Modelling, and Design of a Quadcopter With MATLAB Implementation

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 09 Issue: 09 | Sep 2022 www.irjet.net p-ISSN: 2395-0072
© 2022, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page 103
Modelling, and Design of a Quadcopter With MATLAB Implementation
Omar Horan1, Amarendra Reddy2
1M.Tech Student in control and Automation, Department of Electrical Engineering, AU Engineering college,
Andhra University, Visakhaptnam, Andhra Pradesh, India
2Associate Professor, Department of Electrical Engineering, AU Engineering college, Andhra University,
Visakhaptnam, Andhra Pradesh, India
---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract -: Quadcopter is an unmanned aerial vehicle,
which can be implemented in different applications. In
paper it will be represented a development of a quadcopter
system. In The paper we will demonstrate how to model a
basic physical quadcopter, then increase the complexity of
the plant model, and finally demonstrate how the model can
be implemented in MATLAB/Simulink.). The advantages of
utilizing MATLAB/Simulink to create and implement these
models come from the tool's processing power as well as the
simplicity with which Simulink models may be applied to
hardware.
Key Words: model, state variables, quadcopter, plant,
sensor
1.INTRODUCTION
Quadcopters are an emerging technology gaining in
popularity. Most users purchase quadcopters fully
assembled with everything that’s needed to fly. One of the
most interesting aspects of quadcopters that may make
flying them even more amazing is understanding how they
work. Believe it or not, users have the ability to install
their own control system in order to have the quadcopter
fly exactly how they want, depending on hardware of
course. Many users are limited by the controller that
comes preinstalled on quadcopters. A custom control
system, depending on the complexity of the system, can
allow the quadcopter to have automatic functions. These
functions could include flips, or inverted flight. However,
to do this, a plant model of the quadcopter is needed to
simulate flight to prove the control system works prior to
installing on the quadcopter. Creating a plant model and
simulating it provides a prediction of what the real-world
quadcopter will do. Quadcopter plant models can increase
in accuracy depending on how involved the modelling
process is. The idea of controlling and creating plant
model of a quadcopter is rather complex but can be
broken down significantly so it can be easily understood.
Control systems and plant models can be seen almost
everywhere without knowing it. The three pieces required
to create a model are the control system, a plant model
and sensors. Since the control system and plant model of a
quadcopter can be complex, the paper will demonstrate
how to model the basic physical quadcopter, then increase
the plant model’s complexity and then demonstrate how
the model can be implemented in MATLABSimulink. It
also demonstrates how to build and tune controllers that
re able to control a quadcopter’s roll, pitch, yaw, altitude
and motor dynamic. The benefits of using
MATLABSimulink to create and implement these models
come from the computing power of the software tool,
along with ease of implementation of Simulink models to
hardware. document is template. We ask that authors
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square brackets (e.g. [1]). However, the authors name can
be used along with the reference number in the running
text. The order of reference in the running text should
match with the list of references at the end of the paper.
1.1 Building the Plant Model
With the use of quadcopter in outdoor applications,
researchers are interested in evaluating the performance of
modelling and controlling of the quadcopter. To
understand the quadcopter system modelling the block
seen in Figure 1 can be used.
Figure 1-Plant model realization
The quadcopter model is a system of equations at
generate outputs based upon inputs, parameters, and
system states. Within the broad plant model of the
quadcopter there are separate plant models such as the
body dynamics, motor, and battery models. There is
somewhat of a flow of variables within the plant model.
The system starts with the inputs, which in this case are
motor angular velocity commands. With these commands,
states within the plant model angular velocity, and axial
velocity. Parameters are constants that effect the states,

such as the friction coefficient of the motors. The outputs of
the plant are based on the previous states, parameters, and
inputs. Lastly, it is important to note that the states
dependent on the frame of reference, or where the zero
point is the ideal plant model would simply be the physical
model, assuming instant thrust, no disturbances, and
unlimited state values. A real-world model would include
motor and battery dynamics along with state limitation.
1.2 Quadcopter Plant model Parameters
Quadcopters inherently have six degrees of freedom.
This means the device can travel in the X, Y and Z
direction, as well as rotate in roll, pitch and yaw
directions. These terms are generally related to most
aircraft, subs, and robotics as not many other platforms
are able to control all 6 degrees. The roll and pitch are
what allow the quadcopter to move in the X and Y
directions. Roll can be defined as the rotation around the X
axis, pitch can be defined as the rotation around the Y axis
and yaw can be defined as the rotation around Z axis. With
this, the axis of rotation for the quadcopter and state
variables can be defined. State variables are the physical
dynamics of the quadcopter that include angles and
positions. Many of the states are dependent on other
states and user input. The reference frame of the
quadcopter will be local NED. Which is why the Z axis is
pointed down.
Table -1: Quadcopter states
Quadcopter states
Symbol Description Unit Observability
Pitch Euler
Angle
𝑟𝑎𝑑 Stereo Vision
̇ Pitch Euler
Angular
Velocity
𝑟𝑎𝑑/𝑠 Gyroscope
̈ Pitch Euler
Angular
Accel.
𝑟𝑎𝑑 /𝑠2 -
𝜙 Roll Euler
Angle
̇ Roll Euler
Angular
Velocity
̈ Roll Euler
Angular
Accel.
𝜓 Yaw Euler
Angle
̇ Yaw Euler
Angular
Velocity
̈ Yaw Euler
Angular
Accel.
𝑋 Position in X 𝑚 Stereo Vision
̇ Velocity in X 𝑚/𝑠 -
̈ Accel. in X 𝑚 /𝑠2 Accelerometer
𝑌 Position in Y 𝑚 Stereo Vision
̇ Velocity in Y 𝑚/𝑠 -
̈ Accel. in Y 𝑚 /𝑠2 Accelerometer
𝑍 Position in Z 𝑚 Stereo Vision
̇ Velocity in Z 𝑚/𝑠 -
̈ Accel. in Z 𝑚 /𝑠2 Accelerometer
Ω1 Motor 1
Angular
Velocity
𝑟𝑎𝑑/𝑠 Voltage/Current/Optical
Ω2 Motor 2
Angular
Velocity
Ω3 Motor 3
Angular
Velocity
Ω4 Motor 4
Angular
Velocity
-Theta (/they-tuh/), 𝜙-Phi (/fi/), 𝜓-Psi (/sai/), Ω-Omega (/oh-
mega/)
Before generating the plant model equations of motion,
Parameter need to be defined. It is important to know the
difference between states and parameters. States are
dynamics of the quadcopter, where parameters are
constants associated with the quadcopter. The first
parameters to be defined are the moment of inertia in
each axis. The moment of inertia is the amount of torque
needed for a desired angular acceleration around a
rotation axis. These relies heavily on the physical design
and components of the quadcopter. These are generated
through various methods which mostly include estimates
from lengths/diameters and masses of the quadcopter. The
rotor inertia, which is the total rotational moment of
inertia around the propeller axis is another parameter
used in the equations of motion. The distance from the
rotor axis to the center of the quadcopter also can be
defined, along with the thrust and drag coefficients. These
can be seen in Table 2 below.

Table -2: Quadcopter parameters
Symbol Description Unit
𝐼𝑥𝑥, 𝐼𝑦𝑦, 𝐼𝑧𝑧 Inertia moments 𝐾𝑔 𝑚2
𝐽𝑟 Rotor inertia 𝐾𝑔 𝑚2
𝑙 Rotor axis to copter center distance 𝑚
𝑏 Thrust coefficient 𝑁/𝑠2
𝑑 Drag coefficient 𝑁𝑚/𝑠2
The next set of parameters have to do with the
quadcopter and environment. In the real world,
quadcopters experience aerodynamical effects, namely
from air resistance and wind. Air resistance can be
thought of as a drag on the quadcopter, opposing the
motion of the quadcopter. This is what makes quadcopters
slow down when no actuator force is acting on them. To
envision air resistance, one can simply drop a piece of
paper, and a pencil from the same height. Since air
resistance effects the quadcopter in all 3 axial directions, a
coefficient is needed for each. Measuring the air resistance
is extremely complex, so in most cases estimates suffice.
The values used for the plant model were 0.1 kg/s for the
resistance in each axis. The parameters can be seen in
Table 3 below.
Table -3: environmental parameters.
Symbol Description Unit
𝐴𝑥, 𝐴𝑦, 𝐴𝑧 Air resistance in each axis 𝑘𝑔/𝑠
1.3 Plant Model Mathematical Equations
Before the quadcopter is modeled in MATLAB/Simulink
the mathematical equations need to be generated. To do
this a few assumptions are made [1].
 The drone is a rigid structure
 The air frame and components are symmetric
 The center of gravity and air frame origin
coincide
 Thrust and drag are proportional to the square
of propeller speed
 There are no external disturbances to the
quadcopter such as wind, temperature, etc.
Next, The physical effects on the quadcopter need to be
considered. These include aerodynamic effects from the
propellers, inertial counter-torques from changes in
propeller speed, gravity and gyroscopic effects from
changes in body orientation. These all play a role in the
equations of motion of the quadcopter. The internal
plant model equations of the system can now be
described. There are four main thrust equations,
vertical thrust, roll thrust, pitch thrust and yaw thrust
with the “+” coordinate which can be seen in table four.
Table -4: input Equations in the “+” orientation
Input Thrust Equation Description
𝑈1+ 𝑏(Ω12 + Ω22 + Ω32 + Ω42) General Thrust
𝑈2+ 𝑏(−Ω22 + Ω42) Roll Thrust
𝑈3+ 𝑏(Ω12 − Ωw32) Pitch Thrust
𝑈4+ 𝑑(Ω12 − Ω22 + Ω32 − Ω42) Yaw Thrust
The quadcopter inputs can also be defined in the “X”
orientation [2]. These input equations can be seen in table
5 below.
Table -5: input Equations in the “x” orientation
Input Thrust Equation Description
𝑈1+ 𝑏(Ω12 + Ω22 + Ω32 + Ω42) General Thrust
𝑈2+ 𝑏𝑠 𝑛 (Ω12 - Ω22 - Ω32 + Ω42) Roll Thrust
𝑈3+ 𝑏𝑠 𝑛 (Ω12 +Ω22 - Ω32 + Ω42) Pitch Thrust
𝑈4+ 𝑑(Ω12 − Ω22 + Ω32 − Ω42) Yaw Thrust
The overall angular velocity of the all propellers can
also be defined. This will be used in angular acceleration
equations. The positives and negatives in equation (9) are
used because of the direction the propeller is spinning. In
most cases the opposite side motors are spinning in the
same direction, and 2 and 4 in the same direction).
Ω𝑟 = Ω1 − Ω2 + Ω3 − Ω4 (9)
Now equations of motion can be defined
Table -6: Plant model equations
Rolling
Torque
𝐼 𝜙̈ ̇𝜓̇(𝐼 𝐼 ) 𝐽 ̇ 𝑙 𝑈
Roll Ang.
Accel 𝜙̈
̇𝜓̇(𝐼 𝐼 ) 𝐽 ̇ 𝑙 𝑈
𝐼
Pitching
Torque
𝐼 ̈ 𝜙̇𝜓̇ 𝐼 𝐼 𝐽 𝜙̇ 𝑙 𝑈
Pitch
Ang. Accl.
̈
𝜙̇𝜓̇ 𝐼 𝐼 𝐽 𝜙̇ 𝑙 𝑈
𝐼
Yawing
Torque
𝐼 𝜓̈ ̇𝜙̇(𝐼 𝐼 ) 𝑈

Yaw Ang.
Accl. 𝜓̈
̇𝜙̇(𝐼 𝐼 ) 𝑈
𝐼
X force 𝑚𝑋̈ 𝜓 𝜙 𝜓 𝜙 𝑈 𝐴 𝑋̇
X Accl.
𝑋̈
𝜓 𝜙 𝜓 𝜙 𝑈 𝐴 𝑋̇
𝑚
Y force 𝑚𝑌̈ 𝜓 𝜙 𝜓 𝜙 𝑈 𝐴 𝑌̇
Y Accl.
𝑌̈
𝜓 𝜙 𝜓 𝜙 𝑈 𝐴 𝑌̇
𝑚
Z force 𝑚𝑍̈ 𝑚𝑔 𝜙 𝑈 𝐴 𝑍̇
Z Accl.
𝑍̈
𝑚𝑔 𝜙 𝑈 𝐴 𝑍̇
𝑚
2.1 Basic Simulink Plant model
Now that the basic equations of motion have been
generated, the Simulink plant model can now be formed.
First, a basic script that defines the inertia moments, rotor
inertia moments, rotor inertia, length, mass, thrust, and
drag coefficients need to be created. Then the Simulink
model is created. Starting a fresh Simulink model block
diagram, First define the solver options. On the top bar of
Simulink hit Simulation. Then Model configuration
parameters. Under solver option, make sure the type is
variable step, and the solver is on Auto. A new subsystem
can now be defined. The subsystem has four inputs (Ω1, Ω2,
Ω3, Ω4) and six outputs (X, Y, Z, roll, pitch, yaw). At this
point the model should be looking like the following figure
below
Figure 1 Quadcopter Plant
After implementing the previous mathematical equations,
we get the following quadcopter plant feedback system
Figure 2 Quadcopter Plant feedback system
2.2 Basic Simulation Results
Figure 3 Roll Plant Simulation
Figure 4 Thrust Plant Simulation

Figure 5 Quadcopter Trajectory Simulation
Figure 6 pitch Plant Simulation
3. CONCLUSIONS
Quadcopters with accelerating industry are an
interesting platform to perform different modelling
strategies. The goal was to develop a simple modelling
approach using MATLAB-Simulink. The vehicle’s Simulink
model was made, based on derived differential Simulink
model simulated perfectly depending on the equations of
motion. This research shows how the original complex
model of quadcopter is modelized taking into account wind
effects.
REFERENCES
[1] B. Fares, P. Apkarian, D. Noll, An Augmented
Lagrangian Method for A class of LMI–Constraint
Problems in Robust Control Theory. Int. J. Control.
2001.
[2] Carrillo, L. R., López, A. E., Lozano, R., & Pégard, C.
(2013). Modeling the Quad-Rotor Mini-Rotorcraft.
Advances in Industrial Control Quad Rotorcraft
Control, 23-34. doi:10.1007/978-1- 4471-4399-4_2
[3] Carrillo, L. R., Lozano, R., & Pégard C. (2013). Modeling
the Quad-Rotor Mini-Rotorcraft. Advances in
Industrial Control Quad Rotorcraft Control 23-34
doi:10.1007/978-1- 4471-4399-4_2.
[4] Horn, G. (2009, March 17). The Aerospace Euler
Angles. Retrieved November06, 2017
[5] P.-J. Bristeau, P. Martin, E. Salaun, and N. Petit, “The
role of propeller aerodynamics in the model of a
quadrotor,” in European Control Conference,
(Budapest, Hungary), 2009.
[6] W. Khan and M. Nahon, “A propeller model for general
forward flight conditions,” Inter J of Intelligent
Systems, 2015.

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