I decided to start on the, “inner loops”. By this I mean the control loops for each propeller drive system. A Brushless DC Motor (BLDC) is standard for hobby-sized quadrotors, so let’s assume the motor type.

I haven’t sized anything yet: propellers or motors, but I can get the dynamic model figured out and put together a propeller speed control design.

Below I start with basic DC motor equations from just about any undergraduate dynamics textbook. I add the Bouabdallah model for the drag term, which for a simple motor would be the back-emf alone, but here I account for propeller and gearbox drag referred to the motor shaft as you can see below.

The motor-propeller speed model is non-linear. The Taylor Series expansion and evaluation about a nominal operating point is used to linearize it. As noted below, I’ll need a plan for sliding these constants and the compensator over the speed range. I can worry about that later.

I’m sharing this here because I want to illustrate more steps in the derivation of the final equation than the Bouabdallah paper has space to do. When I read papers often a final equation is given after it is stated that, “the following equation has been linearized”, for example. That is often a, “leap” for me. I don’t like to keep reading until I work it out for myself. Here I want to share each step.

I’ll leave this post to cover derivation of the propeller-motor equation with drive voltage as the control input.

BLDC