University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2013-01-25 Control of an Unconventional VTOL UAV for Complex Maneuvers Amiri, Nasibeh Amiri, N. (2013). Control of an Unconventional VTOL UAV for Complex Maneuvers (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/25452 http://hdl.handle.net/11023/462 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca
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Control of an Unconventional VTOL UAV for Complex Maneuvers
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University of Calgary
PRISM: University of Calgary's Digital Repository
Graduate Studies The Vault: Electronic Theses and Dissertations
2013-01-25
Control of an Unconventional VTOL UAV for Complex
Maneuvers
Amiri, Nasibeh
Amiri, N. (2013). Control of an Unconventional VTOL UAV for Complex Maneuvers (Unpublished
doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/25452
http://hdl.handle.net/11023/462
doctoral thesis
University of Calgary graduate students retain copyright ownership and moral rights for their
thesis. You may use this material in any way that is permitted by the Copyright Act or through
licensing that has been assigned to the document. For uses that are not allowable under
copyright legislation or licensing, you are required to seek permission.
Downloaded from PRISM: https://prism.ucalgary.ca
UNIVERSITY OF CALGARY
Control of an Unconventional VTOL UAV for Complex Maneuvers
2.1.1 Linear Control Techniques of UAV Flight Control . . . . . . . 182.1.2 Nonlinear Control Techniques of UAV Flight Control . . . . . 212.1.3 Control of eVader Vehicle in the Literature . . . . . . . . . . 24
3.2 Fans tilted longitudinally 90 degrees for high speed forward flight [1]. 373.3 a) Oppositely spinning disks tilted equally towards one another gener-
ating gyroscopic moment τgyro, b) The whole System rotated about yaxis to a new attitude orientation [1]. . . . . . . . . . . . . . . . . . . 39
3.4 Schematic of VTOL aerial vehicle with dual-axis OAT mechanism [1]. 413.5 Schematic of the eVader VTOL with a body fixed frame B and the
inertial frame E. The circular arrows indicate the direction of rotationof each propeller [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Control signals of FL control method in presence of white gaussiannoise with mean = 0 and variance = 0.1 [2]. . . . . . . . . . . . . . . 82
4.2 Regulation of orientation angles of the eVader by FL controller withadditive white noise (φ = 22.5, θ = 15, ψ = 18). . . . . . . . . . . . . 82
4.3 Regulation of position of the evader by FL controller with additivewhite noise (xd = 3, yd = 4, zd = 2). . . . . . . . . . . . . . . . . . . . 82
4.4 Control signals of adaptive FL control method in presence of aerody-namic coefficient uncertainties and unknown mass. . . . . . . . . . . . 83
4.8 The altitude output of FL and AFL controllers when the mass of thesystems is changed. The FL controller failed to reach the desired alti-tude zd = 2 with almost 0.4 m steady state error. . . . . . . . . . . . 84
5.1 Attitude control of evader’s orientation and the corresponding controlinput signals of IB control method (φ = 10, θ = 35, ψ = 5). . . . . 99
vii
5.2 Position (x, y) stabilization of the eVader and the corresponding controlinput signals of IB control method. . . . . . . . . . . . . . . . . . . . 102
5.3 Autonomous take-off, altitude control in hover and landing of theeVader and the effect of tuning the IB controller gains by GradientDescent algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 Stabilization of roll, pitch and yaw angles by IB control method (leftfigure) and pitched stability of the eVader at 25 in hover (right figure). 104
6.1 Chattering due to delay in control switching. . . . . . . . . . . . . . . 1126.2 Control input signals of SMC technique for performing pitched hover
8.1 Control input signals of FL, AFL and SMC controllers for orientationand position regulation in Scenario #1. . . . . . . . . . . . . . . . . . 169
8.2 Attitude outputs of the eVader obtained by applying FL, AFL andSMC controllers in Scenario #1. . . . . . . . . . . . . . . . . . . . . . 170
8.3 Position outputs of the eVader obtained by applying FL, AFL andSMC controllers in Scenario #1. . . . . . . . . . . . . . . . . . . . . . 171
ix
8.4 Control signals of AFL controller without robust modification in pres-ence of wind disturbance. The control signals u1, u2 and u3 go toinfinity and make the eVader unstable. . . . . . . . . . . . . . . . . . 172
8.5 eVader Orientation goes to infinity with AFL controller without robustmodification in presence of wind disturbance. . . . . . . . . . . . . . . 173
8.13 The Cartesian position output of eVader obtained by applying AFLwith robust modification and SMC control in presence of ground effectdisturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.14 Three dimensional position output result obtained by applying RAFLcontrol performing aggressive maneuver. . . . . . . . . . . . . . . . . 183
8.15 Three dimensional position output result obtained by applying SMCcontroller performing aggressive maneuver. . . . . . . . . . . . . . . . 184
8.16 Control signals of RAFL controller in aggressive maneuver scenario. . 1858.17 Control signals of SMC controller in aggressive maneuver scenario. . . 1858.18 Orientation of the eVader performing aggressive maneuver obtained by
AFL Adaptive Feedback LinearizationBP Back PropagationCFD Computational Fluid DynamicCG Center of GravityCMG Control Moment GyroscopeDOF Degree of FreedomdOAT double-axis Oblique Active TiltingFL Feedback LinearizationGE Ground EffectGWE Ground and Wall EffectsIB Integral BacksteppingIMU Inertial Measurement UnitIOL Input Output LinearizationISL Input State LinearizationLQ Linear QuadraticLQR Linear Quadratic RegulatorLP Linear ParameterizableMIMO Multiple Input Multiple OutputMLP Multi Layer PerceptronOAT Oblique Active TiltingOLT Opposed Lateral TiltingPD Proportional DerivativePID Proportional Integral DerivativeRAFL Robust Adaptive Feedback LinearizationSAR Search and RescueSISO Single Input Single OutputSMC Sliding Mode ControlsOAT Single-axis Oblique Active TiltingUAV Unmanned Aerial VehicleUUB Uniformly Ultimately BoundedVTOL Vertical Take-off and Landing
xi
1
List of Terms
Variables:
α1, α2 : Longitudinal tilting angles, rotation of eVader rotor about the vehicle’s
y-axis for right (#1) and left (#2) rotor, respectively
β1, β2 : Lateral tilting angles, rotation of eVader rotor about the vehicle’s x-axis
for right (#1) and left (#2) rotor, respectively
ωi, ω1,
ω2
: Propellers speeds, Rotational velocity of right (#1) and left (#2) rotor,
respectively
φ : Orientation roll angle
θ : Orientation pitch angle
ψ : Orientation yaw angle
r1, r2 : Rotor/disc #1, and #2, respectively
E =
xE, yE, zE
: Right hand inertia frame (earth’s frame)
B =
xB, yB, zB
: Body fixed frame
ζ(t) =
[x(t), y(t), z(t)]T: Position vector of UAV relative to the inertia frame of reference
η(t) =
[φ(t), θ(t), ψ(t)]T: Euler angle vector of UAV relative to the inertia frame of reference E
ζ : Translation velocity vector
η : Rotation velocity vector
Ttot =
[τx, τy, τz]T
: Total torque in Newton-Euler equations applied to the body of vehicle
relative to the body frame B
Ftot : Total force in Newton-Euler equations applied to the body of vehicle
relative to the body fixed frame B
Fgrav : Gravity force
Faero =
[Fax, Fay, Faz]T
: Aerodynamic forces
Fcg : All forces applied to the center of gravity of vehicle’s body relative to
body fixed reference frame B
2
FEcg : All forces applied to the center of gravity of vehicle’s body relative to
inertia reference frame E
FEtot : Total force in Newton-Euler equations applied to the body of vehicle
relative to the inertia frame of reference E
FEaero : Aerodynamic forces relative to inertia reference frame E
Taero =
[Tax, Tay, Taz]T
: Aerodynamic torques relative to inertia reference frame E
Tgyro : Gyroscopic effects of vehicle’s body and propellers
gr(z) : Ground effect function of altitude z
dw(t) : Wind gusts disturbance
τgyro : Gyroscopic pitch moments
τprop : Fan-torque pitch moments
τthrust : Thrust-vectoring pitch moments
τreact : Reactionary moments
τx : Total torque along x-axis
τy : Total torque along y-axis
τz : Total torque along z-axis
ν =
[uv, vv, wv]T
: UAV body linear velocity vector
Ω =
[pv, qv, rv]T
: UAV body angular velocity vector
v : Derivative of UAV body linear velocity vector, Accelerator vector
Ω : Derivative of UAV body angular velocity vector
m : Mass of eVader
J =
diag[Jx, Jy, Jz]
: Vehicle’s body Inertia matrix
Jr : Propeller’s inertia
T1, T2 : Thrust force of right (#1) and left (#2) rotor, respectively
D1, D2 : Drag force of right (#1) and left (#2) rotor, respectively
Q1, Q2 : Net torque applied to right (#1) and left (#2) rotor shaft, respectively
CT : Aerodynamic coefficient in thrust force
CQ : Aerodynamic coefficient in drag force
ρ : Density of air
3
Ar : Rotor blade area
rr : Radius of rotor blade
Rx(βi) : A counterclockwise rotation of a vector through angle βi about the x axis
Ry(αi) : A counterclockwise rotation of a vector through angle αi about the y axis
Rxy(β, α)i : Rotation matrix of vectors in coordinate frame attached to each rotor
about lateral and longitudinal tilting angles
Rx(φ) : A counterclockwise rotation of a vector through angle φ about the x axis
Ry(θ) : A counterclockwise rotation of a vector through angle θ about the y axis
Rz(ψ) : A counterclockwise rotation of a vector through angle ψ about the z axis
Ryxz : Rotation matrix whose Euler angles are φ, θ, ψ with x−y−z convention
cg : Vehicle’s centre of gravity (centre of mass)
O : Aerodynamic venter
lO : Distance from the centre of propeller to the centre of the vehicle (O)
hO : Distance from centre of the vehicle (O) to the centre of gravity (cg)
AGi: Gyroscopic moments
Qi : Propeller torques
di : Translational displacement of the ducts and the vehicles cg
Pi : Reactionary torques
SΩ : Skew-symmetric matrix
Kfax : Friction aerodynamic coefficient along x-axis affects total force
Kfay : Friction aerodynamic coefficient along y-axis affects total force
Kfaz : Friction aerodynamic coefficient along z-axis affects total force
ktax : Friction aerodynamic coefficient along x-axis affects total torque
ktay : Friction aerodynamic coefficient along y-axis affects total torque
ktaz : Friction aerodynamic coefficient along z-axis affects total torque
A : State matrix in linear state-space model
B : Control matrix in linear state-space model
C : Output matrix in linear state-space model
n : System order
x0 : State vector of initial conditions
xd : State vector of desired values
f : Nonlinear function of
g : Nonlinear function of
4
u : Input vector
y2i : Second derivative of ith output yi of a system
req : Relative degree
Γa1 : Positive update gain of the parameter estimation update law in adaptive
control method
eφ1 : Regulation error for φ(t)
eφ2 : Filtered regulation error for φ(t)
eθ1 : Regulation error for θ(t)
eθ2 : Filtered regulation error for θ(t)
eψ1: Regulation error for ψ(t)
eψ2: Filtered regulation error for ψ(t)
d1, ..., d6 : Additive external disturbances
λ1, ..., λ6 : Positive constant gain in integral backstepping control method
χ1, ..., χ6 : Integral of tracking errors
e1, ..., e11 : Roll tracking error
e2 : Angular velocity tracking error corresponding to roll angle
Sφ, Sθ, Sψ : Sliding surfaces of roll, pitch and yaw orientation angles
Sx, Sy, Sz : Sliding surfaces of Euclidean position
eφ, eθ, eψ : Regulation errors of roll, pitch and yaw angles, respectively
ex, ey, ez : Regulation errors of position x, y and z, respectively
V : Lyapunov function
W(1)ik : Connection weight from the kth input to ith neurone in the first layer
W(2)jq : Connection weight from the qth neurone in the first layer to the jth
neurone in the output layer
dref : Neural network desired (reference) output
ek : Error vector of
fu : Estimate of function fu
f−1 : Inverse of function f
yk : Neural network output for kth input point
yj : jth output signal of the second layer of neural network
zi : ith output signal of the first layer of neural network
Chapter 1
Introduction
The use of Unmanned Aerial Vehicles (UAVs) has recently gained extensive interest
due to their diverse potential applications. Different types of UAVs have been uti-
lized in various civil, industrial and military applications such as search and rescue,
weather research and environmental monitoring (e.g., Aerosonde), natural disaster
risk management, pipeline inspection and high altitude military surveillance (e.g.,
MQ-1 Predator). In fact UAVs are becoming more attractive lately as the result of
recent advancements in aerodynamics, propulsion, computers and sensor technology.
However, current UAVs cannot be controlled to navigate autonomously in confined
spaces. Therefore continuous effective improvement is essential in control mecha-
nisms to support and secure multiple tasks being performed with a single airframe
for complex missions in confined spaces.
Current UAVs have different levels of autonomy for operation and control. Some
UAV systems are controlled by an operator through a wireless connection from a
ground control station (remote control). Some systems combine remote control and
computerized automation. Some other systems are capable of semi-autonomous flight
following pre-specified destinations. More sophisticated versions have built-in control
and/or guidance systems to perform low-level human pilot duties such as speed and
flight-path stabilization, and simple scripted navigation functions such as waypoint
following. However only a small group of advanced UAV systems have the ability to
5
6
execute high-level operations in such a way that they can perform only by having
the initial states and desired destinations known. Indeed, from this perspective, early
UAVs are not autonomous at all. In fact, the field of air-vehicle autonomy is a recently
emerging field. Compared to the manufacturing of UAV flight hardware, the market
for autonomous flight technology is fairly immature and undeveloped. Because of
this, autonomy has been and may continue to be the bottleneck for future UAV
developments, and the overall value and rate of expansion of the future UAV market
could be largely driven by advances to be made in the field of autonomy.
Technology development of a fully autonomous UAV, which refers to the technol-
ogy that enables aircrafts to fly with reduced or no human intervention, comprises
the following seven main categories:
1. Task allocation and scheduling: Determining the optimal distribution of tasks
amongst a group of agents, considering different constraints such as time and equip-
ment.
2. Communications: Communication management and coordination between mul-
tiple agents in the presence of imperfect information and missing data.
3. Path planning: Determining an optimal path for vehicle to move while meeting
certain objectives and dealing with constraints, such as obstacles or fuel requirements.
4. Sensor fusion: Combining data from different sensor sources to be used in
vehicle.
5. Trajectory generation (also named motion planning): Determining an optimal
control movement to follow a given path or to go from one position to another.
6. Cooperative tactics: Formulating an optimal algorithm and spatial distribu-
tion of activities between agents in order to maximize chance of success in all given
challenges.
7
7. Trajectory regulation: The specific control mechanisms required to constrain a
vehicle within some deviation from a trajectory.
In the present study the ultimate focus is on developing a control mechanism to
improve trajectory tracking and set point regulation. Thus secure performance is
guaranteed in various possible extreme conditions such as complex agile maneuvers
in confined spaces.
1.1 Background of Unmanned Aerial Vehicles
As briefly discussed in the previous section, the term of UAV refers to aircrafts that
are designed to operate with no human pilot on-board [3]. Consequently, UAVs
have been considered for many applications with the purpose of reducing the human
involvement, and in turn, minimizing mission limitations where human presence is
dangerous, as in a case of searching for people trapped in a fire, or finding sources
of dangerous chemicals at industrial accident sites. Conventional UAVs are typically
classified in two main groups: fixed-wing and rotor crafts. Each of these two types has
advantages and disadvantages depending on the aimed mission and the characteristics
of the environment in which the desired task is to be executed. Conventional fixed-
wing aircrafts are capable of achieving long lasting flights, long distance ranges and
high forward speeds that are not attainable in traditional rotor crafts. However
maneuverability is limited for fixed-wing vehicles. Therefore, they are not suitable for
operations in confined spaces. Conventional fixed-wing aircrafts require a constant
forward speed to generate lift. On the other hand, rotary-wing aircrafts, such as
helicopters, have the advantage of being able to hover and perform Vertical Take Off
and Landing (VTOL) without a need for runways in a limited space. Additionally,
8
rotor crafts have some additional advantages including the ability to fly stationary in
hover, omni-directionality and VTOL capability. However, traditional VTOL vehicles
are usually highly affected by wind and ground effect disturbances. Moreover, their
big rotors decrease maneuverability, causing a limitation in application of rotary-wing
aircrafts in confined spaces. Having rotary-wing aircrafts advantages in mind, from
stability perspective, although fixed-wing aircrafts are generally internally stable [4],
the rotary-wing aircrafts dynamics are naturally unstable without closed-loop control
[5]. This intuitive characteristic makes the control system design more challenging for
rotary-wing UAVs. In what follows and throughout this thesis the term UAV refers
to a rotary-wing UAV.
1.1.1 Examples of Unmanned Aerial Vehicles
The analysis of control methods and the investigation of their performances are fo-
cused on civilian UAV missions in this thesis. Hence, a brief historical development
of the civil UAV sector is presented here. The following UAVs are examples of some
of the more prominent civilian UAV systems that are considered to be operational.
A more complete list of civilian UAVs is presented in [6].
1. AEROSONDE: The AEROSONDE UAV was developed by Aerosonde Pty,
Ltd. of Australia. It was originally designed for meteorological reconnais-
sance and environmental monitoring although it has found additional missions.
AEROSONDEs are currently being operated by NASA Goddard Space Flight
Center for earth science missions.
2. ALTAIR: ALTAIR was built by General Atomics Aeronautical Systems In-
corporated as a high altitude version of the Predator aircraft. It has been
9
designed for increased reliability. It comes with a fault-tolerant flight control
system and triplex avionics. It is operated by General Atomics although NASA
Dryden Flight Research Center maintains an arrangement to conduct Altair
flights.
3. ALTUS I/ALTUS II: The ALTUS aircrafts were developed by General Atom-
ics Aeronautical Systems Incorporated, San Diego, CA, as a civil variant of
the U.S. Air Force Predator. Although ALTUS is similar in appearance with
Predator, it has a slightly longer wingspan and is designed to carry atmospheric
sampling and other instruments for civilian scientific research missions in place
of the military reconnaissance equipment carried by the Predators.
4. CIPRAS: The Office of Naval Research established CIRPAS in the spring of
1996. CIRPAS provides measurements from an array of airborne and ground-
based meteorological, aerosol and cloud particle sensors, and radiation and re-
mote sensors to the scientific community. The data is reduced at the facility
and provided to the user groups as coherent data sets. The measurements are
supported by a ground based calibration facility. CIRPAS conducts payload
integration, reviews flight safety, and provides logistical planning and support
as part of its research and test projects around the world.
5. RMAX: The Yamaha RMAX helicopter has been around since about 1983.
It has been used for both surveillance and crop dusting, and other agricultural
purposes.
6. Quad-rotor: Although the first successful quad-rotors flew in the 1920s
[7], no practical quad-rotor helicopters have been built until recently, largely
10
due to the difficulty of controlling four motors simultaneously with sufficient
bandwidth. Recently, quad-rotor design has become one of the most popular
designs for small UAVs. The dynamic model of the quad-rotor helicopter has
six outputs while it only has four independent inputs. Therefore the quad-rotor
is an under-actuated system and it is not possible to control all its outputs at
the same time.
Due to the fact that new aerial vehicles have no conventional design basis, many
research groups build their own tilt-rotor vehicles according to their desired technical
properties and objectives. Some examples of these tilt-rotor vehicles are large scale
commercial aircrafts like Boeing’s V22 Osprey [8], Bell’s Eagle Eye [9] and smaller
scale vehicles like Arizona State University’s HARVee [10] and Compigne University’s
BIROTAN [11] which consist of two rotors. Some other examples of tilt-rotor vehicles
with quad-rotor configurations are Boeing’s V44 [12] and Chiba University’s QTW
UAV [13]. In fact none of these UAVs can be deployed in confined spaces. The focus
of this research is on developing a control system for an advanced unconventional
VTOL UAV with high maneuverability and capability, named eVader, to provide
secure performance in confined spaces. The eVader is introduced briefly in Section
1.2 and more comprehensively in Chapter 3.
1.2 Introducing the eVader Unmanned Aerial Vehicle
The capability of small UAVs which only need small ground spaces to fly has become
a priority element in the design and development of unmanned vehicles in this modern
era [14]. In this respect, UAVs that are small, autonomous, and have high maneuver-
ability have been considered in recent years. Furthermore the ducted fan configuration
11
has gained more interest (Fig. 1.1). Enclosing the rotors within a frame, ducted fan
rotors, would conclude better rotor protection from breaking during collisions, per-
mit flights in obstacle-dense environments among other aspects beyond the scope of
this thesis such as aerodynamic characteristics. It decreases the risk of damaging
the vehicle, or its surroundings. Moreover, with the helicopters’ limitations in both
flying in closed environments and forward speed, development of alternate VTOL air
vehicles has been increasingly considered by many researchers [15], [16], [17], and [18].
The most popular small helicopter type UAV in the literature is the quad-rotor. Al-
though quad-rotors are small and have diverse advantages over traditional helicopter
designs in the case of small electrically actuated aircraft [8], they do not have high
maneuverability in confined spaces due to their under-actuated property.
The eVader is a novel VTOL UAV which is targeted for operations in confined
spaces. In order to achieve this goal, the vehicle exploits a new mechanism of dual
ducted fans with a lateral and longitudinal rotor tilting mechanism to provide the
agility characteristic needed for missions in confined spaces. The novelty of the design
is the degrees of freedom of the fans which can rotate along both longitudinal and
lateral axes as shown in Fig. 1.2. This mechanism utilizes the inherent gyroscopic
properties of tilting rotors and driving torques of the fans for vehicle pitch control,
and eliminates the need for external control elements or lift devices. The special
characteristics of this new design, which will be discussed in more details in Chapter
3, offer unique capabilities such as inclined hovering, a task which is not theoretically
possible by other type of VTOLs [19]. As a result of the unique characteristics of the
eVader, it is a potential alternative of VTOL for complex maneuvers in urban areas
or inside confined spaces. Throughout present thesis, in order to successfully perform
these missions, an accurate nonlinear model of the vehicle’s dynamics is developed,
12
Figure 1.1: Ducted fans of the unconven-tional highly maneuverable VTOL UAV.
Figure 1.2: Fans rotating around longitu-dinal (y-axis) and lateral (x-aixs) axes.
and control methodologies are designed based on the UAV dynamic model to precisely
track the trajectory (position and orientation) of complex maneuvers.
This research is focused on this kind of VTOL UAV. The weight of the prototype
vehicle, that the simulations of this thesis are based on its parameters, is approxi-
mately 6.5 kg and the fans are 40.64× 25.4 cm. The fans rotational speeds must be
about 6000 rpm to produce enough thrust (equal to the weight of the vehicle) for
hover flight. This 1.7272 m long and 1.1684 m wide aerial vehicle is one of the first
of its kinds among tilt-wing vehicles on that scale range.
1.3 Motivation
As mentioned, the development of fully autonomous and self guided UAVs will re-
sult in minimizing the risk to and the cost of human life. UAVs have been used in
various anti-terrorist and accident-related missions and emergencies, sometimes with
success, but more often just confirming their potential. Although UAVs showed their
potential, they were not completely reliable, accurate and capable of performing the
tasks needed. They were not able to perform diverse tasks in an obstructed unknown
urban environment (e.g., Search and Rescue (SAR) and patrol operations). The main
13
motivation of this work is to deploy UAVs in confined spaces such that they can be
used in operations that may not be possible today such as search for victims in a
collapsed building.
Despite continued research has recently resulted in relative success and consider-
able enhancements in the UAV design, there still remain a number of major challenges
in the mentioned seven fields in Section 1. The main challenges associated with the
UAV controller design that require huge efforts can be listed as follow:
• open loop instability
• High degree of coupling among different state vectors and different variables
• Highly nonlinear behavior
• Diverse sources of noise and disturbances
• Very fast dynamics especially in the case of small model UAVs
Thus, designing a nonlinear control that demonstrates the high performance and ro-
bust stability encounter with significant disturbances is a challenging control problem
for UAVs. In fact, a great amount of innovative work must still be done across a num-
ber of disciplines before the full potential of UAVs would be achieved. A number of
examples of such challenges in the scope of this research interest are:
1. Overcoming the nonlinearity characteristics of UAV flying vehicle such as
open-loop instability and very fast dynamics to achieve a perfect control and
tracking for complex maneuvers.
2. Applying one type of controller for position and orientation trajectory regu-
lation.
14
3. Investigating the effect of changes in aerodynamic of the vehicle on the control
system for UAV Flying in close proximity of solid boundaries (e.g., ground and
walls).
4. Flying autonomously in presence of model inaccuracies (parametric uncer-
tainties), unmodeled dynamics and external disturbances.
5. Flying in confined spaces, which is a challenge for both aerodynamic and
control system design.
6. Performing aggressive maneuvers with agility and stability.
1.4 General Problem Statement
UAVs have been considered for many applications with the purpose of reducing the
human involvement where human presence is dangerous and in turn, reducing mission
limitations. All these applications demand advanced robotics technologies, leading
ultimately to fully autonomous, specialized, and reliable UAVs. In order to achieve
the stated mission, without a need to have an expert pilot, certain levels of auton-
omy are needed for the vehicle to maintain its stability and follow a desired path,
under embedded guidance and control algorithms. The level of autonomy of current
UAV systems, in terms of their control systems for precise trajectory tracking, varies
greatly.
Recent advances in technology, including sensors and micro controllers, now allow
small electrically actuated UAVs and Micro UAVs to be built relatively easily and
cost effectively. These small UAVs, such as small quad-rotors, have completely new
applications and would be able to fly either indoors or outdoors. Indoor flight offers
15
some challenging requirements in terms of size, weight and maneuverability of the
vehicle. Combined indoor and outdoor flying also requires a more advanced on-board
automation system. Inside a building, not much space for maneuvering is available,
but many obstacles exist. Therefore, a very accurate stabilization of the platform,
a highly precise trajectory tracking, and a highly maneuverable UAV are necessary
in order to guarantee a higher degree of autonomy. Certain control systems enable
certain UAVs to operate in cluttered environments, but not for indoor confined spaces
or urban confined environments. A number of other challenges are associated with
small UAV control systems in each of these environments. For instance, wind gusts
in outdoor spaces, the wall effect when flying in close proximity to buildings in urban
areas and the ground effects while flying close to the ground surface, are few examples.
In order to extend the range of current applications of UAVs to areas such as search
and rescue, as well as indoor surveillance, a reliable control methodology and an
agile UAV configuration are required that could facilitate highly complex maneuvers
in confined spaces. To effectively overcome this difficulty, the following problem
statement is formulated in this thesis:
• Develop a control methodology for the new configuration of a small VTOL UAV,
eVader, to successfully maneuver through confined 3D spaces in the presence of
external disturbances such as wind gusts, ground and wall effects and perform
complex tasks such as inclined hover and aggressive maneuvers with agility and
stability.
Motivated by the goal to design a controller which is robust to external distur-
bances and capable of adapting to changes in model parameters as well as sensor
noise, the controller is designed in this thesis for orientation and position regulation
16
and trajectory tracking with capability of independent control of all 6 degrees of free-
dom, including pitch, roll, and yaw angles, and altitude, lateral, and longitudinal
translations.
1.5 Thesis Outline
This thesis is organized as follows: Chapter 2 provides a literature review of the
control techniques and methods that have been studied on UAVs. This chapter also
includes objectives and goals to be addressed in this thesis. Chapter 3 provides
the detailed discussion of the characteristic of the eVader and how it produces the
essential moments, as well as the nonlinear dynamic model of the eVader vehicle. The
following three chapters are focused on the proposed control methodologies and how
these approaches are employed to achieve successful control in various flight scenarios.
This includes a design of feedback linearization, adaptive feedback linearization and
adaptive feedback linearization with robust modification techniques in Chapter 4,
Integral backstepping and adaptive integral backstepping controllers in Chapter 5,
and sliding mode control technique in Chapter 6. Chapter 7 addresses the problem of
approximating the relation between virtual and actual control signals using a neural
network as a nonlinear function approximator. A comprehensive set of simulations
on adverse flight conditions such as windy situations and ground effects, and on
performing aggressive maneuvers are presented in Chapter 8. Finally, Chapter 9
provides a list of contributions and future work of this research.
Chapter 2
Literature Review
Successful implementation of a UAV depends on the level of controllability and flying
capabilities. Throughout the years, different control methods have achieved different
levels of success in controlling UAVs. These methods can be classified in two main
categories: i) linear control and ii) nonlinear control methods. This chapter discusses
these different methods and their limitations.
This chapter starts with a comprehensive survey of works that have been done so
far, related to UAV control, in Section 2.1, followed by the objectives and goals of the
present research in Section 2.2. Definitions of terms used in this thesis to investigate
the problem in hand are introduced in Sections 2.3. The main key points of this
chapter are summarized in Section 2.4.
2.1 Overview of Previous Research on UAV Control
As stated in Chapter 1, UAVs have recently attracted considerable interest for a
wide variety of applications in the civilian world, including monitoring of traffic con-
ditions, recognition and surveillance of vehicles, and search and rescue operations
[20]. In order for a UAV to accomplish its tasks in each of these applications, a
fully autonomous flight system is needed. In addition, fully autonomous flight system
technology requires high-authority control systems, such as position and orientation
control systems and trajectory tracking systems. So far, various control method-
17
18
ologies have been developed for controlling UAVs, ranging from classical linear and
nonlinear techniques to fuzzy control intelligent approaches. However, the fact is that
these techniques have been applied mostly on helicopter type air vehicles.
The existing works can be subdivided into two main categories in term of the
tasks of control systems: the first class addresses the stabilization (regulation) prob-
lem, whereas the second group deals with solving the trajectory tracking issues. In
stabilization problems, a control system, called stabilizer (or a regulator), should be
designed to make the state of the closed-loop system stable around an equilibrium
(operating) point. Example of stabilization task is altitude control of UAVs. In track-
ing control problems, the design objective is to construct a controller, called tracker,
so that the system output tracks a given time-varying trajectory. Making a UAV fly
along a specified path is a tracking control task. In the following sections, some of
these works in the literature on controlling rotary-wing VTOL aircrafts (e.g., heli-
copter and quad-rotor) by linear and nonlinear control approaches for the purpose of
stabilization and trajectory tracking are presented.
2.1.1 Linear Control Techniques of UAV Flight Control
Cranfield University’s Linear Quadratic Regulator (LQR) controller [21], Swiss Fed-
eral Institute of Technology’s Proportional-Integral-Derivative (PID), Linear Quadratic
(LQ) controllers [16] and Lakehead University’s PD2 [22] controller are examples of
the controllers developed on quad-rotors’ linearized dynamic models. The result of
the work of Pounds et. al. shows that linear controls successfully stabilized the pro-
totype X-4 Flyer in the presence of step disturbances [23]. The vehicle uses tuned
plant dynamics with an on-board embedded attitude controller to stabilize flight.
Later the same research group tested a newer Mark II prototype of quad-rotor with
19
a linear single-input single-output (SISO) controller to regulate its attitude without
disturbances [24]. The controller designed by Pounds et. al. stabilized the dominant
decoupled pitch and roll modes, and used a model of disturbance inputs to estimate
the performance of the UAV. The disturbances experienced by the attitude dynamics
were expected to take the form of aerodynamic effects propagated through variations
in the rotor speed. Therefore, the sensitivity model was developed for the motor
speed controller to predict the displacement in position due to a motor speed out-
put disturbance. The desired position variations were in the order of 0.5 m and the
success of the controller to regulate attitude was at low speeds.
The second iteration testbed of the Stanford Testbed of Autonomous Rotorcraft
for Multi-Agent Control (STARMAC-II) quad-rotor prototype achieved free-flight
hovering using PID control [25], where it was noted that wind disturbances caused
the control to fail. In [25], a number of issues were observed in quad-rotor aircrafts,
operating at higher speeds and in the presence of wind disturbances. Hoffman et.
al. in [25] have explored the resulting forces and moments applied to the quad-rotor
related to three aerodynamic effects. The first group of the mentioned effects results
from total thrust variation not only with the power input, but also with the free
stream velocity and the angle of attack in respect to the free stream. This type of
effect would impact altitude control. The second effect results from differing inflow
velocities experienced by the advancing and retreating blades, and it may lead to
blade flapping, which includes roll and pitch moments on the rotor hub as well as a
deflection of the thrust vector. The third group of effects that have been researched
by Hoffman et. al. deals with the interferences caused by the vehicle body in the slip
stream of the rotor. They may result in unsteady attitude-tracking difficulties. The
impact of these type of effects may be significantly reduced by airframe modifications.
20
In summary, according to the result of [25], existing models and control techniques
are inadequate for accurate trajectory tracking at higher speeds and in uncontrolled
(unknown or not-engineered) environments. Later, the same research group worked
on outdoor trajectory tracking [26]. Hoffman et. al in [26] presented a trajectory
tracking algorithm to follow a desired path. The proposed control law in [26] tracks
line segments connected to sequences of waypoints at a desired velocity. The discussed
trajectory tracking algorithm has been experimentally tested to track a path indoors
with 10 cm accuracy and outdoors with 50 cm accuracy. Another prototype called
OS4 achieved autonomous flight where a linear Proportional Derivative (PD) control
maintained stable hover providing robustness to small disturbances [27].
Although, the results of linear control approaches including PD, PD2, PID, LQ
and LQR methods are sufficiently good, either for stabilizing specific operating points
such as hover flight, or small displacements from hover, this stabilization can only
be achieved at low velocities and with small additive aerodynamic disturbances. It
is worth noting here that unstable situations may occur as rotor speed increases and
also in presence of external disturbances such as wind gusts and ground effects. Un-
stable situations makes untethered flight almost impossible for flying vehicles [24].
Moreover, in real conditions the use of a classical linear control is limited to a small
neighbourhood around the operating point. Due to the fact that tracking complex
trajectories involves far away operation from neighbourhood of operating points, per-
forming difficult maneuvers, which require complex trajectory following control, are
not achievable with linear control theory. However, since the goal of this thesis is to
design a controller for the eVader UAV to perform difficult tasks and maneuvers such
as tracking complex trajectories and regulation in presence of external disturbances,
classical linear controllers are not applicable, and nonlinear control approaches are
21
required.
2.1.2 Nonlinear Control Techniques of UAV Flight Control
Nonlinear controls can substantially expand the region of controllable flight angles
compared to linear controls. For instance, Spectrolutions HMX-4 is a tethered quad-
rotor that uses state inputs from a camera fed into a feedback linearization control
without disturbances [28]. In this study, ground and on-board cameras, were used to
estimate the full six degrees of freedom of the helicopter. The pose estimation algo-
rithm is compared through simulation to some other feature based pose estimation
methods and is shown to be less sensitive to feature detection errors. Backstep-
ping controllers have been used to stabilize and perform output tracking control [28].
Nonlinear controls also achieved robustness to impulse disturbances, both in sim-
ulation [29], [30] and using a test-stand experiment [31], [32]. In [33] and [34], a
nested-saturations controller stabilized a Draganfly III in the presence of impulse dis-
turbances, and results were compared to linear feedback controls. An algorithm was
introduced by Hauser et al. to control the VTOL based on an approximate input-
output linearization procedure that achieves bounded tracking [35]. A non-linear
small gain theorem was proposed in [36], for stabilizing a VTOL, which proved the
stability of a controller based on nested saturations. An extension of the algorithm
proposed by Hauser was presented in [37], finding a flat output of the system that was
used for tracking control of the VTOL in the presence of unmodeled dynamics. The
forwarding technique developed in [38] was used in [39] to propose a control algorithm
for the VTOL. This approach leads to a Lyapunov function which ensures asymptotic
stability. Other techniques based on linearization were also proposed in [40]. Marconi
proposed a control algorithm of the VTOL for landing on a ship whose deck oscillates
22
[41]. They designed an internal model-based error feedback dynamic regulator that is
robust with respect to uncertainties. [42] presented an algorithm to stabilize a VTOL
aircraft with a strong input coupling, using a smooth static state feedback. An ap-
proach based on Lyapunov analysis to control the VTOL which can lead to further
developments in nonlinear systems is presented in [43]. The controller has been tested
in numerical simulations, but also in a real-time application. It simplified the tuning
of the controller parameters. [44] proposed control strategy which aims to be both
adaptive to model uncertainty (payloads) as well as robust to disturbances. There is
a summary of UAVs stabilization literature review available in [44]. Reference [45]
provides a review of adaptive intelligent approaches for robust control of a helicopter.
Uncertainties associated with dynamic models lead to a more challenging control
design. Different strategies have been proposed to deal with uncertain quad-rotor
model, such as adaptive control, neural network based control, sliding mode control,
H∞ control and so on. In [46], a direct adaptive control algorithm was designed for
the tracking control of a quad-rotor UAVs roll, pitch, yaw angles, together with alti-
tude while compensating for the model parameter uncertainties. A reference system
corresponding to a virtual UAV, which contains a third order oscillator, was utilized
to track the desired trajectory. In [47], a backstepping based approach was used for
quad-rotor UAV control, while two neural networks were used to approximate the un-
certain aerodynamic components. More literature review for quad-rotor UAV control
can be found in [48].
There are also robust controllers designed for quad-rotor systems. A sliding mode
disturbance observer was presented in [49] to design a robust flight controller for a
quad-rotor vehicle. This controller allowed continuous control, robust to external
disturbance, model uncertainties and actuator failure. Robust adaptive-fuzzy control
23
was applied in [50]. This controller showed a good performance against sinusoidal
wind disturbance. Mokhtari presented robust feedback linearization with a linear
generalized H∞ controller, and the results showed that the overall system was robust
to uncertainties in system parameters and disturbances, when weighting functions
are chosen properly [51]. In [52], a robust dynamic feedback controller of Euler
angles is proposed using estimates of wind parameters. This controller performed well
under wind perturbation and uncertainties in inertia coefficients. In [53], a sliding
mode controller was suggested. Due to the under-actuated property of a quad-rotor
helicopter, they divided a quad-rotor system into two subsystems: a fully-actuated
subsystem and an under-actuated subsystem. Two separate controllers were designed
for these subsystems. A PID controller was applied to the fully actuated subsystem
and a sliding mode controller was designed for the under-actuated subsystem. Because
of the advantage of a sliding mode controller, namely insensitivity to uncertainties, it
robustly stabilized the overall system under parametric uncertainties. A pre-trained
neural network stabilized a Draganfly II quad-rotor in hover without disturbances [54].
Adaptive neural network controls successfully stabilized quad-rotors in simulation
[55], [56].
Most of these methods in prior works have focused on simple trajectories, espe-
cially in case of having uncertainties associated with dynamic model of UAVs. The
simple external disturbances such as impulse signals were studied in most of the pre-
vious works. However, even few studies on sinusoidal wind disturbances have not
achieved the ability of performing complex tasks and executing in confined spaces. In
fact, performing aggressive maneuvers providing agility and stability with rotary-wing
aircrafts have not received much attention in the literature.
24
2.1.3 Control of eVader Vehicle in the Literature
A great deal of research has been done on controlling rotary-wing aerial vehicles such
as helicopters and quad-rotors [57], [46], [58], but controlling VTOLs having a lateral
and longitudinal rotor tilting ability, such as with eVader, is new in the literature.
Previous research on eVader by Gary Gress mainly discussed the eVader mechanism
and the potential of better control responses and independent 6-axis control [59], [1],
[19]. Gress linearized the equation of motion in pitch by assuming small values for
a propeller tilt angle and used a simple linear proportional controller. He investi-
gated the predicted pitch response of the MicroVader UAV to positive control input
(positive angles) of tilt angle for various oblique angles [1]. The feedback propor-
tional controller was applied on the eVader for regulation of the vehicle pitch angle
for different propeller speeds and different longitudinal angles [60].
Although the above-mentioned research by Gress, in eVader control and operation,
investigated the potential of an OAT mechanism to produce sufficient moments to
change the orientation of the UAV, the orientation and position set point regulation
and trajectory tracking of this vehicle to be used as part of an autonomous flight
system, which are the main focus of the present research, were not studied before.
In fact, all the methods presented in Sections 2.1.1 and 2.1.2 have been studied and
applied on helicopter type of UAVs especially on quad-rotor vehicles, but eVader is
still a new subject with lots of room for research and experiment. Hence, this thesis
provides a comprehensive study of different nonlinear control approaches on eVader
aerial vehicles in order to find the best choice of control methodology for this vehicle
in order to implement an autonomous UAV.
25
2.2 Objectives and Goals
In terms of UAV control, most prior works have focused on simple trajectories at
low velocities. Performing complex and aggressive maneuvers providing agility and
stability with rotary-wing aircrafts has not been studied much in literature. Ad-
ditionally, previous treatments of rotary-wing vehicle dynamics have often ignored
known aerodynamic effects of rotor craft vehicles. At slow velocities, such as dur-
ing hovering period, ignoring aerodynamic effects is indeed a reasonable assumption.
However, even at moderate velocities, the impact of the aerodynamic effects resulting
from variation in air speed is significant (e.g., wind gusts). Preliminary results of
the inclusion of aerodynamic phenomena in vehicle and rotor design show promise in
flight tests, although an instability currently occurs as rotor speed increases, making
untethered flight of the vehicle impossible [24]. Although many of the effects have
been discussed in the helicopter literature (e.g, flow simulations of a helicopter in low
speed forward flight in ground effect) [61], [7], [62], their influence on eVader type of
UAV has not been explored.
Considering the shortcomings described above and motivated by the overall ob-
jective of developing an aerial vehicle, capable of performing complex and agile tasks
in confined spaces, this thesis is focused on the following objectives:
1. The first step of every control design is modeling the system in order to
be controlled. The performance of the UAV controller will be dependent on
the availability of a sufficiently accurate vehicle model. Thus, the complete
dynamic model of a VTOL aerial vehicle having a lateral and longitudinal rotor
tilting mechanism (e.g., eVader) is derived based on a first principles approach.
Chapter 3 is devoted to development the complete dynamic model of the eVader.
26
2. The developed 6 Degree of Freedom (DOF) nonlinear dynamic model of the
vehicle in this thesis accounts for various parameters which affect the dynamics
of a flying structure, such as gyroscopic effects and ground effects. The nonlinear
state-space model of the VTOL vehicle under investigation is presented for the
first time in this thesis.
3. There are more advantages associated with the OAT mechanism of eVader
than just stability and controllability in the conventional sense, which have not
been explored yet. Examining these properties by applying a proper choice of
controller to verify the OAT capabilities, is one of the goals of this project. To
achieve this goal, various simulation experiments are tested to investigate the
characteristics of this unique UAV.
4. This project is a comprehensive study on controlling the eVader aerial ve-
hicle with nonlinear control techniques, namely: 1) feedback linearization, 2)
adaptive feedback linearization, 3) adaptive feedback linearizarion with robust
modification (in Chapter 4), 4)integral backstopping, 5) adaptive integral back-
stepping (in Chapter 5), 6) sliding mode approach (in Chapter 6).
5. Along with the presented nonlinear methodologies, another objective of this
thesis is to achieve full six degrees of freedom control including the position
and orientation stabilization and regulation, which is a unique characteristics of
eVader due to the fact that the rotary wing UAVs are usually under-actuated
vehicles, and controlling all six outputs of interest at the same time is impossible.
6. Using the above-mentioned nonlinear control techniques to investigate the
capabilities of the eVader such as performing aggressive and agile maneuvers,
27
maneuvering close to ground and wall surfaces for indoor and outdoor mis-
sions, taking off and landing from sloped surfaces, and tracking an object while
pointing to it that requires the pitched hover capability.
7. Another focus of the present research is to investigate the application of
different control approaches on the eVader to obtain asymptotic stability. Tra-
jectory tracking and set point regulation are desired with complete performance.
For the purpose of this study, complete performance is defined as a performance
that provides the asymptotic stability of the tracking method with structured
(e.g., modelling errors, unmodeled dynamics, sensor noise) and unstructured
uncertainties (e.g., external disturbances).
8. The designed control structure requires achieving both robustness and high
performance in presence of disturbances. The robustness of the flight controller
is defined as its ability to compensate for: 1) external disturbances such as wind
gusts and ground and wall effects, 2) model parameter uncertainties in terms of
changing payload and aerodynamic parameters, and 3) sensor noise for attitude
control signals.
9. Real actuator signals can not be obtained directly as an outcome of a control
algorithm. This problem makes the feasibility of control approaches a difficult
task to investigate. A neural network mapping is utilized to verify the feasibility
issue and to obtain the amount of system inputs including longitudinal and
lateral angles of each duct, and the rotor speeds of each of them.
In summary, the ultimate goal of this project is providing a powerful control
system for a fully autonomous UAV, which would be capable of doing agile and
28
aggressive maneuvers for indoor and outdoor applications. For this purpose, the
focus of this thesis is on the accurate stabilization and precise trajectory tracking
in different flight scenarios with sensor noise, model inaccuracies and unmodeled
dynamics.
2.3 Definitions
A set of fundamental definitions are used for interpretation of a number of terms
to address the problem at hand within this thesis. Some of these definitions are
specifically defined for this thesis.
1. Autonomy: The ability to execute processes or missions using on-board
decision capabilities.
2. Agility: Agility refers to being able to execute controllable maneuvers
under high g forces on complex flight trajectories, very much like piloted fighter
aircrafts do.
3. Aggressive maneuvers: Aggressive maneuverability in this thesis is in
the sense of attitude control: 1) controlling in the whole range of the attitude
angles of the UAV, and 2) tracking a given trajectory at the highest possible
velocity. If aggressive maneuverability in these terms is achieved, the controller
described here executes, in a stable and robust manner: 1) tracking of trajecto-
ries describing curvilinear translational (or horizontal) motion at relatively high
speed and constant altitude, and 2) set-point regulation for fast translational
acceleration/deceleration, hovering, and climb.
4. Asymptotic stability: An equilibrium point 0 is asymptotically stable if it
29
is stable, and if in addition there exists some r > 0 such that ‖x(0)‖ < r implies
that x(t) → 0 as t→ ∞ [63]. Asymptotic stability means that the equilibrium
is stable, and in addition, states started close to 0 actually converge to 0 as
time goes to infinity.
5. Asymptotic tracking: Asymptotic tracking implies that perfect tracking
is asymptotically achieved.
6. Confined environment: Environments with high environment density
measure. In confined environments, the distance between the UAV and obstacles
is usually smaller than in cluttered environments.
7. Exponential stability: An equilibrium point 0 is exponentially stable if
there exist two strictly positive numbers γ and λ such that
∀t > 0, ‖x(t)‖ ≤ γ‖x(0)‖e−λt
in some ball Br around the origin [63]. In other words, it means that the state
vector of an exponentially stable system converges to the origin faster than an
exponential function.
8. Lie derivatives: Let h : Rn → R be a smooth scalar function, and
f : Rn → Rn be a smooth vector field on Rn, then the Lie derivative of h with
respect to f is a scalar function defined by Lf = ∇hf .
9. Lie Bracket: Let f and g be two vector fields on Rn. The Lie bracket of f
30
and g is a third vector field defined by
[f ,g] = ∇gf −∇fg
10. Maneuvering: Maneuver is a tactical move, or series of moves, that
improves or maintains a UAV’s strategic situation in a competitive environment
or avoids a worse situation.
11. Perfect tracking (perfect control): When the closed-loop system is
such that proper initial states imply zero tracking error for all the times, y(t) =
yd(t), ∀t ≥ 0.
12. Relative degree: The number of times the output of a system needs to
be differentiated to generate an explicit relationship between the output y and
the input u.
13. Robustness: The robustness of the flight controller is defined as its ability
to compensate for external disturbances.
14. Reliable control (reliability): The ability of a UAV flight system to
adapt to system or hardware failures is called reliability, which is a key tech-
nology for flying UAVs. Considering that, the most critical system for the
aircraft is the flight control system, having a reliable control is critical. One
approach to improve system reliability is simply to increase the redundancy of
flight systems. This comes with both an initial cost and an on-going weight
penalty. Another approach would be adding on-board intelligence to recognize
and remedy a failure.
31
15. Smooth function: A function that has derivatives of all orders is called a
smooth function.
16. Stability: The equilibrium state x = 0 is said to be stable if, for any
R > 0, there exists r > 0, such that if ‖x(0)‖ < r, then ‖x(t)‖ < R for all
t ≥ 0. Otherwise, the equilibrium point is unstable [63]. Qualitatively, a system
is described as stable if starting the system somewhere near its desired operating
point implies that it will stay around the point ever after.
19. Zero-dynamics (internal dynamics): The zero-dynamics for a nonlinear
system is defined to be the internal dynamics of the system when the system
output is kept at zero by the input.
2.4 Summary
The present study is devoted to developing and discussing a set of perfect control laws
to achieve high maneuverability and high reliability for autonomous flight in highly
cluttered environments. The focus of this research is on a newly built configuration
of small rotary-wing VTOL aerial vehicle with ducted fans, each of which has two
rotors named eVader. A control law needs to be designed to employ the unique fly-
ing capabilities of the eVader UAV using the novel OAT mechanism. By applying
the suggested nonlinear control to the nonlinear dynamics of the eVader, this thesis
pursues the goal of performing aggressive motions such as tracking of trajectories
describing sharp (aggressive) curvilinear translational (or horizontal) motion at rela-
tively high speed, take off and landing from severely sloped surfaces, and stationary
pitched hover, which is not possible in any other manned or unmanned aerial vehicles.
Diverse sources of noise and disturbance plus very fast dynamics, especially in the
32
case of small UAVs such as eVader, adds up to the challenges associated with the
open loop unstable systems (e.g., rotary-wing UAVs). Therefore, a stability problem
should be carefully studied. It is desired to develop a control algorithm that would
simultaneously stabilize the orientation and position of the eVader UAV based on its
nonlinear dynamic model, without making simplification by linearizing the model.
That is to achieve the independent six degrees-of-freedom (DOF) of control which
includes three orientation angles of pitch, roll and yaw and three Cartesian position
variables in 3D space (x, y, z). The model includes frictions due to the aerodynamic
torques, drag forces and gyroscopic effects as well as independent tilting of the right
and left rotors (different tilting angles) and independent (not necessarily the same) ro-
tor speeds. Moreover, motivated by the desire of maintaining stability in the presence
of model uncertainty and external disturbances (such as wind and ground effects gen-
erated by the down wash produced by the vehicle itself), the control strategy is aimed
to be both adaptive to model uncertainties as well as robust to external disturbances.
According to the dynamic model of the vehicle, the resulting control signals from
the output of control methodologies are virtual signals. It is important to note that
actual control signals should be obtainable from virtual control signals, otherwise the
resulting controller would not be a practical design. As a result, another challenge as-
sociated with controlling the eVader is finding the actual control signals corresponding
to actuator signals. Present study takes the advantage of neural network method in
order to calculate actual control signals from virtual control signals by approximating
the mapping relation between virtual and actual control signals of the system.
Chapter 3
Modeling
The first step in designing a control system for a given physical plant (the eVader in
this thesis) is to derive a model that captures the key dynamics of the plant in the
operational range of interest. The presented model in [] by G. Gress was the sOAT
equation of motion for the eVder tilting its propellers by the same angle in the same
but opposite oblique directions. This model only represents the pitching moments
results about the aircraft pitch axis. In this thesis, the highly maneuverable charac-
teristics of the eVader are modeled to represent the autonomous UAV with double
Oblique Active Tilting (dOAT) mechanism for the first time in the literature. This
mechanism allows the vehicle to navigate in confined environments [64], by enabling
the vehicle to respond fast and with agility to obstacles. Following traditional model-
ing approaches, a complete dynamic model of this unconventional UAV is developed
using a Newton-Euler formulation. The dynamic model of the eVader developed in
this research includes reactionary moments, ground effects and aerodynamic friction
effects, and considers the capability of independent movements of each rotor about
both the rotor’s own x-axis and y-axis. When compared to previous models of this
vehicle, the developed new model makes the model more realistic and more reliable
for complex maneuvers. The lateral (β1, β2) and longitudinal (α1, α2) angles, and the
propellers speeds (ω1, ω2) are input signals in the model. The vehicle’s orientation
angles (φ, θ, ψ), the x and y vehicle position, and UAV’s altitude z are the six outputs
33
34
in the model manipulated via α1, α2, β1, β2, ω1 and ω2.
This chapter focuses on a lift-fan dOAT or Opposed Lateral Tilting (OLT) mecha-
nism which is used in the eVader UAV as a control device. The capability of providing
the three required moments for control including pitch, roll, and yaw moments, and
the way in which the dOAT mechanism produces them, are discussed in this chapter.
Moreover, a theoretical analysis of the dOAT vehicle control response is described
3.1 Introduction
The ability of small or medium air vehicles to access and operate in confined and
obstructed environments is required for air mobility to enable aerial transportation
systems and UAV complex mission execution (e.g. search and rescue within collapsed
buildings). Satisfying this condition necessitates VTOL aerial vehicles to be highly
agile, highly maneuverable, and more compact for a given payload, while maintaining
effective vehicle control, which becomes more difficult as the vehicle’s size reduces.
When the moment arms decrease in length, the control of the vehicle requires larger
forces, which conventional control devices (e.g. ailerons) can not provide. To obtain
an effective control for a compact UAV, the vehicle should be able to provide sufficient
moments for control regardless of its dimensions.
A control device that does not rely on moment arms is a gyroscope. It has been
used before in satellites, missile guidance, and space stations [65]. A gyroscope gener-
ates the large moments required to change the attitudes of satellites and space stations
within short time periods [66]. A traditional mechanical Control Moment Gyroscope
(CMG) has been proposed in the literature for attitude stability and control [66],
[67], but weight limitations make it impractical for small aerial vehicles. Along with
35
this, Gary Gress found that utilizing the vehicle’s lift-fans as CMGs offers a powerful
control system with minimal weight and independent of vehicle geometry or scale [1].
He discovered that the tilt rotor-based mechanism can provide hover stability of a
small UAV by using the gyroscopic nature of two tilting rotors [68].
Furthermore, due to the helicopters’ limitations in both close environments and
forward speed, development of alternate VTOL air vehicles has been considered by
many researchers [15], [18], [69]. Therefore, a combination of VTOL capability with
efficient, high-speed cruise flight plus high maneuverable characteristics in tilt-rotor
aircrafts have the potential to revolutionize UAVs. A good example of this type of
VTOL aerial vehicles is VTOL having lateral and longitudinal rotor tilting mecha-
nisms that give them the unique ability to maneuver in confined spaces. This entirely
new system, which uses only the dual lift-fans for control, has been developed recently
[1]. It utilizes the inherent gyroscopic properties and driving torques of the fans for
vehicle pitch control, and it eliminates the need for external control elements or lift
devices.
The system allows for agile and compact VTOL air vehicles by generating pure and
extensive moments rather than just forces. Figure 3.1 shows the University of Calgary
prototype of the eVader UAV that utilizes the dOAT concepts proposed in [68]. From
model simulations verified by several scenarios the dOAT control mechanism has
shown to be very promising.
3.2 Lift-fan OAT Mechanism
In the lift-fan OAT mechanism, unlike other tilt rotor UAVs, propellers can tilt in-
dependently in two directions (lateral and longitudinal with respect to the vehicle’s
36
Figure 3.1: Dual-fan VTOL air vehicle having lateral and longitudinal tilting rotors proto-type (eVader).
frame of reference) providing stability and control in hover mode. This mechanism
provides the required lift and control moments without the need for any helicopter
type cyclic controls. This is unique because most of the UAVs with tilt rotor design
have helicopter type cyclic controls. Cyclic controls are not compact and require rela-
tively slow turning with large diameter rotors which are not desirable for the purpose
of performing aggressive maneuvers, especially when flying in confined spaces. In this
thesis, aggressive maneuver refers to the maneuver with fast changing complex flight
trajectories such as acrobatic maneuvers, like loops and barrel rolls, and cuban eight
maneuvers.
In OAT design, the roll movement is obtained by differential propeller speeds. The
yaw angle can be controlled through differential longitudinal tilting. The gyroscopic
moment issued from opposed lateral tilting, together with the torque generated by
the collective longitudinal tilting, allow to obtain a significant pitching moment. In
the following section, we provide a brief description of how this novel vehicle (Fig.
3.1) operates and how the essential moments are produced.
37
Figure 3.2: Fans tilted longitudinally 90 degrees for high speed forward flight [1].
3.2.1 Special Characteristics of OAT
The OAT design comprises a differential or opposed lateral tilting element for gen-
erating gyroscopic and fan-torque pitching moments, in addition to the collective
longitudinal tilting component, which produces pitch moments from thrust vectors,
as well as forces for controlling horizontal motion. This mechanism can contribute
more than just stability and control in the conventional sense. Using the dual-axis
version makes it possible to have an independent control of all six axes [59].
1. High Speed Flight: Transition to high speed forward flight or airplane
mode is achieved by tilting the fans longitudinally 90 degrees (Fig. 3.2), during
which longitudinal stability is maintained by lateral tilting and by the horizontal
stabilizer at the rear of the aircraft. Because VTOL air vehicles do not require
runways their lifting surface areas need not be as large as those of a conventional
airplane. As a result, there is no need for conventional control surfaces (except the
horizontal stabilizer) and associated dual control system, thereby reducing weight,
complexity, and cost. Furthermore, because the entire wing-halves (fan shrouds) tilt,
and differential longitudinal tilting of the fans generates a gyroscopic rolling moment
(whether in hover or airplane mode), roll rates of the vehicle are substantially higher
than those using a conventional wing with ailerons.
38
2. Gyroscopic pitch moments: Tilting both spinning fans simultaneously to-
wards or away from one another laterally produces gyroscopic moments perpendicular
to their tilt axes in the right angles direction. This moment, τgyro, changes the vehi-
cle’s attitude as illustrated in Fig. 3.3, and this is the moment used to initiate control
and dynamically stabilize the pitch attitude of the developed eVader UAV. Return-
ing the spinning discs to their neutral orientation will stop rotation of the vehicle in
the case of space vehicles, where it will rest at the new attitude. In aerial vehicles,
however, there are aerodynamic and inertial forces which tend to terminate, limit, or
enhance the vehicle’s rotation without returning the fans to neutral.
3. Fan-torque pitch moments: When using lift-fans as CMGs for air vehi-
cle pitch control, there is another pitching moment associated with the fans’ lateral
tilting, which is a fan-torque pitching moment. Unlike the gyroscopic moment, a
fan-torque will remain after the tilting has stopped. Without this moment the fans
would have to be tilting continuously to generate gyroscopic moments in order to
reach a desired pitch angle or to compensate for a pitch disturbance. With the fans’
net torques providing a static pitching moment, these aerial vehicles have the poten-
tial to remain level in hover despite any pitch imbalances. So they have the ability
to pitch while stationary, a particularly advantageous feature allowing direct target-
pointing and VTOL take off and land from sharply inclined surfaces. Till now, only
tandem-rotor helicopters can achieve pitched hover stationary [7].
4. Thrust-vectoring pitch moments: The fan net torque may be insufficient
to provide the static restoration. Therefore, to improve the vehicle’s static stability in
all instances, an additional pitch control moment is obtained by collectively tilting the
fans in the longitudinal direction while simultaneously tilting them laterally. These
improvements all derive from the resulting characteristic of non-vertical thrust vector,
39
Figure 3.3: a) Oppositely spinning disks tilted equally towards one another generatinggyroscopic moment τgyro, b) The whole System rotated about y axis to a new attitudeorientation [1].
40
which also provides more direct horizontal motion control. Therefore, the fans’ tilting
for full and proper pitch control of the UAV will be in oblique direction. Hence the
name of this control method in either of its two executions is single-axis or dual-axis
OAT.
3.2.2 Overview of sOAT and dOAT
1. Single-axis Oblique Active Tilting (sOAT): In the simplest method, called
single-axis OAT or sOAT, the fans or propellers tilt about a fixed and oblique hori-
zontal axis, and the corresponding tilt path lies along a vertical plane oriented at a
fixed angle α from the longitudinal direction. As a result, the rotating disc changes
its lateral and longitudinal tilting following a predefined (fixed) curve. Thus, inde-
pendent lateral and longitudinal disc tilting is not provided. This is due to the fact
that the lateral tilt (β) is coupled with the longitudinal tilt (α). The tilt angle β is
measured along the tilt-path plane, and is zero when the propeller spin axis is verti-
cal. The sOAT mechanism provides full, helicopter-like pitch control of the vehicle.
Moreover, it also improves stability and control in yaw and roll, either by reducing
their high degree of coupling intuitively associated with dual-fan rotor crafts, or by
taking advantage of that coupling. This distinct superiority, together with its sim-
plicity, makes sOAT an exceptional choice of control method for small UAVs (at the
expense of reducing the maneuvering capabilities of the vehicle, e. g. very limited
pitched hover).
2. Dual-axis Oblique Active Tilting (dOAT): In this mechanism the fans
tilt independently about the x and y axis providing lateral (βi) and longitudinal (αi)
tilting angles, respectively (Fig. 3.4). There is much more to be gained by taking
full advantage of the dual-axis OAT capability, like the potential of better control
41
Figure 3.4: Schematic of VTOL aerial vehicle with dual-axis OAT mechanism [1].
response for independent 6-axis control, vertical take offs and landings from severely
sloped terrain, remaining perfectly level in hover, remaining stationary while pitching
and yawing to track a target, and extreme maneuvering in three dimensional space
[59]. The capabilities of dOAT are still an open area of research and exploration. To
investigate these capabilities and verify the characteristics of this control mechanism,
in this thesis, a full model of the dual-fan VTOL aerial vehicle with lateral and
longitudinal tilting rotors is derived, which represents a general dynamic model for
this kind of vehicle and can be used to explore the features of both sOAT and dOAT.
In this thesis, the focus is on dOAT as sOAT is a specific case of dOAT.
3.3 Lateral and Longitudinal Rotor Tilting VTOL Modeling
In this section, the translational and the rotational dynamic equations of the tilt-rotor
aerial vehicle are developed, and a state-space model is suggested for the developed
42
UAV prototype. In this modeling, a general form of the eVader is considered, having a
dOAT ability in which each of its ducts can have independent (different) lateral and
longitudinal angles and different propeller’s speeds that have not been considered
in previous works; see [70] and [1]. Due to the complexity of the vehicle and its
mathematical model, a set of four assumptions are used in this research:
Assumptions
1. The UAV structure is rigid and symmetrical,
2. The centre of mass is fixed below the origin of the body fixed frame B,
3. The propellers are rigid and have a fixed pitch blade, and
4. Thrust and drag forces are proportional to the square of the propeller’s speed.
Let E = xE, yE, zE represent the right hand inertial frame and B = xB, yB, zB
represent the body fixed frame, as can be seen in Fig. 3.5. The eVader is studied
as a vehicle with six Degree of Freedom (DOF). It changes its position along three
coordinate axes, longitudinal x, lateral y and vertical z. Its attitude is described
by three angles, roll φ, pitch θ, and yaw ψ as shown in Fig. 3.5. The Euclidean
position of the UAV with respect to E is represented by ζ(t) = [x(t), y(t), z(t)]T ∈
R3, and the Euler angles of the UAV with respect to E are represented by η(t) =
[φ(t), θ(t), ψ(t)]T ∈ R3 where φ(t) is the roll, θ(t) is the pitch and ψ(t) is the yaw
angle. Rotating the blades around the vehicle’s y axis defines the longitudinal tilting
angle (αi). Considering the right rotor as rotor number one (r1) and the left one as
rotor number two (r2), α1 and α2 are used to refer to the longitudinal angles of r1
(rotor/disc #1) and r2 (rotor/disc #2), respectively. βi, (i = 1, 2 for each rotor) is
the lateral tilting angle, which denotes rotating the blades around the vehicle’s x axis
(Fig. 3.5).
The equations of motion for a rigid body subject to body force Ftot ∈ R3 and
43
Figure 3.5: Schematic of the eVader VTOL with a body fixed frame B and the inertialframe E. The circular arrows indicate the direction of rotation of each propeller [1].
torque Ttot ∈ R3 applied to the center of mass are given by Newton-Euler equations
with respect to the coordinate frame B and can be written as:
mI3×3 0
0 J
V
Ω
+
Ω×mV
Ω× JΩ
=
Ftot
Ttot
where V ∈ R3 is the body linear velocity vector, Ω ∈ R3 is the body angular velocity
vector, m ∈ R specifies the mass, J ∈ R3×3 is the body inertia matrix, and I3×3 is an
identity matrix. A short list of main parameters and effects acting on the eVader is
listed in Table 3.1.
44
Table 3.1: Main physical parameters and effects acting on the eVader VTOL UAV withrespect to the inertial frame E.
Effect Source Symbol used in the modelForces and torques induced The rotation of two propellers FE
tot = Ryxz[Fx, Fy, Fz]on the vehicle Ttot = [τx, τy, τz]
Figure 4.8: The altitude output of FL and AFL controllers when the mass of the systemsis changed. The FL controller failed to reach the desired altitude zd = 2 with almost 0.4 msteady state error.
85
The simulation results obtained show that the proposed controller is able to sta-
bilize the eVader even for relatively critical initial conditions. However, the feedback
linearization method has some important limitations. In feedback linearization an
important assumption is that the model dynamics are perfectly known and can be
canceled entirely. Full state measurement is necessary in implementing the control
law. Many efforts are being made to construct observers for nonlinear systems and to
extend the separation principle to nonlinear systems. Finding convergent observers
for nonlinear systems is difficult. Besides, the lack of a general separation principle,
which would guarantee that the straightforward combination of a stable state feed-
back controller and a stable observer will guarantee the stability of the closed-loop
system, is another difficulty associated to this problem. Moreover, no robustness is
guaranteed in the presence of parameter uncertainty or unmodeled dynamics. This
problem is due to the fact that the exact model of the nonlinear system is not available
in performing feedback linearzation.
Adaptive control based on feedback linearization method has been successfully
developed for eVader. However, adaptive control technique is only applicable for
nonlinear control problems that satisfy the following conditions:
• The nonlinear plant dynamics can be linearly parameterized.
• The full state is measurable.
• Nonlinearities can be canceled stably (i.e., without unstable hidden modes or
dynamics) by the control input if the parameters are known.
However, in next chapters it will be revealed that there are other types of uncertainties
in the eVader dynamic model in addition to the linearly parameterizable nonlineari-
86
ties. For example, there are additive disturbances in control input signal, which make
the adaptive control approach unable to provide asymptotic stability.
Chapter 5
Integral Backstepping Control of eVader
Backstepping is a technique providing a recursive method of designing stabilizing
controls for a class of nonlinear systems that are transformable to a strict feedback
system. Backstepping can force a nonlinear system to behave like a linear system in
a new set of coordinates in the absence of uncertainties. However, backstepping and
other forms of feedback linearization such as IOL and ISL, addressed in the previous
chapter, require cancellations of nonlinearities, even those which are helpful for stabi-
lization and tracking. A major advantage of backstepping over feedback linearization
is that it has the flexibility to avoid cancellations of useful nonlinearities and pursue
the objectives of stabilization and tracking, rather than that of linearization.
In this chapter, an integral backstepping (IB) control technique is proposed to
improve the pitch, yaw, and roll stability of the eVader UAV. The controller is able
to simultaneously stabilize the 6 outputs of the system (i.e., orientation angles and
x, y, z). Position and orientation outputs are regulated to their desired values and all
six of them (φ, θ, ψ, x, y, z) track the desired reference trajectories at the same time.
Thus, unlike backstepping control of VTOL (vertical take off and landing) UAVs such
as quad-rotors, there is no need for inner-loop and outer-loop control. Simulation re-
sults using backstepping verified the potential of the eVader as a small UAV used in
different scenarios such as autonomous take off and landing and tracking time-varying
trajectories in order to maneuver inside obstructed environments. Simulation scenar-
87
88
ios presented in this chapter include attitude and position control, and stabilization
and autonomous take off and landing, which show promising results.
The remainder of this chapter is organized as follows: Section 5.1 discusses the
background of backstepping control technique. The state-space model of the vehicle
for backsteeping design is suggested in Section 5.2, followed by an explanation of the
controller design procedure, first for the vehicle’s attitude and then for its altitude
and position, in Section 5.3. The designed backstepping controller’s gains are tunned
based on gradient descent optimization method in Section 5.4. Simulation results
are also shown. Adaptive backstepping controller design for the eVader dynamic
model with parametric uncertainty is presented in Section 5.5. The conclusion and
discussion are summarized in Section 5.7.
5.1 Overview and Background of Backstepping Control Technique
Backstepping techniques provide an easy method to obtain a control algorithm for
nonlinear systems. Several controllers based on backstepping technique have been
developed for controlling rotary aerial vehicles such as quad-rotors and helicopters
[87], [88]. Madani et. al. designed a full-state backstepping technique based on the
Lyapunov stability theory and backstepping sliding mode control to perform hover
and tracking of desired trajectories [30], [89]. Another controller proposed by Castillo
et. al. used the backstepping technique and saturation functions. Using saturation
function in the control law guaranteed attitude and control inputs boundedness in
the presence of perturbations in the angular displacement [34]. However, this con-
troller was designed for a linear system and can not work properly out of the hover
operation point. Metni et. al. used backstepping techniques to derive an adaptive
89
nonlinear tracking control law for a quad-rotor system, using visual information [90].
This control law uses visual information and defines a desired trajectory by a series of
prerecorded images. The above-mentioned papers, all studied quad-rotor helicopter
control and showed that backstepping control method can effectively deal with the
under-actuated property of quad-rotors. However, due to the under-actuated prop-
erty of quad-rotor UAVs, only four outputs out of six outputs of the system are
controllable independently. For example, the controller can set the quad-rotor tracks
three Cartesian positions (x, y, z) and the yaw angle (ψ) to their desired values and
stabilize the roll (φ) and pitch (θ) angles to zero.
Improvements have been introduced by combining integral action within the con-
trol law, which consequently results in guaranteed asymptotic stability, as well as
steady state errors cancellation due to integral action [88]. The idea of adding in-
tegral action in the backstepping design was first introduced by Kanellakopoulos in
[91] to increase the robustness against external disturbances and model uncertainties.
In this chapter of this thesis, the application of integral backstepping controller pro-
posed in [91] is extended to the eVader UAV for stabilization at hover and trajectory
tracking maneuvers. The goal of using this control method on the eVader is to allow
this vehicle to use the full potential of its flying characteristics, enabled by the OAT
mechanism, in order to maneuver in confined spaces. The backstepping controller
presented in this thesis is a Multi-Input Multi-Output (MIMO) IB controller, which
enables the eVader to track all six outputs of the system including the three Cartesian
positions (x, y, z) and the three orientation angles (φ, θ, ψ). The design methodology
is based on the Lyapunov stability theory. Various simulations on the eVader’s dy-
namic model show that the control law stabilizes the whole system with zero steady
state tracking error.
90
After designing the IB control, the controller gains are tunned by employing a gra-
dient descent technique to avoid trial and error procedure. Optimization-based meth-
ods help to systematically accelerate the multiple-parameter tuning process. With
optimization-based techniques, controller gains are tuned based on optimization of
defined performance indices to improve transient stability, which would enhance the
UAV’s maneuvering performance. That is, the problem of setting nonlinear control
gains is formulated as an optimization problem based on gradient descent optimiza-
tion method including system and control constraints.
5.2 State-Space Model for Control
The state-space form of the model of the targeted OAT mechanism can be written as
in Chapter 3, with the following u input and x state vectors:
Table 5.2: The IB controller gains for autonomous take off and landing scenario. In thistable k1 = 1− c211 + λ6, k2 = c11 + c12, k3 = c11λ6.
The blue curve The red curve The green curvek1 = 20.7476 k1 = 47.7377 k1 = 9322k2 = 7.9543 k2 = 8.1071 k2 = 181.5819k3 = 5.1735 k3 = 5.3916 k3 = 115.9134
103
Figure 5.3: Autonomous take-off, altitude control in hover and landing of the eVader andthe effect of tuning the IB controller gains by Gradient Descent algorithm.
104
Figure 5.4: Stabilization of roll, pitch and yaw angles by IB control method (left figure)and pitched stability of the eVader at 25 in hover (right figure).
cients in control law in (5.24). The gradient descent optimization method was used to
apply the desired constraints to the problem and force the output of the IB controller
to track desired trajectories with the error restrictions within an acceptable range
[92]. Therefore, the coefficients of the controller c1, c2, ..., c12, λ1, ...λ6, have been
adjusted by the gradient descent optimization algorithm. Applying the optimization
algorithm improves the controller design by estimating and tuning its parameters.
Different objectives of the optimization technique help improve system performance
and reduce control efforts. For the purpose of automatically tuning and optimiz-
ing controller gains of this thesis work, the rise-time and overshoot constraints were
chosen as objectives. Whenever the optimization solver finds a solution that meets
the design requirements within the parameter bounds, (e.g controller gains must be
positive), the local minimum is found and the optimization process terminates.
The effect of this optimization can be seen in Fig. 5.3. The position initial value
105
is (x(0), y(0), z(0)) = (1, 1, 0) m and the desired value of altitude is fixed at 2 m.
The effect of controller coefficient optimization is illustrated in this figure as well.
The first and second curves in Fig. 5.3 on top are infeasible answers because they
cause the vehicle crash on the ground. The third curve on top in this figure is a
feasible answer obtained from optimization process. The coefficients of control law
(5.24) for this answer are obtained as follows: c7 = 181.58, c8 = 9140.82, and λ4 = 1.
On bottom of Fig. 5.3 all these three curves are shown in the same coordinate to
provide a better sense of comparison. Figure 5.4 shows the stabilization of roll, pitch
and yaw angles. In this simulation the initial values of Euler angles is considered at
(φ0 = 10, θ0 = 35, ψ0 = 5) and the final values are (φ = 0, θ = 25, ψ = 0). The
vehicle becomes stable at 25 pitch angle while is at hover.
5.7 Summary
In this Chapter, an Integral Backstepping control approach was proposed to control
both position and orientation of an eVader. The IB controller is presented based on
the derived dynamic model of this special UAV, which enables us to design 6 control
laws. The controller design is divided to six subsystems, each of which is designed
through similar procedure. The main advantage of the above design method is that
difficulties of designing a controller for the entire system are avoided.
The recursive Lyapunov methodology in the backstepping technique ensures the
system stability, and the integral action increases the system robustness against dis-
turbances and model uncertainties. Furthermore, gradient descent optimization al-
gorithm was applied on controller to find and adjust the coefficients, which makes
the results to be even more promising. As a design tool, backstepping is less restric-
106
tive than feedback linearization of the pervious Chapter. In some situations, it can
overcome singularities such as lack of controllability [82].
Simulation results show that the proposed algorithm is capable of controlling the
nonlinear model of the dual-fan VTOL air vehicle with lift-fan oblique active tilting
mechanism. It also fulfills the position and orientation stabilization task as well as the
ability of pitched hover. It provides the required stabilization to perform aggressive
maneuvering and reliable navigation. More importantly, the model presented in this
thesis, along with the proposed controller, show the ability of performing 6DOF
control without the need to cope with the under-actuated property of helicopter type
aerial vehicles. This chapter validated the usefulness of this method and verified that
the eVader has a lot more to offer in autonomous flight control of UAVs.
The adaptive IB controller provides asymptotic stability in presence of constant
parametric uncertainties. The parametric uncertainties has to appear linearly in the
dynamic model of the system. Now the question is what would happen to UAV if it
flies outdoor and a strong wind blows, or if, in an indoor flight, the eVader gets too
close to a wall or ground. In these situations the system is under external disturbances
and adaptive control methods can not guarantee the system stability anymore. This
issue is discussed in the next chapter.
Chapter 6
Sliding Mode Control for the eVader
The controllers designed in the preceding chapter guarantee that in the presence of
uncertain bounded nonlinearities the closed-loop state remains bounded. Adaptive
control can deal with uncertainties by tuning of the parameters online, but generally
is able to achieve only asymptotical convergence of the tracking error to zero. As it is
mentioned in previous chapters, adaptive control method is based on the assumption
that the structure of the system model is known with unknown slow-varying system
parameters, where the parameters appear linear. But several issues, such as transient
performance, unmodeled dynamics, disturbances such as wind and ground effects,
and not linear parameterizable uncertainties often complicate the adaptive approach
[93], [94].
In this chapter, the focus is on a useful and powerful robust control scheme to deal
with the uncertainties, nonlinearities, and bounded external disturbances using the
sliding mode control (SMC) scheme. These and other uncertainties may come from
unmodeled dynamics, variations in system parameters, or approximations of complex
plant behaviours. In robust control designs, a fixed control law based on a priori
information of the uncertainties is typically designed to compensate for their effects,
107
108
and exponential convergence of the tracking error to a (small) ball centred at the
origin is obtained (see the definition of exponential stability in Chapter 2). Robust
control has some advantages over the adaptive control, such as its ability to deal with
disturbances, quickly-varying parameters, and unmodeled dynamics [63].
Sliding controller design provides a systematic approach to the problem of main-
taining stability and consistent performance despite modelling imprecisions. An
overview of the sliding-mode nonlinear controller and main concepts of SMC de-
sign, such as sliding surface, are introduced in Section 6.1. Section 6.2 represents the
state-space form of the eVader model to facilitate the SMC design. The following
section describes how to design a SMC for the eVader UAV. Section 6.4 then presents
studies of some simulations to investigate the robustness of SMC methodology for
the eVader maneuvers, such as pitched hover maneuvers, while a sudden wind may
be affecting the vehicle’s behavior. The final section of this chapter summarizes the
main concepts of this chapter.
6.1 Overview of Sliding Mode Control
From the control point of view, modelling inaccuracies can be classified into two ma-
jor kinds: i) structured or parametric uncertainties, which correspond to inaccuracies
in the terms included in the model and ii) unstructured uncertainties or unmodeled
dynamics, which correspond to inaccuracies in the system order. Modelling inaccura-
cies has strong effects on nonlinear control systems. Therefore, practical designs must
109
address them explicitly. Two major approaches to dealing with model uncertainty are
robust control and adaptive control. In Chapter 4 of this thesis, adaptive control for
the eVader was employed. In this chapter the focus is on robust control approaches.
The typical structure of a robust controller is composed of a nominal part, similar
to a feedback linearizing or inverse control law, and of additional terms to cope with
model uncertainty. The structure of an adaptive controller is similar, but the differ-
ence is that the model is actually updated during operation, based on the measured
performance. A simple approach to robust control is the so-called sliding control
methodology. ”Perfect” performance can be achieved in the presence of arbitrary
parameter inaccuracies.
6.1.1 Sliding Surfaces
Sliding mode control is a high-speed switched feedback control. It is also known as
variable structure control (VSC) in the literature. The most important task in the
SMC methodology is to design a switched control that drives the plant state to the
switching surface and maintains it on the surface upon interception. A Lyapunov
approach is used to characterize this task. A generalized Lyapunov function, that
characterizes the motion of the state trajectory to the sliding surface, is defined in
terms of the surface. For each chosen switched control structure, one chooses the gains
so that the derivative of this Lyapunov function is negative definite, thus guaranteeing
motion of the state trajectory to the surface. After proper design of the surface, a
110
switched controller is constructed so that the tangent vectors of the state trajectory
point towards the surface such that the state is driven to and maintained on the
sliding surface. Such controllers result in discontinuous closed-loop systems. To be
more precise, the gains in each feedback path switch between two values according
to a rule that depends on the value of the state at each instant. The purpose of
the switching control law is to drive the nonlinear plant’s state trajectory onto a
prespecified (user-chosen) surface in the state-space and to maintain the plant’s state
trajectory on this surface for subsequent time. The surface is called a switching
surface. When the plant state trajectory is ”above” the surface, a feedback path
would have a specific gain and then, a different gain if the trajectory drops below the
surface. This surface defines the rule for proper switching. This surface is also called
a sliding surface (sliding manifold). Ideally, once intercepted, the switched control
maintains the plant’s state trajectory on the surface for all subsequent time, and the
plant’s state trajectory slides along this surface. The motion of the system as it slides
along boundaries of the control structures is called a sliding mode [95].
Intuitively, sliding mode control uses practically infinite gain to force the tra-
jectories of a dynamic system to slide along the restricted sliding mode subspace.
Trajectories from this reduced-order sliding mode have desirable properties (e.g., the
system naturally slides along it until it comes to rest at a desired equilibrium). The
main strength of sliding mode control is its robustness. Because the control can be as
simple as a switching between two states (e.g., on/off ), it does not need to be precise
111
and will not be sensitive to parameter variations that enter into the control chan-
nel. Additionally, because the control law is not a continuous function, the sliding
mode can be reached in finite time (i.e., better than asymptotic behavior). However,
real implementations of sliding mode control approximate the theoretical behavior
of sliding along the surface with a high-frequency and generally non-deterministic
switching control signal that causes the system to ”chatter” in a tight neighborhood
of the sliding surface [96].
In summary, the motion consists of a reaching phase during which trajectories
starting off the manifold S = 0 move toward it and reach it in finite time, followed by
a sliding phase, during which, the motion is confined to the manifold S = 0 and the
dynamics of the system are represented by a reduced-order model with exponentially
stable error dynamics. The S = 0 is called the sliding mode, and the control law
manifold u = −κ(x)sgn(S) is called sliding control mode.
6.1.2 Chattering
It should be noted that the controller is discontinuous at S = 0. Thus, sliding mode
control must be applied with more care than other forms of nonlinear control that
have more moderate control action. In particular, due to the effects of sampling,
switching and delays in the actuators used to implement the controller, and other
imperfections, the hard SMC action can lead to chatter, energy loss, plant damage,
and excitation of unmodeled dynamics [97]. Figure 6.1 shows how delays can cause
112
Figure 6.1: Chattering due to delay in control switching.
chattering. It depicts a trajectory in the region S > 0 heading toward the sliding
manifold S = 0. The trajectory first hits the manifold at a point Pa. In ideal sliding
mode control, the trajectory should start sliding on the manifold from a point Pa.
In reality, there will be a delay between the time the sign of S changes and the time
the control switches. During this delay period, the trajectory crosses the manifold
into the region S < 0. Chattering results in low control accuracy, high heat losses
in electrical power circuits, and high wear of moving mechanical parts. It may also
excite unmodeled high frequency dynamics, which degrade the performance of the
system and may potentially even lead to instability.
113
One commonly used method to eliminate the effects of chattering is to replace the
switching control law by a saturating approximation within boundary layer around
the sliding surface [63], [98], and [99]. Inside the boundary layer, the discontinuous
switching function κsgn(S) is approximated by a continuous function to avoid dis-
continuity of the control signals. Even though the boundary layer design can alleviate
the chattering phenomenon, this approach, however, provides no guarantee of conver-
gence to the sliding mode, and involves a trade-off between chattering and robustness,
and results in the existence of the steady state error.
6.2 Modeling for Control
From Chapter 3, the vehicle’s model can be written in the following form with additive
external disturbances d1, ..., d6.
φ = 1Jx
[
(Jy − Jz)θψ − ktaxφ2 − JrΩθ + u1 + d1
]
θ = 1Jy
[
(Jz − Jx)ψφ− ktayθ2 − JrΩφ+ u2 + d2
]
ψ = 1Jz
[
(Jx − Jy)θφ− ktazψ2 + u3 + d3
]
x = 1m[−Kfaxx
2 + u4 + d4]
y = 1m[−Kfayy
2 + u5 + d5]
z = 1m[−Kfaz z
2 −mg + u6 + d6]
(6.1)
114
Consider the state-space model presented in (6.1), where the vector y = [φ, θ, ψ, x, y, z]T
is the output of interest, the vector u = [u1, u2, u3, u4, u5, u6]T is the control input,
and x = [x1, x2, ..., x12]T is the vehicle’s state vector such as
x = [φ, φ, θ, θ, ψ, ψ, x, x, y, y, z, z]T .
The state-space model can be written in the following closed form:
x = f(x) + b(x)u+ d(x) (6.2)
with f(x), b(x), d(x) defined as follows:
115
f(x) =
x2
a1x4x6 − a2x22 − a3Ωx4
x4
a4x2x6 − a5x24 − a6Ωx2
x6
a7x2x4 − a8x26
x8
−a9x28
x10
−a10x210
x12
−a11x212 − g
12×1
(6.3)
where
a1 =Jy−JzJx
a2 =ktaxJx
a3 =JrJx
a4 =Jz−JxJy
a5 =ktayJy
a6 =JrJy
a7 =Jx−JyJz
a8 =ktazJz
a9 =Kfax
ma10 =
Kfay
ma11 =
Kfaz
m.
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b(x) =
0 0 0 0 0 0
b1 0 0 0 0 0
0 0 0 0 0 0
0 b2 0 0 0 0
0 0 0 0 0 0
0 0 b3 0 0 0
0 0 0 0 0 0
0 0 0 b4 0 0
0 0 0 0 0 0
0 0 0 0 b4 0
0 0 0 0 0 0
0 0 0 0 0 b4
12×6
(6.4)
where b1 =1Jx, b2 =
1Jy, b3 =
1Jz, and b4 =
1m.
And finally d(x) is defined as:
d(x) =
[
0 d1 0 d2 0 d3 0 d4 0 d5 0 d6
]T
(6.5)
In (6.2) the nonlinear function f(x) is not exactly known, and the control gain b(x)
is of known sign but unknown exact value. f(x) and b(x) are both upper bounded
by known, continuous function of x and external disturbance b(x). The inertia of
117
a mechanical system is only known to a certain accuracy, and friction models only
describe part of the actual friction forces. The control problem is to get the state x
to track a specific time varying state xd in the presence of model imprecision on f(x)
and b(x).
Remark: The disturbance d(x) can describe uncertainties such as inaccurate
torques and lifts of the rotors, the ground effects, wind disturbance, and the bias
between the geometric centre and its centre of gravity.
6.3 Sliding Mode Control based on Backstepping
In this thesis a 2 step approach for the design of the controller is taken. The two
steps are:
1. Defining the sliding mode. This is a surface that is invariant of the controlled
dynamics, where the controlled dynamics are exponentially stable, and where the
system tracks the desired set-point.
2. Defining the control that drives the state to the sliding mode in finite time.
6.3.1 Controller Design
The first step in designing the sliding mode controller is similar to the one used for
the backstepping approach, except Sφ (Surface) as defined in (6.6) is used instead of
e2:
Sφ = λ1e1 + x2 − x1d (6.6)
118
For the second step the following augmented Lyapunov function is considered:
V (e1, Sφ) =1
2(e21 + S2
φ)
The chosen law for the attraction surface is the time derivative of (6.6) satisfying
(SS < 0):
Sφ = −k1sgn(Sφ)− λ1Sφ
= λ1e1 + x2 − x1d
= λ1(x2 − x1d)− x1d + a1x4x6 + b1u1 + d1(t)
(6.7)
As for the backsteppning approach the control u1 is extracted:
u1 =1
b1[−a1x4x6 + x1d − k1sgn(Sφ)− λ21e1 − 2λe1] (6.8)
Using the backstepping approach as a recursive algorithm for the synthesis of
control-law, all the stages of calculation concerning the tracking errors and Lyapunov
functions can be simplified in the following way:
ei =
xi − xid i ∈ 1, 3, 5, 7, 9, 11
λje(i−1) + xi − x(i−1)d i ∈ 2, 4, 6, 8, 10, 12(6.9)
119
with λj > 0,∀j ∈ [1, 6], and
Vi =
12e2i i ∈ 1, 3, 5, 7, 9, 11
12(Vi−1 + e2i ) i ∈ 2, 4, 6, 8, 10, 12
(6.10)
The choice of the sliding surfaces is based upon the synthesized tracking errors
which permitted us the synthesis of stabilizing control laws. Thus, from (6.9) the
dynamic sliding surfaces, Sφ, Sθ, Sψ, Sx, Sy and Sz, are defined as:
Sφ = λ1e1 + e1
Sθ = λ2e3 + e3
Sψ = λ3e5 + e5
Sx = λ4e7 + e7
Sy = λ5e9 + e9
Sz = λ6e11 + e11
(6.11)
As e1 = φ− φd, e3 = θ − θd, e5 = ψ − ψd, e7 = x− xd, e9 = y − yd, and e11 = z − zd,
and for simplicity in notations, the switching surfaces can be rewritten as:
120
Sφ = λ1eφ + eφ
Sθ = λ2eθ + eθ
Sψ = λ3eψ + eψ
Sx = λ4ex + ex
Sy = λ5ey + ey
Sz = λ6ez + ez
(6.12)
To synthesize a stabilizing control law by sliding mode, the necessary sliding con-
dition (SS < 0) must be verified; so the synthesized stabilizing control laws are as
follows:
u1 =1b1
[
−a1x4x6 + φd(t)− 2λ1eφ(t)− λ21eφ(t)− k1sgn(Sφ)]
u2 =1b2
[
−a2x2x6 + θd(t)− 2λ2eθ(t)− λ22eθ(t)− k2sgn(Sθ)]
u3 =1b3
[
−a3x2x4 + ψd(t)− 2λ3eψ(t)− λ23eψ(t)− k3sgn(Sψ)]
u4 =1b4[−a9x
2 + xd(t)− 2λ4ex(t)− λ24ex(t)− k4sgn(Sx)]
u5 =1b4[−a10y
2 + yd(t)− 2λ5ey(t)− λ25ey(t)− k5sgn(Sy)]
u6 =1b4[−a11z
2 − g + zd(t)− 2λ6ez(t)− λ26ez(t)− k6sgn(Sz)]
(6.13)
6.4 Simulation Results
In order to test the developed SMC a number of simulations were tested. Herein,
four of such simulations are presented, which show the performance of the developed
121
SMC. The following four scenarios are presented: 1) pitched hover, 2) pitched hover
under wind disturbances, 3) Scenario # 2 with varying model parameters, and 4)
picking up a load.
1. Pitched hover scenario (Scenario #7): The first simulation is the pitched hover
maneuver, where the vehicle is commanded to hover in a maneuver that no aircraft
known to have performed. In this case study, the system initial condition and the
desired states are chosen as below:
x0 = [22.5, 0, 0, 0, 18, 0, 5, 0, 4, 0, 3, 0]T
xd = [0, 0, 22.5, 0, 0, 0, 0, 0, 0, 0, 3, 0]T
In other words, the controller is commanded to maintain the eVader at the same
altitude at 3 m while pitching its nose from 0 to 22.5, and regulate all other states
to zero. The results are illustrated in Figs. 6.2, 6.3, and 6.4. For this simulation, the
switching functions and controller gains are selected according to Table 6.1. Figure
6.2 shows the six control signals u1, ..., u6. The chattering effect is obviously seen in
this figure. Figure 6.3 shows that the controller is able to regulate the eVader’s roll
and yaw angles to zero, while keeping the vehicle’s pitch angle at 22.5 .
are the same as previous scenario, and it is assumed that after 1.5 s, the wind blows
for 0.2 s. Wind as an external disturbance is formulated with d1(t) = d2(t) = d3(t) =
d4(t) = d5(t) = d6(t) = 5(10+5sin(2πt)) in eVader model in (6.5). It is also assumed
that wind blows in all directions effects on all states. The results in Figs. 6.5 and 6.6
illustrate the robustness of sliding mode controller in presence of unstructured un-
certainties. This property insures the stability of eVader, while external disturbances
such as wind, ground and wall effects affect the vehicle during its maneuvers.
3. Scenario # 2 with varying model parameter (structured uncertainty): The sim-
ulation scenario # 2 used in Chapter 4, Section 4.7, is simulated again here to verify
the robustness of the designed SMC controller to parameter variations. The scenario
is defined as follows: the initial condition is x0 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]T , and
the desired value is xd = [22.5, 0, 15, 0, 18, 0, 3, 0, 4, 0, 2, 0]T . In this simulation the
mass of the eVader is changed with the same amount as in Chapter 4. Moreover,
the inertia matrix of the vehicle is also changed by 20%. Another simulation was
made on the eVader model with mass and inertia matrix uncertainty and the added
123
random noise. This noise can be considered as a measurement noise. The results
were compared with the results of simulation without parameter uncertainty, and
also with the scenario with only the parameter uncertainty but no added noise, in
Table 6.2. As it can be seen easily from this table, after 1 second has passed, the error
for all six outputs are almost zero for all mentioned scenarios. The results in Table
6.2 verify the robustness of the proposed designed controller in presence of struc-
tured (parametric) uncertainty. Figures 6.7 and 6.8 show the control input signals,
the orientation output, and the position output of the eVader, respectively, in this
simulation scenario.
4. Picking-up-a-load scenario (Scenario #11): In this simulation, the eVader is
commanded to pick up a heavy load of 3.5 kg after 1 second into the simulation. It
is assumed that this load adds a sudden disturbance to the pitch (θ) and the pitch
angular velocity (θ) of the vehicle. The results are depicted in Figs. 6.9, 6.10, and
6.11, which show the ability of the controller in handling this situation. The system
pitch returned to a steady state in 0.25 s. The steady state error of altitude is only
4 cm and reached after 0.5 s.
6.5 Summary
A simple approach to robust control is the so-called sliding control methodology. The
aim of a sliding controller is to design a control law to effectively account for parameter
uncertainty, including imprecision on the mass properties or loads, inaccuracies in the
124
Table 6.2: The results of SMC controller for position and orientation regulation simulation(Scenario # 2) in three different conditions: i) the parameters of the dynamic model areknown and do not change, ii) the mass and inertia matrix of the vehicle have uncertaintyand change during the flight, iii) the parameters of the dynamic model have uncertaintyand change plus there is additive white noise due to the sensor measurement noise.
i) Fixed parameters ii) Varying parameters iii) Additive sensor noisetime (sec) error (deg, m) error (deg, m) error (deg, m)
Roll φ 1 0.0089 0.0089 0.0290Pitch θ 1 0.0090 0.0089 0.0379Yaw ψ 1 0.0090 0.009 0.0023Position x 1 0.00003 0.0008 0.0027Position y 1 0.00007 0.0011 0.0033Altitude z 1 0.0003 0.0426 0.0124
0 0.5 1 1.5−5
0
5
10Control Signal u1
time[s]
u1
0 0.5 1 1.5−2000
0
2000
4000Control Signal u4
time[s]
u4
0 0.5 1 1.5−10
0
10Control signal u2
time[s]
u2
0 0.5 1 1.5−2000
0
2000
4000Control signal u5
time[s]
u5
0 0.5 1 1.5−10
0
10Control signal u3
time[s]
u3
0 0.5 1 1.598.0698.08
98.198.1298.14
Control signal u6
time[s]
u6
Figure 6.2: Control input signals of SMC technique for performing pitched hover scenario(Scenario # 7).
Figure 6.4: Position regulation while sta-tionary at pitched hover scenario (Sce-nario # 7).
torque constants of the actuators, friction, and so on.
Although perfect tracking can be achieved in principle in the presence of arbitrary
parameter inaccuracies, such performance is obtained at the cost of extremely high
control activity. The drawback is that this high control activity may excite high
frequency dynamics neglected in the course of modeling. In practice, this corresponds
to modification of control laws by replacing a switching, chattering control law by its
smooth approximation. The switching control laws derived above can be smoothly
interpolated in boundary layers, so as to eliminate chattering, thus leading to a trade-
off between parametric uncertainty and tracking performance.
In this chapter, a robust controller based on SMC methodology was designed for
126
0 0.5 1 1.5 2−20
0
20Control Signal u1
time[s]
u1
0 0.5 1 1.5 2−2000
0
2000
4000Control Signal u4
time[s]
u4
0 0.5 1 1.5 2−40
−20
0
20Control signal u2
time[s]
u2
0 0.5 1 1.5 2−2000
0
2000
4000Control signal u5
time[s]
u50 0.5 1 1.5 2
−20
−10
0
10Control signal u3
time[s]
u3
0 0.5 1 1.5 2−1000
−500
0
500Control signal u6
time[s]u6
Figure 6.5: Control input signals of SMC technique in presence of a strong sudden winddisturbance (unstructured uncertainty, Scenario # 8).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−10
0
10
20The roll angle
time[s]
Ro
ll[d
eg
re
e]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
20
40The pitch angle
time[s]
Pitch
[de
gre
e]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−50
0
50The yaw angle
time[s]
Ya
w[d
eg
re
e]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6The Cartesian position x
time[s]
x[m
]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
5
10The Cartesian position y
time[s]
y[m
]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22.8
3
3.2The Cartesian position z
time[s]
z[m
]
Figure 6.6: Orientation angles and position regulation in hover pitched scenario with astrong sudden wind disturbance with magnitude 5 (Scenario # 8).
127
0 0.2 0.4 0.6 0.8 1−10
0
10Control Signal u1
time[s]
u1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−500
0
500
1000
1500Control Signal u4
time[s]
u4
0 0.2 0.4 0.6 0.8 1−10
0
10Control signal u2
time[s]
u2
0 0.2 0.4 0.6 0.8 1−1000
0
1000
2000Control signal u5
time[s]
u50 0.2 0.4 0.6 0.8 1
−10
0
10Control signal u3
time[s]
u3
0 0.2 0.4 0.6 0.8 1−1000
0
1000
2000Control signal u6
time[s]u6
Figure 6.7: Control input signals of SMC technique when performing Scenario # 2 andsystem parameters (mass and inertia matrix) are varying.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
20
40The roll angle
time[s]
Ro
ll[d
eg
re
e]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
10
20The pitch angle
time[s]
Pitch
[de
gre
e]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
10
20The yaw angle
time[s]
Ya
w[d
eg
re
e]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4The Cartesian position x
time[s]
x[m
]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6The Cartesian position y
time[s]
y[m
]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2The Cartesian position z
time[s]
z[m
]
Figure 6.8: Orientation angles and position regulation with model parameter variationsshow robustness of SMC technique in presence of structured uncertainties.
128
0 0.5 1 1.5−40
−20
0
20Control Signal u1
time[s]
u1
0 0.5 1 1.5−1000
0
1000
2000Control Signal u4
time[s]
u4
0 0.5 1 1.5 2−100
−50
0
50Control signal u2
time[s]
u2
0 0.5 1 1.5−1000
0
1000
2000Control signal u5
time[s]
u50 0.5 1 1.5
−50
0
50Control signal u3
time[s]
u3
0 0.5 1 1.5 260
80
100
120Control signal u6
time[s]u6
Figure 6.9: Control input signals of SMC technique while picking up a heavy load (Scenario# 11).
eVader, which achieves exponential tracking control of a desired trajectory where
the plant dynamics contain uncertainty and bounded non-LP disturbances. Simula-
tion results demonstrate the robustness of the controllers to sensor noise, exogenous
perturbations, parametric uncertainty, and plant nonlinearities, while simultaneously
exhibiting the capability to follow a reference trajectory. However, in order to com-
pensate for uncertainties in the model and external disturbances, large input gains
are required which makes a serious limitation in power-limited systems such as small
UAVs like eVader. Moreover, the chattering phenomena may cause the rotors to
rotate and tilt in a different direction very fast. This may makes using the SMC
impossible for eVader even though it has a robustness property which is necessary for
129
0 0.5 1 1.5−100
0
100The error of roll angle
time[s]
Ro
ll[d
eg
]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−50
0
50The error of pitch angle
time[s]
Pitch
[de
g]
0 0.5 1 1.5−100
0
100The error of yaw angle
time[s]
Ya
w[d
eg
]
Figure 6.10: Orientation angles regula-tion error when the eVader picks up aheavy load (Scenario # 11).
0 0.5 1 1.5−5
0
5The error of x
time[s]
x[m
]
0 0.5 1 1.5−5
0
5The error of y
time[s]
y[m
]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.02
0.04The error of z
time[s]z[m
]
Figure 6.11: Position regulation errorwhen the eVader picks up a heavy loadscenario (Scenario # 11).
control of such a nonlinear dynamics system.
It is now time to investigate the feasibility of the designed controllers including
SMC and other controllers discussed in previous chapters. This issue will be discussed
in the next chapter.
Chapter 7
Neural Network Nonlinear Function Approximation
In previous chapters of this thesis we have derived the dynamic model of eVader UAV
and then designed several control laws to do specific tasks such as regulation to an
arbitrary desired position and orientation, tracking a time varying trajectory in 3D
space, and taking off and landing. All of the designed controllers in previous chapters
are based on six control signals u1, ..., u6, such as six control signals in Equations
(5.3) and (4.8) in Chapters 5 and 4, respectively. However, control signals u1, ..., u6
are not the real actuator signals. They are nonlinear functions of the actual control
signals α1, α2, β1, β2, ω1, ω2, that would be used to actually control the actuators
influencing them. The applicability of previously designed controllers, the eVader
and dOAT mechanism, lies in the existence of a feasible (in range) combination of
longitudinal and lateral angles, and rotor angular speeds, namely α1, α2, β1, β2, ω1, ω2,
to produce the necessary control signals. Thus the question is how to solve equations
of u1, ..., u6 based on α1, α2, β1, β2, ω1, ω2 to obtain actual control signals. Since these
equations do not have a mathematical systematic solution, it is not possible to obtain
the exact values of α1, α2, β1, β2, ω1, ω2. Hence, we proposed to utilize nonlinear
function approximator approaches to find an approximation of these functions in a
130
131
way that by entering the six control signals u1, ..., u6, the function approximator would
give us α1, α2, β1, β2, ω1, ω2 as its outputs. To address this problem, a Multi Layer
Perceptron (MLP) neural network is trained in this thesis as an inverse mapping of the
u1, ..., u6 functions, using supervised learning with back-propagation (BP) algorithm.
By applying this approach, we find an approximation of the actual control signals of
dOAT mechanism, which has not been addressed in the literature before.
The rest of this chapter is organized as follows. The benefits of using neural
networks for the specific problem of this research are first discussed in Section 7.2,
followed by an explanation of feedforward networks in Section 7.3. The MLP networks
are introduced in Section 7.4. Section 7.5 explains the BP learning algorithm. The
application and the results of the MLP neural network with BP learning algorithm
for our specific problem of approximation of the inverse mapping between α1, α2, β1,
β2, ω1, ω2 and u1, ..., u6 is investigated in Section 7.6. Finally, a summary of this
chapter and the main observations and findings of it are discussed in Section 7.7.
7.1 Benefits of Neural Networks
The neural network derives its computing power through, first, its massively par-
allel distributed structure and, second, its ability to learn and therefore generalize.
Generalization refers to the neural network producing reasonable outputs for inputs
not encountered during training (learning). These two information-processing capa-
bilities make it possible for neural networks to solve complex (large-scale) problems
132
that are currently intractable such as the problem of solving the nonlinear functions
of u1, ..., u6 based on six unknown variables α1, α2, β1, β2, ω1, ω2, which does not
have any conventional mathematical solution. The use of neural networks offers the
following useful properties and capabilities:
1. Nonlinearity. A neuron is basically a nonlinear device. Consequently, a neu-
ral network, made up of an interconnection of neurons, is itself nonlinear. Moreover
the nonlinearity is of a special kind in the sense that it is distributed throughout the
network. Nonlinearity is a highly important property, particularly if the underlying
physical mechanism responsible for the generation of an input signal is inherently
nonlinear such as the eVader’s dynamic in the specific problem at hand in this thesis.
2. Input-Output Mapping. A popular concept of learning called supervised
learning (Fig. 7.1) involves the modification of the weights of a neural network by
applying a set of labeled training samples or task examples. Each example consists
of a unique input signal and the corresponding desired response. The network is
presented and an example picked at random from the set, and the weights (free
parameters) of the network are modified so as to minimize the difference between
the desired response and the actual response of the network produced by the input
in accordance with an appropriate statistical criterion. The training of the network
is repeated for many examples in the set until the network reaches a steady state,
where there are no further significant changes in the weights. The previously applied
training examples may be reapplied during the training session but in a different order.
133
Figure 7.1: Supervised learning block diagram.
Thus the network learns from the examples by constructing an input-output mapping
for the problem at hand. Lack of existing systematic mathematical approaches for
the problem of finding actual control signals of UAV flight makes us to think of its
solution as an input-output mapping problem.
3. Adaptivity. Neural networks have a built-in capability to adapt their weights
to changes in the surrounding environment. In particular, a neural network trained to
operate in a specific environment can be easily retrained to deal with minor changes
in the operating environmental conditions. Moreover, when it is operating in a non-
stationary environment (i.e. one whose statistics change with time), a neural network
can be designed to change its weights in real time (adaptive neural network). As a
general rule, it may be said that the more adaptive we make a system in a properly
134
designed fashion, assuming the adaptive system is stable, the more robust its per-
formance will likely be when the system is required to operate in a non-stationary
environment. It should be emphasized, however, that adaptivity does not always
lead to robustness. Indeed, it may do the very opposite. For example, an adaptive
system with short time constants may change rapidly and therefore tend to respond
to spurious disturbances, causing a drastic degradation in system performance.
7.2 Problem Statement
Recalling the equations of u1, ..., u6 from Chapter 3, we have:
• xk, (k = 1, 2, ..., n) input signal of order n (input layer), xk = [x1x2...xn]T , N
samples of input elements k = 1, 2, ..., N ,
• zi, (i = 1, 2, ..., p) output signal of the first layer (hidden layer), zk = [z1z2...zp]T
the outputs of all neurons in the hidden layer,
• yj, (j = 1, 2, ...,m) output signal of second layer (output layer), yk = [y1y2...ym]T
the output vector of neural network,
• W(1)ik , (i = 1, 2, ..., p), (k = 1, 2, ..., n) the weights corresponding to first layer
neurons, the connection weight from the kth input to ith neuron in first layer,
• W(2)jq , (j = 1, 2, ...,m), (q = 1, 2, ..., p) the connection weight from the qth neuron
in the first layer to the jth neuron in the output layer.
144
In this flowchart a set of N input data xk = [x1, ..., xn] is first presented to the input
layer. The output from this layer are then fed as inputs to the hidden layer and
subsequently the outputs from the hidden layer are fed as weighted inputs (i.e., the
outputs from the hidden layer are multiplied by the weights (W(1)ik xk) to the second
layer which is a output layer in this flowchart. The hidden layer has activation
function of sigmoid. Thus, the accumulative weighted inputs (S(1)i =
∑n
k=0W(1)ik xk)
of all neurons in the hidden layer first go into a sigmoid function (zi = σ(S(1)i )) and are
then fed to the next layer as inputs. Despite the fact that the output layer performs
based on a linear function, the same process as for the hidden layer happens in the
output layer, and the output of this layer (yj =∑p
q=0W(2)jq zq) is the response of neural
network to the input xk. The output of neural network is then compared with the
desired ones and the error goes back to update the weights of output and hidden
layers through BP algorithm as illustrated in Modifications of Weights sections in the
flowchart in Fig. 7.7.
7.6 Training the MLP Neural Network for Actual Control Signal
Approximation
As mentioned before, for approximation or fitting problem, a neural network has to
map between a data set of numeric inputs and a set of numeric targets. A two-
layer feed-forward with sigmoid hidden neurons and linear output neurons is trained,
which based on universal approximation theorem, can fit multi-dimensional mapping
145
Figure 7.7: Flowchart of training process in a two-layer perceptron network. This flowchartdoes not include the stopping criteria of the training process.
146
problems arbitrarily well, given consistent data and enough neurons in its hidden
layer. The network is trained with Levenberg-Marquardt back-propagation algorithm.
Assumptions: We have made some assumptions on variables in (7.1), because
their real values are unknowns. For example, the rate of change of the longitudinal
and lateral angles, (αi, βi), depends on the capability of the mechanical system, and
they are assumed to be constant in training the MLP network. As a result the second
derivatives are considered to be zero. Furthermore, by considering Ti = CTω2i , and
Qi = CQω2i , the values of aerodynamic coefficients CT , CQ are needed for generating
data for neural network training. The list of all of the assumptions for neural network
training is as below:
1. The rates of change of α1, α2 are constant, and are equal to 0.05 rad/s.
2. The rates of change of β1, β2 are constant and are equal to 0.05 rad/s.
3. The second derivatives of α1, α2, β1, β2 are zero.
4. T1 = CTω21, T2 = CTω
22 and CT is assumed to be constant and equal to 0.01.
To improve the performance of the approximator and reduce MSE error of training, six
separate MLP networks with six inputs and one output were trained. These networks
are each trained for approximating one of the variables α1, α2, β1, β2, ω1, and ω2.
The progress of each network is shown in Table 7.3. The best training performance
and the regression (R) for the training set of data and a test set of data are shown
in Table 7.4 for all 6 MLP networks. Figures 7.15 to 7.32 show the performance and
error histogram of all trained MLP networks with 15 neurones.
153
Table 7.4: MSE and Regression results for training data and a set of test data for MLPnetworks with 6 inputs and 1 output and 15 neurons in hidden layer
α1 α2 β1 β2 ω1 ω2
Best training 5.0613e-06 1.1154e-05 0.0596 0.486 6.1295e-04 1.6938e-04performance
The simulation results shown in Figs. 8.4 to 8.9, show that although the AFL
control approach without robust modification can handle constant and slow-varying
uncertainties such as model parameters, it is not capable of maintaining the stability
of the UAV in the presence of time-varying external disturbance. On the other hand,
the stability of the eVader is preserved with the RAFL controller (Figs. 8.8 and 8.9),
although there is steady state error, while the wind keeps blowing.
172
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2x 10
215 Control signal u1
time[s]
u1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−200
−150
−100
−50
0Control signal u4
time[s]u
4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−6
−4
−2
0
2x 10
215 Control signal u2
time[s]
u2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−200
−150
−100
−50
0Control signal u5
time[s]
u5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
3
4x 10
238 Control signal u3
time[s]
u3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−40
−20
0
20
40Control signal u6
time[s]
u6
Figure 8.4: Control signals of AFL controller without robust modification in presence ofwind disturbance. The control signals u1, u2 and u3 go to infinity and make the eVaderunstable.
173
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2
0
2
4
6x 10
213 The roll angle
time[s]
Ro
ll[d
eg
ree
]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−15
−10
−5
0
5x 10
213 The pitch angle
time[s]
Pitc
h[d
eg
ree
]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
2
4
6
8
10x 10
236 The yaw angle
time[s]
Ya
w[d
eg
ree
]
Figure 8.5: eVader Orientation goes to infinity with AFL controller without robust modifi-cation in presence of wind disturbance.
8.3 Ground Effect (Scenario #12)
Ground and wall effects are important fluid flow characteristics that aerial vehicles
(e.g., helicopters and other VTOL vehicles) experience. Under ground and wall effects,
it is extremely difficult to control any aerial vehicles. The basic principle of ground
effects is that the closer the rotor’s fan operates to an external surface such as the
ground, said to be in ground effect (GE), the more thrust it produces. In order to
control an aerial vehicle under GE, there is a need to have a representation of such
effects. Previous graduate students working with the eVader UAV in the Autonomous
Reconfigurable Robotic Systems Laboratory at the University of Calgary have studied
174
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−6
−4
−2
0Control signal u1
time[s]
u1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−200
−150
−100
−50
0Control signal u4
time[s]
u4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5
−4
−3
−2
−1Control signal u2
time[s]
u2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−200
−150
−100
−50
0Control signal u5
time[s]
u5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4
−3
−2
−1Control signal u3
time[s]
u3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−40
−20
0
20
40Control signal u6
time[s]u
6
Figure 8.6: Control signals of RAFL controller, with e-modification, in presence of winddisturbance.
the GE experienced by the eVader. Therefore, to model the GE disturbance and add
such disturbances to the model of the eVader, we used Eq. (8.1), which first presented
and explained in Chapter 3, to represent the GE.
gr(z) =
az(z+zcg)2
− az(z0+zcg)2
0 < z ≤ z0
0 else(8.1)
In (8.1) z is the altitude of the eVader’s rotor with respect to ground, zcg is the altitude
of the vehicle’s centre of gravity, z0 is the hight below which the GE is effective (Fig.
8.10), and az is an unknown constant. The value of az is calculated using the result
175
0 1 2 3 4 5−0.3
−0.2
−0.1
0
0.1
0.2
a1
time
a1
0 1 2 3 4 5−1
0
1
2
3
4
a2
time
a2
0 1 2 3 4 5−0.2
−0.1
0
0.1
0.2
a3
time
a3
0 1 2 3 4 5−2
−1
0
1
2
a4
time
a4
0 1 2 3 4 50
0.2
0.4
0.6
0.8
a5
time
a5
0 1 2 3 4 5−2
−1
0
1
2
a6
time
a6
0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
a7
time
a7
0 1 2 3 4 50
0.1
0.2
0.3
0.4
a8
time
a8
0 1 2 3 4 50
0.1
0.2
0.3
0.4
a9
time
a9
Figure 8.7: Parameter estimation of RAFL controller with e-modification in presence ofwind disturbance.
of thrust ratios in constant fan speed, obtained from the CFD (Computational Fluid
Dynamics) simulations for a ducted fan in [75]. The observation made in [75] is that
in constant rotational speed, as the fan gets closer to the ground, the thrust force
increases. The thrust starts to increase at a hight of 1.5-fold rotor diameter. At a
rotor hight of one-half the rotor diameter, the thrust ratio is increased by about 50
percent. This result is compatible with the model in (8.1). Thereby, to calculate the
value of az, Equation (8.1) is solved at point hrr
= 1, with z0 = 1.5, where h is a
fan’s elevation from the ground and rr is the rotor’s radius, which gives a value of
az = 16.7. As a result of the ground effect disturbance, the thrust force on each rotor
of the eVader is increasing as much as modelled in (8.1). To examine the performance
176
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
5
10
15
20
25The roll angle
time[s]
Ro
ll[d
eg
ree
]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
5
10
15The pitch angle
time[s]
Pitch
[de
gre
e]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
5
10
15
20The yaw angle
time[s]
Ya
w[d
eg
ree
]
Figure 8.8: eVader orientation with RAFL controller with e-modification in presence ofwind disturbance.
of the designed robust controllers including RAFL and SMC, a landing simulation
scenario was performed in which the initial and final state conditions are as follows:
xd= [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]T . Thus, the vehicle is commanded to descend from a
height of three meters, while translating and changing its orientation, and land. The
control signal results and position and orientation outputs of the eVader by applying
AFL, with e-modification, and SMC are shown in Figs. 8.11 to 8.13. As it is seen
in Figs. 8.12 and 8.13 the SMC controller is much faster than the RAFL control
technique, as observed in Scenario #1 as well. The AFL controller is also tested in
this scenario, but as in the case of buffeting wind, this controller was not able to
177
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52
2.1
2.2
2.3The Cartesian position x
time[s]
x
system outputdesired x(t)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 53
3.1
3.2
3.3
3.4The Cartesian position y
time[s]
y
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
4.99
5
5.01
5.02
5.03
5.04The altitude z
time[s]
z
Figure 8.9: eVader position with RAFL controller with e-modification in presence of winddisturbance.
maintain the stability of the eVader when it lands on the ground.
8.4 Aggressive Maneuver
In this thesis, aggressive maneuver refers to maneuvers having a fast changing complex
flight trajectories such as acrobatic maneuvers like loops and barrel rolls, and cuban
eight maneuvers. To produce an aggressive trajectory as a function of time and apply
it to the designed controllers, herein, the Mobius strip equations [111] are used in
178
Figure 8.10: Schematic diagram of the eVader which shows z, z0, zcg.
(8.2) as:
x = [R + rcos(γ)] cos(κ)
y = [R + rcos(γ)] sin(κ)
z = rsin(γ)
(8.2)
with R = 4, r = 3, −π < κ < π, and −π/2 < γ < π/2. Equation (8.2) is used
to provide the 3D Cartesian position of the eVader’s path in this flight scenario.
The vehicle’s orientation reference trajectories are chosen arbitrarily consistent with
a trajectory in Fig. 8.14. Therefore, the roll angle is set to be stable at zero, and
the pitch and yaw angles follow the piecewise constant curves with sharp stepwise
patterns, which vary from 25 to −50 for pitch and from zero degree to 175 for yaw
179
angle (Figures 8.18 and 8.19). Both robust controllers are successful in tracking the
reference trajectories of position and orientation. The RAFL controller reaches to
the desired position trajectory after 4 s (Figure 8.20), while it takes only 0.7 s for
the SMC controller (Figure 8.21). The observations are consistent with the obtained
results in the GE simulation scenario. The 3D position tracking of the aggressive
maneuver for RAFL and SMC controllers are shown in Figs. 8.14 and 8.15. The
control signals of RAFL and SMC controls are shown in Figs. 8.16 and 8.17, the
vehicle’s orientation tracking output by RAFL and SMC shown in Figs. 8.18 and
8.19, the eVader’s position tracking curves shown in Figs. 8.20 and 8.21, and the
eVader’s orientation and position tracking errors by SMC control shown in Figs. 8.22
and 8.23, respectively. A comprehensive discussion of the results is presented in the
next section.
8.5 Result Discussion
This chapter presented and demonstrated the capabilities of the proposed nonlinear
model of the eVader and proposed designed nonlinear control approaches proposed
in Chapters 3 to 6. In this chapter, throughout several simulation scenarios, the
objective of choosing the best controller amongst the designed nonlinear controllers
in this thesis for the eVader is investigated. Because, in this thesis, the eVader is
supposed to perform complex agile maneuvers, a fast response controller is desirable.
The results of simulation Scenario # 1, Figs. 8.1 to 8.3, show the SMC controller is the
180
0 1 2 3 4 5−15
−10
−5
0
5
Control Signal u1
u1
0 1 2 3 4 5−3000
−2000
−1000
0
1000
Control Signal u4
u4
0 1 2 3 4 5−15
−10
−5
0
5Control Signal u2
u2
0 1 2 3 4 5−4000
−2000
0
2000Control Signal u5
u50 1 2 3 4 5
−10
−5
0
5Control Signal u3
time (s)
u3
0 1 2 3 4 5−15000
−10000
−5000
0
5000Control Signal u6
time (s)
u6
AFL control with e−modificationSMC control
Figure 8.11: Control signals of AFL with robust modification and SMC control in presenceof ground effect disturbance.
first one that reaches the desired steady states. However, the undesirable chattering
phenomena observed in Figure 8.1, makes us to consider the second fastest controller,
which is the FL controller. However, as addressed in Chapter 4, the FL controller is
not able to provide asymptotic stability in the presence of model uncertainties. That
means, if there are any variations in the parameters of the model, the FL controller is
no longer able to regulate the system to the desired position and orientation without
steady state errors. The other choice of controller is the AFL one, which is capable
of coping with model uncertainties and parameter variations. The next simulation
of this chapter, testing the performance of eVader under strong windy conditions,
shows that the eVader requires a robust controller to execute stable maneuvers in the
181
0 1 2 3 4 5 6−20
0
20
40
The roll angle
Rol
l (de
gree
)
0 1 2 3 4 5 6−20
0
20
40The pitch angle
Pitc
h (d
egre
e)
0 1 2 3 4 5 6−10
0
10
20The yaw angle
time (s)
Yaw
(deg
ree)
AFL + e−modificationDesired outputSMC control
Figure 8.12: Output orientation angles of eVader obtained by applying AFL with robustmodification and SMC control in presence of ground effect disturbance.
presence of external disturbances, and the adaptive control approach without adding
a robust modification term, is not successful in this condition. Figures 8.4 to 8.5 verify
that the AFL control approach without robust modification is not able to perform
in windy situations. On the other hand, the RAFL is able to maintain the eVader’s
stability while a strong wind is blowing on the vehicle along all directions. The
next two simulation scenarios were made to compare the robust controllers, RAFL,
and SMC. Landing on the ground with GE, and performing Mobius strip maneuver,
with a trajectory including a barrel roll maneuver and a half cuban eight aggressive
maneuvers were considered. The SMC controller shows a better performance than
RAFL when the eVader lands on the ground and there is an external disturbance in
182
0 1 2 3 4 5 6−2
0
2
4The Cartesian position x
x (m
)
0 1 2 3 4 5 6−2
0
2
4
6The Cartesian position y
y (m
)
0 1 2 3 4 5 6−5
0
5
10The Cartesian position z
time (s)
z (m
)
AFL+ e−modificationdesired outputSMC control
Figure 8.13: The Cartesian position output of eVader obtained by applying AFL with robustmodification and SMC control in presence of ground effect disturbance.
thrust forces of the two rotors. It takes some time for adaptive control to adapt to
parameters of the system and this makes it act slower. This delay causes the eVader
to pitch-up to 20 before the vehicle lands on the ground. But, the SMC control
prevents the eVader to pitch up too much and regulate the pitch to zero with only a
small pitch up deviation of 5 (Figure 8.12). The roll and yaw angle deviations caused
by the change of thrust as a result of GE is 4 and 5 for SMC and 17 and 18 for
the RAFL, respectively. In the last scenario of this chapter, the aggressive maneuver,
both SMC and RAFL show great performance following a fast time-varying trajectory
of Mobius strip in three-dimensional space.
183
Figure 8.14: Three dimensional position output result obtained by applying RAFL controlperforming aggressive maneuver.
184
Figure 8.15: Three dimensional position output result obtained by applying SMC controllerperforming aggressive maneuver.
185
0 10 20 30−0.1
0
0.1Control signal u1
time[s]
u1
0 1 2 3−1
0
1x 10
4 Control signal u4
time[s]
u4
0 10 20 30−15−10
−505
Control signal u2
time[s]
u2
0 10 20 30−200
0
200Control signal u5
time[s]
u50 10 20 30
0
10
20Control signal u3
time[s]
u3
0 10 20 3095
100
105Control signal u6
time[s]u6
Figure 8.16: Control signals of RAFL controller in aggressive maneuver scenario.
0 10 20 30−1
0
1Control Signal u1
time[s]
u1
0 1 2 3 4 5−5000
0
5000Control Signal u4
time[s]
u4
0 10 20 30−20
0
20Controller u2
time[s]
u2
0 10 20 30−200
0
200Controller u5
time[s]
u5
0 10 20 30−50
0
50Controller u3
time[s]
u3
0 10 20 3050
100
150Controller u6
time[s]
u6
Figure 8.17: Control signals of SMC controller in aggressive maneuver scenario.
186
0 5 10 15 20 25 30−0.1
0
0.1The roll angle
time[s]
Roll[d
egre
e]
0 5 10 15 20 25 30−50
0
50The pitch angle
time[s]
Pitch
[degr
ee]
0 5 10 15 20 25 300
100
200The yaw angle
time[s]
Yaw[
degr
ee]
Figure 8.18: Orientation of the eVader performing aggressive maneuver obtained by apply-ing RAFL controller.
0 1 2 3 4 5 6 7 8 9 10−0.05
0
0.05The roll angle
time[s]
Roll [
degr
ee]
0 5 10 15 20 25 30−100
0
100The pitch angle
time[s]
Pitch
[deg
ree]
0 5 10 15 20 25 300
100
200The yaw angle
time[s]
Yaw
[degr
ee]
Figure 8.19: Orientation of the eVader performing aggressive maneuver obtained by apply-ing SMC controller.
187
0 5 10 15 20 25 30−10
0
10The Cartesian position x
time[s]
x
system outputdesired x(t)
0 5 10 15 20 25 30−10
0
10The Cartesian position y
time[s]
y
0 5 10 15 20 25 30−5
0
5The altitude z
time[s]
z
Figure 8.20: Position of the eVader performing aggressive maneuver obtained by applyingRAFL controller.
0 5 10 15 20 25 30−10
0
10The Cartesian position x
time [s]
x [m]
system outputdesired trajectory
0 5 10 15 20 25 30−10
0
10The Cartesian position y
time [s]
y [m]
0 5 10 15 20 25 30−5
0
5The Cartesian position z
time [s]
z [m]
Figure 8.21: Position of the eVader performing aggressive maneuver obtained by applyingSMC controller.
188
0 1 2 3 4 5 6 7 8 9 10−0.04−0.02
00.02
The error of roll angle
time (s)
Roll (
degr
ee)
0 5 10 15 20 25 30−40−20
020
The error of pitch angle
time (s)
Pitc
h (d
egre
e)
0 5 10 15 20 25 300
100
200The error of yaw angle
time (s)
Yaw
(deg
ree)
Figure 8.22: Orientation tracking error of SMC controller in aggressive maneuver scenario.
0 0.5 1 1.5 2 2.5 30
5
10The error of x
time[s]
x[m
]
0 5 10 15 20 25 30−0.05
0
0.05The error of y
time[s]
y[m
]
0 5 10 15 20 25 30−0.01
0
0.01The error of z
time[s]
z[m
]
Figure 8.23: Position tracking error of SMC controller in aggressive maneuver scenario.
Chapter 9
Conclusion and future work
The objective of this work was to control the eVader so that it could be used to its full
potential and perform complex maneuvers, such as following full state trajectories of
aggressive maneuvers, and executing successful landing in presence of ground effects.
Moreover, we tried to control the eVader to execute tasks that other rotary-wing
UAVs are not able to perform, such as pitched hover, which allows the eVader to take
off and land on sloped surfaces. In this thesis, throughout the process of deriving
the nonlinear dynamic model and designing nonlinear controllers to obtain the stated
objectives, we contributed to the literature of small UAVs’ autonomous flight in some
ways. These contributions are listed in the next section.
9.1 Contribution
A set of ten contributions from this thesis were accomplished. These contributions
are listed below:
1. Developing a complete dynamic model of the VTOL UAV with
dOAT mechanism, which includes all torques applyed to the eVader.
189
190
This version of the dynamic model of this vehicle is complete in the sense that
for the first time it takes into account all torques applying on the eVader,
including pitch gyroscopic moments and reactionary torques. In fact, this is the
first model in the literature considering the dOAT mechanism for the eVader,
in which the lateral and longitudinal tilting angles of the eVader vehicle are
different, and each fan rotates and tilts independently. Moreover, the model
also includes air drag, friction aerodynamic forces, and ground effects, which
are typically not included in UAV models for the purpose of control. The main
achievement of developing this model is obtaining the nonlinear model with six
DOF. The developed dynamic model is valid throughout the whole flight range
and covers all flight modes (e.g., hover, pitched hover, following trajectories,
and transition between these modes). Each flight mode can be represented by
a 12-dimensional vector of the initial and final states. The 12 states include
position, linear velocity, pitch, two angles of orientation, and their derivatives.
2. Independent 6 degree-of-freedom control of the UAV.
One of the main advantages of the eVader and its developed nonlinear dynamic
model is that they offer a great property of 6 DOF of control. Usually with
other rotary-wing UAVs such as helicopters and quad-rotors you can only set
4 desired states and stabilize the other 2 states to zero. Independent 6 DOF
control of the eVader enhanced the maneuverability capability of this UAV in
191
such a way that it is able to go from any arbitrary initial state to any arbitrary
final position in 3D space with an arbitrary orientation in the range of (−π, π).
Moreover, performing any possible maneuver within the vehicle’s capabilities in
terms of translations and orientations happens simultaneously, not by having
a sequence of controls in time like what happens in under-actuated vehicles.
For instance, to control a quad-rotor it is only possible to select a set of four
variables to be controlled (e.g., position and yaw angle), and the desired values
of the other two variables (in this case pitch and roll angles) would be forced
by the controller and are not arbitrary. As a result of the independent control
of all outputs of interest, the eVader is able to make difficult tasks and do
maneuvers that other similar UAVs are not able to perform. Maneuvers such
as pitched hover and precisely tracking a desired trajectory and orientation are
now possible.
3. Applying a single type of control technique for full control of the
UAV.
Utilizing single type of control methodology is an original approach as usually
all other UAV control systems combine several control techniques. In each chap-
ter of this thesis on controller design, a single control technique was designed
for all six desirable outputs of the vehicle. Each individual proposed single tech-
nique approach brings simplicity, flexibility and a clearer view of the interaction
192
between the different controllers.
4. Increasing reliability of control system to adapt to real time changes
in aerodynamic parameters and vehicle’s payload.
Due to the fact that the exact model of the nonlinear system is not available in
practice, it is necessary for a reliable controller to be able to adapt to parametric
uncertainties (structured disturbances). Thereby, the designed adaptive con-
troller in this thesis maintained uncertainties in aerodynamic parameters, such
as aerodynamic friction coefficients and the mass of the eVader. The designed
adaptive controller was able to cope with uncertainties in model parameters
and preserved the asymptotic stability of the system under constant external
disturbances and slow-varying parameters, which have not been studied before
for the eVader.
5. Increasing the robustness of the eVader and enabling it to perform
in presence of time-varying disturbances.
The robustness issue of UAVs is very important to be studied as more robust-
ness results in more system reliability in presence of time-varying disturbances.
Although the designed adaptive controller was able to maintain asymptotic
stability of the eVader under constant external disturbances and slow-varying
parameters, it was unable to provide asymptotic stability in presence of time-
varying disturbances such as wind gusts, ground effects and additive distur-
193
bances in control input signal (due to the error of neural network mapping
approximation). In realistic environmental flight conditions, the ground effects
and buffeting wind affect the eVader flight. Thus, the controller is required
to be robust to both unstructured uncertainties as well as structured ones, to
be reliable in practical situations. In order to increase reliability of flight con-
trol system in such situations, two robust controllers were proposed in this
thesis: i) the adaptive control technique with e-modification and ii) the SMC
approach. These controllers both achieved robustness to external unstructured
uncertainties (e.g. unmodeled dynamics) as well as structured model param-
eter uncertainties, both linear and non-linear in parameters. The AFL with
robust modification was designed for the eVader in Chapter 4, and the SMC
control was designed in Chapter 6. Simulation results, in Chapters 4, 6 and
8, demonstrated the robustness of designed controllers to sensor noise, exoge-
nous perturbations, parametric uncertainties, and plant nonlinearities, while
simultaneously exhibiting their capability to follow a reference trajectory.
6. Effectively controling the eVader when operating in the presence
of ground effects.
One of the major external disturbances, which affects the eVader when maneu-
vering in confined spaces is the flow ground effects. This issue was investigated
in this thesis as related to the eVader UAV. The ground effects on the eVader
194
are modeled in this thesis based on the CFD study of ducted fans. Despite
the lack of research in studying UAV flight control under ground effects, the
ground effects should be considered a major concern for applications in confined
spaces. Due to the importance of this issue and to investigate the performance
of the eVader, two robust controllers, RAFL and SMC, were designed to ef-
fectively control the eVader, when operating in the presence of ground effects.
The eVader demonstrated significant results, which have not been obtained and
investigated before.
7. Investigating the eVader performance in presence of buffeting wind
disturbance.
Studies of a constant wind velocity and very small wind gusts have appeared
before in the literature of UAV control. However, the study of buffeting (sinu-
soidal) wind disturbances, which cause a tough scenario for small UAV control,
has not been touched much. In fact, buffeting wind makes UAV control very
difficult because such a disturbance throws off angular and Cartesian estima-
tions. Tough, windy condition for the eVader was investigated in this thesis
for the first time. As presented in Section 8.2, simulation results using the
developed control laws in Chapter 4 and 6 are promising and show robustness
of the proposed control for the eVader flying in windy situations. In addition,
simulation results demonstrate that the eVader with new controller tolerates
195
much higher magnitude of wind (as an external disturbance) than other VTOL
vehicles, without becoming unstable.
8. Enabling the eVader to execute successful aggressive maneuvers
with a single control approach.
Successful performance of aggressive and complex maneuvers by a small eVader
UAV was achieved in this thesis, which was not possible before. Aggressive ma-
neuvers with aerial robots is an area of active research with considerable effort
focusing on strategies for generating sequences of controllers that stabilize the
robot to a desired state. The common approach in the literature for controlling
a complex aggressive maneuver is the Multi Modal control framework and the
hybrid control [112], [113]. In contrast to existing approaches in the literature,
a single controller for maneuver tracking over the full flight envelope, instead
of tracking maneuver mode sequence, was developed in this thesis. With the
proposed controller the eVader is able to handle aggressive maneuver tracking,
similar to such maneuvers with piloted aircrafts.
9. Developing the eVader control system toward flying in confined
environments.
Enabling the eVader to fly in confined environments was listed as one of the
main objectives of this thesis. Although the UAV needs to be equipped with
a navigation system to autonomously fly in high cluttered environments, the
196
key point is to develop a control system that provides high maneuverability and
agility along with maintaining the UAV stability. As results show in Chapter
8, the SMC controller designed for the eVader offers fast response and precise
trajectory tracking ability, which are essential for autonomous fly in confined
spaces. The aggressive maneuver, pitched hover, and ground effects simula-
tions established the efficiency of the proposed SMC and RAFL controllers for
the eVader, performing complex missions required to fly in confined spaces. It
should be mentioned, however, that even thought both SMC and RFAL con-
trollers provide satisfactory responses, due to the faster response of SMC in
comparison with RAFL under the same conditions, SMC is the better choice
for eVader control in such situations.
10. Achieving the estimation of actual control signals by approximat-
ing the nonlinear function relation between virtual control signals in
the model of the vehicle (u1, ..., u6) and the actual controls (α1, α2, β1,
β2, ω1, ω2).
According to the dynamic model of the vehicle, input signals obtained from
the proposed control methodologies are the virtual signals. Therefore, another
challenge associated with controlling the eVader is finding the real actuator
signals and investigating the feasibility of the associated control techniques.
This problem was addressed in this thesis by designing a neural network to
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match the mapping between the virtual control signals and the real inputs of
the system. Thereby, in other words this thesis provided a good approximation
of the actual six control signals of the eVader dOAT mechanism, which leads to
the feasibility of full controllability. The MLP neural network with BP learning
algorithm was utilized to find the inverse mapping between variables u1, ..., u6
and α1, α2, β1, β2, ω1, ω2. From the results of this thesis work (Chapter 7), it
was observed that spanning the domain range of each variable is very important
for proper training. Spanning the domain range has a direct impact on the
performance of the network. Splitting domain range of each actual control
signal into smaller segments results in a more accurate function approximator.
Hence, dividing the domain range of each variable, may be done by separating
the maneuvers from each other. Different maneuvers may span a different part
of the range of longitudinal and lateral angles. Training specific networks for
each, gives us a better accuracy.
9.2 Future Work
Despite achieving the control of a UAV with dOAT capabilities (which was not pos-
sible before) there are a number of challenges that need to be resolved. Some of the
future tasks that we envision are the next logical steps in controlling highly manoeu-
vrable UAVs, specially the eVader, are listed below:
1. As demonstrated in Chapter 8, in presence of uncertain disturbances such as
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ground effect and wind, the robust controllers are able to stabilize the system,
but do not achieve the asymptotic stability. Consequently, obtaining asymptotic
stability despite the presence of general uncertain disturbances is left for future
study.
2. Modification of control laws of the robust SMC controller is required to
reduce the observed chattering and the extremely high control activity, which
occurs in SMC technique.
3. The eVader’s rotors (ducts) tilt longitudinally and laterally to perform ma-
neuvers and follow complex flight trajectories. As a result of this tilting, the
mass distribution of the vehicle changes, and consequently the center of mass
of the UAV also changes. This is specially true when heavy ducts are used to
enhance the desired gyroscopic effects, that are used in controlling the vehicle.
However, the developed dynamic model in Chapter 3 is based on the fixed cen-
ter of gravity. Thus, investigating the effect of changing the center of gravity on
dynamics of the system during the flight is another study that needs investiga-
tion. In view of this fact, it is worth to mention again that the eVader requires
an adaptive or robust control law due to the fact that the center of gravity is
changing because of tilting ducted-fans mechanism.
4. Aggressive maneuver tracking trajectories are not simple to obtain. A motion
planning technique should be developed to make the trajectory planning for the
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vehicle’s desired states as a function of time.
5. Executing experimental tests and performing different flight modes and ma-
neuvers is of great interest to study the proposed controllers in real flight. There-
fore, the proposed controller based on the developed nonlinear model should be
implemented on the prototype of the eVader.
6. As stated in Chapter 7, the neural network function approximator should
be trained on-line because there are pitch, roll and yaw angles in the equations
of control signals, u4, u5, and u6. Another approach for taking into account the
orientation angles is to train the network based on the real data collected by
executing experiments. Therefore, another future study is training the neural
network with real data to approximate the actual mapping between the virtual
and actual control signals.
7. The further step is to develop a fully autonomous and self guided eVader
based on our model, which would need navigation, motion, and path planning
system.
Bibliography
[1] G. R. Gress, “Lift fans as gyroscopes for controlling compact vtol air vehi-
cles: Overview and development status of obliques active tilting,” American
Helicopter Society, 63rd Annual Forum, May 2007.
[2] M. Wis and I. Colomina, “Dynamic dependent imu stochastic modeling for
enhanced ins/gnss navigation,” 5th ESA Workshop on Satellite Navigation
Technologies and European Workshop on GNSS Signals and Signal Processing
(NAVITEC), 2010.
[3] P. Castillo, R. Lozano, and A. Dzul, “Modelling and control of mini-flying
machines,” Springer-Verlag in Advances in Industrial Control, July 2005, iSBN:
1-85233-957-8.
[4] D. P. Raymer, Aircraft Design: A Conceptual Approach. AIAA Education
Series, 2004.
[5] A. Filippone, Flight Performance of Fixed and Rotary Wing Aircraft. Elsevier
Science, 2006.
[6] A. Week and S. Technology, 2005 Aerospace Source Book, 2005.
[7] J. G. Leishman, Principles of Helicopter Aerodynamics. Cambridge University
Press, 2006, second edition.
[8] “Boeing, v-22 osprey,” Available: http://www.boeing.com, 2008, september 13.
200
201
[9] “The bell eagle eye usa,” Available: http://www.bellhelicopter.com, 2008,
september 13.
[10] J. J. Dickeson, D. Miles, O. Cifdaloz, V. L. Wells, and A. A. Rodriguez, “Robust
lpv h gain-scheduled hover-to-cruise conversion for a tilt-wing rotorcraft in the
presence of cg variations,” American Control Conference (ACC2007), pp. 5266
– 5271, 2007, 9-13 July.
[11] I. Kendoul, F.and Fantoni and R. Lorenzo, “Modeling and control of a small
autonomous aircraft having two tilting rotors,” Proceedings of the 44th IEEE
Conference on Decision and Control, and the European Control Conference,
2005, 12-15 December.
[12] D. Snyder, “The quad tiltrotor: Its beginning and evolution,” Proceedings of
the 56th Annual Forum, American Helicopter Society, May 2000.
[13] K. Nonami, “Prospect and recent research & development for civil use au-
tonomous unmanned aircraft as uav and mav,” Journal of System Design and
Dynamics, vol. 1, no. 2, 2007.
[14] D. Murphy and J. Cycon, “Applications for mini vtol uav for law enforce-