A Hovering Control Strategy for a Tail-Sitter VTOL UAV that Increases Stability Against Large Disturbance

Conference Paper (PDF Available)inProceedings - IEEE International Conference on Robotics and Automation · June 2010with703 Reads
DOI: 10.1109/ROBOT.2010.5509183 · Source: IEEE Xplore
Conference: Robotics and Automation (ICRA), 2010 IEEE International Conference on
Abstract
The application range of UAVs (unmanned aerial vehicles) is expanding along with performance upgrades. Vertical take-off and landing (VTOL) aircraft has the merits of both fixed-wing and rotary-wing aircraft. Tail-sitting is the simplest way for the VTOL maneuver since it does not need extra actuators. However, conventional hovering control for a tail-sitter UAV is not robust enough against large disturbance such as a blast of wind, a bird strike, and so on. It is experimentally observed that the conventional quaternion feedback hovering control often fails to keep stability when the control compensates large attitude errors. This paper proposes a novel hovering control strategy for a tail-sitter VTOL UAV that increases stability against large disturbance. In order to verify the proposed hovering control strategy, simulations and experiments on hovering of the UAV are performed giving large attitude errors. The results show that the proposed control strategy successfully compensates initial large attitude errors keeping stability, while the conventional quaternion feedback controller fails.
A Hovering Control Strategy for a Tail-Sitter VTOL UAV that
Increases Stability Against Large Disturbance
Takaaki Matsumoto
1
, Koichi Kita
2
, Ren Suzuki
1
, Atsushi Oosedo
1
,
Kenta Go
1
, Yuta Hoshino
1
, Atsushi Konno
1
and Masaru Uchiyama
1
1
Department of Aerospace Engineering, Tohoku University.
Aoba-yama 6-6-01, Sendai, Miyagi, 980-8579, Japan
2
Fuji Heavy Industries Ltd., 1-1-11
Yonan, Utsunomiya, Tochigi, 320-8564, Japan
{takaaki, konno, uchiyama}@space.mech.tohoku.ac.jp, KitaK@uae.subaru-fhi.co.jp
Abstract The application range of UAVs (unmanned aerial
vehicles) is expanding along with performance upgrades. Ver-
tical take-off and landing (VTOL) aircraft has the merits of
both fixed-wing and rotary-wing aircraft. Tail-sitting is the
simplest way for the VTOL maneuver since it does not need
extra actuators. However, conventional hovering control for
a tail-sitter UAV is not robust enough against large distur-
bance such as a blast of wind, a bird strike, and so on. It
is experimentally observed that the conventional quaternion
feedback hovering control often fails to keep stability when the
control compensates large attitude errors. This paper proposes
a novel hovering control strategy for a tail-sitter VTOL UAV
that increases stability against large disturbance. In order to
verify the proposed hovering control strategy, simulations and
experiments on hovering of the UAV are performed giving large
attitude errors. The results show that the proposed control
strategy successfully compensates initial large attitude errors
keeping stability, while the conventional quaternion feedback
controller fails.
I. INTRODUCTION
VTOL UAVs make missions possible which are normally
impossible to accomplish using either fixed-wing or rotary-
wing UAVs alone; for example, search and rescue operations
covering a broad area located at the rooftop of a building.
There are several ways to perform VTOL maneuvers such as
tilting-rotor, tilting-wing, thrust-vectoring and tail-sitting etc.
The simplest way is tail-sitting since it does not need extra
actuators for the VTOL maneuver. A simple mechanism is
preferable for UAVs, because weight saving is crucial for the
VTOL maneuver and has the advantage of cost saving. Tail-
sitter VTOL aircraft switches between level flight mode and
hover mode by changing the pitch attitude of the fuselage
by 90 ° as shown in Fig. 1.
US Air Force Research Lab and AeroVironment Inc.
have developed “SkyTote” which is equipped with a coaxial
contra-rotating propeller [1]. The Defense Advanced Re-
search Projects Agency (DARPA) and Aurora Flight Sciences
have developed “GoldenEye” which is equipped with a
ducted fan. It uses fins outside the duct during level flight
and fins in the duct during hovering [2]. Stone developed “T-
Wing” which has a canard wing and tandem rotors [3],[4].
Kubo and Suzuki proposed a twin-fuselage plane [5]. Green
and Oh developed a micro air vehicle [6], and added two
Transition to LandingTransition from Takeoff
Hovering
Fig. 1. Takeoff and landing of the tail-sitter VTOL aircraft.
wingtip rotors which generate a rotational force countering
the motor torque to their MAV [7].
However, those tail-sitter UAVs have some complex equip-
ments such as a coaxial contra-rotating propeller [1], a ducted
fan and fins [2], side-by-side rotors [4],[5], and wingtip rotors
[7] for the tail-sitting VTOL maneuver.
Only few attempts have been made to develop tail-sitter
UAVs without any extra equipment so far. However, since
these simple tail-sitter UAVs have no extra equipment for
countering the motor torque, robust stationary hovering is
more difficult than other robots with complex equipments.
Large disturbances in hovering such as strong wind or bird
impact are major problems to overcome.
Frank et al. have succeeded in the indoor flight experiment
using a commercially available R/C acrobatic airplane and
the motion capture system [8]. However, since the flight
experiments were performed in a room, there was no dis-
turbance such as strong wind which is a major problem to
overcome [8]. Knoebel et al. proposed a new airframe design
[9]. They are working on flight tests using a commercially
available single propeller R/C model which represents the
“XFY1”. However, hovering performance with large distur-
bance were not reported in [9]. Johnson et al. have developed
“GTEdge” which is a large scale R/C airplane weighting
about 15 kg and studied the tail-sitter maneuver [10],[11].
They have succeeded transition flight and hovering; however,
the robustness of hovering control was not discussed.
This paper is intended to propose a novel control strategy
for robust hovering when large attitude errors are generated
by some disturbances. Simulations and experiments on hov-
ering control of the UAV are performed giving large attitude
2010 IEEE International Conference on Robotics and Automation
Anchorage Convention District
May 3-8, 2010, Anchorage, Alaska, USA
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Fig. 2. Tail-Sitter UAV.
errors to verify the robustness against the errors. The results
show that the proposed strategy successfully compensates the
large errors, while the conventional control strategy failed to
stabilize the UAV. The UAV used in this paper is equipped
with all necessary sensors and computers on the fuselage
[12].
II. SYSTEM CONFIGURATION
An overview of the tail-sitter VTOL UAV is shown in
Fig. 2. The main wingspan is 1.0 m, and the weight is
0.75 kg. The main and tail wings are parts of commercially
available R/C airplane (Hyperion Co., Sniper 3D), and other
parts such as the body are newly developed. The motor and
propeller, of which the static thrust amounts to 120 % of the
fuselage weight at a continuous maximum motor load are
selected.
The UAV is equipped with the following processors and
sensors.
A microcomputer board (Alpha Project Co., STK-7125)
that has an SH2 microcomputer made by Renesass
Technology Co. The microcomputer calculates control
input based on each sensor data, and sends pulse-width
modulated (PWM) signals to servo motors to control
surfaces (aileron, elevator, and rudder) and the thrust
motor.
An attitude sensor module (Microstrain Co., 3DM-
GX1). This module provides the attitude, azimuth,
three-axis angular velocity and acceleration. The sen-
sor’s datasheet gives its attitude angle accuracy as ±2 °.
An ultrasonic sensor to detect altitude when the air-
craft’s distance from the ground is less than 6 m.
An atmospheric pressure sensor to detect altitude when
aircraft distance from the ground is more than 6 m.
A global positioning system (GPS) receiver module
(Garmin Co., GPS 18-5Hz). The GPS module obtains
absolute position on the earth and absolute velocity of
the three axes.
A micro-SD card module to record flight data and other
information for postexperiment analysis.
A R/C receiver to control aircraft by a human in emer-
gency. The main computer receives commands from
an R/C transmitter, but these are not used in control
calculation.
A configuration diagram of the electronic system is shown
in Fig. 3.
Pressure Altitude
Sensor
Micro-SD Card Module
R/C Transmitter
Microcomputer
GPS Module
Ultrasonic Sensor
Attitude Sensor
Module
Thrust Motor
Motor Controller
Aileron Servo
Elevator Servo
Rudder Servo
Ground
Aerial Robot
R/C Receiver
Fig. 3. On-board electronics system.
Z
Y
X
Aileron
Rudder
Elevator
Fig. 4. Aircraft body coordinates and control surfaces.
III. HOVERING CONTROL
A. Quaternion Feedback Control
The earth fixed coordinate system defines X axis as true
north, Y axis as east, and Z axis as perpendicular downward.
The fuselage fixed coordinate system is defined as shown
in Fig. 4 as a principal axis of inertia. The attitude of
the fuselage is expressed with respect to the earth fixed
coordinate system.
Because the tail-sitter maneuver covers a wide range of
attitudes, quaternion expression which theoretically has no
singularity is used as a method of describing the attitude.
Quaternion expresses the attitude by a three dimensional unit
vector r and its rotation angle
ζ
, as follows:
q =
cos(
ζ
/2)
rsin(
ζ
/2)
=[q
0
q
1
q
2
q
3
]
T
. (1)
Quaternion feedback is generally used for UAVs. In con-
trolling the attitude of the UAV, following three quaternions
are defined: q
r
that shows the desired reference attitude, q
c
that shows the current attitude, and q
e
that shows the error
or deviation between q
r
and q
c
. The q
e
is shown as follows
55
Propeller
Downwash
Center of Gravity
Control Surface
mg
mg
mg tanθ
≈ mgθ
T
θ
Fig. 5. Operating principle for hovering.
by using q
c
and q
r
[13]:
q
e
=
q
r0
q
r1
q
r2
q
r3
q
r1
q
r0
q
r3
q
r2
q
r2
q
r3
q
r0
q
r1
q
r3
q
r2
q
r1
q
r0
q
c
, (2)
where q
r
=[q
r0
q
r1
q
r2
q
r3
]
T
. The vector part of q
e
(q
e1
,q
e2
,q
e3
) calculated by (2) shows amount of error about
each axis in the body coordinates.
Each three axes are controlled by a PID controller. The
control command is sent to control surfaces corresponding
to each axis as follows:
δ
i
= 2(K
P
q
ei
+ K
I
q
ei
dt + K
D
˙q
ei
), (3)
where
δ
1
,
δ
2
and
δ
3
are the aileron angle, elevator angle
and rudder angle, respectively. The PID gains are provided
by the ultimate sensitivity method, and tuned by trial and
error. The attitude is operated by blowing a slip stream of
the propeller to each control surface as shown in Fig. 5.
B. Resolved Tilt-Twist Angle Feedback Control
Quaternion feedback works well when attitude errors are
not very large. However, when the rolling error is large, the
quaternion feedback control presented in the previous section
may fail to stabilize the UAV.
For example, let (
αβγ
) = (0 90 0) ° in a reference attitude
and let (
αβγ
)=(180800)° be the current attitude, where
α
,
β
, and
γ
are ZYX Euler angles (yaw, pitch, and roll,
respectively). In this case, the error quaternion is calculated
as [0 0.57 0 0.34]
T
. This error quaternion derives no
error around the Y axis of the aircraft body coordinates.
Therefore, the pitch error (error around Y axis) is not
compensated in the beginning of the quaternion feedback
control.
We propose a novel hovering control strategy based on an
analogy of inverted pendulum to achieve robustness against
large attitude errors. The proposed hovering control strategy
is named Resolved tilt-twist angle control. In this control,
attitude error is resolved into the tilt and twist angles. The tilt
angle is composed of two angles of orthogonal axes. Fig. 6
shows the concept of the control. The resolved tilt-twist angle
control is composed of the following four steps.
X
C
X
N
v
X
N
θ
twist
θ
tilt
X
N
Y
N
(a) Step 1. Tilt
(Current Attitude)
(c) Reference Attitude(b) Step 2. Twist
Fig. 6. Concept of the resolved tilt-twist angle control.
Step 1 Derive pitch and yaw errors based on an analogy
of inverted pendulum
The first step derives the pitch and yaw errors.
Current attitude
O
C
R and reference attitude
O
N
R of the
UAV are defined as follows:
O
C
R [e
x
C
e
y
C
e
z
C
], (4)
O
N
R [e
x
N
e
y
N
e
z
N
], (5)
where e
jC
and e
jN
( j = x,y,z) are the unit vectors
along j axis of the body coordinate frame with
respect to the world coordinate frame at current
attitude and reference attitude, respectively.
Considering the UAV as an inverted pendulum, its er-
ror angles can be calculated. The attitude of inverted
pendulum is defined as follows,
R
E
=
O
N
R
TO
C
R =
r
11
E
r
12
E
r
13
E
r
21
E
r
22
E
r
23
E
r
31
E
r
32
E
r
33
E
. (6)
The X axis elements of R
E
gives pitch and yaw errors
as follows,
θ
Y
= atan2(r
31
E
,r
11
E
), (7)
θ
Z
= atan2(r
21
E
,r
11
E
), (8)
where atan2(y, x) is a function that calculates
tan
1
(y/x).
θ
Y
and
θ
Z
define the tilt angle of inverted
pendulum
θ
tilt
as follows:
θ
tilt
=
θ
2
Y
+
θ
2
Z
. (9)
Step 2 Derive roll error
The second step derives the roll error. The rotation
of
θ
tilt
is given by Rodrigues’ rotation formula as
follows,
R
v
=
E + ˆv sin
θ
tilt
+ ˆv
2
(1 cos
θ
tilt
), for R
E
= E
(10a)
E, for R
E
= E (10b)
where E is a 3 × 3 identity matrix, v is the rotation
axis vector given by the normalized cross product of
e
x
C
and e
x
N
as follows,
v =
e
x
C
× e
x
N
|e
x
C
× e
x
N
|
[v
x
v
y
v
z
]
T
. (11)
56
The hat operator transforms a vector v into a skew-
symmetric matrix as follows,
ˆv =
0 v
z
v
y
v
z
0 v
x
v
y
v
x
0
. (12)
The UAV attitude after compensating
θ
tilt
(see the
Fig. 6(b)), is given using R
v
as follows:
R
P
= R
v
O
C
R [e
x
P
e
y
P
e
z
P
], (13)
where e
jP
( j = x,y, z) are the unit vectors along j
axis of the body coordinate frame after compensating
θ
tilt
with respect to the world coordinate frame. The
absolute roll error is defined as follows,
θ
twist
= cos
1
e
z
P
· e
z
N
|e
z
P
||e
z
N
|
. (14)
Since aircraft roll angle range is 180 ° 180 °, the
sign of the roll error must be identified. In order to
identify the sign of the roll error
θ
X
,
θ
sign
is defined
as follows:
θ
sign
= cos
1
e
y
P
· e
z
N
|e
y
P
||e
z
N
|
. (15)
Using
θ
sign
, the roll error
θ
X
of the UAV is identified
as follows:
θ
X
=
θ
twist
, for
θ
sign
π
2
(16a)
θ
twist
. for
θ
sign
>
π
2
(16b)
Step 3 Projection of pitch and yaw errors onto the rolling
body coordinate frame
In order to simultaneously compensate pitch, yaw,
and roll errors, the pitch and yaw errors must be
projected onto the body coordinate frame which is
rolling with respect to the world coordinate frame.
Errors around each axis in the aircraft body coordi-
nates are given as follows:
d
1
d
2
d
3
=
10 0
0 cos
θ
X
sin
θ
X
0 sin
θ
X
cos
θ
X
θ
X
θ
Y
θ
Z
. (17)
Step 4 Feedback control for each control surface
Control command is sent to control surfaces based
on individual axes as follows:
δ
i
= (K
P
d
i
+ K
I
d
i
dt + K
D
˙
d
i
), (18)
where
δ
1
,
δ
2
and
δ
3
are the aileron angle, elevator
angle and rudder angle, respectively. d
1
d
3
are cal-
culated by (17). PID gains are same as the quaternion
PID feedback gains.
C. Altitude Control
The altitude controller is independently designed. The
desired propeller reference rotation speed is calculated from
the reference and current altitudes. Altitude control is gener-
ally possible without propeller rotation speed feedback, but
control performance is deteriorated by changes in battery
conditions and motor load due to disturbance. Therefore, a
feedback control of propeller rotation speed is introduced in
altitude control system to enhance robustness against these
changes. Control gains of the altitude control system were
determined through simulation.
IV. SIMULATION
A. Mathematical Model
To evaluate the hovering algorithms, a two-dimensional
tail-sitter UAV simulator was developed. The translational
mathematical model of the UAV in the aircraft body coordi-
nates is represented as follows,
m
˙
U + QW
= L sin
α
Dcos
α
mgsin
θ
+ T D
P
, (19)
m
˙
W QU
= L cos
α
Dsin
α
+ mgcos
θ
, (20)
where U and W are velocities along the X and Z axes in the
aircraft body coordinates, L and D are lift and drag forces,
α
is the attack angle,
θ
is the pitch angle, m is the fuselage
mass, g is the gravitational acceleration, T is the thrust force,
D
P
is the propeller drag force, and Q is the angular velocity
of the Y axis around the aircraft body coordinates.
The rotational mathematical model of the UAV is repre-
sented as follows,
I
xx
˙
P + C
p
P = M
a
+ M
p
, (21)
I
yy
˙
Q +C
q
Q = M
t
+ M
e
, (22)
where P and Q are angular velocities around the X and Y
axes of the aircraft body coordinates, I
xx
and I
yy
are inertia
around the X and Y axes of the aircraft body coordinates, C
p
and C
q
are viscous resistance coefcients, M
a
and M
p
are the
aileron and propeller rolling momentum around the X axis
of the aircraft body coordinates, M
t
and M
e
are fuselage
and elevator pitching momentums around the Y axis of the
aircraft body coordinates.
To identify aerodynamic forces (L,D,D
p
,M
a
,M
p
,M
t
,M
e
),
experiments including wind tunnel test are performed with
scale model of the UAV. Coefficients of main wing aerody-
namic forces (C
L
,C
D
,C
M
t
) are measured in all attack angle
range (180 ° 180 °). Inherent parameters of the propeller
are measured through wind tunnel test. The momentum the-
ory is used for its aerodynamic force calculation. Electrical
and mechanical time constants of the DC motor are identified
by experiment.
B. Simulation Results
A typical hovering simulation result of quaternion feed-
back is shown in Fig. 7. The initial attitude is (
αβγ
)=
(0 0 90) ° and the reference attitude is (
αβγ
)=(1700
80) ° , where
α
,
β
, and
γ
are ZXY Euler angles. The error
angle around Z axis decreased rapidly. However, note that
the error angle around Y axis increased in the early stage
57
01234
−180
−150
−120
−90
−60
−30
0
30
60
90
120
150
Time [s]
ZXY Euler angle [°]
Z result
Z reference
Y result
Y reference
Fig. 7. Quaternion feedback control simulation.
01234
−180
−150
−120
−90
−60
−30
0
30
60
90
120
150
Time [s]
ZXY Euler angle [°]
Z result
Z reference
Y result
Y reference
Fig. 8. Resolved tilt-twist angle control simulation.
0
1
2
3
4
5
6
7
8
0 20 40 60 80 100 120 140 160 180
Time [s]
Yaw [°]
Resolved tilt-twist angle feedback control
Quaternion feedback control
Fig. 9. Simulated recovery time comparison resolved tilt-twist angle control
with quaternion feedback control.
of simulation. This error increase causes a long horizontal
movement.
Fig. 8 is the result of simulation on a resolved tilt-twist
angle control hovering. Same conditions are given in both
simulations. The deceleration in error angle around Z axis is
slightly slower than quaternion feedback. Nevertheless, the
error angle around Y axis deceleration is very fast. As a
result, with short horizontal movement, stable hovering is
realized.
Fig. 9 shows a comparison of recovery times of both the
strategies. In quaternion feedback control, when the error
angle around Z axis surpasses approx 70 °, the recovering
time increased exponentially. On the other hand, in resolved
tilt-twist angle control, the rate of increase of recovering
time is linear. Therefore, resolved tilt-twist angle control has
superior stability against the large error angle around Z axis.
Furthermore, the error angle around Y axis was converged
very quickly in resolved tilt-twist angle feedback in all
error angle ranges around Z axis. However, in quaternion
feedback, the larger the error angle around Z axis exists, the
longer the error angle around Y axis converge time is needed.
01234
−180
−150
−120
−90
−60
−30
0
30
60
90
120
150
Time [s]
ZXY Euler angle [°]
Z result
Z reference
Y result
Y reference
X result
X reference
Fig. 12. Quaternion feedback control experiment.
01234
−180
−150
−120
−90
−60
−30
0
30
60
90
120
150
Time [s]
ZXY Euler angle [°]
Z result
Z reference
Y result
Y reference
X result
X reference
Fig. 13. Resolved tilt-twist angle control experiment.
V. EXPERIMENTAL RESULTS
A. Hovering with Quaternion Feedback Control
Fig. 10 shows snapshots of one of the hovering experi-
ments with quaternion feedback control. In this experiment,
the reference and initial attitudes are about the same as
simulation. The result of the experiment is shown in Fig. 12.
In the beginning of experiment, the error angle around Y
axis increased and the UAV lost stability. This result is the
same as the computer simulation. Moreover, the error angle
around Y axis increase and the error angle around Z axis
decrease caused the error angle around X axis. As a result,
the UAV couldn’t continue hovering.
It is notable that quaternion feedback control works well
when errors are not very large. However, in some cases
as shown in Fig. 10, quaternion feedback control causes
problem.
B. Hovering with Resolved Tilt-Twist Angle Feedback Control
Fig. 11 shows snapshots of one of the hovering experi-
ments with resolved tilt-twist angle control. The experiment
conditions are largely similar to the quaternion feedback
expriment. The result of the experiment is shown in Fig. 13.
The angles around Y and X kept reference values, respec-
tively. The error angle around Z axis decreases smoothly.
This arises from independent calculation steps for tilt and
twist angles in resolved tilt-twist angle control.
Additionally, even when a human inflicted large distur-
bance during hovering the UAV continued stable flight and
errors were converged (Figs. 14 and 15). This robustness will
be effective in order to overcome dynamic disturbance like
a bird strike during hovering. These flights are experimented
indoors, but the strategies brought out same performance in
the open air.
58
Fig. 10. Quaternion feedback control experiment. The UAV couldn’t continue hovering.
Fig. 11. Resolved tilt-twist angle control experiment. Since large aileron angle caused drag force, the UAV lost altitude slightly. However, the UAV could
continue hovering stably.
Fig. 14. Human inflicted rotational disturbance while hovering with
resolved tilt-twist angle control, but the UAV continued hovering stably.
Fig. 15. Human inflicted translational disturbance while hovering with
resolved tilt-twist angle control, but the UAV continued hovering stably.
VI. CONCLUSIONS
In this paper, we presented a novel hovering control
strategy and applied it to PID controller to realize robust
UAV hovering. The hovering control strategy is based on
the analogy of an inverted pendulum model and composed
of four steps. The two-dimensional UAV simulator was de-
veloped to evaluate the strategy. The resolved tilt-twist angle
control achieves superior stability to quaternion feedback
control when aircraft has large error angle around Z axis
through simulation and experiment.
The application of the resolved tilt-twist angle feedback
control for UAVs is not limited in hovering motion. It doesn’t
depend on any aircraft current and reference attitude. We be-
lieve it can be extended for many kind of aircraft maneuvers
which dynamically shift attitude with stall condition, not just
normal motion such as level flight.
VII. ACKNOWLEDGMENTS
This work was supported by Grant-in-Aid for Exploratory
Research (No. 21656219), and Grant-in-Aid for JSPS Fel-
lows (21-6015).
R
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    • "In recent years, there has been a considerable attention towards the propeller-pushing and flapping-wing aircrafts which can not only take off vertically, but also fly forward with high speed. A successful example includes V-22 aircraft [8] as well as tail-sitter designs91011121314151617. The T-wing is a VTOL UAV that is capable of both wing-born horizontal flight and propeller born vertical mode flight including hover and descent. "
    [Show abstract] [Hide abstract] ABSTRACT: This paper presents a model of an agile tail-sitter aircraft, which can operate as a helicopter as well as capable of transition to fixed-wing flight. Aerodynamics of the co-axial counter-rotating propellers with quad rotors are analysed under the condition that the co-axial is operated at equal rotor torque (power). A finite-time convergent observer based on Lyapunov function is presented to estimate the unknown nonlinear terms in co-axial counter-rotating propellers, the uncertainties and external disturbances during mode transition. Furthermore, a simple controller based on the finite-time convergent observer and quaternion method is designed to implement mode transition.
    Full-text · Article · Sep 2015
    • "However, PID controller is applicable only for SISO systems, therefore it does not account for the cross coupling effects present in UAVs. For such cases, multiple independent PID controllers are usually utilized in the hybrid UAVs such as in [35], [43], [52], [72], [55], [73], [76], [80], [82], [85]. 2) Linear Quadratic Regulator (LQR) Controller: LQR controllers goal is to find a control input of the form, that minimizes the performance index, ℑ, which is given by "
    Full-text · Dataset · Jun 2015 · Journal of the Franklin Institute
    • "However, PID controller is applicable only for SISO systems, therefore it does not account for the cross coupling effects present in UAVs. For such cases, multiple independent PID controllers are usually utilized in the hybrid UAVs such as in [35], [43], [52], [72], [55], [73], [76], [80], [82], [85]. 2) Linear Quadratic Regulator (LQR) Controller: LQR controllers goal is to find a control input of the form, that minimizes the performance index, ℑ, which is given by "
    Full-text · Dataset · Jun 2015 · Journal of the Franklin Institute
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