Mathematical Model¶
\[\begin{split}\begin{eqnarray} %
m {\mathbf{\ddot{x}}} &=&
-m g\, \mathbf{e}_3+ {}^W\mathbf{R} {f} \mathbf{e}_3,\\
%
\mathbf{J}\dot{\boldsymbol{\omega}} &=&
-\boldsymbol{\omega} \times \mathbf{J}\boldsymbol{\omega}
+ \boldsymbol{\tau},\\
\dot{\mathbf{R}}^W &=&\mathbf{R}^W\hat{\boldsymbol{\omega} },
\end{eqnarray}\end{split}\]
Where the rotation matrix is,
\[\begin{split}\boldsymbol{R}_{xzy} = \left [
\begin{array}
c\psi c\theta - s\phi s\psi s\theta & -c\phi s\psi & c\psi s\theta + c\theta s\phi s\psi\\
c\theta s\psi + c\psi s\phi s\theta & c\phi s\psi & s\psi s\theta + c\psi c\theta s\phi\\
- c\phi s\theta & s\phi & c\phi c\theta
\end{array} \right ],\end{split}\]
Each rotor has an angular velocity \(\boldsymbol{\omega}\) and produce a thrust \(\boldsymbol{f_i} = k_F \boldsymbol{\omega_i}^2\), Also The rotors produce a moment by \(\boldsymbol{M_i} = k_M \boldsymbol{\omega_i}^2\)