Attitude Dynamics

Model attitude dynamics of one or more spacecraft

Since R2023a

• 

Libraries:
Aerospace Blockset / Spacecraft / Spacecraft Dynamics

Description

The Attitude Dynamics block models rotational dynamics of spacecraft using numerical integration. The block computes the attitude and angular velocity of one or more spacecraft over time using provided position and velocity states. For the most accurate results, use a variable-step solver with low tolerance settings (less than 1e-8). To trade off accuracy for speed, use larger tolerances, depending on your mission requirements.

For more information on the coordinate systems and translational and rotation dynamics that the Attitude Dynamics block uses, see Algorithms.

Ports

Input

X — Position of spacecraft

3-element vector | numSat-by-3 array

Position of the spacecraft with respect to the state vector coordinate frame, specified as a 3-element vector or numSat-by-3 array at the current time step. numSat is the number of spacecraft.

Dependencies

• To change the coordinate frame for this port, set the State vector output coordinate frame parameter.

• The size of the initial conditions provided in the Mass or Attitude tab control the port dimension.

Data Types: double

V — Velocity

3-element vector | numSat-by-3 array

Velocity of the spacecraft with respect to the state vector coordinate frame, specified as a 3-element vector or numSat-by-3 array at the current time step. numSat is the number of spacecraft.

Dependencies

• To change the coordinate frame for this port, set the State vector output coordinate frame parameter.

• The size of the initial conditions provided in the Mass or Attitude tab control the port dimension.

Data Types: double

M_b — External moments

3-element vector | numSat-by-3 array

Moment applied to the spacecraft in the body frame, specified as a 3-element vector or numSat-by-3 array at the current time step. numSat is the number of spacecraft.

Dependencies

To enable this port, select the Input body moments check box.

Data Types: double

dm/dt — Rate of change of mass

scalar | 1D array of size numSat

Rate of change of mass (positive if accreted, negative if ablated) at the current timestep, specified as a scalar or 1D array of size numSat. numSat is the number of spacecraft.

Dependencies

To enable this port, set Mass type to Simple Variable.

Data Types: double

m — Spacecraft mass

scalar | 1D array of size numSat

Spacecraft mass at the current timestep. numSat is the number of spacecraft.

Dependencies

To enable this port, set Mass type to Custom Variable.

Data Types: double

I — Spacecraft inertia tensor

3-by-3 array | 3-by-3-by-numSat array

Spacecraft inertia tensor, specified as a 3-by-3 array or 3-by-3-by-numSat array at the current timestep. numSat is the number of spacecraft.

Dependencies

To enable this port, set Mass type to Custom Variable.

Data Types: double

dI/dt — Rate of change of inertia tensor matrix

3-by-3 array | 3-by-3-by-numSat array

Rate of change of inertia tensor matrix, specified as a 3-by-3 array or 3-by-3-by-numSat array at the current time step. numSat is the number of spacecraft.

Dependencies

To enable this port, set Mass type to Custom Variable.

Data Types: double

V_re — Relative velocity

3-element vector | numSat-by-3 array

Relative velocity at which the mass is accreted to or ablated from the body in body-fixed axes, specified as a 3-element vector or numSat-by-3 array. numSat is the number of spacecraft.

Dependencies

To enable this port:

• Set Mass type to Custom Variable or Simple Variable.

• Select the Include mass flow relative velocity check box.

Data Types: double

φθψ — Moon libration angles

3-element vector

Moon libration angles for transformation between the International Celestial Reference Frame (ICRF) and Moon-centric fixed-frame using the Moon-centric Principal Axis (PA) system, specified as a 3-element vector. To get these values, use the Moon Libration block.

Note

The fixed-frame used by this block when Central body is set to Moon is the Mean Earth/pole axis (ME) system. For more information, see Algorithms.

Dependencies

To enable this port:

• Set Central body to Moon.

• Select the Input Moon libration angles check box.

Data Types: double

αδW — Right ascension, declination, and rotation angle

3-element vector

Central body spin axis instantaneous right ascension, declination, and rotation angle, specified as a 3-element vector. This port is available only for custom central bodies.

Dependencies

To enable this port:

• Set Central body to Custom.

• Set Central body spin axis source to Port.

Data Types: double

Output

q_body2icrf — Spacecraft attitude quaternion

4-element quaternion | numSat-by-4 array

Spacecraft attitude quaternion, returned as a scalar-first 4-element quaternion rotation from the body axis to the outport frame or scalar-first numSat-by-4 array at the current time step. numSat is the number of spacecraft.

Dependencies

The coordinate frame and attitude format of this port depends on these settings:

• To specify the attitude reference coordinate frame, set the Attitude reference coordinate frame parameter.

• Set Attitude representation to Quaternion.

Data Types: double

DCM — Spacecraft attitude direction cosine matrix

3-by-3 array | numSat-by-3-by-3 array

Spacecraft attitude direction cosine matrix (DCM), returned as a 3-by-3 array or numSat-by-3-by-3 array. numSat is the number of spacecraft.

Dependencies

The coordinate frame and attitude format of this port depends on these settings:

• To specify the attitude reference coordinate frame, set the Attitude reference coordinate frame parameter.

• Set Attitude representation to DCM.

Data Types: double

R1,R2,R3 — Spacecraft attitude Euler angles

3-element vector | numSat-by-3 array

Spacecraft attitude Euler angles, returned as a 3-element vector or numSat-by-3 array. numSat is the number of spacecraft.

Dependencies

The coordinate frame and attitude format of this port depend on these settings:

• To specify the attitude reference coordinate frame, set the Attitude reference coordinate frame parameter.

• Set Attitude representation to Euler angles.

Data Types: double

ω — Angular rate of spacecraft

3-element vector | numSat-by-3 array

Angular rate of the spacecraft relative to the attitude reference coordinate frame, returned as a 3-element vector or numSat-by-3 array, expressed as body axis angular rates PQR. numSat is the number of spacecraft.

Dependencies

The attitude reference coordinate frame depends on the Attitude reference coordinate frame parameter.

Data Types: double

q_icrf2ff — Coordinate system transformation

4-element array

Coordinate system transformation between the ICRF and fixed-frame coordinate system at the current timestep, returned as a 4-element array.

Dependencies

To enable this port, select the Output quaternion (ICRF to Fixed-frame) check box.

Data Types: double

t_utc — Time at current time step

scalar | 6-element array

Time at current time step, returned as a:

• scalar — If you specify the Start data/time parameter as a Julian date.

• 6-element array — If you specify the Start data/time parameter as a Gregorian date with six elements: year, month, day, hours, minutes, seconds.

This value equals the Start date/time parameter value plus the elapsed simulation time.

Dependencies

To enable this parameter, select the Output current date/time (UTC Julian date) check box.

Data Types: double

Fuel Status — Fuel status

scalar | numSat-element array

Fuel tank status at the current timestep, returned as a scalar or numSat-element array:

• 1 — Tank is full.

• 0 — Tank is not full or empty.

• -1 — Tank is empty.

numSat is the number of spacecraft.

Dependencies

To enable this port:,

• Set Mass type to Simple Variable.

• Select the Output fuel tank status check box.

Data Types: double

Parameters

Main

Input body moments — Option to enable external moments

on (default) | off

To enable external moments to be included in the integration of the spacecraft equations of motion in the body frame, select this check box. Otherwise, clear this check box.

Programmatic Use

Block Parameter: momentsIn

Type: character vector

Values: 'off' | 'on'

Default: 'off'

State vector coordinate frame — Position and velocity state port coordinate frame

ICRF (default) | Fixed-frame

Position and velocity state port coordinate frame setup, specified as ICRF or Fixed-frame.

Programmatic Use

Block Parameter: outportFrame

Type: character vector

Values: 'ICRF' | 'Fixed-frame'

Default: 'ICRF'

Start date/time (UTC Julian date) — Initial start time for simulation

juliandate (2020, 1, 1, 12, 0, 0) (default) | valid scalar Julian date | valid Gregorian date

Initial start date and time of simulation, specified as a Julian or Gregorian date. A valid Gregorian date must include the year, month, day, hours, minutes, and seconds as a 1D or 6-element array. The block defines initial conditions using this value.

Tip

To calculate the Julian date, use the juliandate function.

Tunable: Yes

Dependencies

The data format for this parameter is controlled by the Time format parameter.

Programmatic Use

Block Parameter: startDate

Type: character vector

Values: 'juliandate(2020, 1, 1, 12, 0, 0)' | valid scalar Julian date | valid Gregorian date

Default: 'juliandate(2020, 1, 1, 12, 0, 0)'

Output current date/time (UTC Julian date) — Option to add output port t_utc

on (default) | off

To output the current date or time, select this check box. Otherwise, clear this check box.

Dependencies

The data format for this parameter is controlled by the Time format parameter.

Programmatic Use

Block Parameter: dateOut

Type: character vector

Values: 'off' | 'on'

Default: 'off'

Action for out-of-range input — Out-of-range block behavior

Warning (default) | Error | None

Out-of-range block behavior action. Specify one of these options.

Action	Description
None	No action.
Warning	Warning in the Diagnostic Viewer, model simulation continues.
Error (default)	Error in the Diagnostic Viewer, model simulation stops.

Programmatic Use

Block Parameter: action

Type: character vector

Values: 'None' | 'Warning' | 'Error'

Default: 'Warning'

Mass

Mass type — Spacecraft mass type

Fixed (default) | Simple Variable | Custom Variable

Spacecraft mass type, specified as:

• Fixed — Mass and inertia are constant throughout the simulation.

• Simple Variable — Mass and inertia vary linearly as a function of mass rate.

• Custom Variable — Instantaneous mass, inertia, and inertia rate are inputs to the block.

Programmatic Use

Block Parameter: massType

Type: character vector

Values: 'Fixed' | 'Simple Variable' | 'Custom Variable'

Default: 'Fixed'

Mass — Initial mass of rigid body spacecraft

4.0 (default) | scalar | vector of size numSat

Initial mass of rigid body spacecraft, specified as scalar or vector of size numSat. numSat is the number of spacecraft.

Tunable: Yes

Dependencies

To enable this parameter, set the Mass type parameter to either Fixed or Simple variable.

Programmatic Use

Block Parameter: mass

Type: character vector

Values: scalar | vector of size numSat

Default: '4.0'

Empty mass — Spacecraft empty mass

3.5 (default) | scalar | vector of size numSat

Spacecraft empty (dry) mass, specified as a scalar or as a vector of size numSat. numSat is the number of spacecraft.

Tunable: Yes

Dependencies

To enable this parameter, set Mass type to Simple variable.

Programmatic Use

Block Parameter: emptyMass

Type: character vector

Values: 1D array of size numSat | 1D array of size numSat

Default: '3.5'

Full mass — Spacecraft full mass

4.0 (default) | scalar | vector of size numSat

Spacecraft full (wet) mass, specified as a scalar or as a vector of size numSat. numSat is the number of spacecraft.

Tunable: Yes

Dependencies

To enable this parameter, set Mass type to Simple variable.

Programmatic Use

Block Parameter: fullMass

Type: character vector

Values: scalar | vector of size numSat

Default: '4.0'

Inertia tensor — Inertia tensor matrix

[0.2273, 0, 0; 0 0.2273 0; 0 0 .0040] (default) | 3-by-3 array | 3-by-3-by-numSat array

Initial inertia tensor matrix of the spacecraft, specified, as a 3-by-3 array for a single spacecraft or a 3-by-3-by-numSat array for multiple spacecraft. numSat is the number of spacecraft.

Tunable: Yes

Dependencies

To enable this parameter, set Mass type to Fixed.

Programmatic Use

Block Parameter: inertia

Type: character vector

Values: '[0.2273, 0, 0; 0 0.2273 0; 0 0 .0040]' | 3-by-3 array | 3-by-3-by-numSat array

Default: '[0.2273, 0, 0; 0 0.2273 0; 0 0 .0040]'

Empty inertia tensor — Empty inertia tensor matrix

[0.1989, 0, 0; 0 0.1989 0; 0 0 .0035] (default) | 3-by-3 array | 3-by-3-by-numSat array

Empty (dry) inertia tensor matrix, specified as a 3-by-3 array for a single spacecraft or a 3-by-3-by-numSat array for multiple spacecraft. numSat is the number of spacecraft.

Tunable: Yes

Dependencies

To enable this parameter, set Mass type to Simple variable.

Programmatic Use

Block Parameter: emptyInertia

Type: character vector

Values: 3-by-3 array | 3-by-3-by-numSat array

Default: [0.1989, 0, 0; 0 0.1989 0; 0 0 .0035]

Full inertia tensor — Full inertia tensor matrix

[0.2273, 0, 0; 0, 0.2273, 0; 0, 0, .0040] (default) | 3-by-3 array | 3-by-3-by-numSat array

Full (wet) inertia tensor matrix, specified as a 3-by-3 array for a single spacecraft or a 3-by-3-by-numSat array for multiple spacecraft. numSat is the number of spacecraft.

Tunable: Yes

Dependencies

To enable this parameter, set Mass type to Simple variable.

Programmatic Use

Block Parameter: fullInertia

Type: character vector

Values: 3-by-3 array | 3-by-3-by-numSat array

Default: [0.2273, 0, 0; 0, 0.2273, 0; 0, 0, .0040]

Include mass flow relative velocity — Option to enable mass flow velocity

off (default) | on

To enable mass flow velocity to the block, select this check box. The mass flow velocity is the relative velocity in the body frame at which the mass is accreted or ablated. To disable mass flow velocity to the block, clear this check box.

Dependencies

To enable this parameter, set Mass type to Simple variable or Custom variable.

Programmatic Use

Block Parameter: useMassFlowRelativeVelocity

Type: character vector

Values: 'on' | 'off'

Default: 'off'

Limit mass flow when mass is empty or full — Option to limit mass flow

on (default) | off

To limit the mass flow when the spacecraft mass is full or empty, select this check box. Otherwise, clear this check box.

Dependencies

To enable this parameter, set Mass type to Simple variable.

Programmatic Use

Block Parameter: limitMassFlow

Type: character vector

Values: 'on' | 'off'

Default: 'on'

Output fuel tank status — Option to enable fuel tank status

on (default) | off

To enable fuel tank status, select this check box. Otherwise, clear this check box.

Dependencies

To enable this parameter, set Mass type to Simple variable.

Programmatic Use

Block Parameter: outputFuelStatus

Type: character vector

Values: 'on' | 'off'

Default: 'on'

Attitude

Attitude reference coordinate frame — Attitude and angular rate coordinate frame

ICRF (default) | Fixed-frame | NED | LVLH

Attitude and angular rate coordinate frame with respect to the attitude and angular rate initial conditions, specified as:

• ICRF

• Fixed-frame

• NED

• LVLH

Programmatic Use

Block Parameter: attitudeFrame

Type: character vector

Values: 'ICRF' | 'Fixed-frame' | 'NED' | 'LVLH'

Default: 'ICRF'

Attitude representation — Orientation format

Quaternion (default) | DCM | Euler angles

Orientation format for spacecraft attitude (initial condition and output port), specified as Quaternion, DCM, or Euler angles.

Programmatic Use

Block Parameter: attitudeFrame

Type: character vector

Values: 'Quaternion' | 'DCM' | 'Euler angles'

Default: 'Quaternion'

Initial body attitude — Spacecraft initial attitude

[1, 0, 0, 0] (default) | 4-element vector | numSat-by-4 array | 3-by-3 array | numSat-by-3-by-3 array | 3-element vector | numSat-by-3 array

Spacecraft initial attitude (orientation) of the spacecraft provided as either a quaternion, DCM, or Euler angle set with respect to Attitude representation.

Tunable: Yes

Dependencies

This parameter name and value format changes depending on the Attitude representation parameter.

Parameter Name	Attitude Representation Setting	Value Format
Initial quaternion	Quaternion	4-element vector numSat-by-4 array
Initial DCM	DCM	3-by-3 array numSat-by-3-by-3 array
Initial Euler angles	Euler angles	3-element vector numSat-by-3 array

Programmatic Use

Block Parameter: attitude

Type: character vector

Values: 4-element vector | numSat-by-4 array | 3-by-3 array | numSat-by-3-by-3 array | 3-element array | numSat-by-3 array

Default: '[1, 0, 0, 0]'

Angle rotation order — Angle rotation order

ZYX (default) | ZYX | ZYZ | ZXY | ZXZ | YXZ | YXY | YZX | YZY | XYZ | XYX | XZY | XZX

Rotation angle sequence for Euler angle attitude representation.

Tunable: Yes

Dependencies

To enable this parameter, set Attitude representation to Euler angles.

Programmatic Use

Block Parameter: rotationOrder

Type: character vector

Values: 'ZYX' | 'ZYZ' |'ZXY' | 'ZXZ' | 'YXZ' | 'YXY' | 'YZX' | 'YZY' | 'XYZ' | 'XYX' | 'XZY' | 'XZX'

Default: 'ZYX'

Initial body angular rates PQR — Initial body-fixed angular rates

[0, 0, 0] (default) | 3-element vector | numSat-by-3 array

Initial body-fixed angular rates (PQR) with respect to Attitude reference coordinate frame.

Tunable: Yes

Programmatic Use

Block Parameter: attitudeRate

Type: character vector

Values: | 3-element vector | numSat-by-3 array

Default: [0, 0, 0]

Include gravity gradient torque — Option to enable gravity gradient torque

on (default) | off

Select this check box to enable the use of the gravity gradient torque in the block rotational dynamics equations. Otherwise, clear this check box.

Tunable: Yes

Programmatic Use

Block Parameter: angAccelOut

Type: character vector

Values: 'on' | 'off'

Default: 'on'

Central Body

Central body — Celestial body around which spacecraft orbits

Earth (default) | Moon | Mercury | Venus | Mars | Jupiter | Saturn | Uranus | Neptune | Custom

Celestial body around which the spacecraft orbits, specified as Earth, Moon, Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune, or Custom.

Programmatic Use

Block Parameter: centralBody

Type: character vector

Values: 'Earth' | 'Moon' |'Mercury' | 'Venus' | 'Mars' | 'Jupiter' | 'Saturn' | 'Uranus' | 'Neptune' | 'Custom' |

Default: 'Earth'

Rotational rate — Rotational rate

4.06124975e-3 (default) | scalar

Rotational rate of a custom central body, specified as a scalar.

Dependencies

To enable this parameter, set Central body to Custom.

Programmatic Use

Block Parameter: 'customOmega'

Type: character vector

Values: '4.06124975e-3' | scalar

Default: '4.06124975e-3'

Use Earth orientation parameters (EOPs) — Option to use Earth orientation parameters

on (default) | off

Select this check box to use Earth orientation parameters for the transformation between the ICRF and fixed-frame coordinate systems. Otherwise, clear this check box.

Dependencies

To enable this parameter, set Central body to Earth.

Additionally, it must satisfy one of these criteria:

• State vector output coordinate frame is set to Fixed-frame.

• Attitude reference coordinate frame is set to Fixed-frame or NED.

Programmatic Use

Block Parameter: useEOPs

Type: character vector

Values: 'on' | 'off'

Default: 'on'

IERS EOP data file — Earth orientation data

aeroiersdata.mat (default) | MAT-file

Custom list of Earth orientation data, specified in a MAT-file.

Dependencies

To enable this parameter:

• Select the Use Earth orientation parameters (EOPs) check box.

• Set Central body to Earth.

Programmatic Use

Block Parameter: eopFile

Type: character vector

Values: 'aeroiersdata.mat' | MAT-file

Default: 'aeroiersdata.mat'

Input Moon libration angles — Moon libration Euler angle rate

off (default) | on

To specify Euler libration angles (φ θ ψ) for Moon orientation, select this check box. Otherwise, clear this check box.

Dependencies

To enable this parameter, set Central body to Moon.

Programmatic Use

Block Parameter: useMoonLib

Type: character vector

Values: 'off' | 'on'

Default: 'off'

Output quaternion (ICRF to Fixed-frame) — Option to add output transformation quaternion port

off (default) | on

To add an output transformation quaternion port for the quaternion transformation from the ICRF to the fixed-frame coordinate system, select this check box. Otherwise, clear this check box.

Programmatic Use

Block Parameter: outputTransform

Type: character vector

Values: 'off' | 'on'

Default: 'off'

Central body spin axis source — Central body spin source

Port (default) | Dialog

Central body spin axis source, specified as Port or Dialog. The block uses the spin axis to calculate the transformation from the ICRF to the fixed-frame coordinate system for the custom central body.

Dependencies

To enable this parameter, set Central body to Custom.

Programmatic Use

Block Parameter: cbPoleSrc

Type: character vector

Values: 'Port' | 'Dialog'

Default: 'Port'

Spin axis right ascension (RA) at J2000 — Right ascension of central body spin axis at J2000

317.68143 (default) | double scalar

Right ascension of central body spin axis at J2000 (2451545.0 JD, 2000 Jan 1 12:00:00 TT), specified as a double scalar.

Tunable: Yes

Dependencies

To enable this parameter:

• Set Central body to Custom.

• Set Central body spin axis source to Dialog.

Programmatic Use

Block Parameter: cbRA

Type: character vector

Values: '317.68143' | double scalar

Default: '317.68143'

Spin axis RA rate (deg/century) — Right ascension rate of central body spin axis

-0.1061 (default) | double scalar

Right ascension rate of the central body spin axis, specified as a double scalar, in specified angle units per century.

Tunable: Yes

Dependencies

To enable this parameter:

• Set Central body to Custom.

• Set Central body spin axis source to Dialog.

Programmatic Use

Block Parameter: cbRARate

Type: character vector

Values: '-0.1061' | double scalar

Default: '-0.1061'

Spin axis declination (Dec) at J2000 — Declination of central body spin axis at J2000

52.88650 (default) | double scalar

Declination of the central body spin axis at J2000 (2451545.0 JD, 2000 Jan 1 12:00:00 TT), specified as a double scalar.

Tunable: Yes

Dependencies

To enable this parameter:

• Set Central body to Custom.

• Set Central body spin axis source to Dialog.

Programmatic Use

Block Parameter: cbDec

Type: character vector

Values: '52.88650' | double scalar

Default: '52.88650'

Spin axis Dec rate (deg/century) — Declination rate of central body spin axis

-0.0609 (default) | double scalar

Declination rate of the central body spin axis, specified as a double scalar, in specified angle units per century.

Tunable: Yes

Dependencies

To enable this parameter:

• Set Central body to Custom.

• Set Central body spin axis source to Dialog.

Programmatic Use

Block Parameter: cbDecRate

Type: character vector

Values: '-0.0609' | double scalar

Default: '-0.0609'

Initial rotation angle at J2000 — Rotation angle of central body x-axis

176.630 (default) | double scalar

Rotation angle of the central body x-axis with respect to the ICRF x-axis at J2000 (2451545.0 JD, 2000 Jan 1 12:00:00 TT), specified as a double scalar, in specified angle units.

Tunable: Yes

Dependencies

To enable this parameter:

• Set Central body to Custom.

• Set Central body spin axis source to Dialog.

Programmatic Use

Block Parameter: cbRotAngle

Type: character vector

Values: '176.630' | double scalar

Default: '176.630'

Rotation rate (deg/day) — Rotation rate of central body x-axis

350.89198226 (default) | double scalar

Rotation rate of the central body x-axis with respect to the ICRF x-axis (2451545.0 JD, 2000 Jan 1 12:00:00 UTC), specified as a double scalar, in angle units per day.

Tunable: Yes

Dependencies

To enable this parameter:

• Set Central body to Custom.

• Set Central body spin axis source to Dialog.

Programmatic Use

Block Parameter: cbRotRate

Type: character vector

Values: '350.89198226' | double scalar

Default: '350.89198226'

Equatorial radius — Equatorial radius

3396200 (default) | double scalar

Equatorial radius for a custom central body, specified as a double scalar.

Tunable: Yes

Dependencies

To enable this parameter, set Central Body to Custom.

Programmatic Use

Block Parameter: customR

Type: character vector

Values: '3396200' | double scalar

Default: '3396200'

Flattening — Flattening of planet

0.00589 (default) | double scalar

Flattening of planet for custom central body, specified as a double scalar.

Tunable: Yes

Dependencies

To enable this parameter, set Central body to Custom.

Programmatic Use

Block Parameter: customF

Type: character vector

Values: '0.00589' | double scalar

Default: '0.00589'

Gravitational parameter — Gravitational parameter

4.305e13 (default) | double scalar

Gravitational parameter for a custom central body, specified as a double scalar.

Tunable: Yes

Dependencies

To enable this parameter, set Central body to Custom.

Programmatic Use

Block Parameter: customMu

Type: character vector

Values: '4.305e13' | double scalar

Default: '4.305e13'

Units

Units — Parameter and port units

Metric (m/s) (default) | Metric (km/s) | Metric (km/h) | English (ft/s) | English (kts)

Parameter and port units, specified as shown in the table.

Units	Moment	Mass	Inertia	Distance	Velocity	Acceleration
Metric (m/s)	Newton-meter	Kilograms	Kilogram m2	meters	meters/sec	meters/sec2
Metric (km/s)	Newton-meter	Kilograms	Kilogram m2	kilometers	kilometers/sec	kilometers/sec2
Metric (km/h)	Newton-meter	Kilograms	Kilogram m2	kilometers	kilometers/hour	kilometers/hour2
English (ft/s)	Foot-pound	Slugs	Slug ft2	feet	feet/sec	feet/sec2
English (kts)	Foot-pound	Slugs	Slug ft2	nautical mile	knots	knots/sec

Programmatic Use

Block Parameter: units

Type: character vector

Values: 'Metric (m/s)' | 'Metric (km/s)' | 'Metric (km/h)' | 'English (ft/s)' | 'English (kts)'

Default: 'Metric (m/s)'

Angle units — Angle units

Degrees (default) | Radians

Parameter and port units for angles, specified as Degrees or Radians.

Programmatic Use

Block Parameter: angleUnits

Type: character vector

Values: 'Degrees' | 'Radians'

Default: 'Degrees'

Time format — Time format for start date and time output

Julian date (default) | Gregorian

Time format for Start date/time (UTC Julian date) and output port t_utc, specified as Julian date or Gregorian.

Programmatic Use

Block Parameter: timeFormat

Type: character vector

Values: 'Julian date' | 'Gregorian'

Default: 'Julian date'

Algorithms

Earth-Centric Coordinate Systems

The Attitude Dynamics block works in the ICRF and fixed-frame coordinate systems.

• ICRF — International Celestial Reference Frame. This frame can be treated as equal to the ECI coordinate system realized at J2000 (Jan 1 2000 12:00:00 TT). For more information, see ECI Coordinates.

• Fixed-frame — Fixed-frame is a generic term for the coordinate system that is fixed to the central body. The axes of the system rotate with the central body and are not fixed in inertial space. If the Use Earth orientation parameters (EOPs) check box is not selected, the block still uses the IAU2000/2005 reduction, but with Earth orientation parameters set to 0.

  • When Central Body is Earth and the Use Earth orientation parameters (EOPs) check box is selected, the fixed-frame coordinate system for the Moon is the Mean Earth/pole axis frame (ME). This frame is realized by two transformations. First, the values in the ICRF frame are transformed into the Principal Axis system (PA), which is the axis defined by the libration angles provided as inputs to the block (for more information, see Moon Libration). The states are then transformed into the ME system using a fixed rotation from the "Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2006" [7].

  • When Central Body is Moon and the Input Moon libration angles check box is selected, the fixed-frame coordinate system for the Moon is the coordinate system defined by the libration angles provided as inputs to the block (for more information, see Moon Libration).

  • When Central Body is Custom, the fixed-frame coordinate system is defined by the poles of rotation and prime meridian defined by the block input α, δ, W, or the spin axis properties. In all other cases, the fixed-frame for each central body is defined by the directions of the poles of rotation and prime meridians defined in the "Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2006" [7].

Vehicle-Centric Coordinate Systems

The Attitude Dynamics block system works in the Body frame, north-east-down (NED), and local vertical, local horizontal (LVLH) coordinate systems.

• Body frame — Fixed in both origin and orientation to the moving craft. For more information, see Body Coordinates.

• North-east-down — Noninertial system with its origin fixed at the aircraft or spacecraft center of gravity. For more information, see ECI Coordinates.

• Local vertical, local horizontal — Also known as the spacecraft coordinate system, Gaussian coordinate system, or the orbit frame. LVLH is a rotating, accelerating frame commonly used in studies of relative motion, such as vehicle maneuvering. The axes of this frame are:

  • R-axis — Points outward from the spacecraft origin along its position vectors (with respect to the center of Earth). Measurements along this axis are referred to as radial.

  • S-axis — Completes the right hand coordinate system. This axis points in the direction of the velocity vector, but is only parallel to it for circular orbits. Measurements along this axis are referred to as along-track or transverse.

  • W-axis — Points normal to the orbital plane. Measurements along this axis are referred to as cross-track.

Rotational Dynamics

Rotational dynamics are governed by:

$${\dot{\overrightarrow{\omega }}}_{b_{icrf}}=\left[\overrightarrow{M_b}-{\overrightarrow{\omega }}_{b_{icrf}}\times \left(I_{mom}{\overrightarrow{\omega }}_{b_{icrf}}\right)-{\dot{I}}_{mom}{\overrightarrow{\omega }}_{b_{icrf}}\right]\text{inv}\left(I_{mom}\right)$$

$${\dot{\overrightarrow{\omega }}}_{b_{icrf}}\underset{\text{integrate}}{\Rightarrow }{\overrightarrow{q}}_{b_{icrf}},{\overrightarrow{\omega }}_{b_{icrf}},$$

where:

• $${\overrightarrow{M}}_b$$ are the body moment components.

• $$I_{mom}$$ is the spacecraft inertia tensor matrix.

When Mass type is Fixed, $${\dot{I}}_{mom}$$ equals 0.

When Mass type is Simple Variable, this equation estimates the rate of change of the inertia tensor:

$${\dot{I}}_{mom}=\frac{I_{full}-I_{empty}}{m_{full}-m_{empty}}\dot{m}$$

This equation gives the rate of change of the quaternion vector:

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

References

[1] Vallado, David. Fundamentals of Astrodynamics and Applications. 4th ed. Hawthorne, CA: Microcosm Press, 2013.

[2] Vepa, Ranjan. Dynamics and Control of Autonomous Space Vehicles and Robotics. New York: Cambridge University Press, 2019.

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

[4] Gottlieb, R. G. Fast Gravity, Gravity Partials, Normalized Gravity, Gravity Gradient Torque and Magnetic Field: Derivation, Code and Data. NASA Contractor Report 188243. Houston: NASA, February 1993.

[5] Seidelmann, P. Kenneth et al. "Report of the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements: 2006." Celestial Mechanics and Dynamical Astronomy 98 (2007): 155.

[6] Standish, E. M. "JPL Planetary and Lunar Ephemerides." DE405/LE405. Interoffice memorandum. JPL IOM 312.F-98-048. August 26, 1998.

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Version History

Introduced in R2023a