Quadrotors

Non-Parametric Dynamics

$$ \begin{aligned} \dot{\mathbf{p}} &= \mathbf{v} \\
\dot{\mathbf{q}} &= \frac{1}{2} \begin{bmatrix} 0 & -\omega_x & -\omega_y & -\omega_z \\
\omega_x & 0 & \omega_z & -\omega_y \\
\omega_y & -\omega_z & 0 & \omega_x \\
\omega_z & \omega_y & -\omega_x & 0 \end{bmatrix} \begin{bmatrix} q_w \\ q_x \\ q_y \\ q_z \end{bmatrix} \\
\dot{\mathbf{v}} &= \mathbf{g} + \begin{bmatrix} 2 \left( q_w q_y + q_x q_z \right) \\
2 \left( q_y q_z - q_w q_x \right) \\
1 - 2 q_x^2 - 2 q_y^2 \end{bmatrix} T_n \end{aligned} $$

Parametric Dynamics without aerodynamic influence

$$ \begin{aligned} \dot{\mathbf{p}} &= \mathbf{v} \\
\dot{\mathbf{q}} &= \frac{1}{2} \begin{bmatrix} 0 & -\omega_x & -\omega_y & -\omega_z \\
\omega_x & 0 & \omega_z & -\omega_y \\
\omega_y & -\omega_z & 0 & \omega_x \\
\omega_z & \omega_y & -\omega_x & 0 \end{bmatrix} \begin{bmatrix} q_w \\ q_x \\ q_y \\ q_z \end{bmatrix} \\
\dot{\mathbf{v}} &= \mathbf{g} + \begin{bmatrix} 2 \left( q_w q_y + q_x q_z \right) \\
2 \left( q_y q_z - q_w q_x \right) \\
1 - 2 q_x^2 - 2 q_y^2 \end{bmatrix} \frac{T}{m} \\
\dot{\mathbf{\omega}} &= \mathbf{J}^{-1} \left( \mathbf{\tau} - \mathbf{\omega} \times \mathbf{J} \mathbf{\omega} \right) \end{aligned} $$

Parametric Dynamics including linear aerodynamics

$$ \begin{aligned} \dot{\mathbf{p}} &= \mathbf{v} \\
\dot{\mathbf{q}} &= \frac{1}{2} \begin{bmatrix} 0 & -\omega_x & -\omega_y & -\omega_z \\
\omega_x & 0 & \omega_z & -\omega_y \\
\omega_y & -\omega_z & 0 & \omega_x \\
\omega_z & \omega_y & -\omega_x & 0 \end{bmatrix} \begin{bmatrix} q_w \\ q_x \\ q_y \\ q_z \end{bmatrix} \\
\dot{\mathbf{v}} &= \mathbf{g} + \begin{bmatrix} 2 \left( q_w q_y + q_x q_z \right) \\
2 \left( q_y q_z - q_w q_x \right) \\
1 - 2 q_x^2 - 2 q_y^2 \end{bmatrix} \frac{T}{m} + \mathbf{R} \mathbf{D} \mathbf{R}^T \mathbf{v} \\
\dot{\mathbf{\omega}} &= \mathbf{J}^{-1} \left( \mathbf{\tau} - \mathbf{\omega} \times \mathbf{J} \mathbf{\omega} - \mathbf{\tau}_g - \mathbf{A} \mathbf{R} \mathbf{v} - \mathbf{B} \mathbf{\omega} \right) \end{aligned} $$

Parametric Simplified Dynamics

$$ \begin{aligned} \dot{\mathbf{p}} &= \mathbf{v} \\
\dot{\mathbf{q}} &= \mathbf{\Lambda}(\mathbf{\omega}) \mathbf{q} \\
\dot{\mathbf{v}} &= \mathbf{g} + \mathbf{q} \odot \mathbf{e}_z \frac{T}{m} \\
\dot{\mathbf{\omega}} &= \mathbf{J}^{-1} \left( \mathbf{\tau} - \mathbf{\omega} \times \mathbf{J} \mathbf{\omega} \right) \\
&=\begin{bmatrix} J^{-1}_x & 0 & 0 \\ 0 & J^{-1}_y & 0 \\ 0 & 0 & J^{-1}_z \end{bmatrix} \left( \begin{bmatrix} \tau_x \\ \tau_y \\ \tau_z \end{bmatrix} - \begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z \end{bmatrix} \times \begin{bmatrix} J_x \omega_x \\ J_y \omega_y \\ J_z \omega_z \end{bmatrix} \right) \\
&=\begin{bmatrix} J^{-1}_x & 0 & 0 \\ 0 & J^{-1}_y & 0 \\ 0 & 0 & J^{-1}_z \end{bmatrix} \left( \begin{bmatrix} \tau_x \\ \tau_y \\ \tau_z \end{bmatrix} - \begin{bmatrix} (J_z - J_y) \omega_y\omega_z \\
(J_x - J_z) \omega_x\omega_z \\
(J_y - J_x) \omega_x\omega_y \end{bmatrix} \right) \\
&= \begin{bmatrix} \frac{\tau_x}{J_x} \\ \frac{\tau_y}{J_y} \\ \frac{\tau_z}{J_z} \end{bmatrix} - \begin{bmatrix} \frac{J_z - J_y}{J_x} \omega_y\omega_z \\
\frac{J_x - J_z}{J_y} \omega_x\omega_z \\
\frac{J_y - J_x}{J_z} \omega_x\omega_y \end{bmatrix} \\
&= \begin{bmatrix} \frac{c_{ux}}{J_x} (u^2_1 - u^2_2 - u^2_3 + u^2_4) \\
\frac{c_{uy}}{J_y} (- u^2_1 - u^2_2 + u^2_3 + u^2_4) \\
\frac{c_{uz}}{J_z} (u^2_1 - u^2_2 + u^2_3 - u^2_4) \end{bmatrix} - \begin{bmatrix} \frac{J_z - J_y}{J_x} \omega_y\omega_z \\
\frac{J_x - J_z}{J_y} \omega_x\omega_z \\
\frac{J_y - J_x}{J_z} \omega_x\omega_y \end{bmatrix} \end{aligned} $$

if simplified with $J_x = J_y = J_{xy}$:

$$ \begin{aligned} \dot{\mathbf{\omega}} &= \begin{bmatrix} \frac{c_{uxy}}{J_{xy}} (u^2_1 - u^2_2 - u^2_3 + u^2_4) \\
\frac{c_{uxy}}{J_{xy}} (- u^2_1 - u^2_2 + u^2_3 + u^2_4) \\
\frac{c_{uz}}{J_z} (u^2_1 - u^2_2 + u^2_3 - u^2_4) \end{bmatrix} - \begin{bmatrix} \frac{J_z - J_{xy}}{J_{xy}} \omega_y\omega_z \\
\frac{J_{xy} - J_z}{J_{xy}} \omega_x\omega_z \\
0 \end{bmatrix} \\
&= \begin{bmatrix} \frac{c_{uxy}}{J_{xy}} (u^2_1 - u^2_2 - u^2_3 + u^2_4) \\
\frac{c_{uxy}}{J_{xz}} (- u^2_1 - u^2_2 + u^2_3 + u^2_4) \\
\frac{c_{uz}}{J_z} (u^2_1 - u^2_2 + u^2_3 - u^2_4) \end{bmatrix} - \begin{bmatrix} (\frac{J_z}{J_{xy}} - 1) \omega_y\omega_z \\
(1 - \frac{J_z}{J_{xy}}) \omega_x\omega_z \\
0 \end{bmatrix} \end{aligned} $$

and with $c_{uxy} = c_u$ and $c_{uz} = k_z c_{u}$:

$$ \begin{aligned} \dot{\mathbf{\omega}} &= \begin{bmatrix} \frac{c_u}{J_{xy}} (u^2_1 - u^2_2 - u^2_3 + u^2_4) \\
\frac{c_u}{J_{xz}} (- u^2_1 - u^2_2 + u^2_3 + u^2_4) \\
\frac{k_z c_u}{J_z} (u^2_1 - u^2_2 + u^2_3 - u^2_4) \end{bmatrix} - \begin{bmatrix} (\frac{J_z}{J_{xy}} - 1) \omega_y\omega_z \\
(1 - \frac{J_z}{J_{xy}}) \omega_x\omega_z \\
0 \end{bmatrix} \end{aligned} $$

where

$$ \begin{aligned} T &= \sum f_i = c_t \sum \omega^2_i = c_{ut} \sum u_i \\
\tau &= \begin{bmatrix} l_y (f_1 - f_2 - f_3 + f_4) \\
l_x (- f_1 - f_2 + f_3 + f_4) \\ \kappa (f_1 - f_2 + f_3 - f_4) \end{bmatrix} = \begin{bmatrix} l_y c_\tau (\omega^2_1 - \omega^2_2 - \omega^2_3 + \omega^2_4) \\
l_x c_\tau (- \omega^2_1 - \omega^2_2 + \omega^2_3 + \omega^2_4) \\ c_\kappa (\omega^2_1 - \omega^2_2 + \omega^2_3 - \omega^2_4) \end{bmatrix} \
&= \begin{bmatrix} c_{u x} (u^2_1 - u^2_2 - u^2_3 + u^2_4) \\
c_{u y} (- u^2_1 - u^2_2 + u^2_3 + u^2_4) \\
c_{u z} (u^2_1 - u^2_2 + u^2_3 - u^2_4) \end{bmatrix} \end{aligned} $$

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