7.4 Understanding rotation matrices

Chapter 7 of the textbook describes the Z–Y–X body-fixed rotation sequence (see Figure 7.12 and Equation 7.10). Another popular convention is the X–Y–Z body-fixed rotation sequence, where the orientation of one frame relative to another is described by the following rotations:

• First, rotate about the X axis by angle ;
• Next, rotate about the new Y axis by angle ;
• Finally, rotate about the newest Z axis by angle .

Recall that a "body-fixed" rotation sequence is one in which each rotation is performed about an axis that is glued to the rotating body (as opposed to a "space-fixed" rotation sequence, where each rotation would be performed about an axis that is glued to the ground). The rotation matrix relating the original frame (A) to the final frame (B) is the following:

where  and .

(a) Multiply three elementary rotation matrices to verify the matrix above.

(b) Substitute , , and  into the matrix above and clearly sketch the reference frames corresponding to each rotation in this sequence. Circle the underlined words that best complete the following sentence: The first column of this rotation matrix tells us that the X / Y / Z axis of the original / final frame is pointed in the same / opposite direction as the X / Y / Z axis of the original / final frame.

(c) As shown in Equations 7.10 and 7.11 of the textbook, we can use the two-argument arctangent function to compute rotation angles from the numerical entries of a given rotation matrix. Derive expressions similar to those shown in Equation 7.11 to compute , , and  from the symbolic rotation matrix shown above. Your answers should be expressed in terms of elements of the rotation matrix (, , etc.). Please state any assumptions that you make.

(d) Use Equation 7.11 in the textbook to compute the angles , , and  corresponding to the Z–Y–X body-fixed rotation sequence for the following matrix:

Sketch the reference frames corresponding to this sequence of rotations. How does this sequence of rotations compare to your sketch from part (b)?