where X and $ have the same meaning as in Sec. 1.1, and a,, a2, and b,, b2, I>3 are dextral sets of orthogonal unit vectors fixed in A and B, respec-
tively, with a, = b, (/ = 1, 2, 3) prior to the rotation. (When a discussion involves more than two bodies or reference frames, notations such as ApB and APiB will be used.)
The Rodrigues parameters are intimately related to the Euler parameters (see Sec. 1.3):
An advantage of the Rodrigues parameters over the Euler parameters is that they are fewer in number; but this advantage is at times offset by the fact that the Rodrigues parameters can become infinite, whereas the absolute value of any Euler parameter cannot exceed unity.
Expressed in terms of Rodrigues parameters, the direction cosine matrix C (see Sec. 1.2) assumes the form
The Rodrigues vector can be used to establish a simple relationship between the difference and the sum of the vectors a and b defined in Sec. 1.1:
This relationship will be found useful in connection with a number of derivations, such as the one showing that the following is an expression for a Rodrigues vector that characterizes a simple rotation by means of which a specified change in the relative orientation of A and B can be produced:
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