Mathematical Model of QCopter

Coordinate Systems

To describe the dynamics of the quadro copter two coordinate systems are defined. An inertial system $\large XYZ$ and a body fixed system $\large xyz$ . The location of the center of gravity is parameterized by the vector $\large\vec{r}$ and the orientation of the copter is parameterized by the euler angles $\large\phi_z, \phi_y, \phi_x$ which correspond to the angles for a rotation on the Z-axis, then a rotation on the resulting y'-axis and finally on x' '-axis (same as x-axis).

attachment:copter_ks.gif

Using this euler parameterization yields in the following transformation matrix which transforms coordinates from body to inertial system:

$\underline{\underline{C}}_{ib} = \left[ \begin {array}{ccc} \cos \left( { \phi_z} \right) \cos \left( { \phi_y} \right) &-\sin \left( { \phi_z} \right) \cos \left( { \phi_x} \right) +\cos \left( { \phi_z} \right) \sin \left( { \phi_y} \right) \sin \left( { \phi_x} \right) &\sin \left( { \phi_z} \right) \sin \left( { \phi_x} \right) +\cos \left( { \phi_z} \right) \sin \left( { \phi_y} \right) \cos \left( { \phi_x} \right) \\\noalign{\medskip}\sin \left( { \phi_z} \right) \cos \left( { \phi_y} \right) &\cos \left( { \phi_z} \right) \cos \left( { \phi_x} \right) +\sin \left( { \phi_z} \right) \sin \left( { \phi_y} \right) \sin \left( { \phi_x} \right) &-\cos \left( { \phi_z} \right) \sin \left( { \phi_x} \right) +\sin \left( { \phi_z} \right) \sin \left( { \phi_y} \right) \cos \left( { \phi_x} \right) \\\noalign{\medskip}-\sin \left( { \phi_y} \right) &\cos \left( { \phi_y} \right) \sin \left( { \phi_x} \right) &\cos \left( { \phi_y} \right) \cos \left( { \phi_x} \right) \end {array} \right]$

Due to the symmetry the inverse can be calculated as the transposed matrix which transform coordinates from inertial to body system:

$\large\underline{\underline{C}}_{bi} = \underline{\underline{C}}_{ib}^T$

By a partial transformation of the change rate $\large\dot{\underline{\phi}}$ of the euler angles to body system and setting them equal to the angular velocities $\large\underline{\omega}_b$ expressed in body system, the following dependency can be calculated:

$\large\dot{\underline{\phi}}=\underbrace{ \left[ \begin {array}{ccc} 1&{\sin \left( { \phi_x} \right) \tan \left( { \phi_y} \right) }&{\cos \left( { \phi_x} \right) \tan \left( { \phi_y} \right) }\\\noalign{\medskip}0&\cos \left( { \phi_x} \right) &-\sin \left( { \phi_x} \right) \\\noalign{\medskip}0&{\frac {\sin \left( { \phi_x} \right) }{\cos \left( { \phi_y} \right) }}&{\frac {\cos \left( { \phi_x} \right) }{\cos \left( { \phi_y} \right) }}\end {array} \right] }_{\underline{\underline{\psi}}}\underline{\omega}_b$

Forces and Torques

Cutting the copter free and drawing all forces and torques acting on the rigid body yields:

attachment:copter_forces.gif

where $\vec{F}_i, \vec{M}_i$ are the forces and torques generated by the motors, $m\vec{g}$ is the weight and $\vec{R}$ is the air drag. The resulting force and torque of the motors acting on the center of mass can be calculated with:

$\large\begin{array}{lll}\vec{F} &=& \sum_i F_i \\ \vec{M} &=& \sum_i \vec{M}_i + \sum_i \vec{r}_{m,i} \times \vec{F_i} \end{array}$

Written in coordinates of the copter system yields:

$\large\begin{array}{lll} \underline{F}_b &=& \left[\begin{array}{c}0\\0\\F_1+F_2+F_3+F_4\end{array}\right]_b \\ \underline{M}_b &=& \left[\begin{array}{c}\ell(F_4-F_3)\\\ell(F_2-F_1)\\M_1+M_2-M_3-M_4\end{array}\right]_b \end{array}$

The air drag is assumed to be proportional to the the squared velocity but in opposit direction:

$\large{\vec{R} = \underbrace{ - \frac{\vec{v}}{\left|\vec{v}\right|}}_{\rm{direction}} \underbrace{k \left|\vec{v}\right|^2}_{\rm{magnitude}} = - k \left|\vec{v}\right| \vec{v}}$

Equations of Motion

The conservation of linear and angular momentum, set up in the center of gravity yield:

$\large\begin{array}{rllll}m \vec{a} &=& \sum \rm{Forces} &=& \vec{F}+\vec{R}+m\vec{g}\\ \vec{\vec{\Theta}}~\dot{\vec{\omega}} + \vec{\omega} \times \vec{\vec{\Theta}}~\vec{\omega} &=& \sum \rm{Torques} &=& \vec{M}\end{array}$

Expressed in coordinates:

$\large\begin{array}{lll} \dot{\underline{r}}_i &=& \underline{r}_i \\ \dot{\underline{v}}_i &=& \frac{1}{m}\underline{\underline{C}}_{ib}~\underline{F}_b + \frac{1}{m}\underline{R}_i + \underline{g}_i \\ \large\dot{\underline{\phi}}&=&\underline{\underline{\psi}}~\underline{\omega}_b \\ \dot{\underline{\omega}}_b &=& \underline{\underline{\Theta}}^{-1}\left( \underline{M}_b - \underline{\omega}_b \times \underline{\underline{\Theta}}~\underline{\omega}_b \right)\end{array}$

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Mathematical Model of QCopter

Coordinate Systems

Forces and Torques

Equations of Motion