Attitude Control of a Helicopter

Attached image depicts the model of this helicopter. The dynamics of this model simulate the attitude dynamics of a helicopter. Using Lagrange’s equations, one may readily show that the dynamical model of this system is given by:

$$\begin{align*} T_p &= I_x\ddot{\phi}-(I_y-I_z)\dot{\psi}^2S_{\phi}C_{\phi}+mgAC_{\phi} \\ T_y &= (I_yS_{\phi}^2 + I_zC_{\phi}^2)\ddot{\psi} + 1(I_y - I_z)\dot{\psi}\dot{\phi}S_{\phi}C_{\phi}, \end{align*}$$

where $\phi$ is the pitch angle in radians, $\psi$ is the yaw angle in radians, $I_x, I_y, I_z$ are inertia constants about the point of rotation, m is the total mass of the system, and $T_p, T_y$ are the pitch and yaw control torques, respectively.

The above can be written in general form:

$u = M(q,\delta)\ddot{q}+C(q,\dot{q},\delta)\dot{q}+G(q,\dot{q},\delta),$

where

$M(q,\delta) = \begin{bmatrix} I_x & 0\\ 0 & I_yS_{\phi}^2+I_zC_{\phi}^2\end{bmatrix}$

$G(q,\delta)=\begin{bmatrix}mgAC_{\phi}\\0\end{bmatrix}$

$C(q,\dot{q},\delta)=\begin{bmatrix}0&-(I_y-I_z)\dot{\psi}S_{\phi}C_{\phi} \\ (I_y-I_z)\dot{\psi}S_{\phi}C_{\phi} &(I_y-I_z)\dot{\phi}S_{\phi}C_{\phi}\end{bmatrix} $

$q=\begin{bmatrix}\phi \\ \psi\end{bmatrix}$

$u=\begin{bmatrix}T_p \\ T_y\end{bmatrix}$

The nominal values of the model parameters are $m=0.5719 kg$, $A = 0.0801 m^2$, $I_x = 0.0762 kg\cdot m^2$, $I_y = 3.86\times10^{-4}kg\cdot m^2$, $I_z = 0.0766 kg\cdot m^2,$ and $g = 9.81 m/sec^2$. It is straight forward to show that $[\dot{M}(q)-2C(q,\dot{q})]$ is skew symmetric.