Quadrotor Dynamics and Motor Mixing

A quadrotor has no control surfaces. It steers only by changing the speeds of four propellers. Spinning all four faster lifts it. Speeding up one side tilts it. And because neighboring propellers spin opposite ways, speeding up one diagonal pair turns it.

Thrust Roll command Pitch command Yaw command

Drag the 3D view to look from another angle.

The mixer

For an X-shaped quadrotor with motors front-left, front-right, rear-right, and rear-left, the commands mix into motor outputs like this, with each motor clamped to its range:

$$\begin{bmatrix} u_{FL} \\ u_{FR} \\ u_{RR} \\ u_{RL} \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & -1 \\ 1 & -1 & 1 & 1 \\ 1 & -1 & -1 & -1 \\ 1 & 1 & -1 & 1 \end{bmatrix} \begin{bmatrix} T \\ \tau_{\text{roll}} \\ \tau_{\text{pitch}} \\ \tau_{\text{yaw}} \end{bmatrix}$$

Each propeller's thrust and its drag torque both grow with the square of its spin rate, $f = k_f\,\omega^2$ and $\tau = k_m\,\omega^2$.

What to notice

Roll speeds up the left pair and slows the right pair, tipping the drone right. Yaw speeds up one diagonal pair and slows the other: total thrust stays the same, but the propeller drag torques no longer cancel, so the body turns. Push thrust near the top and some motors hit their limit, so the commands can no longer all be met. That saturation is why drones lose control authority at full throttle. To move sideways, a quadrotor must first tilt, which is why its position control runs through its attitude control.