Fundamental Coordinate System Concepts

Coordinate systems allow you to track an aircraft or spacecraft position and orientation in space. The Aerospace Blockset™ 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 Blockset software uses right-handed (RH) Cartesian coordinate systems. The right hand rule establishes the x-y-z sequence of coordinate axes.

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 Blockset makes two 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 only noninertial effect in Earth-fixed coordinates is due to the Earth rotation about its axis. This system is a rotating, geocentric system. The Aerospace Blockset does not take into account the acceleration of the Sun within the galaxy or the acceleration of the galaxy itself through the cosmos.

Passive Transformations

All quaternions in Aerospace Blockset 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 Blockset 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.

For information on coordinate systems used for modeling, navigation and display in Aerospace Blockset, see:

• Coordinate Systems for Modeling

• Coordinate Systems for Navigation

• Coordinate Systems for Display

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.