Three Parametrizations of Rotations

A rotation can be parameterized in several ways. This Demonstration compares three popular parametrizations:
• Euler angles about the axes
• the angle of rotation about an arbitrary axis
• the roll, pitch and yaw about the world axes
The progress slider rotates a teapot shape through these rotations, from an initial orientation in green to a final orientation in red. The intermediate configurations of the teapot depend on the parametrization chosen, but the final configuration is always the same.

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This Demonstration shows three parametrizations to describe rotations between a fixed coordinate frame and a rotated frame .
The first parametrization uses Euler angles. There are many Euler angle conventions. This Demonstration uses the convention, which specifies the orientation of frame by three successive rotations. The first rotates about the axis by the angle . Next, we rotate about the current axis by the angle . Finally, we rotate about the current axis by the angle .
The composite rotation, using the shorthand convention of for and for , is
.
Euler's rotation theorem states that any combination of rotations of a rigid body, such that a point in the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. The second parametrization uses axis/angle parametrization, a rotation of about the unit axis . This again gives only three parameters, by representing by the two-parameter latitude/longitude pair: . Using the convention :
.
The final parametrization uses roll, pitch, and yaw angles, denoted as , and . The order of rotation in this Demonstration is around the fixed coordinate frame axes: first, a yaw about through an angle ; second, a pitch about by an angle ; and third, a roll about by an angle . Because the rotations are about the fixed coordinate frame, the successive rotations pre-multiply, giving the composite rotation
.
Reference
[1] M. W. Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control, Hoboken, NJ: John Wiley & Sons, 2006.
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