Welcome to AE Resources

Converted document

ROTATION MATRIX

Figure 0.1 shows two frame of reference. A frame rotates to B frame. A’ and B’s origins are at the same place. These two frames are shown in the figure separately for clarity. In other word, this operation does not include traslation.

Figure 0.1: Rotation operator

General formulation

Figure 0.2 shows a more general case, where B frame (the blue one) rotates with respect to A frame about unit vector (n). One can write the direction cosine matrix which brings A frame to B as below
$CBA = Δ - sin Φ˜n + (1 - cosΦ)˜n ˜n$ (0.1)

where Φ is the rotation angle. This is the result of Euler’s thorem on finite rotations. Euler’s thorem on finite rotations: Any arbitrary finite rotation that leaves a point fixed can be viewed as a single rotation of magnitude Φ about a unit vector [?].

Figure 0.2: Rotation operator, the general case