3.3 Euler Angles

As mentioned earlier, the Euler angles approach forms the DCM through a sequence of simple rotations. Although there is no unique sequence for forming the Euler angles, there are widely acceptable sets for specific applications. One of them is the rotation sequence about aircraft body axes. In this frame, the axes are defined so that the x-axis points to the nose along the symmetric longitudinal axis of the craft, the y-axis is orthogonal to the x-axis along the right wing, and the z-axis is perpendicular to the x-y plane and points downwards. In this configuration it is assumed that when the craft is level, the x-y plane will be horizontal and hence the z-axis will be vertical to the ground. As shown in Fig 3.1, the Euler sequence is performed such that the reference frame first rotates an angle ? (yaw) about the z-axis to form x' y' z' frame. Since the rotation is about the z-axis, then z and z' will be the same. Second, the new frame is rotated an angle ? (pitch) about the y'-axis to form x" y" z". Finally this frame is rotated an angle (roll) about its x"-axis to form frame. We observe that the DCMs that describe these rotations are those special matrices given in the previous section.

Figure 3.1: Euler Angles Sequence

The DCM product of the...