In this chapter, we will explore how quadrotors fly. This chapter will cover some basic mechanics and how to design a drone.

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Robotics: Aerial Robotics (air robot) notes (1): Introduction

Basic Mechanics

A quadrotor has four motors to support the weight of the aircraft $mg$ . Each motor will rotate and generate an upward thrust. At the same time, the rotation of the motor needs to overcome the resistance torque, so we need to observe the two quantities of thrust and resistance torque and the motor speed relationship.

If you don't know the concept of torque here, you can look here:

Author: A Reed
Link: https://www.zhihu.com/question/20260034/answer/287592720
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When studying a physical phenomenon, we hope to find out the factors that affect it, and then quantify these factors, and then summarize the laws and draw the relationship between them and the phenomenon.

For example, when we push a box of things on the ground, how much force do we need to push it?

Through experimental observation we found that the heavier the box, the harder it is to push. Also, the rougher the ground, the harder it is to push. That is to say, for Sokoban, there are two influencing factors, one is the weight M of the box itself, and the other is the roughness of the ground. A problem encountered here is how to express the degree of roughness. We try to quantify it into a coefficient k, which is called the friction coefficient. The rougher the ground, the greater the k. Then the force required to push the box:

F=kM

In this way, the force required to push the box is found. In the future, for a box of any weight, on any ground, as long as we know the coefficient of friction k, we can use this formula to calculate the force required to push it.

torque is similar. When we turn something, such as turning a screw with a wrench. Through experiments, we found that if you want to turn a screw, one is to use as much force as possible, and the other is to use a longer wrench. That is to say, there are two factors that affect whether the screw can be twisted, one is the torque F, and the other is the length L of the arm, which is the length of the wrench. How to express the effect of this twisting effect? That is, how to quantify this effect? Then there is the torque, which is expressed as:

M = F X L

The three quantities here are all vectors, and the reason why they are represented by cross product is that the operation of cross product coincides with this physical phenomenon. That is to say, the magnitude of the moment is not only related to the magnitude of the force F and the moment arm L, but also related to the angle between them. The direction of the moment is perpendicular to the plane of the force and moment arm.

If we plot thrust versus motor speed:

We can see that the relationship between thrust, $F=k_fw^2$ resistance torque $M=k_Mw^2$ and rotational speed can be regarded as a quadratic equation, where k is a coefficient, which is equivalent to the friction coefficient of the ground when we calculate the friction force generated by pushing an object horizontally.

By observing the relationship curve between thrust and rotational speed, it can be determined that the rotational speed required to produce a thrust equal to a quarter of the weight $w_0$ , and at the same time this needs to overcome the corresponding resistance torque, so how do we overcome it? By adjusting the size of the motor, the motor can generate torque to overcome the aforementioned resistance torque, which is the one in the figure $\ can$ (describing the relationship between motor torque and speed).

So, if we want to make the quadrotor UAV hover, the thrust and torque generated must satisfy the following formula:

When we know the corresponding coefficients $k_f$ and $k_M$ the position of the UAV's center of mass, we can calculate the corresponding resultant force and moment, where r is the distance from the center of mass to each rotor.

In equilibrium, the resultant force is obviously zero, and the resultant moment is also zero.

But what happens when the resultant force is instantaneously non-zero? It will get accelerated. For the sake of simplicity, let’s talk about the acceleration in the vertical direction first. When the sum of the thrust of the rotor (mainly determined by the speed of the rotor) is greater than the weight of the quadrotor, an upward acceleration will be generated, otherwise, a downward acceleration will be generated. The formula is as follows :

Dynamics and 1-D Linear Control

This part mainly discusses the movement in the vertical direction. Our purpose is to control the quadrotor robot to a required vertical height, that is, it needs to move up or down for a certain distance $x$ .

Obviously, we can find the acceleration by taking the second derivative of the displacement:

$a=\frac{\mathrm{d^2x} }{\mathrm{d} t^2}=\ddot{x}$

We denote the resultant force divided by mass by u:

$u=\frac{1}{m}[\sum_{i=1}^{4}k_fw_{i}^2+m\textbf{g}]$

We can get a second-order differential equation:

$u=\ddot{x}$

Our goal of controlling this quadrotor is actually to determine the function $u$ so that the quadrotor reaches the desired position $x$ .

As mentioned above, this is a very simple second-order linear system, and we find that the input functions $u$ as a function of time can drive to the desired position . $t$ $u(t)$ $x$ $x(t)\rightarrow x^{des}(t)$

For this purpose, an error function is defined $e(t) = x^{des}(t) - x(t)$ , which represents the amount of error between the desired trajectory and the actual trajectory,

Obviously, the larger the error (the system), the greater the deviation from the actual trajectory. What we have to do is to eliminate the error to zero, and converge to zero exponentially (faster and more stable), that is, to find the satisfaction of $u$ :

The first formula is to let the error converge to zero exponentially. For details, please refer to the link below:

Another explanation and application of PD controller

Derivation of the second formula:

$\ddot{e}(t) = \ddot{x}^{des}(t) - \ddot{x}(t) = \ddot{x}^{des}(t) - u(t)$

$\ddot{x}^{des}(t) - u(t) + K_v\dot{e}(t) + K_pe(t) =$

$u(t) = \ddot{x}^{des}(t) + K_v\dot{e}(t) + K_pe(t)$

Among them, $K_v$ is the differential gain (derivative gain), $K_p$ is the proportional gain (proportional gain), $\ddot{x}^{des}(t)$ called the feedforward term (feedforward term). When $K_v$ both and $K_p$ are positive, it is guaranteed that the error term tends to zero at an exponential rate.

The higher the proportional gain, the more resilient the system becomes and the more likely it is to overshoot. For example, the current state is 2, the desired state is 4, and the error is 2. If the current proportional gain may be adjusted to +1, then the state becomes 3 and the error is 1. If the proportional gain is increased, the adjusted amount may become + 2. The error is 0. If the proportional gain is increased, the adjustment amount may become +3, and the error becomes -1, that is, overshoot.

The higher the differential gain, the system essentially has more buffering. We can roughly understand this item as the energy decay rate. If it $K_v$ is 0, $K_p$ > 0, the system is in a state of constant oscillation. By increasing the derivative gain, we can make the system converge more smoothly. But if the differential gain is too large, the system will be in an under-damped state, and the system will converge slowly, but will not vibrate too much.

In special cases, you may consider using a more complex version of PD control, also known as PID control. For example, when the specific model of the system is not known or there are some unknown interference factors, an additional term proportional to the error integral can be introduced:

Although introducing this term would make the differential equation third order, making it difficult to solve, it would make the error term eventually zero.

Design Considerations

Thrust-to-weight ratio

In the previous analysis, we always assumed that the motor can generate the thrust required by the controller, but the thrust of the motor is limited, because the motor speed will have a peak value, which will cause the quadrotor to have a maximum thrust and maximum $T_{max}$ acceleration $a_{max}$ .

We know from the previous formula that the control input is actually determined $u$ by the thrust T and the gravity of the quadrotor . $mg$

Assuming you know $T_{max}$ , by simply adding up max thrust and weight, you get $u_{max}$ :

So now when we do PD control and PID control, $u(t)$ it is also determined by the maximum thrust:

The ratio of the maximum thrust to weight is the thrust to weight ratio . We can adjust the control performance of the quadrotor by adjusting the thrust to weight ratio.

energy consumption

If we want the quadrotor to fly, we must make the thrust produce enough thrust to resist gravity. This will generate basic power consumption. The basic power consumption of the quadrotor is: 200 W/kg. Fortunately, the energy provided by the lithium battery is basically > 200 W/kg, so it can actually be satisfied.

In addition to considering power consumption, you also need to consider the total energy carried by the battery. Most lithium batteries generate about 200 Whr/kg of electricity:

Lithium batteries are the most used now, but now lithium batteries don’t carry a lot of energy, so how we can make drones fly longer is usually to reduce weight. Generally speaking, the battery accounts for about 35% of the total weight, and the motor plus propeller accounts for about 25% of the total mass.

The sensors carried by the quadrotors also consume energy, and generally they consume more energy due to added weight than they consume to operate.

Agility and Maneuverability

This part mainly discusses how to make the quadrotor move flexibly.

Stop at maximum speed (Maximum velocity to rest)

When the quadrotor is flying forward at maximum speed, when you decide to let it stop at a certain position, you have to let it tilt back and reverse the direction of thrust so that it can slow down. But as a result, this will cause the height of the quadrotor to decrease in the vertical direction, because the thrust in the vertical direction is less than the current weight.

Turn quickly without slowing down

The other thing is to give the quadrotor the ability to turn quickly. The purpose is to let the quadrotor fly forward at the maximum speed, and then turn as fast as possible, mainly by reducing the turning radius.

motion analysis

We first analyze the movement of the quadrotor when it stops at the maximum speed. We can see in the following figure as an example that the quadrotor is tilted at this time, and the thrust generated by the quadrotor rotor is perpendicular to the quadrotor, so we will The resultant force and resultant moment are divided into two directions of z-axis and y-axis. $\phi$ If $u_1$ the angle between the quadrotor and the y-axis is Yes $u_1cos\phi$ and $-u_1sin\phi$ (the direction is opposite to the direction of motion, so it is decelerating at this time).

Then we can find the acceleration and angular acceleration of the quadrotor on the y-axis, z-axis:

$\ddot{y} = \frac{\vec{F}}{m}=-\frac{1}{m}u_1sin\phi$

in the same way $\ddot{z} = \frac{1}{m}u_1cos\phi - g$

$\ddot{\phi} = \frac{M}{I_{xx}} = \frac{u_2}{I_{xx}}$ (Angular acceleration equals torque divided by moment of inertia)

Written in linear algebra it is:

We want to accelerate and decelerate faster, spin faster. Want to:

Maximize linear acceleration $a$ : Maximize $\frac{u_{1,max}}{m}$

Maximize angular acceleration $\alpha$ : Maximize $\frac{u_{2,max}}{I_{xx}}$

Component Selection

Since we want the UAV to be able to fly flexibly and at the same time to achieve some functions (such as autonomous flight), then we need to carry some equipment to complete these functions. For this reason, we must pay great attention to the weight of these devices (from the above analysis, the weight will also affect the flexibility of flight), and power consumption (if the power consumption is too large, you have to consider a large battery, which will also affect the weight).

Effects of Size

The size of the quadrotor will also affect its flexibility. Let's look at the plan view of a quadrotor, $l$ which is the characteristic length of the quadrotor (the characteristic length usually refers to the representative length of the length of the object, which is abstracted. Here can be roughly considered the size of the quadrotor), the mass of the quadrotor is $m$ , and the radius of the propeller is $r$ :

The mass $m$ is proportional to $l$ the third power (mass is related to the volume, and the volume is equivalent to the length multiplied three times), and the moment of inertia is proportional to the $l$ fifth power (the moment of inertia is equal to the mass multiplied by the square of the length):

$m \sim l^3$ $I \sim l^5$

Thrust $F$ is proportional to the area turned by the propeller times the square of the linear velocity of the rotor tip:

$F \sim \pi r^2 \times (\omega r)^2 \rightarrow F \sim r^2v^2$

Moment $M$ is proportional to thrust times length:

$M \sim Fl \rightarrow M \sim lr^2v^2$

Now, assuming that the size of the blade is proportional to the characteristic length $r \sim l$ , this assumption satisfies the geometric constraints, so there are:

$F \sim l^2v^2$ $M \sim l^3v^2$

There is:

$a \sim \frac{F}{m} \rightarrow a \sim \frac{v^2}{l}$ $\alpha \sim \frac{M}{I} \rightarrow \alpha \sim \frac{v^2}{l^2}$

So the maximum linear acceleration and maximum angular acceleration will change with the change of $l$ and . $v$

In fact , $v$ it is also $l$ related to aerodynamics. There are two kinds of scaling (scaling).

Froude scaling ：

$v \sim \sqrt{l} \rightarrow a \sim 1,\alpha \sim \frac{1}{l}$

Mach scaling ：

$v\sim1\rightarrow a\sim \frac{1}{l},\alpha \sim \frac{1}{l^2}$

So based on these two assumptions, we can get that the smaller the size of the quadrotor, the greater the maximum angular acceleration it can obtain, and the more flexible it is.

Next chapter link:

Robotics: Aerial Robotics (air robot) notes (3): UAV kinematics

Robotics: Aerial Robotics Notes (2): How to Design a Quadcopter UAV