What is the intuition of the math of the aerodynamics?

[PrasadNR]

I am creating this question as several people have asked this question to me.

https://github.com/PrasadNR/Webots-Quadcopter-Python-SITL/blob/master/controllers/mavic2proController/mavic2proController.py#L93-L96

front_left_motor_input = float(params["k_vertical_thrust"]) + vertical_input - roll_input - pitch_input + yaw_input
front_right_motor_input = float(params["k_vertical_thrust"]) + vertical_input + roll_input - pitch_input - yaw_input
rear_left_motor_input = float(params["k_vertical_thrust"]) + vertical_input - roll_input + pitch_input - yaw_input
rear_right_motor_input = float(params["k_vertical_thrust"]) + vertical_input + roll_input + pitch_input + yaw_input

I did not create this equation by myself; these were derived from the C++ equations I saw in Webots software https://github.com/cyberbotics/webots/blob/master/projects/robots/dji/mavic/controllers/mavic2pro/mavic2pro.c#L173-L180 (where I saw these equations without explanations or comments):

const double front_left_motor_input = k_vertical_thrust + vertical_input - roll_input - pitch_input + yaw_input;
const double front_right_motor_input = k_vertical_thrust + vertical_input + roll_input - pitch_input - yaw_input;
const double rear_left_motor_input = k_vertical_thrust + vertical_input - roll_input + pitch_input - yaw_input;
const double rear_right_motor_input = k_vertical_thrust + vertical_input + roll_input + pitch_input + yaw_input;
wb_motor_set_velocity(front_left_motor, front_left_motor_input);
wb_motor_set_velocity(front_right_motor, -front_right_motor_input);
wb_motor_set_velocity(rear_left_motor, -rear_left_motor_input);
wb_motor_set_velocity(rear_right_motor, rear_right_motor_input);

The intuition is like the one below:
k_vertical_thrust is for the quadcopter to just stay in the air without losing altitude. So, just for this discussion, we can assume that the quadcopter just stays in the air even when k_vertical_thrust = 0 (may be by magnetic fields that lets quadcopter "hover" without having to have any air-power). Then, if M₁, M₂, M₃ and M₄ are the four motors of the quadcopter, we can conveniently represent the 4 channel input mapped to these four motors' inputs.

[PrasadNR]

latex_source.zip
LaTeX source if any person is not able to view the image properly.