We consider the attached image where the VARIO helicopter mounted on an experimental platform is represented. It is important to say that in this particular case the helicopter is in an OGE condition. The effects of the compressed air in take-off and landing are then neglected. The model has the form
$$M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=Q(u),$$
where $M(q)\in\mathbb{R}^{3\times3}$ is the inertia matrix, $C(q,\dot{q})\in\mathbb{R}^{3\times3}$ is the Coriolis matrix, $G(q)\in\mathbb{R}^3$ is the vector of conservative forces, $Q(u)=\begin{bmatrix}f_z & \tau_z & \tau_\gamma \end{bmatrix}^{\mathrm T}$ is the vector of generalized forces, $q = \begin{bmatrix} z & \phi & \gamma \end{bmatrix}^{\mathrm T}$ is the vector of generalized coordinates and $u=\begin{bmatrix}h_M & h_T \end{bmatrix}^{\mathrm T}$ is the vector of control inputs. Here $f_Z, \tau_Z$ and $\tau_{\gamma}$ are the vertical forces, the yaw torque and the main rotor torque, respectively. The height $z < 0$ upwards, $\phi$ is the yaw angle and $\gamma$ is the main rotor azimuth angle.
$M(q)=\begin{bmatrix}
c_0 & 0 & 0 \\
0 & c_1 + c_2 \cos^2{(c_3\gamma)} & c_4\\
0 & c_4 & c_5 \end{bmatrix},$
$C(q,\dot{q})=\begin{bmatrix}
0 & 0 & 0\\
0 & c_6\sin{(2c_3\gamma)}\dot{\gamma} & c_6\sin{(2c_3\gamma)}\dot{\phi} \\
0 & -c_6\sin{(2c_3\gamma)}\dot{\phi} & 0\end{bmatrix},$
$G(q)=\begin{bmatrix}c_7 \\ 0 \\ 0 \end{bmatrix},$
where $c_i$'s $i = 0, ..., 7$ are the physical constants given in the table below.
The generalized forces vector is given by
$Q(u)=\begin{bmatrix}
c_8\dot{\gamma}^2u_1 + c_9\dot{\gamma} + c_{10} \\
c_{11}\dot{\gamma}^2u_2\\
(c_{12}\dot{\gamma}^2 + c_{13})u_1 + c_{14}\dot{\gamma}^2 + c_{15}
\end{bmatrix}$
| $c_i$ |
Numerical value |
| $c_0$ |
$7.5$ $kg$ |
| $c_1$ |
$0.4305$ $kg\times m^2$ |
| $c_2$ |
$3 \times 10^{-4}$ $kg\times m^2$ |
| $c_3$ |
$-4.143$ |
| $c_4$ |
$0.108$ $kg\times m^2$ |
| $c_5$ |
$0.4993$ $kg\times m^2$ |
| $c_6$ |
$-6.214 \times 10^{-4}$ $kg\times m^2$ |
| $c_7$ |
$-73.58$ $N$ |
| $c_8$ |
$3.411$ $kg$ |
| $c_9$ |
$0.6004$ $kg \times m/s$ |
| $c_{10}$ |
$3.679$ $N$ |
| $c_{11}$ |
$-0.1525$ $mg \times m$ |
| $c_{12}$ |
$12.01$ $kg \times m/s$ |
| $c_{13}$ |
$1 \times 10^{5}$ $N$ |
| $c_{14}$ |
$1.206 \times 10^{-4}$ $kg \times m^2$ |
| $c_{15}$ |
$2.642$ $N$ |
