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MAE 6245 (Spring 2022)

Robotic Systems

Assignment # 1

1)   For the given frames (0 and 1), find the rotation matrix   specifying the orientation of frame 1 relative to frame 0.

[3 points]


2)   Show that the dot product of two free vectors does not depend on the choice of frames in which their coordinates are defined. [Hint: use the definition of the dot product (xTx)]              [5 points]

 

3)   Imagine two unit vectors, v1  and v2, embedded in a rigid body. Note that, no matter how the      body is rotated, the geometric angle between these two vectors is preserved (i.e., rigid-body     rotation is an "angle-preserving" operation). Use this fact to give a concise (four- or five-line)     proof that the inverse of a rotation matrix must equal its transpose and that a rotation matrix is orthonormal.                                                                                                                                  [5 points]


This proves orthonormality as well.

4)   2 points awarded to anybody who has indicated their topic of interest.