Coordinate Frame Transforms

Introduction

The satkit.frametransform module provides functions for transforming between various coordinate frames used in satellite tracking and orbit determination. These include multiple variations of “inertial” coordinate frames, and multiple verisons of “Earth-fixed” coordinate frames.

Some notes:

Most of the algorithms in this module are from the book `”Fundamentals of Astrodynamics and Applications”`` by David Vallado.
The frame transforms are defined as arbitrary rotations in a 3-dimensional space. The rotations are a function of time, and are represented as Quaternions
The rotation from the Geocentric Celestial Reference Frame (GCRF) to the Earth-Centered Inertial (ECI) frame is defined by the International Astronomical Union (IAU), available here. See IERS Technical Note 36 for the latest values.

API Reference

Transformations between coordinate frames, and associated utility functions

Coordinate frame transforms are mostly pulled from Vallado: https://www.google.com/books/edition/Fundamentals_of_satkitdynamics_and_Applic/PJLlWzMBKjkC?hl=en&gbpv=0

or the IERS: https://www.iers.org/

satkit.frametransform.earth_orientation_params(time: earth_orientation_params.time)

Get Earth Orientation Parameters at given instant

Parameters:

time (satkit.time) – Instant at which to query parameters

Returns:

Tuple with following elements:: 0 : (UT1 - UTC) in seconds 1 : X polar motion in arcsecs 2 : Y polar motion in arcsecs 3 : LOD: instantaneous rate of change in (UT1-UTC), msec/day 4 : dX wrt IAU-2000A nutation, milli-arcsecs 5 : dY wrt IAU-2000A nutation, milli-arcsecs

Return type:

(float, float, float, float, float, float) | None

satkit.frametransform.earth_rotation_angle(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Earth Rotation Angle

Notes

See: IERS Technical Note 36, Chapter 5, Equation 5.15
Calculation Details:
- Let t be UT1 Julian date
- let f be fractional component of t (fraction of day)
- ERA = 2𝜋 ((0.7790572732640 + f + 0.00273781191135448 * (t - 2451545.0))

Parameters:: tm (satkit.time|datetime.datetime|list|numpy.array) – Time[s] at which to calculate Earth Rotation Angle
Returns:: Earth Rotation Angle at input time[s] in radians
Return type:: float|numpy.array

satkit.frametransform.gast(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Greenwich Apparent Sideral Time

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – scalar, list, or numpy array of astro.time or datetime.datetime representing time at which to calculate output
Returns:: Greenwich apparant sidereal time, radians, at input time(s)
Return type:: float | npt.ArrayLike[np.float]

satkit.frametransform.gmst(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Greenwich Mean Sidereal Time

Notes

GMST is the angle between the vernal equinox and the Greenwich meridian
Vallado algorithm 15
GMST = 67310.5481 + (876600h + 8640184.812866) * tᵤₜ₁ * (0.983104 + tᵤₜ₁ * −6.2e−6)

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – scalar, list, or numpy array of astro.time or datetime.datetime representing time at which to calculate output
Returns:: Greenwich Mean Sideral Time, radians, at intput time(s)
Return type:: float | npt.ArrayLike[np.float]

satkit.frametransform.qcirs2gcrf(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Rotation from Celestial Intermediate Reference System to Geocentric Celestial Reference Frame

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – Time[s] at which to calculate the quaternion
Returns:: Quaternion representing rotation from CIRS to GCRF at input time(s)
Return type:: quaternion | npt.ArrayLike[quaternion]

satkit.frametransform.qgcrf2itrf(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Quaternion representing rotation from the Geocentric Celestial Reference Frame (GCRF) to the International Terrestrial Reference Frame (ITRF)

Notes

Uses full IAU2010 Reduction
See IERS Technical Note 36, Chapter 5
Does not include solid tides, ocean tides
Very computationally expensive

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – Time[s] at which to calculate the quaternion
Returns:: Quaternion representing rotation from GCRF to ITRF at input time(s)
Return type:: quaternion | npt.ArrayLike[quaternion]

satkit.frametransform.qgcrf2itrf_approx(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Quaternion representing approximate rotation from the Geocentric Celestial Reference Frame (GCRF) to the International Terrestrial Reference Frame (ITRF)

Notes

Accurate to approx. 1 arcsec

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – Time[s] at which to calculate the quaternion
Returns:: Quaternion representing rotation from GCRF to ITRF at input time(s)
Return type:: quaternion | npt.ArrayLike[quaternion]

satkit.frametransform.qitrf2gcrf(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Quaternion representing rotation from the International Terrestrial Reference Frame (ITRF) to the Geocentric Celestial Reference Frame (GCRF)

Notes

Uses full IAU2010 Reduction
See IERS Technical Note 36, Chapter 5
Does not include solid tides, ocean tides
Very computationally expensive

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – Time[s] at which to calculate the quaternion
Returns:: Quaternion representing rotation from ITRF to GCRF at input time(s)
Return type:: quaternion | npt.ArrayLike[quaternion]

satkit.frametransform.qitrf2gcrf_approx(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Quaternion representing approximate rotation from the International Terrestrial Reference Frame (ITRF) to the Geocentric Celestial Reference Frame (GCRF)

Notes

Accurate to approx. 1 arcsec

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – Time[s] at which to calculate the quaternion
Returns:: Quaternion representing rotation from ITRF to GCRF at input time(s)
Return type:: quaternion | npt.ArrayLike[quaternion]

satkit.frametransform.qitrf2tirs(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Rotation from Terrestrial Intermediate Reference System to Celestial Intermediate Reference Systems

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – Time[s] at which to calculate the quaternion
Returns:: Quaternion representing rotation from ITRF to TIRS at input time(s)
Return type:: quaternion | npt.ArrayLike[quaternion]

satkit.frametransform.qteme2itrf(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Quaternion representing rotation from the True Equator Mean Equinox (TEME) frame to the International Terrestrial Reference Frame (ITRF)

Notes

This is equation 3-90 in Vallado
TEME is the output frame of the SGP4 propagator used to compute position from two-line element sets.

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – Time[s] at which to calculate the quaternion
Returns:: Quaternion representing rotation from TEME to ITRF at input time(s)
Return type:: quaternion | npt.ArrayLike[quaternion]

satkit.frametransform.qtirs2cirs(tm: time | numpy.typing.ArrayLike[time] | datetime.datetime | numpy.typing.ArrayLike[datetime.datetime])

Rotation from Terrestrial Intermediate Reference System (TIRS) to the Celestial Intermediate Reference System (CIRS)

Parameters:: tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime]) – Time[s] at which to calculate the quaternion
Returns:: Quaternion representing rotation from TIRS to CIRS at input time(s)
Return type:: quaternion | npt.ArrayLike[quaternion]