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}_{C}**C*** ^{A}_{B}* =

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 ...

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