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Simulation of Quad-rotor Flight Dynamics for the Analysis of Control, Spatial
Navigation and Obstacle Avoidance
Gyula Mester1, Aleksandar Rodic2
1University of Szeged, Faculty of Engineering, Robotics Laboratory, Szeged, Hungary, drmestergyula@gmail.com.
2University of Belgrade, Institue Mihajlo Pupin, Robotics Laboratory, Belgrade, Serbia, aleksandar.rodic@pupin.rs .
Abstract – Autonomous outdoor quadrotor helicopters
increasingly attract the attention of potential researchers.
Several structures and configurations have been
developed to allow 3D movements. The autonomous
quadrotor architecture has been chosen for this research
for its low dimension, good maneuverability, simple
mechanics and payload capability. This paper presents
the navigation of an autonomous outdoor quadrotor
helicopter. The paper is organized as follows: Section
1:Introduction. In Section 2, the modeling of the
Quadrotor helicopter are presented. In Section 3 the
control strategy are presented. In Section 4, the GPS
navigation of the autonomous quadrotor helicopter is
illustrated. Conclusions are given in Section 5.
1. INTRODUCTION
The quadrotor helicopter configuration is well known and
has been studied since the beginning of 1900s. In 1907,
the first known quadrotor helicopter, Gyroplane No. I, fli-
ed. Autonomous quadrotor helicopters increasingly attract
the attention of potential researchers. In fact, several indu-
stries require robots to replace men in dangerous, boring
or onerous situations. A wide area of this research is dedi-
cated to aerial platforms. Several structures and configu-
rations have been developed to allow 3D movements [1]-
[12], there are blimps, fixed-wing planes, single rotor hel-
icopters, bird-like prototypes, quadrotors, etc. Each of
these has advantages and drawbacks. The vertical take-off
and landing requirements exclude some of the aforementi-
oned configurations. However, the platforms which show
these characteristics have a unique ability for vertical, sta-
tionary and low speed flight. The electrically powered
four-rotor quadrotor helicopter architecture has been cho-
sen for this research for its low dimension, good maneuv-
erability, simple mechanics and payload capability (Fig.
1).
Figure 1: Quadrotor helicopter
This structure can be attractive in several applications, in
particular for surveillance, for imaging dangerous
environments and for outdoor navigation and mapping.
The paper is organized as follows: Section 1:Introduction.
In Section 2, the modeling of the Quadrotor helicopter are
presented. In Section 3 the control strategy are presented.
In Section 4, the GPS navigation of the quadrotor helicop-
ter is illustrated. Conclusions are given in Section 5.
2. MODELING OF THE QUADROTOR
HELICOPTER
The model of the quadrotor helicopter and the rotational
directions of the propellers can be see in Figure 2. The
rotor pair 2 and 4 rotates clockwise direction and the rotor
pair 1 and 3, anticlockwise direction. A quad-rotor heli-
copter has fixed pitch angle rotors, and the rotor speeds
are controlled in order to produce the desired lift forces.
Figure 2: The model of the quadrotor helicopter
2.1 Actuators of the Quadrotor Helicopter
The quadrotor helicopter has four actuators - brushless
DC motors wich exert lift forces F1, F2, F3, F4
proportional to the square of the angular velocities of the
rotors. Actually, four motor driver boards are needed to
amplify the power delivered to the motors. Their rotation
is transmitted to the propellers which move the entire
structure.
2.2 Sensor System of the Quadrotor Helicopter
Two types of sensors are used for measuring the robot
attitude and for measuring its height from the ground.
For the first, an Inertial Measurement Unit (IMU) was
adopted, while the distance was estimated with a SOund
Navigation And Ranging (SONAR) and an InfraRed (IR)
modules. There are: accelerometers and angular velocity
sensors ont he board of the quadrotor helicopter. The
concept of the vision system is originated from motion-
stereo approach. The camera is attached to the quadrotor
helicopter.
2
The data processing and the control algorithm are handled
in the Micro Control Unit (MCU) which provides the
signals to the motors.
2.3 Coordinate Systems for Navigation
To describe the motion of a 6 DOF rigid body it is usual
to define two reference frames [1]:
• the earth inertial frame (E-frame), and
• the body-fixed frame (B-frame)
The equations of motion are more conveniently formula-
ted in the B-frame because of 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 (OXYZ) is chosen as the inertial right-hand
reference. Y points toward the North, X points toward the
East, Z points upwards with respect to the Earth, and O is
the axis origin. This frame is used to define the linear
position (in meters) and the angular position (in radians)
of the quad-rotor. The B-frame (oxyz) is attached to the
body. x points toward the quad-rotor front, y points
toward the quad-rotor left, z points upwards and o is the
axis origin. The origin o is chosen to coincide with the
center of the quad-rotor cross structure. This reference is
righthand too. The linear velocity v (m/s), the angular
velocity Ω (rad/s), the forces F (N) and the torques T
(Nm) are defined in this frame. The linear position of the
helicopter (X, Y, Z) is determined by the coordinates of
the vector between the origin of the B-frame and the
origin of the E-frame according to the equation.
The angular position (or attitude) of the helicopter (φ ,θ
,ψ) is defined by the orientation of the B-frame with
respect to the E-frame. This is given by three consecutive
rotations about the main axes which take the E-frame into
the Bframe. In this paper, the “roll-pitch-yaw” set of
Euler angles were used. The vector that describes the
quad-rotor position and orientation with respect to the
Eframe can be written in the form:
s = [X Y Z φ θ ψ ]T (1)
The rotation matrix between the E- and B-frames has the
following form:
ccscs
cssscssscccs
cscssssccscc
R
(2)
The corresponding transfer matrix has the form:
cccs
sc
tcts
T
//0
0
1
(3)
In the previous two equations (and in the following) this
notation has been adopted:
sin(.)
(.) s
,
cos(.)
(.) c
,
tan(.)
(.) t
.
2.4 Kinematic Model of the Quadrotor Helicopter
The system Jacobian matrix, taking (2) and (3), can be
written in the form:
T
R
J
x
x
33
33
0
0
(4)
where
33
0x
is a zero-matrix. The generalized quad-rotor
velocity in the B-frame has a form [1]:
T
zyxv
(5)
Finally, the kinematical model of the quadrotor helicopter
can be defined in the following way:
vJs
(6)
2.5 Dynamic Model of the Quadrotor Helicopter
Dynamic modelling of the quadrotor helicopter is a well
elaborated field of aeronautics. The dynamics of a generic
6 DOF rigid-body system takes into account the mass of
the body m and its inertia matrix I.
Two assumptions have been done in this approach:
• The first one states that the origin of the body-fixed
frame is coincident with the center of mass (COM) of the
body. Otherwise, another point (COM) should be taken
into account, which could make the body equations
considerably more complicated without significantly
improving model accuracy.
• The second one specifies that the axes of the B-frame
coincide with the body principal axes of inertia. In this
case the inertia matrix I is diagonal and, once again, the
body equations become simpler.
The dynamic model of a quad-rotor can be defined in the
following matrix form:
BBB GvvCvM )(
(7)
where
B
M
is the system Inertia matrix,
B
C
represents
the matrix of Coriolis and centrifugal forces and
B
G
is
the gravity matrix. The mentioned matrices have the
known forms as presented in [6]. A generalized force
vector
has the form [3]:
2
)( BB EvO
(8)
where:
0000
0000
0000
0000
TPBJO
(9)
3
is the gyroscopic propeller matrix and
TP
J
is the total
rotational moment of inertia around the propeller axis.
The movement aerodynamic matrix has the form [3]:
dddd
lblb
lblb
bbbb
EB
00
00
0000
0000
(10)
where:
b
(
2
sN
) and
d
(
2
smN
) are thrust and drag
factors [6] and l (m) is the distance between the center of
the quad-rotor and the center of the propeller. Equation
(11) defines the overall propellers’ speed (rad s−1) and the
propellers’ speed vector (rad s−1) used in equation (8).
4321
(11)
T
4321
(12)
Equations (1)-(12) take into account the entire quadrotor
non-linear model including the most influential effects.
3. MODELING OF THE CONTROL STRATEGY
Together with modeling, the determination of the control
algorithm structure is very important for improving
stabilization. Controlling a autonomous quadrotor
helicopter is basically dealing with highly unstable
dynamics and strong axes coupling. In addition to this,
any additional on-board sensor increases the autonomous
quadrotor helicopter total weight and therefore decreases
its operation time. The control system of the autonomous
quadrotor helicopter requires accurate position and
orientation information [4], [5], [7] [8] [9]. In this section
we present a control strategy to stabilize of the quad-rotor.
Figure 3 shows the block diagram of the quad-rotor
control system.
Figure 3: The block diagram of the quadrotor helicopter control system
4. GPS NAVIGATION OF THE AUTONOMOUS
QUADROTOR HELICOPTER
The trajectory of the autonomous quadrotor can be
introduced by GPS coordinates (e.g.
)( jPGPS
) as shown
in Figure 4.
Figure 4: Quadrotor helicopter localization and navigation
with respect to the imposed GPS coordinates
The autonomous quadrotor helicopter is requested to track
the imposed trajectory between the particular points
(
nj ,...,1
) with satisfactory precision, keeping the
desired attitude and height of flight [10], [11]. The
autonomous quadrotor helicopter checks for the current
position: X and Y by use of a GPS sensor and/or
electronic compass. Also, the altitude is measured by a
barometric sensor. An on-board microcontroller calculates
the actual position deviation from the imposed trajectory
given by successive GPS positions
)( jPGPS
. It localizes
itself with respect to the nearest trajectory segment, by
calculation of the distances:
1
or
2
.
Gyroscopes provide angular velocity measurements with
respect to inertial space. With recent developments in
gyroscope technology, their usage in various fields is
observably increasing. In combination with accelero-
meters, gyroscopes are used in position, velocity, and
attitude computation in a variety of navigation and motion
tracking applications for aircraft and robots [13-19]. By
providing angular velocity measurements, gyroscopes can
also be used in angular orientation estimation.
Using the gyroscope, the autonomous quadrotor
helicopter determines desired azimuth of flight
(Figure
4) and keeps the desired direction of flight. The height of
flight is also controlled to enable the performance of
the imposed mission (task).
4
5. CONCLUSIONS
We presented the modeling and navigation of an
autonomous quadrotor helicopter in a outdoor scenario.
The main aspects of modeling of rotorcraft kinematics
and rigid body dynamics, spatial system localization and
navigation of autonomous quadrotor helicopter in
outdoor scenario are considered in the paper. The control
strategy are presented. The GPS navigation of the
autonomous quadrotor helicopter is illustrated.
Acknowledgement
This work was supported by the innovation project
‘Research and Development of Ambientally Intelligent
Service Robots’, TR-35003,. 2011-2014, funded by the
Ministry of Science of the Republic Serbia and partially
supported by the TÁMOP-4.2.2/08/1/2008-0008 program
of the Hungarian National Development Agency.
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