# Quadrotor dynamics: How is torque created

I'm a software engineer and I've recently engaged into Robotics field. I've started by a course on Coursera which talks about Aerial robotics and thus quad-rotors. I'm trying to understand how quad-rotors dynamic works and i'm struggling understanding some key points.

I'm going to explain and make some assumptions based on what i've understood by studying till now, and will ask a specific question at the end, please correct me if i'm wrong.

A quad-rotor is a robot of 6 degrees of freedom, which involved translations in the 3 different axis (X, Y, Z) and rotating over the same axis, which defines the 3 famous movements: Roll, Pitch and Yaw.

The quad-rotor has 4 motors rotating with specific angular velocities, and the previously discussed motions are achieved by regulating these angular velocities. More specifically, when rotating, the motor blades create two type of forces, commonly known as Thrust and Torque. The Thrust permit the robot to maintain a specific altitude, however the Torque is used to allow the robot to rotate.

My question is: how are these forces generated ? what's the physical explanation behind it ?

My assumptions are as follow:

1- The Thrust is generated because of the existence of a flowing air which flows from the bottom to the top passing through the motor's blade due to the high velocity of this latter

2- The b is generated due to the existing of Thrust and depends on the blade's length. What i basically understood is that, when an object is rotating over an axe of rotation, a moment is generated (also denoted Torque when the involved quantity is a force), which is no more than the cross-product of both the force and the length between a considered point of that object and the axis of rotation. (Here i'm assuming this force is the Thrust, but my mind is doubting on my assumption, correct me if i'm wrong)

To illustrate my assumptions, here's a picture depicting what i said:

From this picture you can see the flowing air, creating the Thurst, and the Torque represented on the blade's tip, created by the previous Thrust and depends on the blade's length (or radius). More specifically:

$$\overrightarrow{Torque} = \overrightarrow{Thrust} * \overrightarrow{Blade's length}$$

The above operation is the Vector Cross-Product

One more question, is about the Torque's direction, i've been reading across multiple sources that it should be in the inverse direction of the angular velocity, but i don't exactly know why.

Thanks and sorry for the long post.

You're right that each spinning prop creates both a force and a torque.

The force is created opposite the direction of the flowing air; each action has an equal and opposite reaction. The same way you can push away from a wall by pushing onto it, the propeller pushes "away" from the air (up) by pushing the air down.

That same force also exerts a torque on the body of the quadrotor, since typically the line of action does not go through the center of mass. This is separate from the torque generated by the propeller itself. The torque about the body center of mass generated by the force is used to tilt the quadrotor and initiate motion in the horizontal plane. When the quadrotor is hovering, each of the four rotors is carefully balancing the torques they generate.

The torque generated by the propeller itself is, as @billmcc mentioned, produced in opposite direction to the torque exerted by the motor on the prop (the same equal and opposite reaction principle as with the force). This torque is around the axis of the motor shaft, and tends to spin the body of the quadrotor around its z-axis. When the rotor is spinning at a constant angular velocity, this torque is needed to fight the drag the propeller experiences as it moves through the air, as well as friction from the motor itself. If there were no air and no friction, a prop spinning at constant angular velocity would not generate any torque on the body.

So the torque generated by the propeller is not quite coming from the force generated by the propeller, but incidentally, many quadrotor controllers model this relationship linearly: Torque = C * Thrust. This tends to hold roughly after empirical analysis. By your "r x F" reasoning and diagram, the torque would be generated coming out of the screen, and since the prop is rotating, the torque on one side is equally cancelled by an opposite torque on the other side.

There is a second type of torque generated around the z-axis when the rotor speed is changing due to the conservation of angular momentum. This torque is proportional to the rate of change of the angular velocity of the rotor, but is typically ignored in quadrotor controllers since it's quite small for small propellers relative to the other torque (the torque due to propeller drag).

Hope this helps and let me know if you have questions!

• Thanks for such a detailed explanation. Let me summaries your informations and correct me if i'm wrong. We can basically distinguish 2 influencing Torques, and an additional one which is ignored because of its small effect. The first Torque is generated due to the motor's Thrust, happens around the center of mass and depends on the Thrust magnitude as well as the distance between the center of mass and the rotor. The second Torque is around one rotor's axis of rotation and its magnitude is calculated empirically. However i'm not very sure of the origin of this latter.
– Mssm
Feb 15, 2021 at 22:50
• Yeah that's right, the origin of the second torque you describe (around the rotor's axis of rotation) is the aerodynamic drag and friction as the rotor spins. This torque is equal in magnitude and opposite in direction to the torque the motor applies to the rotor. The helicopter example is a good one. Another one is a boat propeller, the boat motor needs to apply torque to spin the propeller through the water, and the boat will tend to rotate in the opposite direction (although for a boat this is in practice negligible).
– Alex
Feb 16, 2021 at 3:48
• @Mssm - Sit in an office chair, pick your legs up off the ground. Then spin. You can't. In order to spin yourself, you must push off against the ground. In the case of the quadcopter, the motor must push off the frame of the quadcopter. A force or a torque can only exist between objects. The same torque that is spinning a blade is also trying to spin the motor casing. If you held the quadcopter by the blade, the frame would spin. You don't notice the reaction force you're applying to the Earth because your torque is negligible relative to the moment of inertia of the Earth. Feb 16, 2021 at 17:36
• Thank you very much guys, i think i have now a better idea about the origin of the blade's torque. I was not aware of that "Aerodynamic drag" force .. after making a research about it, i found out that if an object is moving in a fluid with a specific velocity (either angular or linear) a force will be generated going exactly in the opposite direction of the velocity. A running car is a good example about it. So here i can assume the Torque applied on the blade is the fruit of the blade's rotation in the fluid (here it's the air) that depends on its the angular velocity, and that makes sense !
– Mssm
Feb 16, 2021 at 19:56

Each motor generates torque to spin it's fan. As the fan turns, the quadrotor will experience the same torque on the frame, only in the opposite direction.

This is similar to a helicopter - the tail rotor is needed to counteract the torque from the main engine. If the tail rotor fails, the helicopter will start to rotate in the opposite direction from it's airfoil.

This is why drones always have an even number of fans - half of them spin counterclockwise and half spin clockwise to keep the torque balanced.

The fans convert most of the torque into thrust by pushing air down. The relationship between torque and thrust isn't going to be a simple linear relationship as you suggest. Fan/propeller performance is usually modelled empirically.