# Three Degree of Freedom Helicopter Model

## Model description:

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 $c_0$ $7.5$ $kg$ $c_1$ $0.4305$ $kg\times m^2$ $c_2$ $3 \times 10^{-4}$ $kg\times m^2$ $c_3$ $-4.143$ $c_4$ $0.108$ $kg\times m^2$ $c_5$ $0.4993$ $kg\times m^2$ $c_6$ $-6.214 \times 10^{-4}$ $kg\times m^2$ $c_7$ $-73.58$ $N$ $c_8$ $3.411$ $kg$ $c_9$ $0.6004$ $kg \times m/s$ $c_{10}$ $3.679$ $N$ $c_{11}$ $-0.1525$ $mg \times m$ $c_{12}$ $12.01$ $kg \times m/s$ $c_{13}$ $1 \times 10^{5}$ $N$ $c_{14}$ $1.206 \times 10^{-4}$ $kg \times m^2$ $c_{15}$ $2.642$ $N$

## Publication details:

 Title Nonlinear modelling and control of helicopters Publication Type Journal Article Year of Publication 2003 Authors Vilchis, J.C. Avila, Brogliato B., Dzul A., and Lozano R. Journal Automatica Volume 39 Pagination 1583-1596 Date Published 09/2003 ISSN 0005-1098 Keywords Aerodynamics, Helicopter; Drone, Nonlinear control, nonlinear systems, Underactuated Abstract This paper presents the development of a nonlinear model and of a nonlinear control strategy for a VARIO scale model helicopter. Our global interest is a 7-DOF (degree-of-freedom) general model to be used for the autonomous forward-flight of helicopter drones. However, in this paper we focus on the particular case of a reduced-order model (3-DOF) representing the scale model helicopter mounted on an experimental platform. Both cases represent underactuated systems ($u \in \mathbb{R}^4$ for the 7-DOF model and $u \in \mathbb{R}^2$ for the 3-DOF model studied in this paper). The proposed nonlinear model possesses quite specific features which make its study an interesting challenge, even in the 3-DOF case. In particular aerodynamical forces result in input signals and matrices which significantly differ from what is usually considered in the literature on mechanical systems control. Numerical results and experiments on a scale model helicopter illustrate the theoretical developments, and robustness with respect to parameter uncertainties is studied. DOI 10.1016/s0005-1098(03)00168-7