Drone 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.Join the discussion on X, Reddit, HackerNewsFor more interactive explainers, see Poor Man’s Autograd and Surprise! A Derivation of Entropy.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 vehicleA 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.Linear representationThe body-fixed reference frame $b$ is affixed to the center of mass of the vehicle. That is, the displacement of the vehicle from the body frame is always 0. The position of the vehicle is represented by the displacement of the body frame from the inertial frame, $\hat{r}^n$.The velocity of the vehicle is the velocity of the body frame relative to the inertial frame. Here, we choose to represent velocity in coordinates of the body frame.Angular representationOrientation of a body in the inertial reference frame follows the Tait-Bryan angles convention. That is, orientation can be described by three sequential rotations: yaw ($\psi$), pitch ($\theta$), and roll ($\phi$) - in that order. The order of rotations matters. Starting from the inertial frame $\hat{n}$, yaw $\psi$ is rotation $R(\psi)$ of the body frame about the inertial $z$ axis. Starting from this yaw-ed frame $\hat{n}_\psi$, pitch $\theta$ is rotation $R(\theta)$ about the new $y$ axis. And starting from this yaw-ed and pitch-ed frame $\hat{n}_{\psi,\theta}$, roll $\phi$ is the final rotation $R(\phi)$ about the new $x$ axis. The final product, $\hat{n}_{\psi,\theta,\phi}$ is the body reference frame.The angular velocity of the vehicle, $\hat{w}^T=[\omega_x,\omega_y,\omega_z]$, is the instantaneous rotational rate of the body frame about itself. Whereas each angle of orientation is defined in its own reference frame ($\hat{n}, \hat{n}_{\psi}, \hat{n}_{\psi,\theta}=\hat{b}$) during a sequence of ordered rotations from the inertial axes to the body frame (yaw, then pitch, then roll).Roll rotation happens last and the result is the body frame. Therefore, $\omega_x$, the rate about the x-axis equals the roll rate, $\dot{\phi}$. Pitch happens after the initial yaw, but before roll. Therefore, $\omega_y$, the rate about the y-axis, is the pitch rate in the frame $\hat{n}_{\psi}$ rotated by roll $\phi$ to get the the body frame. Finally, yaw rotation happens first in the inertial frame.

Therefore, $\omega_z$, the rate about the body frame’s z-axis, is the yaw rate in the inertial frame rotated by pitch $\theta$ and roll $\phi$. This can be expressed as a matrix equation:$$ \begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z \end{bmatrix} = R(\phi)\cdot R(\theta) \begin{bmatrix} 0 \\ 0 \\ \dot{\psi} \end{bmatrix} + R(\phi) \begin{bmatrix} 0 \\ \dot{\theta} \\ 0 \end{bmatrix} + \begin{bmatrix} \dot{\phi} \\ 0 \\ 0 \end{bmatrix} $$$$ \begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z \end{bmatrix} = \begin{bmatrix} \dot{\phi} + \dot{\psi} \sin{\theta}\\ - \dot{\psi} \sin{\phi} \cos{\theta} + \dot{\theta}\\ \dot{\psi} \cos{\phi} \cos{\theta} \end{bmatrix} $$Reconciling inertial and body framesA body frame may be different from the intertial frame due to (1) displacement and (2) rotation. The body frame’s origin is fixed to the origin of the UAV. Therefore, the displacement of the body frame from the inertial frame is the inertial position of the UAV: $\hat{r}^n=[x,y,z]^T$.A vector relative to the origin of the body frame will appear rotated if displaced to the origin of the inertial frame. Given a vector in the inertial frame $\hat{\mathcal{V}}^n=[x^n,y^n,z^n]^T$ and the same vector at the origin of the body frame $\hat{\mathcal{V}}^b=[x^b,y^b,z^b]^T$, the rotation matrix from the body to inertial reference frames $R_b^n$ is defined as, where each rotation matrix is:$$ \begin{align} R(\phi) =

\begin{bmatrix} 1 & 0 & 0 \\ 0 & c\phi & -s\phi \\ 0 & s\phi & c\phi \end{bmatrix} \end{align} $$$$ \begin{align} R(\theta) = \begin{bmatrix} c\theta & 0 & s\theta \\ 0 & 1 & 0 \\ -s\theta & 0 & c\theta \end{bmatrix} \end{align} $$$$ \begin{align} R(\psi) = \begin{bmatrix} c\psi & -s\psi & 0 \\ s\psi & c\psi & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{align} $$$$ \begin{align} \hat{\mathcal{V}}^b &= R(\phi)\cdot R(\theta) \cdot R(\psi) \cdot \hat{\mathcal{V}}^n \\ \hat{\mathcal{V}}^b &= R_n^b \hat{\mathcal{V}}^n \\ \begin{bmatrix} x^b \\ y^b \\ z^b \end{bmatrix} &= \begin{bmatrix} c\psi c\theta & - s\psi c\theta & s\theta\\ s\phi s\theta c\psi + s\psi c\phi & - s\phi s\psi s\theta + c\phi c\psi & - s\phi c\theta\\ s\phi s\psi - s\theta c\phi c\psi & s\phi c\psi + s\psi s\theta c\phi & c\phi c\theta \end{bmatrix} \begin{bmatrix} x^n \\ y^n \\ z^n \end{bmatrix} \end{align} $$Here $c | s$ of $\phi | \theta | \psi$ refer to the cosine and sine respectively. This rotation is useful when looking at forces acting in the body frame from an inertial viewpoint. The above transforms a vector instantaneously between rotated reference frames.Reconciling derivatives in rotating framesIf, however, the vector is changing while the reference frame is rotating, then the rate of change is not as simple. The rate of change of a vector is due to (1) the change of the vector quantity itself in the body frame, and (2) the change of the frame coordinates.

That is, in a body frame rotating with instantaneous angular velocity $\hat{\omega}=[\omega_x,\omega_y,\omega_z]$ about its axes, the time-derivative of a vector in the body frame $\hat{\mathcal{V}}^b = \hat{b} \cdot \hat{\mathcal{V}}$ as measured in a co-located inertial frame is given by the Transport theorem:$$ \begin{align} \frac{d \hat{\mathcal{V}}^b}{d t} &= \hat{b} \cdot \frac{d \hat{\mathcal{V}}}{d t} + \frac{d \hat{b}}{d t} \cdot \hat{\mathcal{V}} \\ &= \hat{b} \cdot \frac{d \hat{\mathcal{V}}}{d t} + \hat{\omega} \times \hat{\mathcal{V}}^b \\ &= \hat{b} \cdot \frac{d \hat{\mathcal{V}}}{d t} + \begin{bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{bmatrix} \hat{\mathcal{V}} \end{align} $$Transport theorem and the cross productAssume the body frame has instantaneous angular velocity $\hat{\omega}$ about each axis. Each basis vector will see a rotation over a time interval $dt$. For small values of $dt$, the angular displacement is small, $\hat{\omega} dt$, and the arc drawn by the tip of the unit vector can be approximated as a straight line of magnitude $1 \cdot \hat{\omega} dt$ (arc length equals radius times angle in radians):$$ \hat{x}^b \rightarrow \hat{x}^b + 0\hat{x}^b + \omega_z dt \hat{y}^b - \omega_y dt \hat{z}^b \\ \hat{y}^b \rightarrow \hat{y}^b - \omega_z dt \hat{x}^b + 0 \hat{y}^b + \omega_x dt \hat{z}^b \\ \hat{z}^b \rightarrow \hat{z}^b + \omega_y dt \hat{x}^b - \omega_x dt \hat{y}^b + 0 \hat{z}^b $$In

matrix form, each column vector represents the change to the basis vector: $$ \hat{b}_{t+dt} = \hat{b}_{t} + \begin{bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{bmatrix} dt $$The rate of change of the basis vectors of the reference frame is:$$ \frac{d \hat{b}}{dt} = \begin{bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{bmatrix} $$The matrix multiplication is equivalent to a cross product with the vector $[\omega_x, \omega_y, \omega_z]^T$.The additive term is a fictitious force that accounts for non-inertial frame. Once that is accounted for, the dynamics can be solved for like an inertial frame. This can be used to integrate accelerations in the body frame to find body-frame velocity.The state of the vehicleThe vehicle dynamics are completely represented by 12 state variables: linear and angular displacement in each dimension, and their time derivatives:$\hat{r}^n=[x,y,z]^T$ are the navigation coordinates in the inertial frame. This is the linear displacement of the body frame from the inertial frame.$\hat{v^b}=[v^b_x, v^b_y, v^b_z]^T$ is the velocity of the vehicle along the body frame axes.$\hat{\Phi}=[\phi, \theta, \psi]^T $ is the orientation of the body reference frame $b$ in Euler angles (roll, pitch, yaw) with reference to the inertial reference frame.$\hat{\omega}=[\omega_x, \omega_y, \omega_z]^T$ is the instantaneous angular velocity along each of the body frame axes.Dynamics of multi-rotor UAVsThe previous section enumerated the variables that describe the vehicle. This section describes how variables change over time. Since state is split into linear and angular variables, the dynamics equations will be split into linear and rotational equations.