Quadrotors

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Quadcopters, also known as quadrotors, are a special type of aerial vehicle (or drone) that consist of four driven propellers placed in a square formation with equal distance from the center of mass of the vehicle. The propellers in this particular formation allows the drone to move about in three dimensional space through select control of the rotation speed (or angular velocities) of the individual propellers. Quadcopters are commonly used as small unmanned aerial vehicles (UAV) due to their low cost and simple structure in applications including, but certainly not limited to, surveillance, search and rescue, and small material delivery.

Before we begin on the basics of quadcopter modeling and flight control we must first understand the components that define a quadcopter. As mentioned in the introduction, quadcopters are a special type of rotorcraft comprised of four propellers arranged in a square formation. Each propeller is driven by an individual electric motor that is controlled by a small computer called a Flight Control Unit (FCU). The FCU is a special type of computer that is responsible for translating user input into aerial flight using information from the vehicle’s on-board sensors and connected motors using pre-programmed flight controllers and temporal logic. The last but certainly not least component of a drone is the battery as a drone simply cannot fly without a means to power its motors and FCU. Figure 1 provides more details below.

As mentioned, the FCU is responsible for translating user input to aerial flight using pre-programmed flight controllers. However, in order to design the flight controllers we must first understand the physics that govern the aerial flight and more importantly we need a coordinate system to assist in describing such aerial flight. In Figure 2 below we find two coordinate frames, a body reference frame for the quadrotor and an inertial reference frame that describes the space in which the quadrotor moves. The quadrotor has six degrees of freedom or six ways in which it can move. We have three degrees of translational motion, up/down (zb), right/left (yb), and forward/back (xb) and three degrees of rotational motion about each aforementioned axis of translational motion as marked by ψ (yaw), θ (pitch), and φ (roll).


We have now reached the point where we can begin to describe how the quadrotor is able to move using its four motor driven propellers. Figure 3 provides a free-body diagram of the quadrotor showing the applied forces, moments, and resulting reactions on the quadrotor body in a simple hover condition. As can be seen, we have four forces Fi provided by the propellers to counteract the force of gravity given by mg where m represents the mass of the vehicle and g represents the acceleration due to gravity. Each force exerted by the individual propellers is given by a function of their rotational speed or angular velocity ωi where

Fi = kF ω 2 i

(1) Here, kF is a constant parameter that accounts for a number physical parameters relating to motor torques and aerodynamics that you don’t need to concern yourself with at the moment. In addition to the forces exerted by the propellers to counteract gravity, each motor produces a moment or torque τMi

that gives the quadrotor a tendency to spin about the zb axis or yaw (ψ). This

torque is also a function of the angular velocity of the respective propeller and is given by

τMi = kMω 2 i

(2) where kM is a physical constant parameter that can be ignored for now. To counteract this tendency, the motors in a quadrotor are configured such that the motors on opposing arms spin in the same direction while the adjacent motors spin in the opposite direction. If the motors are spinning at the same speed in their respective directions, the torques of the two motors spinning in the same direction should counteract the torques of the motors spinning in the opposite directions. This is also highlighted in Figure 3 by noting the direction of the arrows for both the angular velocities (ωi) and torque moments (τMi ).

Let’s now move on to understand how these equations and components translate to motion of a quadrotor. To start, we will begin with the one-dimensional model.