Euler Angles

The orientation of an airplane, relative to local axes, can be specified by the three sequential rotations about the body axes. Starting with the body axes aligned with the local axes, the first rotation is about the z-axis through an angle $\Psi$, followed by a rotation about the y-axis through an angle $\Theta$, followed by a rotation about the x-axis through an angle $\Phi$. These angles of rotation are the Euler angles, and can represent any possible orientation of the airplane.

Equation 16 lists the airplane's direction cosine matrix constructed from the Euler angles.

$$[C]= \left[ \begin{array}{ccc} \cos\Theta\cos\Psi & \sin\Phi\... ...a & \sin\Phi\cos\Theta & \cos\Phi\cos\Theta \end{array}\right]$$ (16)

Equations 17-19 represent the kinematic equations for the Euler angles, relating the Euler angle time derivatives to the angular velocity.

   

$$\dot\Phi p + q\sin\Phi\tan\Theta + r\cos\Phi\tan\Theta$$ = (17)

$$\dot\Theta q\cos\Phi - r\sin\Phi$$ = (18)

$$\dot\Psi q\sin\Phi\sec\Theta + r\cos\Phi\sec\Theta$$ = (19)