## Coordinate Systems for Navigation

Modeling aerospace trajectories requires positioning and orienting the aircraft or spacecraft with respect to the rotating Earth. Navigation coordinates are defined with respect to the center and surface of the Earth.

### Geocentric and Geodetic Latitudes

The *geocentric latitude* λ on the Earth surface is defined by the angle
subtended by the radius vector from the Earth center to the surface point with the
equatorial plane. This definition assumes Earth as a perfect sphere, considering all
points on the surface to be equidistant from the center of the Earth. However, because
Earth is actually an oblate spheroid, the geocentric latitude does not accurately
reflect the surface curvature in terms of the actual distance from the equator or the
poles.

The *geodetic latitude* μ on the Earth surface is defined by the angle
between the equatorial plane and the normal to the reference ellipsoid that best
approximates Earth's surface at that point. The reference ellipsoid has a smaller radius
at the poles and a larger radius at the equator, matching Earth's shape more closely
than a sphere would. Geodetic latitude is commonly used in mapping and navigation
because it provides a more accurate representation of the location of a point on Earth's
surface.

### NED Coordinates

The north-east-down (NED) system is a noninertial system with its origin fixed at the center of gravity of the ground station, aircraft, or spacecraft. The NED axes are oriented along the geodetic directions defined by the Earth surface. The NED system is a type of local tangent plane coordinate system, which means it is defined relative to a specific location on Earth, defined by latitude, longitude and altitude, typically where the measurement is being made or the vehicle, such as an aircraft or drone, is operating.

The

*x*-axis points north parallel to the geoid surface, in the polar direction.The

*y*-axis points east parallel to the geoid surface, along a latitude curve.The

*z*-axis points downward, toward the Earth surface, antiparallel to the outward normal*n*of the surface.Flying at a constant altitude means flying at a constant

above the Earth's surface.`z`

### 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.

#### 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.