Coordinate Systems for Modeling

Modeling aircraft and spacecraft are simplest if you use a coordinate system fixed in the body itself. In the case of aircraft, the forward direction is modified by the presence of wind, and the craft's motion through the air is not the same as its motion relative to the ground.

ECI and ECEF Coordinates

ECI Coordinates

The Earth-centered inertial (ECI) system is considered non-rotating and is generally treated as inertial for most applications, despite the equinox and equatorial plane experiencing very slight movements over time. For high-precision orbit calculations, the ECI system is recognized as truly inertial, especially when the equator and equinox are defined for a specific epoch, such as J2000. Aerospace functions and blocks that use a specific realization of the ECI coordinate system include this information in their documentation. The origin of the ECI system is established at the Earth's center, making this system particularly useful for describing the orbits of satellites and other celestial bodies in space. The reference plane for the ECI system is the mean equatorial plane.

The x-axis is aligned with the celestial sphere, pointing towards the vernal equinox (the first point of Aries ♈), which is the imaginary point in space found at the intersection of the Earth's equatorial plane and the plane of the Earth's orbit around the sun (the ecliptic plane).
The y-axis extends 90 degrees east from the x-axis within the equatorial plane.
The z-axis extends upwards from the North Pole, completing the right-handed coordinate system.
The International Celestial Reference Frame (ICRF) can be treated as equal to the ECI coordinate system realized at J2000 (Jan 1 2000 12:00:00 TT).
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.
The origin of the ICRF is located at the barycenter of the solar system. The ICRF axes are oriented in a space-fixed manner, meaning they do not kinematically rotate relative to distant objects in the universe.

ECI Applications

The ECI system does not rotate with the Earth, making it simpler to model the motion of satellites and other celestial bodies in space.
The ECI system is useful for analyzing and predicting the orbits of objects in an inertial space framework.

The inertial and Earth-fixed z-axes are not perfectly aligned due to the effects of polar motion, precession, and nutation.

Comparison of Earth centered and Earth Inertial coordinates

ICRF and Fixed Frame

The origin of the ICRF and Fixed Frame coordinate systems are at the center of the central body.

ICRF — International Celestial Reference Frame (ICRF). This frame can be treated as equal to the ECI coordinate system with its origin at the center of the central body 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". 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".
- 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.

ECEF Coordinates

The Earth-centered, Earth-fixed (ECEF) system is a non-inertial frame that rotates alongside the Earth, with its origin stationed at the Earth's center. The reference plane for this system is the mean equatorial plane. In the ECEF system:

The x-axis extends outward from the point where the Earth's equatorial plane intersects the prime meridian (0° longitude).
The z-axis projects upward from the North Pole.
The y-axis extends eastward, perpendicular to the x-z plane, adhering to the right-hand (RH) rule for coordinate systems.

ECEF Applications

The ECEF system adheres to the right-hand (RH) rule for coordinate systems, making it ideal for mapping, navigation, and positioning tasks on the planet.
ECEF coordinates change with time for a given point on the Earth's surface due to Earth's rotation, which is useful for real-time applications like GPS.

Precession, Nutation and Rotation

Precession

Precession refers to the slow movement of the axis of the planet's rotation. More specifically, precession is the gradual shift in the direction of Earth's axis of rotation, which causes the position of the celestial poles to change over time. This movement is similar to the wobble seen in a spinning top as its speed decreases. The precession of the Earth's axis takes about 26,000 years to complete a full cycle. This phenomenon affects the timing of the seasons and the visibility of stars from Earth over long periods.

Nutation

Nutation is a smaller, more complex motion superimposed on the precession of the Earth's rotation axis. Nutation is a slight "nodding" or oscillation in the Earth's axis of rotation. Nutation causes the Earth's tilt to vary slightly over a period of 18.6 years. This variation is due to the gravitational influences of the Moon and, to a lesser extent, the Sun on the Earth's equatorial bulge. Nutation affects the precise position of the celestial poles and equinoxes, but its effects are much smaller compared to precession.

Rotation

Rotation refers to the spinning of the Earth around its own axis. This rotation defines day and night. The Earth rotates from west to east, which is why the Sun appears to rise in the east and set in the west. The Earth completes one full rotation approximately every 24 hours. The speed of the Earth's rotation is not perfectly constant. The rotational speed varies slightly due to complex interactions with the Moon and the Sun, as well as geophysical processes within the Earth itself. However, these variations are so small that, for most practical purposes, you can consider the Earth's rotation period as constant.

Comparison Between ECI and ECEF Coordinates

The ECI and ECEF coordinate systems are two fundamental frameworks used to represent the positions and velocities of objects in space and on the Earth's surface. Each system is uniquely characterized and applied in various fields such as satellite navigation, aerospace engineering, and geodesy. The ECI system is inertial, meaning it does not rotate with the Earth, whereas the ECEF system rotates in alignment with the Earth's surface.

Both the ECI and ECEF systems have their origins at the Earth's center of mass. A key difference between them is their angular orientation in the xy plane. This difference is quantified by the Earth rotation angle (ERA) or Greenwich sidereal angle, which measures the angle between the x-axis of the ECI and ECEF systems.

Satellite positions are often defined relative to the ECI frame, which aligns with the ICRF for satellite applications. The ICRF is essential because the Earth's rotation axis changes over time due to precession, nutation, and rotation. Therefore, an inertial frame with fixed axes in inertial space, typically defined at the epoch of J2000 (January 1, 2000, 12:00), is used. The z-axis of the ICRF frame aligns with the Earth's rotation axis as of the J2000 epoch. In contrast, the z-axis of the ECEF frame represents the rotation axis of Earth at the current time.

The transformation from ECI to ECEF is achieved through the IAU2000/2006 reduction, which accounts for axial precession and nutation, providing an adequate adjustment for most applications. For enhanced fidelity, time-varying terms such as Earth orientation parameters (EOPs) may be included. Using functions like dcmeci2ecef and eci2ecef can significantly improve the accuracy of these transformations.

Comparison of coordinate systems

ICRF to GCRF — GCRF is the Earth-centered version of the ICRF, used for applications that require a geocentric perspective.
GCRF to ECI — ECI frame is aligned with the GCRF and is used for practical satellite navigation and space operations.
Fixed Frame to ECEF — ECEF is a specific implementation of a Fixed Frame, providing a rotating reference frame that remains aligned with the Earth's surface. ECEF is the Earth-centered version of Fixed-frame.

Body Coordinates

The noninertial body coordinate system is fixed in both origin and orientation to the moving craft. The craft is assumed to be rigid.

The orientation of the body coordinate axes is fixed in the shape of body.

The x-axis points through the nose of the craft.
The y-axis points to the right of the x-axis (facing in the pilot's direction of view), perpendicular to the x-axis.
The z-axis points down through the bottom of the craft, perpendicular to the x-y plane and satisfying the RH rule.

Translational Degrees of Freedom

Translations are defined by moving along these axes by distances x, y, and z from the origin.

Rotational Degrees of Freedom

Rotations are defined by the Euler angles P, Q, R or Φ, Θ, Ψ. They are

P or Φ: Roll about the x-axis
Q or Θ: Pitch about the y-axis
R or Ψ: Yaw about the z-axis

Unless otherwise specified, by default the software uses ZYX rotation order for Euler angles.

Wind Coordinates

The noninertial wind coordinate system has its origin fixed in the rigid aircraft. The coordinate system orientation is defined relative to the craft's velocity V.

The orientation of the wind coordinate axes is fixed by the velocity V.

The x-axis points in the direction of V.
The y-axis points to the right of the x-axis (facing in the direction of V), perpendicular to the x-axis.
The z-axis points perpendicular to the x-y plane in whatever way needed to satisfy the RH rule with respect to the x- and y-axes.

Translational Degrees of Freedom

Translations are defined by moving along these axes by distances x, y, and z from the origin.

Rotational Degrees of Freedom

Rotations are defined by the Euler angles Φ, γ, χ. They are

Φ: Bank angle about the x-axis
γ: Flight path about the y-axis
χ: Heading angle about the z-axis

Unless otherwise specified, by default the software uses ZYX rotation order for Euler angles.

References

[1] 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).