## Fundamental Coordinate System Concepts

Coordinate systems allow you to track an aircraft or spacecraft position and orientation in space. The Aerospace Toolbox coordinate systems are based on these underlying concepts from geodesy, astronomy, and physics. For more information on geographic information, see Mapping Toolbox.

### Definitions

The Aerospace Toolbox software uses *right-handed* (RH) Cartesian coordinate systems. The *rightmost rule* establishes the * x*-

*-*

`y`

*sequence of coordinate axes.*

`z`

An *inertial frame* is a nonaccelerating motion reference frame. Loosely speaking, acceleration is defined with respect to the distant cosmos. In an inertial frame, Newton's second law (force = mass X acceleration) holds.

Strictly defined, an inertial frame is a member of the set of all frames not accelerating relative to one another. A *noninertial frame* is any frame accelerating relative to an inertial frame. Its acceleration, in general, includes both translational and rotational components, resulting in *pseudoforces* (*pseudogravity*, as well as *Coriolis* and *centrifugal forces*).

The toolbox models the Earth shape (the *geoid*) as an oblate spheroid, a special type of ellipsoid with two longer axes equal (defining the *equatorial plane*) and a third, slightly shorter (*geopolar*) axis of symmetry. The equator is the intersection of the equatorial plane and the Earth surface. The geographic poles are the intersection of the Earth surface and the geopolar axis. In general, the Earth geopolar and rotation axes are not identical.

Latitudes parallel the equator. Longitudes parallel the geopolar axis. The *zero longitude* or *prime meridian* passes through Greenwich, England.

### Approximations

The Aerospace Toolbox software makes three standard approximations in defining coordinate systems relative to the Earth.

The Earth surface or geoid is an oblate spheroid, defined by its longer equatorial and shorter geopolar axes. In reality, the Earth is slightly deformed with respect to the standard geoid.

The Earth rotation axis and equatorial plane are perpendicular, so that the rotation and geopolar axes are identical. In reality, these axes are slightly misaligned, and the equatorial plane wobbles as the Earth rotates. This effect is negligible in most applications.

The only noninertial effect in Earth-fixed coordinates is due to the Earth rotation about its axis. This is a

*rotating*,*geocentric*system. The toolbox ignores the Earth motion around the Sun, the Sun motion in the Galaxy, and the Galaxy's motion through cosmos. In most applications, only the Earth rotation matters.This approximation must be changed for spacecraft sent into deep space, that is, outside the Earth-Moon system, and a heliocentric system is preferred.

### Passive Transformations

All quaternions in Aerospace Toolbox are passive transformations. In a passive transformation, the vector is unchanged and the coordinate system in which it is defined is rotated. For more information on transformations, see Active and passive transformations.

### Motion with Respect to Other Planets

The Aerospace Toolbox software uses the standard WGS-84 geoid to model the Earth. You can change the equatorial axis length, the flattening, and the rotation rate.

You can represent the motion of spacecraft with respect to any celestial body that is well approximated by an oblate spheroid by changing the spheroid size, flattening, and rotation rate. If the celestial body is rotating westward (retrogradely), make the rotation rate negative.

## References

[1] *Recommended Practice for Atmospheric and Space Flight Vehicle Coordinate Systems*, R-004-1992, ANSI/AIAA, February 1992.

[2] Rogers, R. M., *Applied Mathematics in Integrated Navigation Systems*, AIAA, Reston, Virginia, 2000.

[3] Stevens, B. L., and F. L. Lewis, *Aircraft Control, and Simulation*, 2nd ed., Wiley-Interscience, New York, 2003.

[4] Thomson, W. T., *Introduction to Space Dynamics*, John Wiley & Sons, New York, 1961/Dover Publications, Mineola, New York, 1986.