TABLE OF CONTENTS Transformation matrices are perceived as means of transforming a vector from one frame to another. Thus far, both the vector and the DCM have been assumed static; that is both are stationary. But one might ask, what if one frame is continuously changing its direction, can we still construct the transformation matrix? Specifically we consider the case when one frame is rotating relative to another frame at some angular velocity.
We would like to determine how the transformation matrix will vary with time; that is, to derive its derivative with respect to time. This derivation is extended to the DCM, quaternion, rotation vector and Euler angles.
Consider the case of two initially coincident frames a and b and in which frame b rotates relative to fixed frame a. Let's adopt this notation: at time t, the transformation matrix from b to a will be given by
Therefore the transformation matrix at time t+dt, will be
With reference to Fig. 2.2, we define the following terms: u is a unit vector along the axis about which vectors rotate and w is the angular rate of the vector frame about u. The instantaneous angular rate vector, ?, is then
Therefore in the time interval dt, frame b will rotate an angle given by
From the above and from Eq. (3.15), the DCM that govern this rotation is given...
