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1. Dynamic modeling and control of a Hexacopter using
PID and Back Stepping controllers
Kattupalli Venkateswara Rao
Instrumentation and Control student in Electrical
Engineering
National Institute of Technology
Calicut, Kerala, India
Venkykattupalli12@gmail.com
Dr.Abraham T.Mathew
Professor, Department of Electrical Engineering
National Institute of Technology
Calicut, Kerala, India
atm@nitc.ac.in
Abstract—A HexaRotor UAV is a class of helicopter, more
specifically of multi-rotors. The Hexarotor has several
characteristics (vertical takeoff and landing, hovering capacities)
that give more operational advantages over other types of UAVs.
But the Hexarotor has highly nonlinear dynamics, coupled and
underactuated which makes it impossible to operate without a
feedback controller action.
In this work, a detailed mathematical model of the Hexrotor has
been presented. With the help of Newtons Euler method the
nonlinear mathematical model of the hexacopter was formulated;
by including rotor dynamics and aerodynamic effects the model was
formulated in detail. In order to control the attitude and altitude of
the hexarotor in space, two control schemes were proposed i.e, PID
and BACK STEPPING Controllers. For evaluating and comparing
the performance of both proposed control techniques in the aspect
of stability, the effect of possible disturbances and dynamic
performances, experiments were done in the Simulink environment.
Keywords—Hexarotor; Nonlinear control; Newton-Euler
method; PID; Back Stepping;
I. INTRODUCTION
This work mainly focuses on modeling and controlling of the
hexacopter. the Hexacopter possesses better performance when
compared with another type unnamed aerial vehicles, which makes it
suitable for this field. Still controlling of the hexacopter is difficult in
research, as it is the nonlinear, underactuated and multivariable
system.
Underactuated systems having a less number of control inputs
compared to the system’s output i.e, degrees of freedom. Because of
the nonlinearity in coupling between the degrees of freedom and
control inputs, the controlling is difficult. Although from
literature[1],[2],[3], some flight linear control algorithms were found,
which can only perform when the Hexacopter is hovering. If the
hexacopter leaves the nominal conditions, they suffer from the huge
performance degradation. To overcome this one, use nonlinear
control methods[4],[5].
The contributions of this work are modeling the hexacopter along
with developing linear and nonlinear control techniques, and the
same has been implemented in computer-based simulations. The
work has been concluded by comparing both the control techniques
in terms of their performance and stability.
II. SYSTEM MODELING
The attitude of the hexarotor has been described by the
mathematical model. All the six propellers of the unnamed aerial
vehicle are placed orthogonally along with the body frame. There are
3 movements that describe the attitude: Roll angle (spinning around
X-axis) was obtained when the balance of rotors1, 2 and 3(or 6, 5 and
4) was changed (speed increases or decreases), lateral acceleration
was obtained, by changing the angle; pitch movement (spinning
around the Y-axis) was obtained when the balance of the speed of the
rotors 1 and 6(or 3 and 4) was changed. The longitudinal acceleration
was obtained by changing the angle; yaw angle (spinning about the
Z-axis) was obtained by a continuous change of speed of the motors
(1,3,5) or (2,4,6).
A. Rotation matrix R
This section describes the dynamical models of the six rotors.
The schematic structure of the hexarotor and the rotational directions
of the propellers are illustrated in figure 1. In order to describe the
hexarotor motion, only two reference systems are necessary: earth
inertial frame (RI-frame) and the body fixed frame (RB-frame).
y
z
x
Figure 1. Structure of the hexacopter
The vehicle-1 frame has been obtained by the yaw angle ψ with the
rotation of inertial frame about its z-axis. The vehicle-1 frame has
been transformed from the inertial frame by












1
0
0
0
cos
sin
0
sin
cos
)
(
1





v
i
R
indicates a spinning from frame fi which is
the inertial frame to frame fv1 which is the vehicle-1 frame. The
vehicle-2 frame has been obtained with the rotation of vehicle-1

2. about its y-axis with pitch angle θ. The vehicle-2 frame has been
transformed from vehicle-1 frame by









 






cos
0
sin
0
1
0
sin
0
cos
)
(
2
1
v
v
R
The body frame has been found from the rotation of the vehicle –2
frames about its x-axis. The body frame has been transformed from
the vehicle-2 frame by

















cos
sin
0
sin
cos
0
0
0
1
)
(
2
b
v
R
the transformation from inertial to body
frame is given by )
(
)
(
)
( 

 i
v
b
b
i R
R
R
R 













































cos
cos
sin
cos
sin
sin
cos
sin
sin
cos
sin
cos
cos
sin
cos
cos
sin
sin
sin
cos
sin
sin
sin
cos
sin
cos
sin
cos
cos
b
i
R
The rotation matrix which has been used for transforming the body
frame to inertial frame is given below













































cos
cos
cos
sin
sin
sin
cos
sin
sin
cos
cos
cos
sin
sin
sin
cos
sin
sin
sin
cos
sin
cos
cos
sin
sin
sin
cos
cos
cos
i
b
R
The orientation vector T
]
[ 


  has been formed with the 3
Euler angles, namely yaw angle ψ, pitch angle θ, and roll angle ϕ.
The vector T
z
y
x ]
[

 denotes the position of the vehicle in an
inertial frame.
B. Kinematics and dynamics
In this part, dynamics of a rigid body and kinematics have been
derived below.
1. Hexarotor kinematics
By taking the inertial frame quantities as state variables x, y and z
and the body frame quantities as velocities u, v and w, the
relationship between the position and velocities can be found out by





















w
v
u
R
z
y
x
dt
d i
b
=












































cos
cos
cos
sin
sin
sin
cos
sin
sin
cos
cos
cos
sin
sin
sin
cos
sin
sin
sin
cos
sin
cos
cos
sin
sin
sin
cos
cos
cos










w
v
u
The relationship between Euler angles and the angular rates p, q, and
r is


















































cos
cos
sin
0
cos
sin
cos
0
sin
0
1
r
q
p where 
 
 r
R
2. Rigid body dynamics
By taking the velocity of the hexarotor as V, after applying
Newton’s laws to the translational motion can be founded by,
f
dt
dV
m  where m=mass, f= net force, d/dt=time derivative in
inertial frame. Where  is the angular velocity w r to the inertial
frame T
r
q
p ]
[

 ,  T
w
v
u
V  the equation is








































z
y
x
f
f
f
m
pv
qu
ru
pw
qw
rv
w
v
u
1
for rotational motion, Newton’s
second law states that m
dt
dh
i
b
 where h and m are defined as
angular momentum and applied torque respectively. From Coriolis
equation, we can write Xh
dt
dh
dt
dh
b
i


 again the equation most
easily resolved in body coordinates where hb
=Jωb
, inertia matrix J is
given by

















z
yz
xz
yz
y
xy
xz
xy
x
J
J
J
J
J
J
J
J
J
J
The hexarotor is completely symmetric about all three axes, so
Jxx=Jyy=Jzz=0











z
y
x
J
J
J
J
0
0
0
0
0
0
C. Applied forces and torques
The model has been made more realistic by including the analysis
of air friction and rotor drag along with the force of gravity and thrust
force of rotor. The unnamed aerial vehicle movements are governed
by aerodynamic or mechanical effects, which make the UAV more
complex.
For deriving the mathematical model of the hexarotor, the Newton
formalism is used. Therefore the following equations are obtained:


































M
F
J
mV
V
J
mI
X
X
X




3
3
3
3
3
3
0
0 (2.1)
Where F=net force acting on the center of mass, m=mass of the body,
V=velocity of the center of mass, the M=resultant torque acting on
the center of mass, ω=angular velocity of the body, J=Moment of
inertia about the center of mass.
mV
V
m
Fb




  

 J
J
M 




1. Forces
(i). Gravitational force: the gravitational force vector acting on the
hexarotor center of gravity in the body coordinate frame can be
expressed as T
g mg
F ]
0
0
[ 
 where m=mass of the hexarotor,
g=gravitational acceleration.
(ii). Thrust force let Ωi be the thrust produced by the propeller i, the
total force Fi to lift the hexarotor is
 
2
6
2
5
2
4
2
3
2
2
2
1 











i
F . The thrust from the
propellers acting on the hexarotor along the z-axis on the body
coordinate frame can be expressed as
T
i
i
i
b
T
i
i
i
b
p b
R
F
R
F ]
0
0
[
]
0
0
[
6
1
2
6
1

 



 where b=thrust coefficient
factor.
(iii). Rotor drag: the drag equation=
2
2
A
C
F D
D

 where FD=drag
force, ρ=mass density, μ=flow velocity relative to the object,
CD=drag coefficient, A= ref area.



 
T
ftz
fty
ftx
X
ft
t k
k
k
I
V
k
F ]
[
3
3
, T
z
y
x ]
[



 
V
The vector of the drag forces, )
,
,
( ftz
fty
ftx
ft k
k
k
diag
k 
(iv). Air resistance

3. 2
2
2
i
i
i d
r
CA 


 
 , where C = drag coefficient of the
propeller, A = area of the blade, ρ = air density, r = blade radius and
Ωi = propeller angular velocity.
2. Torques
(i). actuator action: the roll torque Mx is the torque produced around
x-axis with propeller thrust Ti is,
Mx=-sin300
LT1-LT2-sin300
LT3+sin300
LT4+LT5+sin300
LT6
2
2
2 6
5
4
3
2
1 lT
lT
lT
lT
lT
lT
Mx






 where L = arm length. The
pitch torque My is the torque produced around y-axis is,
My=sin600
LT1-sin600
LT3-sin600
LT4+sin600
LT6
2
3
3
3
3 6
4
3
1 lT
lT
lT
lT
M y



 the yaw torque Mz is the
torque produced in the z-axis in the body-fixed frame. Every
individual propeller shaft generates the torque when DC-motor
accelerates and maintains the rotation motion of the propellers. As
per Newton’s third law, the motor subject to produce equal torque in
the opposite direction from the propeller’s shaft. These propellers are
mounted on the body of hexarotor. So the torque generated by them
will propagate to the airframe. The torque produced by propeller by
propeller blades is often named as reaction torque and is given by τi
for propeller i is 6
5
4
3
2
1 




 






z
M .The angular
rotational speed of propeller i is denoted as Ωi, for generated thrust Ti
and generated torque τi is 2
i
i b
T 
 , 2
i
i d

 , 
 4
p
T r
C
b  ,

 5
p
Q r
C
d  ,ρ is air density, rp is the propeller radius, CT thrust
coefficient, CQ torque of the propeller. The vector Mf can be written
as T
z
y
x
f M
M
M
M ]
[
 .
(ii). Torque aerodynamic resistance
T
faz
fay
fax
a k
k
k
M ]
[ 2
2
2



 

 where kf aerodynamic force
constant.
(iii). Gyroscopic effect
The rotational motion of the propeller rotor combination generates a
gyroscopic effect that acts on the hexarotor in the body coordinate
frame. The gyroscopic effect is contributed by the rotor’s moment of
inertia, the rotor’s angular velocity and the body attitude rate, which
can be expressed by
r
r
r
r
gh X
r
q
p
J
X
J
M 





















































1
0
0
1
0
0
 where 6
4
3
2
1 5













r
r
r
r
r
gh J
p
q
J
M 



























0
0


D. Hexarotor mathematical model
The rotational and translational motion, known as equations of
motion of the hexarotor with respect to body frame is
1. Translational dynamics






F
F
F
F
m t
g
p

m
x
k
F
x i
ftx
i








6
1
)
sin
sin
sin
cos
(cos 




(2.2)
m
y
k
F
y i
fty
i








6
1
)
cos
sin
sin
sin
cos 




(2.3)
g
m
z
k
F
z i
ftz
i








6
1
)
cos
(cos 

(2.4)
2. Rotational dynamics
gh
a
f M
M
M
J
J 









)
2
)
(
(
)
(
2
6
2
4
2
3
2
1
2
5
2
2
2 
























bl
J
k
J
J
J r
r
fax
zz
yy
xx 




(2.5)
2
)
(
3
)
(
2
6
2
4
2
3
2
1
2 



















bl
J
k
J
J
J r
r
fay
xx
zz
yy 




(2.6)
)
(
)
( 2
6
2
5
2
4
2
3
2
2
2
1
2





















d
k
J
J
J faz
yy
xx
zz 


 (2.7)
The vector of control input variables, UT
=[u1, u2, u3, u4] can be
found out by relating the net thrust force and torque control inputs u1,
u2, u3, u4 with the six motor’s speed, which is given below































































2
6
2
5
2
4
2
3
2
2
2
1
4
3
2
1
2
3
0
2
3
2
3
0
2
3
2
2
2
2
d
d
d
d
d
d
bl
bl
bl
bl
bl
bl
bl
bl
bl
bl
b
b
b
b
b
b
u
u
u
u (2.8)
3. Total system model
Finally, this derivation provides the 2nd
order differential equations
for the aircraft’s position and orientation in space. Applying relation
(2.1 to 2.8) and rewriting the matrix equation in form of the system,
we obtain the following:
xx
r
r
fax
zz
yy
J
u
J
k
J
J ]
)
(
[ 2
2











 



 (2.9)
yy
r
r
fay
xx
zz
J
u
J
k
J
J ]
)
(
[ 3
2











 



 (2.10)
zz
faz
yy
xx
J
u
k
J
J ]
)
(
[ 4
2








 



(2.11)
m
u
u
x
k
x
x
ftx 1






(2.12)
m
u
u
y
k
y
y
fty 1






(2.13)
g
m
z
k
z
ftz






 
 cos
cos (2.14)
Where 



 sin
sin
sin
cos
cos 

x
u ,




 cos
sin
sin
sin
cos 

y
u
The dynamic mathematical model presented in equation set (2.9-
2.14) can be redefined in the state space model )
,
( U
X
f
X 

.

4. 12
R
X  = state variables vector













z
z
y
y
x
x
X T



























5
6
5
3
4
3
1
2
1
x
x
x
x
x
x
x
x
x















z
x
x
z
x
y
x
x
y
x
x
x
x
x
x
11
12
11
9
10
9
7
8
7
2
1
4
3
2
2
2
6
4
1
2 u
b
x
a
x
a
x
x
a
x r 








 (2.15)
3
2
2
6
2
4
5
6
2
4
4 u
b
x
a
x
a
x
x
a
x r 








 (2.16)
4
3
2
6
8
2
4
7
6 u
b
x
a
x
x
a
x 






 (2.17)
x
u
u
b
x
a
x
x 1
4
8
9
8 





(2.18)
y
u
u
b
x
a
y
x 1
4
10
10
10 





(2.19)
g
u
m
x
a
z
x 






1
12
11
12
cos
cos 
 (2.20)
For simplification,
 
xx
zz
yy
J
J
J
a


1
 
xx
fax
J
K
a


2
 
yy
fay
J
K
a


5
 
zz
faz
J
K
a


8
 
xx
J
l
b 
1
 
xx
r
J
J
a


3
 
yy
xx
zz
J
J
J
a


4
 
m
K
a ftx


9
 
yy
r
J
J
a


6
 
yy
J
l
b 
2
 
m
K
a fty


10
 
m
K
a ftz


11
 
zz
J
l
b 
3
 
zz
yy
xx
J
J
J
a


7
m
b
1
4 
III. CONTROL OF HEXAROTOR
In this chapter, it was discussed about the linear and nonlinear
control operations for the hexarotor and these operations are used to
control the hexarotor at hovering condition, controlling the attitude
and expressing stability in the aspect of overshoot and settling time.
A. Pid controller
The main advantage of using PID controller is that parameter
gains are easily adjustable and it has a simple structure. Owing to the
nonlinearity and inaccuracy in a dynamic model of system dynamics,
hexacopter faces many challenges. Hence the performance of the
hexacopter gets limited by the use of PID controller. In order to get a
satisfactory degree of tracking performance in Euler angles, PID
controller has to be designed efficiently. The aspired control inputs
for the hexacopter are being generated by PID controller. The PID
controller basic block diagram has been shown in below figure.
Kp e(t)
Kp ∫e(t)
Kp( de(t)/dt)
process
-
Figure 2.block diagram of PID controller
1. Altitude controller
 







)
(
)
(
)
( ,
,
,
1 d
z
i
d
z
d
d
z
p z
z
k
z
z
k
z
z
k
U where zd and

d
z
Are desired altitude and altitude rate of change.
2. Attitude controller
The prime objective of the controller is to fix the hexacopter at
hovering position. For controlling ϕ, θ, ψ dynamics, PID controller
laws can be given as
 







)
(
)
(
)
( ,
,
,
2 d
z
i
d
z
d
d
z
p k
k
k
U 





 







)
(
)
(
)
( ,
,
,
3 d
z
i
d
z
d
d
z
p k
k
k
U 





 







)
(
)
(
)
( ,
,
,
4 d
z
i
d
z
d
d
z
p k
k
k
U 




 where ϕd desired
roll, θd desired pitch, ψd desired yaw. In order to maintain the
stability, the nonlinear dynamics of the hexacopter are linearized
around hovering point, which can be formulated by
xx
J
lu2




yy
J
lu3




zz
J
lu4



 
g
x 



g
y 



m
u
z 1



Apply Laplace transform, we get
xx
J
S
l
s
U
s
2
2 )
(
)
(


yy
J
S
l
s
U
s
2
3 )
(
)
(


zz
J
S
l
s
U
s
2
4 )
(
)
(


m
s
U
s
Z
)
(
)
( 1

3. Attitude control design
Kp e(t)
Kp ∫e(t)
Kp( de(t)/dt)
230/(7.5S2)
-
Figure 3.control diagram of the system
From the block diagram
S
k
Sk
k
S
s
E
s
U i
p
d 


2
2
)
(
)
(
2
2 5
.
7
230
)
(
)
(
S
s
U
s


The overall transfer function of the system is
i
p
d
i
p
d
d k
Sk
k
S
S
k
Sk
k
S
s
s
67
.
30
67
.
30
67
.
30
)
(
67
.
30
)
(
)
(
2
3
2







 the desired
conditions are Mp=15%, ts=2sec. we get Kp=1.46, Ki=2.151,
Kd=0.326.
4. Results
(i). At hovering condition Zd=5, θd=ϕd=ψd=0
(a)

5. (b)
(c)
(d)
Figure 4. Altitude & attitude responses at hovering. (a)
altitude response. (b),(c),(d) attitude responses at hovering.
The above figure shows the responses of the hexacopter system at
hovering condition. And the figure shows the system altitude
response. The system is stabilized at 5 meters height with 36%
overshoot and settles at 1.75 sec with the help of PID controller.
(ii). Euler angle responses ϕd=θd=ψd=0.1 radians
(a)
(b)
(c)
Figure 5. Euler angle responses (a). roll angle (b). pitch
angle (c). the yaw angle
The PID controller stabilizes the Euler angles at desired condition 0.1
rad and it gives the 40% overshoot and settles at 3.2 sec with 2%
tolerance.
B. Backstepping controller
The backstepping controller was developed for nonlinear
dynamical systems. These systems are built from subsystems that
radiate out from an irreducible subsystem that can be stabilized. The
backstepping control algorithm works by dividing the controller into
small subsystems and subsequently stabilizes each and every
subsystem. The major benefit of using a backstepping controller is
that it cancels the nonlinearities in the system.
For controlling the attitude and altitude of the hexacopter
backstepping controller has been designed. Based on the state space
model(2.15-2.20), this controller is derived. Consider the following
system
     













 )
12
.
,.........
2
(
,
0
),
12
,
10
,
8
,
6
,
4
(
,
)
11
,
9
,
7
,
5
,
3
(
,
1
1
1 i
i
x
e
x
i
x
x
e
i
i
i
i
d
i
i
id
i


1. Backstepping control of the rotational subsystem
For designing the controller, the below-given steps are to be followed
At first, the tracking error (ei=xid-xi) is considered as the Lyapunov
function by using Lyapunov theorem while considering Vi as positive
definite and its time derivative is negative semidefinite.









 )
12
,
10
,
8
,
6
,
4
(
,
2
1
)
11
,
9
,
7
,
5
,
3
(
,
2
1
2
1
2
i
e
V
i
e
V
i
i
i
i
consider the Lyapunov
function V1=1/2(e1
2
) and )
( 2
1
1
1 x
e
e
e
V d 





 , by
inserting a virtual control input x2, stabilization of e1 can be
found out: 0
, 2
1
1
1
1
1
2 






e
V
e
x d 

 , in the next step
the augmented Lyapunov function is considered as











2
2
2
1
2
2
1
1
2
2
1
2
1
e
e
V
x
e
e d 
 and it’s time derivative is




 2
2
1
1
2 e
e
e
e
V
)
(
)
( 2
1
4
3
2
2
2
6
4
1
1
1
2
2
1
1
1
2 u
b
x
a
x
a
x
x
a
e
e
e
e
e
V r
d 















 The
control input U2 is then obtained, satisfying





 2
2
2
2
2
1
1
2 ,
0 V
e
e
V 
 is
negative semi-definite, e1&e2 converge to zero according to
Lyapunov theory.



















1
2
2
2
1
1
1
4
3
2
2
2
6
4
1
1
2 )
(
1
e
e
e
e
x
a
x
a
x
x
a
b
U d
r 



Follow the same steps for other Euler angles and translational
dynamics control inputs,

6. 


















3
4
4
4
3
3
3
2
6
2
4
5
6
2
4
2
3 )
(
1
e
e
e
e
x
a
x
a
x
x
a
b
U d
r 




















5
6
6
6
5
5
5
2
6
8
4
2
7
3
4 )
(
1
e
e
e
e
x
a
x
x
a
b
U d 



















11
12
12
12
11
11
11
12
11
3
1
1 )
(
cos
cos
e
e
e
e
z
x
a
g
x
x
m
U d 


















7
8
8
8
7
7
7
8
9
1
)
( e
e
e
e
x
x
a
u
m
u d
x 


















9
10
10
10
9
9
9
10
10
1
)
( e
e
e
e
y
x
a
u
m
u d
y 


2. Results
(i) At hovering condition Zd=5, ϕd=θd=ψd=0
(a)
(b)
(c)
(d)
Figure 6. (a). altitude response (b),(c),(d).attitude
responses at hovering
(ii). Euler angle responses ϕd=θd=ψd=1 radians
(a)
(b)
(c)
Figure 7. attitude responses (a). roll angle (b).
pitch angle (c). yaw angle.
Table 1: comparison of performance analysis

7. In the above table, comparison regarding the performances of
the two techniques i.e, PID and Backstepping controllers has
been detailed clearly. By using PID controller peak overshoot
36% with a settling time of 1.75 sec at hovering, whereas with
the backstepping controller the peak overshoot is reduced to
0.4% with a settling time of 9 sec. Though the Backstepping
controller is a nonlinear controller, it always gives better
results irrespective of the nonlinearity present in the system.
Therefore it is clear that by using Backstepping controller
system performances have been improved greatly.
IV. CONCLUSIONS
A. Conclusions and Future work
The main aim of this work is to derive the mathematical model
and to control the hexacopter by using linear and nonlinear control
methods. The mathematical model of the hexarotor unnamed aerial
vehicle was developed in detailed including the aerodynamic and
rotor dynamic effects. Two control methods were developed namely
PID and Backstepping for controlling the hexarotor system at
hovering position and to control the attitude. The simulation
environment was used to evaluate the PID and Backstepping
controller performance in the aspect of settling time and overshoot.
The backstepping controller gives the better performance even
outside the linear region compared to PID controller.
Under the future work context, one practical hexacopter system
with PID and backstepping controllers can be designed. The same
can be implemented in hardware environment for a field test. After
implementing both control techniques in hardware platform, the best-
suited control technique can be found. Simulink results can be
validated with hardware results.
References
[1] Mostafa Mousse, Adil Sayouti, Hicham Medromi, “dynamic modeling
and control of a hexacopter using linear and nonlinear methods”, an
international journal of applied information systems. vol. 9, August
2015. (references)
[2] C.balas, “modeling& linear control of quadrotor”, MSc thesis Cranfield
university 2007.
[3] A.alaimo, V.artale, C.milazzo, A.ricciardello, “mathematical modeling
and control of hexarotor”, an international conference on unnamed
aircraft systems, may 2013.
[4] H.bolandi, M.rezaei, R.mohsenipour, “attitude control of quadrotor with
optimized PID controller”, intelligent control and automation, August
2013
[5] M.A.M.basil, K.A.danapalasingam, A.R.husain,“design and
optimization of the backstepping controller for autonomous quadrotor
UAV,” Issn,2014.