3.6 Simple Quaternions

In the following we shall use the shorthand notation of Q _x( ) to describe a quaternion that causes a rotation of angle about the x-axis.

1. x-axis: Eq. (3.16) show that the quaternion that rotates an angle about u = [1 0 0] ? is

   
   

Using Eq. (2.27) we see that it corresponds to the DCM in Eq. (3.1).

Thus this quaternion will perform a rotation about the x-axis

1. y-axis: Eq. (3.16) show that the quaternion that rotates an angle ? about u = [0 1 0] ? is

   
   

which corresponds to the DCM given in Eq. (3.2)

1. z-axis: Eq. (3.16) show that the quaternion that rotates an angle ? about u = [0 0 1] ? is

   
   

which corresponds to the DCM given in Eq. (3.3)

Therefore, the quaternion that performs the Euler angles sequence is given by

When expanded it becomes