EULER ANGLES

Although we will not be concerned with them to any extent, it might be well at this point to define the so-called Euler angles. Finite angular rotations of an airplane about its own body axes are not commutative. The final orientation of the airplane will depend on the order in which the rotations are performed. To illustrate this, hold a small model in front of you, heading directly away from you with wings level. Now rotate the model 90° about its x-axis, then 90° about its у-axis, and then 90° around its 2-axis, all directions positive in accordance with the right-hand rule. The model will now be pointing nose-down with its top toward you. Now reverse the order. Rotate it 90° about its 2-axis, then the у-axis, and then the x-axis. In this case, the model’s final orientation will be nose-up with its top toward you.

The Euler angles, starting with a given airplane orientation, are rotations denoted by ф, в, and ф about the x, y, and 2-axes, respectively. However, the order of rotation is first about the 2-axis, then about the у-axis, and then about the x-axis. In other words, the airplane is first yawed, then pitched, and then rolled.