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

.

[1] M. W. Spong, S. Hutchinson and M. Vidyasagar,

*Robot Modeling and Control*, Hoboken, NJ: John Wiley & Sons, 2006.