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Universit´
e libre de Bruxelles
´
Ecole polytechnique de Bruxelles
Robotic hummingbird:
Design of a control mechanism for a hovering
flapping wing micro air vehicle
Matˇej KAR ´
ASEK
Thesis submitted in candidature for the
degree of Doctor in Engineering Sciences November 2014
Active Structures Laboratory
Department of Mechanical Engineering and Robotics
Jury
Supervisor: Prof. Andr´e Preumont (ULB)
President: Prof. Patrick Hendrick (ULB)
Secretary: Prof. Johan Gyselinck (ULB)
Membres:
Dr. Guido de Croon (Delft University of Technology)
Dr. Franck Ruffier (CNRS / Aix-Marseille University)
Prof. Jean-Louis Deneubourg (ULB)
Prof. Emanuele Garone (ULB)
iii
To my grandfather, Stano...
Acknowledgements
First of all, I would like to thank to my supervisor, professor Andr´e Preumont, for
inviting me to join the Active Structures Laboratory (ASL) as a visiting researcher
and, after one year, for giving me the opportunity to stay and continue with my
PhD on the exciting project of robotic hummingbird for another nearly four years.
I am extremely grateful for his guidance, for his many ideas as well as challenging
questions and, last but not least, for his permanent availability.
I remain also deeply obliged to my former supervisor, professor Michael Val´aˇsek from
Czech Technical University in Prague, who gave me the chance to go to Belgium
while taking the risk of losing me.
Many thanks belong to all my ASL colleagues. In particular, I would like to thank
to Laurent Gelbgras for introducing me to the hummingbird project and for all his
work at the project beginning. I really enjoyed our collaboration and the numerous
discussions we had. Many thanks go to Yanghai Nan and Mohamed Lalami, who
joined the project later, but quickly became valuable members of the team with
whom it was a pleasure to cooperate. They deserve special thanks for assembling
and testing the prototypes, designing and manufacturing the wings and for their
work on the control design. I am also grateful to Renaud Bastaits for helping me a
lot with my FRIA fellowship proposal, among other things. I want to thank to Iu-
lian Romanescu together with Mihaita Horodinca and Ioan Burda for their expertise
in manufacturing, technology and electronics that was behind all the experimental
setups used in this work. I would like to thank to Geoffrey Warniez for manufac-
turing many parts. Hussein Altartouri deserves my thanks for helping me with the
final version corrections. And I thank to all other lab members (Martin, Jos´e, Bi-
lal, Christophe, David A., Elodie, Isabelle, Pierre, David T., Gon¸calo, ...) for their
advices and for the friendly atmosphere there was among us.
I must not forget to thank to all the interns and students whose projects and master
theses were, closely or remotely, related to this work, namely to Servane Le N´eel,
Lin Jin, Yassine Loudad, Michael Ngoy Kabange, El Habib Damani, Arnaud Ronse
vii
De Craene, Beatriz Aldea Pueyo, Ilias El Makrini, Raphael Girault, Mathieu Du-
mas, Neda Nourshamsi, Roger Tilmans, Nicolas Cormond, Alexandre Hua, Romain
Hamel, Malgorzata Sudol, Tristan de Crombrugghe, Hava ¨
Ozdemir and others that
I might have already forgotten.
I want to thank to the BEAMS department for lending us their high speed camera
many times.
I am very grateful to Marie Currie Research Training Networks, which financed my
first year as a visiting researcher at ASL, and to F.R.S.-FNRS for the F.R.I.A. fel-
lowship (FC 89554) financing most of my PhD studies.
I would like to thank to all my Czech friends for paying me numerous visits in Brus-
sels and for finding some time to meet me whenever I went back home. Many thanks
goes also to my friends in Brussels for all the outdoor activities, which worked great
for taking my mind of this work every now and then.
Finally, I want to thank all my family, my brother and sister, my grandparents and
my parents for their love, continuous support and encouragements! And to Barney,
the dog, of course...
viii
Abstract
The use of drones, also called unmanned aerial vehicles (UAVs), is increasing every
day. These aircraft are piloted either remotely by a human pilot or completely au-
tonomously by an on-board computer. UAVs are typically equipped with a video
camera providing a live video feed to the operator. While they were originally devel-
oped mainly for military purposes, many civil applications start to emerge as they
become more affordable.
Micro air vehicles are a subgroup of UAVs with a size and weight limitation; many
are designed also for indoor use. Designs with rotary wings are generally preferred
over fixed wings as they can take off vertically and operate at low speeds or even
hover. At small scales, designs with flapping wings are being explored to try to
mimic the exceptional flight capabilities of birds and insects.
The objective of this thesis is to develop a control mechanism for a robotic humming-
bird, a bio-inspired tail-less hovering flapping wing MAV. The mechanism should
generate moments necessary for flight stabilization and steering by an independent
control of flapping motion of each wing.
The theoretical part of this work uses a quasi-steady modelling approach to approx-
imate the flapping wing aerodynamics. The model is linearised and further reduced
to study the flight stability near hovering, identify the wing motion parameters suit-
able for control and finally design a flight controller. Validity of this approach is
demonstrated by simulations with the original, non-linear mathematical model.
A robotic hummingbird prototype is developed in the second, practical part. Details
are given on the flapping linkage mechanism and wing design, together with tests
performed on a custom built force balance and with a high speed camera. Finally,
two possible control mechanisms are proposed: the first one is based on wing twist
modulation via wing root bars flexing; the second modulates the flapping amplitude
and offset via flapping mechanism joint displacements. The performance of the
control mechanism prototypes is demonstrated experimentally.
ix
Glossary
List of abbreviations
ASL Active Structures Laboratory
BL DC Brushless Direct Current electric motor
BR DC Brushed Direct Current electric motor
CFRP Carbon Fibre Reinforced Polymer
COG Centre of Gravity
CP Centre of Pressure
DARPA Defense Advanced Research Projects Agency, United States
DOF Degree of Freedom
FDM Fused Deposition Modelling
IMU Inertial Measurement Unit
LQR Linear-Quadratic Regulator
MAV Micro Air Vehicle
RC Radio-Controlled
SISO Single-Input Single-Output (system)
SMA Shape Memory Alloy
SLS Selective Laser Sintering
UAV Unmanned Aerial Vehicle
ULB Universit´e Libre de Bruxelles
Nomenclature
˙xFirst derivative of xwith respect to time
¨xSecond derivative of xwith respect to time
¯xCycle averaged value of x(xrepresents forces, moments, speed),
average value of x(xrepresents dimensions)
ˆx x divided by mass (xrepresents forces), by inertia (moments) or
by characteristic length (dimensions)
xeEquilibrium value of x
∆xDifference from equilibrium value of x
xi
F0Wing section force
Ftr, Fr, FaQuasi-steady components of force Fdue to translation,
rotation and added mass
Fx=∂F
∂x ,Mx=∂ M
∂x Stability derivatives (partial derivatives of force F/
moment Mwith respect to system state x)
Fp=∂F
∂p , Mp=∂ M
∂p Control derivatives (partial derivatives of force F/
moment Mwith respect to control parameter p)
A(s)/B(s) Transfer function of Laplace transforms of input b(t)
to output a(t)
List of symbols
α, αgAngle of attack, geometric angle of attack
α0, αmAngle of attack offset and magnitude around mid-stroke
α34 Angle of intermediary link 34 of the flapping mechanism
α∗Wing inclination angle
βMean stroke plane angle
δWing deviation angle
δm1, δm2Amplitudes of oval and figure-of-eight deviation patterns
∆x, ∆yx and y distance from the nominal position of the displaced
joints
L, RLeft and right offset servo angle
ηRoll servo angle
ηmMotor efficiency
γWing root bar angle
Γ Circulation
λSystem pole
νKinematic viscosity
ωBody angular velocity vector
ωmMotor angular velocity
ωW, ωwx, ωwy, ωwz Wing angular velocity vector and its components
ΩWWing angular velocity skew-symmetric matrix
φ, φ0, φmSweep angle, sweep angle offset and amplitude
φmax, φmin Maximal and minimal measured flapping angle
φroot, φtip Flapping angle measured at wing root and at wing tip
Φ Flapping amplitude
ϕ, ϑ, ψ Roll, pitch and yaw body angles
ϕαPhase shift between wing inclination and wing sweep
Ψ, ψ3Intermediary link 34 amplitude and angle
ρAir density
xii
θFlapping mechanism input angle
AWing aspect ratio
A1, A2Flapping mechanism dimensions
Along,Alat System matrices of longitudinal and lateral dynamics
BDistance between force balance sensors
Blong,Blat Input matrices of longitudinal and lateral dynamics
c, ¯c, ˆcWing chord, mean wing chord and normalized wing chord
CL, CDLift and drag coefficients
CN, CTNormal and tangential force coefficients
eChest width (distance between wing shoulders)
fFlapping frequency
FL, FDLift and drag forces
FN, FTNormal and tangential forces
gGravity acceleration
HVertical distance of the prototype from the force sensors
Ixx, Iyy, Iz z , Ixz Moments of inertia and inertia product in body axes
JAdvance ratio
J,JS,JA,Jred Matrix of control derivatives, for symmetric, asymmetric
and reduced set of wing motion parameter changes
kα, kφWing inclination and sweep angle function shape parameters
khover Reduced frequency in hover
kp, kqGains of roll and pitch rate feedback
L, M, N Moments around body axes xB, yBand zB
L1, ..., L6Flapping mechanism link dimensions
Lext, Mext , Next External moments around body axes xB, yBand zB
mbody mass
OB, xB, yB, zBBody coordinate system
OG, xG, yG, zGGlobal coordinate system
OW, xW, yW, zWWing coordinate system
OSP, xSP , ySP, zSP Stroke plane coordinate system
p, q, r Body angular velocity components around xB, yBand zBaxes
p, piWing motion parameters vector, its ith element
Pel Motor electrical power
Pmech Mechanical power at the motor output
r, ˆrRadial distance from the wing root, absolute and normalized
rCentre of pressure position vector in body frame
RWing length
RRotation matrix
ˆr2Normalized radial centre of pressure position
rcCentre of pressure position vector in wing frame
xiii
RCP Radial centre of pressure position
rwWing shoulder position vector in body frame
RxForce balance reaction
Rx,Ry,RyRotation matrices for rotations around x, y and z axes
Re Reynolds number
sLaplace transformation parameter
SWing area
S1, S2Sensor 1 and 2 forces
St Strouhal number
t, t+Time, nondimensional cycle time
TmMotor torque
u, v, w Body velocity components around xB, yBand zBaxes
U, U∞, UCP Wing speed, free stream speed, centre of pressure speed
U, Ux, Uy, UzWing speed vector and its components in body axes
vBody velocity vector
X, Y, Z Forces along body axes xB, yBand zB
xState vector
ˆx0Non-dimensional position of wing rotation axis
Xext, Yext , Zext External forces along body axes xB, yBand zB
xW, zWWing shoulder position in body frame
zTransfer function zero
xiv
Contents
Jury iii
Acknowledgements vii
Abstract ix
Glossary xi
1 Introduction 1
1.1 UAVapplications............................. 1
1.2 UAVtypes ................................ 3
1.3 FlappingwingMAVs........................... 5
1.3.1 Actuators and flapping mechanisms . . . . . . . . . . . . . . 5
1.3.2 Tail stabilized and passively stable MAVs . . . . . . . . . . . 7
1.3.3 MAVs controlled by wing motion . . . . . . . . . . . . . . . . 9
1.4 Motivation and outline . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 References................................. 13
2 Flapping flight 17
2.1 Fixed wing aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Flightinnature.............................. 21
2.2.1 Glidingflight ........................... 21
2.2.2 Flapping forward flight . . . . . . . . . . . . . . . . . . . . . 24
2.2.3 Hoveringflight .......................... 25
2.3 Hovering flapping flight aerodynamics . . . . . . . . . . . . . . . . . 30
2.3.1 Dynamic scaling . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 Lift enhancing aerodynamic mechanisms . . . . . . . . . . . . 31
2.3.3 Flight stability . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.4 Attitude stabilization . . . . . . . . . . . . . . . . . . . . . . 37
2.3.5 Flightcontrol........................... 40
2.4 References................................. 40
xv
xvi CONTENTS
3 Mathematical modelling 47
3.1 Flapping flight aerodynamics . . . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Wingkinematics ......................... 48
3.1.2 Quasi-steady aerodynamics . . . . . . . . . . . . . . . . . . . 50
3.1.2.1 Force due to wing translation . . . . . . . . . . . . . 51
3.1.2.2 Force due to wing rotation . . . . . . . . . . . . . . 52
3.1.2.3 Force due to the inertia of added mass . . . . . . . . 53
3.1.2.4 Total force . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.3 Centre of pressure velocity and angle of attack . . . . . . . . 55
3.1.4 Comparison with CFD . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Bodydynamics.............................. 60
3.2.1 System linearisation . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.2 Stability and control derivatives . . . . . . . . . . . . . . . . 64
3.3 Reduced model of a flapping wing MAV . . . . . . . . . . . . . . . . 65
3.4 Stability predicted by various aerodynamic models . . . . . . . . . . 66
3.4.1 Stability derivatives . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.2 Longitudinal system poles . . . . . . . . . . . . . . . . . . . . 68
3.4.3 Lateral system poles . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.4 Effect of derivatives ˆ
Xqand ˆ
Yp................. 72
3.4.5 Effect of inertia product Ixz ................... 72
3.4.6 Conclusion ............................ 74
3.5 References................................. 74
4 Stability of near-hover flapping flight 77
4.1 Hummingbird robot parameters . . . . . . . . . . . . . . . . . . . . . 77
4.2 Pitchdynamics .............................. 78
4.2.1 Pitch stability derivatives . . . . . . . . . . . . . . . . . . . . 78
4.2.2 Systempoles ........................... 83
4.2.3 Active stabilization . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Rolldynamics............................... 89
4.3.1 Roll stability derivatives . . . . . . . . . . . . . . . . . . . . . 89
4.3.2 Systempoles ........................... 93
4.3.3 Active stabilization . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 Vertical and yaw dynamics stability . . . . . . . . . . . . . . . . . . 96
4.5 Wingpositionchoice........................... 98
4.6 Ratefeedbackgains ........................... 99
4.7 Conclusion ................................ 101
4.8 References................................. 101
CONTENTS xvii
5 Flapping flight control 103
5.1 Controldesign .............................. 103
5.1.1 Pitchdynamics.......................... 104
5.1.2 Rolldynamics........................... 109
5.1.3 Yaw and vertical dynamics . . . . . . . . . . . . . . . . . . . 112
5.1.4 Complete controller . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Control moment generation . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.1 Control derivatives . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.2 Choice of control parameters . . . . . . . . . . . . . . . . . . 117
5.3 Simulationresults ............................ 118
5.4 Conclusion ................................ 122
5.5 References................................. 122
6 Flapping mechanism 125
6.1 Flapping mechanism concept . . . . . . . . . . . . . . . . . . . . . . 125
6.1.1 Kinematics ............................ 126
6.1.2 Mechanism design . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1.3 Wingdesign............................ 130
6.2 Experiments................................ 133
6.2.1 Wingkinematics ......................... 133
6.2.2 Forcebalance........................... 137
6.2.3 Lift production . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.2.4 Wing design optimization . . . . . . . . . . . . . . . . . . . . 140
6.3 References................................. 146
7 Control mechanism 147
7.1 Moment generation via wing twist modulation . . . . . . . . . . . . 148
7.1.1 Moment generation principle . . . . . . . . . . . . . . . . . . 149
7.1.2 Manually operated control mechanism performance . . . . . . 150
7.1.3 SMA actuated control mechanism . . . . . . . . . . . . . . . 154
7.1.4 SMA driven control mechanism performance . . . . . . . . . 156
7.1.5 Conclusion on wing twist modulation . . . . . . . . . . . . . 158
7.2 Moment generation via amplitude and offset modulation . . . . . . . 158
7.2.1 Amplitude and offset modulation . . . . . . . . . . . . . . . . 159
7.2.2 Control mechanism prototype . . . . . . . . . . . . . . . . . . 162
7.2.3 Wingkinematics ......................... 162
7.2.4 Control mechanism dynamics . . . . . . . . . . . . . . . . . . 166
7.2.5 Pitch moment and lift generation . . . . . . . . . . . . . . . . 168
7.2.6 Combined commands . . . . . . . . . . . . . . . . . . . . . . 171
7.2.7 Conclusion on amplitude and offset modulation . . . . . . . . 174
7.3 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 174
Chapter 1
Introduction
Drones, also called unmanned aerial vehicles (UAVs), are aircraft without a human
pilot on board that can be either remotely piloted or completely autonomous. UAVs
are slowly becoming part of our daily lives. While 20 years ago they were almost
exclusively used by the military, the recent technological advancements made them
accesible even to the general public. Nowadays, UAVs are being used in many fields
ranging from aerial photography to remote inspection and small drones can be found
in hobby stores for less than e150, including a live video link.
Micro air vehicles (MAVs) are a class of UAVs restricted in size. DARPA origi-
nally defined an MAV as a micro-drone of no more than 15 cm. The term, however,
started to be used more broadly and refers to smaller UAVs. Thus, palm sized UAVs
are sometimes called nano air vehicles. Most MAVs can perform hovering flight and
operate indoors, although this is not a requirement. Their popularity over larger
UAVs increases as they are easily portable, more discreet and less dangerous in case
of a crash.
1.1 UAV applications
UAVs are being used in various fields and their number is growing (Figure 1.1).
Traditionally, UAVs are equipped with an on-board camera and provide a live video
feed to the operator or to the ground station. They can be, however, also equipped
with other sensor types (chemical, biological, radiation, ...).
The obvious application of camera equipped UAVs is video surveillance and recon-
naissance. Apart from military use, the UAVs are starting to be employed by police
and fire brigades. No men aboard and much lower costs compared to traditional
1
21 Introduction
Figure 1.1: UAV applications: monitoring of crops, inspection of power lines, transport of
packages or police air reconnaissance are just a few examples.
aircraft allows their use even in risky conditions. The UAVs can be deployed during
natural catastrophes or after terrorism acts to quickly map the situation, find access
routes, identify potential dangers, look for victims, ... Their small size allows them
even to enter into buildings through windows and fly through confined spaces.
Another field of application is aerial photography. Images from a bird’s-eye view
are used for cartography, but also in archaeology, biology or in urbanism. UAVs
were also quickly adopted in sports-photography and cinematography to shoot ac-
tion scenes from unusual perspectives.
UAVs are further used for remote inspection of pipelines or power lines, as well as by
farmers for inspecting their fields and choosing the optimal moment for fertilization
or harvest. Security applications like patrolling around private properties or along
the borderlines are also emerging. Last but not least, the use of UAVs for goods
deliveries is being explored.
1.2 UAV types 3
1.2 UAV types
UAVs can be split into three groups according to the way they generate lift force.
The fixed wing UAVs are similar to aeroplanes. To produce enough lift, the wing
needs to keep moving above certain minimal speed. This limits the use of fixed wing
UAVs mostly to outdoors. On the other hand, it makes them efficient, as most en-
ergy is spent to overcome drag, and thus suitable for applications where maximum
flight time is the key factor. The forward thrust is usually produced by one or several
propellers. Designs with tail wings are passively stable, however, smaller MAVs are
often built as flying wings, which usually require some stability augmentation by an
on-board computer. Autonomous flight requires sophisticated trajectory planning,
as all the manoeuvres need to stay within the aircraft’s flight envelope. Some ex-
amples of fixed wing UAVs are shown in Figure 1.2.
Figure 1.2: Examples of fixed wing UAVs.
Rotary wing UAVs generate the lift by one or several rotating bladed rotors. Nowa-
days, four and more rotor designs, known as quadrocopters and multicopters, are
the most popular UAV platform. These designs are inherently unstable and require
an on-board autopilot for attitude control. This, however, makes the designs also
very manoeuvrable and agile yet relatively insensitive to disturbances. Nevertheless,
the smallest commercially available professional MAV, the Black Hornet by Prox-
dynamics, uses a traditional helicopter design with one main rotor and a stabilizing
tail rotor. Examples of rotary wing UAVs are shown in Figure 1.3.
The advantage of rotary wing UAVs over fixed wing designs is the capability of
vertical take-off, hovering and slow flight in any direction, which makes them useful
especially in confined urban environments or even indoors. Autonomous opera-
41 Introduction
Ladybird V2 FPV
Walkera
12.5 cm, 35 g, 6 min.
Figure 1.3: Examples of rotary wing UAVs: The Black Hornet is the smallest UAV used
by the British military, the firefly represents a commercial hex-rotor. For comparison, the
Ladybird is a e150 mini-quadrotor for hobbyist equipped with a live video feed.
tion represents much smaller challenge: a trajectory between two waypoints can be
planned as a straight line, limited only by the maximal speed. On the other hand,
the rotary wing designs are less efficient compared to fixed wing UAVs and thus
their flight times are limited. Hybrid designs combining the vertical take-off and
low speed flight of rotary wings with flight endurance of fixed wings are also being
developed.
The last group, the flapping wing UAVs, takes the inspiration in birds and insects.
The existing designs are still immature and their performance is, for the moment,
worse compared to the previous two groups. However, natural flyers with flapping
wings exhibit long flight times (without any energy income) together with acrobatic
skills which are far superior to any man made aircraft, especially at small scales.
Thus, the vision of potentially achieving these exceptional flight qualities is what
drives the active research field of flapping flight. More details on the current state
of the art will be given in the next section.
Bio-mimetic approach is used also in other fields of UAV and MAV design in par-
ticular. Optical flow sensors and attitude sensors inspired by the insect compound
eyes and simple eyes (Ruffier and Franceschini, 2005; Fuller et al., 2014) or MAVs
flying and communicating in swarms (Hauert et al., 2009) are just a few examples.
1.3 Flapping wing MAVs 5
1.3 Flapping wing MAVs
People have always been fascinated by flying animals. A sketch of one of the first
flying machines with flapping wings, although human powered, can be found in
Leonardo da Vinci’s Paris Manuscript B dated 1488-1490 (Figure 1.4). However, it
took another five centuries to reach a sufficient technology level that allowed us to
build first bio-inspired MAVs. The field of flapping wing MAVs is still very young
and provides plenty of space for improvement. The biggest challenge of flapping
wing MAV design remains the integration of relatively complex flapping and control
mechanisms into a small and lightweight package that can be lifted by the thrust
produced.
Figure 1.4: Sketches of human powered flying machines with flapping wings by Leonardo
da Vinci from Paris Manuscript B, 1488-1490.
1.3.1 Actuators and flapping mechanisms
When designing flying machines that mimic nature we need to find a replacement
for the animal’s powerful flight muscles as well as for their rapid metabolism sup-
plying energy at high rates. Thanks to recent technological advancements in mobile
electronic devices, batteries with high energy densities emerged. Their high capac-
ity to mass ratio made them a very attractive power source for MAVs. Thus, the
61 Introduction
Figure 1.5: Examples of flapping mechanisms. a) DelFly Micro mechanism (Bruggeman,
2010), b)+c) Nano Hummingbird linkage and cable mechanism (Keennon et al., 2012), d)
direct drive mechanism (Hines et al., 2014), e) compliant mechanism of Harvard robotic fly
(Finio and Wood, 2010), f) flexible resonant wing (Vanneste et al., 2011), g) resonant thorax
(Goosen et al., 2013).
majority of existing MAVs uses electric actuators driven by Lithium-ion (Li-ion) or
Lithium-ion polymer (Li-Pol) batteries.
The most common actuator is a DC motor, either with brushes (e.g. Keennon et al.,
2012) or brushless (e.g. de Croon et al., 2009). It is usually combined with a reduc-
tion gearbox and a transmission mechanism producing the flapping motion, which
can either be a linkage mechanism as in Figures 1.5 a)+b) or a cable mechanism as
in Figure 1.5 c). However, a direct drive option exploiting resonance is also being
explored (Hines et al., 2014), see Figure 1.5 d).
Piezo-actuators are another option as they can be directly operated at the flapping
frequency. They are usually combined with a (compliant) linkage mechanism (Finio
and Wood, 2010), Figure 1.5 e). Some flapping mechanisms try to mimic the res-
1.3 Flapping wing MAVs 7
onant thorax of insects, as this should provide high flapping amplitudes with low
energy expenditure (Vanneste et al., 2011; Goosen et al., 2013). These designs are
driven by electro-magnetic actuators, see Figures 1.5 f)+g).
Apart from electric drives, an internal combustion engine was successfully used to
drive a larger flapping wing MAV in the past (Zdunich et al., 2007). Also a rather
exotic chemical micro engine has been considered to drive a resonant thorax mech-
anism (Meskers, 2010).
1.3.2 Tail stabilized and passively stable MAVs
The first designed flapping wing MAVs were either built passively stable, or used
tail surfaces for attitude stabilization. Majority of these designs have four wings
and take advantage of the clap-and-fling lift enhancement mechanism, which will be
explained in Section 2.3.2.
One of the first flapping wing MAVs was the Mentor shown in Figure 1.6 a), devel-
oped under a DARPA program and presented in 2002. The first generation of the
vehicle had a wingspan of 36 cm and weighted, from today’s perspective enormous,
580 g due to an internal combustion engine driving the device. Nevertheless the
vehicle was able to take-off and hover at a flapping frequency of 30 Hz. Two pairs of
wings were located at the top of the vehicle, flapping with 90◦amplitude and using
the clap-and-fling at both extremities. The flight was stabilized and actively con-
trolled by fins exposed to the airflow coming from the wings. The flight endurance
was up to 6 minutes. The second generation used a brushless motor and was a bit
smaller and lighter (30 cm, 440 g). Its flight time was limited to only 20 s by the
discharge rate of the batteries available at that time.
DelFly was developed in 2005 by TU Delft (Lentink and Dickinson, 2009) and rep-
resents one of many ornithopter projects (e.g. Park and Yoon, 2008; Yang et al.,
2009). Unlike most ornithopters that fly only forward, the DelFly can also operate
near hovering or even fly slowly backwards, all controlled by tail control surfaces
operated by servos. Its two pairs of flapping wings are driven by a brushless motor
and a linkage mechanism. It takes advantage of the clap-and-fling mechanism twice:
when the lower wings meet the upper wings and also when the upper wings touch
each other. The current version, the DelFly Explorer shown in Figure 1.6 b), has a
wingspan of 28 cm, weights 20 g and has a flight endurance of 9 minutes (Wagter
et al., 2014). It is capable of fully autonomous flight thanks to an on-board stereo-
vision system (Tijmons et al., 2013). A smaller DelFly Micro shown in Figure 1.6 c)
has a wingspan of 10 cm and weights only 3.07 g, including an on-board camera.
81 Introduction
Figure 1.6: Examples of flapping wing MAVs that are stabilized by tail or that are passively
stable.
1.3 Flapping wing MAVs 9
The first hovering passively stable MAV was built at Cornell University by van
Breugel et al. (2008), see Figure 1.6 e). It uses four pairs of wings, that clap and
fling at both extremities. The vehicle has a 45 cm wingspan, although the wings
themselves are rather short (around 85 mm). The flapping is driven by 4 DC pager
motors, one for each wing pair, and the total weight is 24.2 g. The passive stability is
achieved by two lightweight sails, one above the wings and one at a greater distance
below the wings. The vehicle can stay in the air, without any control, for 33 s. An
updated version of the previous flyer, using only 2 wing pairs and a single motor
was presented by Richter and Lipson (2011). The major part of the robot, shown
in Figure 1.6 f), is 3D printed, including the wings. With sails for passive stability
the robot weight is 3.89 g and it can fly for 85 s.
The last passively stable MAV has been presented recently by Ristroph and Chil-
dress (2014). Unlike the previous robots, the inspiration comes from a swimming
jellyfish, see Figure 1.6 d). The vehicle has four wings, one on each side. The wings
do not flap horizontally like in insects, but rather vertically. The opposing wings
flap together while the neighbouring wings are in anti-phase. The vehicle is very
small (10 cm) and very light (2.1 g). It carries only a DC pager motor but no power
source; flying was demonstrated at 19 Hz flapping frequency while being tethered
to an external power source. The jelly-fish-like wings make the vehicle inherently
stable and thus it doesn’t need any additional stabilizing surfaces.
1.3.3 MAVs controlled by wing motion
Compared to the majority of MAVs from the previous section, designs that are
stabilized and controlled by adjusting the wing motion are much closer in function
to their biological counterparts, insects and hummingbirds. However, they are also
more complex because of the necessary control mechanisms that modify the wing
kinematics. First MAV stabilized and controlled through wing motion was presented
in 2011 and only three designs have demonstrated stable hovering flight so far.
The Nano Hummingbird shown in Figure 1.7 a) is an MAV funded by DARPA,
presented in 2011 by AeroVironment, mimicking a hummingbird (Keennon et al.,
2012). It is the only flapping wing MAV capable of true hovering as well as of flight
in any direction while carrying an on-board camera with live video feed. All this
is integrated into a robot with 16.5 cm wingspan weighting 19 g that has a flight
endurance of up to 4 minutes. The necessary control moments are generated by
independent modulation of the wing twist.
10 1 Introduction
Figure 1.7: Examples of flapping wing MAVs that are actively controlled by wing motion.
The Harvard RoboBee with a wingspan of only 3 cm and weight of 80 mg is the
smallest and lightest MAV, see Figure 1.7 b). It took off for the first time in 2008
while using guide wires for stabilization (Wood, 2008) and performed first controlled
hovering flight five years later (Ma et al., 2013). It mimics insects of the Diptera
order, the true flies. It has a single pair of wings that are driven independently
by a pair of piezoelectric bimorph actuators. Each wing can be operated with dif-
ferent amplitude, different mean position and different speed in each half-stroke,
so that moments along the three body axis can be produced to stabilize the robot
in air. The power source as well as flight controller remain off-board for the moment.
The BionicOpter, Figure 1.7 c), was built as a technology demonstrator of Festo
company (Festo, 2013). It mimics a dragonfly, although it is much larger (63 cm
wingspan) and heavier (175 g). It uses four flapping wings that are driven by a single
motor and that beat at a frequency of 15 Hz to 20 Hz. Their amplitude and flapping
1.4 Motivation and outline 11
plane inclination can be controlled independently by 8 servo motors in total, which
allows independent drag and lift modulation of each wing. Thus, the vehicle can
hover as well as fly in any direction without the need to pitch or roll.
1.4 Motivation and outline
The goal of our project is to develop a tail-less flapping wing MAV capable of hov-
ering flight. The flight should be stabilized and controlled by adapting the wing
motion. Looking into the nature, only insects and hummingbirds are capable of sus-
tained hovering. Their wing beat frequency and total mass are linearly correlated
with the wing length, see Figure 1.8. Interestingly, existing flapping wing MAVs
also follow this trend.
To make our lives easier, we have chosen to mimic larger hummingbirds, which
should allow us to use, at least to some extent, some of the off-the-shelf components
as well as traditional technologies. Thus, the target specification for the designed
robotic hummingbird was set to: 20 g total mass, 25 cm wingspan and flapping
frequency between 20 and 30 Hz.
The aim of this thesis is to design a working prototype of the wing motion control
mechanism that generates the control moments necessary to stabilize and control
the flight. The thesis was split into two parts, theoretical (Chapters 2 - 5) and
practical (Chapters 6 - 7).
Chapter 2 recalls the basics of fixed wing aerodynamics. Then, different types of
flight observed in nature are explained. More details are given on hovering flapping
flight, its aerodynamic mechanisms as well as control mechanisms observed in nature.
Chapter 3 introduces a mathematical model of flapping flight, which combines quasi-
steady aerodynamics and rigid body dynamics. Further, the model is linearised and
reduced and its validity is demonstrated by comparisons to other models, including
a CFD study.
Chapter 4 is devoted to near-hover flapping flight stability. The damping effects
coming from the flapping wings are explained, with a special attention given to the
effect of wing position, and a simplified solution of the stability problem is proposed.
Chapter 5 describes the control design for the developed MAV, based on the lin-
earised mathematical model. Wing kinematics parameters suitable for flight control
are identified and the control performance is demonstrated on numerical simulations.
1.5 References 13
Chapter 6 gives details on the development of the flapping mechanism and of the
wing shape and presents experimental results obtained with a high speed camera
and a custom built force balance.
Finally, Chapter 7 describes the development of two control mechanisms and their
implementation to the robot prototype. Their performance is demonstrated by
force and moment measurements and by high speed camera wing kinematics mea-
surements.
1.5 References
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design and driving mechanism. Master’s thesis, Delft University of Technology,
2010.
G. de Croon, K. de Clerq, R. Ruijsink, B. Remes, and C. de Wagter. Design,
aerodynamics, and vision-based control of the DelFly. International Journal of
Micro Air Vehicles, 1(2):71–97, Jun. 2009. doi:10.1260/175682909789498288.
Festo. BionicOpter. http://www.festo.com/cms/en_corp/13165.htm, 2013. Ac-
cessed: 18/08/2014.
B. M. Finio and R. J. Wood. Distributed power and control actuation in the thoracic
mechanics of a robotic insect. Bioinspiration & biomimetics, 5(4):045006, 2010.
doi:10.1088/1748-3182/5/4/045006.
S. B. Fuller, M. Karpelson, A. Censi, K. Y. Ma, and R. J. Wood. Controlling free
flight of a robotic fly using an onboard vision sensor inspired by insect ocelli.
Journal of The Royal Society Interface, 11(97), 2014. doi:10.1098/rsif.2014.0281.
J. F. Goosen, H. J. Peters, Q. Wang, P. Tiso, and F. van Keulen. Resonance based
flapping wing micro air vehicle. In International Micro Air Vehicle Conference
and Flight Competition (IMAV2013), Toulouse, France, September 17-20, page 8,
2013.
C. H. Greenewalt. Hummingbirds. Dover Publications, 1990.
S. Hauert, J.-C. Zufferey, and D. Floreano. Evolved swarming without positioning
information: anapplication in aerial communication relay. Autonomous Robots,
26(1):21–32, 2009. ISSN 0929-5593. doi:10.1007/s10514-008-9104-9.
L. Hines, D. Campolo, and M. Sitti. Liftoff of a motor-driven, flapping-wing mi-
croaerial vehicle capable of resonance. IEEE Transactions on Robotics, 30(1):
220–231, 2014. doi:10.1109/TRO.2013.2280057.
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M. T. Keennon, K. R. Klingebiel, H. Won, and A. Andriukov. Development of
the nano hummingbird: A tailless flapping wing micro air vehicle. AIAA paper
2012-0588, pages 1–24, 2012.
D. Lentink and M. H. Dickinson. Biofluiddynamic scaling of flapping, spinning and
translating fins and wings. Journal of Experimental Biology, 212(16):2691–2704,
2009. doi:10.1242/jeb.022251.
K. Y. Ma, P. Chirarattananon, S. B. Fuller, and R. J. Wood. Controlled
flight of a biologically inspired, insect-scale robot. Science, 340:603–607, 2013.
doi:10.1126/science.1231806.
A. Meskers. High energy density micro-actuation based on gas generation by means
of catalysis of liquid chemical energy. Master’s thesis, Delft University of Tech-
nology, 2010.
J. H. Park and K.-J. Yoon. Designing a biomimetic ornithopter capable of sustained
and controlled flight. Journal of Bionic Engineering, 5(1):39–47, 2008.
C. Richter and H. Lipson. Untethered hovering flapping flight of a 3d-printed me-
chanical insect. Artificial life, 17(2):73–86, 2011. doi:10.1162/ artl a 00020.
L. Ristroph and S. Childress. Stable hovering of a jellyfish-like flying machine. Jour-
nal of The Royal Society Interface, 11(92):1–13, 2014. doi:10.1098/rsif.2013.0992.
F. Ruffier and N. Franceschini. Optic flow regulation: the key to aircraft au-
tomatic guidance. Robotics and Autonomous Systems, 50(4):177–194, 2005.
doi:10.1016/j.robot.2004.09.016.
S. Tijmons, G. Croon, B. Remes, C. Wagter, R. Ruijsink, E.-J. Kampen, and Q. Chu.
Stereo vision based obstacle avoidance on flapping wing mavs. In Q. Chu, B. Mul-
der, D. Choukroun, E.-J. Kampen, C. Visser, and G. Looye, editors, Advances in
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Automation Magazine, IEEE, 15(4):68–74, 2008. doi:10.1109/MRA.2008.929923.
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gram flapping wing mav with a 4-gram onboard stereo vision system. Accepted
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16 References
Chapter 2
Flapping flight
This chapter introduces the reader to the problematic of flapping flight. Because
the majority of man made aircraft uses fixed wings, basic concepts of fixed wing
aerodynamics are reviewed first. Then, three flight types observed in nature (glid-
ing, flapping and hovering flapping flight) are described. An extra attention is given
to the hovering flapping flight and its aerodynamic mechanisms enhancing the lift
production as well as to flight control in nature.
2.1 Fixed wing aerodynamics
Considering an incompressible steady flow around a 2D airfoil according to Figure
2.1, the lift production can be explained by two basic laws of physics: the conti-
nuity of mass and the conservation of energy. As the flow approaches the airfoil
the streamlines above the airfoil get closer to each other and, because of the mass
continuity (AU = const.), the flow velocity increases. On the contrary, below the
airfoil the streamlines get further away and as a result the flow velocity drops. To
comply with the Bernoulli equation (p+1
2ρU2= const.), which can be derived from
upper
lower
Figure 2.1: Steady flow around a 2D airfoil. Nomenclature: αangle of attack, FLlift, FD
drag, cchord width and U∞free-stream velocity (left) and typical streamlines (right).
17
18 2 Flapping flight
the conservation of energy, the static pressure needs to drop above the wing and rise
below the wing. This pressure difference results in a suction force lifting the airfoil:
the lift force.
Figure 2.2: Pressure distribution around an airfoil (left) and the resulting lift force distri-
bution (right), from Whitford (1987).
The distribution of the lift force is given by the pressure distribution, as in Figure
2.2. Thus, the resultant lift force vector is placed at the centre of pressure (CP).
For an airfoil of a general shape, the CP location is varying with the angle of attack.
However, it can be demonstrated by employing the thin airfoil theory that for a
symmetric airfoil the CP lies in 1/4 of the chord from the leading edge.
If we start moving an airfoil from rest the flow pattern at the very beginning will
look like in Figure 2.3 (left). A circulation Γ will develop around the airfoil to fulfil
the Kutta condition, i.e. move the stagnation point to the trailing edge as in Figure
2.3 (right). This circulation is associated to a vortex that remains bound to the
wing and is thus called the bound vortex. According to Kelvin’s theorem, stating
that the total circulation is conserved, the bound vortex needs to be compensated
Figure 2.3: Circulation theory of lift: inviscid flow around an airfoil producing zero lift
(left), circulation according to the Kutta condition (centre) and combined flow fulfilling the
Kutta condition and generating lift (right). Figure adapted from Ellington (1984c).
2.1 Fixed wing aerodynamics 19
by a vortex with an opposite circulation. This vortex is formed near the trailing
edge due to high velocity gradients and is called the starting vortex. Once the
trailing edge is reached the flow reaches steady conditions: the bound vortex stops
growing and the starting vortex is shed into the wake. Similar transverse vortex is
shed whenever the bound circulation changes, e.g. due to a change of angle of attack.
In a finite wing another pair of counter-rotating vortices is present in the wake be-
hind the wing tips, one on each side. They are called the wing-tip vortices and are
caused by an opposite spanwise flow above and below the wing. The whole vortex
system of a finite wing is shown in Figure 2.4. Stopping the wing suddenly makes
also the bound vortex shed, forming a ring vortex. It will be shown in the next
section that similar vortex systems can be observed around flapping wings.
Figure 2.4: Vortex system of a finite span wing: a) Development of the starting vortex
as the wing starts moving, b) steady state, c) vortex ring shed when the wing is stopped.
Figure adapted from Lehmann (2004).
For a steady flow around a flat 2D airfoil the circulation of the bound vortex can be
expressed using the thin airfoil theory as
Γ = παcU∞,(2.1)
where αis the angle of attack, cthe chord width and U∞the free stream velocity.
Combining the result with the Kutta-Joukowski theorem, F0
L=ρU∞Γ, we obtain
the lift force as
F0
L=ρU∞Γ = παcρU 2
∞,(2.2)
where ρis the fluid density.
20 2 Flapping flight
The lift coefficient is a dimensionless characteristic of the airfoil profile defined as
CL=FL
1
2ρU2
∞S=F0
L(1)
1
2ρU2
∞c(1),(2.3)
where S=c(1) is the surface of a unity span wing with a chord c. By combining
the last two equations we obtain the lift coefficient for a flat airfoil as CL= 2πα.
According to equation (2.2) the lift of a theoretical flat airfoil increases linearly with
angle of attack. In reality the lift drops above certain value of angle of attack, see
Figure 2.5. The pressure gradients on the upper airfoil side become too high, which
results into flow separation due to viscosity. The pressure in the separated region
does not drop any more and as a consequence the lift is reduced. This phenomenon
is called stall.
Figure 2.5: Stall: Lift coefficient drops at high angles of attack due to flow separation,
from Whitford (1987).
While the lift force is given by the pressure distribution around the airfoil, there are
several sources of drag force. The total drag of a 2D airfoil is called the profile drag.
It is given by terms due to viscosity (skin friction drag) and due to pressure and
subsequent separation (form drag). The total drag coefficient is defined similarly to
the lift coefficient
CD=FD
1
2ρU2
∞S=F0
D(1)
1
2ρU2
∞c(1).(2.4)
In a finite wing, a small downward flow component, called the downwash w, is su-
perposed to the flow around the wing, coming from the wing tip vortices (Figure
2.6). The downwash varies along the wingspan and as a result the local angle of
attack changes. The original, geometric, angle of attack αgdecreases by an induced
2.2 Flight in nature 21
Figure 2.6: Wing tip vortices formed in clouds behind Boeing B-757 (left) and the effects
of the resulting downwash on a local section of a finite length wing (right).
angle of attack αi. The local lift is then produced according to an effective angle of
attack αeff =αg−αimeasured with respect to the relative flow.
Since the local lift vector FLis perpendicular to the relative flow vector, it has a
component in the direction of U∞called the induced drag FDi. The induced drag
increases with increasing angle of attack. It is proportional to the inverse of the
square of velocity and so it is mostly important at low speeds. The induced drag
can be reduced by several design means (high aspect ratio wing, tapered wing,
twisted wing, winglets, ...).
2.2 Flight in nature
No matter how big progress has been made in aviation since the first powered flight of
the Wright brothers in 1903, the flight qualities and agility of modern aircraft remain
incomparable to flying animals that have evolved over several hundred millions of
years. When looking into nature, three types of flight can be observed: gliding flight,
flapping flight and hovering.
2.2.1 Gliding flight
In gliding flight the animal is moving forward and descending at the same time. The
necessary thrust to maintain the forward speed is produced by the gravity force. It
is a common flight technique for bats and larger birds, but gliding flight was also
observed among certain fish, frogs, reptiles or even squirrels. The aerodynamics of
most gliding animals can be described by the theory for fixed wing aircraft.
The ratio between the lift and drag is equal to the glide ratio, which relates the trav-
elled horizontal distance to the vertical descent. The best natural gliders, vultures
22 2 Flapping flight
CLmax trailing-edge flap (a)
a
b
c
d
leading-edge slat (b)
split flap (d)
slotted trailing edge flap (c)
symmetric airfoil
Figure 2.7: High lift devices used in aircraft and their equivalents in flying animals, from
Norberg (2002).
and albatrosses, can achieve glide ratios higher than 20:1 (Pennycuick, 1971). For
comparison, the best man made glider has a glide ratio of 70:1 (Flugtechnik & Le-
ichtbau, 2001). To keep the altitude some animals glide in ascending air that rises
due to convenient atmospheric conditions or due to terrain relief. Such flight is
called soaring.
Several mechanisms are used to increase the maximal lift coefficient of fixed wing
aircraft. These high lift devices, consisting of flaps and slats, are used particularly at
low flight speeds, i.e. during take-off and landing, see Figure 2.7 (top). Equivalent
mechanisms can be observed in natural flyers (Norberg, 2002). Bats can actively
control the camber of their wing (Figure 2.7 a), which increases the lift coefficient
and delays the stall. It is equivalent to trailing edge flaps and Kreuger flaps or
drooped leading edges used in aircraft. Stall can be further delayed by leading-edge
slats and slotted trailing edge flaps. The role of these devices, which deviate part
of the flow from below the wing above the wing, is to delay the stall by modifying
the pressure distribution above the wing and energizing the upper surface boundary
2.2 Flight in nature 23
layer. Birds can achieve similar effect by lifting their “thumb” with several feathers
on the leading edge (Figure 2.7 b). Equivalent to the slotted trailing edge flaps can
be observed in birds with long forked tail, which they can spread wide to help keep-
ing the flow attached even at high angles of attack (Figure 2.7 c). During landings,
a raised covert feathers can be observed in many birds (Figure 2.7 d). This self-
activated mechanism prevents backward flow of the turbulent air and delays flow
separation. It is similar to split flaps, used during aircraft landings to increase the
gliding angle.
Vortex generators in Sea Harrier
Wikimedia Commons, commons.wikimedia.org
de.academic.ru
a) Protruding digit in a bat wing
b) Serrated leading edge feather of an owl
c) Corrugated dragonfly wing
Figure 2.8: Vortex generators used in aircraft to introduce turbulence into the boundary
layer (left) and their equivalents in flying animals, from Norberg (2002); Neuweiler (2000),
(right).
Another way to delay the stall is to introduce turbulence in the boundary layer of the
upper wing surface. The turbulence helps to maintain an interchange of momentum
between the slow layers close to the wing and the free flow, so the flow separation
occurs at higher angles of attack. In aircraft, this is done by vortex generators, which
are typically placed close to the thickest part of the wing and distributed along the
span, see Figure 2.8 (left). A protruded digit on bat wing, serrated feathers at the
wing leading edge in owls and corrugated wings of dragonflies have the same role,
see Figure 2.8 (right). Apart from delaying the stall, this solution also reduces the
flight noise.
24 2 Flapping flight
Figure 2.9: Flow structures behind the wings in different flight modes, from Norberg
(1985).
2.2.2 Flapping forward flight
Apart from soaring, the animals need to actively produce thrust to stay airborne
without loosing altitude. They do so by flapping their wings. The flow around a
flapping wing is unsteady as the bound vortex gets distorted by the wing motion.
The unsteady effects get more important as the forward speed decreases compared
to the flapping speed.
This can be observed on the flow structures behind the wings (Figure 2.9). In gliding
flight a pair of tip vortices is observed as in fixed-wing aircraft (Figure 2.9 a). For
fast flapping flight the tip vortices remain continuous but undulate due to flapping
2.2 Flight in nature 25
(Figure 2.9 b). For slower speeds the downstroke becomes dominant in thrust gener-
ation. Transverse vortices are being periodically created on the trailing edge at the
beginning and at the end of each downstroke (Shyy et al., 2013). It is similar to the
starting vortex and to the shedding of bound vortex when a fixed wing starts and
stops to move, respectively. These transverse vortices connect with the two wing tip
vortices and a vortex ring is shed at the end of each downstroke (Figure 2.9 c).
With decreasing flight speed the animals adapt the flapping motion direction. The
flapping plane is almost vertical for cruising speeds but it inclines backwards as the
speed decreases (Figure 2.10). The body posture is also adapted.
6 m s–1 14 m s–1
10 m s–1 12 m s–1
8 m s–1
Figure 2.10: Wing-tip path of a pigeon flying at speeds of 6-14 m.s-1 . Adapted from
Tobalske and Dial (1996).
2.2.3 Hovering flight
Hovering flight can be mostly observed in insects and hummingbirds. While bats
(Muijres et al., 2008) and other birds (Tobalske et al., 1999) are also capable of
hovering, they only use it in transitions (taking off, landing, perching) as hovering
can require more than twice the power necessary for cruising (Dial et al., 1997).
Apart from hummingbirds, birds generate most of the lift during downstroke when
their wing is fully extended; they flex their wings in upstroke to reduce drag (Figure
2.11). We call this type of hovering asymmetric hovering (Norberg, 2002) or avian
stroke (Azuma, 2006).
Hummingbirds and many insects can hover for much longer periods as they use
symmetric hovering, also called insect stroke (Figures 2.12). The wings remain fully
extended throughout the wingbeat, but rotate and twist at the end of each half
stroke. Hummingbirds flap their wings almost horizontally (Figure 2.13 left) and
produce lift also during upstroke, about 25%-33%. The flapping plane of two-winged
insects can be slightly inclined (Figure 2.13 right); nevertheless, the upstroke gener-
ates up to 50% (Warrick et al., 2005, 2012).
26 2 Flapping flight
Downstroke
Upstroke
Time
Figure 2.11: Asymmetric hovering typical for birds and bats. Adapted from Azuma (2006).
Downstroke
Upstroke
Time
Figure 2.12: Symmetric hovering typical for hummingbirds (left) and insects (right).
Adapted from Greenewalt (1990).
Figure 2.13: Wingtip tra jectory in hovering hummingbirds and insects. Adapted from
Ellington (1984a,b).
2.2 Flight in nature 27
Hummingbird wing morphology differs from other birds as the the upper arm and
forearm bones are significantly shorter (Figure 2.14) and so the “hand” part of the
wing, called the handwing, is much larger: over 75% of wing area in hummingbirds
compared to about 50% in most birds (Warrick et al., 2012). On top of that the
wrist and the elbow cannot articulate; all the motion comes from the very mobile
shoulder. The wing is being moved by a pair of powerful muscles: a depressor mus-
cle powers the downstroke and an elevator the upstroke. The depressor is twice
as heavy as the elevator, which corresponds to the uneven lift production between
downstroke and upstroke mentioned earlier. The hummingbird muscles form up to
30% of the body weight (Greenewalt, 1990, p116).
Figure 2.14: Wing morphology: size of forelimb bones with human arm as a reference
(left) and handwing size (right, handwing in grey). The handwing is significantly larger in
hummingbirds. Adapted from Dial (1992); Warrick et al. (2012) and www.aokainc.com.
There are two ways how flapping motion is produced in insects (Dudley, 2002). Phy-
logenetically older insects use direct muscles to flap their wings (Figure 2.15 left).
They have two groups of muscles, the depressors and the elevators, that contract to
move the wing in downstroke and in upstroke, respectively. Direct drive is typical
for four-winged insects like dragonflies and damselflies. The brain controls each wing
independently, which makes their flight very agile, but it also limits their flapping
frequency, which is relatively low.
In phylogenetically modern insects, e.g. flies and bees, the wings are driven indi-
rectly by the deformation of thorax and by displacing the dorsal part of the thorax
called the notum (Figure 2.15 right). The upstroke is effected by contracting the
vertical muscles and lowering the notum. Longitudinal muscles are contracted in
downstroke to deform the thorax in longitudinal direction and subsequently raise
the notum. The thorax acts as a resonant system, so the animals can flap at much
higher frequencies, with greater amplitudes and the wings are always synchronized.
28 2 Flapping flight
Figure 2.15: Direct (left) and indirect insect flight muscles (right). From Hill et al. (2012).
Surprisingly, the maximum lift to muscle-weight ratio is constant among insects and
birds, despite their different evolution paths (Marden, 1987).
Hummingbirds and insects can combine precise hovering flight with fast cruising as
well as with backward flight. Figure 2.16 shows wing-tip path and body positions
of a hummingbird in all the mentioned flight modes. As in other birds, the flapping
plane is nearly vertical in cruising. It inclines backwards as the flight speed de-
creases becoming approximately horizontal when hovering. It inclines further back
to fly backwards. The wing-tip follows an oval pattern in most situations, but a
figure-of-eight pattern is used near hovering.
Figure 2.17 shows wing-tip paths of a bumblebee flying at different speeds. The
paths in this figure combine the flapping velocity with the downwash (the air moved
by the interaction with the flapping wings). The shape of these paths can be charac-
terized by a dimensionless ratio between the flight velocity and the (average) flapping
velocity called the advance ratio (Ellington, 1984b)
J=U
2ΦfR ,(2.5)
2.2 Flight in nature 29
Figure 2.16: Wing-tip paths of a hummingbird in forward, hovering and backward flight.
Adapted from Greenewalt (1990).
Figure 2.17: Wing-tip paths of a bumblebee at different flight speeds composed of flapping
velocity and downwash. The arrows represent the generated forces. The imbalance between
upstroke and downstroke path lengths and forces is characterized by the advance ratio J.
Figure from Ellington (1999).
30 2 Flapping flight
where Φ is the flapping amplitude, fthe flapping frequency and Rthe wing length.
Ellington defines hovering as flight with advance ratio Jbelow 0.1, where both up-
stroke and downstroke produce approximately equal amount of lift. We can observe
that as the advance ratio increases, the upstroke paths become more vertical and
shorter while downstroke paths more horizontal and longer. This signifies that for
higher advance ratios downstroke generates higher force, which is directed upwards
to provide lift, while the force during upstroke is smaller and is directed forwards to
provide thrust.
2.3 Hovering flapping flight aerodynamics
Because of the scope of this work, only symmetric hovering flight (ie. flight with
an advanced ratio Jless than 0.1) with a single pair of flapping wings is considered
further.
2.3.1 Dynamic scaling
The flow patterns over flapping wings can be characterized by several dimensionless
numbers (Shyy et al., 2013). The most important is the Reynolds number which
relates the inertial and viscous forces. For hovering flight it is defined as
Rehover =Uref Lref
ν=2ΦfR¯c
ν=4ΦfR2
νA,(2.6)
where νis the kinematic viscosity of air, the mean tip velocity, calculated as 2Φf R,
is taken as the reference speed Uref and mean chord ¯cas the reference length Lr ef .
The definition was also rewritten using the wing aspect ratio A=2R
¯c. We can see
that for flyers with similar flapping amplitudes Φ and aspect ratios Athe Reynolds
number is proportional to fR2. Typical Reynolds numbers for hovering flapping
flight lie between ∼10 and ∼10 000.
For forward flight the forward velocity U∞is usually taken as the reference velocity
and the definition becomes independent of flapping
Ref orward =U∞¯c
ν.(2.7)
Another dimensionless quantity, characterizing an oscillating flow, is the Strouhal
number. It relates the oscillatory and forward motion and is thus only applicable to
forward flight. It is defined as
St =f Lref
Uref
=ΦfR
U∞
,(2.8)
2.3 Hovering flapping flight aerodynamics 31
where the reference length is the distance travelled by the wing tip over one half-
stroke ΦR. The Strouhal numbers typical for most swimming and flying animals
are between 0.2 and 0.4 (Taylor et al., 2003).
The level of unsteadiness associated with flapping wings in hover can be character-
ized by the reduced frequency used for pitching and plunging airfoils. It is defined
as
khover =2πf Lref
Uref
=πf ¯c
2ΦfR =π
ΦA,(2.9)
where the reference velocity is again the mean tip velocity and the reference length is
half of the mean chord ¯c/2. The higher is the value of reduced frequency the greater
is the role of the unsteady effects on the aerodynamic force production. Morpholog-
ical data and dimensionless numbers of several hovering animals are in Table 2.1.
Chalcid Fruit Hawkmoth Rufous Giant
wasp fly hummingbird hummingbird
m[g] 2.6e-7 0.002 1.6 3.4 20
R[mm] 0.7 2.39 48.3 47 130
¯c[mm] 0.33 0.78 18.3 12 43
f[Hz] 370 218 26.1 43 15
Φ [◦] 120 140 115 116 120
A[-] 4.2 6.1 5.3 7.8 6
Re [-] 23 126 5885 6249 22353
k[-] 0.35 0.21 0.30 0.20 0.25
Table 2.1: Morphological parameters and dimensionless numbers of hovering flapping flight
in nature. Data taken from Shyy et al. (2010); Weis-Fogh (1973).
2.3.2 Lift enhancing aerodynamic mechanisms
When researchers tried to model insect aerodynamics using traditional, steady state
formulations in a quasi steady manner the models would grossly under-predict the
generated lift; some of the animals would hardly be able to take-off (Ellington,
1984a). This induced further research that revealed that the produced forces are en-
hanced by flow patterns of highly unsteady nature. The flow structures around the
wings involve periodic formation and shedding of vortices; they are still under active
research. Many key mechanisms were observed and identified both experimentally
and numerically, including the delayed stall of leading edge vortex, Kramer effect,
wake capture and clap-and-fling (Sane, 2003; Lehmann, 2004).
32 2 Flapping flight
Delayed stall of the leading edge vortex is the most important feature of the flow
around flapping wings. Insects operate their wings at very high angles of attack. As
it was described in Section 2.1 a fixed 2D airfoil operated at high angles of attack
stalls. The flow separates on the upper surface of the airfoil and the airfoil loses
lift due to smaller pressure difference between the regions above and below the wing.
Figure 2.18: Difference between a 2D translating wing (left) and 3D flapping wing in the
translation phase (right), from Sun and Wu (2003).
However, if an airfoil is started from still the leading edge vortex remains attached
during the first couple of chord lengths, which results in very high lift. This mech-
anism, first identified by Walker (1932), is called the delayed stall. A fixed wing
operated at low Reynolds numbers, which are typical for insects, would then start
to alternately shed trailing edge and leading edge vortices forming a wake pattern
known as the von Karman vortex street (Figure 2.18 left).
On the contrary, for a flapping wing the leading edge vortex remains stably attached
(Figure 2.18 right) as was first shown by Ellington et al. (1996) and confirmed by
Dickinson et al. (1999). The stability of the leading edge vortex is attributed to the
observed axial flow from the wing root to the wing tip (Ellington et al., 1996; van den
2.3 Hovering flapping flight aerodynamics 33
Figure 2.19: Leading edge vortex stabilized by axial flow, from van den Berg and Ellington
(1997)
Berg and Ellington, 1997; Usherwood and Ellington, 2002), see Figure 2.19, similarly
to low aspect ratio delta wings. This flow tends to be important for higher Reynolds
numbers, while being rather weak for lower Reynolds numbers (Birch et al., 2004),
nevertheless this seems to be sufficient for the leading edge vortex stability (Shyy
and Liu, 2007).
The leading edge vortex enhances not only the lift but also the drag force. It con-
tributes by about 7 to 16% to the bound circulation of a hummingbird wing (Warrick
et al., 2009) and by up to 40% to the circulation of slow flying bats (Muijres et al.,
2008).
The second mechanism that can enhance the lift production is the Kramer effect,
sometimes called the rapid pitch rotation (Shyy et al., 2010) or rotational forces
(Sane and Dickinson, 2002). As it was first demonstrated by Kramer (1932) in the
context of wing flutter, the lift of a fixed wing in steady flow will increase if the
wing rotates from low to high angle of attack. The span-wise rotation of the wing
causes that the stagnation point moves away from the trailing edge and as a result
additional circulation is generated to restore the Kutta condition. Depending on the
sense of rotation, this circulation is added to or subtracted from the bound vortex
circulation which results into positive or negative change of lift force, respectively.
34 2 Flapping flight
Similar mechanism occurs in flapping wings at the reversal point between strokes,
where the wing rotates rapidly along its span-wise axis. Studies of this phenomena
carried out by Dickinson et al. (1999); Sane and Dickinson (2002) showed that an
advanced rotation will enhance the lift force whereas a delayed rotation will cause
the lift to drop. Insects take advantage of this phenomena by timing the rotation
during manoeuvres (Dickinson et al., 1993).
Another mechanism related to the stroke reversal is the wake capture, sometimes
also called the wing-wake interaction. It was first demonstrated by Dickinson et al.
(1999) and further investigated by Birch and Dickinson (2003). As the wing reverses
it interacts with the shed vortices from the previous strokes (Figure 2.20). This
increases the relative flow speed and the transferred momentum results in higher
aerodynamic force just after reversal. The magnitude of this enhancement depends
strongly on the wing kinematics just before and just after the reversal.
Figure 2.20: Wake capture mechanism. Light blue arrows represent the generated force,
dark blue arrows show the flow direction, from Sane (2003)
The last aerodynamic mechanism enhancing the lift is called the clap-and-fling or
clap-and-peel (Weis-Fogh, 1973; Ellington, 1984b). It occurs only in animals that
touch their wings dorsally at the end of upstroke. In the ’clap’ phase the wings touch
first with their leading edges and keep rotating until also the trailing edges touch,
pushing the trapped air downwards, which generates additional thrust (Figure 2.21).
Once the wings start to ’fling’ apart a gap opens between the leading edges. The air
is sucked in which boosts the circulation build-up around the wings. Also, the start-
ing vortices eliminate each other which further enhances the circulation development.
Clap-and-fling mechanism was observed in multiple insect species (e.g. Weis-Fogh
(1973); Ellington (1984b); Zanker (1990)) and can enhance the lift by up to 25%
(Marden, 1987).
2.3 Hovering flapping flight aerodynamics 35
Figure 2.21: Clap and fling mechanism. Light blue arrows represent the generated force,
dark blue arrows show the flow direction, from Sane (2003).
Apart from purely aerodynamic mechanisms, the interaction of the flow and the
wing structure can also have a positive effect on the lift production (Shyy et al.,
2013; Tanaka et al., 2013). For example, an appropriate combination of chord- and
span-wise flexibility leads to a relative phase-advance of the wing rotation, resulting
into lift increase due to Kramer effect.
2.3.3 Flight stability
While lift generation is of a primary importance for flying animals they also need to
balance their body when facing perturbations coming from the wind or when ma-
noeuvring. Many works tried to identify whether this stability is inherent or whether
it is augmented by the sensory systems. To control the flight, insects can rely on
their vision (compound eyes and ocelli) as well as on airflow sensors (antennae and
wind sensitive hairs) and on inertial sensors (halteres), see Taylor and Krapp (2008).
Studying passive stability experimentally is complicated as breaking the feedback
loops by “deactivation” of the sensory systems leads to abnormal behaviour of the
animal. Thus, numerical treatment was preferred by most authors.
36 2 Flapping flight
The numerical studies employed aerodynamic models with various complexities
(CFD, quasi-steady aerodynamics). The studies considered hovering or forward
flight and covered both longitudinal and lateral directions of various insect species
differing in size and in wing kinematics (Sun et al., 2007; Xiong and Sun, 2008;
Zhang and Sun, 2010; Faruque and Humbert, 2010a,b; Orlowski and Girard, 2011;
Cheng and Deng, 2011). While minor differences in the predicted behaviour exist
especially in the lateral direction (Karasek and Preumont, 2012), the common con-
clusion is that the hovering flapping flight is inherently unstable and needs to be
actively controlled.
Experimental studies of near-hover flapping flight stability are sparse. Taylor and
Thomas (2003) performed experiments on a desert locust in forward flight and found
that it was unstable. However, the animal was tethered and it could use its sensory
systems. Hedrick et al. (2009) studied yaw turns in animals ranging from fruit flies
to large birds. They showed that the deceleration phase of a yaw turn can be ac-
complished passively, without any active control of the animal, thanks to damping
coming from the flapping motion, which they termed Flapping Counter Torque.
Recent works of Ristroph et al. (2010, 2013) studied the response of free flying fruit-
flies to an external disturbance in yaw and pitch, respectively. Tiny ferromagnets
were glued to the fly’s body so that it could be reoriented by a magnetic field while
being recorded by three high speed cameras. They observed that the fly used the
same wing kinematic changes as it would use for a voluntary manoeuvre, suggesting
it employs active auto-stabilization.
Figure 2.22: Demonstration of insect flight inherent instability: a) Halteres, biological vi-
brating gyroscopes in fruit-flies sensing the angular rates, before and after their deactivation,
b) histogram of flight trajectory angles of flies without halteres, before and after increasing
the passive damping, c) passive damping increased by fibres. From Ristroph et al. (2013).
2.3 Hovering flapping flight aerodynamics 37
In the next experiment they deactivated the halteres (insect gyroscopes), leaving
the fly with only the visual feedback, whose reaction is about four times slower.
This made the fly unable to fly as it fell nearly straight down, suggesting that the
flight is indeed inherently unstable. Nevertheless, it was possible to restore the
insects stability by attaching light dandelion fibres to its abdomen. This generated
sufficient damping, so that the insect could keep more or less the same orientation,
see Figure 2.22.
2.3.4 Attitude stabilization
Due to the inherent instability of flapping flight, hovering animals need to balance
their bodies actively. The attitude stabilization requires independent control of body
rotation around the roll (longitudinal) and pitch (lateral) axis. On top of that, turn-
ing requires control of rotation around the yaw (vertical) body axis. The necessary
moments are produced by introducing small asymmetries into otherwise symmetric
wing motion.
Figure 2.23: Pitch moment generation in insects: a) via angle of attack asymmetry, b) via
mean wing position. Adapted from Conn et al. (2011).
For pitching the animal needs to shift the centre of lift in fore/aft direction (Elling-
ton, 1999). This shift can be realized by moving the maximal/minimal wing stroke
positions (the mean stroke angle) and/or by a difference in the angle of attack during
upstroke and downstroke (Dudley, 2002), see Figure 2.23. The former was observed
in free flying fruit flies during the auto-stabilization after an externally triggered
pitch perturbation (Ristroph et al., 2013) and also in tethered fruit flies (Zanker,
1988).
38 2 Flapping flight
Figure 2.24: Roll moment generation in insects: a) via angle of attack difference, b) via
flapping amplitude difference. Adapted from Conn et al. (2011).
Roll can be initiated by introducing an asymmetry between the lift of the left and
right wing. Insects achieve this by increasing the flapping amplitude and/or by
modifying the angle of attack on one wing (Ellington, 1999), as it was observed in
tethered fruit-flyes (Hengstenberg et al., 1986), see Figure 2.24.
Finally yawing can be effected by increasing the drag force on one of the wings. A
difference in angle of attack while yawing was observed by Ellington (1999), see Fig-
ure 2.25. The same was reported for fruitflies by Bergou et al. (2010), together with
significant asymmetry of mean stroke angles. Nevertheless, they attributed 98% of
the yaw moment to the angle of attack difference. Interestingly, a different strategy
in fruit-fly yaw turns was reported by Fry et al. (2003). They observed a backward
tilt of stroke plane together with an increase of amplitude on the outside wing.
Figure 2.25: Yaw moment generation in insects via angle of attack asymmetry. Adapted
from Conn et al. (2011).
2.3 Hovering flapping flight aerodynamics 39
Apart from wings also other body parts contribute to the overall torques produced.
Zanker (1988) observed lateral deflection of the abdomen that should increase the
drag on one side during visually simulated yaw turns in tethered fruit-flies. A
(smaller) dorso-ventral deflection was reported while pitching. The drag can be
increased further by hindlegs (Zanker et al., 1991). Video footages of flying hum-
mingbirds also reveal that many species use their tail, in addition to wing motion
and stroke plane changes, to control their body rotation when manoeuvring, see
Figure 2.26.
Figure 2.26: Rufous hummingbird flying backwards. The interval between the dis-
played positions is 3 wingbeats, each position is a composite of two frames to show
the wing limit positions and the stroke plane direction. Original video footage,
http://youtu.be/Cly6Y69WOYk, courtesy of JCM Digital Imaging (http://jcmdi.com).
40 References
2.3.5 Flight control
Flying animals control their speed by modulating the thrust force. Hummingbirds
can increase the produced thrust by increasing the stroke amplitude and at a smaller
rate also the flapping frequency, as it was documented by Altshuler and Dudley
(2003) by load lifting. The same was observed by Chai et al. (1996) by reducing
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0303-1.
Chapter 3
Mathematical modelling
In this chapter a dynamic model of hovering flapping flight is introduced. This model
serves as a basis for stability study of flapping flight as well as for control design
and flight simulations of the designed hummingbird robot. Its goal is to provide
reasonable estimates of the generated aerodynamic forces and moments rather than
a detailed analysis of the flow around the flapping wings. Thus, a computationally
efficient model with low complexity was selected. The robot is modelled as a rigid
body and a quasi-steady modelling approach is used to evaluate the aerodynamic
forces of the flapping wings.
3.1 Flapping flight aerodynamics
The unsteady nature of the flow mechanisms in flapping flight has already been dis-
cussed in the previous chapter. Hence, proper modelling of flapping flight requires
a numerical solution of Navier-Stokes equations. Such a solution is usually very
complex due to moving boundaries and 3D geometry. On top of that real wings
are being twisted and flexed under the aerodynamic loads, which means the Navier-
Stokes model should be coupled with a model of structural dynamics.
Many CFD simulations were used to study the aerodynamics of hovering insects of
different sizes, some considering also flexible wings. A thorough review of published
results was carried out by Shyy et al. (2013). However, these models require signif-
icant computational power and their implementation is very complex, which limits
their use in flight simulations and in control design.
An alternative to complex models can be the quasi steady modelling approach (Sane
and Dickinson, 2002). Unlike in a proper unsteady treatment, the quasi-steady
model assumes that the instantaneous forces depend only on the wing motion at
47
48 3 Mathematical modelling
the given instant. The model was derived from steady flow thin airfoil theory and
from theory originally developed for flutter analyses (Sedov, 1965; Fung, 1969). It
consists of relatively simple analytic equations with force coefficients from Dickinson
et al. (1999) that partially include the effect of delayed stall, which was discussed
in Section 2.3.2. It is thus possible to predict the time history of wing forces for
any kinematic pattern with low demands on computation power. Despite many sim-
plifications, it will be demonstrated that this model provides cycle-averaged results
comparable to a CFD simulation when assessing hovering flight stability.
3.1.1 Wing kinematics
The flapping motion of hummingbird and insect wings has three DOFs (Figure 3.1).
The principal motion is an oscillatory motion with frequency f(the wingbeat fre-
quency) occurring in the stroke plane xSPySP . The stroke plane is inclined from the
α*
δ
αg
xSP
OB
OSP
zSP
xB
zB
stroke plane
xSPySP
β
zWαg
α*
stroke plane
xSPySP
ϕ
ϕ
xSP
ySP
ϕ
xSP
OB
OSP
zSP
β
ySP
xW
Figure 3.1: Angles defining the wing position.
3.1 Flapping flight aerodynamics 49
body horizontal plane xByBby angle β. The wing position inside the stroke plane
is defined by the sweep angle φ. One stroke consists of two consecutive half-strokes:
the downstroke (dorso-ventral motion, ˙
φ < 0) and the upstroke (ventro-dorsal mo-
tion, ˙
φ > 0). At the reversal between each half-stroke the wing rotates along its
longitudinal axis, its orientation is given by the wing inclination angle α∗measured
between the wing chord and the normal to the stroke plane. The geometric angle of
attack αg, measured from the stroke plane, is thus defined as αg=π/2− |α∗|. The
rotation is termed pronation and supination on the dorsal and ventral side, respec-
tively. The wings can also deviate from the stroke plane by the deviation angle δ,
which is measured with respect to wing longitudinal axis.
The wing motion can be parametrised, similar to Berman and Wang (2007), in the
following way:
φ=φ0+φm
arcsin(kφ)arcsin [kφcos (2πf t)]
α∗=α0+π/2−αm
tanh(kα)tanh [kαsin(2πf t −ϕα)] (3.1)
δ=δm1sin(2πf t) + δm2sin(4πf t),
tis time, φ0sweep offset (mean position), φmsweep (flapping) amplitude, αmgeo-
metric angle of attack around mid-stroke, α0inclination offset, ϕαthe phase shift
between the sweep and the inclination, δ1deviation amplitude of an oval pattern and
δ2deviation amplitude of a figure-of-eight pattern. The parameter kφdefines the
shape of sweep angle function φ(t) from harmonic (kφ→0) to triangular (kφ= 1).
Similarly, the parameter kαchanges the shape of the inclination angle function
α∗(t) from harmonic (kα→0) to square wave (kα→ ∞). The parameter effects are
demonstrated in Figures 3.2 and 3.3.
δm2 ϕ0
ϕm
β
αm+α0
αm-α0
Downstroke
Upstroke
δm1
Figure 3.2: Parameters defining the wing kinematics.
50 3 Mathematical modelling
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1.5
−1
−0.5
0
0.5
1
stroke cycle [−]
φ/φm [−]
Sweep angle − shape parameter kφ
0 0.9 0.99 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−1.5
−1
−0.5
0
0.5
1
stroke cycle [−]
α*/α*m [−]
Inclination angle − shape parameter kα
0 2 5 inf
Figure 3.3: Parameters kφand kαdefining the shape of sweep and inclination angle
functions.
3.1.2 Quasi-steady aerodynamics
The quasi-steady model consists of three force components: force due to wing trans-
lation with normal and tangential part FNtr and FT tr , force due to wing rotation
FNr and force due to the inertia of added mass of the fluid FNa. The directions of
the normal and tangential force components and their relation to the lift and drag
are shown in Figure 3.4.
Figure 3.4: Different components of the total aerodynamic force.
In this work a flat and rigid wing of an arbitrary form is assumed. The important
wing geometry parameters are displayed in Figure 3.5. Ris the wing length, cis
the mean chord length, S=Rc is the surface of a single wing. A chord section at
the distance rfrom the wing root has a lenght c. ˆx0is the non-dimensional posi-
tion of the rotational axis. The aspect ratio of the wing is defined as A=2R2
S=2R
¯c.
3.1 Flapping flight aerodynamics 51
dr
r̂
2R
Centre of pressure
Wing centroid
c̄/2
c̄/2 c̄/4
Axis of wing rotation
R
x̂
0c̄
r
c(r)
Wing root
Figure 3.5: Wing geometry parameters
3.1.2.1 Force due to wing translation
The lift force due to translation of a 2D airfoil can be expressed according to an
equation for fixed wing in steady flow as
F0
Ltr =1
2ρCL(α)cU2
∞,(3.2)
where ρis the air density and CLis the lift coefficient.
If we consider hovering flapping flight and wing motion with 2 DOF the section
speed can be expressed as U∞(r) = r˙
φ. For a wing with length Rand mean chord
¯cwe introduce the non-dimensional section position and the non-dimensional chord
length as ˆr=r
Rand ˆc=c
¯c, respectively (Ellington, 1984a). The force of the whole
wing is given by an integral of the section force over the wing span
FLtr =ZR
0
F0
Ltr(r)dr =
=1
2ρCL(α)˙
φ2ZR
0
c(r)r2dr =
=1
2ρCL(α)˙
φ2R3¯cZ1
0
ˆc(ˆr)ˆr2dˆr=
=1
2ρCL(α)Sˆr2
2R2˙
φ2,(3.3)
where ˆr2
2=R1
0ˆc(ˆr)ˆr2dˆris the non-dimensional radius of the second moment of iner-
tia. It defines the span-wise position of the centre of pressure where the forces with
52 3 Mathematical modelling
circulatory origin act. Its chord-wise position is assumed, similar to other studies,
to be at one quarter of the chord from the leading edge (Figure 3.5), which is a
simplification as Dickson et al. (2006) showed experimentally on a model wing that
the location of the centre of pressure depends on the angle of attack.
The lift can also be formulated as a function of the centre of pressure speed UCP =
ˆr2
2R2˙
φ2as
FLtr =1
2ρCL(α)SU 2
CP .(3.4)
Similar formulation can be written for the drag force
FDtr =1
2ρCD(α)SU 2
CP .(3.5)
Since the remaining force components to be introduced in the next sections are all
of a pressure origin and thus act normally to the wing chord it is convenient to
transform the above lift and drag forces also into normal and tangential component
(Deng et al., 2006)
FNtr =1
2ρCN(α)SU 2
CP (3.6)
FT tr =1
2ρCT(α)SU 2
CP .(3.7)
The normal and tangential force coefficients CNand CTwere obtained by trans-
formation of the empirical coefficients of Dickinson et al. (1999), which include the
effect of the delayed stall of the leading edge vortex, and have the following form
CN(α)=3.4 sin(α) (3.8)
CT(α) =
0.4 cos2(2α) 0 ≤ |α|<π
4
0π
4≤ |α|<3π
4
−0.4 cos2(2α)3π
4≤ |α|< π.
(3.9)
3.1.2.2 Force due to wing rotation
Another force component that contributes to the total force generated by a flapping
wing is the force due to wing rotation. It was derived from quasi-steady equations
used for flutter analysis (e.g. Fung, 1969). A translating wing that also starts to
rotate will develop additional circulation around itself to satisfy the Kutta condition
(Ellington, 1984b). With the notation of Figure 3.5 this circulation can be expressed
as
Γr=π˙αc23
4−ˆx0.(3.10)
3.1 Flapping flight aerodynamics 53
For a 2D airfoil the force is defined as
F0
Nr(r) = ρU∞Γr=πρc23
4−ˆx0U∞˙α. (3.11)
As a force of circulatory origin it acts at the centre of pressure and normally to the
wing.
The total wing force is again obtained by integration over the wingspan while sub-
stituting U∞(r) = r˙
φand introducing the non-dimensional position ˆrand chord ˆc
FNr =ZR
0
F0
Nrdr =
=πρ 3
4−ˆx0˙α˙
φR2¯c2Z1
0
ˆc2(r)ˆrdˆr. (3.12)
This can be rewritten further using the centre of pressure speed UCP as
FNr =ρπ 3
4−ˆx0˙αUCP
R¯c2
ˆr2Z1
0
ˆc2(r)ˆrdˆr. (3.13)
3.1.2.3 Force due to the inertia of added mass
The force due to the inertia of the added mass off the fluid can be formulated with
the help of equations originally introduced for flutter analysis. According to Fung
(1969) this force consists of two components: a force due to the acceleration of an
apparent mass and a centrifugal force.
The apparent mass acceleration force acts at the mid-chord (Figure 3.6). It is defined
as the apparent mass of a cylindrical section ρπ c2
4times the normal acceleration
c/2
c/2
xr
C
rϕ
-rϕ
.
A
C'
A'
α
α*
aN
FNa
Figure 3.6: Wing section and parameters for calculation of the added mass force.
54 3 Mathematical modelling
aNof the mid-chord. Assuming hovering flight and wing motion with 2 DOF the
acceleration of a wing section at a distance rfrom the wing root can be written as
aN(r) = xr¨α−r¨
φsin α, (3.14)
where xris the distance of the axis of rotation A to the mid-chord C (Figure 3.6).
Assuming further that this distance is equal to a quarter chord (xr=c
4) the force
becomes
F0
Na(r) = ρπ c2(r)
4aN(r) = ρπ c2(r)
4c(r)
4¨α−r¨
φsin α.(3.15)
The centrifugal force is a force of circulatory origin and is located at 3/4 of the
chord from the leading edge. It is defined as the apparent mass times U˙α, where U
is the component of relative wind speed in chord direction. For hovering flight with
2 DOF the wing section force can be written as
F0
Nc(r) = ρπ c2(r)
4−r˙
φcos α˙α. (3.16)
The total force is obtained by integration of the section forces over the wingspan
while also introducing the non-dimensional position ˆrand chord ˆc
FNa =ZR
0F0
Na(r) + F0
Nc(r)dr =
=−1
4ρπR2¯c2¨
φsin α+˙
φ˙αcos αZ1
0
ˆc2ˆrdˆr+1
16ρπR¯c3¨αZ1
0
ˆc3dˆr. (3.17)
This formulation was presented by Maybury and Lehmann (2004), which is the
formulation from Sane and Dickinson (2001) with additional corrections. Compared
to Maybury and Lehmann (2004) the terms including derivations of sweep angle φ
in our formulation differ in sign. This is due to an opposite definition of the positive
direction of φ.
3.1.2.4 Total force
Since the effects of the added mass inertia are relatively small compared to the other
two components it will be neglected further in the text for simplicity. However, its
effects will be demonstrated at the end of this section. Thus, the total force of the
flapping wing is a sum of forces due to translation and rotation
FN=FNtr +FNr =1
2ρCN(α)SU 2
CP +ρπ 3
4−ˆx0˙αUCP
R¯c2
ˆr2Z1
0
ˆc2(ˆr)ˆrdˆr
FT=FT tr =1
2ρCT(α)SU 2
CP .(3.18)
3.1 Flapping flight aerodynamics 55
For a wing with given form the force varies only with the velocity of the centre of
pressure UCP and with the angle of attack α. The angle of attack is defined in Figure
3.7 as the angle between the negative xWaxis of the wing and the UCP velocity vec-
tor. It can be either positive or negative, with values from the interval h−π, πi. The
normal component FNacts in zWdirection (thus for hovering flight we get positive
values in downstroke and negative in upstroke) and the tangential component FT
acts in xWdirection (in hover it is always positive or zero). The expressions for the
centre of pressure velocity and for the angle of attack for arbitrary flight conditions
will be derived in the following section.
xW
zW
UCP
FNα
+α
FT
Downstroke Upstroke
α*
+α*
stroke plane
xSPySP CP zB
xB
OB
+α +α*
α
α*
UCP
zW
xW
FT
FN
CP
stroke plane
xSPySP
Figure 3.7: Angle of attack αand orientation of normal, FN, and tangential, FT, force
component in upstroke and downstroke (hovering flight). The figure displays right wing
section and considers β= 0 and δ= 0 for clarity.
3.1.3 Centre of pressure velocity and angle of attack
For hovering flight (body velocities and angular velocities are zero) and zero devi-
ation from the stroke plane the aerodynamic angle of attack can be approximated
as α= sign (UCP )π/2−α∗(see Figure 3.7). Assuming further that the centre of
pressure (CP) lies on the span-wise rotation axis, its velocity is given by relation
UCP = ˆr2R˙
φ.
For a general case, when the body is in motion, the CP velocity is modified. Since
not only the magnitude but also the direction of CP velocity changes, the aerody-
namic angle of attack is also affected as can be seen in Figure 3.8 a). Situations,
where angle of attack is negative or greater than 90◦are sketched in 3.8 b) and 3.8
c) respectively. Things get even more complicated when the wing deviates from the
stroke plane and the rotation axis is not going through the CP.
56 3 Mathematical modelling
xW
zW
CP
UCP
α
+α
velocity due
to flapping
velocity due
to body motion
stroke plane
zWxW
CP
+α
α
UCP
zW
xW
CP
+α
α
UCP
a) b) c)
Figure 3.8: Influence of body motion on the angle of attack; the figures represent the
right wing cross-section and consider β= 0 and δ= 0 for clarity. a) Decomposition of the
wing velocity into components due to flapping and due to body motion. b) Situation with
negative angle of attack. c) Situation with angle of attack greater than 90◦.
The motion of the wing with respect to the surrounding air can be decomposed into
a series of simultaneous motions. For their description we introduce three coordi-
nate frames displayed in Figure 3.9 a) - global (inertial) frame G, body-fixed frame
B and wing-fixed frame W. A left superscript with the frame letter will be used to
indicate the coordinate system in which a vector or matrix is expressed. The origin
of the B frame is in the centre of gravity of the bird. In hover its axes are parallel
to global frame axes with zBaxis pointing towards the sky, xBaxis in the backward
flight direction and yBaxis in the direction of the right wing. The origin of the
wing frame W is placed into the wing-root. The yWaxis is aligned with the wing
spanwise rotation axis, xWaxis is parallel to the chord and pointing towards the
trailing edge of the wing and zWaxis is normal to the wing surface, positive on the
dorsal side of the wing.
OB
CP
xG
zG
yG
OG
U
rcOW
rw
v
ω
r
ωw
OB
xW
zW
yW
CP
xB
zB
yB
xG
zG
yG
OG
OW
a) b)
Figure 3.9: Wing kinematics description. a) Introduced coordinate frames: global frame
G, body-fixed frame B and wing-fixed frame W. b) Centre of pressure position and velocity
as a result of simultaneous motions.
3.1 Flapping flight aerodynamics 57
Any vector in the W frame can be transformed to the B frame by the rotation matrix
R, which is a function of the wing position angles from Section 3.1.1. For the right
wing the matrix is defined as
R=Ry(−β)Rz(−φ)Rx(δ)Ry(π
2−α∗).(3.19)
The opposite vector transformation (from B frame to W frame) is given by a trans-
pose of the above matrix RT. The angular velocity between the wing and the body
is, in skew-symmetric form, given by the derivation of the rotation matrix
Ωw=
0−ωwz ωwy
ωwz 0−ωwx
−ωwy ωwx 0
=˙
R.(3.20)
For wings flapping in a horizontal flapping plane with no deviation (β= 0, δ= 0)
we obtain
R=
cos φsin α∗sin φcos φcos α∗
−sin φsin α∗cos φ−sin φcos α∗
−cos α∗0 sin α∗
(3.21)
ωw=
ωwx
ωwy
ωwz
=
˙
φcos α∗
−˙α∗
−˙
φsin α∗
.(3.22)
The CP velocity is a resultant of body absolute motion (with velocity v= [u, v, w]
and angular velocity ω= [p, q, r]) and the wing rotation relative to the body ωw,
see Figure 3.9 b). By using the theory for simultaneous motion we can express the
velocity of CP of the wing in W frame as
U=RT(v+ω×r) + ωw×rc,(3.23)
where rc=(1
4−ˆx0)c, ±ˆr2R, 0Tis the position of the CP in the right/left wing
frame, respectively, and ris the CP position in the body frame defined as
r=rw+Rrc,(3.24)
where rw= [xw,±e/2, zw]Tis the right/left shoulder position in the body frame,
respectively. eis the width of the chest, i.e. the distance between the shoulders.
The angle of attack of the wing is measured between the wing chord and the relative
velocity vector of the wing with respect to the surrounding air (Figure 3.8). Here
the angle of attack is approximated to be constant along the wingspan and the value
58 3 Mathematical modelling
for the CP is used. The magnitude of the CP velocity vector in xWzWplane of the
wing is
UCP =pU2
x+U2
z.(3.25)
The effect of the spanwise component Uyon the aerodynamics is not considered.
According to Figure 3.8 the angle of attack can be computed as
α= atan2 (−Uz,−Ux),(3.26)
where the atan2 is a four-quadrant inverse tangent function returning values between
−πand π(MathWorks, 2014). The time derivative of the angle of attack can be
derived from the above equation as
˙α=˙
UzUx−Uz˙
Ux
U2
x+U2
z
.(3.27)
The velocity derivations in the formula represent the change of CP velocity vector
in the wing frame and thus need to be carried out in that frame.
3.1.4 Comparison with CFD
To justify the use of quasi-steady aerodynamics and to show the importance of the
force component due to rotation, the instantaneous forces calculated by our model
are compared to CFD data. Lift, drag and side force curves from CFD study by
Zhang and Sun (2010a) are plotted in Figure 3.10 together with the curves result-
ing from the quasi-steady model. To demonstrate the degree of importance of the
quasi-steady model components, first only the force due to translation (3.6-3.7) is
considered and the model is called ASL (tr). In model ASL (tr+rot) the force due to
rotation (3.13) is also included. Finally in model ASL (tr+rot+add) the last com-
ponent due to the inertia of the added mass, similar to (3.17), is considered. The
details on the wing kinematics and morphology used are in (Karasek and Preumont,
2012).
The forces are presented in the form of non-dimensional force coefficients defined as
CL=FL/(0.5ρS ¯
U2
CP ) and similarly for CDand CY.¯
UCP is the mean CP velocity
defined as ¯
UCP = 4φmfˆr2R. Lift FLis the force that is normal to the stroke plane.
Drag FDlies inside the stroke plane and represents the force in an opposite direction
to the wing instantaneous motion. Finally the side force FYrepresents the yBaxis
component of the drag force. The curves start at cycle time t+= 0, which is the
point of reversal between upstroke and downstroke, the cycle is completed at t+= 1.
3.1 Flapping flight aerodynamics 59
Lift Drag Y force
ASL (tr)ASL (tr+rot)
ASL (tr+rot+add)
CFD - Zhang&Sun
0 0.25 0.5 0.75 1
1
0
1
2
3
4
CL[ ]
0 0.25 0.5 0.75 1
2
1
0
1
2
3
CD[ ]
0 0.25 0.5 0.75 1
2
1
0
1
2
CY[ ]
0 0.25 0.5 0.75 1
1
0
1
2
3
4
CL[ ]
0 0.25 0.5 0.75 1
2
1
0
1
2
3
CD[ ]
0 0.25 0.5 0.75 1
2
1
0
1
2
CY[ ]
0 0.25 0.5 0.75 1
1
0
1
2
3
4
CL[ ]
0 0.25 0.5 0.75 1
2
1
0
1
2
3
CD[ ]
Flight Sideways Left Wing Flight Sideways Right Wing Reference Flight
0 0.25 0.5 0.75 1
2
1
0
1
2
CY[ ]
0 0.25 0.5 0.75 1
1
0
1
2
3
4
CL[ ]
t+[ ]
0 0.25 0.5 0.75 1
2
1
0
1
2
3
t+[]
CD[ ]
0 0.25 0.5 0.75 1
2
1
0
1
2
CY[ ]
t+[ ]
Figure 3.10: Instantaneous forces over one flapping cycle predicted with quasi-steady
model with increasing number of components and compared to CFD results of Zhang and
Sun (2010a).
60 3 Mathematical modelling
Apart from the wing forces in hovering conditions (dash-dotted line, lift and drag
identical for both wings), the left and right wing forces in sideways-flight with speed
v= 0.15 ¯
UCP are also displayed. Good agreement of the force traces when the sys-
tem is slightly deviated from its equilibrium is important for assessing the system
stability, which will be studied in the following section.
When we compare the CFD results (Figure 3.10, 4th row) to the force curves from the
translational component model ASL (tr) (Figure 3.10, 1st row) it is clear, that such
a model is insufficient. On the contrary, the curves obtained by the ASL (tr+rot)
model (Figure 3.10, 2nd row) show much better agreement with the CFD results.
The rotational force component is responsible for the positive lift peak before and
negative lift peak after the stroke reversal. Similar peaks can be observed in drag
and in the side force (both positive). Major differences occur after the reversal,
where the forces predicted by the ASL (tr+rot) model drop, while positive peaks
can be observed in the CFD force traces. The differences might be, at least partially,
explained by the wake capture mechanism that is included only in the CFD model.
The addition of the added mass inertia component increases the magnitude of the
curves and further modifies the behaviour around the stroke reversal (Figure 3.10,
3rd row). The accordance with the CFD results is slightly improved in some parts
(lift peak and side force sign after the reversal) while in other parts the accordance
gets worse (non-zero lift and drag force at the reversal). Thus, the overall improve-
ment over the ASL (tr+rot) model force traces is insignificant, if any.
All the quasi-steady models over predict the CFD force magnitudes by about 20%.
Considering that the used quasi-steady force coefficients were determined empirically
for a wing with different geometry and under different conditions, this difference is
still acceptable. Regardless the magnitudes, the shape of the force traces and the
trend of their changes, important for studying the stability and control of flapping
flight, are captured with good accordance when the force due to rotation is used.
3.2 Body dynamics
The dynamics of the flying robot can be described, under rigid body assumption,
by Newton-Euler motion equations. Similar to an aircraft we obtain 12 ordinary
differential equations with 12 unknown coordinates - velocity (u, v, w), angular ve-
locity (p, q, r), position (x, y, z) and orientation expressed by Roll-Pitch-Yaw angles
(ϕ, ϑ, ψ) - see Figure 3.11. By omitting the equations for position and heading (yaw)
3.2 Body dynamics 61
angle ψthe system is reduced to 8 equations
˙u=−(wq −vr) + X/m +gsin ϑ
˙v=−(ur −wp) + Y/m −gcos ϑsin ϕ
˙w=−(vp −uq) + Z/m −gcos ϑcos ϕ
Ixx ˙p= (Iyy −Izz)qr +Ixz ( ˙r+pq) + L
Iyy ˙q= (Izz −Ixx)pr +Ixz r2−p2+M
Izz ˙r= (Ixx −Iyy )pq +Ixz ( ˙p−qr) + N
˙ϕ=p+qsin ϕtan ϑ+rcos ϕtan ϑ
˙
ϑ=qcos ϕ−rsin ϕ, (3.28)
where mis the body mass and Ixx ,Iyy ,Izz and Ixz are the non-zero moments and
product of inertia in body frame (products Ixy and Iyz are both zero due to body
symmetry). Aerodynamic forces and moments are represented by vectors (X, Y, Z )
and (L, M, N ) respectively.
xB
zB
yB
u, X
v, Y
w, Z
φ
p, L
ψ
r, N
ϑ
q, M
OB
Yaw
Roll
Pitch
Figure 3.11: Definition of body coordinates
There are two main types of aerodynamic forces to be considered: wing forces and
drag of body, legs and tail. According to Ristroph et al. (2013) the body drag force
becomes important only in miligram-scale insects and legs and tail need to be con-
sidered only in specific species. Xiong and Sun (2008) showed, that the body effects
become significant only for higher forward speeds. Thus, all drag-type forces are
neglected and only the wing forces are considered further.
62 3 Mathematical modelling
The wing forces (3.18) are transformed into the body frame as follows
[X, Y, Z]T=X
i
[Xi, Yi, Zi]T=X
i
Ri[FT i,0, FN i ]T(3.29)
[L, M, N ]T=X
i
rci ×[Xi, Yi, Zi]T,(3.30)
where index istands for the left and the right wing.
3.2.1 System linearisation
The mathematical model introduced in the previous sections is nonlinear and was
used in flight simulations. For stability studies and control design a linear model is
preferred.
The system dynamics (3.28) include aerodynamic forces and moments (3.29, 3.30)
that are functions of wing motion parameters p= [f,βL,φmL,φ0L,αmL ,α0L,ϕαL,
δm1L,δm2L,βR,···,δm2R]T, system state x= [u,v,w,p,q,r,ϕ,ϑ]Tand time t.
Assuming the flapping frequency is much higher than the bandwidth of the system,
the aerodynamic forces can be replaced by their cycle averaged values (mean values
over one wingbeat), e.g.
X=Z1
f
0
X(x,p, t)dt = X(x,p),(3.31)
that depend only on xand p. Small perturbation theory is used to rewrite the states
and wing motion parameters as
x=xe+δx,p=pe+δp,(3.32)
where subscript esignifies the equilibrium values and δsymbol stands for the per-
turbation. The aerodynamic forces and moments are approximated by the linear
terms of Taylor’s expansion. For the x-axis force we obtain
X(x,p) = Xe(xe,pe) +
6
X
i=1
∂X
∂xi
(xe,pe)δxi+
n
X
j=1
∂X
∂pj
(xe,pe)δpj,(3.33)
where Xeis the cycle averaged force generated in equilibrium and nis the number
of wing kinematic parameters. The terms of the first summation are the derivatives
with respect to body velocities and angular velocities called the stability derivatives.
If taken with an opposite sign they represent aerodynamic damping. The second
summation terms are the derivatives with respect to changes in wing motion. They
3.2 Body dynamics 63
are called the control derivatives. Further the overbar notation for cycle averages is
dropped and the notation of the derivatives is shortened in the following manner
∂X
∂u =Xu,∂X
∂v =Xv, . . . , ∂X
∂f =Xf,∂X
∂β =Xβ, . . . (3.34)
Since only the near hover flight is considered, all the equilibrium states are zero
(ue=ve=we=pe=qe=re=ϕe=ϑe= 0) and the perturbed states are equal to
their absolute values (δx=x). The wing motion parameters pemust ensure that
the z-force is in balance with the gravity force (Ze=mg), while the remaining forces
and moments are zero (Xe=Ye=Le=Me=Ne= 0).
First it is supposed that the wing kinematics does not change (δp=0). Instead, an
arbitrary external force [Xext, Yext , Zext] or moment [Mext, Next, Lext] can be applied
on the body. According to previous works on passive stability, e.g. Taylor and
Thomas (2003); Zhang and Sun (2010a), as well as to our results there exists no
aerodynamic coupling between the longitudinal and lateral system. By neglecting
the second order terms, the equations can be rewritten as two linear subsystems
represented in state space as
˙u
˙w
˙q
˙
ϑ
=Along
u
w
q
ϑ
+Blong
Xext
Zext
Mext
(3.35)
˙v
˙p
˙r
˙ϕ
=Alat
v
p
r
ϕ
+Blat
Yext
Lext
Next
,(3.36)
where the system matrices are
Along =
Xu
m
Xw
m
Xq
mg
Zu
m
Zw
m
Zq
m0
Mu
Iyy
Mw
Iyy
Mq
Iyy 0
0 0 1 0
(3.37)
Alat =
Yv
m
Yp
m
Yr
m−g
LvIzz +NvIxz
IxxIzz −I2
xz
LpIzz +NpIxz
IxxIzz −I2
xz
LrIzz +NrIxz
IxxIzz −I2
xz 0
LvIxz+NvIxx
IxxIzz −I2
xz
LpIxz+NpIxx
IxxIzz −I2
xz
LrIxz+NrIxx
IxxIzz −I2
xz 0
0 1 0 0
(3.38)
64 3 Mathematical modelling
and the control matrices are
Blong =
1
m0 0
01
m0
0 0 1
Iyy
0 0 0
(3.39)
Blat =
1
m0 0
0Izz
IxxIzz −I2
xz
Ixz
IxxIzz −I2
xz
0Ixz
IxxIzz −I2
xz
Ixx
IxxIzz −I2
xz
0 0 0
.(3.40)
In reality, the control forces and moments will be generated by the wings. From the
approximation in (3.33) a relation between the vector of cycle averaged forces and
moments and the modifications of wing kinematics parameters ∆p can be written
as
[X, Y, Z, L, M, N ]T=J∆p,(3.41)
where Jis the matrix of control derivatives defined as
J=
Xp1Xp2··· Xpn
Yp1Yp2··· Ypn
.
.
..
.
.....
.
.
Np1Np2··· Npn
.(3.42)
The kinematic parameters modifications that should produce desired forces/moments
are estimated by a pseudoinverse of the previous relation
∆p =J+[X, Y, Z, L, M, N ]T.(3.43)
3.2.2 Stability and control derivatives
The stability and control derivatives are computed numerically. For example, the
value of Xucan be obtained from the mean force ¯
Xcomputed for small positive and
for small negative forward velocity ∆uand −∆u, respectively, while keeping other
states xand wing motion parameters in their equilibrium values xeand pe.
Xu =∂¯
X
∂u ∼
=¯
X(xe+∆u,pe)−¯
X(xe−∆u,pe)
2∆u,(3.44)
where ∆u = [∆u, 0,0,0,0,0]. Other stability and control derivatives are obtained
in a similar manner.
3.3 Reduced model of a flapping wing MAV 65
3.3 Reduced model of a flapping wing MAV
In hovering flapping flight, many of the stability derivatives can be dropped. As-
suming symmetric wing motion and wing shoulders placed in the zByBplane of the
body, derivatives Xw,Zu,Zqand Mwwill be zero in the longitudinal system and
Yr,Lr,Nvand Npin the lateral system. This simplifies the system matrices Along
and Alat substantially as many of the cross-coupling terms disappear.
In natural fliers the forward flight is the preferred motion. Thus, even in hovering
their body posture stays inclined slightly forward with respect to the vertical axis.
Due to the definition of the body axes (Figure 3.11) the product of inertia Ixz is
non-zero and is responsible for the coupling of roll and yaw. Most hovering MAVs,
on the other hand, are designed so that hovering is the nominal operating point and
so their body posture is vertical and Ixz = 0, which decouples yaw from the rest of
the system.
Thus, the flapping MAV dynamics can be modelled by four subsystems:
•the pitch dynamics (u,q,ϑ)
˙u
˙q
˙
ϑ
=
ˆ
Xuˆ
Xqg
ˆ
Muˆ
Mq0
0 1 0
u
q
ϑ
+
1 0
0 1
0 0
ˆ
Xext
ˆ
Mext (3.45)
•the vertical dynamics (w)
˙w=ˆ
Zww+ˆ
Zext (3.46)
•the roll dynamics (v,p,ϕ)
˙v
˙p
˙ϕ
=
ˆ
Yvˆ
Yp−g
ˆ
Lvˆ
Lp0
0 1 0
v
p
ϕ
+
1 0
0 1
0 0
ˆ
Yext
ˆ
Lext (3.47)
•the yaw dynamics (r)
˙r=ˆ
Nrr+ˆ
Next,(3.48)
where the following shortened notation was used: ˆ
Xu=Xu/m, etc. for force deriva-
tives, ˆ
Mu=Mu/Iyy , etc. for moment derivatives, ˆ
Xext =Xext/m, etc. for external
forces and ˆ
Lext =Lext/Ixx for external moments.
66 3 Mathematical modelling
3.4 Stability predicted by various aerodynamic models
The objective of this section is to compare the pole locations of various models,
including CFD, to justify the use of quasi-steady aerodynamics and cycle-averaging
approximations near hovering. This section is based on the results published in
(Karasek and Preumont, 2012).
Similar to the model developed in the preceding sections, all the models considered
in the comparison employ the rigid body assumption: the periodic effects due to flap-
ping are neglected and only the cycle averaged forces are applied on a rigid body,
whose equations of motion are linearised, subsequently. The validity of the rigid
body assumption was confirmed by Zhang and Sun (2010b) who coupled Navier-
Stokes equations with equations of motion and, for small amplitude disturbances,
observed no major difference from the cycle-averaged linearised model even for a
hawkmoth with relatively low flapping frequency (26.1 Hz).
Because the studied systems (3.35-3.36) are reachable and, assuming all the states
being accessible, also observable, the poles and respective modes of motion of the
insect can be computed as eigenvalues and eigenvectors of the linearised system ma-
trices Along (3.37) and Alat (3.38). The stability derivatives that appear in the
linearised model are calculated based on the wing aerodynamics in all the models;
the body drag is not considered as it was found to be negligible near hover (Sun and
Xiong, 2005).
It is difficult to measure the stability derivatives directly on flying animals as it
would require “deactivation” of all their control system to obtain correct data (Tay-
lor and Thomas, 2003). Thus, most of the published data come from numerical
studies. The model animal that will be studied in this section was chosen to be a
drone-fly, because CFD-based results are available both for longitudinal (Wu and
Sun, 2009) and lateral dynamics (Zhang and Sun, 2010a) for this insect. The im-
portant morphological and wing kinematics parameters are in Table 3.1.
Models from Cheng and Deng (2011) and Orlowski and Girard (2011) based on
quasi-steady aerodynamics are also included in the comparison. They consider only
the force component due to translation, equivalent to (3.6-3.7), but provide analytic
expressions of the stability derivatives that can be evaluated for any animal or MAV.
Similar to the comparison of force traces in Section 3.1.4 the quasi-steady model
from this chapter is represented three times: ASL (tr) includes only the force due to
translation (3.6-3.7), ASL (tr+rot) adds the force due to rotation (3.13) and finally
ASL (tr+rot+add) adds also the component due to the inertia of the added mass,
which is similar to (3.17).
3.4 Stability predicted by various aerodynamic models 67
f φmm Ixx Iyy Izz Ixz RA
(Hz) (◦) (mg) (kg.m2) (kg.m2) (kg.m2) (kg.m2) (mm) (-)
164 53.55 87.76 6.58e-10 1.31e-9 8.35e-10 -5.50e-10 11.2 7.52
Table 3.1: Dronefly wing and morphological parameters.
3.4.1 Stability derivatives
The stability derivatives resulting from all the considered models are summarised
in Table 3.2 for longitudinal and in Table 3.3 for lateral system. As in the case of
the reduced MAV model from the previous section, many of the derivatives are zero
or negligible (compared to derivatives of the same type, i.e. with the same units)
no matter what aerodynamic model is used, namely ˆ
Xw,ˆ
Zu,ˆ
Zqand ˆ
Mwin the
longitudinal system and ˆ
Yr,ˆ
Lr,ˆ
Nvand ˆ
Npin the lateral system. Thus, the vertical
motion wis aerodynamically decoupled from the rest of the longitudinal system
(u,q,ϑ) and the yaw rotation ris aerodynamically decoupled from the rest of the
lateral system (v,p,ϕ). Nevertheless, there still exists coupling of roll pand yaw r
through the inertia product Ixz . All the coordinates were defined in Figure 3.11.
Model ˆ
Xuˆ
Xqˆ
Zwˆ
Muˆ
Mq
(s-1) (ms-1) (s-1) (m-1 s-1) (s-1 )
Wu and Sun (2009) (CFD)*-2.22 0.0095 -2.03 -638 -1.65
ASL (tr+rot+add) -3 -0.0641 -4.7 -691 -9.94
ASL (tr+rot) -3 -0.0539 -4.7 -607 -9.58
ASL (tr) -3 -0.00549 -4.7 -369 -3.18
Cheng and Deng (2011) -2.67 0 -3.63 -328 -2.87
Orlowski and Girard (2011) -2.74 0 -3.8 -243 -3.11
Table 3.2: Stability derivatives - longitudinal system. ∗Further non-zero derivatives were
found by the CFD study of Wu and Sun (2009): ˆ
Xw= 0.179 s-1,ˆ
Zu=−0.108 s-1,ˆ
Zq=
0.0012 ms-1,ˆ
Mw= 43.0 m-1s-1 .
When comparing the stability derivatives from the quasi-steady models to the CFD
results the conclusions are ambiguous: for some derivatives we get the same sign
and similar magnitude for all the models (e.g. ˆ
Xu,ˆ
Yv,ˆ
Nr) while for many others
there exists no clear correspondence as the magnitudes and sometimes even the signs
differ. On the other hand, not all the stability derivatives affect the stability (system
poles) in the similar manner. As has been shown in Sun et al. (2007) and Zhang
and Sun (2010a) the most important derivatives with respect to stability are ˆ
Xu,
ˆ
Zw,ˆ
Mu,ˆ
Mq(longitudinal system) and ˆ
Lv,ˆ
Lp,ˆ
Nr(lateral system).
68 3 Mathematical modelling
Model ˆ
Yvˆ
Ypˆ
Lvˆ
Lpˆ
Nr
(s-1) (ms-1) (m-1s-1 ) (s-1) (s-1 )
Zhang and Sun (2010a)*(CFD) -1.57 0.0043 -575 -19.7 -25.8
ASL (tr+rot+add) -1.26 -0.0606 -692 -21.4 -29.6
ASL (tr+rot) -1.26 -0.0609 -429 -20.8 -29.2
ASL (tr) -1.26 0.00228 310 -37.6 -29.2
Cheng and Deng (2011) -1.11 0 272 -20.6 -21.5
Table 3.3: Stability derivatives - lateral system. ∗Further non-zero derivatives were found
by the CFD study of Zhang and Sun (2010a): ˆ
Lr= 0.0493 s-1,ˆ
Nv=−21.4 m-1s-1 ,ˆ
Np=
−1.54 s-1.
3.4.2 Longitudinal system poles
If it was difficult to make some conclusions from the control derivatives values, the
pole maps make the situation much easier to understand. The longitudinal system
poles are given in Table 3.4. The pole map in Figure 3.12 shows that although
the pole locations vary among the different models, the general distribution remains
the same: we always see a pair of complex conjugate poles with positive real part,
resulting into an unstable oscillatory natural mode, and two negative real poles,
representing fast and slow stable natural modes, called subsidence modes by Wu
and Sun (2009).
It can be observed that even the simplest models, Cheng & Deng, Orlowski & Girard
and ASL (tr), with only translational component give us poles which are reasonably
close to the CFD study. If we add the components due to rotation and added mass
inertia, the oscillatory mode poles move towards the CFD poles, on the other hand
the fast stable mode pole moves away from its CFD counterpart.
Model Longitudinal system
λ1,2(Mode 1) λ3(Mode 2) λ4(Mode 3)
Wu and Sun (2009) 7.88 ±16i -19.6 -2.03
ASL (tr+rot+add) 5.87 ±15.5i -24.7 -4.7
ASL (tr+rot) 5.47 ±14.9i -23.5 -4.7
ASL (tr) 5.68 ±13.2i -17.5 -4.7
Cheng and Deng (2011) 5.57 ±12.7i -16.7 -3.63
Orlowski and Girard (2011) 4.76 ±11.5i -15.4 -3.8
Table 3.4: Poles of the longitudinal system
3.4 Stability predicted by various aerodynamic models 69
The eigenvectors (Table 3.6) further show that all the natural modes are estimated
reasonably well both in amplitude as well as in phase for all the models. The slow
stable subsidence mode (Mode 3), representing the motion in vertical direction w,
is decoupled from the rest. The remaining two modes are dominated by horizontal
motion ucoupled with pitch rotation q. These two motions are in phase for Mode
2 (fast subsidence), but almost in anti-phase (phase difference always greater than
120◦) for Mode 1 (unstable oscillatory).
−25 −20 −15 −10 −5 0 5
−20
−15
−10
−5
0
5
10
15
Re (λ)
Im (λ)
Wu (CFD)
ASL(tr+rot+add)
ASL(tr+rot)
ASL(tr)
Cheng
Orlowski*
Dronefly − longitudinal stability
Figure 3.12: Pole map - longitudinal system. All the models show similar dynamic be-
haviour.
3.4.3 Lateral system poles
In the lateral direction the situation is different, see Table 3.5 and Figure 3.13. The
quasi steady based models with only translational force, Cheng & Deng and ASL
(tr), predict one pair of complex conjugate poles with positive real part, resulting
into unstable oscillatory mode, and two negative real poles, representing fast and
slow stable subsidence modes. On the contrary the models where rotational force
is included as well as the CFD model predict a positive real pole, representing a
slow unstable divergence mode, a pair of complex conjugate poles with negative real
parts (a stable oscillatory mode) and a negative real pole (fast subsidence mode).
This shows the importance of the rotational force component on the stability results.
70 3 Mathematical modelling
Model Lateral system
λ5(5,6) (Mode 4) λ6,7(7) (Mode 5) λ8(Mode 6)
Zhang and Sun (2010a) 12.9 -14.6 ±9.32i -83.7
ASL (tr+rot+add) 14.3 -15.1 ±9.33i -98.8
ASL (tr+rot) 11.5 -13.5 ±7.87i -97.2
ASL (tr) 0.274 ±8.64i -20.3 -130
Cheng and Deng (2011) 1.25 ±10.1i -14.9 -82.5
Table 3.5: Poles of the lateral system
−140 −120 −100 −80 −60 −40 −20 0
−15
−10
−5
0
5
10
Re (λ)
Im (λ) Dronefly − lateral stability
Zhang (CFD)
ASL(tr+rot+add)
ASL(tr+rot)
−140 −120 −100 −80 −60 −40 −20 0
−15
−10
−5
0
5
10
Re (λ)
Im (λ)
ASL(tr)
Cheng
Figure 3.13: Pole map - lateral dynamics. Results divided into two figures according to
the pole location structure: CFD study Zhang and Sun (2010a) and quasi-steady models
that include rotational force (top) and quasi-steady models based only on translational force
(bottom).
3.4 Stability predicted by various aerodynamic models 71
Wu and
Sun (2009), ASL ASL ASL Cheng and Orlowski and
Zhang and (tr+rot+add) (tr+rot) (tr) Deng (2011) Girard (2011)
Sun (2010a)
Mode 1
u0.522 (56.9◦) 0.531 (66.3◦) 0.556 (65.3◦) 0.619 (57.1◦) 0.647 (57.1◦) 0.715 (56.9◦)
w0.004 (-49.1◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦)
q17.859 (-63.8◦) 16.578 (-69.3◦) 15.910 (-69.9◦) 14.369 (-66.7◦) 13.901 (-66.4◦) 12.444 (-67.5◦)
ϑ1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦)
Mode 2
u0.553 (180◦) 0.526 (180◦) 0.540 (180◦) 0.682 (180◦) 0.701 (180◦) 0.777 (180◦)
w0.002 (180◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦)
q19.626 (180◦) 24.667 (180◦) 23.513 (180◦) 17.527 (180◦) 16.668 (180◦) 15.373 (180◦)
ϑ1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦)
Mode 3
u3.965 (180◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦)
w58.792 (180◦) 1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦)
q2.034 (180◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦)
ϑ1 (0◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦) 0 (0◦)
Mode 4
v0.675 (180◦) 0.686 (180◦) 0.821 (180◦) 1.118 (100◦) 0.945 (103◦) -
p12.888 (0◦) 14.310 (0◦) 11.534 (0◦) 8.644 (88.2◦) 10.186 (82.9◦) -
r2.972 (180◦) 3.076 (180◦) 2.152 (180◦) 1.603 (20◦) 2.743 (38◦) -
ϕ1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) -
Mode 5
v0.616 (35.3◦) 0.533 (37.6◦) 0.619 (35.8◦) 0.519 (0◦) 0.710 (0◦) -
p17.325 (147◦) 17.775 (148◦) 15.623 (150◦) 20.263 (180◦) 14.926 (180◦) -
r11.892 (74.6◦) 12.120 (83.8◦) 9.158 (92.9◦) 30.264 (180◦) 22.222 (180◦) -
ϕ1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) -
Mode 6
v0.124 (0◦) 0.039 (0◦) 0.041 (0◦) 0.078 (0◦) 0.121 (0◦) -
p83.682 (180◦) 98.839 (180◦) 97.176 (180◦) 130.466 (180◦) 82.518 (180◦) -
r77.503 (0◦) 92.924 (0◦) 91.553 (0◦) 110.773 (0◦) 73.580 (0◦) -
ϕ1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) 1 (0◦) -
Table 3.6: Eigenvectors of the longitudinal (Modes 1-3) and lateral system (modes 4-6) showing the amplitude and phase (in
parenthesis). Longitudinal system values are normalized by pitch ϑ, lateral by roll ϕ.
72 3 Mathematical modelling
The addition of the third component due to added mass inertia further moves the
complex pair of poles towards the CFD results.
From the eigenvectors in Table 3.6 we see that the dominant motion in Mode 4 is
the sideways motion v. For the first three models it is in anti-phase with roll p
but in phase with yaw r. In the remaining models, ASL(tr) and Cheng & Deng,
the sideways motion and roll are nearly in phase, the phase difference from yaw is
smaller than 90◦. Mode 5 is a combination of the three motions v,p,r. In the
first three models the sideways motion and roll are out of phase by about 110◦and
the yaw phase lies approximately in between. In the remaining models the sideways
motion is in anti-phase with roll and yaw rotation. Mode 6, similar in all the models,
represents the out of phase coupling of roll and yaw.
3.4.4 Effect of derivatives ˆ
Xqand ˆ
Yp
In Section 3.4.1 it was shown that various models predict very similar dynamic be-
haviour, even though the sign of stability derivatives ˆ
Xqand ˆ
Ypvaries among the
models. This suggests that these derivatives have very small effect on the dynamics.
The comparison of pole positions for the full system and for the system with ˆ
Xq= 0
and ˆ
Yp= 0 is in Table 3.7. The differences are indeed very small and, therefore, the
derivatives ˆ
Xqand ˆ
Ypcan be neglected in the preliminary analyses.
Model Longitudinal system Lateral system
λ1,2λ3λ5(5,6) λ6,7(7) λ8
Wu&Sun (2009) / 7.88 ±16i -19.6 12.9 -14.6 ±9.32i -83.7
Zhang&Sun (2010a) 7.94 ±15.9i -19.7 12.9 -14.6 ±9.28i -83.7
ASL (tr+rot+add) 5.87 ±15.5i -24.7 14.3 -15.1 ±9.33i -98.8
5.39 ±16i -23.7 13.8 -15.3 ±9.9i -98
ASL (tr+rot) 5.47 ±14.9i -23.5 11.5 -13.5 ±7.87i -97.2
5.1 ±15.3i -22.8 11.2 -13.6 ±8.32i -96.7
ASL (tr) 5.68 ±13.2i -17.5 0.274 ±8.64i -20.3 -130
5.65 ±13.2i -17.5 0.266 ±8.64i -20.3 -130
Table 3.7: Effect of derivatives ˆ
Xqand ˆ
Ypon the system poles: complete system (black),
ˆ
Xqand ˆ
Yvneglected (red). Only the poles and models that were affected are displayed.
3.4.5 Effect of inertia product Ixz
While in longitudinal system the vertical dynamics is decoupled from the pitch, a
coupling between yaw and roll exists in the lateral system due to the inertia product
3.4 Stability predicted by various aerodynamic models 73
Ixz. Assuming the dronefly body posture in hovering is vertical (as in most hovering
MAVs), the inertia product Ixz becomes zero and the coupling disappears. The
resulting pole locations are shown in Figure 3.14 and listed in Table 3.8.
We can observe some differences compared to the nominal body posture (inclined by
48◦from the vertical axis), nevertheless the pole distribution remains the same. The
biggest difference is in the fastest stable real pole, which after decoupling represents
the yaw dynamics. On the other hand, the remaining poles (now representing the
roll and lateral dynamics) represent still a good approximation of the original sys-
tem. Hence, the roll and lateral dynamics can be in the first approximation treated
independently of yaw, which greatly simplifies the problem of stability and control
design, as will be shown in the next chapter.
− 40−120 −100 −80 −60 −40 −20 0
−15
−10
−5
0
5
10
Re ( )
Im ( )
140 120 100 80 60 40 20 0
15
10
5
0
5
10
Re ( )
Im ( ) Dronefly lateral stability
Zhang (CFD)
ASL(tr+rot+add)
ASL(tr+rot)
ASL(tr)
Cheng
Ixz≠0Ixz=0
Figure 3.14: Effect of inertia product Ixz on the lateral poles: nominal body posture
(Ixz 6= 0) in blue, vertical posture (Ixz = 0) in red. Results split into two parts according
to the pole location structure: CFD study Zhang and Sun (2010a) and quasi-steady models
that include rotational force (top), quasi-steady models based only on translational force
(bottom).
74 References
Model Lateral system
λ5(5,6) λ6,7(7) λ8
Zhang and Sun (2010a) 12.9 -14.6 ±9.32i -83.7
10.9 -11.2 ±11.6i -113
ASL (tr+rot+add) 14.3 -15.1 ±9.33i -98.8
11.7 -11.9 ±12.3i -130
ASL (tr+rot) 11.5 -13.5 ±7.87i -97.2
9.65 -10.7 ±10.3i -128
ASL (tr) 0.274 ±8.64i -20.3 -130
1±8.23i -22.3 -128
Cheng and Deng (2011) 1.25 ±10.1i -14.9 -82.5
2.26 ±8.88i -16 -94.6
Table 3.8: Effect of inertia product Ixz on the system poles: original body posture (Ixz 6= 0,
black), vertical body posture (Ixz = 0, red).
3.4.6 Conclusion
It was shown that the stability of the flapping flight in hover can be successfully
estimated while using a quasi-steady based model that includes components due
to rotation and translation. Despite the simplifications used (the effective angle of
attack and relative wing velocity were considered constant along the wingspan, the
centre of pressure was placed on the wing spanwise rotation axis) the results are
comparable to the ones obtained by CFD modelling, while the necessary compu-
tation power is significantly reduced. Therefore such a quasi-steady model can be
advantageously used in flapping-wing MAV parameter and control design. However,
the model, assuming rigid wings, will still remain a rather gross approximation as
the wings of many natural fliers as well as of most MAVs are flexible.
The linearised model was further reduced by neglecting the inertia product Ixz and
derivatives ˆ
Xqand ˆ
Yp, whose effects on the pole locations were shown to be very
small. Thus, the dynamics of hovering flapping flight can be described by only 8
stability derivatives and splits into 4 subsystems (pitch, roll, yaw and vertical dy-
namics) that can be treated separately in the control design.
3.5 References
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doi:10.1098/rsif.2013.0237.
S. P. Sane and M. H. Dickinson. The control of flight force by a flapping wing: lift
and drag production. Journal of Experimental Biology, 204:2607–2626, 2001.
S. P. Sane and M. H. Dickinson. The aerodynamic effects of wing rotation and a
revised quasi-steady model of flapping flight. Journal of Experimental Biology,
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Experimental Biology, 208:447–459, 2005.
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Chapter 4
Stability of near-hover flapping
flight
This chapter studies the stability of the developed hummingbird-like MAV around
hovering. The linearised model of near-hover flapping flight from the previous chap-
ter is employed. The choice of wing position for an easily controllable MAV is
discussed and a simplified solution of the pitch and roll stability problem is pro-
posed.
4.1 Hummingbird robot parameters
Since the existing models predict that the hovering flapping flight of natural fliers is
unstable, it is very unlikely that a passively stable MAV design would exist (with-
out additional stabilizing surfaces like sails). However, the mathematical model can
help us identify a design that would be easily controllable. In this section a 20 g
hummingbird-like MAV with parameters according to Table 4.1 will be studied. The
wing parameters were selected according to the robot prototype that will be pre-
sented in Chapter 6, the mass properties are an estimate based on a CAD model.
Vertical body orientation is assumed while hovering so the inertia product Ixz is
equal to zero.
m Ixx Iyy Izz e R Aˆr2ˆx0R1
0c2ˆrdˆr*
(g) (kg.m2) (kg.m2) (kg.m2) (mm) (mm) (-) (-) (-) (-)
20 1e-5 1e-5 1e-3 31.6 90 9.33 0.531 0.25 0.418
Table 4.1: Hummingbird robot parameters, according to definitions in Chapter 3. *Wing
shape characteristic in equation (3.13).
77
78 4 Stability of near-hover flapping flight
OB
xB
zB
yB
xB
zw
xw
e
Figure 4.1: The position of the wing shoulders with respect to the COG.
Based on experiments that will be presented in Section 6.2 the wing operating con-
ditions were estimated to a flapping amplitude φm= 80◦and frequency f= 26 Hz.
The value of the angle of attack in mid-stroke that provides an equilibrium between
the weight and the average lift force, αm= 33◦, was found by iterations. The last
but very important design parameter is the position of the wings with respect to the
centre of gravity (Figure 4.1). Our intuition tells us that the wings should be higher
than the centre of gravity, which is also what can be observed in nature. To see its
effect the vertical wing position zwwas kept as a free parameter. The longitudinal
position of the wings was set to xw= 0.
4.2 Pitch dynamics
The pitch dynamics is represented by the state space model
˙u
˙q
˙
ϑ
=
ˆ
Xuˆ
Xqg
ˆ
Muˆ
Mq0
0 1 0
u
q
ϑ
+
1 0
0 1
0 0
ˆ
Xext
ˆ
Mext .(4.1)
The positive directions of the coordinates are displayed in Figure 4.2.
4.2.1 Pitch stability derivatives
The stability of the pitch dynamics is determined by the stability derivatives ˆ
Xu,
ˆ
Xq,ˆ
Muand ˆ
Mq. Their values can be evaluated numerically by the aerodynamic
4.2 Pitch dynamics 79
xB
zB
OB
M
X
u
ϑ
q
g
Figure 4.2: Coordinates of pitch dynamics.
model developed in the previous chapter. Such approach, however, does not provide
much insight into the aerodynamic mechanisms that generate the damping effects.
Here, first a simplistic approach is presented with the aim to find an (aproximative)
relationship between the stability derivatives (in particular their signs) and the ver-
tical wing position zw. The findings are then compared to the derivatives obtained
numerically.
zW= 0
Z̄
M=0
̄
X=0
̄
Upstroke
αFD
FL
U
Downstroke
FD
FL
α
U
Figure 4.3: Symmetrical lift and drag forces generated in hover (left) and the resulting
cycle averaged efforts (right).
Let’s consider a wing that, in hovering, flaps with a constant CP speed Uand a
constant angle of attack α(Figure 4.3). It is assumed that the wing lift and drag in-
crease/decrease when either the wing speed Uor angle of attack αincrease/decrease.
When the body moves backward with speed ∆u, the wing speed decreases in down-
stroke to U−∆u, but increases in upstroke to U+ ∆u, which creates an imbalance
in both lift and drag (Figure 4.4). While the average lift force ¯
Zis not affected, the
average longitudinal force ¯
Xopposes the motion u. Thus, the stability derivative
ˆ
Xushould be negative, representing a damping term.
80 4 Stability of near-hover flapping flight
zW= 0
Z̄
M=0
̄
X<0
̄
Δu
Δu
Upstroke
α↑FD
↑FL
↑U
Downstroke
↓FD
↓FL
α
↓U
Figure 4.4: Effects of a disturbance in longitudinal direction ∆uwhen the wing shoulders
are placed at the COG (zw= 0). A wing speed difference between upstroke and downstroke
induces a cycle-averaged damping force ¯
Xthat opposes the disturbance.
zW<0
Upstroke Downstroke
α↓FD
↓FL
↑FD
↑FL
α
zW>0
Z̄
M<0
̄
X<0
̄
Z̄
M>0
̄
ΔuΔu
α↓FD
↓FL
↑FD
↑FL
αX<0
̄
Δu
Δu
↑U
↑U
↓U
↓U
Figure 4.5: Effects of a disturbance in longitudinal direction ∆ufor a general wing shoulder
position (zw6= 0). The cycle-averaged drag force acts at a lever zwand induces a pitch
moment ¯
M, whose sign is opposite to the sign of zw.
For a general wing shoulder position (zW6= 0) a pitch moment ¯
M=¯
Xzwarises due
to the lever arm from the wing shoulder to the COG (Figure 4.5). Therefore, ˆ
Mu
should be positive below the COG and negative above the COG.
Rotating the body around the pitch axis with an angular rate ∆qaffects both the
wing speed and the angle of attack. Considering first the wing shoulders placed at
the COG (zW= 0), the principal effect is on the angle of attack α(Figure 4.6). The
angle increases for wing positions behind the body but decreases in front of the body
because of a vertical velocity component due to the body rotation. The resulting
difference in lift and drag averages out over one wingbeat, but the lift imbalance
4.2 Pitch dynamics 81
↓α
Upstroke
Δq
zW= 0
Downstroke
Δq
zW= 0
↑α↑FD
↑FL
↓FD
↓FL
↓α
↑α
↑FD
↑FL
↓FD
↓FL
Z̄
M<0
̄
Δq
X=0
̄
Back Front
Figure 4.6: Effects of a pitch disturbance ∆qwhen the wing shoulders are placed at the
COG (zw= 0). The angle of attack increases behind the body, but decreases in front of the
body, which causes an uneven distribution of the lift force and results into a cycle-averaged
pitch moment ¯
Mthat opposes the disturbance.
α↓FD
↓FL
↑FD
↑FLα
zW<0
Upstroke
Δq
Downstroke
α↓FD
↓FL
Δq
↑FD
↑FL
α
zW>0
Z̄
M<0
̄
Δq
X<0
̄
Z̄
M<0
̄
Δq
X>0
̄
↑U
↑U
↓U
↓U
Figure 4.7: Effects of a pitch disturbance ∆qfor a general wing shoulder position (zw6=
0). On top of the angle of attack changes the speed increases/decreases in upstroke and
decreases/increases in downstroke when the wings shoulders are placed above/below the
COG, respectively. This induces a non-zero cycle averaged force ¯
X, whose sign is opposite
to the sign of zw, and decreases further the negative stabilizing pitch moment ¯
Mfor both
positive and negative zW.
82 4 Stability of near-hover flapping flight
gives rise to a pitch moment Mopposing the rotation ∆q. Thus, the derivative ˆ
Mq
should be negative.
If the wing shoulders are placed above or below the COG, the wing speed is also
modified in the central wing position (Figure 4.7). The speed increases in upstroke
and decreases in downstroke for zw>0; the opposite is true for zw<0. This cre-
ates a drag asymmetry between upstroke and downstroke, which translates into the
cycle averaged longitudinal force ¯
X. Therefore, the derivative ˆ
Xqshould be positive
for zw<0 and negative for zw>0. Further, the pitch moment gets modified to
¯
M=¯
M0+¯
Xzw, where the first term represents the pitch damping moment for wing
shoulders at COG (zw= 0). The second term is always smaller or equal to zero due
to opposite signs of ¯
Xand zw. Thus, the derivative ˆ
Mqshould stay negative for any
wing shoulder position.
−1.5
−1
−0.5
0
−0.1
0
0.1
−100
0
100
−30 −20 −10 0 10 20 30
−10
−5
0
ASL(tr+rot) ASL(tr)
Xu [s-1]
̂
Xq [m.s-1]
̂
Mq [s-1]
̂
Mu [m-1s-1]
̂
zw [mm]
Figure 4.8: Stability derivatives of pitch dynamics for varying wing position zW.
The stability derivatives given by the model presented in the previous chapter for
robot parameters given in Section 4.1 are plotted in Figure 4.8. Two quasi-steady
models were used, a simpler model including only the force due to translation, ASL
(tr), and a more complex model that includes also the force due to rotation, ASL
4.2 Pitch dynamics 83
(tr+rot). It has been shown in Section 3.4 that the latter gives comparable stability
derivatives to a CFD model.
The simpler model ASL (tr) plotted in red dashed lines in Figure 4.8 is in complete
agreement with the findings about the derivative signs based on Figures 4.4-4.7 made
earlier. The derivative ˆ
Xurepresenting the damping of the forward translation is
always negative and independent on the wing position. The derivative ˆ
Mq, the pitch
rotation damping, remains always negative, varying with square of wing position z2
w
with a maximum at zw= 0. The cross-coupling derivatives ˆ
Muand ˆ
Xqare zero for
zw= 0 and vary linearly, with a negative slope, with the wing position zw.
The addition of the force due to rotation in ASL (tr+rot) has no effect on the
derivative ˆ
Xu(blue solid lines in Figure 4.8). We can observe some differences
among the other three derivatives, but the general behaviour remains similar. The
pitch damping ˆ
Mqremains approximately quadratic function of z2
w. Its maximum
value, now at zw≈ −18 mm, slightly increases, but stays negative. The derivatives
ˆ
Xqand ˆ
Muare slightly offset from the simpler model, but keep the same slope. ˆ
Xq
stays always negative in the studied range, while ˆ
Mucrosses zero at around -4 mm.
4.2.2 System poles
Assuming a reachable and observable system with a system matrix A, the system
poles are defined as roots of the characteristic equation |λI−A|= 0. For pitch
dynamics, we obtain a cubic equation
λ3−ˆ
Xu+ˆ
Mqλ2+ˆ
Xuˆ
Mq−ˆ
Muˆ
Xqλ−ˆ
Mug= 0 (4.2)
with three roots. Figure 4.9 shows position of the poles (root locus) for wing po-
sition zwranging from -30 mm to 30 mm, calculated from the stability derivatives
shown in Figure 4.8. In both models, ASL (tr) in red-dashed lines and ASL (tr+rot)
in blue lines, we can observe 3 different pole configurations. For positive values of
zw(wings placed above the centre of gravity), we have a pair of complex unstable
poles and one real stable pole. For negative zw, there is one stable real pole and an
unstable pair of complex poles. Finally, there is a narrow transient region around
the COG (zw≈0) for ASL (tr) and below the COG (zw≈ −4mm) for ASL (tr+rot)
where 3 real poles exist.
It can be demonstrated that the pole configuration is determined by the derivative
ˆ
Mu. By neglecting the derivative ˆ
Xq(whose effect on the dynamics has been shown
to be very small, see Section 3.4.4) the characteristic equation becomes
λ3−ˆ
Xu+ˆ
Mqλ2+ˆ
Xuˆ
Mqλ−ˆ
Mug= 0,(4.3)
84 4 Stability of near-hover flapping flight
−15 −10 −5 0 5 10
−10
−5
0
5
10
−15 −10 −5 0 5 10
−30
0
30
ASL (tr+rot)
ASL (tr)
Im (λ)zw [mm]
Figure 4.9: Influence of wing position zwon the pitch dynamics: pole positions in complex
plane (top) and real part of the poles against the wing position (bottom).
which can be rewritten as
λλ−ˆ
Xuλ−ˆ
Mq−ˆ
Mug= 0 (4.4)
or
1 + ˆ
Mu−g
λλ−ˆ
Xuλ−ˆ
Mq= 0.(4.5)
The equation has the same form as a characteristic equation of system H(s) =
−g/(λ(λ−ˆ
Xu)(λ−ˆ
Mq)) in closed loop with a feedback gain ˆ
Mu. Thus, a root locus
method can be used to study the effect of the derivative ˆ
Muon the pitch dynamics
poles.
The system H(s) has three poles, one at the origin and two on the real axis given
by the derivatives ˆ
Xuand ˆ
Mq, respectively. Since H(s) has no zeros and a negative
numerator, the root locus will follow asymptotes at ±60◦and 180◦for ˆ
Mu→ −∞
and asymptotes at 0◦and ±120◦for ˆ
Mu→ ∞. The start point of the asymptotes
4.2 Pitch dynamics 85
Re (λ)
Im (λ)
0
0
^
Mq^
Xu
^
Mu → +∞
0
^
Mu → −∞60°
120°
Mq + Xu
3
^ ^
Figure 4.10: Root locus showing the effect of derivative ˆ
Muon the system dynamics.
lies at ( ˆ
Xu+ˆ
Mq)/3. The resulting root locus is shown in Figure 4.10 and looks
very similar to the locus for varying wing position zw(Figure 4.9): negative ˆ
Mu
results into a pair of unstable complex poles and a stable real pole, while positive
ˆ
Muresults into an unstable real pole and a pair of stable complex poles.
Provided that both ˆ
Xuand ˆ
Mqare negative, a small region of stability exists for
small negative values of ˆ
Mu. This region, with one fast real pole and a pair of slow
lightly damped complex poles, can only be enlarged by increasing the damping of
forward translation or pitch rotation. This can be done either passively by adding
some damping surfaces (tail, sails) or actively by feedback control.
4.2.3 Active stabilization
Flying animals are equipped with sensory systems that provide feedback for active
stabilization of their flight. As it has been mentioned already in Section 2.3.3, flies
use two types of sensors to sense angular velocities: the compound eyes and the hal-
teres. Compared to compound eyes, which are most sensitive to rates of the order
of 100◦/s, the halteres can sense rates up to 1000◦/s (Taylor and Krapp, 2008) and
their latency is about four times shorter (Ristroph et al., 2013). Sherman and Dick-
inson (2004) showed that the wing response simply follows a weighted sum of the two
sensory signals. Thus, the aim of this section is to show under which conditions can
the pitch dynamics be stabilized using a simple angular rate feedback with a gain kq.
86 4 Stability of near-hover flapping flight
Such control will introduce a stabilizing moment ˆ
M=−kqq. By plugging the
moment into the state space model we get
˙u
˙q
˙
ϑ
=
ˆ
Xuˆ
Xqg
ˆ
Muˆ
Mq−kq0
0 1 0
u
q
ϑ
+
1 0
0 1
0 0
ˆ
Xext
ˆ
Mext ,(4.6)
which yields the characteristic equation
λ3+kq−ˆ
Xu−ˆ
Mqλ2+hˆ
Xuˆ
Mq−kq−ˆ
Muˆ
Xqiλ−ˆ
Mug= 0.(4.7)
The necessary (but not sufficient) condition for stability requires that all coefficients
of the characteristic equation are positive. This already shows that the pitch rate
feedback can stabilize the system only if the derivative ˆ
Muis negative. By neglecting
again the derivative ˆ
Xqand rewriting the characteristic equation into the root locus
form with parameter ˆ
Muwe obtain
1 + ˆ
Mu−g
λλ−ˆ
Xuλ+kq−ˆ
Mq= 0.(4.8)
We can observe, that the feedback gain moves the pole related to the pitch rate
damping towards minus infinity (see Figure 4.11). This results into a shift of the
Re (λ)
Im (λ)
0
0
^
Mq^
Xu
^
Mu → +∞
0
^
Mu → −∞
Mq − kq + Xu
3
^ ^
kq
CL
CL
OL
OL
Figure 4.11: The effect of derivative ˆ
Muon the pitch dynamics in open loop (OL) and in
closed loop (CL) with pitch rate feedback kq.
4.2 Pitch dynamics 87
asymptotes origin to the left and causes an “opening” of the root locus for negative
values of ˆ
Mu. Thus, the system becomes stable for a much larger interval of the
stability derivative ˆ
Mu.
The minimal necessary gain for stability for a given ˆ
Mucan be expressed by using the
Routh-Hurwitz stability criterion (e.g. Franklin et al., 2002, p. 158). The criterion
yields three conditions
kq−ˆ
Xu−ˆ
Mqhˆ
Xuˆ
Mq−kq−ˆ
Muˆ
Xqi+ˆ
Mug > 0 (4.9)
kq−ˆ
Xu−ˆ
Mq>0 (4.10)
−ˆ
Mug > 0.(4.11)
The third condition confirms that ˆ
Mumust be negative. The second condition is
usually fulfilled as both derivatives ˆ
Xuand ˆ
Mqare typically negative (see Table 4.2).
Thus, the minimal gain can be expressed from the first condition. Since the gain
should be positive we obtain
kq,min =
ˆ
Muˆ
Xq−2ˆ
Xuˆ
Mq−ˆ
X2
u+rˆ
X2
u+ˆ
Muˆ
Xq2+ 4 ˆ
Xuˆ
Mug
−2ˆ
Xu
.(4.12)
For a typical hovering animal the minimal necessary gain is much higher than the
remaining terms in the brackets on the left side of relation (4.9), namely kq,min
−ˆ
Xu−ˆ
Mqand kq,min −ˆ
Mq+ˆ
Muˆ
Xq
ˆ
Xu. By neglecting these terms, and keep-
ing in mind that ˆ
Xu<0, we obtain much simpler condition
kq>sˆ
Mu
ˆ
Xu
g, (4.13)
which can serve as a good first estimate of minimal necessary gain. Moreover, if
ˆ
Mq<ˆ
Muˆ
Xq
ˆ
Xu, all the terms neglected on the left side of the relation (4.9) are posi-
tive. In that case the relation (4.13) becomes a sufficient condition for stability.
Table 4.3 shows the poles computed for several animals using stability derivatives
from Table 4.2. It also compares the values of the minimal feedback gain kcal-
culated with the exact formula (4.12) and the values estimated by relation (4.13).
We can see that in most cases the approximative formula is very close to the exact
solution.
88 4 Stability of near-hover flapping flight
Animal Model ˆ
Xuˆ
Xqˆ
Muˆ
Mqˆ
Zw
(s-1) (ms-1) (s-1) (m-1 s-1) (s-1 )
Bumblebee 1 CFD1-1.35 0.00456 -1360 -3.34 -1.78
quasi-steady -3.08 0 -988 -3.17 -2.69
Bumblebee 2 morphology -4.19 -0.0163 -445 -1.9 -
Hawkmoth 1 CFD2-3.61 0.0342 -908 -8.76 -2.05
Hawkmoth 2 quasi-steady -3.02 0 -185 -1.79 -2.32
Hawkmoth 3 morphology -3.04 -0.0265 -151 -2.85 -
Rufous hummingbird
morphology
-0.624 -0.00437 -9.01 -0.432 -
Blue-throated hum. -0.717 -0.01 -8.43 -0.566 -
Magnificent hum. -0.758 -0.00833 -5.61 -0.362 -
Black-chinned hum. -0.741 -0.00815 -13.6 -0.242 -
Table 4.2: Stability derivatives computed using CFD, quasi-steady modelling and esti-
mated from animal morphology. Data taken from the following studies: CFD1- Sun and
Xiong (2005) with correction from Xiong and Sun (2008), CFD2- Zhang and Sun (2010),
quasi-steady - Cheng and Deng (2011), morphology - Ristroph et al. (2013).
Animal Model λ1λ2,3kq,min k∗
q,min
Bumblebee 1 CFD1-25.2 10.2 ±20.6 i 92.9 99.2
quasi-steady -23.5 8.6 ±18.4 i 51.4 56.1
Bumblebee 2 morphology -18.6 6.3 ±14 i 29.3 32.3
Hawkmoth 1 CFD2-24.6 6.1 ±18.0 i 34.8 49.7
Hawkmoth 2 quasi-steady -13.9 4.5 ±10.5 i 21.2 24.5
Hawkmoth 3 morphology -13.6 3.8 ±9.7 i 18.5 22.1
Rufous hummingbird
morphology
-4.8 1.9 ±3.8 i 11.2 11.9
Blue-throated hum. -4.8 1.8 ±3.8 i 9.9 10.7
Magnificent hum. -4.2 1.5 ±3.3 i 7.8 8.5
Black-chinned hum. -5.5 2.2 ±4.4 i 12.9 13.4
Table 4.3: System poles as roots of (4.2), minimal feedback gain for stability kq,min from
(4.12) and its estimate k∗
q,min according to (4.13). All the values computed for the data in
Table 4.2.
4.3 Roll dynamics 89
4.3 Roll dynamics
The roll dynamics is represented by the state space model
˙v
˙p
˙ϕ
=
ˆ
Yvˆ
Yp−g
ˆ
Lvˆ
Lp0
0 1 0
+
1 0
0 1
0 0
ˆ
Yext
ˆ
Lext .(4.14)
The positive directions of the coordinates are shown in Figure 4.12. The system
has the same structure as the pitch dynamics (4.1). Both systems have 3 DOF and
coupling exists between the translation and rotation rate. The different sign next
to the gravity acceleration gcomes from the frame orientation, the effect on the
dynamics remains the same as in pitch.
OB
yB
zB
φ
Y
g
vp
L
Figure 4.12: Coordinates of roll dynamics.
4.3.1 Roll stability derivatives
The roll dynamics stability is determined by the stability derivatives ˆ
Yv,ˆ
Yp,ˆ
Lvand
ˆ
Lp. As in previous section, a simplistic approach is used to approximate the rela-
tionship between the signs of the stability derivatives and the vertical wing position
zw. The findings are again compared to the derivatives obtained numerically.
In the simplistic model the wing forces depend on the angle of attack αand the CP
velocity U, which is the velocity component tangential to the flapping motion. It is
assumed, that the velocity component in the direction of the wing longitudinal axis
has no effect on the produced forces.
90 4 Stability of near-hover flapping flight
A lateral body motion ∆vaffects the velocity Umostly close to the extremal wing
positions (Figure 4.13). Uis reduced when the wing moves in the direction of lat-
eral disturbance, but it is increased when moving in the opposite direction. This
affects the drag force distribution over the cycle; its dominant direction determines
the cycle averaged lateral force ¯
Y, which is negative and opposes the disturbance ∆v.
For a general wing position (zw6= 0) the averaged drag force, acting at the wing
shoulder, also produces a roll moment ¯
L=¯
Y zw(Figure 4.14). Therefore, the deriva-
tive ˆ
Yvshould be negative and independent on zw.ˆ
Lvshould be positive for wing
shoulders placed below the COG and negative for wing shoulders above the COG.
↑U
↓U
↑FD
↓FD
Δv
↑U
↓U
↑FD
↓FD
↓FD
↓U
↑U
↑FD
Upstroke
Δv
↓FD
↓U
↑U
↑FD
Downstroke
Δv
Y<0
̄
L=0
̄
zW= 0
Figure 4.13: Effects of a disturbance in lateral direction ∆vwhen the wing shoulders are
placed at the COG (zw= 0). The tangential component of the wing speed Uincreases
when the wing moves in the same direction as the body, but decreases when the wing moves
opposite to the body. This induces an asymmetry in the drag forces and results into a
cycle-averaged damping force ¯
Ythat opposes the lateral disturbance.
Z̄
Δv
L<0
̄L<0
̄
Y<0
̄
Y<0
̄
Z̄
Δv
L>0
̄
FD
̄
FD
̄
zW<0
zW>0
Figure 4.14: Effects of a disturbance in lateral direction ∆vfor a general wing shoulder
position (zw6= 0). The cycle-averaged drag force acts at a lever zwand induces a roll
moment ¯
L, whose sign is opposite to the sign of zw.
4.3 Roll dynamics 91
Δp
zW= 0
⇧α⇧FD
⇧FL
⇩α
⇩FD
⇩FL
Right Left
Δp
Z̄
L<0
̄
Y=0
̄
Figure 4.15: Effects of a roll disturbance ∆pwhen the wing shoulders are placed at the
COG (zw= 0). The angle of attack decreases on the right wing and increases on the left
wing, causing a lift asymmetry and resulting into a roll moment ¯
Lopposing the roll rotation.
A roll disturbance ∆phas a dominant effect on the angle of attack, which increases
on the left wing and decreases on the right wing. This causes a lift difference result-
ing into a cycle-averaged roll moment ¯
Lopposing the disturbance direction (Figure
4.15). Therefore, the derivative ˆ
Lpshould be negative.
For a general wing position (zw6= 0), the body roll introduces also a speed compo-
nent in the lateral direction (Figure 4.16). Similar to the lateral disturbance, the
drag distribution is modified resulting into a cycle averaged lateral force ¯
Y. Thus,
the derivative ˆ
Ypshould be positive for wing shoulders above the COG and negative
for wing shoulders below the COG. On top of that, the lateral force modifies further
the roll moment to L=L0+Y zwm/Ixx, where L0is the roll moment at zw= 0 and
the second term is always smaller or equal to zero due to opposite signs of Yand
zw. Therefore, the derivative ˆ
Lpshould always stay negative.
The roll stability derivatives given by the model from the previous chapter are plot-
ted in Figure 4.17. Again two quasi-steady models were used, a simpler model
including only the force due to translation ASL (tr) and a more complex model that
includes also the force due to rotation, ASL (tr+rot).
As in pitch dynamics, the simpler model ASL (tr) plotted in red dashed lines in Fig-
ure 4.17 is coherent with the findings based on Figures 4.13-4.16. The derivative ˆ
Yv
is negative and thus represents the lateral motion damping. It is independent on the
wing position zw. The derivative ˆ
Lpstays also negative and represents the damping
of the roll rotation. Its maximum is at zw= 0 and it decreases with a square of wing
position z2
w. The derivatives ˆ
Lvand ˆ
Yprepresent the cross-coupling terms. They
are zero at zw= 0 and vary linearly, with a positive slope, with the wing position zw.
92 4 Stability of near-hover flapping flight
↑U
↓U
↓U
↑U
Downstroke
↑U
↓U
↓U
↑U
Upstroke
Z̄
L<0
̄
Y>0
̄
Δp
zW<0
↑U
↓U
↓U
↑U
↑U
↓U
⇩↑FD
⇩↓FD
⇧↓FD
↓U
↑U
⇧↑FD
Δp
Z̄
L<0
̄
Y>0
̄
Δp
zW>0
⇧FD
⇩FD
⇧FD
⇩FD
⇩↑FD
⇧↑FD
⇩↓FD
⇧↓FD
⇧FD
⇩FD
⇩↑FD
⇩↓FD⇧↑FD
⇧↓FD
⇧↑FD
⇧FD
⇧↓FD
⇩↑FD
⇩FD
⇩↓FD
Figure 4.16: Effects of a roll disturbance ∆pfor a general wing shoulder position (zw6=
0). On top of the angle of attack changes, the wing speed due to body rotation has a
lateral component, which is negative/positive for wing shoulders above/below the COG,
respectively. This component introduces a drag force asymmetry similar to the lateral
disturbance ∆vand results into a lateral force with the same sign as zw. Moreover, the roll
moment decreases further for both positive and negative zw.
The more complex model ASL (tr+rot), which includes also the force due to rota-
tion, is plotted in blue lines in Figure 4.17. There is no change in the derivative ˆ
Yv.
The roll damping ˆ
Lpkeeps its quadratic form, but the maximum slightly increases
and moves to zw≈22mm, while staying negative. The cross/coupling derivatives
ˆ
Lvand ˆ
Ypkeep the same slope, but are offset. The former crosses zero at around 5
mm, the latter stays negative in the studied range.
We can observe that while the aerodynamic mechanisms behind the stability deriva-
tives of pitch and roll dynamics are different, they follow the same trends with
respect to wing shoulder position zw. The translation derivatives ˆ
Xuand ˆ
Yvare
both negative and constant, the rotation derivatives ˆ
Mqand ˆ
Lpare also negative
and vary with the square of wing position z2
w, and the cross-terms ˆ
Mu,ˆ
Xqand ˆ
Lv,
4.3 Roll dynamics 93
−2
−1
0
−0.1
0
0.1
−100
0
100
−15
−10
−5
ASL(tr+rot) ASL(tr)
Yv [s-1]
̂
Yp [m.s-1]
̂
Lv [m-1s-1]
̂
Lp [s-1]
̂
−30 −20 −10 0 10 20 30
zw [mm]
Figure 4.17: Stability derivatives of roll dynamics for varying wing position zW.
ˆ
Ypall vary with wing position zw. The different slope directions are caused by frame
definitions, the effects on the system are the same. Thus, the conclusions made
about stability of the pitch dynamics closely apply also to the roll system.
4.3.2 System poles
System poles of roll dynamics are given by characteristic equation
λ3−ˆ
Yv+ˆ
Lqλ2+ˆ
Yvˆ
Lp−ˆ
Lvˆ
Ypλ+ˆ
Lvg= 0.(4.15)
As expected, the observed root locus (Figure 4.18) is very similar to the one of pitch
dynamics. Again, 3 different pole configurations exist in both models, ASL(tr) in red
dashed lines and ASL(tr+rot) in blue lines. A pair of unstable complex conjugate
poles and one stable real pole exists for wing positions well above the COG. Wing
positions well below the COG yield a pair of stable complex conjugate poles and
one unstable real pole. A transient region with 3 real poles exists around the COG
(zw≈0) for ASL(tr) and slightly above the COG (approximately for zw∈(3,5)
mm) for ASL(tr+rot).
94 4 Stability of near-hover flapping flight
−20 −15 −10 −5 0 5 10
−10
−5
0
5
10
−20 −15 −10 −5 0 5 10
−30
0
30
ASL (tr+rot)
ASL (tr)
Im (λ)zw [mm]
Figure 4.18: Influence of wing position zwon the roll dynamics: pole positions in complex
plane (top) and real part of the poles against the wing position (bottom).
The pole configuration in the pitch dynamics was determined by the derivative
ˆ
Mu, the pitch moment induced by longitudinal translation. In roll the important
derivative is ˆ
Lv, the roll moment induced by the lateral translation. Assuming the
derivative ˆ
Ypis negligible and following the same steps as in pitch, the characteristic
equation can be rewritten to
1 + ˆ
Lv
g
λλ−ˆ
Yvλ−ˆ
Lp= 0,(4.16)
which is a root locus form with virtual feedback gain ˆ
Lv. The root locus is shown in
Figure 4.19. Because the numerator of the virtual open loop system is positive, the
situation is opposite to the pitch dynamics. We observe a pair of unstable complex
poles and a stable real pole for positive ˆ
Lv, while negative ˆ
Lvresults into an unstable
real pole and a pair of stable complex poles.
However, because positive wing shoulder positions zw>0 yield positive ˆ
Lv, but
negative ˆ
Mu, the difference in the root locus positive directions of pitch and of roll
4.3 Roll dynamics 95
Re (λ)
Im (λ)
0
0
^
Lp^
Yv
^
Lv → +∞
0
^
Lv → −∞60°
120°
Lp + Yv
3
^ ^
Figure 4.19: Root locus showing the effect of derivative ˆ
Lvon the roll dynamics.
is compensated; for a given wing position zwthe two systems behave similarly. A
small region of stability exists in roll for small positive values of ˆ
Lv, provided that
the derivatives ˆ
Yvand ˆ
Lpremain negative. The stable region (with one fast real
pole and a pair of slow lightly damped complex poles) can be enlarged by increasing
the damping of lateral translation or roll rotation. The roll can be damped similar
to pitch: either passively by damping surfaces (tail, sails) or by an active feedback.
Due to the different predictions of the derivative ˆ
Lvby the two models ASL(tr)
and ASL(tr+rot) there exists an interval of positive wing positions, where ASL(tr)
already predicts an unstable pair of complex poles and a stable real pole, while
ASL(tr+rot) still gives a stable pair of complex poles and an unstable real pole.
This can explain the discrepancy in the dronefly stability results (Section 3.4.3)
among the considered models. Although the more complex models predict the pole
configuration with an unstable real pole, the following section will show that the
other configuration with an unstable complex pole-pair is more probable as it is
easier to control.
4.3.3 Active stabilization
The active stabilization of roll via rate feedback is analogous to pitch. A proportional
gain kpintroduces a stabilizing moment ˆ
L=−kpp. The characteristic equation
becomes
λ3+kp−ˆ
Yv−ˆ
Lpλ2+hˆ
Yvˆ
Lp−kp−ˆ
Lvˆ
Ypiλ+ˆ
Lvg= 0 (4.17)
96 4 Stability of near-hover flapping flight
and can be rewritten further while neglecting ˆ
Ypas
1 + ˆ
Lv
g
λλ−ˆ
Yvλ+kp−ˆ
Lp= 0.(4.18)
The feedback gain moves the pole related to the roll rate damping towards minus
infinity which “opens” the root locus for positive values of ˆ
Lvand enlarges the in-
terval of stability.
The Routh-Hurwitz stability criterion yields three conditions
kp−ˆ
Yv−ˆ
Lphˆ
Yvˆ
Lp−kp−ˆ
Lvˆ
Ypi−ˆ
Lvg > 0 (4.19)
kp−ˆ
Yv−ˆ
Lp>0 (4.20)
ˆ
Lvg > 0.(4.21)
The third condition requires that ˆ
Lvmust be positive. The second condition is
usually met as both derivatives ˆ
Yvand ˆ
Lpare typically negative. Finally, the first
condition can be used to express the minimum gain value for stability. Assuming
the gain is positive we obtain
kp,min =
ˆ
Lvˆ
Yp−2ˆ
Yvˆ
Lp−ˆ
Y2
v+rˆ
Y2
v+ˆ
Lvˆ
Yp2−4ˆ
Yvˆ
Lvg
−2ˆ
Yv
.(4.22)
4.4 Vertical and yaw dynamics stability
The vertical and yaw dynamics are decoupled from the rest of the dynamics (as long
as Ixz = 0) and each can be treated separately. They are modelled by two first-order
differential equations
˙w=ˆ
Zww+ˆ
Zext (4.23)
˙r=ˆ
Nrr+ˆ
Next (4.24)
with coordinates according to Figure 4.20. The system poles can be expressed
directly from the characteristic equation as
λw=ˆ
Zw(4.25)
λr=ˆ
Nr.(4.26)
4.4 Vertical and yaw dynamics stability 97
xB
zB
OB
Z
wr
N
Figure 4.20: Cordinates of vertical and yaw dynamics.
↓α
Upstroke
Δw
Downstroke
↓FD
↓FL
↓α
↓FD
↓FL
↓Z̄
Δw
Figure 4.21: Effects of a disturbance in vertical direction ∆w. A decreased angle of attack
in both upstroke and downstroke causes the cycle-averaged vertical force ¯
Zto drop, opposing
the disturbance.
For a stable system, the stability derivatives ˆ
Zwand ˆ
Nrneed to be negative.
Figure 4.21 shows that a vertical disturbance ∆wdecreases the angle of attack in
both upstroke and downstroke resulting into a drop of cycle-averaged vertical force
¯
Z. Thus, the derivative ˆ
Zwshould be negative, representing a damping of the ver-
tical motion.
The effect of a yaw disturbance ∆ron the wing aerodynamic forces is captured in
Figure 4.22. In downstroke, the right wing moves in the sense of the disturbance,
which increases the wing velocity and results into an increase of drag. The left wing
moves against the disturbance, which reduces the wing velocity and subsequently the
drag. Opposite situation can be observed in upstroke. Overall, the drag produced
98 4 Stability of near-hover flapping flight
↓U↑U
Upstroke
Downstroke
Δr
↑FD
↑U
↑U
↓U
↓FD
↓U
↓U
↑U
Δr
↑U
↑U
↓U
↓U
Δr
N<0
̄
↓FD
↓FD
↓FD
↓FD
↓FD
↑FD
↑FD
↑FD
↑FD
↑FD
Figure 4.22: Effects of a yaw disturbance ∆r. In downstroke, the wing speed of the right
wing increases as it moves in the sense of the disturbance, while the speed of the left wing
decreases as it moves against the disturbance. The opposite is true for the upstroke. The
drag force distribution over the cycle is affected, resulting into a negative cycle-averaged
yaw moment ¯
Nthat counteracts the disturbance.
in the direction opposing the disturbance dominates, which results into a negative
cycle-averaged yaw moment ¯
N. Thus, the derivative ˆ
Nrshould also be negative
meaning the yaw disturbance is being damped passively.
Because both considered disturbances are related to the vertical body axis zB, the
derivatives ˆ
Zwand ˆ
Nrshould stay independent of wing position zw. Indeed, the
models ASL (tr) and ASL (tr+rot) give the same and constant values for any wing
position of the robot with parameters from Section 4.1: ˆ
Zw=−1.37 s-1 and ˆ
Nr=
−54.9 m-1s-1 . Thus, both vertical and yaw motion should be passively stable.
4.5 Wing position choice
In the previous paragraphs it has been shown that out of the four subsystems (pitch,
roll, yaw and vertical dynamics) only the vertical and yaw dynamics are inherently
stable for any wing position zw. Moreover, the latter is true only for flyers with a
vertical posture in hovering, as the system gets coupled with the roll dynamics via
non-zero inertia product Ixz otherwise.
While there exists an interval of theoretical stability in both pitch and roll dynam-
ics, the intervals of wing positions are very small and in general do not intersect.
Therefore, it is impossible to find a wing position where the hummingbird robot
would be, even theoretically, passively stable. The wing position could be chosen
from one of the two intervals, which would mean that only one of the systems needs
4.6 Rate feedback gains 99
to be stabilized actively. However, even if the model predictions were correct, the
poles would still lie very close to the imaginary axis so some stability augmentation
would be necessary.
Since the flight needs to be stabilized in any case, a wing position should be chosen
so that the controller can be as simple as possible yet robust. It has been shown
that if ˆ
Muis positive and ˆ
Lvis negative, both pitch and roll systems have a pair of
complex unstable poles and a real stable pole. Such a system can be easily stabilized
by a simple rate feedback. On the other hand, negative ˆ
Muand positive ˆ
Lvyields
a pair of stable complex poles and an unstable real pole, a configuration that will
remain unstable even with the rate feedback.
Re (λ)
Im (λ)
0
0
Re (λ)
Im (λ)
0
0
zw ≫ 0
zw ≪ 0
OB
OB
→ not controllable by rate feedback → controllable by rate feedback
Figure 4.23: Pole configurations observed in both pitch and roll systems: situation for
ˆ
Mu0 or ˆ
Lv0 (wing positions well below the COG) cannot be stabilized by rate
feedback (left); situation for ˆ
Mu0 or ˆ
Lv0 (wing positions well above the COG) is
controllable by rate feedback (right).
The quasi-steady based aerodynamic model predicts that positive ˆ
Muand negative
ˆ
Lvcan only be obtained if the wing shoulders are placed (sufficiently high) above
the COG. For the MAV considered in this chapter the wings should be placed above
zw>5 mm (see Figures 4.8 and 4.17). To keep a certain safety margin, zw= 10
mm was selected.
4.6 Rate feedback gains
The minimal necessary gains for stable pitch and roll rate feedback can be com-
puted from equations (4.12) and (4.22), respectively. Considering zw= 10 mm we
100 4 Stability of near-hover flapping flight
will obtain kq,min ≈13.8 and kp,min ≈1.5. As expected, the pitch dynamics require
slightly higher gain as the chosen wing position is further from the passively stable
interval in pitch than in roll.
The root locus can be obtained by rewriting the characteristic equations 4.7 and
4.17 (while neglecting ˆ
Xqand ˆ
Yp) into the following form
1 + kq
λλ−ˆ
Xu
λλ−ˆ
Xuλ−ˆ
Mq−ˆ
Mug
= 0 (4.27)
1 + kp
λλ−ˆ
Yv
λλ−ˆ
Yvλ−ˆ
Lp+ˆ
Lvg
= 0.(4.28)
where the terms multiplied by the gains kqand kpare the open-loop transfer func-
tions Q(s)/ˆ
M(s) and P(s)/ˆ
L(s), respectively. The root loci for gains kqand kpgoing
from zero to infinity are plotted in Figure 4.24. We can observe that, no matter how
big the gain is, the real part of the originally unstable pair of poles will always stay
“trapped” between the pair of zeros, one at the origin and the other given by ˆ
Xu
and ˆ
Yvfor pitch and roll, respectively. Thus, a more complex controller needs to be
used if faster response is required. Alternatively, the derivatives ˆ
Xuand ˆ
Yvcould be
decreased by adding passive damping surfaces. However, this would also influence
the remaining stability derivatives that define the open loop poles, so the effect on
the whole root locus is not apparent.
6
Im (λ)
Re (λ)
4
2
0
-2
-4
-6
-20 -15 -10 -5 0 5
Pitch dynamics
6
Im (λ)
Re (λ)
4
2
0
-2
-4
-6
-20 -15 -10 -5 0 5
Roll dynamics
Figure 4.24: Root locus for a rate feedback gain in pitch (left) and in roll systems (right).
4.7 Conclusion 101
4.7 Conclusion
The inherent instability of pitch and roll dynamics was demonstrated through a sim-
plistic approach, which is coherent with quasi-steady modelling results. The systems
can be actively stabilized by a simple angular rate feedback, provided that the wing
shoulders are placed sufficiently above the COG. This conclusion is in accordance
with morphology of hovering animals as well as with observations of basic motoric
response in fruit-flies. Of course a more sophisticated control law can be used (and
is probably used to some extent also by flying animals). The design of a controller
for the hummingbird robot will be described in more detail in the next chapter.
4.8 References
B. Cheng and X. Deng. Translational and rotational damping of flapping flight and
its dynamics and stability at hovering. IEEE Transactions on Robotics, 27(5):
849–864, Oct. 2011. doi:10.1109/TRO.2011.2156170.
G. F. Franklin, J. Powel, and A. Emami-Naeini. Feedback Control of Dynamic
Systems. Prentice Hall, 4th edition, 2002.
L. Ristroph, G. Ristroph, S. Morozova, A. J. Bergou, S. Chang, J. Gucken-
heimer, Z. J. Wang, and I. Cohen. Active and passive stabilization of body
pitch in insect flight. Journal of The Royal Society Interface, 10(85):1–13, 2013.
doi:10.1098/rsif.2013.0237.
A. Sherman and M. H. Dickinson. Summation of visual and mechanosensory feed-
back in drosophila flight control. Journal of Experimental Biology, 207(1):133–142,
2004. doi:10.1242/jeb.00731.
M. Sun and Y. Xiong. Dynamic flight stability of a hovering bumblebee. Journal of
Experimental Biology, 208:447–459, 2005.
G. K. Taylor and H. G. Krapp. Sensory systems and flight stability: what do
insects measure and why? Advances in insect physiology, 34(231-316), 2008.
doi:10.1016/S0065-2806(07)34005-8.
Y. Xiong and M. Sun. Dynamic flight stability of a bumblebee in forward flight.
Acta Mech. Sinica, 24(1):25–36, Feb. 2008. doi:DOI 10.1007/s10409-007-0121-2.
Y. Zhang and M. Sun. Dynamic flight stability of a hovering model insect: theory
versus simulation using equations of motion coupled with navier-stokes equation.
Acta Mech. Sinica, 26(4):509–520, 2010. doi:10.1007/s10409-010-0360-5.
102 References
Chapter 5
Flapping flight control
Since the hovering flapping flight is inherently unstable, it needs to be stabilized
passively (by additional damping surfaces like tails or sails) or actively (by wing
motion). In this chapter an active attitude controller is developed based on the
linearised cycle-averaged model from the previous chapter. On top of that, a 4
DOF flight controller is added to control the heading and the flight velocity in any
direction. Wing motion parameters suitable for thrust and control moment mod-
ulation are selected. Finally, the controller performance is demonstrated through
simulations with the full non-linear model from Chapter 3.
5.1 Control design
Many strategies of tail-less flapping flight control can be found in the literature. It
has been demonstrated on a real insect-sized MAV that a simple angular rate feed-
back succeeds to stabilize the attitude (Fuller et al., 2014), confirming the previous
chapter conclusions. For flight control, most works use design techniques based on
output or state feedback with decentralized (Doman et al., 2010) or centralized ar-
chitectures (Deng et al., 2006b,a; Rifai et al., 2009; Rakotomamonjy et al., 2010; Ma
et al., 2013). Nevertheless, more elaborate control techniques have also been em-
ployed (Chung and Dorothy, 2010). Most of the controllers were tested numerically
in non-linear simulations, often based on quasi-steady aerodynamics. However, the
controller of Ma et al. (2013) succeeded to control near hover flight of a real insect-
sized MAV inside a flight arena with a position tracking system.
The control design presented here uses the linearised model developed in the pre-
vious chapters. The controllers are chosen and tuned by pole placement method.
The MAV parameters were given in Section 4.1. According to the conclusions of the
previous chapter, the wing shoulders are placed at zw= 10 mm. Nevertheless, the
103
104 5 Flapping flight control
described approach should be applicable to any MAV with vertical body posture in
hovering (Ixz = 0) and with wing shoulders placed sufficiently high above the COG.
The dynamics of such an MAV splits into 4 subsystems (pitch, roll, yaw and vertical
dynamics) that can be treated separately. Pitch and roll dynamics are unstable,
therefore the attitude is stabilized first. The second step is to control the flight
velocity (u, v, w) and the steering (r). In this section it is assumed that an arbitrary
external moment and force can be applied on the MAV body. The generation of
these control moments and forces by wing motion will be discussed in Section 5.2.
From now on, the subscript ext of external efforts will be dropped, i.e. Lext =L,
etc.
5.1.1 Pitch dynamics
The pitch dynamics has 2 DOFs: the pitch angle ϑand the longitudinal velocity u.
However, they do not need to be controlled independently. Similar to quadrocopters,
e.g. Michael et al. (2010), the flight speed can be controlled by pitching the body
forward and backward, which modulates the longitudinal component of the total
thrust vector. Thus, the pitch dynamics can be controlled through a single input,
the pitch moment M, and the state space model (3.45) can be rewritten as a set of
transfer functions
U(s)
ˆ
M(s)=ˆ
Xqs+g
s3−(ˆ
Xu+ˆ
Mq)s2+ ( ˆ
Xuˆ
Mq−ˆ
Muˆ
Xq)s−ˆ
Mug(5.1)
Θ(s)
ˆ
M(s)=s−ˆ
Xu
s3−(ˆ
Xu+ˆ
Mq)s2+ ( ˆ
Xuˆ
Mq−ˆ
Muˆ
Xq)s−ˆ
Mug.(5.2)
The system poles are given by the denominator roots. For wings placed above the
COG (zw>0), the derivative ˆ
Muis negative, which results into a stable real pole
and an unstable pair of complex-conjugate poles (see previous chapter). The zeros
are given by the numerator roots and are different for the two transfer functions:
z=−g/ ˆ
Xqfor forward motion and z=ˆ
Xufor pitch (another zero at z= 0 would
appear for transfer function to pitch rate qdue to derivation).
For the hummingbird robot parameters and zw= 10 mm we get the poles λ1=−8.2
and λ2,3= 2.11±5.65i and zeros z= 215 for forward motion and z=−1.1 for pitch.
The open loop pole-zero maps are displayed in Figure 5.1.
The proposed controller uses a cascade structure according to Figure 5.2. The in-
nermost loop 1 controls the pitch rate q, the middle loop 2 controls the pitch angle ϑ
and the outer loop 3 controls the forward speed u. The first two loops can be imple-
mented on-board of an MAV equipped with an Inertial Measurement Unit (IMU),
5.1 Control design 105
−50 0 50 100 150 200 250
−6
−4
−2
0
2
4
6
U(s)/M(s) − open loop
Real Axis
Imag Axis
−10 −5 0
−6
−4
−2
0
2
4
6
Θ(s)/M(s) − open loop
Real Axis
Imag Axis
Figure 5.1: Open loop pole-zero maps of pitch dynamics: forward motion (left) and pitch
angle (right).
which senses the attitude and angular velocities. The third loop represents the pilot
operating the MAV or an autopilot, if an external measurement system like motion
tracking or GPS is available. Each loop compensator was tuned by pole-placement,
starting from the innermost loop. As it will be shown in Section 5.3, the output of
the full non-linear model needs to be cycle-averaged to remove the oscillation due
to flapping from the feedback. This introduces a time delay not included in the
linearised model. Thus, an emphasis was put on keeping the phase margin large in
order to make the controller more robust.
It has been shown in the previous chapter that the pitch rate can be stabilized by a
rate feedback. However, much better performance is achieved by adding an integra-
tor (cancelling the system zero at the origin) and a negative zero, i.e. a PI controller.
The resulting root-locus and Bode plot for the selected gain are shown in Figure 5.3.
Such a compensator reduces the steady state error and has a relatively high phase
margin of 77◦. The step response of the linear system is plotted in Figure 5.6 (left).
The steady state error is still present but it will be compensated by the outer loops.
+
-
M
+
-
PI
uref
+
-
PIP
qϑu
Q(s)/M(s) 1/s
ϑref qref
LOOP 2
LOOP 3
LOOP 1
U(s)/Θ(s)
Figure 5.2: Cascade control of pitch dynamics: loop 1 controls the pitch rate q, loop 2 the
pitch angle ϑand loop 3 the longitudinal velocity u.
106 5 Flapping flight control
10 210 1100101
360
180
0
P.M.: 77 deg
Freq: 4.99 Hz
Frequency (Hz)
Phase (deg)
10
5
0
5
10
15
20
G.M.: 18 dB
Freq: 1.05 Hz
Magnitude (dB)
20 15 10 5 0 5
8
6
4
2
0
2
4
6
8
Real Axis
Imag Axis
Figure 5.3: Loop 1 - pitch rate q: root locus (left) and Bode plot1for the selected gain
(right).
10 210 1100101102
180
135
90
45
P.M.: 74.1 deg
Freq: 1.86 Hz
Frequency (Hz)
Phase (deg)
60
40
20
0
20
40
G.M.: inf
Freq: Inf
Magnitude (dB)
15 10 5 0
50
40
30
20
10
0
10
20
30
40
50
Real Axis
Imag Axis
Figure 5.4: Loop 2 - pitch angle ϑ: root locus (left) and Bode plot for the selected gain
(right).
1In non-minimum phase systems (with poles/zeros in the right hand plane) the Bode plot follows
less common rules and the stability condition gets modified. If an increasing gain brings the unstable
system to stability and if its magnitude crosses 0 dB once, the stability criterion requires that
magnitude is greater than 0 dB at −180◦phase. The definitions of the gain and phase margins are
modified accordingly.
5.1 Control design 107
10 210 1100101102103
P.M.: 65.5 deg
Freq: 0.696 Hz
Frequency (Hz)
Phase (deg)
150
100
50
0
50
G.M.: 14.3 dB
Freq: 2.85 Hz
Magnitude (dB)
0 50 100 150 200 250 300
100
50
0
50
100
Real Axis
Imag Axis
2 1.5 1 0.5 0
1
0.5
0
0.5
1
Imag Axis
270
180
90
360
Figure 5.5: Loop 3 - longitudinal velocity u: root locus (top, left) with detail views
(bottom) and Bode plot for the selected gain (top, right).
The output of the first closed loop is integrated to obtain the pitch angle ϑ, which
results into an additional system pole at the origin. The second loop is controlled
by a simple proportional feedback. The root locus and the Bode plot are shown
in Figure 5.4. The phase margin is 74.1◦and the step response (Figure 5.6 centre)
shows that closed loop system converges quickly to the desired value.
The output of the second closed loop is multiplied by the transfer function U(s)/Θ(s) =
(ˆ
Xqs+g)/(s−ˆ
Xu) to get the longitudinal speed u. Thus, the small negative zero of
the pitch motion is cancelled and a large positive zero of forward motion is added.
A PI controller (an integrator and a negative zero) is used to control uin the last
loop. The root locus and bode plot is shown in Figure 5.5. The chosen controller
gain results into a phase margin of 65.5◦. The step response is displayed in Figure
5.6 (right) showing a good performance with very little overshoot.
108 5 Flapping flight control
0 1 2 3
1
0
1
2
3
x 10 3
0 1 2 3
1
0
1
2
3
x 10 3
0 1 2 3
1
0
1
2
3
x 10 3
0 1 2 3
0.5
0
0.5
1
1.5
0 1 2 3
2
0
2
4
6
8
0 1 2 3
2
0
2
4
0 1 2 3
0.5
0
0.5
1
1.5
0 1 2 3
0.5
0
0.5
1
1.5
0 1 2 3
0.2
0
0.2
0.4
0 1 2 3
2
0
2
4
6
8
0 1 2 3
2
0
2
4
6
8
0 1 2 3
0.5
0
0.5
1
1.5
Loop 1: qref = 1
xB
zB
OB
M
u
ϑ
q
Loop 2: ϑref = 1 Loop 3: uref = 1
u [m.s-1]ϑ [rad] q [s-1]M [Nm]
time [s] time [s] time [s]
Figure 5.6: Step response of the linearized pitch dynamics in closed loop: loop 1 - pitch
rate control (left), loop 2 - pitch angle control (center) and loop 3 - longitudinal speed control
(right).
5.1 Control design 109
5.1.2 Roll dynamics
The control of the roll dynamics is nearly identical to pitch. The lateral flight speed
is controlled by rolling the body left or right. Thus, only one input, the roll moment
L, is considered. The system can be represented by a pair of transfer functions
V(s)
ˆ
L(s)=ˆ
Yps−g
s3−(ˆ
Yv+ˆ
Lp)s2+ ( ˆ
Yvˆ
Lp−ˆ
Lvˆ
Yp)s+ˆ
Lvg(5.3)
Φ(s)
ˆ
L(s)=s−ˆ
Yv
s3−(ˆ
Yv+ˆ
Lp)s2+ ( ˆ
Yvˆ
Lp−ˆ
Lvˆ
Yp)s+ˆ
Lvg.(5.4)
For wings well above the COG the denominator roots give one stable real pole and
an unstable pair of complex-conjugate poles. Unlike in longitudinal direction, the
lateral motion zero is negative, placed at z=g/ ˆ
Yp(derivative ˆ
Ypis always neg-
ative). The roll motion zero lies at z=ˆ
Yv(another zero would lie at z= 0 for
transfer function to roll rate p). For the hummingbird robot parameters we get
the system poles λ1=−9.2 and λ2,3= 0.2±3.8i and zeros z=−258 for lateral
motion and z=−1.4 for roll. The open loop pole-zero maps are shown in Figure 5.7.
V(s)/L(s) − open loop
−300 −250 −200 −150 −100 −50 0 50
−4
−2
0
2
4
Real Axis
Imag Axis
Φ(s)/L(s) − open loop
−10 −8 −6 −4 −2 0 2
−4
−2
0
2
4
Real Axis
Imag Axis
Figure 5.7: Open loop pole-zero maps of roll dynamics: sideways motion (left) and roll
angle (right).
Again a cascade controller is used, where the innermost loop 1 controls the roll rate
p, the middle loop 2 controls the roll angle ϕand the outer loop 3 controls the
lateral speed v(Figure 5.8). The compensators were chosen the same as in the pitch
system: PI for loops 1 and 3 and P for loop 2. They were tuned by pole placement.
110 5 Flapping flight control
+
-
L
+
-
PI
vref
+
-
PIP
pφv
Q(s)/M(s) 1/s
φref pref
LOOP 2
LOOP 3
LOOP 1
U(s)/Θ(s)
Figure 5.8: Cascade control of roll dynamics: loop 1 controls the roll rate p, loop 2 the
pitch angle ϕand loop 3 the lateral velocity v.
10 210 1100101
360
270
180
90
P.M.: 83.4 deg
Freq: 3.34 Hz
Frequency (Hz)
Phase (deg)
10
0
10
20
30
40
G.M.: 34.7 dB
Freq: 0.618 Hz
Magnitude (dB)
15 10 5 0 5
5
4
3
2
1
0
1
2
3
4
5
Real Axis
Imag Axis
Figure 5.9: Loop 1 - roll rate p: root locus (left) and Bode plot for the selected gain (right).
10 210 1100101102
180
135
90
45
P.M.: 69 deg
Freq: 1.46 Hz
Frequency (Hz)
Phase (deg)
80
60
40
20
0
20
40
G.M.: inf
Freq: Inf
Magnitude (dB)
15 10 5 0
30
20
10
0
10
20
30
Real Axis
Imag Axis
Figure 5.10: Loop 2 - roll angle ϕ: root locus (left) and Bode plot for the selected gain
(right).
5.1 Control design 111
10 1100101102103
270
225
180
135
90
P.M.: 77.4 deg
Freq: 0.423 Hz
Frequency (Hz)
Phase (deg)
150
100
50
0
50
G.M.: 17.5 dB
Freq: 2.37 Hz
Magnitude (dB)
250 200 150 100 50 0
20
15
10
5
0
5
10
15
20
Real Axis
Imag Axis
10 5 0
15
10
5
0
5
10
15
Real Axis
Imag Axis
1.5 1 0.5 0
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
Real Axis
Imag Axis
Figure 5.11: Loop 3 - lateral velocity v: negative root locus (top, left) with detail views
(bottom) and Bode plot for the selected gain (top, right).
The root loci and the bode plots for the three loops are shown in Figures 5.9-5.11.
The plots of loops 1 and 2 are equivalent to those in pitch. The loop 3, however,
is different. A negative gain is required because the body needs to roll in negative
ϕdirection to move in positive vdirection and also both system zeros are negative.
The phase margins of the loops 1, 2 and 3 are 83.4◦, 69◦and 77.4◦, respectively.
The step responses for all three compensators are plotted in Figure 5.12. The per-
formance is comparable to the pitch system. The lateral speed response is slightly
slower due to a smaller gain, because a higher phase margin was preferred.
112 5 Flapping flight control
0 1 2 3
5
0
5
10
15
x 10 4
0123
0
10
20
x 10 4
0 1 2 3
10
5
0
5
x 10 4
0 1 2 3
0.5
0
0.5
1
1.5
0123
2
0
2
4
6
8
0 1 2 3
3
2
1
0
1
0 1 2 3
0.5
0
0.5
1
1.5
0123
0.5
0
0.5
1
1.5
0 1 2 3
0.4
0.2
0
0.2
0 1 2 3
8
6
4
2
0
2
0123
8
6
4
2
0
2
0 1 2 3
0.5
0
0.5
1
1.5
Loop 1: pref = 1 Loop 2: φref = 1 Loop 3: vref = 1
time [s] time [s] time [s]
v [m.s-1]φ [rad] p [s-1]L [Nm]
OB
yB
zB
φ
vp
L
Figure 5.12: Step response of the linearized roll dynamics in closed loop: loop 1 - roll rate
control (left), loop 2 - roll angle control (center) and loop 3 - lateral speed control (right).
5.1.3 Yaw and vertical dynamics
The yaw dynamics is a SISO (single input single output) system that can be repre-
sented by the transfer function from yaw moment Nto yaw rate r
5.1 Control design 113
R(s)
ˆ
N(s)=1
s−ˆ
Nr
.(5.5)
Because ˆ
Nris negative, the system pole λ=ˆ
Nris always stable (see Section 4.4).
For the considered MAV parameters we get λ=−54.9. While the yaw could be
controlled by a simple proportional feedback, a pure integral feedback results into
a zero steady state error. The root locus showing the selected gain and the closed
loop step response are displayed in Figure 5.13 (left).
The vertical dynamics is also a SISO system with a transfer function from vertical
force Zto vertical speed w
W(s)
ˆ
Z(s)=1
s−ˆ
Zw
.(5.6)
The system is also passively stable, because the derivative ˆ
Zwis negative, but the
aerodynamic damping is much smaller than in yaw (λ=−0.027 for the considered
MAV). Thus, a PI compensator is used for vertical speed control. The root locus
and step response are shown in Figure 5.13 (right).
−60 −50 −40 −30 −20 −10 0
−20
−15
−10
−5
0
5
10
15
20
Real Axis
Imag Axis
Yaw rate
−12 −10 −8 −6 −4 −2 0
−5
0
5
Real Axis
Imag Axis
Vertical speed
time [s]
r [s-1]
0 0.05 0.1 0.15 0.2 0.25
0
0.5
1
w [m.s-1]
0 0.2 0.4 0.6 0.8
0
0.5
1
time [s]
Figure 5.13: Root locus (top row) and step response (bottom row) of the yaw rate (left)
and vertical speed (right) controllers. The vertical speed system pole is very close to the
origin and so it is overlapped by the integrator pole in the root locus.
114 5 Flapping flight control
5.1.4 Complete controller
The scheme of the complete controller with all 4 subsystems is shown in Figure
5.14. The right dashed rectangle highlights the attitude controller, the velocity
controller is inside the dashed rectangle on the left and represents the (auto)pilot.
The compensator parameters are summarized in Table 5.1.
+
-
M
+
-
PI
uref
+
-
PIP
qϑu
Q(s)/M(s) 1/s U(s)/Θ(s)
ϑref
+
-
Z
PI
wref w
qref
+
-
N
I
rref r
+
-
L
+
-
PI
vref
+
-
PIP
pφv
P(s)/L(s) 1/s V(s)/Φ(s)
φref pref
W(s)/Z(s)
R(s)/N(s)
VELOCITY
CONTROL ATTITUDE CONTROL LINEARISED DYNAMICS
Figure 5.14: Cascade control of the linearised system.
Transfer function KPKI
q0.0003s+ 10
s0.0003 0.003
Pitch dynamics ϑ10 10 -
u0.5s+ 1.3
s0.5 0.65
p0.0002s+ 10
s0.0002 0.002
Roll dynamics ϕ9 9 -
v-0.3s+ 1.2
s-0.3 -0.36
Yaw dynamics r0.003 1
s- 0.003
Vertical dynamics w0.3s+ 5
s0.3 1.5
Table 5.1: Control parameters: the compensator transfer function and an equivalent PI
controller with a proportional gain KPand an integral gain KI.
5.2 Control moment generation 115
5.2 Control moment generation
The proposed controller can stabilize the attitude and control the flight in 4 DOF by
modulating an external vertical force Zand moments around the three body axes
L,Mand N. In reality, these control efforts need to be generated through changes
of wing kinematics.
5.2.1 Control derivatives
As it was already presented in Section 3.2.1, the effects of the wing motion parameter
changes ∆p on the generated efforts are given by the control derivatives matrix J
defined as
J=
Xp1Xp2··· Xpn
Yp1Yp2··· Ypn
.
.
..
.
.....
.
.
Np1Np2··· Npn
.(5.7)
Each matrix element represents a partial derivative of a cycle averaged force / mo-
ment component with respect to one of the wing motion parameters, pi. The deriva-
tives are calculated in the equilibrium with states xeand wing motion parameters
pe; a shortened notation is used as in the previous sections, i.e. Xpi=∂X
∂pi(xe,pe).
The wing motion parameter changes ∆p, necessary to generate the desired combi-
nation of forces and moments, can be computed by relation
∆p =J+[X, Y, Z, L, M, N ]T,(5.8)
where J+is a pseudo-inverse of the matrix J. Finally, these changes are added to
the equilibrium values
p=pe+∆p.(5.9)
The control derivatives were evaluated numerically using the approach described in
Section 3.2.2, assuming the set of wing kinematics parameters paccording to Sec-
tion 3.1.1. The results are split into two parts. First, we change the parameters
symmetrically, i.e. the parameter changes made on both wings are equal and have
identical signs. Second, we change the parameters asymmetrically, i.e. the changes
done on the two wings have equal magnitudes but opposite signs.
116 5 Flapping flight control
When applying the wing kinematic changes symmetrically (subscript S) only the
longitudinal system efforts (X,Zand M) are modified. The linearised relationship
can be written as
[X, Z, M ]T=JS[φmS , φ0S, αmS , α0S, ϕαS, δm1S, δm2S, βS, fS]T.(5.10)
The matrix JSwas normalized to have comparable values between the forces and
moments: the row belonging to the moment Mwas divided by a characteristic
length RCP =Rˆr2(the CP position on the wing). The normalized matrix with
units mN/◦, or mN/Hz in case of the flapping frequency, was evaluated as
˜
JS=
φmS φ0SαmS α0SϕαS δm1Sδm2SβSfS
X0 0.59 0 −4.8 0 0.29 0 −3.4 0
Z4.9 0 1.4 0 −0.6 0 4.3 0 15
M
RCP 0−2.5 0 −0.41 0 2.9 0 −0.44 0
.(5.11)
For asymmetric changes of wing kinematics (subscript A) only the lateral system
force and moments Y,Land Nare affected. The relationship can be written as
[Y, L, N ]T=JA[φmA, φ0A, αmA , α0A, ϕαA, δm1A, δm2A, βA]T,(5.12)
where the control derivatives matrix, normalized as before, is
˜
JA=
φmA φ0AαmA α0AϕαA δm1Aδm2AβA
Y−0.66 0 0.31 0 1.9 0 −0.29 0
L
RCP −4.3 0 −2 0 0.47 0 −7.2 0
N
RCP 0 0.2 0 −7.800.094 0 −3.8
.(5.13)
By studying the matrices ˜
JSand ˜
JAwe can identify two groups of parameters ac-
cording to their effect on the generated forces and moments. The first group includes
flapping frequency f, sweep amplitude φm, angle of attack amplitude αm, phase shift
ϕαand amplitude of figure eight-like deviation δm2(Figure 5.15 left). If we modify
these parameters symmetrically on both wings, we control the vertical force Z. If
we modify these parameters, excluding the flapping frequency, asymmetrically (with
positive sign on the left wing and negative on the right wing) we modulate the roll
moment Land lateral force Y.
The second group includes sweep angle offset φ0, angle of attack offset α0, amplitude
of oval-like deviation δm1and mean stroke plane inclination β(Figure 5.15 right).
Symmetric changes of these parameters result into pitch moment Mand longitu-
dinal force Xmodulation. Same parameters taken asymmetrically modify the yaw
moment N.
5.2 Control moment generation 117
ϕm
δm2
αm
β
δm1
ϕ0
αm+α0
αm-α0
Δsymmetric → Z
Δasymmetric → L & Y
Δsymmetric → M & X
Δasymmetric → N
flapping
frequency
f
phase
shift
φα
Figure 5.15: Wing kinematics parameters, split into two groups according to their effect
on the cycle averaged forces: parameters affecting Z,Land Y(left) and M,Xand N
(right).
5.2.2 Choice of control parameters
In the MAV design the number of parameters needed for control needs to be min-
imized. The matrices (5.11) and (5.13) show that two parameters modified sym-
metrically and two modified asymmetrically are sufficient to generate independently
the four control forces/moments Z,L,Mand N. While this leaves no control of
the ”parasite” forces Xand Y, it allows a simpler design of the MAV wing motion
control mechanism. Another two parameters, one symmetric and one asymmetric,
would be necessary to assure that Xand Yis zero.
For the selected 4 parameters (pS1controlling Z,pS2controlling M,pA1controlling
Land pA2controlling N) the control derivatives matrix reduces to Jred and the
control force and moments can be expressed as
Z
M
L
N
=Jred
∆pS1
∆pS2
∆pA1
∆pA2
=
ZpS10 0 0
0 0 LpA10
0MpS20 0
0 0 0 NpA2
∆pS1
∆pS2
∆pA1
∆pA2
.(5.14)
118 5 Flapping flight control
The relationship shows that the control system is, in theory, fully decoupled. Finally,
the control force and moments are transformed into the wing kinematic parameters
as
[∆pS1,∆pS2,∆pA1,∆pA2]T=J−1
red [X, L, M, N ]T.(5.15)
There are many possible choices of the control parameters set. Since the effect on
Xand Yforces is ignored, a successful control is not guaranteed. Thus, the control
performance of each selected combination needs to be tested in a non-linear simu-
lation. Finally, the choice of the control parameters is heavily constrained by the
design feasibility of the wing motion control mechanism.
Because the designed MAV is powered by a DC motor, the flapping frequency can be
controlled easily. It was chosen as the symmetric parameter controlling the vertical
force Z. The choice of the remaining parameters reflects the control mechanism that
will be presented in Section 7.2 and is sketched in Figure 5.16. The roll moment
Lis controlled by asymmetric modulation of the flapping amplitude φm, the pitch
moment Mby symmetric modulation of the mean wing position (offset) φ0and the
yaw moment by asymmetric modulation of the offset φ0.
L (roll) M (pitch) N (yaw)
−Δϕm+Δϕm−Δϕ0−Δϕ0+Δϕ0−Δϕ0
Figure 5.16: Control moments generated through wing kinematics parameters φmand φ0.
5.3 Simulation results
Figure 5.17 shows the control scheme with the full, non-linear model that has been
implemented in Matlab/Simulink. The model includes the non-linear aerodynamics
and body dynamics from Chapter 3. A moving average over the last wingbeat is
used to remove the periodic oscillation due to flapping from the feedback. To test
the controller performance, a trajectory with step commands in all four controlled
DOFs (velocities u,v,w, angular velocity r) was used. First a step input was applied
successively on each DOF, one at a time, to show the control performance with the
coupling effects of the rest of the nonlinear system. Then, the step input was applied
to all the 4 DOFs at once to reveal any control cross-coupling effects and evaluate
5.3 Simulation results 119
VELOCITY
CONTROL
+
-
AERO-
DYNAMICS
+
BODY
DYNAMICS
M
Z
w
ϑ
u
+
-
PI
+-PI
CONTROL
DERIVATIVES
MATRIX
J+
Δp
wref
uref
+
-L
N
+
-
PI
+-
rref
vref
I
v
r
φ
wing
kinematics
parameters
p
q
+
-
+
-PI
PI
P
P
ATTITUDE CONTROL
ū
w
̄
v̄
r̄
φ̄
ϑ̄q̄
p̄
CYCLE
AVERAGING
CYCLE
AVERAGING
++
p0
Figure 5.17: Cascade control of the full non-linear system.
the potential decrease of control performance. The control commands were set to
0.1 m/s and 10◦/s for the linear and angular velocities, respectively, representing
slow flight around hovering.
The simulation results are shown in Figure 5.18 where the green solid lines repre-
sent the full non-linear model, the black dotted lines the linearised model and the
red dashed lines the command. Unlike in the linearised model, there exists small
coupling between the pitch and vertical dynamics and between the roll and yaw
dynamics in the non-linear model. Nevertheless, the longitudinal (pitch + vertical
dynamics) and lateral (roll + yaw dynamics) systems stay independent.
The high frequency oscillation in the longitudinal DOFs (u,w,ϑ) of the non-linear
system is caused by the periodic aerodynamic forces due to the wing flapping. The
mean values, however, are closely following the linear results. In the lateral DOFs
(v,r,ϕ) the oscillation appears only when the left and right wing kinematics or flow
conditions differ. It is in particular pronounced during the lateral motion, where
an asymmetrically varying wing drag induces high yaw oscillation (see Figure 4.13).
Yet again, the non-linear system response is reasonably close to the linearised model.
The time behaviour of the control parameters (Figure 5.19) shows that the neces-
sary changes in the wing kinematics are very small. The vertical flight controller
varies the flapping frequency fby approximately ±2 Hz. The lateral flight control
requires changes of flapping amplitude φmby only ±0.5◦. The wing offset φ0is used
for both longitudinal flight and yaw control. While variation of ±1.5◦is enough
for longitudinal speed control, up to ±4◦is required for yaw rotation. This is in
120 5 Flapping flight control
0 2 4 6 8 10 12 14 16 18 20
0.1
0
0.1
0 2 4 6 8 10 12 14 16 18 20
0
0.05
0.1
0 2 4 6 8 10 12 14 16 18 20
0.05
0
0.05
0.1
0.15
0 2 4 6 8 10 12 14 16 18 20
40
20
0
20
40
0 2 4 6 8 10 12 14 16 18 20
2
1
0
1
0 2 4 6 8 10 12 14 16 18 20
5
0
5
0 2 4 6 8 10 12 14 16 18 20
0
20
40
Longitudinal
speed
u [m.s-1]
Lateral
speed
v [m.s-1]
Vertical
speed
w [m.s-1]
Yaw
rate
r [°.s-1]
Roll
angle
φ [°]
Pitch
angle
ϑ [°]
Yaw
angle
ψ [°]
Time [s]
Full model Command Linear model
Individual
steps Combined
steps
Figure 5.18: Simulation results: response to a sequence of step inputs and a combined
step in the 4 controlled DOFs (u,v,wand r). Comparison of the full, non-linear model
with the cycle-averaged linearised model.
.
5.3 Simulation results 121
0 2 4 6 8 10 12 14 16 18 20
24
26
28
0 2 4 6 8 10 12 14 16 18 20
79.5
80
80.5
0 2 4 6 8 10 12 14 16 18 20
−5
0
5left right
Flapping
frequency
f [Hz]
Flapping
amplitude
ϕm [°]
Flapping
offset
ϕ0 [°]
Time [s]
Figure 5.19: Simulation results: wing kinematic parameters controlling the flight during
the test sequence in Figure 5.18.
0 2 4 6 8 10 12 14 16 18 20
0.05
0
0.05
0 2 4 6 8 10 12 14 16 18 20
0.1
0
0.1
0 2 4 6 8 10 12 14 16 18 20
0.04
0.02
0
0.02
0.04
Roll moment
L [mNm]
Time [s]
Pitch moment
M [mNm]
Yaw moment
N [mNm]
Figure 5.20: Simulation results: estimates of cycle averaged control moments generated
by the wings during the test sequence in Figure 5.18.
122 References
accordance with the control derivatives matrix (5.13), which shows that the effect of
φ0on yaw moment Nis relatively low. A more effective parameter for yaw control
could be chosen, but a solution using the same parameter for both pitch and roll
was preferred as it reduces the MAV design complexity. No significant change in
the overall performance is observed in the combined command and also the wing
kinematics parameters remain in a similar range.
Figure 5.20 displays the control moments generated by the wings with the modified
wing kinematics. They were obtained by averaging the wing aerodynamic efforts
over each flapping cycle while removing the component due to aerodynamic damping
(estimated by the linearised model). Their maximum absolute values are |L|max ≈
0.05 mNm (roll), |M|max ≈0.13 mNm (pitch) and |N|max ≈0.04 mNm (yaw).
5.4 Conclusion
The controller of near-hover flapping flight presented in this chapter can stabilize
and control the flight in 4 DOF by altering 4 parameters: the flapping frequency, the
difference between the left and right wing flapping amplitude and, independently,
the left and right wing offset. The controller was tuned for the linearised, cycle-
averaged model. Its performance was tested in a non-linear simulation that showed
good agreement between the two models. The study provides an initial estimate of
the necessary wing kinematics changes as well as of control moments these changes
should generate. A prototype designed based on these results will be presented in
the following chapters.
5.5 References
S. Chung and M. Dorothy. Neurobiologically inspired control of engineered flapping
flight. Journal of Guidance, Control, and Dynamics, 33(2):440–453, 2010.
X. Deng, L. Schenato, and S. Sastry. Flapping flight for biomimetic robotic insects:
part II-flight control design. IEEE Transactions on Robotics, 22(4):789–803, Aug.
2006a. doi:10.1109/TRO.2006.875483.
X. Deng, L. Schenato, W. C. Wu, and S. S. Sastry. Flapping flight for biomimetic
robotic insects: part I-system modeling. IEEE Transactions on Robotics, 22(4):
776–788, Aug. 2006b. doi:10.1109/TRO.2006.875480.
D. B. Doman, M. W. Oppenheimer, and D. O. Sigthorsson. Wingbeat shape modula-
tion for flapping-wing micro-air-vehicle control during hover. Journal of Guidance,
Control, and Dynamics, 33(3):724–739, 2010. doi:10.2514/1.47146.
References 123
G. F. Franklin, J. Powel, and A. Emami-Naeini. Feedback Control of Dynamic
Systems. Prentice Hall, 4th edition, 2002.
S. B. Fuller, M. Karpelson, A. Censi, K. Y. Ma, and R. J. Wood. Controlling free
flight of a robotic fly using an onboard vision sensor inspired by insect ocelli.
Journal of The Royal Society Interface, 11(97), 2014. doi:10.1098/rsif.2014.0281.
K. Y. Ma, P. Chirarattananon, S. B. Fuller, and R. J. Wood. Controlled
flight of a biologically inspired, insect-scale robot. Science, 340:603–607, 2013.
doi:10.1126/science.1231806.
N. Michael, D. Mellinger, Q. Lindsey, and V. Kumar. The GRASP Multi-
ple Micro UAV Testbed. IEEE Robotics and Automation, 17(3):56–65, 2010.
doi:10.1109/MRA.2010.937855.
T. Rakotomamonjy, M. Ouladsine, and T. L. Moing. Longitudinal modelling and
control of a flapping-wing micro aerial vehicle. Control engineering practice, 18
(7):679–690, Jul. 2010. doi:10.1016/j.conengprac.2010.02.002.
H. Rifai, J.-F. Guerrero-Castellanos, N. Marchand, and G. Poulin. Bounded at-
titude control of a flapping wing micro aerial vehicle using direct sensors mea-
surements. In 2009 IEEE International Conference on Robotics and Automation,
Kobe, Japan, pages 3644–3650. 2009 IEEE International Conference on Robotics
and Automation, Kobe, Japan, 2009.
124 References
Chapter 6
Flapping mechanism
While the theoretical basis of the hovering flapping flight has been introduced in
the previous text, the remaining chapters are dedicated to the robotic humming-
bird design. Our project aims at a tail-less flapping wing MAV with a wingspan of
around 20 cm and a mass around 20 g. This chapter presents the development of an
uncontrolled robot prototype generating the lift force, consisting of a flapping mech-
anism and a pair of wings. Details are given on its design as well as on conducted
experiments that lead to a successful tethered take-off demonstration.
6.1 Flapping mechanism concept
The lift production is crucial for any air vehicle. The designed hummingbird-like
robot generates the lift by flapping its wings. They need to be driven at a frequency
that is high enough to counteract the gravity and stay airborne. There are many
ways how to obtain the flapping motion - the design of the transmission mechanism
depends strongly on the actuator choice. The vast majority of existing MAV designs
employed rotary electric motors (e.g. de Croon et al., 2009; Keennon et al., 2012).
However, other actuators were also used, including piezo-actuators (Wood, 2008),
SMA actuators (Furst et al., 2013), magnetic actuators (Vanneste et al., 2011) and
even combustion engines (Zdunich et al., 2007).
In this project an electric DC motor was chosen for three main reasons:
1. There is a relatively wide choice of electric motors in the 2 g to 7 g range (10 %
to 35 % of expected total weight).
2. DC motors can be directly powered by off-the-shelf speed controllers and Li-Po
batteries.
125
126 6 Flapping mechanism
3. The motors should have enough power to lift our robot as they are being used
in Radio Controlled (RC) helicopters and multicopters of similar sizes and
weights.
Employing a rotary actuator requires a transmission mechanism that transforms the
rotating motion into the flapping motion of the wings. A linkage mechanism has
been developed for this purpose. It consists of two stages: a slider crank based mech-
anism that generates a rocker motion of (low) amplitude Ψ, and a four-bar linkage
that amplifies the motion to the desired amplitude of Φ = 120◦(Figure 6.1). While
a single stage mechanism could also be used to achieve such an amplitude, the speed
profile of the resulting motion would not be symmetric. Also, such a mechanism
would operate close to its singular positions, which in reality could result in mecha-
nism blocking due to compliance and backlashes. The advantage of using two stages
is that the asymmetries coming from individual stages can be compensated by an
appropriate choice of dimensions and at the same time mechanism singularities can
be further away from the working range.
θ
Ψ
Φ
"slot" ≡
Figure 6.1: The flapping mechanism is composed of two stages: slider crank with a rocker
producing oscillating motion and a four-bar mechanism for motion amplification. Mechanism
model (left) and scheme displaying the motor input θ, intermediate amplitude Ψ and flapping
amplitude Φ at the output (right).
6.1.1 Kinematics
The kinematics of the proposed mechanism can be treated separately for each mech-
anism stage. Using the notation of Figure 6.2 the kinematics of the first stage can
be expressed analytically as
ψ3= arctan
A1−L1cos θ−qL2
2−L2
1sin2θ
L3
+π
2,(6.1)
6.1 Flapping mechanism concept 127
where θis the input angle and ψ3is the angle of the intermediary link 3-4. The
analytic solution of the second (amplification) stage is a classical solution of a four
bar mechanism
φ= arctan a
b−arccos c
√a2+b2,(6.2)
where
a=−2L4L6sin (ψ3−α34)
b= 2A2L6−2L4L6cos (ψ3−α34) (6.3)
c=L2
5−A2
2−L2
4−L2
6+ 2A2L4cos (ψ3−α34).
The dimensions were optimized numerically by minimizing a cost function, which
consisted of a difference from the desired amplitude and a difference between up-
stroke and downstroke velocity profiles; the final dimensions are in Table 6.1. The
relationship of the output angle φon the input angle θis very close to a harmonic
function (Figure 6.3).
Figure 6.2: Kinematic scheme of the flapping mechanism.
L1L2L3L4L5L6A1A2α34
2.25 12.00 8.00 14.00 7.53 3.57 11.00 -9.20 -70
Table 6.1: Mechanism dimensions (lengths in mm, angles in ◦)
The final structure of the flapping mechanism was rearranged (compared to Figure
6.2) to minimize the overall mechanism dimensions. The mutual orientation of the
two stages was adapted as is shown in Figure 6.1 and the wing bar was connected
128 6 Flapping mechanism
0 90 180 270 360
60
30
0
30
60
[°]
[°]
Mechanism
Harmonic
ϕ
θ
Figure 6.3: Kinematics of the flapping mechanism compared to a harmonic function.
to the output link at an angle so that the mean wing position is parallel with the
lateral body axis.
6.1.2 Mechanism design
The design of the flapping mechanism has undergone a lot of development. Some of
the prototype versions are listed in Table 6.2 and displayed in Figure 6.4. The frame
and the links are all 3D printed. Several technologies (FDM, SLS, PolyJet) and var-
ious materials were tested. Finally, the PolyJet technology was chosen because the
parts do not require any further processing thanks to its fine resolution (42 µm in
xy, 16 µm in z - Objet Eden series). The originally used material, DurusWhite,
was replaced by the Digital ABS composite photo-polymer, which is considerably
stronger. The links are connected together and to the frame by steel and aluminium
rivets.
The first mechanism generations (A to E) were designed for bench tests. They were
easy to repair in case of a failure as all the links were accessible. However, the posi-
tion of the motor was not ideal for flight because the COG of the system was not on
the axis between the shoulders. Moving the motor above the mechanism in later gen-
erations (G to J) allowed its placement exactly in the centre. Although this solution
is more complex, harder to repair and means an extra penalty in weight, the frame
is stronger and the wing shoulders are more robust thanks to ball bearings (version
G) or brass bearings (version J) that were integrated into the shoulder design. On
top of that, this configuration allows a relatively easy access to the mechanism from
below, which is necessary for control mechanism implementation (Chapter 7).
Several DC motors of diameters between 6 mm and 8.5 mm were used to drive the
mechanism, see Table 6.3. The majority of them are brushed pager motors adopted
6.1 Flapping mechanism concept 129
Ver. Date Material Motora
Motor Mechanism Total Max. Wing Flapping
Weight Weight Weight Lift Length Frequency
[g] [g] [g] [g] [mm] [Hz]
A2/2012 FDM ABS A-6 1.77 3.43 5.2 - - -
C2 5/2012 DurusWhite A-6 1.77 4.03 5.8 6.4 70 21.1
E2 10/2012 DurusWhite D-7 2.78 4.72 7.5 9.6 70 23.5
E4 1/2014 Digital ABS F-8 5.20 4.90 10.1 16.1 90 26.2
G2 4/2013 DurusWhite A-7 2.69 6.28 9.0 9.6 70 24.5
J2 1/2014 Digital ABS F-8 5.20 7.30 12.5 16 90 23.8
Table 6.2: Evolution of the flapping mechanism. aSee Table 6.3 for motor details.
A C2 E4
G2 J2
Figure 6.4: Evolution of the flapping mechanism - generations A (2/2012) to J (1/2014).
130 6 Flapping mechanism
by RC modellers (BR DC), but professional brushless motors (BL DC) were also
successfully used. The factors limiting the maximal speed of small DC motors are
usually the temperature (the motor is heated by current and friction) and, in case
of brushed motors, the wear of the brushes. It is the latter case that also limits the
lifetime of the brushed motors.
The highest lift was obtained with the 8mm BL DC from Faulhaber. Its advantage
is that the performance remains constant over time, but it needs to be driven by
a speed controller whose stock version (SC 1801P) is too large and heavy (23 mm
x 32 mm, 3.6 g). Thus, a smaller and lighter speed controller needs to be either
found on the market or developed to use this motor in flight. An alternative motor,
giving nearly the same performance, is the 8.5 mm BR DC from AEO-RC. The
performance, however, decreases with time as the motor heats up. The motor lasts
only about 5 minutes when operated at the maximal flapping frequency before its
brushes wear out. Currently, the BL motor is used for bench tests and the BR motor
for take-off demonstrations.
A two stage gearbox is used to reduce the motor speed. The limited space requires
that gears with a small module of 0.3 mm are used. The choice of the reduction
ratio was limited by the gears available on the market. The final gearbox consists
of a 9t pinion, 40t/9t double spur gear and 40t spur gear giving a reduction ratio of
approximately 19.75:1. A custom built gear set with a gear ratio optimized for the
load and motor characteristic combination is being considered.
An exploded view of the most recent flapping mechanism (version J) is shown in
Figure 6.5. Its weight, without the motor, is 7.3 g.
6.1.3 Wing design
The design of the wings is inspired by the Nano Hummingbird (Keennon et al., 2012)
and by the DelFly (de Croon et al., 2009). The wing is made of a thin flat mem-
brane. It has two sleeves, one on the leading edge and one on the root edge (close to
the body), that accommodate the leading edge and root edge CFRP (carbon-fibre-
reinforced polymer) bars. Since the angle between the sleeves is greater than the
angle between the bars the wing becomes cambered and twisted after the assembly
(Figure 6.6). The camber is bistable - it flips passively from one side to another
depending on the direction of motion, as can be seen in Figure 6.7.
The wings are hand built - a reasonable accuracy and repeatability is achieved by
printing the desired shape on a sheet of paper that is attached under the membrane
6.1 Flapping mechanism concept 131
Motor Type m D L kVR Pmax +
[g] [mm] [mm] [rpm/V] [Ω] [W]
A-6 BR*AEO-RC GPS6 1.77 6 14 13000 2.8 1.2
A-7 BR*AEO-RC GPS7 2.69 7 16 11000 2.1 1.5
A-8 BR*AEO-RC EPS8 4.85 8.5 20 12600 0.7 2.5
D-7 BR*Didel MK07-1.7 2.78 7 17 11000 2.1 1.6
F-8 BL Faulhaber 0824-006B 5.20 8 24 5753 3.1 >2.5
Table 6.3: DC motors used along the project. BR = brushed motor, BL = brushless
motor. All motors have output shaft of 1 mm in diameter. *No data-sheets provided,
parameters estimated experimentally. +Maximal output power achieved during experiments
with flapping prototype E4 without motor overheating (approximate values).
DC Motor
Motor adapter
(optional)
Brass bearings
Leading edge bar
Root edge bar
Top frame
Bottom frame
Gearbox
Flapping mechanism
Connecting rivets
Shoulder hinge axis
Motor pinion
Figure 6.5: Exploded view of the flapping mechanism J2.
132 6 Flapping mechanism
BoPET (Mylar) Polyester (Icarex) CFRP
Figure 6.6: Polyester film wing becomes cambered after assembly.
0 ms 2 ms 4 ms 6 ms 8 ms 10 ms
Figure 6.7: High speed camera sequence of a wing flipping at the stroke reversal (top view)
while flapping at 15 Hz. The whole sequence represents about 15 % of the flapping cycle.
and used as a template for cutting. The best results (in terms of lift and durability)
were obtained with a 15 micron thick polyester film. 1 mm x 0.12 mm CFRP bands
are used as stiffeners. The sleeves at the leading edge and at the root edge are
reinforced with Icarex to increase their durability. They allow an easy assembly and
disassembly as well as free rotation around the 0.8 mm leading edge and 0.5 mm
root edge CFRP bars.
In the early prototypes the whole wing frame, including the leading edge and root
edge bars, was mobile, as displayed in Figure 6.8 (left). An adapted version of
this solution has an adjustable angle between the leading edge and root edge bars
(Figure 6.8 centre) and is used to find an optimal angle between the sleeves. Finally
the design was simplified and the mass being moved reduced by fixing the root bar
directly to the frame, making it coincident with the wing shoulder axis (Figure 6.8
right).
6.2 Experiments 133
Figure 6.8: Different solutions of the wing frame: mobile root bar (left), mobile root bar
with adjustable angle (center), fixed root bar (right).
6.2 Experiments
Many tests were carried out throughout the development process. While the main
goal was to maximize the generated lift force by mechanism and wing design opti-
misations, mechanism reliability and wing robustness were also important factors.
A high speed camera (Photron FASTCAM SA3, resolution 1024 x 1024 pixels) was
used to observe the wing behaviour throughout the cycle and to track the mecha-
nism and wing kinematics. The generated forces were measured on a custom built
force balance.
6.2.1 Wing kinematics
While the flapping mechanism was designed for a nominal flapping amplitude Φ0=
120◦, the amplitude will increase with frequency due to compliance of the wing bars
and of the mechanism itself. The high speed camera was employed to quantify these
ϕroot R
ϕR
xB
yB
Figure 6.9: Definition of the tracked angles.
134 6 Flapping mechanism
effects as well as to identify any imperfections compared to the mechanism theoret-
ical kinematics.
The sweep angle was tracked at the wing root (φroot) and at the wing tip (φtip, see
Figure 6.9). The camera was set to record at 2000 fps with a shutter speed of 10
000 fps to remove any motion blur. The amplitude was subsequently calculated as
Φ = φmax −φmin and offset as φ0= (φmax +φmin)/2, where φmax and φmin are the
maximal and minimal observed angles φ, respectively.
Figure 6.10 shows the traces of wing angles of prototype E4 over two wingbeats
at a flapping frequency of 25Hz. The increase of the tracked flapping amplitude
compared to the theoretical model is significant. The amplitude measured at the
wing roots includes the effect of backlashes in the linkage mechanism as well as the
compliance of its links. The compliance of the leading edge bars results into a fur-
ther amplification of the amplitude, measured at the wing tip. It can be observed
that the bars flex in particular during the stroke reversal when the accelerations are
high. Thus, the sweep angle at the wing tip becomes more triangular compared to
the theoretical, nearly harmonic curve.
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
90
60
30
0
30
60
90
Wingbeat [ ]
Sweep angle [°]
Flapping frequency: 25 Hz
tip R
tip L
root R
root L
theory
Figure 6.10: Tracked wing kinematics of prototype E4 at f= 25 Hz. Left wing traces are
in blue, right wing traces in red. The sweep angle was measured at the wing tip (φtip) and
at the output of the mechanism (φroot). Black dash-dotted line represents the theoretical
kinematics, without compliance and backlashes.
6.2 Experiments 135
t= 0 ms
t=20 mst=20 ms
01
2
3
4
5
6
7
8
0
1
2
3
4
5
6
78
t=40 ms
t=20 mst=20 ms
16 15 14
13
12
11
10
9
8
8
9
10
11
12
13
14
15
DOWNSTROKE
UPSTROKE
Figure 6.11: High speed camera measurement: the image displays the wings every 2.5 ms
over one complete cycle at 25 Hz. The sequence is split into the downstroke (frames 0 →8,
top) and the upstroke (frames 8 →16, bottom).
136 6 Flapping mechanism
The small differences in the left and right wing amplitudes are caused by the mech-
anism imperfections. There is, however, a small difference in the timing of the wing
reversal, which is an inherent property coming from the design of the joints between
the slider crank stage and the amplification stages. The right wing reverses before
the left wing when behind the body (φ > 0), while the opposite happens in front
of the body (φ < 0). This can produce a small oscillation around the yaw axis,
nevertheless, the effects should average out over one wingbeat.
A composite image of high speed camera frame sequence showing the wing positions
every 2.5 ms over the full wingbeat is shown in Figure 6.11. We can observe the
leading edge bar deformation close to the reversal, the differences between upstroke
and downstroke of left and right wing as well as the positions of mechanism joints
throughout the wingbeat.
The relationship between the flapping amplitude and offset and the flapping fre-
quency is displayed in Figure 6.12. For the measured range of frequencies (15 to
25 Hz) the mechanism amplitude increases from above 140◦to almost 150◦. The
increase of the wing tip amplitude is more significant, from below 160◦to almost
180◦. As will be shown in Section 6.2.3, this can be considered as beneficial, be-
cause an increase of flapping amplitude means a lower frequency is needed to take
off and a lower frequency means lower accelerations and lower inertial forces on the
mechanism links.
15 20 25
140
150
160
170
180
Frequency [Hz]
Flapping amplitude [°]
wingtip R
wingtip L
wingroot R
wingroot L
Φ
15 20 25
0
2
4
6
8
10
Frequency [Hz]
Mean wing position [°]
wingtip R
wingtip L
wingroot R
wingroot L
ϕ0
Figure 6.12: The tracked flapping amplitude (left) and offset (right) as a function of
flapping frequency. Results measured for prototype E4.
6.2 Experiments 137
The offset remains slightly positive and nearly constant in the studied range (around
4◦on the left and between 4◦and 7◦on the right wing); there is almost no difference
between the wing roots and tips. The small non-zero offset does not represent a
major problem. The generated pitch moment can be compensated by a small shift
of the prototype COG.
6.2.2 Force balance
Measuring the efforts of a flapping wing robot is a challenging task. The generated
forces are relatively small (order of 0.01 N) which requires high sensitivity. On top
of that these efforts are of a periodic nature where not only the flapping frequency
but also the higher harmonics are present. Hence the sensor should have a high
resonance frequency.
The most frequently used commercial 6DOF force-torque sensor in the flapping wing
research is the Nano17 Titanium (ATI, 2014). It is the only sensor on the market
that is compact and has a high sensitivity to forces (resolution up to 1/682 N ≈
0.15 g) and moments (resolution up to 3/364 mNm ≈0.008 mNm) while keeping a
high resonance frequency in all DOFs (3 kHz). However, its price was well above the
project budget so a decision was made to build a custom force balance. To keep the
design simple the balance should be able to measure only lift Zand pitch moment
M. Moreover, only the cycle averaged efforts were of our primary interest.
In the past we already used a precision pocket scale to evaluate the mean lift with
acceptable results. The sensors used in the scales are usually double beam can-
tilevers with strain gages in full bridge configuration. Their advantage is that they
are insensitive to the axial force as well as to the bending moment. The experimen-
tally determined resonance frequency of a sensor extracted from one of the scales
was 210 Hz, roughly 8 times the flapping frequency of the robot prototype. That is
not enough to measure the time histories within one flapping cycle, but sufficient to
evaluate the cycle averaged values. Thus these sensors were selected as inexpensive
yet reasonably precise base components for the designed force balance.
The balance uses two of these single axis force sensors in a configuration that is
shown in Figure 6.13. Applying lift FL, drag FDand moment Mon the balance
results in the following sensor forces
S1= (FDH−M)/B −FL/2
S2= (M−FDH)/B −FL/2 (6.4)
Rx=−FD.
138 6 Flapping mechanism
Figure 6.13: Schematics of the force balance (left) and its free body diagram (right).
Originally a third sensor was used to measure the drag force FD. However, the
first tests showed that the cycle averaged drag force was very low and its effect
on the moment was negligible. Thus the third sensor has been dismounted for the
measurements presented here. This allows to mount the robot closer to the rotation
joints, which increases the resonant frequency of the whole system. The moment at
the robot COG, M, is approximated by the moment at the centre between the two
rotation joints, MC. The measured efforts can be expressed as
FL=−S1−S2
MC=M−FDH= (S2−S1)B/2.(6.5)
The sensitivity of sensors 1 and 2 to the moment can be tuned by the selection of
distance Bbetween the two sensor joints. It was set to 50 mm, giving a good sensi-
tivity yet enough space in between to fix the robot prototype. For small distance H
and small cycle averaged drag force FDthe moment MCis a good approximation of
the true moment M, with an error that can be expressed from equation (6.5). This
has no effect on the lift force precision.
The assembled force balance is shown in Figure 6.14. Each sensor is connected to
a custom build electronic circuit that provides stabilized power to the bridge and
amplifies the bridge output. The sensors have been calibrated one at a time. The
rotational joints in the system should have as little friction as possible. The joint
on the left is constructed as a blade inside a groove. The joint is held together by
a magnetic force of a NdFeB cylindrical magnet that attracts the blade inside the
groove. Both the blade and the groove are from soft magnetic steel. The joint on
the right is built in a similar manner. Since it should also allow displacement to the
sides to have an isostatic system, the blade was replaced by a steel ball that is touch-
ing a flat steel plate. Since the contact of the spherical ball and flat plate is only
in one point, another magnet was attached on the top to increase the attractive force.
6.2 Experiments 139
Steel
blade
Steel
groove
Magnet Magnet
Steel
plate
Steel
ball
Sensor 1 Sensor 2
Sensor 1 Sensor 2
Flapper attachment
Tested prototype
Amplifiers
Figure 6.14: Force balance overview and detail views on sensors and magnetic joints
The force balance signals are processed with a dSpace 1103 digital signal proces-
sor, together with the voltage and current readings of the DC motor. The flapping
frequency can be detected from the motor current, because the motor torque is con-
stantly changing due to the periodic aerodynamic and inertial forces. Nevertheless,
the setup is also equipped with an optional infrared barrier providing a trigger signal
at every wing pass. In case of the brushless DC the frequency is calculated from the
signals of Hall sensors, which are integrated in the motor.
The measured resonant frequency of the balance is 173 Hz for lift and 297 Hz for
pitch moment (Figure 6.15). As expected, this is too low to measure detailed force
and moment traces over a flapping cycle, but the system provides enough bandwidth
to evaluate the cycle averages for the expected flapping frequencies (below 30 Hz).
50 100 150 200 250 300 350 400
10
20
30
40
50
Frequency [Hz]
Magnitude [dB]
Lift force
fpeak = 173 Hz
flapping
frequency
50 100 150 200 250 300 350 400
10
20
30
40
50
Frequency [Hz]
Magnitude [dB]
Pitch moment
fpeak = 297 Hz
flapping
frequency
Figure 6.15: Force balance: measured frequency response gain magnitude of lift (left) and
moment (right) showing the resonance frequencies.
140 6 Flapping mechanism
The averaging is done online and is always calculated over a finite number of cycles.
The averaging interval can be adjusted and was set to 3 seconds for the measure-
ments presented here. The other readings (voltage, current, frequency) are averaged
in the same way. The repeatability of the lift measurements is good, with a typi-
cal standard deviation below 1 mN. We observe a bigger dispersion in the moment
measurements, where a typical standard deviation is 0.05 mNm. This includes ran-
dom effects coming from both the measurement and processing system and from the
tested prototypes.
6.2.3 Lift production
The mean lift force of flapping wings can be approximated by the classical theory
for a fixed wing in steady flow while employing cycle-averaged quantities. A flat
and rigid wing is assumed and, for simplicity, the centre of pressure is placed at the
mid-length RCP =R/2. When a wing flaps at a frequency fwith an amplitude
Φ=2φmits CP moves with a mean velocity
¯
UCP = 2Φf RCP = Φf R. (6.6)
Then, the cycle averaged lift force of a pair of wings can be written as
¯
FL=1
2ρ¯
CL(2S)¯
U2
CP =ρ¯
CLS(ΦfR)2,(6.7)
where Sis the surface of a single wing and ¯
CLis the mean lift coefficient that, if we
accept the quasi-steady assumption, depends on the wing geometry and the angle of
attack variation over one wingbeat. By including the definition of the wing aspect
ratio A= 2R2/S we can rewrite the expression to
¯
FL=1
2ρ¯
CLA(SΦf)2.(6.8)
Since the real wing is flexible it deforms more under higher aerodynamic loads at
higher frequencies and the deformation affects also the angle of attack, so the coeffi-
cient ¯
CLis not likely to stay constant. Nevertheless, we see that the lift force should
primarily vary with the wing lift coefficient, the aspect ratio and with the squares
of wing surface, flapping amplitude and frequency. The validity of this simplified
relation was studied experimentally and will be discussed in the following text.
6.2.4 Wing design optimization
Many wing designs were tested in order to find the size and shape producing the
highest lift force. The parameters studied included the material of the wing (5, 10
6.2 Experiments 141
Figure 6.16: A sample of over 70 wing designs that were built and tested.
and 15 micron polyester film, Icarex, Chikara, ...), the stiffeners size and placement,
the overall wing shape, the length L, the surface S, the aspect ratio A, the taper or
the angle between the sleeves βW. Some of the tested designs are shown in Figure
6.16.
Several difficulties had to be faced during the experiments. The early prototypes
suffered from short lifetime and performance variations due to wear. The same was
true for the brushed motors used. As the flapping mechanism evolved, its reliability
improved significantly and its performance got more constant over time. Using a
quality brushless motor added further to a better consistency of the results.
Nevertheless, the lift force is always determined by the combination of the motor, the
flapping mechanism and the wings. Any change in the whole chain of components
resulted in a change of the maximal force. For these reasons results of each measure-
ment set were related to the results of a nominal wing design (measured together
with each set) rather than compared with absolute values of previous measurements.
Equation (6.8) showed that the mean lift force approximation depends on the wing
geometry (A, S2,¯
CL) and also on the squares of flapping frequency f2and amplitude
Φ2. The flapping amplitude is determined by the flapping mechanism (although it
can still be modified by the choice of the leading edge bars that flex while flapping),
while the flapping frequency remains free. However, keeping it low was preferred
in order to reduce the inertial effects in the flapping mechanism. This implies that
wings with higher aspect ratio, higher surface and of course higher lift coefficients
should be searched. Some examples of the conducted tests are given next.
142 6 Flapping mechanism
4 6 8 10 12 14 16
20
30
40
50
60
70
80
90
Frequency*Current (~ Mech. Power) [A/s]
Lift [mN]
12 14 16 18 20 22 24
20
30
40
50
60
70
80
90
Frequency [Hz]
Lift [mN]
AR = 5.6
AR = 6.4
AR = 7.3
AR = 8.3
AR = 9.3
AR
S = const.
AR AR
Figure 6.17: Lift force of wings with constant surface, but varying aspect ratio and taper
(bottom), plotted against the flapping frequency (top-left) and against the motor current
multiplied by the frequency (top-right) which can be related to the mechanical power driving
the flapping mechanism. The tests were conducted with the prototype E4 and motor A-7.
Figure 6.17 displays the effect of the wing aspect ratio Afor wings with constant
surface (S= 1750 mm2). The wing length was being increased (R= 70 →90 mm)
while decreasing the tip chord. The root chord was kept constant (25 mm) so the
wing transformed from a rectangular towards a triangular shape. Figure 6.17 (top-
left) shows the measured lift plotted against the flapping frequency. As predicted,
the highest lift was produced with the highest aspect ratio wing, at any frequency.
The mechanical power on the motor output Pmech is defined as the output torque
times the angular velocity of the output shaft, Pmech =Tmωm. It represents the
power consumption of the flapping mechanism and wings combination: it includes
the effects of the wing drag, the gearbox efficiency, the mechanism inertia and fric-
tion, but excludes the motor itself. Because the torque on the motor output shaft
Tmis proportional to the motor current I, the current multiplied by the flapping
frequency fcan be related to the motor output mechanical power that drives the
flapping mechanism. Figure 6.17 (top-right) shows that the highest aspect ratio
wing is also the most efficient, on the tested prototype, as it produces the highest
lift per mechanical power unit.
6.2 Experiments 143
10 15 20 25 30
20
40
60
80
100
120
140
160
Frequency [Hz]
Lift [mN]
S = 1750 mm2
S = 2054 mm2
S = 2382 mm2
0.58 0.6 0.62 0.64 0.66 0.68
20
40
60
80
100
120
140
160
Motor efficiency [−]
Lift [mN]
S
AR = const.
SS
Figure 6.18: Lift force of wings with constant aspect ratio, but varying surface. The
measured lift is plotted against the flapping frequency (top-left) and against the efficiency of
the motor driving the flapping mechanism (top-right). The tests were terminated when the
motor recommended heat limit was reached. The tests were conducted with the prototype
E4 and motor F-8.
The effect of the wing area Son the lift force should be even more pronounced as
it appears in the approximation in the second power. The experimental results for
wings of constant aspect ratio (A= 9.3) are shown in Figure 6.18. The surface was
being increased while preserving the overall shape. Indeed, for a constant frequency,
the highest lift was produced by the largest wing (Figure 6.18 top-left).
All the wings produce comparable amount of lift per mechanical watt. To include
also the electric drive, the lift is this time plotted against the motor efficiency ηm
defined as the ratio between the mechanical and electrical power
ηm=Pmech
Pel
=Tmωm
UI ,(6.9)
where Uand Iare the motor voltage and current, respectively. Figure 6.18 (top-
right) shows that the most efficient wing with the used drive (BL DC Faulhaber
0824) is the smallest wing. On top of that, this allows to reach a higher mechanical
power without a risk of motor overheating and thus also a higher absolute lift, al-
though at a cost of increased flapping frequency. These results show that the wings
need to be optimized for every new combination of motor and flapping mechanism
in order to achieve maximal efficiency, which will be crucial for flight endurance.
144 6 Flapping mechanism
βw = 16°
30° cmin = 14 mm
L = 90 mm
cmax = 25 mm
30°
Figure 6.19: Shape and dimensions of the wing producing the highest lift. Its aspect ratio
is A= 9.3.
90 mm
32 mm
Faulhaber
0824
Figure 6.20: Prototype J2 with the best wing producing almost 160 mN of lift at 24 Hz.
It needs to be said that not all the tests were as conclusive as those presented here.
The effects of the stiffeners placement were much more subtle. Some results would
even contradict previous findings due to differences among the used motors and pro-
totypes. Nevertheless, the experimental approach lead to a gradual lift increase.
The highest generated lift so far, almost 160 mN ≈16 g, was produced with the
wings with dimensions shown in Figure 6.19 attached to the mechanism J2 driven
by the 8 mm brushless motor F-8 (Figure 6.20). The measured lift curve is plotted
in Figure 6.21. The grey dash-dotted line represents a trend-line assuming the lift
force FLis proportional to the second power of frequency f, while the lift coef-
ficient ¯
CLand flapping amplitude Φ in equation (6.8) remain constant. The real
measured curve scales with slightly higher power, because the flapping amplitude
also increases with frequency (see Figure 6.12) and the lift coefficient changes due
to wing flexibility. Nevertheless, this shows that FL≈const.f2can be used as a
(conservative) rule of thumb when extrapolating the results.
6.2 Experiments 145
5 10 15 20 25
0
20
40
60
80
100
120
140
160
Frequency [Hz]
Lift [mN]
ftake-off = 21.5 Hz
prototype weight 12.5 g
FL
* = const.f2
0 1 2 3 4
0
20
40
60
80
100
120
140
160
Power [W]
Lift [mN]
mechanical power
electrical power
Figure 6.21: Measured lift of the best wing: relationship on the flapping frequency (left)
and on the mechanical and electrical power of the motor (right). The black dotted line dis-
plays the prototype weight, grey dash-dotted line is a fitted curve assuming FL=const.f2.
Individual measurements are plotted with crosses, the lines connect the average values.
The force necessary to lift the prototype weight (12.5 g) is displayed by the black
dotted line, showing that the prototype would take-off around f= 21.5 Hz. How-
ever, the power source is still off-board and the prototype lacks any control.
Due to the hovering flapping flight inherent instability, a take-off was demonstrated
with a help of guide-wire that fixes the prototype attitude, but allows a free move-
ment along the vertical axis as well as free yaw rotation (Figure 6.22). Older proto-
Figure 6.22: Take-off demonstration: a guide wire was used to stabilize the attitude of
the uncontrolled prototype E4. Power was off-board, brought by 200 microns thin copper
wires and controlled by hand to maintain a constant altitude after the take-off. Time of
each frame is displayed in the upper left corner.
146 References
type E4 was used for this test, because it was easier to attach it to the guide wire.
The power was still off-board and was brought by 200 microns thin copper wires.
A control mechanism needed for active stabilization is being developed and will be
described in the next chapter.
6.3 References
ATI Industrial Automation. Nano17 titanium. https://www.ati-ia.
com/products/ft/ft_models.aspx?id=Nano17+Titanium, 2014. Accessed:
18/08/2014.
G. de Croon, K. de Clerq, R. Ruijsink, B. Remes, and C. de Wagter. Design,
aerodynamics, and vision-based control of the DelFly. International Journal of
Micro Air Vehicles, 1(2):71–97, Jun. 2009. doi:10.1260/175682909789498288.
S. J. Furst, G. Bunget, and S. Seelecke. Design and fabrication of a bat-inspired
flapping-flight platform using shape memory alloy muscles and joints. Smart Ma-
terials and Structures, 22(1):014011, 2013. doi:10.1088/0964-1726/22/1/014011.
M. T. Keennon, K. R. Klingebiel, H. Won, and A. Andriukov. Development of
the nano hummingbird: A tailless flapping wing micro air vehicle. AIAA paper
2012-0588, pages 1–24, 2012.
T. Vanneste, A. Bontemps, X. Q. Bao, S. Grondel, J.-B. Paquet, and E. Cattan.
Polymer-based flapping-wing robotic insects: Progresses in wing fabrication, con-
ception and simulation. In ASME 2011 International Mechanical Engineering
Congress and Exposition, pages 771–778. American Society of Mechanical Engi-
neers, 2011.
R. J. Wood. The first takeoff of a biologically inspired at-scale robotic
insect. IEEE Transactions on Robotics, 24(2):341–347, Apr. 2008.
doi:10.1109/TRO.2008.916997.
P. Zdunich, D. Bilyk, M. MacMaster, D. Loewen, J. DeLaurier, R. Kornbluh, T. Low,
S. Stanford, and D. Holeman. Development and testing of the Mentor flapping-
wing micro air vehicle. Journal of Aircraft, 44(5):1701–1711, 2007.
Chapter 7
Control mechanism
The primary role of the flapping wings is the thrust production. In a tail-less design
the wings have a second, yet equally important, role: the active flight stabilization
and control. To stabilize the body attitude and to steer, the wings need to generate
moments around the three body axes: roll moment Laround xB, pitch moment M
around yBand yaw moment Naround zBaxis.
To produce the control moments the robot needs to be able to modulate the mean
lift and drag forces of each wing independently and also to control the placement of
the cycle-averaged lift force with respect to the centre of gravity, see Figure 7.1. In
such case the roll moment can be generated through a difference between the mean
lift force of the left and of the right wing. The pitch moment can be produced by
placing the mean lift force of both wings in front of or behind the centre of gravity.
Finally, the yaw moment can be generated by a difference between the mean drag
of the left and of the right wing.
xB
OBM
2FL
yB
zB
L
stroke plane
↑FL
↓FL
yB
xB
N−FD
+FD
Roll moment Pitch moment Yaw moment
stroke plane
zB
Figure 7.1: Principle of control moment generation. FLand FDare cycle averaged lift
and drag forces, respectively, generated by a single wing.
147
148 7 Control mechanism
Two possible solutions of the control mechanism, published in (Karasek et al., 2013,
2014), are presented in the following text. The first is based on the wing twist
modulation according to Keennon et al. (2012). The second generates the necessary
moments by modulating the wing flapping amplitude and offset (mean position),
a strategy similar to Ma et al. (2013) and Truong et al. (2014).
7.1 Moment generation via wing twist modulation
The wing design presented in Section 6.1.3 consists of a membrane attached between
the leading edge and root edge bars. At rest the membrane is slightly loose. The
wing becomes cambered and twisted when moved due to a pressure difference. For
a wing with specifically optimized geometry (Figure 7.2 left), the lift force can be
increased by moving the root bar away from the membrane (membrane is stretched,
γ > 0) and decreased by moving it towards the membrane (membrane is loosened,
γ < 0 in Figure 7.2 right). This concept is used in the Nano Hummingbird and is
called the Variable Wing Twist Modulation (Keennon et al., 2012).
γ
γ
Figure 7.2: Wing twist modulation principle adapted from Keennon et al. (2012) (left)
and measurement of lift as a function of angle γfor various flapping frequencies (right).
7.1 Moment generation via wing twist modulation 149
7.1.1 Moment generation principle
If the root bars are attached to the body frame and are displaced with respect to
the frame the angle γwill not stay constant during the wingbeat. It will vary with
the wing position given by the sweep angle φ. Thanks to that not only the mean
lift force but also its position can be controlled.
If the root bar end is displaced backwards the γangle is negative (and the lift is re-
duced) when the wing is behind the body, but γis positive (and the lift is increased)
when in front of the body. If both left and right wing root bars are displaced in the
same sense this results into a nose up pitch moment (Figure 7.3). If the root bars
are displaced in the opposite sense a yaw moment is generated because the drag
forces are also affected.
Lateral displacement of the root bars is used to generate a roll moment (Figure 7.4).
A displacement of one of the bars towards the body causes a positive γand thus a
lift increase compared to the other wing, whose root bar is moved away from the
body, which results into a lift reduction.
Figure 7.3: Pitch moment generation via wing twist modulation: displacing the bars
longitudinally creates a front-back lift asymmetry.
150 7 Control mechanism
Figure 7.4: Roll moment generation via wing twist modulation: displacing the bars later-
ally creates a left-right lift asymmetry.
7.1.2 Manually operated control mechanism performance
To test the concept a plastic plate with a regular grid of holes was fixed below the
flapping mechanism. The root bar ends could be manually placed and fixed in any
of the equally spaced holes (Figure 7.5). Instead of a universal joint, which would
give the root bars the necessary 2DOFs, flexible CFRP bars clamped in the body
frame were used. This simplified the design substantially as no joints were necessary.
The bar ends were fixed in 5 positions at 0, ±2 and ±4 mm, measured from the
central position where the root bars are straight. The fixing plate was 32 mm be-
low the leading edge bars. The measurements were carried out at constant motor
voltages (2, 2.5, 3 and 3.5 V) and repeated 3 times for each position.
The results for the pitch moment are plotted in Figure 7.6. The individual mea-
surements are plotted as crosses, the lines represent the average values. The bar
deformation has a negligible effect on the average lift force. The modulation of
moment is approximately linear, but we can observe slightly different trend in the
positive and negative directions. This might be caused by the asymmetric wing
design where the stiffeners are glued only on one of the faces. When operating at
7.1 Moment generation via wing twist modulation 151
32 mm
Pitch Roll
Figure 7.5: Testing prototype allowing fixing the bar ends in a grid of holes: hover position
(top), pitch command (bottom-left) and roll command (bottom-right).
3.5 V, displacing the bars by ±4 mm generates a pitch moment between -0.59 mNm
and 0.5 mNm.
While our force balance cannot measure the roll moment, it was possible to estimate
it indirectly (Figure 7.7). The roll moment is generated by increasing the lift force
of one wing (by moving the root bar end towards the body) and decreasing the lift
force of the other wing (by moving the bar end away from the body); the total lift
should remain nearly constant. If both bars are moved symmetrically (e.g. towards
the body) the roll moment is zero, but the total lift will change. Thus, the relation-
ship between the bar deformation and the lift force can be evaluated and used to
estimate the roll moment for asymmetric bar deformations.
The measured relationship between the total lift and the symmetric bar deformation
is shown in Figure 7.7 a). Ideally, this relationship should be linear, however, the
measurement shows a maximum at -2 mm and the lift starts to drop again for lower
values. The linearity could be improved by a modified wing design.
152 7 Control mechanism
15 17.5 20 22.5 25
0
20
40
60
80
100
Flapping frequency [Hz]
Lift [mN]
15 17.5 20 22.5 25
0.5
0.25
0
0.25
0.5
Flapping frequency [Hz]
4 mm
2 mm
0 mm
2 mm
4 mm
Pitch moment [mNm]
− −2 0 2 4
0
20
40
60
80
100
Bar position [mm]
Lift [mN]
−4−2 0 2 4
−0.5
−0.25
0
0.25
0.5
Bar position [mm]
2V 2.5V 3V 3.5V
Pitch moment [mNm]
−4 mm
0 mm 4 mm
32 mm
Figure 7.6: Lift and pitch moment at various positions of root bar ends. The bars were
displaced manually.
The roll moment is estimated from the lift difference between positive and negative
wing bar positions. The results, together with the estimated total lift, are plotted
in Figure 7.7 c). The total lift decreases slightly with increasing absolute value of
roll moment due to the nonlinear relation between the bar deformation and lift force
mentioned earlier. The modulation of the roll moment is approximately linear (for
higher motor voltages), although a small moment decrease can be observed in the
extremities, again due the nonlinearity. At 3.5 V, a roll moment of ±0.43 mNm is
estimated for the maximal bar displacement of ±4 mm.
7.1 Moment generation via wing twist modulation 153
Left and right bar position [mm]
−4 −2 0 2 4
20
40
60
80
100
Lift [mN]
2V 2.5V 3V 3.5V
20
40
60
80
100
−0.5
−0.25
0
0.25
0.5
15 17.5 20 22.5 25
20
40
60
80
100
Flapping frequency [Hz]
15 17.5 20 22.5 25
20
40
60
80
100
Flapping frequency [Hz]
15 17.5 20 22.5 25
−0.5
−0.25
0
0.25
0.5
Flapping frequency [Hz]
Roll Moment
Estimation of
Lift and Roll Moment
4 2 0 2 4
±
∓ ∓ ±
4 2 0 2 4
±
∓ ∓ ±
4 mm
2 mm
0 mm
−2 mm
−4 mm
2V 2.5V 3V 3.5V
4 mm
2 mm
0 mm
2 mm
4 mm
∓
±
±
∓
Lift
Measurement
32 mm
0 mm
−4 mm
4 mm
0 mm
4 mm
−4 mm
estimate [mNm] Lift estimate [mN]
Roll Moment
estimate [mNm] Lift estimate [mN]
Lift [mN]
Left / right bar position [mm]
Left / right bar position [mm]
a)
b)
c)
Figure 7.7: Roll moment indirect measurement: a) Lift measured for symmetric bar de-
formations, b) symmetric bar end positions during the lift measurement (blue arrows) and
asymmetric positions for roll moment estimation (red arrows), c) estimated lift force and
roll moment for asymmetric bar deformations.
154 7 Control mechanism
7.1.3 SMA actuated control mechanism
One of the concepts considered for active displacement of the bar ends is using Shape
Memory Alloys (SMA) wires as actuators. The material uses a shape memory ef-
fect: when heated above certain temperature the crystal structure changes and, if
the material is under stress, we observe contraction of the wire. After cooling the
original shape is restored (Lagoudas, 2008).
The use, advantages and limitations of SMAs were thoroughly reviewed by Mohd Jani
et al. (2014). We chose this actuator because it is very lightweight and provides di-
rectly a displacement. It can be heated simply by Joule effect. But it also has some
limitations that need to be considered in the design: the maximal stroke is only
about 5% of the wire length. The necessary (passive) cooling reduces significantly
the bandwidth. Moreover, the material has a hysteretic behaviour due to phase
transformation: the heating follows different characteristics than the cooling. It can
also suffer from fatigue, so operation at smaller strains (under 3.5%) and limited
stresses (under 160MPa) is recommended (SAES Getters, 2009).
The use of flexible bars instead of joints not only reduces the control mechanism com-
plexity, but the bar deformation also creates the stress necessary for SMA proper
function. The small stroke achievable with an SMA wire can be overcome by the
kinematics. The SMA wire is attached between two supports; the distance between
these is just slightly shorter than the length of the SMA wire itself. Thus, a small
contraction of the wire results in a relatively large displacement in the normal di-
rection (Figure 7.8 left). The downside of this approach is that the maximal force
is also reduced.
COLD
HOT Front wire on
(nose-down
pitch)
Rear wire on
(nose-up
pitch)
Both wires on
(roll)
Both wires off
(hover)
SMA wire
Figure 7.8: SMA driven control mechanism: kinematics for larger stroke (left), bottom
view of the mechanism model (middle) with corresponding stress distribution for the front
wire (right). The two supports are on top of each other in the bottom view.
7.1 Moment generation via wing twist modulation 155
The designed system uses one pair of SMAs per wing that can displace the bar end
in both longitudinal and lateral directions (Figure 7.8 middle). If only one of the
wires is heated the bar moves diagonally in forward or backward direction, heating
the two wires at the same time moves the bar laterally closer to the body.
Heating the rear wires on both wings results into backward displacement of the bar
ends and thus into a nose-up moment (as in Figure 7.3). Similarly heating the front
wires on both wings results into a nose-down moment. Heating both wires on one
wing while keeping them relaxed on the other wing results into a roll moment (sim-
ilar to Figure 7.4).
The dimensions were selected to maximize the workspace while keeping the SMA
wire stress under the maximal recommended value yet high enough to assure proper
phase transformation (Figure 7.8 right).
An important aspect that determines the actuator bandwidth is the cycle time.
While the heating phase can be accelerated by increasing the current, the cooling
phase usually takes longer because the heat needs to be dissipated into the environ-
ment. The cooling is faster for wires with smaller diameter as the surface to volume
ratio is higher. However, thinner wires mean also smaller maximal forces. The
thinnest wire to withstand the estimated stress levels has a diameter of 50 microns.
The complete robot with the control mechanism is shown in Figure 7.9. The used
SMA wires are SmartFlex R
50µm, SAES Getters (2009). Their active section is 53
31.6 mm 70 mm
32 mm
SMA
wires
SMA
supports
Wing
root bars
SMA
power
Figure 7.9: Robot prototype with SMA actuated control mechanism.
156 7 Control mechanism
mm long, the distance between the supports is 47 mm. The system to attach the
SMA wires consists of two washers under the head of a bolt. The SMA wire goes
around the bolt and is pressed between the washers. The power is brought by an-
other cable, pressed by the second washer to the support.
7.1.4 SMA driven control mechanism performance
To determine the mechanism bandwidth the front and rear pair of wires were peri-
odically heated and cooled in an alternating manner, i.e. the bars were alternating
between the positions for nose-down and nose-up pitch moment. The achieved dis-
placement was measured from a long exposure camera image. The duty cycle was
50 % and the frequency was being changed from 1 Hz to 5Hz. The current was
constant during the heating phase, a value of 110 mA was identified as optimal (no
overheating). The airflow from the wings accelerated the cooling process. The re-
sults presented in Figure 7.10 were measured at a moderate flapping frequency of
about 16 Hz. The maximal displacement of 2.9 mm at 1 Hz decreases significantly as
the command becomes faster. According to the results from the previous section the
maximal displacement would generate a pitch moment of approximately ±0.2 mNm.
1 2 3 4 5
0
0.5
1
1.5
2
2.5
3
Driving frequency [Hz]
∆ [mm]
Figure 7.10: Long exposure images of root bar displacement by the SMA actuators driven
at various frequencies (top) and processed results (bottom).
7.1 Moment generation via wing twist modulation 157
15 17.5 20 22.5
0
20
40
60
Flapping frequency [Hz]
Lift [mN]
Hover Nose down Nose up
15 17.5 20 22.5
0.15
0.075
0
0.075
0.15
Flapping frequency [Hz]
Pitch moment [mNm]
Nose down Hover Nose up
0
20
40
60
Lift [mN]
Nose down Hover Nose up
0.15
0.075
0
0.075
0.15
Bar position [ ]
~2V ~2.5V ~3V
Bar position [ ]
Pitch moment [mNm]
Figure 7.11: Lift and moment measurement results - SMA actuated bar displacement.
A direct pitch moment measurement was carried out in three (steady) positions ac-
cording to Figure 7.8 (right): 1) in the hover position with all the wires relaxed,
2) in the nose-up moment position with the rear pair of wires heated and 3) in the
nose-down moment position with the front pair of wires heated. The measurement
was repeated five times in each position. The results are plotted in Figure 7.11;
individual measurements are displayed as crosses and the lines represent the average
values. The lift in hover position is slightly lower when compared to both nose-up
and nose-down positions. This is in accordance with our expectations, because to
generate a moment the wing root bar moves also in lateral direction which stretches
the wing membrane and results in a lift increase.
The maximum generated pitch moment is approximately -0.11 mNm (nose-up) and
0.06 mNm (nose-down), which is already at the limit of the resolution of the force
balance. The moments are lower than ±0.2 mNm estimated from the results in
previous section, probably due to smaller bar displacements at higher flapping fre-
quencies that involve higher stress and higher cooling rates. However, direct com-
parison is not completely correct as the wing design in the SMA actuated prototype
had to be modified to compensate for the wing root bar deformation, that needs
to be present even in hover position to create pre-stress. The asymmetry between
nose-up and nose-down moments might come from imperfections of the hand built
prototype (slight misalignment of the SMA supports, small variations of the SMA
158 7 Control mechanism
wires lengths, ...). The indirect measurement of the roll moment estimated even
lower moment values, which were already at the level of variation among individual
measurements, because the SMA wires didn’t provide enough displacement.
7.1.5 Conclusion on wing twist modulation
The presented concept, combining the wing twist modulation with flexible wing
root bars, proved to be a feasible solution for generating the control moments. The
experiments showed that a (manual) bar displacement of ±4 mm can generate a
pitch moment of ±0.5 mNm and roll moment of ±0.4 mNm, which is about 4 times
and 8 times the necessary value, respectively, predicted by the flight simulation in
Section 5.3. The maximal generated lift produced during the experiments (about
85 mN) was close to the maximal performance of the flapping mechanism at the
time of the experiment. Because the control mechanism only acts on the root bar
ends, it is completely independent on the flapping mechanism. The newest flapping
mechanism version produces almost twice as much lift and so a similar increase in
the pitch moment can be expected.
The actively controlled mechanism driven by SMA wires was attractive due to its
low weight and relatively low complexity. However, several weak points of this so-
lution were identified. Most importantly, the bandwidth as well as the achievable
stroke (and thus the moments) were too low. Apart from that, it was cumbersome
to adjust the initial stress in the SMA wires and many wire failures were experienced
due to stress concentration in the attachment points. Thus, the full potential of the
concept should be exploited by an alternative solution using actuators with larger
stroke and higher bandwidth, such as micro-servomotors.
7.2 Moment generation via amplitude and offset
modulation
The second concept generates the control moments by modulating the flapping am-
plitude and offset (mean wing position) according to Figure 7.12. This strategy
was also used in the simulation presented in Chapter 5. Compared to wing twist
modulation this strategy is more straight forward and works with any wing design.
However, it requires a modification of the flapping mechanism. The roll moment is
produced by increasing the amplitude of one wing and decreasing the amplitude of
the other wing, which introduces a lift imbalance between the two wings. Moving
the mean wing position of both wings forward or backward results into a nose-up or
7.2 Moment generation via amplitude and offset modulation 159
Figure 7.12: Moment generation via flapping amplitude and offset modulation.
nose-down pitch moment as the mean lift force origin moves forward or backward
from the COG, respectively. Yaw moment generation is not so evident, but the
mathematical model in Section 5.2 predicts that the necessary drag imbalance can
also be introduced by changing the mean wing position asymmetrically, see equation
(5.13).
7.2.1 Amplitude and offset modulation
During the flapping mechanism optimization process it has been found that both
the flapping amplitude and the offset can be controlled by displacing the mechanism
joints A and B, highlighted in Figure 7.13 a). The map in Figure 7.13 b) shows
the relation between the position of the joint and the wing amplitude Φ and offset
φ0. The blue lines connect positions with constant offset and the red lines con-
nect positions with constant amplitude. Thus, moving the joint along a blue line
will modify the amplitude, but the offset will remain constant. Similarly a displace-
ment along a red line will only affect the offset while keeping the amplitude constant.
It can be noticed that the two sets of curves cross each other at high angles (above
70◦) meaning the two parameters can be controlled independently. Moreover, the
lines of constant offset are almost straight and nearly parallel; the curves of constant
160 7 Control mechanism
−1−0.5 0 0.5 1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
ΦL = 120°
ΦL = 110°
ΦL = 130°
ΦL = 140°
ΦL = 150°
ΦL = 100°
ϕ0L = 0°
ϕ0L = 10°
ϕ0L = 20°
ϕ0L = 30°
ϕ0L = -10°
ϕ0L = -20°
ϕ0L = -30°
ΔxA [mm]
ΔyA [mm]
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
ΦR = 120°
ΦR = 110°
ΦR = 130°
ΦR = 140°
ΦR = 150°
ΦR = 100°
ϕ0R = 0°
ϕ0R = 10°
ϕ0R = 20°
ϕ0R = 30°
ϕ0R = -10°
ϕ0R = -20°
ϕ0R = -30°
ΔxB [mm]
ΔyB [mm]
θ
ΦL
ϕ0L ΦR
ϕ0R
−1−0.5 0 0.5 1
AB
Φ control (ϕ0 = const.)
ϕ0 control (Φ = const.)
a)
b)
displaced joints
Figure 7.13: Amplitude and offset modulation via joint displacement: a) the flapping mechanism and the displaced joints A
and B, b) lines of constant amplitude Φ and offset φ0for varying joint position. Joint motion along the blue lines controls the
amplitude Φ, motion along the red lines controls the offset φ0.
7.2 Moment generation via amplitude and offset modulation 161
−1 −0.5 0 0.5 1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
ΔxA [mm]
ΔyA [mm]
η = const
εL = const
Δη = 0°
Δη = -4°
Δη = -8°
Δη = 4°
Δη = 8°
ΔεL = 0°
ΔεL = 4°
ΔεL = 8°
ΔεL = -4°
ΔεL = -8°
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
ΔxB [mm]
ΔyB [mm]
η = const
εR = const
Δη = 0°
Δη = 4°
Δη = 8°
Δη = -4°
Δη = -8°
ΔεR = 0°
ΔεR = 4°
ΔεR = 8°
ΔεR = -4°
ΔεR = -8°
ηεLεRη
ROLL CONTROL
(ΔΦL = -ΔΦR)
PITCH CONTROL (εL = εR)
AB
ϕ0L ϕ0R
ΦR
ΦL
YAW CONTROL (εL = -εR)
a)
b)
−1 −0.5 0 0.5 1
Figure 7.14: Amplitude and offset modulation via joint displacement: a) the control mechanism with 3DOF (L,R,η) defining
the position of joints A and B by an intersection of two mobile slots, b) lines of constant commands and ηapproximating
well the lines of constant amplitude Φ and offset φ0in Figure 7.13 b). Joint motion along the magenta lines controls the roll
moment, motion along the green lines controls the pitch (∆L= ∆R) or yaw (∆L=−∆R) moments.
162 7 Control mechanism
amplitude are also equally spaced and can be approximated by straight lines around
the nominal position. This allows to design a joint displacement mechanism with
two DOFs where the control is decoupled - one DOF controls directly the amplitude
and the other controls the offset.
7.2.2 Control mechanism prototype
The scheme of the proposed mechanism for joint displacement is shown in Figure
7.14 a). Each joint is displaced by two arms with slots that rotate with respect to the
frame by angles and η, respectively. The joint position is defined by an intersection
of the two slots. The arm hinges are located on the lines of nominal amplitude and
of the nominal offset, respectively. If one of the arms is blocked and the other one
is moving, the joint moves along a line defined by the slot of the blocked arm. If
the hinges are placed far enough from the nominal position of the displaced joint
(∆x= 0, ∆y= 0), these lines appear nearly parallel in the region of interest and
the joint paths approximate well the theoretical curves of constant amplitude and
offset, see Figure 7.14 b).
The control of left and right wing offset needs to be independent, operated by sep-
arate actuators: a symmetric offset change (∆L= ∆R) produces pitch moment
while an asymmetric change (∆L=−∆R) produces yaw moment. However, the
amplitude can be controlled by a single actuator (η) since only asymmetric ampli-
tude changes are needed for roll. This can be achieved by a parallelogram linking
the two arms responsible for amplitude control as is shown in Figure 7.14 a).
The final mechanical solution of the joint displacement mechanism is shown in Figure
7.15. The rivets of the joints to be displaced are fixed from the top to ”anchors” that
are free to slide in the horizontal plane of the frame. The displacement is limited
to the zone considered in Figure 7.14 b) by the shape of the frame cut-out. All the
parts were 3D printed, a photograph of an assembled prototype is shown in Figure
7.16. The mechanism is actuated by three micro servos (HobbyKing 5330) with a
weight of 2.0 g each. The total weight of the controlled prototype including the
servos (6.0 g) and the propulsion motor (5.2 g) is 21.4 g.
7.2.3 Wing kinematics
A high speed camera (Photron FASTCAM SA3) was used to study the changes in the
wing kinematics of the control prototype. The wing motion was recorded at 500 fps
under different control commands and the sweep angle φwas tracked in the record-
ings (see Figure 6.9 in the previous chapter for its definition). As already mentioned
in Section 6.2.1 the observed amplitudes were much larger than the design value of
7.2 Moment generation via amplitude and offset modulation 163
Roll servo arm
Top
view
Section
view
Bottom
view
Exploded
view
Displaced joints
Right offset
servo arm
Left offset
servo arm
Roll control
parallelogram
Control
servos
Control
mechanism
Flapping
mechanism
Propulsion
DC motor
"Anchor" sliding
freely inside
the frame
Figure 7.15: CAD model of the control prototype. Details of the flapping mechanism with
the joint displacement system (left) and an exploded view of the whole prototype (right).
Figure 7.16: The assembled prototype with pitch, roll and yaw control via joint displace-
ment.
164 7 Control mechanism
+ Roll
+ Pitch
+ Yaw
Hover
− Roll
− Pitch
− Yaw
Figure 7.17: Extremal positions of the wings recorded with the high-speed camera in the
neutral position as well as for the maximum and minimum commands in the 3 DOF. Each
image was obtained by blending up to 4 frames with the extremal positions of each wing.
7.2 Moment generation via amplitude and offset modulation 165
120◦due to the wing bars compliance and partly also due to the mechanism back-
lash. Unless mentioned otherwise, the tests were carried out at a moderate flapping
frequency of 15Hz in order to have consistent results along all the tests. For higher
frequencies the prototype performance can slightly deteriorate over time due to wear.
The wing extremal positions under maximal and minimal commands of pitch, roll
and yaw are displayed in Figure 7.17. The mechanism succeeds to modify the flap-
ping amplitude and offset in the desired manner. The roll and pitch command will
be studied in more detail in the following text. The performance in yaw is the same
as in pitch, only the left and right wing offset commands have opposite signs.
Figure 7.18 shows the results of amplitude difference control. The servos controlling
the wing offset were kept in their nominal position and the roll control servo was
commanded from the minimal to the maximal position with a step of 10% of the full
range. The left and right wing amplitudes, ΦLand ΦR, are approximately equal for
zero servo position. As intended, their difference ∆Φ = ΦR−ΦLincreases/decreases
approximately linearly as the servo moves towards the positive/negative limit, where
the difference is +52◦and −44◦, respectively. Thus, a good control authority of the
roll moment is achieved. The wing offset remains relatively close to zero and the
average amplitude ¯
Φ = (ΦR+ ΦL)/2 stays approximately constant (around 155◦)
in the central part of the servo range. There exist some imperfections, in particular
close to the limits, but these should be compensated by the flight controller feedback
in future.
−1−0.5 0 0.5 1
120
140
160
180
Flapping amplitude [°]
−1−0.5 0 0.5 1
−20
−10
0
10
20
Servo position [−]
Wing offset [°]
Average Right wing Left wing
Figure 7.18: Amplitude control with the roll servo.
166 7 Control mechanism
−1−0.5 0 0.5 1
120
140
160
180
Flapping amplitude [°]
−1−0.5 0 0.5 1
−20
−10
0
10
20
Servo position [−]
Wing offset [°]
Average Right wing Left wing
Figure 7.19: Offset control with the pair of pitch servos.
The offset control is presented in Figure 7.19. The left and right offset servos
were commanded together over the full range, again with a step of 10%. The roll
servo was kept at zero. The relationship between the offset servos position and
the wing offset is linear and shows a good control authority, even though the slope
slightly decreases closer to the servo limits. The maximal and minimal average offset
¯
φ0= (φ0R+φ0L)/2 is +17.7◦and −14.5◦, respectively. The amplitude of the left and
right wing varies quite a lot, but the average ¯
Φ stays close to 155◦. A simultaneous
control of the amplitude difference would be necessary to achieve zero roll moment.
The combined commands and resulting coupling effects will be discussed at the end
of this section.
7.2.4 Control mechanism dynamics
Figures 7.20 and 7.21 show the dynamics of the transition from minimal to maximal
command of pitch and roll, respectively. For this experiment the flapping frequency
was around 17Hz. The figures display the wing tip angles, their extremal positions
are connected with a full/dashed line and the average position (over the last wing-
beat) is displayed as dash-dotted/dotted line for the left/right wing, respectively.
An LED was placed on the prototype to indicate the moment of the step command
(black line).
As can be seen in Figure 7.20 the transition from maximal to minimal offset occurs
within 2 wingbeats. The transition from negative to positive amplitude difference
7.2 Moment generation via amplitude and offset modulation 167
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25
−150
−100
−50
0
50
100
Time (s)
Sweep angle (°)
Flapping frequency 17 Hz
left right trigger
Figure 7.20: Pitch up →down command dynamics. Full/dashed line connects the extremal
positions, dash-dotted/dotted line represents the mean position (over the last wingbeat).
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−150
−100
−50
0
50
100
Time (s)
Sweep angle (°)
Flapping frequency 17 Hz
left right trigger
Figure 7.21: Roll left →right command dynamics. Full/dashed line connects the extremal
positions, dash-dotted/dotted line represents the mean position (over the last wingbeat).
takes around 4 wingbeats (Figure 7.21), however an opposite sign of the difference
is achieved already after 2 wingbeats. The same could be observed for the step
commands in opposite directions.
The faster pitch response can be explained by a combination of two effects. First,
in offset control two servos are employed, one acting on each joint, while in the roll
control only a single servo is displacing the two joints. The second reason is that the
reaction due to the flapping motion on the displaced joints has a major component
in the direction, where the joints are displaced for offset control. This speeds up
the offset transition when a change is desired, but has an adverse effect on the
mechanism efficiency as the joints keep shaking in this direction during operation.
168 7 Control mechanism
7.2.5 Pitch moment and lift generation
Apart from the wing kinematics measurements, the generated lift and pitch moment
was measured directly on the force balance presented in Section 6.2.2. Figure 7.22
shows the measured pitch moment, lift, frequency and motor current for the full
range of the pitch servos. The motor voltage was kept at 4.2V, which gives a flap-
ping frequency of approximately 15Hz for the nominal servo position. The measured
pitch moment ranged from -0.5 mNm to 0.8 mNm. The lift force and the flapping
frequency increase and the current decreases when the servos approach the servo
limits. This is caused by the shaking of the displaced joints due to flapping (already
mentioned earlier) that happens particularly for the central servo positions. The
joints get a better fix in the limit positions, where the servo pushes the displaced
joints against a wall of the frame, and thus the efficiency increases. The lift varies
between 53 and 59 mN and the frequency between 14.9 and 16.2 Hz.
−1 −0.5 0 0.5 1
50
52
54
56
58
60
62
Pitch servo [−]
Lift [mN]
−1 −0.5 0 0.5 1
−0.5
0
0.5
1
Pitch servo [−]
−1 −0.5 0 0.5 1
0.22
0.24
0.26
0.28
0.3
0.32
Pitch servo [−]
Current [A]
−1 −0.5 0 0.5 1
14.5
15
15.5
16
16.5
Pitch servo [−]
Frequency [Hz]
Figure 7.22: Lift, pitch moment, motor current and flapping frequency against pitch
servos position. Measurement done at 4.2 V leading to 15 Hz at the nominal position.
Individual measurements are displayed as crosses, the solid line represents an average of
four measurements.
7.2 Moment generation via amplitude and offset modulation 169
Figure 7.23 combines the moment measurement with the wing kinematics measure-
ment from the previous section. The relationship is approximately linear with a
slope of 0.04 mNm per degree of the offset φ0. The non-zero moment produced at
zero offset can be explained by a combination of asymmetric wing design (stiffeners
glued only on one side of the membrane), different velocity profiles in upstroke and
downstroke and by an imperfect alignment of the prototype on the balance. Never-
theless, a compensating moment can be easily introduced by an offset of the COG
from the shoulders in the longitudinal direction.
−20 −10 0 10 20
−1
−0.5
0
0.5
1
Offset [°]
Figure 7.23: Pitch moment against wing offset φ0.
Figure 7.24 shows the pitch moment, lift, frequency and motor current for motor
voltages up to 6V. The three curves represent the measurements for servo positions
-1, 0 and 1, respectively, related to the full range. The behaviour corresponds to the
one observed at 15Hz. At the highest tested voltage the mechanism produces pitch
moments from -0.7 mNm to 1.1 mNm while the lift ranges between 90 and 100 mN.
The lift vs frequency characteristic is close to the one of the uncontrolled prototype
(with fixed joints), see Figure 7.25, with only small differences caused by a variation
of the flapping amplitude. However, the electrical power of the controlled version
is almost twice as high due to losses in the shaking joints. Thus, the mechanical
design of the joint displacement mechanism needs to be improved in order to get a
better fix of the joints and, subsequently, a better efficiency.
170 7 Control mechanism
3 4 5 6
0
25
50
75
100
Motor voltage [V]
Lift [mN]
3 4 5 6
Figure 7.24: Lift, pitch moment, motor current and flapping frequency measured for
increasing motor voltage. Black lines represent the zero pitch servos position, red and blue
lines represent the minimal and maximal pitch command. The crosses represent individual
measurements.
10 12 14 16 18 20
2
4
6
8
10
12
Frequency [Hz]
Lift [g]
0 1 2 3
2
4
6
8
10
12
Electrical power [W]
Lift [g]
Joints fixed
Pitch 1
Pitch 0
Pitch −1
Figure 7.25: Pitch moment against wing offset.
7.2 Moment generation via amplitude and offset modulation 171
7.2.6 Combined commands
Finally, combined pitch and roll commands were tested to identify the amount of
cross-coupling. Again all the measurements were carried out at a constant motor
voltage (4.2V) giving a flapping frequency of around 15Hz at the nominal position
of the servos. The measurements were taken at servo positions -0.8, -0.4, 0, 0.4 and
0.8 of the full range for both pitch and roll servos. Thus, 25 measurements were
taken in total.
Figure 7.26 shows the amplitude and offset maps of the left and the right wing from
the high speed recordings. The camera images combining the extremal wing posi-
tions for maximal commands (corners of the maps) are shown in Figure 7.27. We
can see that while a small cross-coupling always exists, the roll servo has a domi-
nant effect on the amplitude and the pitch servo has a dominant effect on the offset.
Moreover, the relation between the amplitude/offset and roll/pitch servo positions
a) b)
c) d)
Figure 7.26: Experimental results: Kinematics for combined pitch and roll servo com-
mands. a) Left amplitude, b) right amplitude, c) left offset, d) right offset.
172 7 Control mechanism
+ Roll − Roll
− Pitch
+ Pitch + Pitch
− Roll
+ Roll
− Pitch
Figure 7.27: Extremal positions of the wings recorded with the high-speed camera for the
combined commands in pitch and roll. Each image was obtained by blending up to 4 frames
with the extremal positions of each wing.
stays always monotonic. Thus, a feedback controller should be able to compensate
the coupling effects and the small differences between the left and right wing be-
haviour, caused by the mechanism imperfections.
The same experiment was repeated with the force balance, the results are shown in
Figure 7.28. The maps show the pitch moment and lift force for the selected servo
input combinations. The lift map keeps a valley-like shape in the pitch servo direc-
tion, similar to what was observed for the pure pitch command and what can be
explained by an improper joint fixation in the central pitch servo positions. There is
also a smaller increase of lift in the positive roll direction, which most likely comes
from the imperfections of the prototype. The minimum and maximum lift is 48 and
59 mN, respectively. The lift force variation is related to the square of frequency
times amplitude displayed in Figure 7.29 c), in accordance with equation (6.8).
The pitch moment depends mostly on pitch servo positions, while there are only
minor differences when the roll servo position changes. Thus, the minimal and max-
imal pitch moment values stay at the same levels as for the pure pitch command,
-0.4 mNm and 0.8 mNm respectively. The pitch moment corresponds closely to the
mean offset shown in Figure 7.29 a).
7.2 Moment generation via amplitude and offset modulation 173
a) b)
Figure 7.28: Experimental results: Force balance measurements for combined pitch and
roll servo commands. a) Lift, b) pitch moment. Lift is tied to the square of mean amplitude
times frequency in Figure 7.29 a), pitch moment to the mean offset in Figure 7.29 b).
a) b)
c)
Figure 7.29: Experimental results: Kinematics for combined pitch and roll servo com-
mands. a) Square of mean amplitude times frequency, b) mean wing offset, c) flapping
amplitude difference.
174 7 Control mechanism
The roll moment could not be measured directly, but it should correspond to the
amplitude difference shown in Figure 7.29 c).
7.2.7 Conclusion on amplitude and offset modulation
The second presented control mechanism generates the necessary control moments
by modulating the flapping wing amplitude and offset. The wing kinematics mod-
ifications are achieved by displacing the joints of the flapping linkage mechanism.
It was demonstrated experimentally that sufficient offset (±15◦) and amplitude dif-
ferences (above ±40◦) for pitch and roll control can be introduced by very small
displacements of the linkage joints (below ±1mm) in two directions. The transi-
tions between maximum and minimum command takes less than two wingbeats in
pitch and about four wingbeats in roll. A very low level of cross-coupling exists
for combined commands. The prototype can produce pitch moments between -0.7
mNm and 1.1 mNm (at least 6 times the moment predicted by the flight simulation
in Section 5.3) while flapping at frequencies around 18Hz and producing a lift of
at least 90mN. The roll moment could not be measured, but should be more than
sufficient as the measured amplitude differences are 40 times higher than those pre-
dicted by flight simulation.
While the control mechanism succeeds in modifying the wing kinematics and, subse-
quently, in the moment generation with sufficient dynamics, the prototype efficiency
drops significantly compared to the uncontrolled prototype. The drawback of the
proposed solution is that the displaced joints need to hold rather large and oscillating
reaction forces due to flapping. This causes the joints to shake, which reduces signif-
icantly the mechanism performance and at higher frequencies also its lifespan. The
motor draws up to twice the electrical power compared to an uncontrolled prototype
with fixed joints. Thus, an alternative mechanical solution, which would reduce the
effect of the oscillating reaction forces on the servos displacing the joints, should be
found.
7.3 Discussion and conclusions
Two control mechanisms generating moments around the three body axis (roll, pitch,
yaw) were developed and tested. The first solution, based on wing twist modulation,
is inspired by the control mechanism of the Nano Hummingbird (Keennon et al.,
2012). The second, original solution generates the moments by flapping amplitude
and offset modulation through displacements of the flapping mechanism linkage
joints. Both control mechanisms succeeded in generating moments that were several
7.3 Discussion and conclusions 175
times higher than the moments needed in the flight simulation in Chapter 5. Nev-
ertheless, the experiments revealed that each of the mechanisms has several strong
points, but also some weaknesses that require further development.
The advantage of the wing twist modulation mechanism is that it is fully inde-
pendent on the flapping mechanism. It acts at the wing root bar ends, where the
reaction forces, that it needs to hold, are relatively small (compared to the second
solution). This is advantageous as it permits the use of smaller and lighter actuators.
Moreover, the use of flexible root bars instead of universal joints greatly reduces the
mechanism complexity. On the other hand, the wing twist modulation concept re-
quires a specific wing design, whose lift force varies approximately linearly with the
root bar deformation. This requires that the wing at the nominal bar position is
operated below its maximal lift.
The second mechanism modulating the flapping amplitude and offset works with
any wing design. It is an integral part of the flapping mechanism as it displaces
one of the linkage joints. This can be considered as beneficial as the whole design
is compact and the actuators can be placed close to the flapping mechanism part.
On the other hand, any design change of the flapping mechanism requires also a
redesign of the joint displacement system. Another drawback of this solution is that
the system needs to hold the reaction forces from the displaced joints, which are in
particular high in the pitch control direction. This made the joints shake during the
operation and resulted into efficiency degradation.
A remaining challenge for both solutions are the actuators. The most fitting actu-
ators for this application on the market, the smallest available micro servos, have
an acceptable power, but remain still relatively large and heavy. While they showed
sufficient dynamics in the joint displacement system, they were unable to fix the
joints completely. Nevertheless, a modified mechanical solution might succeed in
reducing the joint shaking.
As an alternative, a shape memory alloy actuator was used in the wing twist mod-
ulation system. This actuator is very light and provides satisfactory stroke in static
experiments, however its performance decreases significantly under dynamic tests
and thus it is not a good candidate for the control actuator.
To conclude, both presented control systems succeed in moment generation, but
require further development and testing, focused mostly on efficiency improvements
and weight reduction. In future, a combination of both systems might also be
considered in order to solve potential cross-coupling of the three generated control
moments.
176 References
7.4 References
M. Karasek, Y. Nan, I. Romanescu, and A. Preumont. Pitch moment generation
and measurement in a robotic hummingbird. International Journal of Micro Air
Vehicles, 5(4):299–310, 2013. doi:10.1260/1756-8293.5.4.299.
M. Karasek, A. Hua, Y. Nan, M. Lalami, and A. Preumont. Pitch and roll control
mechanism for a hovering flapping wing MAV. In International Micro Air Vehicle
Conference and Flight Competition (IMAV2014), Delft, The Netherlands, August
12-15, pages 118–125, 2014.
M. T. Keennon, K. R. Klingebiel, H. Won, and A. Andriukov. Development of
the nano hummingbird: A tailless flapping wing micro air vehicle. AIAA paper
2012-0588, pages 1–24, 2012.
D. C. Lagoudas. Shape Memory Alloys: Modeling and Engineering Applications.
Springer, 2008. doi:10.1007/978-0-387-47685-8.
K. Y. Ma, P. Chirarattananon, S. B. Fuller, and R. J. Wood. Controlled
flight of a biologically inspired, insect-scale robot. Science, 340:603–607, 2013.
doi:10.1126/science.1231806.
J. Mohd Jani, M. Leary, A. Subic, and M. A. Gibson. A review of shape memory
alloy research, applications and opportunities. Materials & Design, 56:1078–1113,
2014. doi:10.1016/j.matdes.2013.11.084.
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default/files/SmartFlex%20Wire%20%26%20Spring%20datasheets_1.pdf,
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T. Q. Truong, V. H. Phan, S. P. Sane, and H. C. Park. Pitching moment generation
in an insect-mimicking flapping-wing system. Journal of Bionic Engineering, 11
(1):36–51, 2014.
Chapter 8
Summary and conclusions
This work dealt with development of a hummingbird-sized tail-less flapping wing
micro air vehicle capable of hovering. More specifically, the aim of this thesis was to
develop a control mechanism that generates the moments controlling the flight by
varying the wing motion of each wing independently.
First, the problem was studied theoretically. In Chapter 3, a mathematical model
was developed combining quasi-steady aerodynamics with rigid body dynamics. The
model was linearised and further reduced. It has been shown that the model gives
comparable results to CFD data from the literature. In Chapter 4, stability of the
designed MAV was studied showing, in accordance with other studies, that hovering
flapping flight is inherently unstable. It has been demonstrated that stability can
be achieved with an angular rate feedback, which is most likely used by insects, as
long as the wings are placed sufficiently high above the centre of gravity. A flight
controller with cascade structure has been designed in Chapter 5. Several ways of
generation of the necessary control moments by wing motion changes have been
identified. A good control performance has been shown in simulation (using the
full, non-linear model) for an example of flapping amplitude and offset (mean wing
position) modulation.
A prototype of the robotic hummingbird was designed and tested in Chapter 6.
The flapping mechanism has undergone a lot of development. Its dimensions were
optimized for symmetric wing motion, which was confirmed by high speed camera
measurements. A force balance capable of measuring cycle averaged lift force and
pitch moment was built to evaluate the prototype performance. The wing shape
was optimized experimentally for maximal lift. The latest prototype, with off-board
power and no control, has a lift to mass ratio of nearly 1.3. A take off was demon-
strated while the prototype was stabilized by a guide cable.
177
178 8 Summary and conclusions
Last but no least, two types of control mechanisms generating the necessary con-
trol moments were designed in Chapter 7. The first solution generates the control
moments by modulating the twist of the wings by flexing the wing root bars. The
second solution modulates the flapping amplitude and offset via displacements of
the flapping mechanism joints in specific directions. Shape memory alloy actuators
were employed to drive the first solution, however, the performance was not suffi-
cient due to low bandwidth. Micro-servomotors were used for the second solution
and satisfactory response times were achieved. Prototypes of both solutions were
tested on the force balance, both generating control moments several times larger
than the maximum control moments estimated in the flight control simulation.
8.1 Original aspects
The following parts of this work represent an original contribution:
•It has been shown, by comparing several aerodynamic models, that the com-
plex non-linear dynamics of flapping flight, once cycle averaged, can be repre-
sented by a reduced linear model, where the decoupled pitch and roll dynamics
are characterized by three poles each. The pole configuration depends on the
wing position. If the wings are well above the centre of gravity, a simple an-
gular rate feedback stabilizes the system.
•A new flapping mechanism with high amplitude consisting of two stages, a
slider-crank and four-bar mechanisms, has been developed. The mechanism
output has nearly symmetric velocity profile, which is ideal for hovering flight
and a big advantage over single stage linkage mechanisms.
•A new control mechanism concept generating the necessary control moments
by amplitude and offset modulation via displacements of the flapping mecha-
nism joints has been developed. The mechanism is tied to a specific flapping
mechanism, but is fully independent of the wing design.
8.2 Future work
The developed robotic hummingbird demonstrated that it can generate sufficient lift
to take-off or sufficient moments to stabilize itself in the air. However, the weight of
the current design of the controlled prototype is too high (i) due to the used servo
8.3 Publications 179
actuators, (ii) because it was designed for bench tests, where robustness and long life
were a priority and (iii) due to a relatively low strength to weight ratio of the photo-
polymer material used in the 3D printing process. Thus, further development of the
robot prototype is necessary, concentrating mainly on improving the lift production
and reducing the weight:
•The control mechanism should be redesigned to better fix the displaced joints.
Currently, a significant amount of energy is lost because the joints shake when
flapping. A possible solution that uses cams to reduce the forces transferred
to the servo actuators is already under development.
•Lightweight actuators with sufficient stroke and power need to be found or
developed. The shape memory alloy wires used in this study are light enough,
but their response times are too slow and the stroke too short. The micro servo-
motors have the necessary performance, but they are too heavy, representing
almost 30% of the total mass.
•The generated lift should be increased by optimizing further the wing design
and by optimizing the gear-ratio for maximal efficiency of a specific motor-
mechanism-wing combination.
•The robot structure mass needs to be reduced. It should be redesigned for pos-
sible weight savings and a different manufacturing technology, using materials
with higher strength to mass ratio, should be considered.
Apart from the mechanical design, robot avionics, including a radio, an attitude
sensor, a motor speed controller and a micro-controller needs to be integrated into
a sufficiently small and lightweight package, before a stable hovering flight can be
achieved.
8.3 Publications
The work presented in this thesis has led to the following publications:
Journal papers
M. Karasek and A. Preumont. Flapping flight stability in hover: A comparison of
various aerodynamic models. International Journal of Micro Air Vehicles, 4(3):
203–226, 2012. doi:10.1260/1756-8293.4.3.203.
M. Karasek and A. Preumont. Simulation of flight control of a hummingbird like
robot near hover. Acta Technica, 58(2):119–139, 2013.
180 8 Summary and conclusions
M. Karasek, Y. Nan, I. Romanescu, and A. Preumont. Pitch moment generation
and measurement in a robotic hummingbird. International Journal of Micro Air
Vehicles, 5(4):299–310, 2013. doi:10.1260/1756-8293.5.4.299. The paper received
the ”Best Paper Award” at IMAV2013 conference.
Conference proceedings
M. Karasek and A. Preumont. Control of longitudinal flight of a robotic humming-
bird model. In 5th ECCOMAS Thematic Conference on Smart Structures and
Materials SMART11, conference proceedings CD-ROM, Saarbrucken, Germany,
July 6-8, 2011.
M. Karasek, A. Hua, Y. Nan, M. Lalami, and A. Preumont. Pitch and roll control
mechanism for a hovering flapping wing mav. In International Micro Air Vehicle
Conference and Flight Competition (IMAV2014), Delft, The Netherlands, August
12-15, pages 118–125, 2014.
Oral presentations and posters
M. Karasek and A. Preumont. Robotic hummingbird: Simulation model and longi-
tudinal flight control (poster and oral presentation). International Workshop on
Bio-Inspired Robots, Nantes, France, April 6-8, 2011.
M. Karasek and A. Preumont. Flight simulation and control of a tailless flapping
wing MAV near hover (poster). International Micro Air Vehicle Conference and
Flight Competition (IMAV2012), Braunschweig, Germany, July 3-6, 2012.
M. Karasek and A. Preumont. Simulation of flight control of a hummingbird like
robot near hover (oral presentation). Engineering Mechanics 2012, Svratka, Czech
Republic, May 14-17, 2012.
M. Karasek, Y. Nan, I. Romanescu, and A. Preumont. Pitch moment generation and
measurement in a robotic hummingbird (oral presentation). International Micro
Air Vehicle Conference and Flight Competition (IMAV2013), Toulouse, France,
September 17-20, 2013.
M. Karasek, I. Romanescu, and A. Preumont. Development of a robotic hum-
mingbird (oral presentation). International Conference on Manufacturing Sys-
tems (ICMS2013), Iasi, Romania, October 24-25, 2013.
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