Overview

Even though a DCM is a 3 3 array with nine elements, it is only characterized by three parameters. The reason is that a DCM is a rotation matrix that indicates a unit vector an axis of rotation along which a frame rotates and an angle of rotation around this axis. Our objective is to characterize errors in the DCM in terms of these three fundamental parameters. This will be more efficient to process three elements rather than nine elements.

Errors in DCM could be classified as static or dynamic. Static sources can be attributed to initialization. Alternatively, dynamic errors result from temporal errors in the rotation vector that drives the DCM. We shall derive the equations that govern the behavior of the three parameters in both cases

Suppose is the ideal DCM that transforms frame b to a. The actual DCM will transform b to frame ? that is a bit different from the intended frame a and is denoted . Supposing that the ideal and actual DCM's are related by a multiplicative matrix, E, i.e.

and the error difference between the two DCM's is given by

From Eq. (G.1), E is given by

Since the product of two DCM's is another DCM, then E can be represented by a product of three independent rotations ( ??, ??, ??). These angular rotations are assumed to be so small that small angle approximation...