6DOF ECEF (Quaternion)

Implement quaternion representation of six-degrees-of-freedom equations of motion in Earth-centered Earth-fixed (ECEF) coordinates

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Libraries:
Aerospace Blockset / Equations of Motion / 6DOF

Description

The 6DOF ECEF (Quaternion) block Implement quaternion representation of six-degrees-of-freedom equations of motion in Earth-centered Earth-fixed (ECEF) coordinates. It considers the rotation of a Earth-centered Earth-fixed (ECEF) coordinate frame (X_ECEF, Y_ECEF, Z_ECEF) about an Earth-centered inertial (ECI) reference frame (X_ECI, Y_ECI, Z_ECI). The origin of the ECEF coordinate frame is the center of the Earth. For more information on the ECEF coordinate frame, see Algorithms.

Examples

Developing the Apollo Lunar Module Digital Autopilot

Develop Apollo Lunar Module digital autopilot using Simulink^® and Aerospace Blockset™.

• Open Live Script

Limitations

• This implementation assumes that the applied forces act at the center of gravity of the body, and that the mass and inertia are constant.

• This implementation generates a geodetic latitude that lies between ±90 degrees, and longitude that lies between ±180 degrees. Additionally, the MSL altitude is approximate.

• The Earth is assumed to be ellipsoidal. By setting flattening to 0.0, a spherical planet can be achieved. The Earth's precession, nutation, and polar motion are neglected. The celestial longitude of Greenwich is Greenwich Mean Sidereal Time (GMST) and provides a rough approximation to the sidereal time.

• The implementation of the ECEF coordinate system assumes that the origin is at the center of the planet, the x-axis intersects the Greenwich meridian and the equator, the z-axis is the mean spin axis of the planet, positive to the north, and the y-axis completes the right-handed system.

• The implementation of the ECI coordinate system assumes that the origin is at the center of the planet, the x-axis is the continuation of the line from the center of the Earth toward the vernal equinox, the z-axis points in the direction of the mean equatorial plane's north pole, positive to the north, and the y-axis completes the right-handed system.

Ports

Input

F_xyz — Force applied to center of mass in body frame

three-element vector

Force applied to the center of mass in the body frame, specified as a three-element vector, in the units specified in Units.

Data Types: double

M_xyz — Applied moments

three-element vector

Moment applied to the body with respect to center of mass in the body frame, specified as a three-element vector, in the units specified in Units.

Data Types: double

L_G(0) — Initial celestial longitude of Greenwich

scalar

Greenwich meridian initial celestial longitude angle, specified as a scalar.

Dependencies

To enable this port

• Set Celestial longitude of Greenwich to External.

• set Planet model to Earth.

Data Types: double

L_PM(0) — Prime meridian initial celestial longitude angle

scalar

Prime meridian initial celestial longitude angle, specified as a scalar.

Dependencies

To enable this port

• Set Celestial longitude of prime meridian to External.

• Set Planet model to Custom.

Data Types: double

Output

V_ff — Velocity of body with respect to fixed frame in fixed-frame axes

three-element vector

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

Data Types: double

X_ff — Position in fixed-frame

three-element vector

Position in fixed-frame, returned as a 3-element vector.

Data Types: double

μ l h — Position in geodetic latitude, longitude, and altitude

three-element vector | M-by-3 array

Position in geodetic latitude, longitude, and altitude, in degrees, returned as a three-element vector or M-by-3 array, in selected units of length, respectively.

Data Types: double

φ θ ψ (rad) — Euler rotation angles

three-element vector

Euler rotation angles for the x, y, z axes [roll, pitch, yaw], as a three-element vector, in radians. Yaw, pitch, and roll angles are applied using the z-y-x right-hand intrinsic passive transformation from frame A to frame B, such as angle2dcm(yaw,pitch,roll,"ZYX").

Data Types: double

DCM_bi — Direction cosine matrix representing the coordinate transformation from inertial to body-fixed axes

3-by-3 matrix

Direction cosine matrix representing the coordinate transformation from inertial to body-fixed axes, returned as a 3-by-3 matrix.

Data Types: double

DCM_be — Direction cosine matrix representing the coordinate transformation from NED axes to body-fixed axes

3-by-3 matrix

Direction cosine matrix representing the coordinate transformation from NED axes to body-fixed axes, returned as a 3-by-3 matrix.

Data Types: double

DCM_ef — Direction cosine matrix representing the coordinate transformation from fixed-frame axes to NED axes

3-by-3 matrix

Direction cosine matrix representing the coordinate transformation from fixed-frame axes to NED axes, returned as a 3-by-3 matrix.

Data Types: double

V_b — Velocity of body with respect to fixed-frame in body-fixed axes

three-element vector

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

Data Types: double

ω_rel — Relative angular rates of body with respect to NED frame

three-element vector

Relative angular rates of body with respect to NED frame, expressed in body frame and returned as a three-element vector, in radians per second.

Data Types: double

ω_b — Angular rates of body with respect to inertial frame

three-element vector

Angular rates of the body with respect to inertial frame, expressed in body frame and returned as a three-element vector, in radians per second.

Data Types: double

dω_b/dt — Angular accelerations of the body with respect to inertial frame

three-element vector

Angular accelerations of the body with respect to inertial frame, expressed in body frame and returned as a three-element vector, in radians per second squared.

Data Types: double

A_bb — Acceleration of body with respect to body-fixed frame in body-fixed axes

three-element vector

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

Data Types: double

A_bi — Acceleration of body with respect to inertial frame in body-fixed axes

three-element vector

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

Dependencies

To enable this point, select Include inertial acceleration.

Data Types: double

Parameters

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

Select the type of mass to use:

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 Fixed 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'

Initial position in geodetic latitude, longitude and altitude [mu,l,h] — Initial location of rigid body

[0 0 0] (default) | three-element vector

Initial location of the rigid body in the geodetic reference frame, specified as a three-element vector. Latitude and longitude values can be any value. However, latitude values of +90 and -90 may return unexpected values because of singularity at the poles.

Programmatic Use

Block Parameter: xg_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. Specify roll, pitch, and yaw angles in the z-y-x rotation sequence.

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_bi port, select this parameter.

Programmatic Use

Block Parameter: abi_flag

Type: character vector

Values: 'off' | 'on'

Default: off

Planet

Planet model — Planet model

Earth (WGS84) (default) | Custom

Planet model to use, Custom or Earth (WGS84).

Programmatic Use

Block Parameter: ptype

Type: character vector

Values: 'Earth (WGS84)' | 'Custom'

Default: 'Earth (WGS84)'

Equatorial radius of planet — Radius of planet at equator

6378137 (default) | scalar

Radius of the planet at its equator, specified as a double scalar, in the same units as the desired units for the ECEF position.

Dependencies

To enable this parameter, set Planet model to Custom.

Programmatic Use

Block Parameter: R

Type: character vector

Values: double scalar

Default: '6378137'

Flattening — Flattening of planet

1/298.257223563 (default) | scalar

Flattening of the planet, specified as a double scalar.

Dependencies

To enable this parameter, set Planet model to Custom.

Programmatic Use

Block Parameter: F

Type: character vector

Values: double scalar

Default: '1/298.257223563'

Rotational rate — Rotational rate

7292115e-11 (default) | scalar

Rotational rate of the planet, specified as a scalar, in rad/s.

Dependencies

To enable this parameter, set Planet model to Custom.

Programmatic Use

Block Parameter: w_E

Type: character vector

Values: double scalar

Default: '7292115e-11'

Celestial longitude of Greenwich source — Source of Greenwich meridian initial celestial longitude

Internal (default) | External

Source of Greenwich meridian initial celestial longitude, specified as:

Internal	Use celestial longitude value from Celestial longitude of Greenwich.
External	Use external input for celestial longitude value.

Dependencies

• To enable this parameter, set Planet model to Earth.

• Setting this parameter to External enables the L_G(0) port.

• If Planet model is set to Custom, the parameter name changes to Celestial longitude of prime meridian source.

Programmatic Use

Block Parameter: angle_in

Type: character vector

Values: 'Internal' | 'External'

Default: 'Internal'

Celestial longitude of Greenwich [deg] — Initial angle

0 (default) | scalar

Initial angle between Greenwich meridian and the x-axis of the inertial frame, specified as a double scalar.

Dependencies

• To enable this parameter, set:

  • Celestial longitude of Greenwich source to Internal.

  • Planet model to Earth (WGS84).

• If Planet model is set to Custom, the parameter name changes to Celestial longitude of prime meridian [deg].

Programmatic Use

Block Parameter: LPM0

Type: character vector

Values: double scalar

Default: '0'

Celestial longitude of prime meridian source — Source of prime meridian initial celestial longitude

Internal (default) | External

Source of prime meridian initial celestial longitude, specified as:

Internal	Use celestial longitude value from Celestial longitude of prime meridian.
External	Use external input for celestial longitude value.

Dependencies

• To enable this parameter, set Planet model to Custom.

• Setting this parameter to External enables the L_PM(0) port.

• If Planet model is set to Earth (WGS84), the parameter name changes to Celestial longitude of Greenwich source.

Programmatic Use

Block Parameter: angle_in

Type: character vector

Values: 'Internal' | 'External'

Default: 'Internal'

Celestial longitude of prime meridian [deg] — Initial angle

0 (default) | scalar

Initial angle between prime meridian and the x-axis of the ECI frame, specified as a double scalar.

Dependencies

• To enable this parameter, set:

  • Celestial longitude of prime meridian source to Internal.

  • Planet model to Custom.

• If Planet model is set to Earth (WGS84), the parameter name changes to Celestial longitude of Greenwich [deg].

Programmatic Use

Block Parameter: LPM0

Type: character vector

Values: double scalar

Default: '0'

State Attributes

Assign a 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-separated list surrounded by braces, for example, {'a', 'b', 'c'}. Each name must be unique.

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

• 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.

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

'' (default) | comma-separated list surrounded by braces

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

Programmatic Use

Block Parameter: quat_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: ''

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: ''

Fixed frame position: e.g., {'Xff', 'Yff', 'Zff'} — ECEF position state name

'' (default) | comma-separated list surrounded by braces

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

Programmatic Use

Block Parameter: posfixedframe_statename

Type: character vector

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

Default: ''

Inertial position: e.g., {'Xinertial', 'Yinertial', 'Zinertial'} — Inertial position state names

'' (default) | comma-separated list surrounded by braces

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

Default value is ''.

Programmatic Use

Block Parameter: posinertial_statename

Type: character vector

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

Default: ''

Celestial longitude of Greenwich: e.g., 'LG' — Celestial longitude state name

'' (default) | character vector

Celestial longitude of Greenwich state name, specified as a character vector.

Dependencies

• To enable this parameter, set:Planet model to Earth (WGS84).

• If Planet model is set to Custom, the parameter name changes to Celestial longitude of prime meridian: e.g., 'LPM'.

Programmatic Use

Block Parameter: LPM_statename

Type: character vector

Values: '' | scalar

Default: ''

Celestial longitude of prime meridian: e.g., 'LPM' — Celestial longitude prime meridian

'' (default) | character vector

Celestial longitude of prime meridian state name, specified as a character vector.

Dependencies

• To enable this parameter, set:Planet model to Custom.

• If Planet model is set to Earth (WGS84), the parameter name changes to Celestial longitude of Greenwich: e.g., 'LG'.

Programmatic Use

Block Parameter: LPM_statename

Type: character vector

Values: '' | scalar

Default: ''

Algorithms

The 6DOF ECEF (Quaternion) block uses these equations to describe the translational motion in the fixed (ECEF) coordinate frame.

• The applied forces F_xyz = [F_x F _y F_z]^T are given in body-fixed axes.

  $$F_{xyz}=\left[\begin{array}{c}{F}_x\\ F{}_y\\ F_z\end{array}\right]=m{A}_{bi}$$

  The mass of the body m is assumed constant.

• The direction cosine matrices in this block use a subscript convention of two indices indicating target and source axes.

  • The be subscript in DCM_be indicates a rotation from NED axes e to body axes b.

  • The bf subscript in DCM_bf indicates a rotation from fixed-frame axes f to body axes b.

• The inertial acceleration of the body A_bi is the input force divided by the mass, calculated with:

  $$A_{bi}=\frac{F_{xyz}}{m}$$

• The acceleration of the body in the body-fixed frame of reference A_bb accounts for

  • Coriolis and centrifugal accelerations

  • Derivative of the body-fixed coordinate system relative to the fixed-frame coordinate system

  A_bb is defined as:

  $$A_{bb}=\left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\end{array}\right]=\frac{F_{xyz}}{m}-\left({\omega }_{bi}+DC{M}_{bf}{\omega }_{ei}\right)\times V_{ff}-DC{M}_{bf}\left({\omega }_{ei}\times \left({\omega }_{ei}\times X_{ff}\right)\right)$$

  where

  Equation	Description
  $$V_{ff}=\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$$	Body velocity relative to the fixed-frame, expressed in body axes.
  $${\omega }_{be}$$	Body rotation rate relative to NED frame, in body axes.
  $${\omega }_{bi}={\omega }_{be}+DC{M}_{bf}{\omega }_{ei}+DC{M}_{be}{\omega }_{NED}$$	Body rotation rate relative to intertial frame.
  $${\omega }_{ei}$$	Earth rotation rate.
  $${\text{X}}_{ff}$$	Fixed-frame position of body relative to origin.

• The rotation rate of the body relative to the fixed-frame contains:

  • Earth rotation rate

  • Body frame rotation rate

  • Relative angular rates with respect to the north-east-down (NED) frame, defined as:

    $${\omega }_{NED}=\left[\begin{array}{c}\dot{l}\cos \mu \\ -\dot{\mu }\\ -\dot{l}\sin \mu \end{array}\right]=\left[\begin{array}{c}\raisebox{1ex}{$V_E$}\!\left/ \!\raisebox{-1ex}{$\left(N+h\right)$}\right.\\ \raisebox{1ex}{$-V_N$}\!\left/ \!\raisebox{-1ex}{$\left(M+h\right)$}\right.\\ \raisebox{1ex}{$-V_E\tan \mu $}\!\left/ \!\raisebox{-1ex}{$\left(N+h\right)$}\right.\end{array}\right]$$

• The rotational dynamics of the body defined in body-fixed frame are given, where the applied moments are [L M N]^T, and the inertia tensor I is with respect to the origin O.

  $$\begin{array}{l}{A}_{bb}=\left[\begin{array}{c}{\dot{u}}_b\\ {\dot{v}}_b\\ {\dot{\omega }}_b\end{array}\right]=\frac{1}{m}{\overline{F}}_b-\left[{\overline{\omega }}_b\times {\overline{V}}_b+DC{M}_{bf}{\overline{\omega }}_e\times {\overline{V}}_b+DC{M}_{bf}\left({\overline{\omega }}_e\times \left({\overline{\omega }}_e\times {\overline{X}}_f\right)\right)\right]\\ A_b{\text{}}_{ecef}=\frac{F_b}{m}\\ {\overline{M}}_b=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I{\dot{\overline{\omega }}}_b+{\overline{\omega }}_b\times (I{\overline{\omega }}_b)\\ I=\left[\begin{array}{ccc}{I}_{xx}& -I_{xy}& -I_{xz}\\ -I_{yx}& I_{yy}& -I_{yz}\\ -I_{zx}& -I_{zy}& I_{zz}\end{array}\right]\end{array}$$

  The integration of the rate of change of the quaternion vector is given:

  $$\left[\begin{array}{c}{\dot{q}}_0\\ {\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]=-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left[\begin{array}{cccc}0& {\omega }_b\left(1\right)& {\omega }_b\left(2\right)& {\omega }_b\left(3\right)\\ -{\omega }_b\left(1\right)& 0& -{\omega }_b\left(3\right)& {\omega }_b\left(2\right)\\ -{\omega }_b\left(2\right)& {\omega }_b\left(3\right)& 0& -{\omega }_b\left(1\right)\\ -{\omega }_b\left(3\right)& -{\omega }_b\left(2\right)& {\omega }_b\left(1\right)& 0\end{array}\right]\left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]$$

  Aerospace Blockset uses quaternions that are defined using the scalar-first convention.

References

[1] Stevens, Brian, and Frank Lewis. Aircraft Control and Simulation, 2nd ed. Hoboken, NJ: John Wiley & Sons, 2003.

[2] McFarland, Richard E. "A Standard Kinematic Model for Flight simulation at NASA-Ames." NASA CR-2497.

[3] "Supplement to Department of Defense World Geodetic System 1984 Technical Report: Part I - Methods, Techniques and Data Used in WGS84 Development." DMA TR8350.2-A.

Extended Capabilities

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

Version History

Introduced in R2006a

R2023b: 6DOF ECEF (Quaternion) Block Coordinate Frame Changes

These updates better clarify the coordinate frame of the 6DOF ECEF (Quaternion) block when Central body is Custom:

• New block icon.

  

• Port name subscripts of ecef have changed to ff.

• The Celestial longitude of Greenwich source programmatic name has changed from LG0 to LPM. Existing scripts continue to work.

• Updated block parameter names, but models from previous releases continue to work.

  Old Parameter Name or Setting	New Parameter Name
  ECEF position...	Fixed frame position...
  When Planet model is Earth	Celestial longitude of Greenwich source and Celestial longitude of Greenwich [deg]
  When Planet model is Custom	Celestial longitude of prime meridian source and Celestial longitude of prime meridian [deg]