specified as a three-element vector in body-fixed axes. For more information on the frame, see Body Coordinates.

Data Types: double

M_xyz(N-m) — Applied moments
three-element vector

Applied moments, specified as a three-element vector in body-fixed axes. For more information on the frame, see Body Coordinates.

Data Types: double

Output

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V_e — Velocity in flat Earth reference frame
three-element vector

Velocity in the flat Earth reference frame, returned as a three-element vector.

Data Types: double

X_e — Position in flat Earth reference frame
three-element vector

Position in the flat Earth reference frame, returned as a three-element vector.

Data Types: double

φ θ ψ (rad) — Euler rotation angles
three-element vector

Euler rotation angles [roll, pitch, yaw] defining an intrinsic x-y-z rotation, as a three-element vector, in radians. Yaw, pitch, and roll angles are applied using the z-y-x rotation sequence, such as angle2dcm(yaw,pitch,roll,"ZYX").

Data Types: double

DCM_be — Coordinate transformation
3-by-3 matrix

V_b — Velocity in the body-fixed frame
three-element vector

Velocity in the body-fixed frame, returned as a three-element vector.

Data Types: double

ω_b (rad/s) — Angular rates in body-fixed axes
three-element vector

Angular rates in body-fixed axes, returned as a three-element vector, in radians per second.

Data Types: double

dω_b/dt — Angular accelerations
three-element vector

Angular accelerations in body-fixed axes, returned as a three-element vector, in radians per second squared.

Data Types: double

A_bb — Accelerations in body-fixed axes
three-element vector

Accelerations in body-fixed axes with respect to body frame, returned as a three-element vector.

Data Types: double

A_be — Accelerations with respect to inertial frame
three-element vector

Accelerations in body-fixed axes with respect to inertial frame (flat Earth), returned as a three-element vector. You typically connect this signal to the accelerometer.

Dependencies

This port appears only when the Include inertial acceleration check box is selected.

Data Types: double

Parameters

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Main

Units — Input and output units
`Metric (MKS)` (default) | `English (Velocity in ft/s)` | `English (Velocity in kts)`

Input and output units, specified as Metric (MKS), English (Velocity in ft/s), or English (Velocity in kts).

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

Programmatic Use

Block Parameter: units

Type: character vector

Values: Metric (MKS) | English (Velocity in ft/s) | English (Velocity in kts)

Default: Metric (MKS)

Mass Type — Mass type
`Fixed` (default) | `Simple Variable` | `Custom Variable`

Mass type, specified according to the following table.

Mass Type	Description	Default for
`Fixed`	Mass is constant throughout the simulation.	6DOF (Euler Angles) 6DOF (Quaternion) 6DOF Wind (Wind Angles) 6DOF Wind (Quaternion) 6DOF ECEF (Quaternion)
`Simple Variable`	Mass and inertia vary linearly as a function of mass rate.	Simple Variable Mass 6DOF (Euler Angles) Simple Variable Mass 6DOF (Quaternion) Simple Variable Mass 6DOF Wind (Wind Angles) Simple Variable Mass 6DOF Wind (Quaternion) Simple Variable Mass 6DOF ECEF (Quaternion)
`Custom Variable`	Mass and inertia variations are customizable.	Custom Variable Mass 6DOF (Euler Angles) Custom Variable Mass 6DOF (Quaternion) Custom Variable Mass 6DOF Wind (Wind Angles) Custom Variable Mass 6DOF Wind (Quaternion) Custom Variable Mass 6DOF ECEF (Quaternion)

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

Programmatic Use

Block Parameter: mtype

Type: character vector

Values: Fixed | Simple Variable | Custom Variable

Default: Simple Variable

Representation — Equations of motion representation
`Euler Angles` (default) | `Quaternion`

Equations of motion representation, specified according to the following table.

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

Programmatic Use

Block Parameter: rep

Type: character vector

Values: Euler Angles | Quaternion

Default: 'Euler Angles'

Initial position in inertial axes [Xe,Ye,Ze] — Position in inertial axes
`[0 0 0]` (default) | three-element vector

Initial location of the body in the flat Earth reference frame, specified as a three-element vector.

Programmatic Use

Block Parameter: xme_0

Type: character vector

Values: '[0 0 0]' | three-element vector

Default: '[0 0 0]'

Initial velocity in body axes [U,v,w] — Velocity in body axes
`[0 0 0]` (default) | three-element vector

Initial velocity in body axes, specified as a three-element vector, in the body-fixed coordinate frame.

Programmatic Use

Block Parameter: Vm_0

Type: character vector

Values: '[0 0 0]' | three-element vector

Default: '[0 0 0]'

Initial Euler orientation [roll, pitch, yaw] — Initial Euler orientation
`[0 0 0]` (default) | three-element vector

Initial Euler orientation angles [roll, pitch, yaw], specified as a three-element vector, in radians. Euler rotation angles are those between the body and north-east-down (NED) coordinate systems.

Programmatic Use

Block Parameter: eul_0

Type: character vector

Values: '[0 0 0]' | three-element vector

Default: '[0 0 0]'

Initial body rotation rates [p,q,r] — Initial body rotation
`[0 0 0]` (default) | three-element vector

Initial body-fixed angular rates with respect to the NED frame, specified as a three-element vector, in radians per second.

Programmatic Use

Block Parameter: pm_0

Type: character vector

Values: '[0 0 0]' | three-element vector

Default: '[0 0 0]'

Initial mass — Initial mass
`1.0` (default) | scalar

Initial mass of the rigid body, specified as a double scalar.

Programmatic Use

Block Parameter: mass_0

Type: character vector

Values: '1.0' | double scalar

Default: '1.0'

Inertia — Inertia
`eye(3)` (default) | scalar

Inertia of the body, specified as a double scalar.

Dependencies

To enable this parameter, set Mass type to Fixed.

Programmatic Use

Block Parameter: inertia

Type: character vector

Values: eye(3) | double scalar

Default: eye(3)

Include inertial acceleration — Include inertial acceleration port
`off` (default) | `on`

Select this check box to add an inertial acceleration port.

Dependencies

To enable the A_{b ff} port, select this parameter.

Programmatic Use

Block Parameter: abi_flag

Type: character vector

Values: 'off' | 'on'

Default: off

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'} — Position state name
`''` (default) | comma-separated list surrounded by braces

Position state names, specified as a comma-separated list surrounded by braces.

Programmatic Use

Block Parameter: xme_statename

Type: character vector

Values: '' | comma-separated list surrounded by braces

Default: ''

Velocity: e.g., {'U', 'v', 'w'} — Velocity state name
`''` (default) | comma-separated list surrounded by braces

Velocity state names, specified as comma-separated list surrounded by braces.

Programmatic Use

Block Parameter: Vm_statename

Type: character vector

Values: '' | comma-separated list surrounded by braces

Default: ''

Euler rotation angles: e.g., {'phi', 'theta', 'psi'} — Euler rotation state name
`''` (default) | comma-separated list surrounded by braces

Euler rotation angle state names, specified as a comma-separated list surrounded by braces.

Programmatic Use

Block Parameter: eul_statename

Type: character vector

Values: '' | comma-separated list surrounded by braces

Default: ''

Body rotation rates: e.g., {'p', 'q', 'r'} — Body rotation state names
`''` (default) | comma-separated list surrounded by braces

Body rotation rate state names, specified comma-separated list surrounded by braces.

Programmatic Use

Block Parameter: pm_statename

Type: character vector

Values: '' | comma-separated list surrounded by braces

Default: ''

Algorithms

The 6DOF (Euler Angles) block uses these reference frame concepts.

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 flat Earth reference frame is considered inertial, an excellent approximation that allows the forces due to the Earth motion relative to the "fixed stars" to be neglected.
Translational motion of the body-fixed coordinate frame, where the applied forces [F_x F_y F_z]^T are in the body-fixed frame, and the mass of the body m is assumed constant.
$\begin{array}{l} {\bar{F}}_{b} = [\begin{matrix} F_{x} \\ F_{y} \\ F_{z} \end{matrix}] = m ({\dot{\bar{V}}}_{b} + \bar{ω} \times {\bar{V}}_{b}) \\ A_{b b} = [\begin{matrix} {\dot{u}}_{b} \\ {\dot{v}}_{b} \\ {\dot{w}}_{b} \end{matrix}] = \frac{1}{m} {\bar{F}}_{b} - \bar{ω} \times {\bar{V}}_{b} \\ A_{b e} = \frac{1}{m} F_{b} \\ {\bar{V}}_{b} = [\begin{matrix} u_{b} \\ v_{b} \\ w_{b} \end{matrix}], \bar{ω} = [\begin{matrix} p \\ q \\ r \end{matrix}] \end{array}$
The rotational dynamics of the body-fixed frame, where the applied moments are [L M N]^T, and the inertia tensor I is with respect to the origin O.
$\begin{array}{l} {\bar{M}}_{B} = [\begin{matrix} L \\ M \\ N \end{matrix}] = I \dot{\bar{ω}} + \bar{ω} \times (I \bar{ω}) \\ I = [\begin{matrix} I_{x x} & - I_{x y} & - I_{x z} \\ - I_{y x} & I_{y y} & - I_{y z} \\ - I_{z x} & - I_{z y} & I_{z z} \end{matrix}] \end{array}$
The relationship between the body-fixed angular velocity vector, [p q r]^T, and the rate of change of the Euler angles, $[\begin{matrix} \dot{ϕ} & \dot{θ} & \dot{ψ} \end{matrix}]^{T}$ , are determined by resolving the Euler rates into the body-fixed coordinate frame.
$[\begin{matrix} p \\ q \\ r \end{matrix}] = [\begin{matrix} \dot{ϕ} \\ 0 \\ 0 \end{matrix}] + [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos ϕ & \sin ϕ \\ 0 & - \sin ϕ & \cos ϕ \end{matrix}] [\begin{matrix} 0 \\ \dot{θ} \\ 0 \end{matrix}] + [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos ϕ & \sin ϕ \\ 0 & - \sin ϕ & \cos ϕ \end{matrix}] [\begin{matrix} \cos θ & 0 & - \sin θ \\ 0 & 1 & 0 \\ \sin θ & 0 & \cos θ \end{matrix}] [\begin{matrix} 0 \\ 0 \\ \dot{ψ} \end{matrix}] \equiv J^{- 1} [\begin{matrix} \dot{ϕ} \\ \dot{θ} \\ \dot{ψ} \end{matrix}]$
Inverting J then gives the required relationship to determine the Euler rate vector.
$[\begin{matrix} \dot{ϕ} \\ \dot{θ} \\ \dot{ψ} \end{matrix}] = J [\begin{matrix} p \\ q \\ r \end{matrix}] = [\begin{matrix} 1 & (\sin ϕ \tan θ) & (\cos ϕ \tan θ) \\ 0 & \cos ϕ & - \sin ϕ \\ 0 & \frac{\sin ϕ}{\cos θ} & \frac{\cos ϕ}{\cos θ} \end{matrix}] [\begin{matrix} p \\ q \\ r \end{matrix}]$

References

[1] Stevens, Brian, and Frank Lewis, Aircraft Control and Simulation. Hoboken, NJ: Second Edition, John Wiley & Sons, 2003.

[2] Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Reston, Va: Second Edition, AIAA Education Series, 2007.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2006a

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Implement Euler angle representation of six-degrees-of-freedom equations of
motion (2024)

FAQs

What is the 6DOF Euler angle? ›

The 6DOF (Euler Angles) block implements the Euler angle representation of six-degrees-of-freedom equations of motion, taking into consideration the rotation of a body-fixed coordinate frame (X_b, Y_b, Z_b) about a flat Earth reference frame (X_e, Y_e, Z_e). For more information about these reference points, see Algorithms.

Show Me More ›

What is the 6DOF algorithm? ›

The 6DOF algorithm is used to simulate the motion of the wedge. The geometry of the wedge is defined by a primitive triangular surface mesh resembling the STL format used as a standard in design and meshing software.

Read The Full Story ›

What is the formula for the Euler angle? ›

Given a rotation matrix R, we can compute the Euler angles, ψ, θ, and φ by equating each element in R with the corresponding element in the matrix product Rz(φ)Ry(θ)Rx(ψ). This results in nine equations that can be used to find the Euler angles. Starting with R31, we find R31 = − sin θ.

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What is the Euler angle sequence? ›

Euler angles can be defined by intrinsic rotations. The rotated frame XYZ may be imagined to be initially aligned with xyz, before undergoing the three elemental rotations represented by Euler angles. Its successive orientations may be denoted as follows: x-y-z or x₀-y₀-z₀ (initial)

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What is the name of the 6DOF? ›

Six degrees of freedom (6DOF), or sometimes six degrees of movement, refers to the six mechanical degrees of freedom of movement of a rigid body in three-dimensional space.

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What are the 12 Euler angles? ›

Consequently, of all these 216 combinations, there exist only twelve unique meaningful ordered sequences of rotations, or twelve Euler angle conventions: XYX, XYZ, XZX, XZY, YXY, YXZ, YZX, YZY, ZXY, ZXZ, ZYX, ZYZ.

Get More Info Here ›

What is A 6 degree of freedom? ›

The 6 degrees of freedom is a representation of how an object moves through 3D space by either translating linearly or rotating axially. A single degree of freedom on an object is controlled by the up/down, forward/back, left/right, pitch, roll, or yaw.

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What is 6 degrees of freedom vibration? ›

Six degree of freedom (DOF) shock and vibration testing provides an avenue for improved mechanical qualification of a system or component. Six DOF testing allows for application of a test input that is more representative of actual operational environments.

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What is A robot with 6 degrees of freedom? ›

6 DOF (Six Degrees of Freedom): This is considered full articulation for most industrial robots. They can achieve all three linear movements (forward/back, left/right, up/down) and all three rotational movements (twisting, tilting, and rotating the arm itself).

Learn More Now ›

How do you write the Euler formula? ›

Definition. Euler's Formula states that for any real x , eix=cosx+isinx, e i x = cos ⁡ x + i sin ⁡ where i is the imaginary unit, i=√−1 .

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What is the Euler's angle theorem? ›

Euler angle/axis parameter is derived via Euler's Rotation Theorem of rigid body, i.e., any rotation around a fixed point could be transformed to a rotation around a fixed axis with an angle, and the axis is called Euler axis e, and the angle is called Euler angle φ.

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What is the famous Euler formula? ›

Euler's formula], e^ix = cos x + i sin x. Moreover, it seems to be unknown who first stated the result explicitly....

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How to prove Euler's theorem? ›

The proof of Euler's Theorem relies on several different mathematical concepts, including modular arithmetic, the Chinese remainder theorem, and group theory. However, the key insight that allows us to prove Euler's Theorem is the observation that aφ(m) ≡ 1 (mod p) for any prime p that divides m.

How many Euler angles are there? ›

Correspondingly, there are 24 different sets of Euler angles. Why 24? It's easy to enumerate them: The z axis can point in any of the six principal axis direction of an ellipsoid, and the xy axis pair can have 4 possible orientations for each z orientation, giving 24 in total.

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Are Euler angles in degrees or radians? ›

If you're working with Euler rotations, they're in radians.

What are the 3 1 3 Euler angles? ›

The Euler angle defines the angle the frame is rotated around the specified axis. For instance, with an Euler Sequence of "3-1-3" as set above, the first Euler Angle is a rotation about the z-axis, the second rotation is about the x-axis, and the final rotation is about the new z-axis.

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What is the range of Euler angles? ›

7 Summary

Representation	Parameters
Euler angles	3 parameters (ϕ,θ,ψ), in range [0,2π)×[−π/2,π/2]×[0,2π).
Axis-angle	3 + 1 parameters (a,θ), in range S2×[0,π) with 1 d.o.f. removed via unit vector constraint \|Va\|=1.
Rot. vector	3 parameters m, in range \|m\|≤π.

20 more rows

What is the PSI Euler angle? ›

Euler angles are typically representes as phi (φ) for x-axis rotation, theta (θ) for y-axis rotation, and psi (ψ) for z-axis rotation. Any orientation can be described through a combination of these angles. Figure 1 represents the Euler angles for a multirotor aerial robot.

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Does order of Euler angles matter? ›

tl;dr The order in which you perform rotations matters. When you perform rotations in order x-y-z vs. z-y-x, you get different orientations. Proper Euler angles are in orders such as z-x-z or x-y-x and Cardan angles are in orders such as x-y-z, y-z-x.

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