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11.04.2021

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
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.

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.

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
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.

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.

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

1. Teppo Luukkonen, Modelling and control of quadcopter, School of Science, Espoo, August 22, 2011.
2. Bo HangWang1, Dao BoWang1, Zain Anwar Ali2 , Bai Ting Ting1 and HaoWang, An overview of various kinds of wind e ects on unmanned aerial vehicle, Measurement and Control, April 9, 2019.
3. Aerodynamic Characteristics of Propellers
4. 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.
5. Beal TR, Digital simulation of atmospheric turbulence for Dryden and von Kar- man models, Guid Control Dyn, 1993.
6. Nelson R. C., Flight Stability and Automatic Control, McGraw-Hill Int. 1998.
7. Hughes, P. C., Spacecraft Attitude Dynamics, John Wiley & Sons Ltd. 1986.

11.04.2021

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