How is it possible to maintain the total thrust when controlling yaw of a quadcopter?

Question

I'm working on the control of a quadcopter and I'd like to understand how come controlling the yaw does not increase the overall thrust. My understanding is that the control is carried out through 2 PIDs per axis (roll, pitch and yaw). The output of the last PID is sent as a PWM signal to correct the rotor speeds of the propellers. The mixing looks something like that:

$T_{FrontLeft} = thrust + roll_{pid} + pitch_{pid} + yaw_{pid}$ $T_{FrontRight} = thrust - roll_{pid} + pitch_{pid} - yaw_{pid}$ $T_{RearLeft} = thrust + roll_{pid} - pitch_{pid} - yaw_{pid}$ $T_{RearRight} = thrust - roll_{pid} - pitch_{pid} + yaw_{pid}$

All the quadcopter controls seem to work that way from what I could gather. So the basic idea to control yaw is to add $yaw_{pid}$ to the clockwise motors and substract the same amount $yaw_{pid}$ to the counterclockwise motors to make the quadcopter turns clockwise. Which translates into a increase of speed of clockwise motors and a decrease of speed for counterclockwise motors from the same amount.

But we know that each motor produces thrust and torque according to those equations:

$T = C_T\rho n^2 D^4$

$Q = C_Q\rho n^2 D^5$

where $T$ is thrust, $Q$ is torque, $C_T$ and $C_Q$ are system dependent constants, $ρ$ is the air density, $n$ is rotor speed, and $D$ is rotor diameter. Which means that the thrust produced by each motor is proportional to the propeller speed squared.

So if $n$ is the speed of all propellers before correction, the thrust of the clockwise propellers after correction will be proportional to $(n+\Delta)^2$ and the thrust produced by the counterclockwise propellers to $(n-\Delta)^2$. The total thrust for these 2 propellers will be proportional to:

$(n+\Delta)^2 + (n-\Delta)^2 = 2n^2 + 2\Delta^2$

As you can see, there is an increase of $2\Delta^2$ in the overall thrust produced by those 2 propellers (and $4\Delta^2$ when we take the 4 propellers into account). Of course, in real life, when we control the yaw the quadcopter does not go up.

So what am I missing?

(the same stands for roll and pitch control but since the quadcopter turns around the roll or pitch axis, the total thrust is no longer entirely on the vertical axis and I could imagine that the projection on the vertical axis is not increasing, but that does not work with yaw)

Answer 1

You are correct that theoretically total thrust can not be maintained. However, about hover, it can be assumed that the torque is small and that total thrust can be maintained.

In control, we are usually interested in linear systems. We have more tools and knowledge for dealing with linear systems. In most cases when we are dealing with a nonlinear system we linearize the system around some operating point with the assumption that it will adequately describe the system around that operating point.

In this case, people have linearized the nonlinear equations that describe the quadrotor dynamics. These equations drop the $2\Delta^2n$ term.