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Simulation of the Mathematical Model of a Quad Rotor Control System Using Matlab Simulink

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Simulation of the Mathematical Model of a Quad Rotor Control System Using Matlab Simulink

Abstract and Figures

Quad rotor vehicles are gaining prominence as Unmanned Aerial Vehicles (UAVs) owing to their simplicity in construction and ease of maintenance. They are being widely developed for applications relating to reconnaissance, security, mapping of terrains and buildings, etc. The control of the quad rotor is a complex problem. As a precursor to developing a model based design, the simulation of the mathematical model of the quad rotor is implemented. This will facilitate easier implementation of the model based design using Matlab Simulink (c).
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Simulation of the Mathematical Model of a Quad Rotor Control System
using Matlab Simulink
Akhil M.
1
, M. Krishna Anand
2
, Aditya Sreekumar
3
and P.Hithesan
4
1
Aerospace Engineering Department,
Amrita Vishwa Vidyapeetham, Coimbatore, India
2
Electrical and Electronics Engineering Department,
Amrita Vishwa Vidyapeetham, Coimbatore, India
3
Electrical and Electronics Engineering Department,
Amrita Vishwa Vidyapeetham, Coimbatore, India
4
Electrical and Electronics Engineering Department,
Amrita Vishwa Vidyapeetham, Coimbatore, India
1
akhilmulloth@gmail.com
2
mkanand89@gmail.com
3
aditya.sreekumar@gmail.com;
4
phithesan@gmail.com
Keywords— Quad rotor, Model Based Design, Unmanned Aerial Vehicle
Abstract Quad rotor vehicles are gaining prominence as Unmanned Aerial Vehicles (UAVs)
owing to their simplicity in construction and ease of maintenance. They are being widely developed
for applications relating to reconnaissance, security, mapping of terrains and buildings, etc. The
control of the quad rotor is a complex problem. As a precursor to developing a model based design,
the simulation of the mathematical model of the quad rotor is implemented. This will facilitate
easier implementation of the model based design using Matlab Simulink©.
N
OMENCLATURE
th
F Force due to i rotor
i
I Moment of inertia about x axis
xx
I M oment of inertia about y axis
yy
I Moment of inertia about z axis
zz
J Rotor Moment of ine
rrtia
th
PWM Pulse Width Modulation signal corresponding to i motor
i
U Net rotor thrust in vertical direction
1*
U Effective force in vertical direction
1
U Component of force
2responsible for pure roll
U Component of force responsible for pure pitch
3
U Component of force responsible for pure yaw
4
a x intercept of force vs pwm graph corresponding t
ith
o i motor
th
b Slope of force vs pwm graph corresponding to i motor
i
b Thrust coefficient of rotor rotating in clock wise direction
cw
b Thrust coefficient of rotor rotating in
aw
anti - clock wise direction
b Thrust coefficient of rotor
d Drag coefficient of rotor rotating in clock wise direction
cw
d Drag coefficient of rotor rotating in anti - clock w
aw
ise direction
e Altitude error
altitude
e Roll error
roll
e Pitch error
pitch
e Yaw error
yaw
g Acceleration due to gravity
l Arm length
k Proportional constant
palt for altitude
Applied Mechanics and Materials Vols. 110-116 (2012) pp 2577-2584
© (2012) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.110-116.2577
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,
www.ttp.net. (ID: 110.225.102.136-03/10/11,11:42:43)
k Proportional constant for roll
k Proportional constant for pitch
k Proportional constant for yaw
k Integral constant for altitude
i
k Integral constant for roll
i
ki
proll
ppitch
pyaw
alt
roll
pitch Integral constant for pitch
k Integral constant for yaw
i
k Derivative constant for altitude
d
k Derivative constant for roll
d
k Derivative constant for pitch
d
k Derivative c
d
yaw
alt
roll
pitch
yaw onstant for yaw
m Mass
u Actuation signal from altitude controller
1
u Actuation signal from roll controller
2
u Actuation signal from pitch controller
3
u Act
4uation signal from yaw controller
x Acceleration in the x direction
y Acceleration in the y direction
z Acceleration in the z direction
Αngular movementφ
about the x axis, Roll
θ Αngular movement about the y axis, Pitch
ψ Αngular movement about the x axis, Yaw
th
τ Torque due to the i motor
i
RPM of the
i
th
i motor
Introduction
A quad rotor is an aerial vehicle that generates lift with four rotors. It is an inherently unstable
system in the open loop [1]. The motion in x and y directions is coupled with pitch and roll. Thus,
developing a closed loop control for maintaining the stability of the quad rotor is essential. The
modelling of the system is a precursor to implementing the control strategy. The mathematical
model of the system gives the state of the quad rotor in accordance with the physical parameters
such as velocity, force, thrust, etc.
This paper begins with an introduction to quad rotor motion, model based design and the
mathematical modelling. In the next section, the coefficients from the quad rotor structure are
incorporated into the modelling of the quad rotor. The implementation of the modeling using
Simulink is explained in the subsequent section. The results of the modeling and the implications of
it on the control strategy are discussed next, followed by conclusions from the study.
2578 Mechanical and Aerospace Engineering
Quad rotor modeling
The quad rotor motion is controlled by varying the rpm of the motors and not by using any
mechanical actuators. The craft requires active control of six degrees of freedom to fly.
Figure 1. Quad Rotor Layout top view
The quad rotor layout is shown in Figure 1. There are two arms, each having motors at both their
ends. The motors 1 and 3 rotate in the clockwise direction while the motors 2 and 4 rotate in the
anti-clockwise arrangement [2]. The motors at opposite ends of the same arm rotate in the same
direction to prevent torque imbalance during linear flight.
Model based design is used to develop the control system for the quad rotor.
The quad rotor model is simulated using Matlab Simulink, which represents the change in state of
the quad rotor when any disturbance occurs or input is provided.
A. Mathematical Model
The mathematical model of the quad rotor can be represented with the following equations [2], [3],
[4], [5], [6]:-
U1
x = (cos( )sin(
θ)cos(ψ) + sin( )sin(ψ)) (2
.1)
m
φ φ
U1
y = (cos( )sin(θ)sin(ψ) -sin( )cos(ψ)) (
2.2)
m
φ φ
U1
z = -g + (cos( )cos(θ))
(2.3)
m
φ
I - I J l
yy zz r
=
2
d
I I I
xx xx xx
φ
 
 
 
 
I - I J l
zz xx r
θ = ψ - + U (2.5)
3
d
I I I
yy yy yy
φ φ
 
 
 
 
 
I - I 1
xx yy
ψ = θ + U (2.6)
4
I I
zz yy
φ
 
 
 
 
B. Assumptions and Decoupling
The quad rotor is assumed to have very low linear and angular velocities when in motion and
assumed not to tilt beyond 15
0
in pitch and roll. The quad rotor is always flying at near hovering
conditions and Coriolis and rotor moment of inertia terms can be neglected [7],[4].
Attitude is controlled by manipulating the four degrees of freedom involved – viz. altitude (z), roll
(
φ
), pitch (
θ
) and yaw (
ψ
). The equations representing the four degrees of freedom are [4], [6]:-
Applied Mechanics and Materials Vols. 110-116 2579
U1
z = -g + (cos( )cos(θ))
(2.7)
m
l
= U
(2.8)
2
Ixx
l
θ = U
3
Iyy
φ
φ
(2.9)
1
ψ = U
(2.10)
4
Iyy
Implementation
The mathematical model of the quad rotor was implemented using Matlab Simulink. Matlab
Function Blocks were used for different modules of the simulation.
A. Reference Model Creator Block (Manual input block):
This block is used to obtain the reference values for rotational parameters - roll, pitch and yaw –
and the translational parameter– z, which are manual inputs in the simulation. The block can be
modified to accept input from an outer control loop for position in future.
B. Controller Block
This block is used to compare the reference values of the rotational parameters - roll, pitch and
yaw - and the translational parameter, z, with the actual values obtained as feedback from the quad
rotor’s mathematical model. The error obtained from the comparison is given to a PID control block
which performs the control action and the necessary control signals are generated. Four PID
controllers control the parameters of the system. The outputs from this block are [4]:-
u
1
=actuation signal from altitude controller
u
2
=actuation signal from roll controller
u
3
=actuation signal from pitch controller
u
4
=actuation signal from yaw controller
The PID controller equations obtained from Matlab Simulink are:-
alt alt alt
roll roll roll
pitch pitch pitch
u (s) 1 s
1
= k + k + k (3.1)
pi d
e s 0.1s +1
altitude
u (s) 1 s
2
= k + k + k (3.2)
pi d
e s 0.1s +1
roll
u (s) 1 s
3
= k + k + k (3.3)
pi d
e s 0.1s +1
pitch
u (s)
4
ey
yaw yaw yaw
1 s
= k + k + k (3.4)
pi d
s 0.1s +1
aw
C. PWM Signal Generation Block
The signals from the controller block are given to the PWM Signal Generation block. This block
generates the PWM signals to be given to the four motors of the quad rotor. The equations used for
this calculation is available in literature and are as follows [4]:-
2580 Mechanical and Aerospace Engineering
1 1 b
F = u + u + u
(3.5)
1 1 3 4
4 2 4d
1 1 b
F = u - u + u
(3.6)
2 1 3 4
4 2 4d
1 1 b
F = u + u - u
(3.7)
3 1 2 4
4 2 4d
1 1 b
F = u - u - u
4 1 2 4
4 2 4d
(3.
8)
The thrust factor for the clockwise and anti-clockwise rotating propellers is different, and the
equations were modified as:-
b
1 1 cw
F = u + u + u (3.9
)
1 1 3 4
4 2 4dcw
b
1 1 aw
F = u - u + u (3.1
0)
2 1 3 4
4 2 4daw
b
1 1 cw
F = u + u - u (3.1
1)
3 1 2 4
4 2 4dcw
b
1 1 aw
F = u - u - u
4 1 2 4
4 2 4daw
(3.12)
The PWM signal to be generated based on the input to the controller [4]:-
1
PWM = (F - a )
(3.13)
1 1 1
b1
1
PWM = (F - a )
(3.14)
2 2 2
b2
1
PWM = (F - a )
(3.15)
3 3 3
b3
1
PWM = (F - a
4 4
b
4
)
(3.16)
4
D. Motor Dynamics Block
This block is used for estimating the thrust generated by the motor for a particular input PWM
signal. The block uses a first order algebraic equation, and converts the input PWM signal to
equivalent force.
The relation between the PWM signal and the thrust developed by the motor is [4]:-
F = b .PWM + a
(3.17)
i i i i
The output from the block represents the resultant force acting on the system resolved into
components responsible for altitude, roll, pitch and yaw. To achieve this the equations are [2],[4],
[5], [8],[6]:-
2 2 2 2
U = b(
+ + Ω + Ω ) (3.18
)
1 1 2 3 4
2 2
U = b(-Ω + Ω )
(3.19)
2 2 4
2 2
U = b(-Ω + Ω )
(3.20)
3 1 3
2 2 2 2
U = d(-Ω + Ω - Ω + Ω
4 1 2 3 4
) (3.21)
Applied Mechanics and Materials Vols. 110-116 2581
The equations were modified for implementation as follows.
The force and torque produced by the motors can be related to their RPM,
as:-
Force due to the i
th
motor
2
= F = b
(3.22)
cw
i i
Torque due to the i
th
motor
2
=
τ = d (3.23)
cw
i i
Where i = 1 or 3
Force due to the i
th
motor
(3.24)
aw
2
= F = b
i i
Torque due to the i
th
motor
(3.25)
aw
2
= τ = d
i i
Where i = 2 or 4
The output from the block is:-
2 2 2 2
U = b (
+ ) + b (Ω + Ω ) (3.26)
1 cw 1 3 aw 2 4
*
U = U - mg
(3.27)
1 1
2 2
U = b (-Ω + Ω )
(3.28)
2 aw 2 4
U = b (
3 cw 2 2
-Ω + )
(3.29)
1 3
2 2 2 2
U = -b (
+ ) + b (Ω + ) (3.30)
4 cw 1 3 aw 2 4
E. Mathematical Model Block
The forces from the motor dynamics block are the inputs to the mathematical model block. This
block gives the positional parameters of the system. Three of the decoupled transfer functions that
represent roll, pitch and yaw are [4], [6]:-
(3.31)
(s) l
=2
U (s) s I
xx
2
φ
(3.32)
(s) l
=2
U (s) s Iyy
3
θ
(3.33)
(s) 1
=2
U (s) s Izz
4
ψ
A transfer function for the altitude was introduced. This is:-
(3.34)
(s) cos( )cos(θ)
=2
*s m
U (s)
1
z
φ
F. Inertial Measurement Unit Block
The acceleration and torque of the system is interpreted in terms of rotational and translational
parameters using this block. It is used for providing feedback to the controller.
Results
The PID parameters were tuned and the values are shown in Table I. The plots illustrate the ability of
the control system in stabilizing the quad rotor. The four single input single output loops for altitude,
roll, pitch and yaw, work simultaneously to effectively control the quad rotor.
TABLE I. PID
GAIN VALUES
Proportional
Gain (Kp)
Integral Gain
(Ki)
Derivative
Gain (Kd)
ROLL
0.0763 -0.0070 0.0987
PITCH
0.0763 -0.0070 0.0987
YAW
0.0763 -0.0070 0.0987
ALTITUDE
-0.0195 -0.0099 -0.0831
In each of the plots shown, the x axis represents time. The duration of the simulation is 35s.
The plot in Figure 2 represents the reference input given for the altitude and the graph in Figure 3
represents the actual altitude of the system. Here, the y axis represents the altitude of the quad rotor
in metres.
2582 Mechanical and Aerospace Engineering
Figure 2. Reference Altitude
Figure 3. Actual Altitude
The plots in Figures 4 to 9 represent the reference values and the actual values of the rotational
parameters – viz. roll, pitch and yaw. The y axis represents the angle of the respective parameter in
radians.
Figure 4. Reference Roll
Figure 5. Actual Roll
Figure 6. Reference Pitch
Figure 7. Actual Pitch
Figure 8. Reference Yaw
Figure 9. Actual Yaw
Applied Mechanics and Materials Vols. 110-116 2583
Conclusion
The paper describes the simulation developed as a precursor to a model based design of quad rotor
control system. The equations available in the literature that are relevant to the simulation were
examined and appropriate modifications have been made to represent the system behavior better.
The plots in Figures 2 to 9 prove the ability of the control system to stabilize the quad rotor. Future
work involves comparison of various control algorithms for quad rotor control and conversion of the
simulation to a control system to be used on the actual flying quad rotor.
Acknowle
D
gment
The authors would like to thank Dr. J. Chandrasekhar and Prof. Radhamani Pillay, our project guides
from the Aerospace Engineering and Electrical and Electronics Engineering departments
respectively and Mr.Sharath S. Nair for their valuable guidance, and Smruthi Ranjan, Neerav Harsh,
S. Maitreyi, Preethi Kumar, Diwakar Mandal and Rajesh for their assistance with the whole project.
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2584 Mechanical and Aerospace Engineering
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The paper proposes a complete real-time control algorithm for autonomous collision-free operations of the quadrotor UAV. As opposed to fixed wing vehicles the quadrotor is a small agile vehicle which might be more suitable for the variety of specific applications including search and rescue, surveillance and remote inspection. The developed control system incorporates both trajectory planning and path following. Using a differential flatness property the trajectory planning is posed as a constrained optimization problem in the output space (as opposed to the control space), which simplifies the problem. The trajectory and speed profile are parameterized to reduce the problem to a finite dimensional problem. To optimize the speed profile independently of the trajectory a virtual argument is used as opposed to time. A path following portion of the proposed algorithm uses a standard linear multi-variable control technique. The paper presents the results of simulations to demonstrate the suitability of the proposed control algorithm.
Conference Paper
Full-text available
The latest technological progress in sensors, actuators and energy storage devices enables the developments of miniature VTOL systems. In this paper we present the results of two nonlinear control techniques applied to an autonomous micro helicopter called Quadrotor. A backstepping and a sliding-mode techniques. We performed various simulations in open and closed loop and implemented several experiments on the test-bench to validate the control laws. Finally, we discuss the results of each approach. These developments are part of the OS4 project in our lab.
Conference Paper
This paper presents a predictive and nonlinear robust control strategy to solve the path tracking problem for a quadrotor helicopter. The dynamic motion equations are obtained by the Lagrange-Euler formalism. The control structure is performed through a model-based predictive controller (MPC) to track the reference trajectory and a nonlinear H-infinity controller to stabilize the rotational movements. Simulations results in presence of aerodynamic disturbances and parametric uncertainty are presented to corroborate the effectiveness and the robustness of the proposed strategy.
Article
This paper focuses on modeling and control of a quadrotor type unmanned aerial vehicle (UAV). Low level control issues are addressed, the derivation of the dynamic model is presented, transient and steady state behavior of the propulsion due to motors are elaborated by the aid of artificial neural networks, and a model is developed. Several simulation works have been discussed. These include proportional integral and derivative (PID) control scheme, sliding mode control (SMC), backstepping technique and feedback linearization. A comparison of the approaches is presented in terms of the tracking precision, applicability of control signals and the qualities of the transient response.
Article
This paper presents a predictive and nonlinear robust control strategy to solve the path tracking problem for a quadrotor helicopter. The dynamic motion equations are obtained by the Lagrange-Euler formalism. The control structure is performed through a model-based predictive controller (MPC) to track the reference trajectory and a nonlinear H ∞ controller to stabilize the rotational movements. Simulations results in presence of aerodynamic disturbances and parametric uncertainty are presented to corroborate the effectiveness and the robustness of the proposed strategy. Copyright c 2008 IFAC.
Conference Paper
A simulation environment is developed to assist in the design, development, and validation of complex controllers with applications to mini-UAVs, such as the four-rotor DraganFly RC helicopter (quadrotor). The simulation system is modular and includes interfaces which allow for substitution of software subsystems with hardware components. This approach enables a smooth transition from the design and simulation phases to the implementation phase. The benefits of the proposed simulation environment are examined through the application of model reference adaptive control to a quadrotor UAV in the presence of actuator uncertainties and nonlinearities.
Modelling, identification and control of a quadrotor helicopter
  • Tommaso Bresciani
Tommaso Bresciani, "Modelling, identification and control of a quadrotor helicopter," Master Thesis, Lund University, Lund, Sweden, October, 2008 , www.roboticsclub.org/redmine/attachments/467/Quadrotor_Bible.pdf
Design and Development of a Compact Integrated Telecommand and Telemetry System for an Unmanned Aerial Vehicle, Undergraduate Project Dissertation, Amrita Vishwa Vidyapeetham
  • Preethi Kumar
  • Diwakar Mandal
Preethi Kumar, Diwakar Mandal, "Design and Development of a Compact Integrated Telecommand and Telemetry System for an Unmanned Aerial Vehicle", Undergraduate Project Dissertation, Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India, 2010