Orbit Propagator Kepler (unperturbed)

Propagate orbit of one or more spacecraft using Kepler universal variable formulation

Since R2020b

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Libraries:
Aerospace Blockset / Spacecraft / Spacecraft Dynamics

Alternative Configurations of Orbit Propagator Kepler (unperturbed) Block:
Orbit Propagator Numerical (high precision)

Description

The Orbit Propagator Kepler (unperturbed) block propagates the orbit of one or more spacecraft using the Kepler universal variable formulation. The Orbit Propagator Kepler (unperturbed) and Orbit Propagator Numerical (high precision) blocks are alternative configurations of the same block.

• Orbit Propagator Kepler (unperturbed) — Kepler universal variable formulation (quicker)

• Orbit Propagator Kepler Numerical (high precision) — More accurate

The Orbit Propagator Kepler (unperturbed) block does not take into account atmospheric drag, third body gravity, or solar radiation pressure. For orbit propagations that take these effects into account, see the Orbit Propagator Numerical (high precision) block.

The size of the provided initial conditions determines the number of spacecraft being modeled. If you supply more than one value for a parameter in the Orbit tab, the block outputs a constellation of satellites. Any parameter with a single provided value is expanded and applied to all the satellites in the constellation. For example, if you provide a single value for all the parameters on the block except True anomaly, which contains six values, the block creates a constellation of six satellites, varying true anomaly only.

The block applies the same expansion behavior to input port A_icrf (applied acceleration). This port accepts either a single value expanded to all spacecraft being modeled, or individual values to apply to each spacecraft.

For more information on the propagation methods the Orbit Propagator blocks use, see Orbit Propagation Methods.

You can define initial orbital states in the Orbit tab as:

• A set of orbital elements

• Position and velocity state vectors in International Celestial Reference Frame (ICRF) or fixed-frame coordinate systems.

The block uses quaternions, which are defined using the scalar-first convention.

For more information on the coordinate systems the Orbit Propagator blocks use, see Coordinate Systems.

Examples

Constellation Modeling with the Orbit Propagator Block

Propagate the orbits of a constellation of satellites and compute and visualize access intervals between the individual satellites and several ground stations.

• Open Live Script

Mission Analysis with the Orbit Propagator Block

Compute and visualize line-of-sight access intervals between satellites and a ground station.

• Open Live Script

Lunar Mission Analysis with the Orbit Propagator Block

Compute and visualize access intervals between the Apollo Command and Service module and a rover on the lunar surface. The module's orbit is modeled using Reference Trajectory #2 from the NASA report Variations of the Lunar Orbital Parameters of the Apollo CSM-Module [2]. This is a lunar orbit studied by NASA for the Apollo program. The example uses:

• Open Live Script

High Precision Orbit Propagation of the International Space Station

Propagate the order of the International Space Station (ISS) using high precision numerical orbit propagation.

• Open Live Script

Ports

Output

X_icrf — Position of spacecraft

3-element vector | numSat-by-3

Position of the spacecraft with respect to (ICRF or fixed-frame), returned as a 3-element vector or numSat-by-3 array, where m is number of spacecraft, at the current time step. The size of the initial conditions provided in the Orbit tab control the port dimension. numSat is the number of spacecraft.

Data Types: double

V_icrf — Velocity

3-element vector | numSat-by-3 array

Velocity of the spacecraft with respect to ICRF or fixed-frame, returned as a 3-element vector or numSat-by-3 array, at the current time step. numSat-by-3 array. The size of the initial conditions provided in the Orbit tab control the port dimension.

Data Types: double

t_utc — Time at current time step

scalar | 6-element vector

Time at current time step, returned as a:

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

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

This value is equal to the Start date/time parameter value + the elapsed simulation time.

Dependencies

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

Data Types: double

Parameters

Main

Propagation method — Orbit propagation method

Kepler (unperturbed) | Numerical (high precision)

Orbit propagation method, specified as:

• Kepler (unperturbed) — Uses a universal variable formulation of the Kepler problem to determine the spacecraft position and velocity at each time step. This method is faster than Numerical (high precision).

• Numerical (high precision) — Determine the spacecraft position and velocity at each time step using numerical integration. This option models central body gravity based on the settings in the Central body tab. This method is more accurate than Kepler (unperturbed), but slower.

Programmatic Use

Block Parameter: propagator

Type: character vector

Values: 'Kepler (unperturbed)' | 'Numerical (high precision)'

Default: 'Kepler (unperturbed)'

Input external accelerations — Input additional accelerations

off (default) | on

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

Dependencies

To enable this check box, set Propagation method to Numerical (high precision).

Programmatic Use

Block Parameter: accelIn

Type: character vector

Values: 'off' | 'on'

Default: 'off'

External acceleration coordinate frame — Frame for additional accelerations

ICRF (default) | Fixed-frame

Input additional accelerations, specified as ICRF or Fixed-frame. These accelerations are included in integration of the spacecraft equations of motion.

Dependencies

To enable this parameter:

• Set Propagation method to Numerical (high precision)

• Select the Input external accelerations check box

Programmatic Use

Block Parameter: accelFrame

Type: character vector

Values: 'ICRF' | 'Fixed-frame'

Default: 'ICRF'

State vector output coordinate frame — Port coordinate frame

ICRF (default) | Fixed-frame

Coordinate frame for output ports, specified as ICRF or Fixed-frame. These port labels are affected:

• Output port X

• Output port V

Dependencies

To enable this parameter, set Propagation method to Numerical (high precision).

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 including year, month, day, hours, minutes, seconds as 6-element vector

Initial start date and time of simulation, specified as a Julian or Gregorian date. The block defines initial conditions using this value.

Tip

To calculate the Julian date, use the juliandate function.

Programmatic Use

Block Parameter: startDate

Type: character vector

Values: 'juliandate(2020, 1, 1, 12, 0, 0)' | valid scalar Julian date | valid Gregorian date including year, month, day, hours, minutes, seconds as 6-element vector

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

Output current date/time (UTC Julian date) — Add output port t_utc

on (default) | off

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

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, specified as follows:

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'

Orbit

Define the initial states of the space craft.

Initial state format — Input method for initial states of orbit

Orbital elements (default) | ICRF state vector

Input method for initial states of orbit, specified as Orbital elements or ICRF state vector.

Programmatic Use

Block Parameter stateFormatKep

Type: character vector

Values: 'Orbital elements' | 'ICRF state vector'

Default: 'Orbital elements'

Orbit Type — Orbit classification

Keplerian (default) | Elliptical equatorial | Circular | Circular equatorial

Orbit classification, specified as:

• Keplerian — Model elliptical orbits using six standard Keplerian orbital elements.

• Elliptical equatorial — Define an equatorial orbit, where inclination is 0 or 180 degrees and the right ascension of the ascending node is undefined.

• Circular — Define a circular orbit, where eccentricity is 0 and the argument of periapsis is undefined.

• Circular equatorial — Define a circular orbit, where eccentricity is 0 or 10 degrees. Argument of periapsis and the right ascension of the ascending node are undefined.

Dependencies

To enable this parameter, set Initial state format to Orbital elements.

Programmatic Use

Block Parameter: orbitType

Type: character vector

Values: 'Keplerian' | 'Elliptical equatorial' | 'Circular inclined' | 'Circular equatorial'

Default: 'Keplerian'

Semi-major axis — Half of major axis of ellipse

6786000 (default) | 1D array of size numSat

Half of ellipsis major axis, specified as a 1D array whose size is the number of spacecraft.

• For parabolic orbits, this block interprets this parameter as the periapsis radius (distance from periapsis to the focus point of orbit).

• For hyperbolic orbits, this block interprets this parameter as the distance from periapsis to the hyperbola center.

Dependencies

To enable this parameter, set Initial state format to Orbital elements.

Programmatic Use

Block Parameter: semiMajorAxis

Type: character vector

Values: scalar | 1D array of size m, number of spacecraft

Default: '6786000'

Eccentricity — Deviation of orbit

0.01 (default) | scalar | value between 0 and 1, or greater than 1 for Keplerian orbit type | 1D array of size numSat

Deviation of the orbit from a perfect circle, specified as a scalar or 1D array of size that is number of spacecraft.

If Orbit type is set to Keplerian, value can be:

• 0 for circular orbit

• Between 0 and 1 for elliptical orbit

• 1 for parabolic orbit

• Greater than 1 for hyperbolic orbit

Dependencies

To enable this parameter, set:

• Initial state format to Orbital elements.

• Orbit type to Keplerian or Elliptical equatorial.

Programmatic Use

Block Parameter: eccentricity

Type: character vector

Values: 0.01 | scalar | value between 0 and 1, or greater than 1 for Keplerian orbit type | 1D array of size numSat

Default: '0.01'

Inclination (deg) — Tilt angle of orbital plane

50 (default) | scalar | 1D array of size numSat | degrees between 0 and 180 | radians between 0 and pi

Vertical tilt of the ellipse with respect to the reference plane measured at the ascending node, specified as a scalar or 1D array of size numSat, in specified units. numSat is the number of spacecraft.

Dependencies

To enable this parameter, set:

• Initial state format to Orbital elements

• Orbit type to Keplerian or Circular inclined

Programmatic Use

Block Parameter: inclination

Type: character vector

Values: 50 | scalar | 1D array of size numSat | degrees between 0 and 180 | radians between 0 and pi

Default: '50'

RAAN (deg) — Angular distance in equatorial plane

95 (default) | scalar value between 0 and 360 | 1D array of size numSat

Right ascension of ascending node (RAAN), specified as a scalar value between 0 and 360 or 1D array of size numSat, in specified units. numSat is the number of spacecraft. RAAN is the angular distance along the reference plane from the ICRF x-axis to the location of the ascending node (the point at which the spacecraft crosses the reference plane from south to north).

Dependencies

To enable this parameter, set:

• Initial state format to Orbital elements.

• Orbit type to Keplerian or Circular inclined.

Programmatic Use

Block Parameter: raan

Type: character vector

Values: 95 | scalar value between 0 and 360 | 1D array of size m number of spacecraft

Default: '95'

Argument of periapsis (deg) — Angle from spacecraft ascending node to periapsis

93 (default) | degrees between 0 and 360 | radians between 0 and 2*pi | 1D array of size m, number of spacecraft

Angle from the spacecraft ascending node to periapsis (closest point of orbit to the central body), specified as a 1D array of size m that is number of spacecraft, in specified units.

Dependencies

To enable this parameter, set:

• Initial state format to Orbital elements

• Orbit type to Keplerian

Programmatic Use

Block Parameter: argPeriapsis

Type: character vector

Values: '95' | scalar value between 0 and 360 | 1D array of size numSat

Default: '93'

True anomaly — Angle between periapsis and initial position of spacecraft

203 (default) | scalar | degrees between 0 and 360 | radians between 0 and 2*pi | 1D array of size numSat

Angle between periapsis (closest point of orbit to the central body) and the initial position of spacecraft along its orbit at Start date/time, specified as a scalar or 1D array of size numSat, in specified units. numSat is the number of spacecraft.

Dependencies

To enable this parameter, set:

• Initial state format to Orbital elements.

• Orbit type to Keplerian or Elliptical inclined.

Programmatic Use

Block Parameter: trueAnomaly

Type: character vector

Values: '203' | scalar | degrees between 0 and 360 | radians between 0 and 2*pi | 1D array of size numSat

Default: '203'

Argument of latitude (deg) — Angle between ascending node and initial position of spacecraft

200 (default) | scalar | degrees between 0 and 360 | radians between 0 and 2*pi | 1D array of size numSat

Angle between the ascending node and the initial position of spacecraft along its orbit at Start date/time, specified as a scalar or 3-element vector or 1D array of size number of spacecraft, in specified units.

Dependencies

To enable this parameter, set:

• Initial state format to Orbital elements.

• Orbit Type to Circular inclined.

Programmatic Use

Block Parameter: argLat

Type: character vector

Values: '200' | scalar | degrees between 0 and 360 | radians between 0 and 2*pi | 1D array of size numSat

Default: '200'

Longitude of periapsis (deg) — Angle between ICRF x-axis and eccentricity vector

100 (default) | scalar | degrees between 0 and 360 | radians between 0 and 2*pi | 1D array of size numSat

Angle between the ICRF x-axis and the eccentricity vector, specified as a scalar or 3-element vector or 1D array of size number of spacecraft, in specified units.

Dependencies

To enable this parameter, set:

• Initial state format to Orbital elements.

• Orbit type to Elliptical equatorial.

Programmatic Use

Block Parameter: lonPeriapsis

Type: character vector

Values: 100 | scalar | degrees between 0 and 360 | radians between 0 and 2*pi | 1D array of size m, number of spacecraft

Default: '100'

True longitude (deg) — Angle between ICRF x-axis and initial position of spacecraft

150 (default) | scalar | degrees between 0 and 360 | radians between 0 and 2*pi | 1D array of size numSat

Angle between the ICRF x-axis and the initial position of spacecraft along its orbit at Start date/time, specified as a scalar or 1D array of size numSat, in specified units. numSat is the number of spacecraft.

Dependencies

To enable this parameter, set:

• Initial state format to Orbital elements.

• Orbit type to Circular equatorial.

Programmatic Use

Block Parameter: trueLon

Type: character vector

Values: '150' | scalar | degrees between 0 and 360 | radians between 0 and 2*pi | 1D array of size numSat

Default: '150'

ICRF position — Cartesian position vector of spacecraft

[3649700.0 3308200.0 -4676600.0] (default) | 3-element vector | | numSat-by-3 array

Cartesian position vector of spacecraft in ICRF coordinate system at Start date/time, specified as a 3-element vector for single spacecraft or numSat-by-3 array for multiple spacecraft. numSat is the number of spacecraft.

Dependencies

To enable this parameter, set Initial state format to ICRF state vector.

Programmatic Use

Block Parameter: inertialPosition

Type: character vector

Values: [3649700.0 3308200.0 -4676600.0] | 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft

Default: '[3649700.0 3308200.0 -4676600.0]'

ICRF velocity — Cartesian velocity vector of spacecraft

[-2750.8 6666.4 2573.4] (default) | 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft

Cartesian velocity vector of spacecraft in ICRF coordinate system at Start date/time, specified as a 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft.

Dependencies

To enable this parameter, set Initial state format to ICRF state vector.

Programmatic Use

Block Parameter: inertialVelocity

Type: character vector

Values: [-2750.8 6666.4 2573.4] | 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft

Default: '[-2750.8 6666.4 2573.4]'

Fixed-frame position — Position vector of spacecraft

[-4142689.0 -2676864.7 -4669861.6] (default) | 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft

Cartesian position vector of spacecraft in fixed-frame coordinate system at Start date/time, specified as a 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft.

Dependencies

To enable this parameter, set:

• Propagation method to Numerical (high precision).

• set Initial state format to Fixed-frame state vector.

Programmatic Use

Block Parameter: fixedPosition

Type: character vector

Values: '[-4142689.0 -2676864.7 -4669861.6]' | 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft

Default: '[-2750.8 6666.4 2573.4]'

Fixed-frame velocity — Velocity vector of spacecraft

[1452.7 -6720.7 2568.1] (default) | 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft

Cartesian velocity vector of spacecraft in fixed-frame coordinate system at Start date/time, specified as a 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft.

Dependencies

To enable this parameter, set:

• Propagation method to Numerical (high precision).

• Initial state format to Fixed-frame state vector.

Programmatic Use

Block Parameter: fixedVelocity

Type: character vector

Values: '[1452.7 -6720.7 2568.1]' | 3-element vector for single spacecraft or 2-D array of size m-by-3 array of multiple spacecraft

Default: '[1452.7 -6720.7 2568.1]'

Central Body

Configure the central body environment around which the spacecraft orbits.

Central body — Celestial body around which spacecraft orbits

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

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

Programmatic Use

Block Parameter: centralBody

Type: character vector

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

Default: 'Earth'

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:

Units	Distance	Velocity	Acceleration	Mass	Area	Density
Metric (m/s)	meters	meters/sec	meters/sec2	Kilograms	m2	kg/m3, some density outputs 1/m3
Metric (km/s)	kilometers	kilometers/sec	kilometers/sec2	Kilograms	m2	kg/m3, some density outputs 1/m3
Metric (km/h)	kilometers	kilometers/hour	kilometers/hour2	Kilograms	m2	kg/m3, some density outputs 1/m3
English (ft/s)	feet	feet/sec	feet/sec2	Slugs	feet2	lbm/ft3, some density outputs 1/ft3
English (kts)	nautical mile	knots	knots/sec	Slugs	feet2	lbm/ft3, some density outputs 1/ft3

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'

Alternative Configurations

Orbit Propagator Numerical (high precision) — Propagate orbit of one or more spacecraft using high precision variable formulation

The Orbit Propagator Numerical (high precision) block propagates the orbit of one or more spacecraft by a numerical (high precision) propagation method. To enable this block, set Propagation method to Numerical (high precision).

Libraries:
Aerospace Blockset / Spacecraft / Spacecraft Dynamics

Algorithms

Coordinate Systems

The Orbit Propagator Kepler (unperturbed) block works in the ICRF and fixed-frame coordinate systems:

• ICRF — International Celestial Reference Frame (ICRF). 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).

  To describe a point in space, you need a frame of reference that does not rotate with respect to the stars. The ICRF, with the origin at the center of the Earth and orthogonal vectors I, J, and K, is used as the frame of reference. The fundamental plane is the IJ-plane, which is closely aligned with the equator with a small offset that changes over time because of precession and nutation of the rotation axis of the Earth.

  

• Fixed-frame — Fixed-frame is a generic term for the coordinate system that is fixed to the central body (its axes rotate with the central body and are not fixed in inertial space). For high precision orbit propagation methods

  • When the central body is earth, and the Earth orientation parameters (EOPs) are used, the Fixed-frame for Earth is the International Terrestrial Reference Frame (ITRF). This reference frame is realized by the IAU2000/2006 reduction from the ICRF coordinate system using the earth orientation parameter file provided. If Earth orientation parameters are not used, the block still uses the IAU2000/2006 reduction, but with Earth orientation parameters set to 0.

  • When the central body is moon, and the moon libration angles are provided as input, 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), 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" [5]. If the Moon libration angles input is not provided, the fixed frame 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" [5].

  • When the 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" [5].

Orbit Propagation Methods

The Aerospace Blockset™ supports two top-level orbit propagation methods: Kepler (unperturbed) and Numerical (high precision).

For more information, see Orbit Propagation Methods.

References

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

[2] Gottlieb, R. G., "Fast Gravity, Gravity Partials, Normalized Gravity, Gravity Gradient Torque and Magnetic Field: Derivation, Code and Data," Technical Report NASA Contractor Report 188243, NASA Lyndon B. Johnson Space Center, Houston, Texas, February 1993.

[3] Konopliv, A. S., S. W. Asmar, E. Carranza, W. L. Sjogen, D. N. Yuan., "Recent Gravity Models as a Result of the Lunar Prospector Mission, Icarus", Vol. 150, no. 1, pp 1–18, 2001.

[4] Lemoine, F. G., D. E. Smith, D.D. Rowlands, M.T. Zuber, G. A. Neumann, and D. S. Chinn, "An improved solution of the gravity field of Mars (GMM-2B) from Mars Global Surveyor", Journal Of Geophysical Research, Vol. 106, No. E10, pp 23359-23376, October 25, 2001.

[5] Seidelmann, P.K., Archinal, B.A., A’hearn, M.F. et al. "Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2006." Celestial Mech Dyn Astr 98, 155–180 (2007).

[6] Montenbruck, Oliver, and Gill Eberhard. Satellite Orbits: Models, Methods, and Applications. Springer, 2000.

Extended Capabilities

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

Version History

Introduced in R2020b

R2023b: Orbit Propagator Block Changes

The Orbit Propagator block has been updated to take into account:

• The effects of third body gravity on orbit propagation.

• Space weather data from a data file generated by the aeroReadSpaceWeatherData function.

• Solar radiation pressure (SRP).