GoKawiilDrone physics
A friendly drone capable of awesome feats of aerodynamics.
This article describes drone physics. Audience should have familiarity with introductory linear algebra, introductory calculus, and introductory classical mechanics. This work was adapted from my research in adaptive control and publication at AIAA DASC 2023 about multirotor , a python simulation framework for drones. The notation here borrows heavily from the excellent work by Charles Tytler.
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The following topics are covered, in order:
The coordinate systems used to describe the vehicle. The state variables used to describe the vehicle. The forces and torques acting on the body. Equations of motion (linear and angular). How motors and propellers generate forces and torques. How control systems determine propeller speeds.
Describing the vehicle
A multi-rotor UAV is modeled with six degrees of freedom: the three linear axes for linear motion: $x, y, z$, and the three angular axes for rotational motion: $\phi, \theta, \psi$. To use coordinates, a coordinate system needs to be agreed upon. Typically the the North-East-Down (NED) system is used. The direction of the positive $x$ axis is considered “forward”/North orientation. The positive $y$ axis is “right”/East and the positive $z$ axis is “down”.This is a right-handed coordinate system; positive rotation about an axis in the direction of the thumb is in the direction of the curled fingers. For example, a positive z-rotation goes from +x to +y.
Two reference frames are used for representing the state of the body:
Nominal, inertial, reference frame $n$ is the static frame of reference where the axes are aligned with arbitrary, global directions. They are represented as column vectors $\hat{n} = [\hat{x}^n, \hat{y}^n, \hat{z}^n]^T$. Body-fixed, non-inertial, reference frame $b$ has the axes aligned with respect to the center of gravity of the rigid body in motion. They are represented as column vectors $\hat{b} = [\hat{x}^b, \hat{y}^b, \hat{z}^b]^T$. The body frame moves and rotates with the vehicle. Consider the origin of the body frame attached to the center of mass of the drone.
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