The Physics of Quadcopter Flight

by | Apr 29, 2014 | theory | 19 comments

I believe that when embarking on any project, especially one as complex as multirotor construction and piloting, it is useful to have an understanding of the theoretical underpinnings involved. So, when building and flying multirotors, I think it is valuable to have at least a basic understanding of the physics of quadcopter flight. While it is certainly possible to simply follow a set of directions, like the ones on this site, for building and flying a multirotor, it will be much clearer, and more meaningful, if you can explain to yourself the rationale behind each step.

Now, I just want to give you one caveat before you read further:  there is a huge amount of physics involved in multirotor flight, and I am only going to skim the surface on this page. Here I am going to focus on the physics involved in maneuvering the multirotor which, as you will understand soon, involves adjusting the balance of forces acting on the craft. I am going to avoid talking about the physics of how the props generate lift, the physics involved in the multirotor’s power system, the physics of how brushless motors work, et cetera. If you are interested in getting really in-depth with the topic of multirotor physics, you will find many resources around the Internet (although information is a bit scattered, which  is one of the reasons I wrote this page).

Background

Before we dive into the physics of quadcopter flight, let’s just cover some background information to establish some basic concepts and terminology.

Multirotor Coordinate System

While discussing multirotor construction and piloting, it will certainly be useful to have a way of communicating different movements of the multirotor. Fortunately, mathematicians way back in the 1700s came up with a way of describing the orientation of rigid bodies in space. The system they developed uses a set of three angles to describe, in this case, the orientation of the multirotor around the three spacial dimensions. You have probably heard of these angles before, they are called roll, pitch, and yaw.

 

roll pitch yaw angles

To describe the orientation of the quadcopter, we use three angles: roll, pitch, and yaw.

 

  • The roll angle of the multirotor describes how the craft is tilted side to side. Rotation about the roll axis is like tilting your head towards one of your shoulders. Rolling the multirotor causes it to move sideways.
  • The pitch angle of the multirotor describes how the craft is tilted forwards or backwards. Rotation about the pitch axis is like tilting your head in order to look up or down. Pitching the multirotor causes it to move forwards or backwards.
  • The yaw angle of the multirotor describes its bearing, or, in other words, rotation of the craft as it stays level to the ground. Rotation about the yaw axis is like when you shake your head to say “no.”

There is one final bit of terminology we will use to discuss flying the multirotor, and that is throttle. Throttle simply controls the altitude of the multirotor.

The Physics of Quadcopter Flight

Steering

While flying your multirotor, it is very important to understand how the multirotor moves and how we control it. At the root of all the multirotor’s movements is the rotational speed of the motors. By adjusting the relative speeds of the motors in just the right ways, keeping in mind that the rotational speed of the motors determines how much lift each prop produces, the flight controller is able to cause the multirotor to rotate around any of the directional axes (roll, pitch, and yaw), or make the multirotor gain or lose altitude.

Roll and Pitch

To make the multirotor rotate about the roll or pitch axes, the flight controller makes the motors on one side of the multirotor spin faster than the motors on the other side. This means that one side of the multirotor will have more lift than the other side, causing the multirotor to tilt.

So, for example, to make a quadcopter roll right (or rotate about the roll axis clockwise), the flight controller will make the two motors on the left side of the multirotor spin faster than the two motors on the right side. The left side of the craft will then have more lift than the right side, which causes the multirotor to tilt.

Similarly, to make a quadcopter pitch down (rotate about the pitch axis clockwise) the flight controller will make the two motors on the back of the craft spin faster than the two motors on the front. This makes the craft tilt in the same way that your head tilts when you look down.

Quadcopters roll left or right by altering the relative speeds of the left and right motors.

Quadcopters roll left or right by altering the relative speeds of the left and right motors.

Quadcopters pitch up and down by adjusting the relative speeds of the front and back motors.

Quadcopters pitch up and down by adjusting the relative speeds of the front and back motors.

The same principles apply for multirotors with more than four motors as well.

Yaw

Controlling the multirotor’s rotation about the yaw axis is a bit more complex than controlling its rotation about the roll or pitch axes. First, let’s discuss how we prevent rotation about the yaw axis. When assembling and programming multirotors, we set up the motors so that each motor spins in the opposite direction than its neighbors. In other words, using a quadcopter as an example again, starting from the front-left motor and moving around the multirotor clockwise, the motors’ rotational directions alternate, CW, CCW, CW, CCW. We use this rotational configuration to neutralize, or cancel out, each motor’s tendency to make the multirotor rotate.

 

prop rotational directions

We configure each motor to spin in the opposite direction than its neighbors.

 

You see, when a prop spins, for example, clockwise, conservation of angular momentum means that the body of the multirotor will have a tendency to spin counter-clockwise. This is due to Newton’s third law of motion, “for every action, there is an equal and opposite reaction.” The body of the multirotor will tend to spin in the direction opposite the rotational direction of the propellers.

 

prop rotational directions all same induced yaw

Note that we do not set up our quadcopters, or any other multirotors, this way. If we were to have all the rotors spin clockwise, the multirotor would start spinning uncontrollably in a counterclockwise direction.

 

I know this might be a bit confusing, so let’s discuss helicopters as a simpler example. As you may know, helicopters have two rotors. One big main rotor responsible for lifting the aircraft, and one small rotor on the tail that adjusts how the helicopter spins. Imaging what would happen if in mid-flight, a helicopter’s tail rotor fell off the aircraft while the big main rotor kept spinning (this by the way is something we hope never happens to any helicopter pilots). You can probably imagine that the helicopter would start spinning. Well this rotation would be caused by the rotation of the propeller in the opposite direction, according to the law of conservation of angular momentum.

Bringing it all together now, each of the quadcopter’s four rotors tends to make the multirotor rotate in the opposite direction than their spin. So by using pairs of rotors spinning in opposite directions, we are able to cancel out this effect and the multirotor does not spin about the yaw axis.

So therefore, when we actually want the multirotor to rotate about the yaw axis, the flight controller will slow down opposite pairs of motors relative to the other pair. This means the angular momentum of the two pairs of props will no longer be in balance and the craft rotates. We can make the multirotor rotate in either direction by slowing down different pairs of motors.

 

rotor spin clockwise yaw

By making diametrically opposite motors (that spin in the same direction) spin at different rates, the craft can be made to yaw. In this case, the clockwise-spinning motors are faster than the counterclockwise-spinning motors, so the craft yaws counterclockwise.

Hovering/Altitude Control

Now that we understand how steering the multirotor works, let’s quickly discuss a much simpler maneuver, hovering. To make the multirotor hover, which means the multirotor stays at a constant altitude without rotating in any direction, a balance of forces is needed. The flight controller will need to counteract the force of gravity with the lift produced by the rotors.

Throwing a bit of math into the picture now, the force of gravity acting on the multirotor is equal to the mass of the multirotor times gravitational acceleration (which, as far as we are concerned, is a constant as long as we are staying on Earth). The lift produced by the multirotor is equal to the sum of the lift produced by each of the its rotors. Therefore, if the force of gravity equals the force of the lift produced by the motors, the multirotor will maintain a constant altitude.

To ascend or descend, therefore, the flight controller disrupts this balance. If the lift produced by the multirotor is greater than the force of gravity, the craft will gain altitude. If the opposite is true, that is, if the lift produced by the multirotor is less than the force of gravity acting on the multirotor, then the multirotor will fall.

Movement

So we’ve already discussed how, by adjusting the relative speeds of the motors, the flight controller can make the multirotor tilt. Well, the reason we want to be able to tilt the multirotor is that tilting the multirotor causes it to move. By tilting the multirotor in different directions, it can be made to move forward, backward, left, or right (neither altitude control nor yaw control involve tilting). For example, when the multirotor pitches down (clockwise around the pitch axis) it moves forward.

The reason the multirotor moves when it tilts is because while the multirotor is tilting, some of the lift produced by the rotor is directed horizontally while normally all of the lift is directed downward. This sideways component of the lift pushes the multirotor.

Now, you might have realized the problem that happens when we sacrifice some of the multirotor’s downward thrust to move the craft horizontally. Since less thrust is directed downward while the multirotor is tilting, multirotors tend to lose altitude while moving around. Now some flight controllers have a feature called “altitude hold” which means that the flight controller automatically adjusts the motor speeds in order to make the craft maintain a constant altitude while moving. Unfortunately, the KK2.1 flight controller used in the tutorials on this site lacks this feature. This helps keep costs down, but also means that the pilot must manually adjust the throttle to maintain altitude while maneuvering.