BACKSTEPPING CONTROL FOR HEXA-ROTOR MICROCOPTER

Abstract

Unmanned autonomous aerial vehicles have become a real center of interest. In the last few years, their utilization has significantly increased. During the last decade many research papers have been published on the topic of modeling and control strategies of autonomous multirotors. Today, they are used for multiple tasks such as navigation and transportation. This paper presents the development of a dynamic modeling and control algorithm-backstepping controller of an autonomous hexa-rotor microcopter. The autonomous hexa-rotor microcopter is an under-actuated and dynamically unstable nonlinear system. The model that represents the dynamic behavior of the hexa-rotor microcopter is complex. Unmanned autonomous aerial vehicles applications are commonly associated with exploration, inspection or surveillance tasks.

Full text

ACTA TEHNICA CORVINIENSIS – Bulletin of Engineering
Tome VIII [2015] Fascicule 3 [July – September]
ISSN: 2067 – 3809
© copyright Faculty of Engineering - Hunedoara, University POLITEHNICA Timisoara
Gyula MESTER
BACKSTEPPING CONTROL FOR HEXA-ROTOR
MICROCOPTER
1. Óbuda University, Doctoral School of Safety and Security Sciences, Budapest, HUNGARY
2. University of Szeged, Faculty of Engineering, Institute of Technology, Robotics Laboratory, Szeged, HUNGARY
Abstract:
Unmanned autonomous aerial vehicles have become a real center of interest. In the last few years, their utilization has significantly increased.
During the last decade many research papers have been published on the topic of modeling and control strategies of autonomous multirotors. Today,
they are used for multiple tasks such as navigation and transportation. This paper presents the development of a dynamic modeling and control algorithm
- backstepping controller of an autonomous hexa-rotor microcopter. The autonomous hexa-rotor microcopter is an under-actuated and dynamically
unstable nonlinear system. The model that represents the dynamic behavior of the hexa-rotor microcopter is complex. Unmanned autonomous aerial
vehicles applications are commonly associated with exploration, inspection or surveillance tasks.
Keywords:
dynamic model, dynamic behavior, unmanned autonomous aerial vehicles, autonomous hexa-rotor microcopter, under – actuated,
dynamically unstable nonlinear system, complex, control strategies, backstepping controller
INTRODUCTION
Unmanned autonomous aerial vehicles have become a real center of
interest [1-14]. In the last few years, their utilization has significantly
increased. Today, they are used for civil and military applications, for
multiple tasks such as navigation and transportation. Unmanned
autonomous aerial vehicles applications are commonly associated with
exploration, inspection or surveillance tasks.
During the last decade many research papers have been published on
the topic of modeling and control strategies of autonomous multirotors
[15-30]. One of the unmanned autonomous aerial vehicles with a strong
potential is the hexa-rotor microcopter.
The autonomous hexa-rotor microcopter have numerous advantages
over quadrotors, since they can offer more:
≡
power and
≡
speed, due to the properly position of the rotors.
Figure 1. Hexa-rotor microcopter used for the experiments
The autonomous hexa-rotor microcopter , Figure 1, consists of six rotors
attached to a rigid body frame and these additional two rotors make it
able to carry:
≡
more payload,
≡
the longest flight time and
≡
high maneuverability
compared to quadrotor. The autonomous hexa-rotor microcopters have
additional redundancy over autonomous quad-rotor microcopters.
The control design carried out for an autonomous quad-rotor
microcopter can be applied to the autonomous hexa-rotor microcopter
since they are modeled as a rotating rigid body dynamic system wih six
degres of freedom (6 DOF), Figure 2.
Figure 2. The Hexa-rotor microcopter in hovering conditions
The autonomous hexa-rotor microcopter is an:
≡
under-actuated and
≡
dynamically unstable nonlinear system.
The model that represents the dynamic behavior of the hexa-rotor
microcopter is nonlinear and complex. This paper presents the
development of a dynamic modeling and control algorithm of an
autonomous hexa-rotor microcopter.
The paper is organized as follows: Section 1: Introduction. In Section 2,
the dynamic modeling of a hexa-rotor microcopter is presented. In

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|
Section 3 backstepping controller for hexa-rotor microcopter is
presented. Conclusions are given in Section 4.
DYNAMIC MODELING OF HEXA-ROTOR MICROCOPTER
The model [31-46] of the hexa-rotor helicopter and the rotational
directions of the propellers are presented in Figure 3. This cross structure
is quite thin and light, however it shows robustness by linking
mechanically the motors. Hexa-rotor microcopter body is rigid. The six
rotors are symmetrically distributed around the center. All the propellers
axes of rotation are fixed and parallel. Propellers are rigid. These
considerations point out that the structure is quite rigid and the only
things that can vary are the propeller speeds.
The hexa-rotor microcopter configuration has six rotors which generate
the propeller forces Fi (i = 1,2,3,4,5,6) as it is shown in Figure 3. Control
of quadrotor is achieved [2] by commanding different speeds to
different propellers, which in turn produces differential aerodynamic
forces and moments. In order to increase the altitude of the aircraft it is
necessary to increase the rotor speeds altogether with the same
quantity.
Figure 3. Hexa-rotor microcopter - a non-linear dynamic system
Each rotor consists of a:
≡
brushless DC motor and a
≡
fixed-pitch propeller.
This rotorcraft is constituted by:
≡
three rotors which rotate clockwise (1,3,5), and
≡
three rotating counterclockwise (2,4,6).
Forward motion is accomplished by increasing the speed of the rotors
(3, 4, 5) while simultaneously reducing the same value for forward
rotors (1, 2, 6). For leftward motion the speed of rotors (5 and 6) is
increased while the speed of rotors (2 and 3) is reduced.
Backward and rightward motion can be accomplished similarly. Finally,
yaw motion can be performed by speeding up or slowing down the
clockwise rotors depending on the desired angle direction.
To describe the motion of a 6 DOF rigid body it is usual to define two
reference frames (Figure 2):
≡
the earth inertial frame (E-frame), and
≡
the body-fixed frame (B-frame).
The equations of motion are formulated using the Newton-Euler laws
with the following reasons:
≡
the inertia matrix is time-invariant;
≡
advantage of body symmetry can be taken to simplify the
equations;
≡
measurements taken on-board are easily converted to body-
fixed frame;
≡
control forces are almost always given in body-fixed frame.
The E-frame (OExyz ) is chosen as the inertial right hand reference. This
frame is used to define the linear position (in meters) and the angular
position (in radians) of the quadrotor.
The B-frame is attached to the body. The origin of the B-frame is
chosen to coincide with the center of the hexa-rotor microcopter cross
structure. This reference is right-hand, too.
The linear position of the helicopter (X, Y, Z) is determi-ned by the
coordinates of the vector between the origin of the B frame and the
origin of the E-frame according to equation.
The angular position of the hexa-rotor microcopter (Φ, θ, ψ ) is defined
by the orientation of the EB-frame with respect to the EE-frame. This is
given by three consecutive rotations about the main axes which take
the EE-frame into the EB-frame. In this paper, the ”roll-pitch-yaw” set
of Euler angles (Φ, θ, ψ) were used.
The vector that describes quad-rotor position and orientation with
respect to the E-frame can be written in the form:
s = [x, y, z, Φ, θ, ψ]T (1)
The rotation matrix between the EE- and EB-frames has the following
from:
=  +
 + 
   
(2)
Now, the model of hexa-rotor dynamics can be described by a system
of equations:
6
Z
4
Z
YX
4
Y
3
Y
R
Y
X
Z
2
X
2
X
R
X
ZY
8
1
12
1
10
1
A
I
U
θφ
I
I
I
ψ
Ad
I
U
ωφ
I
J
ψ
φ
I
II
θ
A
d
I
U
ω
θ
I
J
ψθ
I
I
I
φ
A
m
U
cosθosθcg
z
A
m
U
sφφsinψinψsin
cosψosψs(
y
A
m
U
sφφcosψosψsin
(sinψsinψx
++
−
=
++
−
−
=
++−
−
=
++−
=
+
+−=
+
+=

















(3)
BACKSTEPPING CONTROLLER FOR HEXA-ROTOR MICROCOPTER
In this paper, controller design for the hexa-rotor microcopter is
proposed by using backstepping technique. Backstepping is a recursive
design methodology that makes use of Lyapunov stability theory to
force the system to follow a desired trajectory. The hexa-rotor
microcopter is controlled by angular speeds of six motors. Each motor
produces a thrust and a torque, whose combination generates the
main trust, the yaw torque, the pitch torque, and he roll torque acting
on the hexa-rotor microcopter. First, the dynamical model is rewritten
in state-space form:
)U,X(fX
=

(4)

ACTA TEHNICA CORVINIENSIS Fascicule 3 [July – September]
– Bulletin of Engineering Tome VIII [2015]
| 123 |
by introducing :
12
T
12
1
]
x..
x[
X
ℜ∈
=
(5)
as space vector of the system:
19
5
65
21 10 9
7
311
87 12 11
43
xxY
x
xx
xx x xY
xX
xxZ
xxX
xxZ
xx
φψ
ψ
φ
θ
θ
==
=
= =
= = = =
=
==
= = = =
= =








(6)
Next, the x- coordinates are transformed into the new z –
coordinates:
1 1_ 1 7 7_ 7
2 2 1_ 1 1 8 8 7_ 7 7
33_3 99_9
4 4 3 _ 3 3 10 10 9 _ 9 9
5 5_ 5 11 11_ 11
6 6 5_ 5 5 12 12 11 _ 11 11
ref ref
ref ref
ref ref
ref ref
ref ref
ref ref
zx x zx x
z xx z z xx z
zx x zx x
z xx z z x x z
zx x z x x
z xx z z x x z
αα
αα
αα
=−=−
=−− =−−
=−=−
=−− =−−
=−=−
=−− =− −



(7)
By introducing the partial Lyapunov functions [2], to all x – coordinates
results in the following backstepping controller:
)z)zz(xx
I
II
z(IU
)z)zz(x
I
J
xx
I
II
z(IU
)z)zz(x
I
J
xx
I
II
z(IU
)z)zz(gz(
xcosxcos
m
U
)z)zz(z(
U
m
U
)z)zz(z(
U
m
U
66556542
ZZ
YYXX
5ZZ4
443343r2
XX
TP
62
YY
XXZZ
3YY3
221121r4
XX
TP
64
XX
ZZYY
1XX2
12121111121111
31
1
1010991099
1
Y
8877877
1
X
ααα
αααΩ
αααΩ
ααα
ααα
ααα
−+−
−
−=
−+−−
−
−=
−+−+
−
−=
−+−+=
−+−=
−+−=
(8)
The position of the hexa-rotor microcopter in the earth reference frame
is illustrated in Figure 4.
Figure 4. Position of the hexa-rotor microcopter in the earth reference
frame.
CONCLUSIONS
This paper presents the development of a dynamic modeling and
control algorithm of an autonomous hexa-rotor microcopter. During the
last decade many research papers have been published on the topic of
modeling and control strategies of autonomous multirotors.
The autonomous hexa-rotor microcopter is an under-actuated and
dynamically unstable nonlinear system. The model that represents the
dynamic behavior of the hexa-rotor microcopter is complex. Unmanned
autonomous aerial vehicles have become a real center of interest. In the
last few years, their utilization has significantly increased. The
autonomous hexa-rotor microcopters have additional redundancy over
autonomous quad-rotor microcopters.
The control design carried out for an autonomous quad-rotor
microcopter - backstepping controller, can be applied to the
autonomous hexa-rotor microcopter since they are modeled as a
rotating rigid body dynamic system wih six degres of freedom.
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