Global Positioning Systems, Inertial Navigation, and Integration, Second Edition

APPENDIX C

COORDINATE TRANSFORMATIONS

C.1 NOTATION

We use the notation C^from_to to denote a coordinate transformation matrix from one coordinate frame (designated by “from”) to another coordinated frame (designated by “to”). For example,

C^ECI_ENU denotes the coordinate transformation matrix from earth-centered inertial (ECI) coordinates to earth-fixed east-north-up (ENU) local coordinates and

C^RPY_NED denotes the coordinate transformation matrix from vehicle body-fixed roll-pitch-yaw (RPY) coordinates to earth-fixed north-east-down (NED) coordinates.

Coordinate transformation matrices satisfy the composition rule

C^B_CC^A_B = C^A_C,

where A, B, and C represent different coordinate frames.

What we mean by a coordinate transformation matrix is that if a vector v has the representation

in XYZ coordinates and the same vector v has the alternative representation

in UVW coordinates, then

where “XYZ” and “UVW” stand for any two Cartesian coordinate systems in three-dimensional space.

The components of a vector in either coordinate system can be expressed in terms of the vector components along unit vectors parallel to the respective coordinate axes. For example, if one set of coordinate axes is ...