Plane Thermoelastic Waves in Infinite Half-Space Caused

FACTA UNIVERSITATIS
Series: Mechanical Engineering Vol. 15, N

o
2, 2017, pp. 201 - 215

DOI: 10.22190/FUME170203005A

Original scientific paper

MODELLING AND CONTROL OF H-SHAPED RACING

QUADCOPTER WITH TILTING PROPELLERS

UDC 681.5

Ahmed Alkamachi
1,2

, Ergun Ercelebi
2

1
Alkawarizmi College of Engineering, University of Baghdad, Iraq

2
Department of Electric and Electronic Engineering, Gaziantep University, Turkey

Abstract. Traditional quadcopter suffers terribly from its underactuation which implies

the coupling between the rotational and the translational motion. In this paper, we

present a quadcopter with dynamic rotor tilting capability in which the four propellers

are allowed to tilt together around their arm axis. The proposed model provides leveled

forward/backward horizontal motion and therefore, ensures a correct view of the

onboard camera, and increases the vehicle speed by reducing the air drag. The rotor

tilt mechanism also provides an instant high speed in the forward or reverse direction

and offers a quick and solid air brake to restrain that fast moving speed. The nonlinear

dynamical model for the quadcopter under consideration is derived using Newton-

Euler formalization. A control strategy is then proposed aimed to control the altitude,

attitude, and the forward speed of the obtained model. Finally, a numerical simulation

is used to integrate the system model with the controller and to test the system

performance. Simulation results are reported to demonstrate the advantages of the

proposed novel configuration.

Key Words: Genetic Algorithm, Newton-Euler Formalization, PID Controller,

Racing Quadcopter, Tilt Rotor

1. INTRODUCTION

In the recent years, unmanned aerial vehicles (UAVs) have gained a considerable attention

for their applications in scientific, civilian, and military fields. Quadcopter UAV is defined as

a small vehicle with four rotor-propeller sets distributed around its body [1]. It is a highly

nonlinear, multi input multi output (MIMO), extremely coupled and underactuated system [2].

The quadcopter has become one of the most popular UAV designs due to its vertical take-off

Received February 3, 2017 / Accepted April 27, 2017
Corresponding author: Ahmed Alkamachi

Affiliation: Alkawarizmi College of Engineering, University of Baghdad, Katir Alnada St., AlJadiriyah,

Baghdad, Iraq
E-mail: amrk1978@gmail.com

202 A. ALKAMACHI, E. ERCELEBI

and landing (VTOL) property, simplicity in mechanical configuration, and ease of

maintenance. Furthermore, the use of four smaller propellers in the quadcopters reduces the

danger posed by the propellers if they touch an external object as compared with one big

propeller in helicopter or airplane [3]. The racing quadcopters with H-shape configuration

have been gaining an increasing attention due to their use in several critical applications like

surveillance, search, rescue, firefighting, and UAV-based delivery.

Classical quadcopters can control their position and orientation by altering the rotors'

spinning speeds. For example, if the quadcopter is required to roll, it would make a

difference between the right and left propellers' speeds. In a similar manner, the desired

pitch angle can be achieved by changing the relative speeds of the front and rear rotors

[4]. For the translational motion, the quadcopter needs to roll for lateral motion and to

pitch for forward/backward motion. This coupling between the quadcopter states has the

undesired influence of changing the onboard surveillance camera viewing axis, so it

limits the quadcopter ability to do some vision-based tasks [5].

Considerable academic references have been reviewed to understand the current state

of the art in the quadcopter design. For the purpose of coping with the underactuation

problem, several prospects have been suggested through the reviewed literature spanning

different techniques in thrust vectoring concepts, and new mechanical configurations.

Many gaps have been filled and several new novel designs have been proposed aiming to

improve the traditional quadrotor performance.

In [6], the authors propose a quadcopter with four tiltable wings distributed under its

rotors. The modified quadcopter can change its mode from quadcopter to aircraft and vice

versa by tilting its wings all at once. M. K. Mohamed et al. propose a novel tri-tilt-rotor

UAV in which the three rotors can be tilted independently to gain a full authority and

control over the vehicle states [7]. Following a similar concept, the rotor tilt mechanism has

been utilized in several studies to resolve the coupling problem between the quadcopter

position and orientation. For instance, M. Ryll et al. design and construct a quad tilt rotors

mini quadcopter that converts the traditional quadcopter into an overactuated air vehicle [8].

Another solution to resolve the fundamental underactuation limitations of the quadrotor is

made by adding one degree of freedom to each of the quadcopter rotors [9]. Looking for

further improvement, Fernandes presents a quadcopter with 2–axis tilting mechanism added

to two of its rotors [10]. For the purpose of more maneuverability and robustness, the

authors in [5, 11, 12] propose a quadcopter with a novel 2–axis tilting mechanism in which

the rotors have two degrees of freedom. Badr et al. introduce a modified quadcopter that

allows each rotor to tilt about the axes perpendicular to its arm [13].

In this paper, an H-shaped racing quadcopter with tiltable rotors is introduced. To the

authors’ best knowledge this configuration is novel and its dynamical model has not been

obtained yet. The four rotors; which are ordinarily fixed, are allowed to rotate around

their arm axes simultaneously. With this tilting capability, the number of control inputs is

increased to five (the four rotors spinning speeds plus the tilt angle). The additional

control input is used to govern the forward/ backward speed of the proposed model.

The main contributions of this work are: (1) To derive a complete dynamical model

for the H-shaped racing quadcopter with tilting propellers, (2) To propose and design a

tracking control aimed at exploring the rotor tilt advantages, (3) To conceptualize the tilt

mechanism and thrust vectoring concept.

Modelling and Control of H-Shaped Racing Quadcopter with Tilting Propellers 203

The paper is outlined as follows: The mathematical model is derived in Section 2. The

controller is designed and tuned in Section 3. A comprehensive set of numerical simulation

tests is then carried out in Section 4. Finally, the concluding remarks are given in Section 5.

2. SYSTEM MODELING

The aim of this section is to develop a dynamical model for the H-shaped tilt rotors

racing quadcopter so as to define the ratio that relates the quadcopter states with the

propellers spinning speed and the rotor tilt angle. At this stage, it is important to state

some acceptable hypotheses that are used to simplify the process of obtaining the system

dynamics. Without these hypotheses, the system mathematical model will be complicated

and difficult to obtain [14]. These hypotheses are:

1. The system is assumed to be symmetric and rigid.
2. The quadrotor body frame origin and the centre of gravity (CoG) are coinciding.
3. The actuators are assumed to have sufficient fast response, so their dynamics are

neglected [15].

2.1. Model configuration

The tilt rotor H-shaped racing quadcopter will be shortened as (Hcopter) throughout

the following sections. The proposed model consists of an H-shaped frame with four

rotors distributed at the frame tips as shown in Fig. 1a. The rotors are allowed to tilt

simultaneously around the arms connecting them to the main frame in the range of

–π/2 < α < π/2. Fig. 1b shows the rotor tilt angle.

a) b)

Fig. 1 Hcopter configuration: a) Model CAD drawing; b) Tilt rotor angle

2.2. Coordinate Systems

First, let F
E
:{X

E
, Y

E
, Z

E
} being the earth frame and F

B
:{X

B
, Y

B
, Z

B
}being the base

frame (see Fig. 2) with its center coinciding with the vehicle's CoG. It is necessary to

define these frames since some quantities should be expressed in F
B
(for instance the

rotors' generated thrusts) while the other should be defined in F
E
(the gravitational force).

A superscript letter "B" is assigned to the variables that belong to F
B
, while "E" lettered

superscript is used to denote the variables resolved in F
E
.

204 A. ALKAMACHI, E. ERCELEBI

Fig. 2 The system model schematic showing the reference frames, Euler angles, and axes

To go from F
E
to F

B
, a rotation matrix should be introduced. The rotation matrix

(also called the direction cosine matrix), is a result of multiplying three canonical rotation

matrices RX(φ), RY(θ), and RZ(ψ) with a specific sequence [16], where φ (roll), θ (pitch),

and ψ (yaw) are the NASA based Euler angles [17]. It follows that:

( ) ( ) ( )* *
B

E X Y Z
R R R R  

C C C S S

C S S S C C C S S S S C

S S C S C S C C S C C C

    

           

           

 
 
    
 
   
 

(1)

where
B
RE is the orientation matrix of the variables in F

E
with respect to F

B
. The C and S

are the sine and cosine functions, respectively.

For the reverse operation (transferring the variables from F
B
to F

E
), an inverse matrix

E
RB =(

B
RE )

–1
should be used. Since the orientation matrix is a result of orthogonal

matrices multiplication, then its inverse is just its transpose [16].

1
( ) ( )

TE B B

B E E
R R R


  (2)

Modelling and Control of H-Shaped Racing Quadcopter with Tilting Propellers 205

2.3. Static and Dynamic Model

In this section, all the dominant forces and torques that act on the Hcopter body are

discovered.

2.3.1. Forces

Rotors' generated forces
B
FTR: Assume that the i

th
propeller spinning speed is

denoted by i. Then, according to the blade element theory [18, 19], the generated force

in the z–direction is given by k (i)
2
, where k is the rotor thrust coefficient. When the

rotor tilts with an angle α, then the generated force can be resolved into its components

along the x and z axes. It follows that the i
th

rotor generated force is:

Ω sin( )

Ω cos( )

B
Ri



 
 

  
 
 

(3)

and the total generated force due to the four spinning propellers is:

B B

TR Ri

F F K U


  (4)

where

0 0 0 0

Κ 0 0 0 0 0 0 0 0

0 0 0 0

k k k k

 
 


 
  

is the thrust coefficient matrix, and

2 2 2 2 2 2 2 2

1 1 2 2 3 3 4 4
[Ω * , Ω * , Ω * , Ω * ,Ω * , Ω * , Ω * , Ω * ]

T
U S C S C S C S C

       


is the control input vector, Sα and Cα are the sin(α) and cos(α) function respectively.

Gravitational force
E
FG: According to the Newton's law of universal gravitation, this

force tries to pull down the vehicle toward the earth and it can be expressed in F
E
by:

0
E

G
F

 
 


 
  

(5)

where g is the gravitational constant, and m is the vehicle total mass.

Drag reluctant force
B
FD: It is a result of the air friction with the quadcopter body in

motion and it is directly proportional to the vehicle moving speed.

B B

D d
F K P  (6)

where Kd is the 3×3 aerodynamic coefficient matrix, and [ , , ]
B T

P x y z is the Hcopter

velocity vector which represents the time derivative of quadcopter body position vector P
B
.

206 A. ALKAMACHI, E. ERCELEBI

Total force
B
FT : The total force acting on the quadcopter body is the vector sum of

the above three individual forces.

*
B B

TR E G

B B E

T D
F R FF F  (7)

2.3.2. Torques

Rotors' generated torque
B
MTR: It is a result of the four rotor's generated force

around the CoG. At this point, it is assumed that the origin of F
B
and quadcopter CoG

coincide. The total moment due to the rotors' generated forces is:

( )
B

i RiTR

L FM


  (8)

where Li being a vector directed from the CoG to the i
th

rotor, i.e.:

L1=[Lx,Ly,0]
T
, L2=[–Lx,Ly,0]

T
, L3=[–Lx,–Ly,0]

T
, and L4=[Lx, –Ly,0]

T
, and Lx, Ly are

shown in Fig. 3.

Fig. 3 Hcopter schematic

Aerodynamic drag torque
B
MDT : It is the counter rotating torque due to the air drag

caused by propeller spinning [20]. According to the blade element theory [18, 19], the i
th

rotor's drag torque around the z–axis can be expressed as (–1)
i
b(i)

2
, where b is the rotor

drag coefficient and the factor (–1)
i
is negative for the propellers rotating clockwise (CW)

(rotors 1 and 3) and positive for those rotating in a counter clockwise (CCW) direction

(rotors 2 and 4). Recall that the rotors can be tilted with angle α, then the drag torque of

the i
th

propeller is resolved into x and z components as follows:

( 1) Ω sin( )

( 1) Ω cos( )



 
 

  
  

(9)

so the total drag torque for the four propellers is:
4

B B

DT Di

M M


  (10)

Modelling and Control of H-Shaped Racing Quadcopter with Tilting Propellers 207

Total torque
B
MT: Expressed in F

B
, the total torque acting on the Hcopter body is the

vector sum of the above two individual torques.

B B B

T TR DT
M M M B U   (11)

where

Β 0 0 0 0

y y y y

x x x x

y y y y

b kL b kL b kL b kL

kL kL kL kL

kL b kL b kL b kL b

    
 

   
     

is the moment coefficient matrix, and U is the control input vector as before.

2.3.3. Virtual control vector V

At this phase of the mathematical modelling and for the purpose of better

understanding, it is important to define a virtual control vector V =[V1, V2, V3, V4]
T
. First

virtual control input V1 is in charge of controlling the vehicle altitude since it represents

the resultant lifting forces generated by the rotors in the upward positive z direction.

2 2 2 2

1 1 2 3 4
(Ω Ω Ω Ω )cos( )

TRz
V F k      (12)

Second virtual control input V2 is the total torque around x–axis; thus it is responsible

for controlling the roll angle.

2 2 2 2 2 2 2 2

2 1 2 3 4 1 2 3 4
( Ω Ω Ω Ω )sin( ) Ω Ω Ω Ω cos(( ) )

Tx y
V M b kL           (13)

Third virtual control input V3 represents the total torque around y–axis so it controls

the pitch angle of the Hcopter.

2 2 2 2

3 1 2 3 4
Ω Ω Ω Ω cos )( ) (

Ty x
V M kL       (14)

Fourth virtual control input V4 is responsible for adjusting the yaw angle since it

represents the total torque around z–axis.

2 2 2 2 2 2 2 2

4 1 2 3 4 1 2 3 4
Ω Ω Ω Ω sin( ) ( Ω Ω Ω Ω( ) ) cos( )

Tz y
V M kL b            (15)

In addition to the above virtual control signals, rotor tilt angle α is used to control the

forward/backward speed of the vehicle. When the rotors tilt, they provide the required

horizontal force to move the Hcopter along the x–axis direction.

Equations (11) through (14) can be combined into one matrix equation. It follows that:

V U  (16)

where Γ is the coefficient matrix and is equal to:

0 0 0 0

y y y y

x x x x

y y y y

k k k k

b kL b kL b kL b kL

kL kL kL kL

kL b kL b kL b kL b

 
 
   

 
  
 
     

208 A. ALKAMACHI, E. ERCELEBI

2.3.4. Model dynamics

With a view to obtain the quadcopter dynamic and the equations of motion, we exploit the

typical Newton-Euler formalization. Recall that P
B
represents the Hcopter position vector

expressed in F
B
, then the Newton-Euler equations are:

B B
T

mP F (17)

and

( )
B B B B

T
J J M     (18)

where JR
3×3

is the moment of inertia tensor, ωB is the body angular velocity vector, and
B

 is the angular acceleration of the quadcopter body expressed in F
B
.

Substituting Eq. (7) in Eq. (17), we can get the linear acceleration vector expressed in

F
B
as:

1 1
( ) 0

TRx
B B B B B

E G d G

F
P K U R F K P R F K

V
P

m m

  
       
  
  

(19)

Angular acceleration B can be obtained by substituting Eq. (11) in Eq. (18):

2
1 1

( ) (( ))
B B B B B B

O J BU VJ
V

J
V

J    
 

  
        
    

(20)

where O
B

represents the vehicle orientation vector expressed in the body coordinate

( [ , , ]
B T

O    ), therefore ( [ , , ]B TO    ).

To this end, we have obtained the Hcopter dynamical equations that govern its

operation.

3. CONTROLLER DESIGN

In this work, the (Matlab/Simulink) environment is used to integrate and examine the

obtained model. It is also used to tune the PID parameters using the genetic algorithm

(GA) and to carry out all the subsequent tests. The simulation environment is set up on a

personal computer with 2.5 GHz processing speed and 6 GB RAM, running on Windows

10. The complete control system block diagram for the Hcopter is shown in Fig. 4. The

list of model physical parameters used through all the tests is given in Table 1.

Fig. 4 The proposed controller block diagram

Modelling and Control of H-Shaped Racing Quadcopter with Tilting Propellers 209

Table 1 The proposed Hcopter model physical parameters

Parameters Value-

m 1.2 Kg.

LX, LY 20 cm

k, b 3e-6 N.sec
2
/rad

2
, 1e-7 Nm.sec

2
/rad

J diag[0.02, 0.02, 0.04]
*
Kg.m

Kd diag[0.3, 0.3, 0.5]
*

min, max
**

100, 2000 rad/sec
*
diag[ ] is a diagonal matrix

**
It is the rotor's upper and lower speed limit respectively

For efficient surveillance tasks, it is important that the quadcopter has a precise control

on its attitude, speed, and altitude [11]. The control problem addressed in this work is an

output tracking problem. A five output PID control system is proposed to control the

orientation (roll, pitch, and yaw), altitude, and the forward speed of this novel air vehicle.

3.1. Forward/backward speed control

The additional control input (tilt angle α) is used to control Hcopter forward speed which

in turns improves the surveillance based tasks. The speed error signal is obtained by

subtracting desired x speed ( dx ) from real Hcopter speed ( x ). The error signal is then fed to

a PID controller that determines the required tilt angle (α) to achieve the desired speed.

3.2. Altitude control

The altitude error signal is formed by subtracting the measured altitude (z position)

from desired elevation zd. It is then applied to the PID controller that adjusts the value of
control input V1 to achieve the required altitude.

3.3. Orientation control

The orientation controller is the core of the control system and it is of critical importance.

It consists of three PID controllers to keep the quadcopter attitude to the required roll (φd),

pitch (θd) and yaw (ψd) angles by controlling the three virtual control signals V2, V3, and V4,
respectively.

3.4 Virtual control to rotors' speeds

In this block, the determined rotor tilt angle (α) and the virtual control vector (V) are

used to determine the proper rotors' spinning speeds with the aid of Eq. (16).

3.5 PID parameters tuning using GA

Genetic algorithm is a search heuristic process that simulates the natural selection

procedure [21]. The algorithm is used to tune the five PID parameters by minimizing the

following cost function:

(Ω Ω Ω 1,2,3,4)
|

min i max for i
Obj err

  
 (21)

where err is the difference between the desired and the actual output response.

210 A. ALKAMACHI, E. ERCELEBI

The tuning process is subjected to actuator upper and lower constraints as noted in Eq.
(21). It means that if the selected PID parameters lead to excessive control signal output that
cause the actuators to exceed their upper and lower limits, then the cost function value for
these selected parameters is assigned to an extremely large weight so that it will be
excluded from the next iteration. The genetic algorithm takes the response data from the
Simulink model to evaluate the cost function and to select the optimal PID parameters. The
obtained PID parameters with their associated step response characteristics (settling time ts
and percentage peak overshoot MP) are tabulated in Table 2.

Table 2 PID parameters and step response characteristics

Controlled state

(Initial Final)

KP KI KD ts

Sec.

MP
%

(0 1 .)
d

z m 22.2 22.4 7.1 1.18 1.5

(0 5deg.)
d

  17.5 1.04 0.78 0.16 0.3

(0 5deg.)
d
  17.5 1.04 0.78 0.16 0.3

(0 10 deg.)
d

  1.63 0.04 0.25 0.43 0.4

(0 2 / sec.)
d

mx  0.99 0.99 0.05 2.11 7.8

4. SIMULATION RESULTS

In order to check the validity of the proposed model and controller, a Simulink model
is developed. The purpose of the simulation is to highlight the Hcopter capabilities and
the dynamic rotor tilting advantages. The simulation process falls into two phases. The
first simulation assumes the ideal case and the second simulation assumes the existence
of sensor noise, parameters uncertainties, and external disturbances.

4.1. Ideal case

Altitude and attitude tracking: In this test, it is assumed that the Hcopter is initially at
rest (P

E
=[0,0,0]

T
, O

B
=[0,0,0]

T
) and it is then ordered to climb to 1 meter (z = 1m.). The

vehicle is then commanded to follow the following desired attitude (φd =5deg. at t=1sec., θd =
–5deg. at t=2sec., ψd =10deg. at t=3sec.). The simulation results for the desired step inputs are
shown in Fig. 5, while the step input characteristics for this test are given in Table 2.

Hcopter speed control: In this part of the simulation, the Hcopter is assumed to be

hovering at z = 1 meter, and it is required that the body roll, pitch and yaw angles should

be kept at zero degree.

Test 1: The Hcopter in this test is examined for its ability to maintain a constant speed

while keeping its body leveled φd = θd = 0deg. The speed throttle is applied gradually
simulating a ramp input from rest to 10 m/sec(36Km/h) in 5 seconds as shown in Fig. 6a. The

simulation results in Figs. 6b and 6c show that the Hcopter can track the desired speed

efficiently while maintaining zero attitudes which is impossible for conventional quadcopter

to do [22].

Test 2: The aim of this test is to find the maximum forward moving speed that our

proposed Hcopter can reach. The tilt angle is increased gradually from 0 deg to its maximum

Modelling and Control of H-Shaped Racing Quadcopter with Tilting Propellers 211

allowable limit (45 deg.) in 10 sec. It can be seen from Fig. 7 that the maximum reachable

speed is approximately 39 m/sec (140Km/h).

Fig. 5 Step input test simulation results: a) Altitude response; b) Attitude response

Fig. 6 Speed control test simulation results:

a) Desired speed; b) Speed error; c) Attitude behavior

Fig. 7 Hcopter maximum speed test simulation result

212 A. ALKAMACHI, E. ERCELEBI

Air braking: From the previous test, it can be noticed that the proposed quadcopter

could reach a very high forward speed in a relatively short period; thus, a solid brake is

required to rein the vehicle efficiently at the proper time. Fortunately, the rotor tilt

mechanism offers an instant braking system that reduces the vehicle speed to zero in a very

short period. In this test, the Hcopter forward speed is increased to 10 m/s and then suddenly

reduced to zero. From Fig. 8a, it can be seen that the rotors tilted with a positive angle during

the acceleration period and instantaneously flipped to the opposite side to provide the

necessary horizontal opposite force to stop the vehicle. The Hcopter speed behavior is

shown in Fig. 8b.

Fig. 8 Air braking test simulation results: a) Tilt angle behavior; b) Vehicle speed

4.2. Non-ideal case

In the non-ideal case, three tests were made to examine the model validity and the

controller robustness with the existence of white Gaussian sensor noise, external disturbances,

and model parameters uncertainty.

Sensor noise: The Hcopter is assumed to be hovering horizontally at z = 1 meter. A

white Gaussian noise with zero mean (shown in Fig. 9a) is applied to the model in the

feedback loop to simulate the sensor noise.

The simulation results in Figs. 9b and 9c reveal the controller ability of suppressing

the sensor noise successfully. The maximum error in the altitude is just a few millimeters

and the maximum drift in the orientation is bounded by +/– 0.01 degree.

External disturbance: The Hcopter is assumed to be hovering horizontally at z = 1

meter and it is commanded to travel with 10 m/sec(36Km/h) in the forward direction with

the existence of opposite light wind with a fixed speed of 1.2 m/sec(4.5Km/h) [23].

The simulation result shown in Fig. 10 asserts the controller effectiveness in coping

with the external disturbances.

Modelling and Control of H-Shaped Racing Quadcopter with Tilting Propellers 213

Fig. 9 Noise suppression test simulation results:

a) Noise signal; b) Altitude drift; c) Attitude drift

Fig. 10 Hcopter speed error with and without the effect of a light wind disturbance

Model parameters uncertainties: To check the controller effectiveness in dealing

with the model uncertainties, the vehicle is ordered to change its altitude from 1 to 2

meter with and without the existence of a 0.5 Kg. payload. The payload is suspended

from the CoG of the model. A comparison between the results (see Fig. 11) for both cases

shows that the controller performed well against the external additive weight.

Fig. 11 Uncertainty test simulation results

214 A. ALKAMACHI, E. ERCELEBI

5. CONCLUSION

In this paper, we have addressed the modeling and control of a novel tilt rotor H-shaped

racing quadcopter. Compared to the traditional quadcopter, the proposed model offers higher

moving speed with instant air brake. The new design resolves an important part of the

underactuation problem in the traditional quadcopter where the translational and rotational

motion cannot be controlled independently. The Hcopter, in contrast, allows controlling the

rotors thrust direction, therefore granting additional control input results in the uncoupling

between the pitch and the forward motion. Several simulation tests have been carried out and

the results have been demonstrated and discussed to evaluate the proposed controller

validation. By obtaining these encouraging results, the next stage is to design, build, and

control the Hcopter prototype.

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