"Real knowledge is to know the extent of one's ignorance."
-Confucius
Some books question, other books command, and still other books present the facts to the reader. The Pleasure of Finding Things Out questioned everything, and my current research book (more of a research paper really) lays out the facts. Model, Design, and Control of a Quadcopter by Andreas Vikane Hystad, is exactly what it says it is. The paper is a compilation of useful physics, control system design, programming, and electronics, all with the goal of teaching the reader about how quadcopters work. While the paper itself is quite dry, the topics that underlie quadcopter are quite fascinating. Quadcopters themselves can be considered an intriguing engineering problem because they have 4 motors, and many degrees of freedom, making them severely under actuated. However fascinating of a control problem they pose, the beauty of a drone is also present in the vehicle’s ability to fly well. Most of a quadcopter’s design goes into its flight characteristics, and this is primarily what the paper covers. The first third of the paper is mostly related to quadcopter control theory, the second third presents effective control loops for handling a quadcopter’s control, and the final third is much more “fluffy”. That is, it discusses a windows application for controlling the quadcopter, and the 3D printed drone frame. I will admit, the mathematical concepts discussed in the paper were often lost on me, but I did my best to understand them. I have not actually thoroughly gone through every part of the paper due to time constraints (it is close to 200 pages), but I am working to comb through the pages thoroughly.
I think the best thing about reading an engineering research paper is the power that it gives you. As opposed to other pursuits, in engineering everything is geared towards actual use, so with the knowledge contained within the paper, I could feasibly design my own quadcopter control system. What makes the research paper an even more compelling read is the fact that it wastes no time in dissecting the problem at hand, translating to pure power.
One thing that I realized while reading this report is that unknown mathematical equations can be extremely daunting, and that there is a certain lexicon used in all research papers that makes them seem relatively inaccessible. However, the ideas behind many research papers similar to this are very intuitive and simple. Many people get strung up by the math, and use the fancy mathematical equations as an excuse for calling it a day and giving up on pursuing a certain line of inquiry. From my very little experience so far, it seems that if you take the equations one symbol at a time, you can dissect them fairly well. This is not always the case because some equations are foreign because they represent whole branches of math that are potentially unfamiliar, but a lot of the equations used in this paper are actually relatively simple, often using nothing greater than basic linear algebra and calculus, and in some cases, just algebra. I think the role of a deep understanding of applied mathematics is to deepen the already apparent beauty of much engineering. In any case, the moral is clear: don't assume something is incredibly complicated just because it looks like it.
So how do quadcopters work? They have 4 motors, two spinning counterclockwise, and two spinning clockwise. These motors are typically brushless, which means that the permanent magnet rotor of the motor rotates on the outside around a stationary electromagnetic stator that can be energized at different segments, attracting and repelling the external rotor. This sort of motor movement is somewhat complicated because it requires relatively complicated electrical gate switching to make it run efficiently. Generally, we want to just output a simple waveform. So we need something called an ESC (Electronic Speed Controller) to convert commands from a flight controller into electrical pulses suitable for driving a motor. Then, we also need a combination of sensors to tell the quadcopter where it is, where it is going, and how it is oriented. This is accomplished by fusing GPS, accelerometer, gyroscope, and magnetometer data together to get an accurate pose estimation. Then there is a telemetry system for communicating with a ground base station, and another wireless module for communicating with the controller of the drone. Finally, there is a usually a power regulation module that provides power to a power distribution board, both providing the correct input voltage to the flight controller, which usually runs at 3.3 V, and also supplying power to the motors.
Now I think I should tell you a little bit about what I've learned as a result of reading this. I'll make it brief.
There are a variety of different coordinate systems that a quadcopter utilizes in order to accurately describe orientation and position. The body coordinate system consists of coordinate axes fixed to the quadcopter frame. The north east down coordinate system has its origin at the quadcopter launching point, where the x axis points north, the y axis east, and the z axis downwards into the earth. The Earth centered, Earth fixed coordinate system describes location based on an origin centered at the center of the earth, with the z axis going through true north, the x axis through the equator and prime meridian, and the z axis completing the orthogonal(right angled) system. ECEF coordinates aren't so important. Then there is the geodetic coordinate system, which GPS uses, consisting of latitude, longitude, and height. We need to convert from coordinate system to coordinate system in order to get the quadcopter's orientation and position relative to the ground. This primarily means we use NED coordinates and body coordinates. How do we convert between them? Well, the total orientation of the drone can actually be represented by 3 unique angles along the x, y, and z axis called the euler angles (pronounced oiler) of the object (yaw, pitch, and roll). Combined with translational coordinates (the position of the drone), we can have position and orientation. To translate from one coordinate system to another, we then utilize something called a rotation matrix, which looks like this:
It's important to realize that matrix multiplication is noncommutative, so multiplying the rotation matrix for yaw by the matrix for pitch and then by the matrix for roll is not the same as pitch, then roll, then yaw. This basically means that if you yaw something, then roll it, then pitch it, you actually won't end up in the same orientation as if you roll something, then yaw it, then pitch it. Crazy, right?!? Okay, so now that we have our coordinate systems down, let's move on to the physics. The torque of a motor is just torque = proportionality constant * (input current - current with no load on the motor). The voltage across the motor is equal to the current through it times the internal resistance + the back electromotive force generated by the motor. The power consumed by the motor is just the voltage times the current, which eventually simplifies down to just being proportional to the torque produced by the motor times the angular velocity. The thrust of the motor is proportional to the angular velocity squared. To put it simply, by ignoring many different external factors, the torque generated by the propeller is also proportional to the angular velocity squared so then we now have a full representation of torque and thrust on the copter.
We can then take drag and gravity into account in our kinetic model, and derive a full kinematic model of the quadcopter motion with a given moment of inertia vector about the different quadcopter axes. We can use the different rotation matrices we have to convert the pose of the multicopter into any coordinate frame we like. But there's an issue! We need sensor readings to be able to actually figure out our pose in the first place. To make matters worse, these sensors often have tons of unwanted noise. We can eliminate some of this by combining a low pass and high pass filter into a complementary filter and passing accelerometer as well as gyroscopic data into it. The magnetometer can be used to help determine the yaw state of the aircraft. I'll be honest, these filters are quite challenging to understand mathematically, since they are closely related to LC(inductance/capacitance) and RC(resistance/capacitance) circuitry. I'm trying to wrap my head around it. I won't go into the extended kalman filter, as it looks to be more complicated, and I don't really understand it right now. Suffice to say that these filters remove the noise from the sensors. We then are able to pass this information as input into a control loop that accounts for the different sensor readings, and potential error, and provides an output to the motors based off of the estimated position, and also user input. That's where all the different sensor readings and user inputs come together!
Reading Model, Design, and Control of a Quadcopter was pleasurable, but pleasurable in the way that a beautifully designed rocket lifting off is. In order for a drone to fly, so many different things have to go right. An aerial system’s many components working together is itself a beautiful thing. Then there is the visceral beauty of an airborne drone cruising close above you. They exude power. In my family, I’m the weird child. My siblings are quite humanities oriented. My brother is an aspiring journalist and educator, and my sister is double majoring in Spanish and Business. One time, my siblings and I were offered the opportunity to go to either Disney World, or Cape Canaveral. My siblings immediately began shouting “Disney World! Disney World!” Then, being the odd child of the family, I yelled just as loudly “Kennedy Space Center!” In the end, I got my way, since Kennedy Space Center was more educational. As you can probably guess, I didn’t win any points with my siblings for that one. Another example of how I’m the weird child? At airports, I used to stare obsessively out the windows at the planes for sizable amounts of time (from around 4-11 years old) while my siblings occupied themselves in other ways.
Thanks to the information that this book has provided me, I feel like I have become the kid I once was, and I am more excited than ever to experiment with drones!
Works Cited
Hystad, Andreas V. Model, Design, and Control of a Quadcopter. Diss. Norwegian U of Science and Technology, 2015. N.p.: n.p., n.d. Print.