*3.2. Rigid Body Dynamics*

Vehicle dynamics is described by Newton–Euler equations of motion projected in F*B*, namely:

$$
\dot{\mathbf{v}} = -\boldsymbol{\omega} \times \mathbf{v} + \mathbf{F}/m\tag{2}
$$

$$
\dot{\omega} = \mathbf{J}^{-1}[-\omega \times (\mathbf{J}\,\omega) + \mathbf{M}] \tag{3}
$$

where *v* = [*u*, *v*, *w*] *<sup>T</sup>* is linear velocity, *ω* = [*p*, *q*, *r*] *<sup>T</sup>* is angular velocity,

$$J = \begin{bmatrix} J\_{xx} & -J\_{xy} & -J\_{xz} \\ -J\_{xy} & J\_{yy} & -J\_{yz} \\ -J\_{xz} & -J\_{yz} & J\_{zz} \end{bmatrix} \tag{4}$$

is the inertia tensor about *CG* with respect to <sup>F</sup>*B*, and *<sup>m</sup>* is the total mass of the rotorcraft. **F** = [*Fx*, *Fy*, *Fz*] *<sup>T</sup>* and **M** = [*Mx*, *My*, *Mz*] *<sup>T</sup>* are the external force and moment vectors, respectively.

The external force acting on the rotorcraft is made of gravity, *<sup>F</sup>*(*g*) , and aerodynamic, *<sup>F</sup>*(*a*) , contributions. Taking into account Equation (1), gravity force vector expressed in the body frame is

$$F^{(\mathcal{G})} = \mathcal{R}(\mathfrak{a}) \begin{bmatrix} 0 \\ 0 \\ m \, \mathcal{G} \end{bmatrix} = m \, \mathcal{g} \begin{bmatrix} -\sin \theta \\ \sin \phi \cos \theta \\ \cos \phi \, \cos \theta \end{bmatrix} \tag{5}$$

where *g* is gravitational acceleration, described by means of WGS84 Taylor series model [26].

Rotorcraft attitude kinematics, that relates the generalized velocity *α*˙ and the angular velocity *ω* is given by [12]:

$$
\dot{\mathfrak{a}} = \begin{bmatrix}
1 & \sin\phi \tan\theta & \cos\phi \tan\theta \\
0 & \cos\phi & -\sin\phi \\
0 & \sin\phi / \cos\theta & \cos\phi / \cos\theta
\end{bmatrix} \mathfrak{a} \tag{6}
$$

while the position of the helicopter *p<sup>E</sup>* = [*xE*, *yE*, *zE*] *<sup>T</sup>*, with components expressed in the inertial frame F*E*, is obtained from the equation:

$$
\dot{\boldsymbol{p}}\_E = \mathbf{R}(\boldsymbol{\omega})^T \boldsymbol{v} \tag{7}
$$
