An Insight into the Dynamics and State Space Modelling of a 3-D Quadrotor

TL;DR

This paper addresses the problem of deriving a complete nonlinear dynamics description for a $3$-D quadrotor and its state-space representation. It builds a rigorous framework using rigid-body transformations and Euler angles to connect world and body frames, culminating in a $12$-state, $4$-input system with $\dot X = f(X,U)$. The key contributions are explicit translational and rotational dynamics, the inclusion of aerodynamic moments, and a full first-order formulation suitable for trajectory planning and control design. The work lays a foundation for nonlinear control of underactuated UAVs and points to quaternion-based extensions to mitigate gimbal-lock and improve computational robustness.

Abstract

Drones have gained popularity in a wide range of field ranging from aerial photography, aerial mapping, and investigation of electric power lines. Every drone that we know today is carrying out some kind of control algorithm at the low level in order to manoeuvre itself around. For the quadrotor to either control itself autonomously or to develop a high-level user interface for us to control it, we need to understand the basic mathematics behind how it functions. This paper aims to explain the mathematical modelling of the dynamics of a 3 Dimensional quadrotor. As it may seem like a trivial task, it plays a vital role in how we control the drone. Also, additional effort has been taken to explain the transformations of the drone's frame of reference to the inertial frame of reference.

Figures (6)

• Figure 1: Arbitary points $p,q$ and their corresponding vectors V,W are being mapped on a rigid body.
• Figure 2: Mapping of two rigid bodies $\{A\}$ and $\{B\}$ with different frame of references.
• Figure 3: A $x-y$ configuration in which a point from position $V$ is rotated to position $V'$.
• Figure 4: Roll($\phi$), pitch($\theta$), yaw($\psi$) in reference to a glider.
• Figure 5: Force diagram of a conventional quadrotor.