Frames

The org.orekit.frames package provides classes to handle frames and
transforms between them.

Frames Presentation

Frames tree

The Frame class represents a single frame. All frames are organized as a
tree with a single root.

frames tree

Each Frame is defined by a single TransformProvider linking it to one specific frame:
its parent frame. This defining transform provider may provide either fixed or
time-dependent transforms. As an example the Earth related frame ITRF depends on time
due to precession/nutation, Earth rotation and pole motion. The predefined root frame
is the only one with no parent frames.

For each pair of frames, there is one single shortest path from one frame to
the other one.

Transform

The Transform class represents a full transform between two frames. It manages
combined rotation, translation and their first time derivatives to handle kinematics.
Transforms can convert position, directions and velocities from one frame to another
one, including velocity composition effects.

Transforms are used to both:

  • define the relationship from a parent to a child frames.
    This transform (tdef on the following scheme) is stored in the child frame.

  • merge all individual transforms encountered while walking the tree from one
    frame to any other one, however far away they are from each other (trel on the
    following scheme).

transforms between nodes of a frames tree

The transform between any two frames is computed by merging individual transforms
while walking the shortest past between them. The walking/merging operations are
handled transparently by the library. Users only need to select the frames, provide
the date and ask for the transform, without knowing how the frames are related to
each other.

Transformations are defined as operators which, when applied to the coordinates of a
vector expressed in the old frame, provide the coordinates of the same vector expressed
in the new frame. When we say a transform t is from frame A to frame B, we mean
that if the coordinates of some absolute vector (say, the direction of a distant star)
are uA in frame A and uB in frame B, then uB=t.transformVector(uA).
Transforms provide specific methods for vectorial conversions, affine conversions, either
with or without first derivatives (i.e. angular and linear velocities composition).

Transformations can be interpolated using Hermite interpolation, i.e. taking derivatives
into account if desired.

Predefined Frames

The FramesFactory class provides several predefined reference frames.

The user can retrieve them using various static methods: getGCRF(), getEME2000(),
getICRF(), getCIRF(IERSConventions, boolean), getTIRF(IERSConventions, boolean),
getITRF(IERSConventions, boolean), getMOD(IERSConventions), getTOD(IERSConventions),
getGTOD(IERSConventions), getITRFEquinox(IERSConventions, boolean), getTEME(),
and getVeis1950().
One of these reference frames has been arbitrarily chosen as the root of the frames tree:
the Geocentric Celestial Reference Frame (GCRF) which is an inertial reference defined by IERS.

For most purposes, the recommended frames are ITRF for terrestrial frame and
GCRF for celestial frame. EME2000, TOD, Veis1950 could also be used for compatibility
with legacy systems. TEME should be used only for TLE.

There are also a number of planetary frames associated with the predefined celestial bodies.

IERS 2010 Conventions

One predefined set corresponds to the frames from the
IERS conventions (2010).
This set defines the GCRF reference frame on the celestial (i.e. inertial) side,
the ITRF (International Terrestrial Reference Frame) on the terrestrial side and several
intermediate frames between them. Several versions of ITRF have been defined. Orekit
supports several of them thanks to Helmert transformations.

New paradigm: CIO-based transformations (Non-Rotating Origin)

There are several different ways to compute transforms between GCRF and ITRF. Orekit supports
the new paradigm promoted by IERS and defined by IERS conventions 2010, i.e., it uses a
single transform for bias, precession and nutation, computed by precession and nutation models
depending on the IERS conventions choice and the Earth Orientation Parameters (EOP) data published online
by IERS. This single transform links the GCRF to a Celestial Intermediate Reference Frame (CIRF).
The X axis of this frame is the Celestial Intermediate Origin (CIO) and its Z axis is the
Celestial Intermediate Pole (CIP). The CIO is not linked to equinox any more. From CIRF,
the Earth Rotation Angle (including tidal effects) is applied to define a Terrestrial
Intermediate Reference Frame (TIRF) which is a pseudo Earth fixed frame. A last transform adds
the pole motion (both observed and published in IERS frames and modeled effects including tidal
effects) with respect to the Earth crust to reach the real Earth fixed frame: the International
Terrestrial Reference Frame. There are several realizations of the ITRS, each one being a
different ITRF. These realizations are linked together using Helmert transformations which are
very small, slightly time-dependent transformations.

The precession-nutation models for Non-Rotating Origin paradigm available in Orekit are those
defined in either IERS 1996 conventions, IERS 2003 conventions or IERS 2010 conventions.

In summary, four frames are involved along this path, with various precession-nutation
models: GCRF, CIRF, TIRF and ITRF.

Classical paradigm: equinox-based transformations

The classical paradigm used prior to IERS conventions 2003 is equinox-based and uses
more intermediate frames. It is still used in many ground systems and can still be used
with new precession-nutation models.

Starting from GCRF, the first transform is a bias to convert to EME2000, which was the
former reference. The EME2000 frame (which is also known as J2000) is defined using
the mean equinox at epoch J2000.0, i.e. 2000-01-01T12:00:00 in Terrestrial Time (not
UTC!). From this frame, applying precession evolution between J2000.0 and current date
defines a Mean Of Date frame for current date and applying nutation defines
a True Of Date frame, similar in spirit to the CIRF in the new paradigm.
From this, the Greenwich Apparent Sidereal Time is applied to reach a Greenwich True
Of Date frame, similar to the TIRF in the new paradigm. A final transform involving pole
motion leads to the ITRF.

In summary, six frames are involved along this path: GCRF, EME2000, MOD, TOD, GTOD
and equinox-based ITRF.

In addition to these frames, the ecliptic frame which is defined from the MOD by rotating
back to ecliptic plane is also available in Orekit.

The so-called Veis 1950 belongs also to this path, it is defined from the GTOD by the
application of a modified sidereal time.

This whole paradigm is deprecated by IERS. It involves additional complexity, first because
of the larger number of frames and second because these frames are computed by mixed models
with IAU-76 precession, correction terms to match IAU-2000, and a need to convert Earth
Orientation data from the published form to a form suitable for this model.

Despite this deprecation, these frames are very important ones and lots of legacy systems
rely on them. They are therefore supported in Orekit for interoperability purposes (but not
recommended for new systems).

As the classical paradigm uses the same definition for celestial pole (Z axis) but not
the same definition for frame origin (X axis) as the Non-Rotating Origin paradigm, the
TOD frame and the CIRF frame share the same Z axis but differ from each other by a
non-null rotation around Z (the equation of the origin, which should not be confused
with the equation of the equinox), and the TIRF and GTOD should be the same frame, at
model accuracy level (and of course ITRF should also be the same in both paradigms).

Orekit implementation of IERS conventions

In summary, Orekit implements the following frames:

  • those related to the Non-Rotating Origin: GCRF, CIRF, TIRF, ITRF for all precession
    and nutation models from IERS 1996, IERS 2003 and IERS 2010,

  • those related to the equinox-based origin: MOD, TOD, GTOD, equinox-based ITRF
    for all precession and nutation models from IERS 1996, IERS 2003 and IERS 2010
    and Veis 1950.

The frames can be computed with or without Earth Orientation Parameters corrections,
and when these corrections are applied, they can either use a simple interpolation
or an accurate interpolation taking sub-daily tidal effects. It is possible to mix
all frames. It is for example one can easily estimate the difference between an ITRF
computed from equinox based paradigm and IERS 1996 precession-nutation, without EOP
and an ITRF computed from Non-Rotating Origin and IERS 2010 precession-nutation, with
EOP and tidal correction to the EOP interpolation. This is particularly interesting
when exchanging data between ground systems that use different conventions.

CIO-based transformations

Here is a schematic representation of the partial tree containing the supported IERS frames
based on CIO.

IERS frames tree

Since Orekit uses the new paradigm for IERS frames, the IAU-2006 precession and IAU-2000A
nutation model implemented are the complete model with thousands of luni-solar and planetary
terms (1600 terms for the x components, 1275 components for the y component and 66 components
for the s correction). Recomputing all these terms each time the CIRF frame is used would be really
slow. Orekit therefore implements a caching/interpolation feature to improve efficiency. The shortest
period for all the terms is about 5.5 days (it is related to one fifth of the moon revolution
period). The pole motion is therefore quite smooth at the day or week scale. This implies that
this motion can be computed accurately using a few reference points per day or week and
interpolated between these points. The trade-off selected for Orekit implementation is to use
eight points separated by four hours each. The resulting maximal interpolation error
on the frame is about 4e-15 radians. The reference points are cached so the computation cost is
roughly to perform two complete evaluations of the luni-solar and planetary terms per simulation
day, and one interpolation per simulation step, regardless of the step size. This represents huge
savings for steps shorter than one half day, which is the rule in most application (step sizes are
mostly of the range of a few tens of seconds). Note that starting with Orekit 6.0, this caching feature
is thread-safe.

Tidal effects are also taken into account on Earth Rotation angle and on pole motion. The 71-terms
model from IERS is used. Since this model is also computing intensive, a caching/interpolation
algorithm is also used to avoid a massive effect on performance. The trade-off selected for
Orekit implementation is to use 8 points separated by 3/32 day (135 minutes) each. The resulting
maximal interpolation error is about 3 micro-arcseconds. The penalty to use tidal effects is therefore
limited to slightly more than 20%, to be compared with the 550% penalty without this mechanism.

Equinox-based transformations

Here is a schematic representation of the partial tree containing the supported IERS frames
based on equinox

equinox-based frames tree

The path from EME2000 to Veis1950, involving the MOD, TOD and GTOD without EOP correction, is
devoted to some legacy systems, whereas the MOD, TOD and GTOD with EOP correction are for
compatibility with the IERS 2003 convention. The gap between the two branches can reach a few
meters, a rather crude accuracy for many space systems.

The same kind of optimization used for the IAU-2006 precession and IAU-2000A nutation model are
also applied for the older IAU-1980 precession-nutation model, despite it is much simpler.

Solar system frames

All celestial bodies are linked to their own body-centered inertial frame, just
as the Earth is linked to EME2000 and GCRF. Since Orekit provides implementations
of the main solar system celestial bodies, it also provides body-centered frames
for these bodies, one inertially oriented and one body oriented. The orientations
of these frames are compliant with IAU poles and prime meridians definitions. The
predefined frames are the Sun, the Moon, the eight planets and the Pluto dwarf planet.
In addition to these real bodies, two points are supported for convenience as if they
were real bodies: the solar system barycenter and the Earth-Moon barycenter ; in these
cases, the associated frames are aligned with EME2000. One important case is the solar
system barycenter, as its associated frame is the ICRF.

solar system frames

Topocentric Frame

This frame model allows defining the frame associated with any position at
the surface of a body shape, which itself is referenced to a frame, typically
ITRF for Earth. The frame is defined with the following
canonical axes:

  • zenith direction (Z) is defined as the normal to local horizontal plane;

  • north direction (Y) is defined in the horizontal plane
    (normal to zenith direction) and following the local meridian;

  • east direction (X) is defined in the horizontal plane
    in order to complete direct triangle (east, north, zenith).

In such a frame, the user can retrieve azimuth angle, elevation angle,
range and range rate of any point given in any frame, at given date.

Local Orbital Frame

Local orbital frames are bound to an orbiting spacecraft. They move with
the spacecraft so they are time-dependent. Two local orbital frames are provided:
the (t, n, w) frame and the (q, s, w) frame.

The (t, n, w) frame has its X axis along velocity (tangential), its Z axis along orbital
momentum and its Y axis completes the right-handed trihedra (it is roughly pointing towards
the central body). The (q, s, w) frame has its X axis along position (radial), its Z axis
along orbital momentum and its Y axis completes the right-handed trihedra (it is roughly
along velocity).

User-defined frames

The frames tree can be extended by users who can add as many frames as they
want for specific needs. This is done by adding frames one at a time, attaching
each frame to an already built one by specifying the TransformProvider from parent to child.

Transforms may be constant or varying. For simple fixed transforms, using directly the
FixedTransformProvider class is sufficient. For varying transforms (time-dependent or
telemetry-based for example), it may be useful to define specific providers that will implement
getTransform(AbsoluteDate).

A basic example of such an extension is to add a satellite frame representing the satellite
motion and attitude. Such a frame would have an inertial frame as its parent frame (GCRF or
EME2000) and the getTransform(AbsoluteDate) method would compute a transform using the
translation and rotation from orbit and attitude data.

Frames transforms are computed by combining all transforms between parent frame and child frame
along the path from the origin frame to the destination. This implies that when one
TransformProvider locally changes a transform, it basically moves not only the child frame but
all the sub-trees starting at this frame with respect to the rest of the tree. This property can
be used to update easily complex trees without bothering about combining transforms oneself. The
following example explains a practical case.

sub-tree showing how a translation between two frames is related to the defining transform of a third frame, closer to the root

This case is an improvement of the basic previous extension to manage orbit and attitude. In this
case, we introduce several intermediate frames with elementary transforms and need to update the
whole tree. We also want to take into account the offset between the GPS receiver antenna and the
satellite center of mass. When a new GPS measurement is available, we want to update the complete
left subtree. This is done by using the dedicated UpdatableFrame which will do all the conversions.

Package organization

frames class diagram

frametree.png - frames tree (11.9 KB) Luc Maisonobe, 2014-07-01 17:32

transform.png - transforms between nodes of a frames tree (13.7 KB) Luc Maisonobe, 2014-07-01 17:32

iers-tree.png - IERS frames tree (55 KB) Luc Maisonobe, 2014-07-01 17:33

pre-iers-tree.png - equinox-based frames tree (56.5 KB) Luc Maisonobe, 2014-07-01 17:33

solar-system-frames.png - solar system frames (20 KB) Luc Maisonobe, 2014-07-01 17:34

antenna-frames.png - sub-tree showing how a translation between two frames is related to the defining transform of a third frame, closer to the root (24.5 KB) Luc Maisonobe, 2014-07-01 17:34

frames-class-diagram.png - frames class diagram (180 KB) Luc Maisonobe, 2014-07-01 17:34