# Quadcopter Stability in Wind

Данная статья была написана в рамках участия в University Physics Competition 2020, по результатам которого получила бронзовую медаль. К сожалению, вставить полноценные формулы в редактор не удалось, так что местами присутствуют скриншоты, в том числе и с текстом, так как в нем встречались математические обозначения. К статье также будет прикреплен полный текст статьи в pdf и исходники модели. Для запуска необходимо запустить tests/main.m.

The attitude, speed, and position of unmanned aerial vehicles are affected

by the wind. This article analyzes and evaluates the maximum permissible

wind speed at which a ight is safe. Combined with a mathematical model of

an unmanned aerial vehicle, the movement of an unmanned aerial vehicle in a

wind field is illustrated in terms of force. Several simulation tests have been

implemented in various programs (Matlab and SolidWorks) to show the effects

of different types of wind. And then, using the data obtained, the maximum

permissible wind speed was obtained at which the unmanned aerial vehicle does

not deviate from the target point by more than 0.2 m.

**1 Introduction**

Nowadays, poor atmospheric conditions are at the root of a significant number

of accidents. Researchers classify winds leading to air crashes into the following

categories: downdraft, turbulent wind, wind shear, and wake vortex.

With the development of technology, unmanned aerial vehicles (UAVs) appear.

They are widely used due to the fact that they are more compact, do not

put the pilot at risk in a crash, and can also y longer (what eliminates the fact

of the fatigue of the aircraft pilot).

The main disadvantages of UAVs include: lower ight speed compared to

manned aircraft, lower take-off weight and dimensions, as well as lower ight altitude,

what means that such device becomes more susceptible to wind. It is the

disturbance of the atmospheric environment that is the difficulty in controlling

the UAV. Winds of different strengths affect drones to varying degrees. Wind

effect is the process of energy transfer, as a result of which the UAV changes its

ight states. It should be noted that drones must operate at different terrain

features, different temperatures and at different time periods, what means that

the in uence of atmospheric disturbances on them is inevitable.

There are a large number of accidents involving drones, for example, in

June 2016, during an inaugural ight of Facebook's Aquila, the drone suffered

a structural failure caused by a strong gust of wind, and in May 2015, Google's

parent company Alphabet's Solara 50 crashed in a collision with a hot updraft

generated during a ight in the New Mexico desert landing zone.

Many scientists strove to contribute to improving ight characteristics of

UAVs using numerical, theoretical or experimental methods, but at the present

there is no way to take fully into account the effect of the wind field, and

therefore there is no way to use it correctly. It is very important to study

different effects of different types of wind in order to minimize the risk of a

disaster involving UAVs.

In order to succeed, you need to do:

1. A detailed generalization of the categories of winds at low altitudes, which

are most often encountered by UAVs as opposed to manned aircraft, including

their character and mathematical models.

2. Illustrate and analyze the effect of different types of wind on the speed,

orientation and position in space of the UAV, as well as implement some

simulation tests in the programm to track the change in the trajectory of

the UAV

3. Make a conclusion about the maximum wind speed which realize the safe

ight of the quadcopter is carried out

During this work, a quadcopter will be considered - this is a remotely controlled

aircraft, the power section of which is represented by four motors and

the same number of propellers. quadcopters are controlled by adjusting the angular

speeds of the rotors, which rotate with the help of electric motors. So, the

rotors of one pair of slides rotate clockwise, the other two rotate in the opposite

direction. The main advantage of the quadcopter is its simple design, which

allows it to be a typical base for unmanned aerial vehicles (UAVs). This unit

is widely used in both entertainment and work industry, including search and

rescue operations, in various building inspections, etc.

The quadcopter is of interest to many researchers, because its basic dynamic

model is the initial one for all studies, but it also has more complex aerodynamic

properties. There are the following types of control: PID controllers, reverse

step control, LQR controllers and nonlinear controllers

with nested saturations. Control methods require precise determination of the

position of the vehicle and its orientation, which is achieved using a gyroscope,

accelerometer, GPS, as well as hydroacoustic and laser sensors.

The main purpose of this article is to determine the maximum wind speed of

variable strength and direction, at which the safe operation of the quadcopter

is carried out. In this case, the UAV must remain within 20 cm from the target

location.

**2 Mathematical model of quadcopter**

The quadcopter design is shown in Figure 1, including the corresponding angular

velocities, torques and forces generated by the four rotors (numbered 1 to 4).

The transition matrix from a body system to the absolute one is

Assuming that the quadcopter has a symmetric structure over axes, we obtain

a diagonal matrix of the tensor of inertia, consisting only of the main central

moments of inertia. We will also assume that the moments of inertia are known

to us inaccurately.

where l is the distance from the center of mass to the propeller. Therefore,

an increase in roll is achieved by increasing the rotation speed of the 4th motor

(relative to the 2nd), an increase in pitch is achieved by increasing the rotation

speed of the 3rd motor (relative to the 1st), and a change in the yaw angle is

achieved by changing the power of the propellers rotating in the same direction

(relative to the speed the other two, rotating in the opposite direction).

**2.1 Newton-Euler equations**

In the absolute coordinate system, the centrifugal velocity is zeroed, and,

therefore, the acceleration of the quadcopter is affected only by the gravitational

force, the magnitude and direction of the thrust.

**2.2 Euler-Lagrange equations**

Linear and angular components do not affect each other, therefore they can

be used separately. The linear external force is the total thrust of the motors.

Linear Lagrange equation

Rotational energy can be expressed in absolute coordinate system as

Angular forces created by motor torques. Angle Lagrange equations

**2.3 Aerodynamical effects**

This model is obtained with simplification of complex dynamic interactions.

In order to achieve a more realistic behavior of the quadcopter, let's take into

account the force of air resistance.

Where Ax;Ay;Az are the air resistance coefficients in the corresponding

directions of the axes of the absolute coordinate system.

Then there is an overview of various kinds of wind effects on unmanned aerial

vehicle. It allows refine the right part in [10] and build a general, accurate model

of the system.

**3 Mathematical model of wind effects****Constant wind**

A constant wind can be described as an average wind speed in a space-time

continuum. Constant wind is only a model and does not exist in nature. Usually,

constant wind is used for simulation tests of UAVs, but this is not enough to

fully model ight conditions. In order to implement a more real simulation test,

other kinds of wind must be added.

Turbulent ow

Turbulent ow is a ow whose movement is random in time and space and it

is always accompanied by constant wind. The cause of turbulent ow is associated

with many factors, such as wind shear, heat exchange, topographic factors,

and so on.

Turbulence models include Dryden model and the Von Karman model, both of

which depend on a large number of measurements and statistics. The difference

is that Dryden's model establishes a correlation function turbulence to obtain

spectral function, while von Karman, on the contrary, sets spectral function and

then outputs correlation turbulence function. After the relevant studies there is

no significant difference between the two models, so both can be used to solve

engineering problems.

The Dryden spectral function

**Wind shear**

Wind Shear is the vector difference (or gradient) of wind speeds at two points

in space, attributed to the distance between them. In other words, changing

the direction and/or speed of wind in the atmosphere at a very short distance.

[*] Wind shear is a discrete or deterministic wind speed, which often occurs

in a very short time, and it is a strong atmospheric disturbance. The wind shear

model can be divided into several categories according to its profile geometry,

including rectangular model, trapezoidal model, and \1-consine" model. Specific

models for simulation tests are expressed as follows:

where V represents the speed of the wind shear with arbitrary direction, dm

represents the length of the range where exist wind shear, while Vmax is the

strength of the wind shear. According to the previous research, the wind shear

scale dm and the wind shear strength Vmax are in connection with the characteristic

wavelength L and the constant wind speed s mentioned in "Turbulent flow" section.

**The principle of wind effects the UAV **

According to this point of view,

the in uence of air ow is considered as an external force on the UAV. A wellknown

fact is Newton's 3 law of mutual forces, and the force caused by air is

called resistance. In fluid dynamics, resistance is a force acting against the relative

motion of any object that has a relative velocity relative to the surrounding

uid, and its magnitude depends on the properties of the uid and the size,

shape, and velocity of the object. One way to express this using the following

equation:

**4 Trivial solution**

Let's find out what is maximum wind speed for the quadcopter to remain at

the same position. We consider the simplest model with a static constant wind.

This requires that the forces along the horizontal and vertical components are

equal to zero. To compensate for the force of the wind, the quadcopter is tilted

towards it. All motors run with maximum power. In the case of the maximum

possible wind speed the vertical force is Fz = mg - T cos(a) = 0, where a is

the angle of deviation from the vertical axis. Horizontal force

**5 Simulation**

The model obtained in the previous paragraphs was simulated in Matlab 2019b.

The values of the drag force coefficients Ax, Ay and Az are selected such as the

quadcopter will slow down and stop when angles and are stabilised to zero

values.

Values in Table 7 were used in the simulation:

The quadcopter is initially in a stable state in which the values of all positions

and angles are zero, the body frame of the quadcopter is congruent with the

inertial frame. The total thrust is equal to the hover thrust, the thrust equal to

gravity.

The model has the following appearance:

The upper figure shows a 3D model of the copter. Bottom figures display

angular velocity and orientation angles.

After the end of the simulation, the time dependences of all flight parameters

are displayed:

To calculate the maximum wind speed at which the quadcopter does not

leave a certain zone, iterative modeling was carried out. The simulations were

carried out for three types of wind: constant wind, wind shear, and Dryden turbulent

ow. A ready-made block at Simulink was used to generate a turbulent

Dryden ow. Initially, a low wind speed was set. If the copter did not move

more than 0.2 meters from the starting point, the wind speed increased by 0.1

m / s and the model was restarted.

**6 Linearized model of quadcopter**

From the point of view of stability analysis and control synthesis, we are more

interested in the linear approximation of a given system of differential equations.

To ensure that the control does not completely leave the system during

linearization, we take the square of the propeller angular velocity for it - the loss

of sign does not bother us, since the propellers can only rotate in one direction, and their direction of rotation is already taken into account in the equations.

For the equilibrium position, we take the following conditions:

Where u0 is the square of the angular velocity of one motor such that the thrust,

creating four such values, will fully compensate for the force of gravity. In other

words, control, in which the drone hangs motionless in the air. The value is

found from the dynamics equation:

To linearize the system, we expand the right-hand side of the system in deviations

from the equilibrium position in a Taylor series as a function of several

variables in the vicinity of the equilibrium position, taking the square of the angular

velocity of the propellers as the control, and discard the nonlinear terms.

As a result, we get the following mathematical model:

The resulting model will be used to develop regulators and stabilization systems.

**7 Stabilisation of quadcopter**

To stabilise the quadcopter, a PD controller is utilised. Advantages of the PD

controller are the simple structure and easy implementation of the controller.

The general form of the PD controller is

in which u(t) is the control input, e(t) is the difference between the desired

state xd(t) and the present state x(t), and Kp and Kd are the parameters for

the proportional and derivative elements of the PD controller.

The performance of the PD controller is tested by simulating the stabilisation

of a quadcopter. The PD controller parameters are presented in Table 2.

**8 Solidworks model**

To visualize our solution, various scenes were created that re ect possible processes.

It was based on the SolidWorks application software package, especially

SolidWorks Simulation. We have created a model of a quadcopter that meets

the necessary characteristics. This model was exposed to different air flows, and

we compared how much our mathematical model coincided with the physical

one. Moreover, the balance position of the quadcopter can be adjusted using

different algorithms. These days, the most common solution in this area is PID

regulators.

At fig. 4 you can see a visualization of the process of resistance of the quadcopter

to the air ow. on the bottom edge of the imaginary cube is a red button,

which is the target point for the UAV.

**9 Conclusion**

This paper studied mathematical modelling and control of a quadcopter. The

mathematical model of quadcopter dynamics was presented and the differential

equations were derived from the Newton-Euler and the Euler-Lagrange equations.

The model was verified by simulating the ight of a quadcopter with

Matlab. Stabilisation of attitude of the quadcopter was done by utilising a PD

controller.

Also was set the maximum wind speed values at which the copter does not move

more than 0.2 m from the starting point for each type of wind:

The simulation proved the presented mathematical model to be realistic in

modelling the position and attitude of the quadcopter. The simulation results

also showed that the PD controller was efficient in stabilising the quadcopter

to the desired altitude and attitude. The presented mathematical model only

consists of the basic structures of the quadcopter dynamics. Also the electric

motors spinning the fours rotors were not modelled. The behaviour of a motor

is easily included in the model but would require estimation of the parameter

values of the motor. The position and attitude information was assumed to be

accurate in the model and the simulations. However, the measuring devices in

real life are not perfectly accurate as random variations and errors occur. Hence,

the effects of imprecise information to the ight of the quadcopter should be

studied as well. The presented model and control methods were tested only with

simulations. Real experimental prototype of a quadcopter should be constructed

to achieve more realistic and reliable results. Even though the construction of

a real quadcopter and the estimation of all the model parameters are laborious

tasks, a real quadcopter would bring significant benefits to the research. With

a real propotype, the theoretical framework and the simulation results could be

compared to real-life measurements. This paper did not include these higlighted

matters in the study but presented the basics of quadcopter modelling and

control.

**Literature**

- Teppo Luukkonen, Modelling and control of quadcopter, School of Science, Espoo, August 22, 2011.
- Bo HangWang1, Dao BoWang1, Zain Anwar Ali2 , Bai Ting Ting1 and HaoWang, An overview of various kinds of wind eects on unmanned aerial vehicle, Measurement and Control, April 9, 2019.
- Aerodynamic Characteristics of Propellers
- THOR I. FOSSEN, MATHEMATICAL MODELS FOR CONTROL OF AIR- CRAFT AND SATELLITES, Department of Engineering Cybernetics Norwegian University of Science and Technology, 2nd edition, January 2011.
- Beal TR, Digital simulation of atmospheric turbulence for Dryden and von Kar- man models, Guid Control Dyn, 1993.
- Nelson R. C., Flight Stability and Automatic Control, McGraw-Hill Int. 1998.
- Hughes, P. C., Spacecraft Attitude Dynamics, John Wiley & Sons Ltd. 1986.

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