Modelling, Control, and Experimental Evaluation of the Hovering Characteristics
of a Tilt-Wing Unmanned Aerial Vehicle
Elias Small
Space Engineering, masters level 2017
Luleå University of Technology
Department of Computer Science, Electrical and Space Engineering
Master Thesis:
Modelling, Control, and Experimental Evaluation of the Hovering
Characteristics of a Tilt-Wing Unmanned Aerial Vehicle
Elias D. Small
Lule˚ a University of Technology
Dept. of Computer Science and Electrical Engineering Div. of Systems and Interaction
21 December 2016
A BSTRACT
A Tilt-Wing Unmanned Aerial Vehicle (TW-UAV) and the preliminary evaluation of its hovering characteristics in extended simulation studies and experiments are presented in this Master Thesis. In the beginning, an overview of the TW-UAV’s design proper- ties are established, highlighting the novelties of the proposed structure and the overall merits. The TW-UAV’s design and structural properties are mathematically modelled and utilized for the synthesis of a cascaded P-PI and PID based control structure for the regulation of its hovering performance. In addition, extensive simulation trials are performed in order to evaluate the structure’s efficiency in controlling the TW-UAV’s attitude and position under various noise and disturbance scenarios. The model and air- craft are then put through experimental evaluation with an on-board processor, namely the KFly, in a Motion-capture equipped laboratory to evaluate the control structure and physical behaviour of the TW-UAV. The results of these experiments are presented and discussed. The system and control scheme are shown to work well. However, an unfortu- nate crash forced the premature termination of experimentation and thus the conclusion of this thesis. Nevertheless, the reason for the crash is understood and discussed for future work.
iii
P REFACE
Throughout the long and tedious process that is a masters thesis, I have learned so much working with the incredibly interesting field that is robotics. I have been humbled by its immensity and by the exceeding knowledge and friendliness of my colleagues in the Field Robotics Group at LTU. I have felt welcome and I have felt free to attempt to access the vast sea of knowledge my betters possess.
I would like to thank my supervisors Emil Fresk and George Nikolakopoulos for their enduring support and patience, as well as Georgios Andrikopoulos and William Small for their help in the writing process.
Elias Small
v
C ONTENTS
Chapter 1 – Introduction 1
Chapter 2 – Physical Design Properties 3
2.1 Prototype . . . . 3
2.2 Design Properties . . . . 4
2.3 Modifications . . . . 5
Chapter 3 – Modelling and Control 9 3.1 Mathematical Model . . . . 9
3.2 Control Scheme . . . . 11
Chapter 4 – Simulations 15 4.1 Torque Disturbance . . . . 17
4.2 Position Step-Reference Response . . . . 18
4.3 Path Following . . . . 20
Chapter 5 – Electronics and Programming 23 5.1 Hardware . . . . 23
5.1.1 KFly Circuit Board . . . . 23
5.1.2 Other Hardware . . . . 24
5.2 Software . . . . 24
5.2.1 KFly Software . . . . 24
5.2.2 Robot Operating System (ROS) . . . . 25
5.3 Experimental System and Laboratory Set-up . . . . 25
Chapter 6 – Experimental Evaluation 27 6.1 Static Test of Engine and Servo . . . . 27
6.2 Remote-Controlled Angular Velocity Reference Test . . . . 27
6.3 Remote-Controlled Angle Reference Test . . . . 28
6.4 Position Reference Test . . . . 29
Chapter 7 – Conclusions and Discussion 35 7.1 Conclusion from Simulation Studies . . . . 35
7.2 Discussion of Experimental Results . . . . 35
7.3 Future Work . . . . 36
C HAPTER 1 Introduction
The area of Unmanned Aerial Vehicles (UAVs) and more specifically the related research specialized in multirotors has recently seen rapid growth, mainly due to their efficiency in accomplishing complex missions and flight scenarios [1]. The vast application range of this type of UAV includes search and rescue missions, forest fire surveillance, border interdiction, area exploration and mapping, agricultural services, buildings inspection, marine operations, media coverage, and many more e.g. in [2].
Generally, multirotors do not possess the energy efficiency of a fixed wing aircraft for long-distance flight, since the research focus is mainly placed on their Vertical Take-Off and Landing (VTOL) properties rather than the utilization of wings to produce a lifting force. Thus, the goal of developing an aircraft with long-distance efficiency and VTOL capabilities recently increased interest in the tilt-rotor or tilt-wing aircraft, which is a combination of a multirotor and a fixed-wing aircraft [3], [4], [5], [6].
The primary novelty presented in this research stems from the formulation of an in- novative modelling and control scheme for an optimally designed novel Tilt-Wing UAV (TW-UAV). With regards to the optimality of the design, the interested reader can di- rectly refer to [3]. In this research, for consistency, the main characteristics of the second revision of the design mentioned in [3] will be presented in detail. A tilt-wing instead of a tilt-rotor design was chosen for this type of aircraft, since rotating the wing in parallel with the motor while transitioning between flying and hovering modes provides impor- tant mechanical advantages. Specifically, the drag caused by a propeller partially causing air flow over the wing is minimized, and this seemingly small difference in energy con- sumption proves crucial when it comes to the development of small-scale UAVs, where the available power is hugely limited due to their design characteristics [7]. In addition, having fewer moving parts reduces the complexity of the physical structure as well as that of the mathematical model. This in turn translates into simpler control structures.
Thus it is advantageous to base the design on a previously built prototype [3] so that more precise improvements can be evaluated, whilst existing experimental data can still be applied.
1
2 Introduction The second novelty presented in the research comes from the proposed modelling and control scheme based on conventional algorithms for the preliminary evaluation of the hovering characteristics of this innovative design. For this purpose, a cascade P-PI con- troller is utilized to control the attitude of the aircraft, whilst multiple PID loops are used for the position control.
Finally, the results of experiments utilizing the proposed control structure are pre- sented. A series of tests were performed, starting with manual rate control and se- quentially adding more complexity to the controller until a fully autonomous flight with position references was performed. This report will serve as a base for continued work on the implementation of a control scheme for cruising, hovering, and the transition between the two states.
The control structure, programmes and applications used during this research were sug-
gested by the supervisors of the main author. Part of this research was published at the
24th Mediterranean Conference on Control and Automation (MED16) as a preliminary
design evaluation article [8].
C HAPTER 2 Physical Design Properties
2.1 Prototype
The proposed TW-UAV was initially based on the design presented in [3]. The graphical representation of the novel TW-UAV with all the components of interest highlighted is depicted in Figure 2.1.
Battery and Electronics Compartment
Electric Motor
Worm Drive Motor
Aileron
Worm Drive
Electric Ducted Fan (EDF) Tail Fin with Rudder
Tail Wing with Elevators Alluminum Tail Shaft
Servomotors
Figure 2.1: Graphical representation of the TW-UAV with components of interest high- lighted.
3
4 Physical Design Properties
2.2 Design Properties
Figure 2.2: Graphical representation of the TW-UAV during (a) flying, (b) dynamic, and (c) hovering states.
For consistency with [3], in the formulation of the modelling and control framework, each major structural part is described below and any modifications in the revised version of the adopted design for the TW-UAV are discussed in Section 2.3.
Given the complexities introduced when switching between the two flight modes (VTOL and plane modes), a design with as few moving parts as possible, while maintaining structural rigidity and stability, is desirable. One solution to this problem is to have the main wings and propellers joined as one structure that can rotate around an axis perpendicular to the chassis of the aircraft as indicated in Figure 2.2.
In addition, the wing structure is both rotated and held in place by a swivelling mech- anism called a worm drive as indicated in Figure 2.3. In this case, an electric motor drives the worm, which in turn rotates a worm gear to which the entire wing structure is attached. The motor can rotate the wings, but the mechanics of the drive prevent any other rotational forces to be transferred the wings.
The main chassis of the TW-UAV, i.e. the aluminium structure holding the worm
Figure 2.3: Graphical detail of the worm-drive mechanism.
2.3. Modifications 5 drive, the battery, and the electronic components, is designed to hold the worm drive in place effectively, while providing ample space for the battery and electronics.
A fundamental complication in the prototype design of the TW-UAV [3] was that the tail experienced excessive vibration and wobbling. This effect could be traced to the fact that the tail shaft was cylindrical and therefore prone to twisting and bending in an undesirable manner. This problem was solved by exchanging the cylindrical carbon fibre tube with a hollow aluminium shaft with a square profile. This easily implemented exchange provides a sturdier and more efficiently assembled tail section.
The most noticeable change made to the prototype design is the size and width of the wing surfaces. A longer in-length air foil is utilized, which allows the shortening of the wingspan in order to add mechanical stability, while increasing lift capabilities of all the wing surfaces.
Most conventional aeroplane structures are constructed with a tail wing that generally produces down force to stabilize the tail since the Centre of Mass (COM) is located near, or in front of, the centre of lift [9]. In the proposed TW-UAV design, the COM is behind the main wings so as to ensure hovering capabilities, resulting in the tail structure having a substantial relative weight with respect to the rest of the aircraft. For this reason, the tail wings are constructed to provide lift for the entire tail section by utilizing the same wing profile as the main wings while simultaneously having elevators that can vary this lift significantly during flight so as to provide elevation control capabilities.
2.3 Modifications
A landing gear was not designed for the prototype. Consequently, for the experimentation undertaken in this research a landing gear for vertical take-off and landing needed to be designed and constructed. A connecting mechanism for the EFD servo also needed to be constructed. All plastic components in this section were designed with Siemens NX and 3D-printed with Ultimaker II.
Materials available in the Field Robotics Laboratory (FROST) of the Department of Computer Science and Electrical Engineering at Lule University of Technology were identified and used for rapid prototyping of a robust landing gear structure. 3D-printed PLA plastic, carbon fibre poles, and steel wire were used to make it lightweight and durable.
The feet are attached to the struts with a ball joint so that even if the TW-UAV landed
at an angle inclined to the horizontal, the feet would make full horizontal contact with
the landing surface. This joint can clearly be seen in Figure 2.4. The flexibility of the
carbon fibre struts allows for a degree of impact absorption but over-flexing and breakage
is hindered by the steel wires connecting the feet.The struts are rigidly mounted to the
aluminium frame of the TW-UAV with 3D-printed brackets. Figures 2.4 and 2.5 show
these features. This landing gear designed proved to be effective as the landing structure
absorbed the impact of most landings and in cases where damage occurred the effected
6 Physical Design Properties
Figure 2.4: The landing gear with ball joint to the foot and support wires visible.
(a) Bracket for front foot. (b) Bracket for rear feet.
Figure 2.5: 3D-printed brackets for holding the carbon fiber tubes in place.
pieces could be easily replaced.
The other addition that was made was a 3D-printed connecting mechanism between the EDF servo and the swivel arm on which the EDF is mounted and manoeuvred.
Previously copper wire connected to a lateral screw had been used but proved ineffective
as the wire could not tolerate the pulling and pushing forces of the servo, and the wire
blocked the servo from reaching its maximum angle. The plastic arms on the connecting
mechanism are curved so as to allow the servo to reach its maximum extension and
thereby increasing the angle to which the EDF could be rotated. The mechanism can be
seen in Figures and , in horizontal and fully rotated positions respectively.
2.3. Modifications 7
(a) Servo in horizontal position.. (b) Servo fully rotated.
Figure 2.6: Connecting mechanism between servo and EDF shown in horizontal and rotated positions.
Figures 2.7 and 2.8 show the completed prototype with all components installed and
ready for experimentation. The Kfly was placed in the center of gravity to maximize the
efficiency of the gyro.
8 Physical Design Properties
Figure 2.7: The landing gear with ball joint to the foot and support wires visible.
Figure 2.8: The landing gear with ball joint to the foot and support wires visible.
C HAPTER 3 Modelling and Control
3.1 Mathematical Model
This section presents the mathematical model for the hovering mode of the TW-UAV.
Firstly, the force and torque equations are derived from the acting force diagram presented in Figure 3.1 as shown below.
z x y
φz
φx
φy
O
Mr
θw
θe
A C
D B
E
F Ml
Fp
Fs
mg
Fr
Fl
Me
Fe
Figure 3.1: Acting force diagram of the TWUAV in hovering state. F l , F r , and F e are the forces produced by the port and starboard motors, and the EDF, F p and F s are the forces produced by air flow over the wing profile, M l , M r , andM e are momentum caused by the spinning propellers or fan, mg is the force caused by gravitational acceleration and mass, θ w is the angle of the wing to the body (zero along the x-axis), θ e is the angle of the EDF from the z-axis, and the capital letter (O and A through F ) denote points of interest so that distances can be indicated. The COM is in E.
9
10 Modelling and Control
F
xF
yF
zτ
xτ
yτ
z
=
−(F
s+ F
p) · sθ
w+ (F
l+ F
R) · cθ
wF
e· sθ
e(F
r+ F
l) · sθ
w+ F
e· cθ
e+ (F
s+ F
p) · cθ
w(F
l− F
r) · sθ
w· (AO) + (M
r− M
l) · cθ
w+ +(F
p− F
s) · cθ
w−(F
l+ F
r) · sθ
w· (OE) + F
e· cθ
e· (EF )−
−M
e· sθ
eF
e· sθ
e· (EF ) + (M
r− M
l) · sθ
w− M
e· cθ
e+ +(F
p− F
s) · sθ
w+ (F
e− F
l) · cθ
w· (AO)
+ R
0 0
−mg 0 0 0
(3.1)
where sinθ = sθ and cosθ = cθ, (AB) is the (distance between point A and point B), and R 6x6 defined as:
R = R rot 0 3x3
0 3x3 0 3x3
(3.2) where R rot is a three by three rotational matrix, representing the attitude of the body in relation to the direction of the gravitational acceleration.
At this point it should be highlighted that the additional aerodynamic forces exerted on the body and caused by movement through the air are neglected in the hovering state, where the angle of the wing θ w will be fixed at 90 degrees. Furthermore, the forces produced by the flow of air from the propellers over the wing profile – F s and F p – are neglected because the minimal force produced can easily be compensated for by a robust controller. Thus, for the case of hovering, the above Equation 3.1 is simplified to:
F x
F y F z τ x
τ y τ z
=
0 F e · sθ e F r + F l + F e · cθ e
(F l − F r ) · (AO)
−(F l + F r ) · (OE) + F e · cθ e · (EF ) − M e · sθ e F e · sθ e · (EF ) + M r − M l − M e · cθ e
+ R
0 0
−mg 0 0 0
(3.3)
Furthermore, the system model can be derived by using the Newton-Euler kinematics for a rigid body[10]. Thus the Newton-Euler equations are formulated as:
F τ
= mI 3x3 0 0 I cm
a cm
˙ ω
+
0
ω × (I cm ω)
, (3.4)
where F = F x F y F z
T
∈ R 3 is the force acting on the COM, τ = τ x τ y τ z
T
∈ R 3 the torque around the COM, ω = ω x ω y ω z T
∈ R 3 the angular velocity, a cm =
a x a y a z
T
∈ R 3 the acceleration of COM, I 3x3 is the 3x3 identity matrix and I cm the inertia matrix, which is defined as:
I cm =
I xx I xy I xz
I yx I yy I yz I zx I zy I zz
∈ R 3x3
.
3.2. Control Scheme 11 The Newton-Euler equations in Equation 3.4, solved for the acceleration and the angular acceleration yield:
a cm
˙ ω
= mI 3x3 0 0 I cm
−1
F τ
− mI 3x3 0 0 I cm
−1 0 ω × (I cm ω)
, (3.5)
The work done by [11] has shown that both the force and the momentum produced by the propellers are proportional to the square of the input signal to the motor. Con- sequently, the forces exerted by the motors as well as the momentum from the rotating propellers are modelled as a constant multiplied by the square of the input signal as follows:
F i = A F,i · u e 2 i (3.6)
M i = A M,i · u e 2 i (3.7)
e u i = 1
τ s,i + 1 · u i (3.8)
where F i and M i are the force and momentum exerted by the propeller respectively, A F,i and A M,i are the force and momentum constants, τ s,i the time constant for the motors, and u i (0 6 u i 6 1) for i = 1, 2, 3, defines the signals to the motors and servo. Since the left and right propeller motors will have the same physical properties, they will have the same force constant A F,r and momentum constant A M,r , but different input signals.
The signal notations will be u 1 , u 2 , and u 3 for the port motor, starboard motor and EDF respectively. The angle of the EDF servo can be defined as:
θ e = π
3 · u 4 , −1 6 u 4 6 1 (3.9)
where u 4 is the signal to the servo.
The final hovering model calculated from (3.3) and (3.5) can then be derived as:
a
xa
ya
z˙ ω
x˙ ω
y˙ ω
z
=
0 sθ
e· A
F,eu
23A
F,rm
m · ( u e
22+ u e
21) + cθ
e· A
F,e· u e
23A
F,r· ( u e
21− u e
22) · (AO) m
I
xx+ I
xy+ I
xz− ω
2x−A
F,r· ( e u
21+ e u
22) · (OE) + A
F,e· u e
2e· cθ
e· (EF ) − A
M,e· e u
2e· sθ
eI
yx+ I
yy+ I
yz− ω
y2A
F,e· e u
23· sθ
e· (EF ) + A
M,r· e u
22− A
M,r· u e
21− A
M,e· u e
23· cθ
eI
zx+ I
zy+ I
zz− ω
z2
+ R
0 0
−g 0 0 0
(3.10)
where u e 1,2,3 are defined according to (3.8) and the angle θ E defined in (3.9).
3.2 Control Scheme
For the preliminary evaluation of the TW-UAV’s hovering capabilities, a control scheme
based on the utilization of angle (P), angular velocity (PI) and position (PID) controllers
12 Modelling and Control
φz,ref P-PI KP,iKI,iKD,i
[u , u ]x yT
x y
ω ,ω ,ω ]zT [x, y, z]refT
uz
Attitude Control Position Control
[u ,u ]p rT [u ,u ,u ]1 2 3T
PID [x, y, z]T
x y
[φ ,φ ,φ , z SM
u4
TW-UAV
Figure 3.2: Position and attitude control structure for the TWUAV.
PI
P [φ ,
x
φ ,
y
φ ]
z j T
e e
k x y[ω , ω , ω ]
zT+ - + -
K
P,jK
P,kK
I,ku
ju
k[u , u , u ]
x y zTTW-UAV
Figure 3.3: P-PI structure for the attitude control for the TWUAV.
was used as indicated in Figure 3.2. In particular, for the attitude control, a cascade P-PI controller (Fig. 3.3) was implemented, where three separate PI controllers were used to control the angular velocities around the x (roll), y (pitch), and z (yaw) axes, while the reference angle is fed through a P regulator for extracting the reference for the angular velocity. Figure 3.2 presents the whole control system architecture.
The inputs for the controllers are a position and a yaw angle, the position being a three dimensional vector and the yaw a value in radians. All simulations and experiments are designed to use these four values as inputs to be able to vary the TW-UAV’s position in the reference frame.
Attitude Control: P-PI Controller
The specifics of the P-PI controller described here are presented in Figure 3.3 below, with references made to the symbols as shown.
The first aspect of the control scheme is the control of angular velocity around each
axis of the TW-UAV, i.e. roll (x ), pitch (y), and yaw (z ). Each axis is controlled by
an individual P-PI controller. The inputs of the three attitude controllers are two angle
values produced by the PID controller (x and y), and the reference yaw angle ϕ z,ref . The
error (e j ) between each of the reference and measured angles is multiplied by a constant
(K P,j ) to produce a reference angular velocity for each PI controller. This new reference
(u j ) is compared to the measured angular velocity and the error between them (e k ) is
sent to the PI controller. The PI will attempt reduce the error for the next iteration,
using the constants K P,k and K I,k . The PIs output three values (u k ) which are then
used to control the motors and servo. The roll (u r ) and pitch (u p ) values are sent to a
3.2. Control Scheme 13 Signal Mixer (SM) where they are converted to signals to the motors, and the yaw (u 4 ) value is sent directly to the servo controlling the angle of the EDF.
The output values of the controllers are directed though the SM before being used as inputs to the motors. The output of the roll controller is subtracted from one propeller motor and added to the other one, depending on which way the UAV needs to roll.
Similarly, the pitch control output is either subtracted from the EDF signal and added to both the propeller motors or vice versa. The SM also has a saturation filter so that all outputs are limited to between 0 and 1 corresponding to minimum and maximum thrust. As mentioned above, for the yaw control the output is directly sent as an input signal (limited to a value between -1 and 1) to the EDF servo. In equation form the corresponding signals can be defined as follows:
u 1 = u z + u r − u p (3.11)
u 2 = u z − u r − u p (3.12)
u 3 = u z + u p (3.13)
Since the goal of this research was primarily to prove the functionality of design rather than to optimise the controller, the cascaded nature of this control scheme, i.e. the fact that a single signal is calculated by several controllers in a series, and the associated time delays of each controller are not considered significant at this stage in the research.
Position Control: PID Controller
Three PID loops were implemented to control position, since only the position is assumed
to be measurable. This assumption is based on the idea that hovering will be tested in a
camera lab or other position tracking system. Outputs of the x-position and y-position
controllers are directly inserted as references (u x and u y respectively in Figures 3.2 and
3.3) to the roll and pitch P-PI controllers. The height control output (u z ) is inserted as
a throttle signal to both propeller motors and to the EDF as shown in equations 3.11,
3.12, and 3.13. The PIDs use three constants each (K P,i , K I,i , and K D,i ) and have the
measured position in the reference frame (room) as feedback.
C HAPTER 4 Simulations
The TW-UAV’s response to the control structure was evaluated through a series of sim- ulations. The purpose of these was to verify the ability of the control structure to satisfactorily control the TW-UAV in actual experiments. These simulations included a torque disturbance simulation, a position step-reference response simulation, and a path following simulation. The angular references in all the simulations were set to zero, i.e.
the attitude coordinates of the aircraft would always be the same as those of the reference frame. Sensor noise disturbance was added to all the outputs in the simulation to observe the controllers’ ability to handle noise. Table 4.1 shows the size of the noise added to the outputs. It should be noted that all the simulations were run with a sample rate of 50 Hz.
Table 4.1: Noise Disturbance
Output Amplitude of random noise
Position 0.001 2 m
Angular Velocity 0.17 2 rad/s Body Angle 0.0087 2 rad
In these simulations, a Computer Aided Design (CAD) programme was used to produce the inertia matrix I cm . In this approach, proper design materials were added to the different parts in the CAD model so that the weight, centre of mass, and inertia matrix could be estimated for use in the simulations. In the calculations, the wings and holders for motors and servos were considered to be designed with a PLA plastic with 20 percent infill, which is a reasonable density for this type of structure[3]. The wings’ profile was
15
16 Simulations the same as in [3],i.e. NACA4412, but scaled from a 0.15m chord (distance from the front of the wing to the back) to a 0.25m chord. This allows the wingspan to be shortened from a 1.58m wingspan to 1.00m to provide more mechanical stability. In addition, the tubes strengthening the wings and the drive shaft for the worm drive were set to carbon fiber in the design. The chassis and other flat structural parts were set as aluminium, and lastly the motors, battery, and servos were given masses corresponding to commercially available products of their intended size.
In Table 4.2 the parameters and the applicable dimensions of the simulated aircraft are shown. Most parameters are taken from the CAD model except for the weight and motor constants. The weight has a reasonable margin added to the weight calculated by the CAD software for cables and electronics and the motor constants are estimated from commercially available motors and EDFs. The control parameters utilized in the proposed control scheme in Chapter 3 are displayed in Table 4.3.
Table 4.2: Design parameters of the TWUAV
Parameter Value Unit
(OE) 0.15 m
(OA),(OB) 0.56 m
(EF) 0.745 m
m 4.9 Kg
I xx 433.6 · 10 − 3 Kg · m 2 I yy 528.8 · 10 − 3 Kg · m 2 I zz 921.6 · 10 − 3 Kg · m 2 I xy , I yx 7.467 · 10 − 3 Kg · m 2 I xz , I zx 22.573 · 10 − 3 Kg · m 2 I yz , I zy −0.938 · 10 − 3 Kg · m 2
A F,L , A F,R 30 N
A F,E 20 N
A M,L , A M,R , A M,E 1 N · m
τ s,1 , τ s,2 0.15 s
τ s,3 0.2 s
4.1. Torque Disturbance 17
Table 4.3: Control parameters of the TWUAV
Gain Values(x, y, z) K P,i 0.09 0.05 1 K I,i 0.05 0.03 0.35 K D,i 0.2 0.03 1
K P,j 1 1.4 1
K P,k 0.05 0.07 1 K I,k 0.08 0.06 1
4.1 Torque Disturbance
In Figure 4.1 the aircraft has a constant position reference but disturbance torques are added around each of the main axes to observe the attitude control and response. In the one-hundred seconds simulation, a 3 Nm torque pulse is added around the x-axis at 30 seconds(d1), a 3 Nm torque pulse is added around the y-axis at 45 seconds(d2), and a 5 Nm torque pulse disturbance around the z-axis at 60 seconds(d3). The corresponding control signals to the motors (u 1 , u 2 , and u 3 ) and servo (u 4 ) can be seen in Figure 4.2.
-0.4 -0.2 0 0.2 0.4
-1 -0.5 0 0.5 1
-0.2 0 0.2 0.4
50 60 70 80 90 100
0 10 20 30 40
t [sec]
φx [rad]φy [rad]φz [rad]
Reference Angle Response
50 60 70 80 90 100
0 10 20 30 40
t [sec]
50 60 70 80 90 100
0 10 20 30 40
t [sec]
d1
d2
d3
Figure 4.1: Simulated response of the TW-UAV to a constant position reference. Dis-
turbance torques are applied in random intervals to test its hovering capabilities.
18 Simulations
0.5 0.75 1
0.5 0.75 1
0.3 0.65 1
-1 -0.5 0 0.5 u1u2u3u4
50 60 70 80 90 100
0 10 20 30 40
t [sec]
50 60 70 80 90 100
0 10 20 30 40
t [sec]
50 60 70 80 90 100
0 10 20 30 40
t [sec]
50 60 70 80 90 100
0 10 20 30 40
t [sec]
d1 d2 d3
Figure 4.2: Corresponding control effort signals to the TW-UAV motors for the simulation presented in Figure 4.1.
From the results obtained it is obvious that the control scheme is able to track varying references well, even in the case of intense and multiple external disturbances. The disturbances simulated are such that they could potentially cause significant deterioration of the overall performance of the closed loop system, even resulting in instabilities that could cause crashes. The controller also achieves rapid convergence to the reference signal, without intense signal overshoots and oscillations. Therefore, after the introduction of a disturbance, the TW-UAV performs a quick recovery in tracking, of only a few seconds in duration, a characteristic that is highly desirable in real-world environments.
4.2 Position Step-Reference Response
In Figure 4.3 the TW-UAV’s response to a changing position reference is simulated. In
Figure 4.4 the corresponding three dimensional representation of the reference signals
and responses are depicted in order to provide a better understanding of the reference
input in relation to the achieved tracking response. The results show that both the x
and y position responses are slower than the z position response. This is due to the
cascaded nature of their controllers, i.e. the PIDs’ results are directly inputted as the
4.2. Position Step-Reference Response 19 P-PIs’ references, increasing the number of iterations needed for x or y position changes compared to the z position, which does not pass through a P-PI. The scheme does nevertheless track the desired position reference with acceptable settling times and few oscillations. The Root Mean Square Error (RMSE) of the reference and achieved response for the x, y, and z positional translation shown in Figure 4.3 is depicted in Table 4.4. As expected, the RMSE in z is less than those of x and y.
-5 0 5
-2 0 2 4
100 120 140 160 180 200
0 2 4 6
0 20 40 60 80
t [sec]
100 120 140 160 180 200
0 20 40 60 80
t [sec]
100 120 140 160 180 200
0 20 40 60 80
t [sec]
x [m]y [m]z [m]
Reference Position Response
Figure 4.3: Step reference and response for position in x, y, and z.
Table 4.4: Root Mean Square Error in Fig. 4.3
RMSE Value
Error in x 0.3382 m
Error in y 0.3524 m
Error in z 0.3071 m
20 Simulations
0 0.5 3 1 1.5 2 2.5
6 2
3 3.5 4
4 4.5
1 0 0 2
-1 -2
-2 -4 x [m]
y [m]
z [m]
Reference Trajectory Position Response
A(0,1,1)
B(0,1,2) C(1,1,2)
D(1,2,2)
E(1,2,4) F(3,2,4)
G(1,0,4)
H(1,0,1)
Figure 4.4: 3D representation of the reference signals and responses from Fig. 4.3
4.3 Path Following
It should be noted from the simulation discussed in the previous section that for the first 20 seconds of simulation the lack of a simulated floor, the lack of consideration of any ground effect, and the different start-up times for the motors cause the TW-UAV to diverge slightly from both the reference and the starting position. For this reason, the following simulation highlights the TW-UAV’s response behaviour once it has started up, taken off and settled at 0.5m above the ground.
Figure 4.5 shows a position reference and the corresponding position response as in
the previous simulation, but with the first twenty five seconds erased, in order to avoid
consideration of the effects of the initial transient response. As in the previous scenarios
considered, the response of the TW-UAV is fast and accurate. Nevertheless, due to
sudden alterations of the reference, an overshoot-like response at the corresponding time
instances is observed but with a rapid settling time and with no oscillations. The inertia
of the the TW-UAV when rapidly changing direction of travel is partially responsible for
the above mentioned overshoot. In the simulated system, the different input values to
the controllers could be further tuned to minimize the overshoot, but the result would
be only marginally different. By changing the control system to a more complex system
such as a model predictive controller (MPC) the errors could be significantly reduced,
but require a lot more computing power than a PID controller. For the purposes of this
study the PID control system was deemed sufficient. Finally, in Table 4.5 the RMSE
between the reference and the obtained response for the x, y, and z positions for this
simulation is presented.
4.3. Path Following 21
0 3 0.5 1 1.5
2 3
2
2.5 2.5
2 3
1 1.5
3.5
1 4
0 0.5
-0.5 0 -1 -1 A
(0,0,0.5)B
(0,0,2)C
(0, 2, 2)D
(2, 2, 2)E
(2, 0, 2)F
(0.5, 0, 2)Reference Trajectory Position Response
