Quadcopter¶

The quadcopter is a rigid body with 6 degrees of freedom. It can move in the x, y, and z directions, and it can rotate about the x, y, and z axes. The motion of the quadcopter is governed by the following equations:

$$ \begin{aligned} \dot{x} &= v_x \\ \dot{y} &= v_y \\ \dot{z} &= v_z \\ \dot{v}_x &= \frac{1}{m}(T_1 - T_3) \\ \dot{v}_y &= \frac{1}{m}(T_2 - T_4) \\ \dot{v}_z &= \frac{1}{m}(T_3 + T_4 - T_1 - T_2) \\ \dot{\theta}_x &= \omega_x \\ \dot{\theta}_y &= \omega_y \\ \dot{\theta}_z &= \omega_z \\ \dot{\omega}_x &= -\frac{k_x}{I_x}(\theta_x - \theta_r) \\ \dot{\omega}_y &= -\frac{k_y}{I_y}(\theta_y - \theta_r) \\ \dot{\omega}_z &= -\frac{k_z}{I_z}(\theta_z - \theta_r) \end{aligned} $$

where:

• x, y, and z are the position of the quadcopter in the global coordinate system
• $\theta_i$ are the angles of rotation of the quadcopter about the x, y, and z axes
• $\omega_i$ are the angular velocities of the quadcopter about the x, y, and z axes
• $T_i$ are the thrust forces on the four rotors
• $k_i$ are the stiffness coefficients of the rotors
• $\theta_r$ is the desired angle of rotation of the rotors