What can four motors control?
Ignoring drag, the four rotor thrusts point in the same direction, fixed to the body. With the craft level,
↑ T
● center of mass
↓ mg
$m$ is mass and $g$ is gravitational acceleration. Height $z$ is positive upward.
\[m\ddot z = T-mg,\qquad T=T_1+T_2+T_3+T_4.\]Hover requires $T=mg$. To accelerate sideways, the craft must tilt. For a tilt $\theta$ toward a horizontal direction $x$, thrust has horizontal and vertical components.
\[\begin{aligned} m\ddot x &= T\sin\theta, \\ m\ddot z &= T\cos\theta-mg. \end{aligned}\]To keep the same height,
\[T=\frac{mg}{\cos\theta},\qquad \ddot x=g\tan\theta.\]More tilt gives more horizontal acceleration, but requires more thrust to hold height. Keeping $T=mg$ while tilting makes the craft accelerate downward.
Changing the tilt takes torque. For rotation about one symmetry axis,
\[I\ddot\theta=\tau,\]where $I$ is the moment of inertia about that axis. A torque starts the roll. An opposing torque stops the rotation at a bank angle. The craft keeps accelerating sideways at that angle.
Leveling removes horizontal acceleration. The sideways velocity remains. Banking the other way slows it down.
The craft has six degrees of freedom, three position coordinates and three orientation angles. Its four inputs produce total thrust and three torques. At a fixed attitude, it cannot choose a sideways thrust independently.
The required acceleration fixes the thrust vector. For nonzero thrust, rotation about that vector remains free. This is why position and attitude cannot follow arbitrary independent trajectories. [0]
The motor thrusts must supply both the total thrust and the three torques. For a symmetric X frame, viewed from above,
front
1 2
front left front right
\ /
●
/ \
rear left rear right
3 4
Positive roll lowers the right side. Positive pitch raises the nose. Positive yaw turns the nose right.
More thrust on the left starts a positive roll. More at the front starts a positive pitch. Yaw comes from propeller reaction torque. Assume 1 and 4 spin clockwise viewed from above, and 2 and 3 counterclockwise. Increasing 2 and 3 while decreasing 1 and 4 produces positive yaw torque.
The four mixing patterns add together.
\[\begin{aligned} T_1 &= C+R+P-Y, \\ T_2 &= C-R+P+Y, \\ T_3 &= C+R-P+Y, \\ T_4 &= C-R-P-Y. \end{aligned}\]$C=T/4$. $R$, $P$, and $Y$ are thrust increments for roll, pitch, and yaw. All four quantities are in newtons. Converting a requested torque to these increments depends on arm geometry and propeller reaction torque. [1]
Each differential pattern keeps total thrust unchanged. But if all four rotors are already at maximum thrust, one pair cannot increase. Producing roll torque then requires reducing the other pair, reducing total thrust too.
A sideways maneuver requires a thrust history $T(t)$ and tilt history $\theta(t)$. Changing the tilt requires torque $I\ddot\theta(t)$. After mixing, every rotor must be able to produce its assigned thrust.
[0] Lee, Leok, and McClamroch, Geometric Tracking Control of a Quadrotor UAV on SE(3) (2010). Section III explains why changing the thrust direction requires changing attitude. Equation (14) constructs the desired thrust axis. ↩︎
[1] Section II, equation (1), of the same paper maps rotor thrusts to total thrust and three torques. Its rotor layout and numbering differ from ours. Miniflight’s AIGP mixer uses the signs above with scaled control efforts, not calibrated thrusts or physical torques. ↩︎