robotics//drone//multirotor//propeller thrust
Propeller thrust is the force a spinning propeller produces along its axis, and its model, thrust proportional to the square of rotor speed, is how every multirotor autopilot converts the forces it wants into the motor speeds it commands. A propeller accelerates air downwards; the air pushes back. Double the speed and both the mass of air moved per second and its velocity double, so the force goes up four times:
Propeller thrust is the force a spinning propeller produces along its axis, and its model, thrust proportional to the square of rotor speed, is how every multirotor autopilot converts the forces it wants into the motor speeds it commands. A propeller accelerates air downwards; the air pushes back. Double the speed and both the mass of air moved per second and its velocity double, so the force goes up four times:
T=kT ω2,Q=kQ ω2,T=k_T\,\omega^2,\qquad Q=k_Q\,\omega^2,T=kTω2,Q=kQω2,
with TTT the thrust, QQQ the drag torque the propeller exerts back on the frame, ω\omegaω the rotor speed and kTk_TkT, kQk_QkQ coefficients measured on a thrust stand for that motor and propeller. The torque term is what lets a quadcopter yaw: two propellers spin each way, and making one pair faster than the other leaves a net QQQ on the frame (motor mixer).
Thrust does not follow a command instantly. The rotor has inertia, so the motor needs time to change its speed: the brushless motor and propeller behave as a first-order system with a time constant of 10 to 100 ms depending on size. That lag sits inside the fastest loop of the aircraft and costs phase, and it is why large drones, with heavy propellers, cannot be tuned as aggressively as small racing frames.
The square law turns small voltage changes into large thrust changes. Rotor speed is roughly proportional to battery voltage, so a cell sagging from 4.2 to 3.6 V costs about a quarter of the thrust for the same command; and a fixed-pitch propeller cannot pull downwards, so the minimum thrust is positive and a falling drone can only stop pushing up.
The coefficients are measured, and the test decides whether they can be. Fitting T=aω2+bωT=a\omega^2+b\omegaT=aω2+bω with data taken only between 6,000 and 7,000 rpm gives huge coefficients of opposite sign, because over that band ω2\omega^2ω2 and ω\omegaω move together and the least squares problem cannot separate them; the fix is to sweep the full speed range, which makes it a matter of experiment design that no change to the fitting code can solve.
The real thrust drifts from the bench value. Air density (altitude, temperature), forward speed, the ground effect near the floor and a nicked blade all change kTk_TkT, which is why autopilots estimate the hover thrust in flight instead of trusting the stand.
The model is quadratic, and controllers designed by linearization work around the hover speed; far from it, at full climb or near idle, the slope differs and gains tuned at hover feel different.
The motor and its driver are the ESC and the brushless motor; what all actuators share (limits, lag, dead zones) is in actuator.