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Modeling aeroelastic deformation of soft wing membrane kites

Thesis

Modeling aeroelastic deformation of soft wing membrane kites

Abstract and Figures

Several structural models that could serve as building blocks for a next-generation FSI-based kite design tool have been developed. The models represent the leading-edge inflatable V3 soft wing membrane kite of the TU Delft and Kitepower B.V., by using a wireframe multi-plate representation. Each plate has three edges of constant length, each being a rigid representation of a tube segment. Only when including canopy billowing does the plate have a varying trailing edge length. The corners of the plates are represented by bridle line attachment points, as the tubular frame of the V3 is bridled on both the leading and trailing edges. The deformations of the shape are modelled through these attachment points, whose coordinate changes are predicted using bridle line system representations and calculation algorithms. A photogrammetry analysis of in-flight footage provided kite width change upon symmetrical actuation (up) and empirical relations, used to model phenomena like billowing. By including empirical relations sufficient bridle line system detail could be modelled allowing a geometric constraint-based 7-plate model to predict the width change upon symmetrical actuation within 1% of the photogrammetry results. The existence of slack made predicting asymmetrical actuation inaccurate, therefore a force-based particle system model (PSM) was developed. For higher computational costs the PSM was able to predict asymmetrical actuation and the width change accurately. Therefore, the PSM is considered an excellent building block for the next-generation kite design models.
Content may be subject to copyright.
Modelling aeroelastic defor-
mation of soft wing mem-
brane kites
J.A.W. Poland
Modelling aeroelastic
deformation of soft wing
membrane kites
by
J.A.W. Poland
to obtain the degree of Master of Science in
Aerospace Engineering at Delft University of Technology
to be defended publicly on 26th of January, 2022 at 14:00.
Supervisor: Dr.-Ing. R. Schmehl TU Delft
Thesis committee: Dr.-Ing. R. Schmehl TU Delft, chair
Prof. dr. ir. C.J. Simão Ferreira TU Delft
Dr. ir. J. Sodja TU Delft
Ir. M. Schelbergen TU Delft
Ir. J. Breuer Kitepower B.V.
An electronic version of this thesis is available at https://repository.tudelft.nl/
Cover Image: Footage shot on 03/05/2021 during experiments with S. Jonard.
Copyright © J.A.W. Poland
All rights reserved.
Delft University of Technology
The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace
Engineering and Department of Aerodynamics & Wind Energy for acceptance a thesis entitled
“Modelling aeroelastic deformation of soft wing membrane kites”
by
J.A.W. Poland
in partial fulfilment of the requirements for the degree of
Master of Science in Aerospace Engineering at TU Delft
Thesis committee:
Dated: 26 January 2022
Dr.-Ing. R. Schmehl (TU Delft)
Prof. dr. ir. C.J. Simão Ferreira (TU Delft)
Dr. ir. J. Sodja (TU Delft)
Ir. M. Schelbergen (TU Delft)
Ir. J. Breuer (Kitepower B.V.)
Preface
“Airplanes are unanchored kites.” - Louis Ferber (1862 - 1900)
Kite research within the airborne wind energy sector was something I stumbled upon about two years
ago. I never expected to find a research domain that aligns so perfectly with what I find interesting,
what I consider ethically just and what I am good at. Besides the research subject itself, the advice and
company of many around me are what made this particular research so enjoyable.
I would like to express my gratitude towards my daily supervisor R. Schmehl. His input, insights, mo-
tivational quotes, introductions, and overall company was much appreciated. I was especially fond of
our weekly progress meetings over a cup of coffee in a local cafe in Rotterdam.
I would furthermore like to thank M. Schelbergen for always answering my questions and his general
continuous involvement. J. Breuer for providing me with much needed data and answers. J. Blom
for being an ideal colleague, his advice and most welcome company on our many kite field trips. S.
Jonard for advice and requesting me to do experiments with him using one of my kites. D. Eijkelhof for
sharing his experiences and providing me with some early on research tips. Furthermore, R. Kooij and
W. Arink for their critical comments. The airborne wind energy research group for their general advice.
My parents for their unconditional support and advice on where to focus on. My roommates for their
company in our home office, making my research days, and lunches in particular, a lot of fun.
J.A.W. Poland
Rotterdam, January 2022
i
Abstract
Kitepower B.V. (https://thekitepower.com/) is an airborne wind energy (AWE) company, that gener-
ates energy with a leading-edge inflatable (LEI) soft-wing membrane kite. Improving energy generation
can be done by increasing the aerodynamic performance of the kite, which is ideally done using shape-
performance optimization procedures that require fast models. Fast and accurate LEI kite models are
difficult to develop, due to the flexible nature of the kite. The flexibility allows several shape changes,
which substantially change the aerodynamic characteristics [33, 47, 68]. The changing shape forms
a complex aeroelastic problem, which is generally solved using fluid-structure interaction (FSI) mod-
els [7]. The two main deformation modes, symmetrical bending and asymmetrical twist, are used to
control the force production and turn the kite [8, 51]. This research will focus on predicting the shape
deformation, i.e. the structural component of a FSI model, using the V3 LEI kite of Kitepower B.V. as
research subject. The model should be modular, fast and accurate to excel as a building block for the
above-mentioned optimization procedure.
It is important to experimentally validate models, therefore, footage shot in flight was analyzed using
photogrammetry. The width changes were analyzed, because they provide a good indication of the
wing bending. The asymmetric deformation results were not of sufficient quality for quantitative analy-
sis, therefore, only the symmetric deformation results are used quantitatively. The local trailing-edge
(TE) lengths are measured, because its linked to the local membrane billowing, which is expected to
affect the aerodynamics and the global shape.
The kite wing is described as a wireframe representation consisting of multiple plates. The leading-edge
(LE) and side edges of the plate represent the tubular frame and are of constant length. The TE is also
of constant length, except for when billowing effects are included. Because the V3’s tubular frame is
supported on both the LE and TE by the bridle line system, all corners of all plates represent bridle line
attachment points. The point locations determine the shape of the kite wing and are calculated using
a bridle line system representation, calculation algorithm and geometric actuation input. Symmetric
actuation input affects the bending deformation mode and asymmetric actuation, i.e. steering, the twist
deformation mode.
The best formulated geometrical calculation algorithm is called trilateration, which has a runtime of
2ms. To work, it requires empirical relations and a straight bridle line assumption. Because slacking
bridles were observed in the symmetrical state, only 7-plates could be taken into account. Slacking
bridles were furthermore observed in the asymmetrical state, therefore, asymmetrical actuation input
is not simulated. A force-based particle system model (PSM) was formulated to resolve the slacking
bridle-induced problems, which has a runtime of 40 s. It is pseudo-physical due to the non-physical
inputs, used for the spring representation of the bridles and plate edges. The PSM solves a dynamic
problem of which only the steady-state solution is considered relevant and physical. Two symmetric
actuation inputs were used, both 7-plate trilateration-based and 9-plate PSM-based show best result.
The present errors are mainly attributed to modeling incorrect bridle line lengths, starting from the wrong
initial shape and using inaccurate loading conditions.
It has become clear that the PSM can predict the change in width due to symmetrical actuation within 1%
of the experimentally obtained width. Because of its relatively low computational cost, non-dependence
on empirical relations, aerodynamic model coupling potential and its ability to predict both main defor-
mation modes, the PSM is considered an ideal structural building block. Combined with the insights
from this research, the PSM forms a solid foundation for the next generation of FSI based kite de-
sign tools. The main recommendation is to validate the asymmetrical deformation, by acquiring more
accurate input and validation data.
ii
Contents
Preface i
Abstract ii
Nomenclature v
List of Figures viii
List of Tables xi
1 Introduction 1
1.1 TypesofAWESs........................................ 2
1.2 TheV3LEIkite......................................... 3
1.3 Aeroleasticeffects....................................... 4
2 Literature review 6
2.1 Deformations.......................................... 6
2.1.1 Vibrations........................................ 7
2.1.2 Fluid-structure interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Structuralmodels........................................11
2.2.1 Multi-platemodel....................................12
2.3 Experimentaldata .......................................14
2.4 Researchquestions ......................................17
3 Research approach 19
4 Data acquisition 21
4.1 Photogrammetry........................................21
4.1.1 Analysissetup .....................................22
4.1.2 Results.........................................26
4.2 Designgeometry........................................29
5 Wing models 32
5.1 Modellinghypothesis......................................32
5.2 wireframe representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Discretisationlevels ......................................35
5.3.1 Triangular 2-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3.2 Quadrilateral 2-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3.3 Quadrilateral 3-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3.4 Quadrilateral 7-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3.5 Quadrilateral 9-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 Bridle line system models 39
6.1 Tetrahedonmodel .......................................40
6.2 Trilateration...........................................41
6.2.1 Modelworkflow.....................................41
6.2.2 Uniformactuation....................................43
6.2.3 Non-uniform actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Particlesystemmodel.....................................48
6.3.1 Theparticles......................................50
6.3.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.3.3 Dampingforce .....................................51
6.3.4 Springforce.......................................52
6.3.5 Liftforce.........................................52
iii
Contents iv
7 Results 55
7.1 Triangular 2-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Quadrilateral 2-plate and 3-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 Quadrilateral 7-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.4 Quadrilateral 9-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.4.1 Discussion .......................................62
8 Conclusions and recommendations 64
8.1 Conclusions...........................................64
8.2 Recommendations.......................................65
References 71
A Geometry specifications 72
Nomenclature
Abbreviations
Abbreviation Definition
AWE Airborne wind energy
AWES Airborne wind energy system
CAD Computer aided design
Ch. Chapter
CFD Computational fluid dynamics
DOF Degrees of freedom
EOM Equations of motion
FE Finite element method
Fig. Figure
FSI Fluid-structure interaction
HAWT Horizontal axis wind turbine
KCU Kite control unit
LE Leading edge
LEI Leading edge inflatable
PSM Particle system model
QSM Quasi-steady model
Sec. Section
TE Trailing edge
V3 The 25m2kite of Kitepower B.V.
2D Two-dimensional
3D Three-dimensional
Latin symbols
Symbol Definition Unit
aLine between middle LE point and tip m
AtArea of the tetrahedon m2
bLine between KCU and tip m
CDamping constant N s m1
CDDrag coefficient -
CLLift coefficient -
cm,i Vector representing chord of plate im
cref Reference chord m
dLine between KCU and middle LE point m
Di,j Length of diagonal line i, j m
eLine between tip and middle TE point m
Ei,j Length of edge i, j m
fiFrequency is1
FdDamping force N
FrResultant force N
FsSpring force N
KSpring constant N m1
LLift force N
v
Contents vi
Symbol Definition Unit
LLift scaling factor -
ledge,i Edge length of plate im
ldDepower tape length m
lim Imaginary line length m
LpDistance from KCU to pulley m
LpPlate lift force N
l0Spring restitution length m
LEmp Middle point of the leading edge m
LEiVector representing the LE of plate -
NNumber of samples -
mMass kg
Pcross Line-of-sight crossing point on TE m
PiPoint im
Pl,i Plate i
q Positions of the particles m
rDistance from LEmp to Pcross m
RDistance from KCU to Pcross m
SSurface area m2
sSemi-perimeter m
SE Standard error m
St,i Strut ii
u Velocities of the particles m/s
uiUnit vector in direction i-
upPower setting -
usSteering setting -
vaVelocity m/s
VtVolume of the tetrahedon m3
vwWind velocity m/s
WWidth between tip bridle line attachment points m
xPosition m
˙xVelocity m/s
¨xAcceleration m/s2
xx-axis in new frame -
xpx-coordinate of pulley m
x
px-coordinate of pulley in new frame -
ypy-coordinate of pulley m
zpz-coordinate of pulley m
zpCross z-coordinate of Pcross m
Greek symbols
Symbol Definition Unit
αAngle of attack °
αdDepower angle °
αsSide-slip angle °
βElevation angle °
ldldchange between the powered and depowered
state
-
LExtension of Lpm
lExtension of the spring m
θAngle between y-axis and pulley line-of-sight °
µMean of the measured widths m
Contents vii
Symbol Definition Unit
ρDensity kg/m3
σStandard deviation -
List of Figures
1.1 How an AWES uses only the essential parts of an HAWT [58]. . . . . . . . . . . . . . . 1
1.2 An illustration of Kitepower B.V. its AWES, used for the PostNL Innovation Stamp series
[29]. ............................................... 1
1.3 OverviewoftheAWESs[59]. ................................. 2
1.4 Visualisation of the pumping cycle operation with the different phases identified [59]. . . 3
1.5 Video still of the V3, indicating the different components [3]. . . . . . . . . . . . . . . . . 4
1.6 Reel-in and reel-out phase of a pumping cycle operation where the change in kite orien-
tation with respect to the wind becomes clear [68]. . . . . . . . . . . . . . . . . . . . . . 5
1.7 Deformation of a LEI surf-kite during a left-turning maneuver [17]. . . . . . . . . . . . . . 5
2.1 Deformation of the V3 LEI kite, with the arrow indicating the position of pulley that controls
the amount of depower. Figure (a) shows a powered kite during the reel-out phase, (b)
a depowered kite and (c) demonstrates the asymmetric deformation during a turn [47]. . 7
2.2 LEI kite deformation modes ranked on time-scale [40]. . . . . . . . . . . . . . . . . . . . 8
2.3 The rainbow-colors show the deformation mode frequencies where as the grey-tones
show the flight-path frequencies. When estimating the frequencies from flight data, 10%
extra was added on both sides of the spectrum [40]. . . . . . . . . . . . . . . . . . . . . 9
2.4 Altered visualization of the FSI model types their conceptual differences [40]. . . . . . . 10
2.5 Geschiere’s approach to model the V3, by coupling a dynamic particle system to a static
FSImodel[27].......................................... 11
2.6 Currently available structural kite models adapted from [6, 8, 26, 54, 72]. . . . . . . . . . 12
2.7 Overview of the orientation reference frame (xw, yw, zw), yaw, pitch, roll, heading (ψ) and
spherical coordinates (β, ϕ, r)[47]. .............................. 13
2.8 A multi-plate model with 6 plates [72]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.9 Kiteplane experiment showing the flow separation aft of the LE due to the recirculation
region[8]. ............................................ 14
2.10 The 4-point model of Fechner et al. with the definition of the: depower angle (αd) and
the steering angle (αs) use to steer the kite [23]. . . . . . . . . . . . . . . . . . . . . . . 15
2.11 Discretized tether model connecting to the 4-point model [23]. . . . . . . . . . . . . . . . 15
2.12 Lift-to-drag ratio vs the angle of attack with a colour coded relative power setting [47]. . 16
2.13 Relation between the relative power setting (up) and the lift-to-drag ratio [47]. . . . . . . 16
2.14 CLplotted against the angle of attack (AOA). Where the experimental values of the
powered and depowered mode are compared against CFD results [53]. . . . . . . . . . 17
2.15 CDplotted against the angle of attack (AOA). Where the experimental values of the
powered and depowered mode are compared against CFD results [53]. . . . . . . . . . 17
3.1 Researchworkflow. ...................................... 20
4.1 Video still from the KCU of the V3 during a launch on the 24th of March 2017 [1]. . . . . 22
4.2 KCU swing demonstration of an 25 m2LEI kite during flight on the 24th of November
2010. It was filmed from a camera attached to the KCU, which at the time was located
lower down the tether [22]. The left image shows the KCU swinging to the right during
a turn. Upon comparison of the middle and right image, a ‘vertical’ difference is noticed,
alsoindicatingaKCUswing. ................................. 23
4.3 Video stills indicating the height difference of the kite within the image frame [1]. The
pulley point, knot point and kite bridle tip movements from powered to the depowered
state are identified. Furthermore, the red line indicates the present slacking bridle in the
depoweredstate......................................... 24
4.4 Effect of the kite pitching on the line-of-sight. . . . . . . . . . . . . . . . . . . . . . . . . 25
viii
List of Figures ix
4.5 Simplified 2D representation of how deformation can affect the length of the line-of-sight.
The anhedral angle change is taken here as example. . . . . . . . . . . . . . . . . . . . 26
4.6 Video still indicating the numbered struts (St,i ), plates (Pl,i) and the measured lengths [1]. 27
4.7 Sampling distribution of the width. Where the grey band indicates the standard error
(SE) and each dot represents an individual measurement. . . . . . . . . . . . . . . . . . 28
4.8 Video still of a left turn [1]. Where the white dotted lines indicate the slacking bridles and
the full white lines qualitatively show the expected asymmetric deformation. . . . . . . . 29
4.9 CAD drawing of the initial design of the V3 [1]. . . . . . . . . . . . . . . . . . . . . . . . 29
4.10 Adjusted bridle line system representation, with the red square roughly outlying the KCU
bridlelinesystem[47]...................................... 30
4.11 A video still indicating the positions of the knots and pulleys [1]. Where the red line
indicatesthenewline. .................................... 31
5.1 Kitepower B.V.’s kite evolution from left to right one sees the: V2, the Hydra and the V3
[60]. ............................................... 32
5.2 Schematic illustration of a kite with three segments. . . . . . . . . . . . . . . . . . . . . 33
5.3 Two images of the V3.A kite, where the red quadrilateral indicates one of the spanwise
segments and the red dotted circles indicate the bridle fan [60]. The curved lines of the
membrane canopy TE in between the strut tips demonstrate the ballooning. . . . . . . . 34
5.4 wireframe representation of the kite wing, where each kite segment is represented by a
plate. The red arrows indicate the DOF of the plates. . . . . . . . . . . . . . . . . . . . . 34
5.5 Triangular 2-plate wing model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.6 Quadrilateral 2-plate model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.7 Quadrilateral 3-plate model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.8 Quadrilateral 7-plate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.9 Quadrilateral 9-plate model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.1 Schematic representation of the triangular 2-plate model. The blue lines indicate the
lines attached to the LE and the red line the line attached to the TE (lim). . . . . . . . . 40
6.2 2D representation of the 3D sphere intersection problem, with P representing the inter-
sectionpointin2D........................................ 41
6.3 Illustration of the 2D plane spanned by P0,P2and P4, shown in red. . . . . . . . . . . . 42
6.4 Quadrilateral 2-plate model. The blue lines indicate the lines attached to the LE and the
red lines those that are actuated and attached to the TE. . . . . . . . . . . . . . . . . . . 43
6.5 Quadrilateral 3-plate model. The blue lines indicate the lines attached to the LE and the
red lines those that are actuated and attached to the TE. . . . . . . . . . . . . . . . . . . 44
6.6 Quadrilateral 7-plate model. The blue lines indicate the lines attached to the LE and
the red lines those that are actuated and attached to the TE. On the right a front view
is shown with only the TE bridles visible, the thick red lines indicate the KCU bridle line
system representation and the dots the knots and pulleys. . . . . . . . . . . . . . . . . . 45
6.7 7-plate wing model, with 2D plane spanned by LEmp , Pcross and the KCU. . . . . . . . . 46
6.8 Illustrations of the non-uniform calculation method. . . . . . . . . . . . . . . . . . . . . . 47
6.9 Top view of the quadrilateral 7-plate. The red arrows indicate the ballooning causing the
TE length to differ, which alters the diagonal distance. . . . . . . . . . . . . . . . . . . . 48
6.10 Orthographic view of the PSM on the left. On the top right top the bridles attached to the
LE are shown, whereas on the bottom right those attached to the TE. . . . . . . . . . . 49
6.11 Dynamic PSMs, depicted in the same colours for ease of comparison. . . . . . . . . . . 50
6.12 Orthogonal view of the PSM, where each dot represent a particle. . . . . . . . . . . . . 51
6.13 Top view of the PSM, where the diagonal kite wing plate elements are clearly shown. . . 51
6.14 Spring force due to a line extension between two knots. . . . . . . . . . . . . . . . . . . 52
6.15 Spring force due to a line extending between two knots, that goes over a pulley. . . . . 52
6.16 Free body diagrams of the spring force (Fs).......................... 52
6.17 9-plate wing model, where the red arrows indicate each lift vector (
Lp) perpendicular to
itsrespectivepanel. ...................................... 53
List of Figures x
7.1 Photogrammetry results for the width changes, plotted together with the simulation re-
sults of the 2-plate triangular model calculated using the trilateration and tetrahedon
algorithms. ........................................... 56
7.2 2-plate triangular subject to a symmetrical deformation, calculated using the trilateration
algorithm. In black the shape for up= 1 and in red the shape for up= 0. The figure shows
an orthographic view in the top left, a top view in the top right, a side view in the bottom
left and a front view in the bottom right. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 Photogrammetry results for the width changes, plotted together with the simulation re-
sults of the 2-plate and 3-plate quadrilateral model calculated using the trilateration algo-
rithm. .............................................. 57
7.4 2-plate quadrilateral subject to a symmetrical deformation, calculated using the trilatera-
tion algorithm. In black the shape for up= 1 and in red the shape for up= 0. The figure
shows an orthographic view in the top left, a top view in the top right, a side view in the
bottom left and a front view in the bottom right. . . . . . . . . . . . . . . . . . . . . . . . 57
7.5 3-plate quadrilateral subject to a symmetrical deformation, calculated using the trilatera-
tion algorithm. In black the shape for up= 1 and in red the shape for up= 0. The figure
shows an orthographic view in the top left, a top view in the top right, a side view in the
bottom left and a front view in the bottom right. . . . . . . . . . . . . . . . . . . . . . . . 58
7.6 Photogrammetry results for the width changes, plotted together with the simulation re-
sults of the 7 quadrilateral plate model calculated using the non-uniform trilateration al-
gorithm. ............................................. 59
7.7 7-plate quadrilateral subject to a symmetrical deformation, calculated using the non-
uniform trilateration algorithm and ld= 13%. In black the shape for up= 1 and in
red the shape for up= 0. The figure shows an orthographic view in the top left, a top
view in the top right, a side view in the bottom left and a front view in the bottom right. . 60
7.8 Photogrammetry results for the width changes, plotted together with the simulation re-
sults of the 9 quadrilateral plate model calculated using a PSM. . . . . . . . . . . . . . . 61
7.9 9-plate quadrilateral subject to a symmetrical deformation, calculated using the PSM and
ld= 8%. In black the shape for up= 1 and in red the shape for up= 0. The figure shows
an orthographic view in the top left, a top view in the top right, a side view in the bottom
left and a front view in the bottom right. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.10 9-plate quadrilateral subject to a symmetrical deformation, calculated using the PSM and
ld= 8%. In black the shape for up= 1 and in red the shape for up= 0. A bottom view
is shown, from a similar angle as the footage was shot, which shows the inward bending
ofthetips............................................. 62
7.11 9-plate quadrilateral particle system based asymmetrical deformation. In black the shape
for us= 0 and in red the shape for us= 0.5. The figure shows an orthographic view in
the top left, a top view in the top right, a side view in the bottom left and a front view in
thebottomright. ........................................ 62
A.1 KCUbridlelinesystem. .................................... 72
A.2 Altered view of the flat V3 shape, where the bridle attachment points are indicated [60]. 73
A.3 Schematic illustration of the TE bridles, with the particles numbered. . . . . . . . . . . . 73
A.4 Schematic illustration of the LE bridles, with the particles numbered. . . . . . . . . . . . 74
List of Tables
4.1 Photogrammetry results of the relative length changes. . . . . . . . . . . . . . . . . . . 28
4.2 Photogrammetry results governing the changes of the pulley and knot line-of-sight canopy
crossingposition......................................... 28
A.1 Design specifications, with the coordinates of each point given. . . . . . . . . . . . . . . 75
xi
1
Introduction
To counteract global warming, many countries have set ambitious targets to transition from fossil fuels
to renewable energy sources [52].Whether or not wind energy alone could theoretically supply all the
world’s energy needs is an ongoing debate between scientists [36].The horizontal axis wind turbine
(HAWT) is the most commonly used instrument to perform the desired conversion of kinetic energy
present in the atmosphere to electrical energy. Looking at the structural components of a conventional
HAWT one finds that most mass does not directly contribute to the power production, i.e. more than
half of the power is produced by only the outer 30% of the blades [16]. An alternative to wind turbines is
airborne wind energy systems (AWESs), these systems use a kite to harvest kinetic energy. According
to J. Breukels, a kite is defined as:
“A tethered heavier-than-air device able to achieve flight by generating a resulting aerody-
namic force, which is countered by the mass of the device and the tension force in the tether”
[8].
By using kites that resemble the tips of a HAWT, AWESs require less material for the same power
generation (see Fig. 1.1) [39]. Another advantage of AWES is their ability to operate at greater altitudes
where generally more kinetic energy is present [2]. This, in turn, increases the operational bandwidth
and reduces the turbulence experienced by the AWES.
Figure 1.1: How an AWES uses only the essential parts
of an HAWT [58].
Figure 1.2: An illustration of Kitepower B.V. its AWES,
used for the PostNL Innovation Stamp series [29].
The combination of higher power output to required material ratio and increased operating ranges has
led to an increased academic and commercial interest in AWESs over the last decade (see Fig. 1.2).
The number of institutions active in this field worldwide grew from less than five in 2000 to over 60 in
2018 [59]. Considering the immaturity of the airborne wind energy sector it is not surprising that several
1
1.1. Types of AWESs 2
challenges are still present, e.g. proving safe autonomous operation over long periods. Other focus
points are reducing the cost of production, installation and operating costs, reducing the environmental
impact and increasing the energy yield of the system. The latter can be achieved by increasing the op-
erational bandwidth or increasing the system efficiency. Increasing operational bandwidth is achieved
through a more optimized geographical system placing or by developing a lighter system. Increasing
system efficiency can be done by increasing the generator conversion efficiency and increasing the
tether force, possible through flight path optimization and increased aerodynamic performance. The
latter is often expressed as the lift-to-drag ratio and increasing its value can also lead to a lighter system,
making its effect on the energy yield twofold.
1.1. Types of AWESs
In 1980, Loyd analytically showed the large potential of a cross-wind flying kite generation system
by quantifying the tether force and comparing it to a regular flying kite. A part of the potential arises
from the difference in kite velocity. Force generation scales with the observed apparent velocity ( va)
squared, increasing the kite velocity is, therefore, an effective way of increasing the force. Loyd also
proposed two different operating modes called the lift- and drag power mode which covers all variants
of AWESs existent today [39]. In the lift power mode, electricity is generated on the ground by a drum
connected to a generator and thus varying the airborne tether length. In the drag power mode electricity
is generated on the kite itself and then transported to the ground through a tether of constant length.
Loyd concluded that the theoretical performance of the two modes is roughly the same and that one
should maximize the lift-to-drag ratio to maximize the energy extraction. As of today no obvious winner
has been identified and in both lift- and drag power modes multiple institutions are actively working on
implementation solutions (see Fig. 1.3). [41]
Figure 1.3: Overview of the AWESs [59].
The different kite wings can be categorized into soft-wings and fixed-wings. Soft-wings are generally
lighter in comparison to fixed-wings allowing them to operate at lower wind speeds. Another benefit
is that the damage of a crash is generally less severe, to both the system and its surroundings. The
aerodynamics of the fixed-wing AWES is comparable to aircraft aerodynamics and therefore currently
better understood. Soft wing membrane AWES has been shown to have the potential for higher lift-
to-drag ratios compared to fixed-wings and the prime reason is the flexibility of the membrane [4].
Maughmer was more conservative in his statements and concluded that flexible sail wings have very
competitive lift-to-drag ratios compared to fixed wings [42]. The lift-to-drag ratio is relevant because it
determines the speed a kite can obtain which relates to the tether force it can generate [61].
1.2. The V3 LEI kite 3
Figure 1.4: Visualisation of the pumping cycle operation with the different phases identified [59].
Within the soft wing category, there is a soft wing membrane leading edge inflatable (LEI) kite. Kitepower
B.V. (https://thekitepower.com/) amongst other airborne wind energy companies use a LEI kite to
generate energy. Kitepower B.V. is a commercial spin-off from the Delft University of Technology that
operates in the lift power mode where it flies pumping cycle operations (see Fig. 1.4). The operations
go in cycles, first the kite reels out the tether with figure of eight movements and once a certain dis-
tance is reached the reel-in phase begins and the generator starts reeling in the tether [47]. Power is
needed to do so, therefore the pumping cycle operation does not continuously deliver power. This is a
downside that can be resolved by operating multiple kites simultaneously off-phase.
1.2. The V3 LEI kite
The design of the V3 LEI kite is based on the surf-kite, in particular the Generatrix Hydra V7 kite. Oehler
and Schmehl amongst others have previously performed research on the V3 and identified the need
for further research efforts [47]. Research into LEI kite behavior would benefit Kitepower B.V. and other
relevant AWES companies because it could help them improve overall system efficiency. The V3 is
furthermore selected as a research subject because of the existing and available experimental data.
Another reason is the existence of simulation results thanks to the work of other researchers [14, 38, 53].
The V3 kite consists of a leading-edge (LE) tube in spanwise direction and strut tubes in chordwise
direction, which both are pressurized with air (see Fig. 1.5). This tubular frame is connected by a
membrane called the canopy and together they form a LEI kite. The rear end is generally referred to
as the trailing edge (TE). The control pod used for automatic control in flight, also called the kite control
unit (KCU), is placed in between the tether and the bridle line system. The bridles attached to the LE
are often referred to as the power lines and those attached to the TE as the steering lines. Most force
is namely carried by the power lines and actuation is done using the rear steering lines.
1.3. Aeroleastic effects 4
Figure 1.5: Video still of the V3, indicating the different components [3].
To assess how much and where there is room for improvement one could determine the theoretical
maximal power output and on what it is dependent on. Fechner and Schmehl calculated the theoretical
total efficiency for the V3 and found estimates around 50% to 60% [20]. The importance of the generator
efficiency was stressed and due to its increase for increasing kite sizes, larger kites are expected to be
able to have higher total efficiencies. Besides the generator efficiency, the lift-to-drag ratio is consid-
ered another area where improvement possibilities lie. Costello et al. developed a different framework
with which one can assess the maximum average power a general LEI AWES kite can generate [11]. In
this framework, the lift-to-drag ratio comes out as the most effective way to increase power production.
1.3. Aeroleastic effects
For high pumping cycle efficiency, one wants to maximize the generated force during the reel-out phase
and minimize during the reel-in phase. Force generation of any body in a fluid is a function of the fluid
properties, the fluid velocity, the shape of the body, the body’s velocity and the angle of attack (α), i.e.
the orientation it has towards the incoming fluid. Kitepower B.V. achieves force control by changing
the kite body its: velocity, shape and orientation. A change in kite velocity is achieved by not flying
figures of eight during the reel-in phase. The lift-to-drag (CL/CD) ratio is controlled by changing the
power setting and is a function of the shape and angle of attack. The power setting affects the angle
of attack by pitching the kite (see Fig. 1.6) and it affects the shape, mainly through inducing a bending
deformation that changes the anhedral angle. Another relevant deformation mode is the asymmetric
twist of the kite, which is needed, together with roll, for turning maneuvers [8].
1.3. Aeroleastic effects 5
Figure 1.6: Reel-in and reel-out phase of a pumping cycle
operation where the change in kite orientation with respect to
the wind becomes clear [68]. Figure 1.7: Deformation of a LEI surf-kite during a
left-turning maneuver [17].
The deformations are the result of the interaction between aerodynamic force dependent on the shape
and the structural force, arising from resisting the aerodynamic force-induced shape deformation (see
Fig. 1.7). The result is a complex aeroelastic problem, which is generally solved using Fluid-structure
interaction (FSI) models. As stated by Bosch et al. incorporating the relevant macro-scale deforma-
tions, bending and asymmetric twist that is, into a strongly coupled FSI model is required to get accurate
results [6]. Problematic for existing FSI models is that they either have too high computational cost or
don’t include sufficient details to remain realistic and relevant. The latter flaw most often has arisen
from models using rigid body assumptions, which considering the relevance of the discussed deforma-
tions is not representative of the in-flight behavior. Without including deformations one can therefore
not study the aerodynamic performance of kites accurately, thereby also not efficiently develop designs
that increase it.
The objective of this thesis will be towards the development of a fast structural model that can take into
account the relevant macro-scale deformation modes of bending and twist. The model focus will be on
accuracy, computational cost and modularity. The latter is key for the model its potential as a building
block since it must be able to deal with other kites. The low computational cost requirements have led
to a low-fidelity model scope. The V3 kite of Kitepower B.V. will be used as a research subject. The
work will contribute to the scientific community by providing insights into the deformation phenomena
and by developing a building block for a kite design tool. The latter will furthermore contribute to the
airborne wind energy (AWE) community and hence towards the energy transition.
2
Literature review
This research deals with using modelling procedures to improve aerodynamic performance. Alterna-
tively, one could determine the performance of the kite by physically producing it and testing it. This
‘Darwinian process’ as Breukels calls it, is the most common approach in surf-kite development today
and is far from ideal [8]. One reason is that performance is generally assessed subjectively by pro-
fessional athletes, most likely due to a lack of quality measurement instruments. Another reason is
that producing and testing a kite is a time and resource-intensive endeavour, i.e. surf-kite designers
need about a month per design iteration [35]. Therefore, one can’t use design optimization procedures,
which are generally considered ideal for finding an optimal shape.
Modelling the kite gives one the possibility of assessing the performance without flying it. This drasti-
cally decreases the time needed for an iteration, making design optimisation iterations viable for fast
models. Design models generally achieve the required low computational cost by reducing the overall
complexity. Robin van de Putte, the founder of a relatively new player in the surf-kite industry called Van-
tage (https://www.vantagekites.com/), identified this modelling procedure as necessary and highly
beneficial.
Developing a LEI kite model is a complex task, especially when one wants to assess its performance.
One reason is that it does not only require a model of the kite but also a simulation framework. Develop-
ing such a framework is difficult partly due to it relying on flight condition measurements for an accurate
real-world representation and partly due to the deformations. Increasing aerodynamic performance
could for a surf-kite producer like Vantage lead to more sales and for an operating AWES, it could lead
to higher power output.
Deformations are considered the key factor for making modelling soft wing membrane kites, like the V3,
so difficult (see Sec. 2.1 and 2.2). The best insights into the physical behaviour are currently found from
studying experimental data rather than through modelling alone (see Sec. 2.3). The relevant research
opportunities found in the gaps of the literature are addressed as research questions (see Sec. 2.4).
2.1. Deformations
A kite will only fly if the lift is greater than its weight. AWES LEI kites are, therefore, designed to have
a high lift-to-weight ratio enabling operation at low wind speeds. Using little material comes at the
cost of low structural strength, hence the flexible nature of LEI kites. Most of the structural strength
a LEI kite has, comes from the pressure distribution in the canopy resisted by the tensile force in the
connecting bridle lines. Due to its flexible nature, the kite deforms during operation, e.g. the body shows
clear deformation during a turn (see Fig. 2.1c). Modelling deformation is considered highly important
because the deformation itself is what allows the kite to turn according to Breukels’s widely accepted
turning theory [8]. It states that an asymmetric steering input leads to asymmetric deformation. This
asymmetric deformation creates an arm for the aerodynamic forces generated on the outer section
of the kite, which combined provide the moment that turns the kite. One might expect the rear tip
6
2.1. Deformations 7
to counteract the moment, but due to the deformation, the inflow angle is less optimal meaning less
force is produced. Another observed effect contributing to turning is the rolling motion of the kite. Both
the twisting and rolling motion happen simultaneously and their effects are hard to separate. Bosch
amongst others has validated the described turning behaviour of Breukels in simulations [5].
Figure 2.1: Deformation of the V3 LEI kite, with the arrow indicating the position of pulley that controls the amount of depower.
Figure (a) shows a powered kite during the reel-out phase, (b) a depowered kite and (c) demonstrates the asymmetric
deformation during a turn [47].
The difference in kite shape during the reel-out phase where it is ‘powered’ and during the reel-in phase
in which it is ‘depowered’ is another form of deformation (see Fig. 2.1a and 2.1b) . In the powered mode,
the kite is flatter and the canopy billows more compared to the depowered mode where the shape is
more arc-like and less billowing is present. The kite has a different lift-to-drag ratio in both modes, which
is key for the pumping cycle operation’s efficiency because during the retraction phase one wants to
spend as little energy as possible.
Bosch states that due to the sensitivity of the aerodynamic forces the change of external force upon
deformation is substantial [7]. For the North Rhino 16 m2kite Bosch found a 24% change in tip-to-tip
distance upon comparing the maximum and minimum deformation [7]. The change in width is caused
by a change in anhedral angle affecting many parameters like the projected surface area that affect
the aerodynamic load [47]. That aeroelastic deformation changes the aerodynamic performance has
been demonstrated from analyzing experimental data (see Sec. 2.3) [33, 51, 68].
The most pronounced differences are found between the reel-in and reel-out phases and during turning.
During a steering manoeuvre, the lift-to-drag ratio is reduced as a result of the additional drag compo-
nent induced by the tip’s side force [23, 47]. Roullier found an average reduction of 5% in lift-to-drag
ratio during a turn for the V3 kite [53]. The differences between the reel-in and reel-out phases are so
pronounced that separate lift-to-drag ratios are used in models that exclude deformation [57, 68]. As
Oehler et al. put it: “only after understanding the influence of kite shape and deformation its design can
be optimized for aerodynamic performance” [48]. The importance and relevance of modelling deforma-
tion become clear from an example case. Suppose one would be able to predict which aspects of the
kite affect the required steering input for a certain turn rate one could design for lower steering needs
using this information. Lower steering needs would mean one could use a smaller KCU resulting in
less system weight, i.e. higher efficiency and larger operational bandwidth.
2.1.1. Vibrations
Besides actuation induced deformations, there are other deformations that occur due to vibrations.
Leuthold split up the frequency modes, making a distinction between larger-scale and sub-scale defor-
mations (see Fig. 2.2).
2.1. Deformations 8
Figure 2.2: LEI kite deformation modes ranked on time-scale [40].
Relating the deformation modes to operation manoeuvres, one finds that the bunny-ear flapping mode
is the same deformation as used for turning and the collapse deformation mode the same as used for
changing the power setting. The latter is achieved by span-wise bending and the resulting change in
anhedral angle is considered the main contribution enabling one to differ the lift-to-drag ratio settings.
Operational manoeuvres are inferred by the actuation input, which changes the bridle line system lay-
out. An example of a deformation mode that is not actuation induced, is TE fluttering. Vortex shedding
from the TE of the canopy membrane causes the TE to flutter, which results in a local oscillatory motion
with relatively high-frequency [40].
For a complete picture of the frequency modes, one should also consider the following flight path
frequencies: figures of eight (f8), general flight dynamics (fF D) and typical characteristic aerodynamic
frequency (fa). Leuthold estimated a frequency band for the deformation modes and flight path frequen-
cies, which were then visualized using a frequency spectrum (see Fig. 2.3). The identified deformation
modes are: trailing edge fluttering (fT EF ), seam-rippling (fSR ), canopy buckling (fC B ), jelly-fishing
(fJF ), bunny-ear (fB E ), leading-edge indentation (fID M ) and collapse (fCollapse).
2.1. Deformations 9
Figure 2.3: The rainbow-colors show the deformation mode frequencies where as the grey-tones show the flight-path
frequencies. When estimating the frequencies from flight data, 10% extra was added on both sides of the spectrum [40].
When the goal is to dynamically simulate the precise shape of the kite, all the deformation modes are
relevant. In practice, it is possible to neglect some while still getting accurate results. Why this is pos-
sible is best explained by thinking about the aerodynamic load. A deformation mode causes a shape
alteration, which changes the aerodynamic load. If this change is present for a substantial time dura-
tion, the effects can’t be neglected. However, if the modes are periodic and occur for a small duration
they can be neglected [8]. Because the higher-order TE canopy vibrations occur periodically and are
dominated by the lower frequency modes, their effects can be neglected [70]. Compared to the TE,
the rest of the canopy is subject to higher tension loads. Higher tension loads, lead to lower frequency
vibration modes. Neglecting sub-scale deformations for the rest of the canopy is, therefore, deemed
valid [40].
Neglecting the sub-scale frequencies one finds a reduced-order frequency range, which for a quasi-
steady flow assumption holds [40]. Van Kappel neglected the sub-scale frequency modes, therefore,
decided to use a quasi-steady flow assumption [35]. Van Kappel’s model showed reasonable results
compared to experiments, thereby providing evidence for the validity of the quasi-steady state assump-
tion. Within the aeroelasticity domain, one can have a static, quasi-steady or dynamic model. The
quasi-steady model is generally assumed valid when the inertial forces are small compared to the aero-
dynamic and elastic forces, which for the V3 kite holds [53]. Furthermore, assuming a quasi-steady
state is useful because it reduces the computational cost needs [5].
2.1.2. Fluid-structure interaction
The occurring deformations of the kite during flight are caused by the fluid interacting with the kite
structure. This effect can be modelled using fluid-structure interaction (FSI) models. FSI models must
be solved numerically and often an aeroelastic solver is used. FSI models can be subdivided into
partitioned and monolithic solvers. The difference lies in that with a partitioned solver one formulates the
aerodynamic and structural model separately. Partitioned solvers are used more frequently because
they allow the use of already existing structural and aerodynamic models.
2.1. Deformations 10
Figure 2.4: Altered visualization of the FSI model types their conceptual differences [40].
There are one way and two way coupled partitioned solvers. In a two way coupled partitioned solver,
iterations occur between a structural model that calculates the shape deformations based on the ex-
erted forces and an aerodynamic model that calculates the aerodynamic forces based on the shape.
Because the mesh is prone to change throughout the iteration, the aerodynamic and structural mesh
must be coupled and formulated dynamically [49]. Partitioned solver models can either be strongly
coupled or loosely coupled. Loosely coupled FSI models are more efficient, but also more prone to
numerical instability [73]. Strongly coupled models are more robust but require higher computational
costs.
In 2020 Folkersma et al. build a two way coupled partitioned aeroelastic solver to model the deforma-
tion of another type of soft wing membrane kite, called a ram-air kite [25]. Nobody has been successful
in setting up an accurate FSI model for the LEI kite, however, due to its complexity and computational
cost. To illustrate this point, some example studies will be discussed. Schwoll devoted a master (MSc)
thesis to develop only a structural model using a finite element (FE) approach with commercial soft-
ware. The complexity was of such level, that within a year of research not all phenomena of interest
could be modelled [62]. Lebesque devoted a MSc thesis to set up only an aerodynamic model for the
V3 [38]. The run-time of Lebesque’s computational fluid dynamics (CFD) model was in the order of
hours. Because iterations are needed, this makes using CFD in FSI too computationally expensive.
Another reason why FSI modelling of LEI kites using CFD and FE is difficult is the attachment between
the LE and canopy. This attachment results in a sharp corner in the CFD mesh, which makes the
problem numerically unsolvable. Van Kappel worked around this problem in the aerodynamic analy-
sis, by assuming it to be a corner of finite radius [35]. This solution does however not work in LEI FSI
models, because a dynamic mesh formulation is required. An alternative modelling solution is to attach
the canopy to the middle of the LE, as Bosch et al. did. This, however, results in overestimating the
lift-to-drag ratio [6].
Geschiere extended Bosch his partitioned two way coupled FSI model using an adaptation of the work
of Fechner et al. by including bridles and an extendible tether [21, 27]. An extendible tether was cho-
sen because modelling a tether of varying length allows for power production prediction. Due to the
differences in the complexity of the modelled AWES components, Geschiere decided to use different
models and couple their outputs (see Fig. 2.5). By not modelling each component with the same level
of detail, a reduction in computational cost was achieved. Geschiere’s dynamic bridle model is coupled
to the FSI kite model through the bridle constraint points. For stability, it was found best to include 0.3m
of bridle length into the FSI kite model. When comparing the modelled V3 kite with experimental data,
2.2. Structural models 11
it was found that with the current aerodynamic model artificial damping is needed for a stable solution.
Artificial damping is considered the main factor causing the underestimation of the turning rate of the
kite.
Figure 2.5: Geschiere’s approach to model the V3, by coupling a dynamic particle system to a static FSI model [27].
Concluding, it is observed that a deformation model is needed to model LEI kites. A partitioned FSI
model based on computational fluid dynamics (CFD) and FE is not a viable option due to its high
computational cost. Besides too long run times, a mesh is required for both CFD and FE methods.
Formulating a dynamic mesh for the complex shape of a LEI kite is a time- and resource-intensive
procedure which makes assessing new shapes fast difficult, if not infeasible. Because accuracy and
computational cost are inherently a trade-off, a less accurate but faster structural and aerodynamic
model are needed. Furthermore, as Geschiere’s model illustrates, besides the kite model the tether
and bridle lines must also be included and formulated separately, to be able to simulate a pumping
cycle operation [27].
2.2. Structural models
Structural models use material properties and forces as inputs to calculate the deformations of a body.
In this section, several developed structural LEI kite models will be discussed (see Fig. 2.6), which are
ranked based on computational cost and included degrees of freedom (DOF).
Black box models use experimental data to find relations between variables. This has the advantage
of being able to model the behaviour of the kite as was experienced during the experiment. Noom
developed a fast analytical quasi-steady model and modelled the kite as a single point ignoring inertia
and deformation [45]. By introducing some mass and dropping the quasi-steady state one arrives at the
point mass model. Diehl developed a point mass model in 2001, which according to Ruppert formed
the basis for many studies performed since then [15, 54]. When representing the kite as a finite num-
ber of points more DOF are taken into account and these models are called particle system models. In
2.2. Structural models 12
particle system models all points have a mass and the connections have some internal forces. A rigid
body model assumes the kite to be a solid which does not deform. The main difference with the point
mass and particle system models is that it incorporates an orientation allowing it to deal with attitude
dynamics [54]. Attitude is referring to the heading of the kite and is a term often used when describing
an aircraft flight path.
Figure 2.6: Currently available structural kite models adapted from [6, 8, 26, 54, 72].
Williams et al. as well as van Til et al. developed multi-plate models to incorporate deformations [67,
71]. In both models, a finite number of rigid plates is used which are connected with joints that are
providing the necessary degrees of freedom (DOF). Furey extended the multi-plate model towards
a lumped mass model. He used point masses connected with constraints to simulate the kite [26].
Breukels developed a multi-body model and the reason for doing so was to be able to model the defor-
mation occurring due to the flexibility without needing FE methods [8]. The multi-body model is based
on multi-body dynamics that connect rigid bodies with springs, dampers and hinges. The models that
incorporate FE methods are those that can achieve the highest accuracy. Bosch et al. developed a
FSI model that used FE method and Breukels his aerodynamic model [6].
Experimental data is needed for black-box models they, therefore, can’t be used for a design model that
assesses new shapes [13]. Because deformations are deemed essential for a design model the current:
point mass model, particle system and rigid-body model all are not feasible options. The lumped-mass
model has problems with numerical instability due to the presence of large forces acting on small
masses [54]. This numerical instability is resolved by using a small time step, which unfortunately
leads to high computational cost. The multi-body approach of Breukels is about 20 to 30 times slower
than real-time, i.e. too computational expensive [8]. Furthermore, according to Bosch, the model does
not include the real flexibility of the kite by not modelling the axial strains. On top of that, the parameters
used require fitting procedures, making using them for design purposes difficult. FE methods are too
computationally expensive for iterative design procedures, e.g. Bosch et al. reports run-times of 27.5
slower than real-time [6]. The best available option is, therefore, the multi-plate model.
2.2.1. Multi-plate model
The multi-plate model of Williams et al. is a flexible kite model designed to study the dynamics and
control of a kite [72]. Each section determines its forces and moments using plate aerodynamics. To
do so, the plate model is completely flexible about its center line for both torsional and bending mode
deformations. Another DOF comes from the plates ability to hinge over the LE. In reality, this is not
possible, because the plates could collide with one another. This is a non-issue according to Williams
2.2. Structural models 13
et al. because the model is simply designed to be an abstract approximation. The connecting bridle
lines are assumed straight. A constraint is set on that all plates must have the same yaw angle (see
Fig. 2.7). The pitch, roll and yaw angles of the plates are determined by the bridle attachment points
and the tensile and aerodynamic forces. The two-plate model (see Fig. 2.6) is easily extended to more
plates as long as one uses an even number of plates.
Figure 2.7: Overview of the orientation reference frame
(xw, yw, zw), yaw, pitch, roll, heading (ψ) and spherical
coordinates (β, ϕ, r) [47].
Figure 2.8: A multi-plate model with 6 plates [72].
Non-physical forces are needed for the hinges and springs with which the plates are attached. Accord-
ing to Deaves the inclusion of non-physical forces reduces the deformation prediction accuracy and
only part of the flexibility is modelled [13]. Furthermore, according to Geschiere, the dynamics are not
accurately modelled since the kite mass is modelled as a point mass, i.e. the bridle mass point (see
Fig. 2.8) [27].
The multi-plate model developed by van Til et al. is designed for control purposes and runs almost
real-time [67]. It is made from three plates, each connected with gimbal joints and it allows the two
main deformation modes required for turning manoeuvres, i.e. spanwise bending and torsion. Van
Til et al. argue that the existent non multi-plate models are either too computationally expensive, e.g.
the model of Bosch et al., or do not model the phenomena accurately enough, e.g. Fechner et al.’s
model [6, 23]. For future work, it is recommended to include more plates since that would improve the
accuracy of the deformation modelling.
With a fast aerodynamic model and a fast accurate stable deformation model, the two way coupled
FSI model has the potential to have low enough computational cost to be viable for design optimisation
iterations. The existent deformation models are however not accurate, not stable or not fast enough
and it is here where the author sees a possibility to contribute.
Considering the current aerodynamic models suffer from the same problems, it is likely that even with
a fast accurate stable deformation model the computational cost of a two way coupled FSI model is
still too large for iterative design procedures. An alternative solution for modelling the deformations
could be to develop a novel one way coupled FSI model. Such a model would not require iterations
nor dynamic mesh formulation, leading to less computational cost and lower lead times. The structural
model needed would have to be able to calculate the shape without knowing the force output of the
aerodynamic model. Another argument supporting this idea comes from the fact that both the kite ma-
terial and the kites deformation modes are non-linear and modelling non-linear behaviour is complex
thereby generally computationally expensive [27]. A one way coupled FSI LEI kite model has never
been developed yet, because there has never been a deformation model that accurately predicts the
shape without knowing the forces. Therefore, the author sees a possibility to contribute to the body of
science by developing such a model.
2.3. Experimental data 14
2.3. Experimental data
Most of the previously discussed models do not use experimental data, which make them useful for
design purposes but less when trying to accurately simulate the flight trajectory. Modelling difficulties
arise due to deformations and their dependency on historical flight manoeuvres, which make it infeasi-
ble to use well-developed aircraft analysis techniques [7]. Most models that simulate the flight trajectory
solve these issues by not modelling deformation.
Without knowing the operating conditions it is impossible to simulate in-flight behaviour accurately. To
obtain operating conditions experiments are needed. Usually, for fixed-wings a wind tunnel experiment
would be conducted, where a scale model would be used and flow characteristics would be made similar
such that the same phenomena as during flight can be observed and studied. For soft wing membrane
kites problems arise because the occurring FSI phenomena on a scale model are different [69]. Be-
cause they are different, it is hard to extrapolate information [47]. Wind tunnel tests are furthermore
expensive, nonetheless, one has been done for the LEI kite plane of Breukels (see Fig. 2.9) [8].
Alternatively, one could perform experiments outside of the wind tunnel, which brings up difficulties
with measuring and controlling the environment. Hummel et al. developed a tow test for measuring
the dynamic properties of tethered membrane wings to eliminate some of these unknowns [33]. In the
experiment and others: velocities, angles, forces and power settings were measured enabling aerody-
namic identification procedures. Such procedures are deemed by Oehler and Schmehl as the most
optimal way of obtaining aerodynamic coefficients [47]. The main reason is the inclusion of all effects,
one of them being deformation effects. A non-exhaustive overview of the most relevant aerodynamic
identification models will, therefore, be presented here.
Figure 2.9: Kiteplane experiment showing the flow separation aft of the LE due to the recirculation region [8].
Van der Vlugt et al.
Van der Vlugt et al. developed a quasi-steady model (QSM) that can predict the power generation
using a separate set of analytic equations for the traction, retraction and transition phase [69]. Two
pumping cycles flown at different wind speeds were used for aerodynamic identification and validation.
Small time scale dynamic processes, tether elasticity, variations of the aerodynamic properties within
a phase and wind changes over time are all assumed negligible and not modelled. A framework with
and without gravity was developed and the conclusion was drawn that including gravity is necessary to
simulate the performance. Van der Vlugt et al. mention that wind measurements locally would improve
identification efforts and that for studying dynamic flight behaviour a dynamic system model is needed.
Schelbergen and Schmehl performed a validation study of the QSM using experimental data with 87
pumping cycles [57]. They found that a drag coefficient correction is needed, which was partly at-
tributed to not modelling crosswind manoeuvres. Furthermore, neglecting the vertical wind component
2.3. Experimental data 15
or assuming a straight non-slacking tether was found to cause substantial errors. The QSM under pre-
dicted the power production by 26.4%. From both studies, it can be concluded that for a performance
model to make accurate predictions the simulation must include the effects of gravity, slacking tether,
crosswind manoeuvres and vertical wind speeds.
Fechner et al.
Fechner et al. developed a real-time capable robust model that allows flight path optimisation and can
fly a similar trajectory as during the experiments [21]. The model bases its aerodynamic coefficients on
two-dimensional (2D) sail wing sections as a function of the angle of attack. Without rotational inertia,
unphysical effects occur on the predicted yaw rate. Therefore, it was decided to build a four-point kite
model (see Fig. 2.10) that parametrizes the shape of the kite by the width, height and distance to the
KCU. Spring-damper systems are used for the connections between the four points to model the struc-
tural properties of a LEI kite. The springs were made so stiff that no flexibility of the kite is explicitly
modelled, making this in essence a rigid body model. Not modelling deformations of the kite through
upfront identified material properties, meant that experimental identification of the steering sensitivity
parameters was needed. To clarify, steering sensitivity is the rate at which a kite turns when encoun-
tering an asymmetric steering input. The steering sensitivity can’t be modelled without deformations,
since a kite can’t turn without deformations. The experimental identification was used to alter the kite
width to make the modelled steering fit the measure steering behaviour, which resulted in less than
2% conversion error. This part of the work is revolutionary because even though he uses a rigid body
model, changing the width of the kite model in steps simulates the real flexibility of the kite. By doing so
the span-wise bending deformation mode is represented, which others had not been able to do while
predicting the aerodynamic coefficients during flight.
To be able to validate the four-point model Fechner et al. coupled the model to a control system and a
discretized particle tether model (see Fig. 2.11). Furthermore, Fechner et al. showed that the turn rate
law derived for simulating turns of non-deforming body models holds for LEI kites [18, 19].
Figure 2.10: The 4-point model of Fechner et al. with the
definition of the: depower angle (αd) and the steering angle
(αs) use to steer the kite [23].
Figure 2.11: Discretized tether model connecting to the 4-point
model [23].
Oehler and Schmehl
Oehler and Schmehl developed an aerodynamic characterization method for the V3 from which one
can derive the aerodynamic characteristics accurately [47]. A novel experimental setup was developed,
which as recommended by Hummet et al. included an in-flight wind measurement sensor [33]. The
setup can measure larger-scale systems during operation. A self-aligning pitot tube is mounted in the
plane of the power lines. This plane is chosen because the geometry stays relatively constant due to
the lines staying straight, i.e. they only changed with 0.1to 0.2during the experiment. During flight
the load distribution between the power and steering lines shifts when the angle of attack changes,
2.3. Experimental data 16
due to a tilting of the aerodynamic load vector [32, 48]. This effect is neglected because the bridle
line force is not measured, instead, a constant distribution is assumed. The angle between the kite
and the ground is called the elevation angle (β) and is often derived from the tether angle of attack
measured at the ground [32]. Oehler and Schmehl however defined the elevation angle separately
from the tether angle to the kite (see Fig. 2.7), which means that sagging does not have a direct impact
on the measured lift-to-drag ratio.
Figure 2.12: Lift-to-drag ratio vs the angle of attack with a
colour coded relative power setting [47].
Figure 2.13: Relation between the relative power setting (up)
and the lift-to-drag ratio [47].
A quasi-steady flight is assumed and smoothing was, therefore, applied to remove sub-scale processes.
It was shown that the aerodynamic characteristic time scale of the V3 is an order of magnitude smaller
than the turning timescale, which is a positive indication of the validity of the used quasi-steady flight
assumption. The lift-to-drag ratio was related to the relative power setting instead of only to the angle
of attack, which increased the accuracy compared to the models of Fechner et al. and Ruppert, who
reported needing major adjustment to fit the trajectories [21, 54].
Roullier
Roullier [53] used experimental data of the V3 to perform an analysis on the effects of the KCU, the
relation between model complexity, aerodynamic parameter identifications and more. Roullier found
that the KCU drag and inertial effects are substantial. Furthermore, the KCU swings because of the
inability of the tether to resist bending, the effort required to compensate for the effects are, however,
only 1% to 2% of the lift. Roullier developed a more complex model than Oehler et al. did, by includ-
ing additional orientation angle and the contributions of weight, drag and inertia [47]. This was partly
achieved by using a three-plate model. Each plate has an aerodynamic centre that generates lift and
drag, which are defined as functions of the: angle of attack, power input and steering input. Roullier
compared his three-plate model results, to Oehler et al.’s model and the work of Demkowicz who per-
formed rigid wing CFD analysis (see Fig. 2.14 and 2.15) [14, 47]. The lift coefficient (CL) predictions
of Roullier and Oehler et al. were in good agreement whereas the drag coefficient (CD) was less so,
which is attributed to the added inclusion of pitch and roll. Differences with respect to the CFD results
are present for both slope and magnitude and arise due to not taking into account deformations of
the kite. These differences are most pronounced in the drag polar, which is caused by not modelling
the frontal area increase and with it the increase in drag. Roullier tried to model turning using the mo-
ment caused by the difference in force creation of the side plates. This force was however found not
sufficient, therefore, turning was modelled using a turning rate law.
2.4. Research questions 17
Figure 2.14: CLplotted against the angle of attack (AOA). Where the experimental values of the powered and depowered
mode are compared against CFD results [53].
Figure 2.15: CDplotted against the angle of attack (AOA). Where the experimental values of the powered and depowered
mode are compared against CFD results [53].
2.4. Research questions
This literature review covered an exploration of relevant structural modelling approaches of a soft-wing
membrane kite, in particular the V3. It was decided that developing a model that could be used in
iterative procedures would be ideal, because it could enable kite designers to increase their aerody-
namic performance. The design model itself would, besides the fast structural model, consists of a fast
aerodynamic model, a FSI coupling algorithm and a simulation framework. The focus of this research
is towards the development of a structural model. Due to the low computational cost requirement, a
high fidelity model will not be developed.
Deformation modelling was found essential, to accurately model the aerodynamics and the in flight
behaviour. Most current structural LEI models do not include deformations and those that do, have too
high computational costs to be useful for iterative design purposes. According to the reviewed literature,
a model that predicts deformation without force input has never been developed. Such a model would
drastically reduce computational cost by alleviating the necessity of iterations between the aerodynamic
2.4. Research questions 18
and structural model. The author, therefore, sees potential to contribute to the scientific community by
developing a structural model that does not need a force input. Another option, is a faster yet stable
and accurate structural model. The work of Geschiere is considered the current state of the art, since
the model was able to take into account deformations by using a spring representation of the bridle line
system [27]. The multi-plate models are also considered, due to their ability of representing the shape
deformation for low computational cost.
Several discussions with other airborne-wind energy researchers, this literature review and with it the
identified gaps in the scientific body of knowledge has let to the following research objective:
Assess the aeroelastic deformation effects on a soft wing membrane kite by developing a
fast structural model.
Based on the research objective and literature review, several research questions came up each with
their own sub questions.
ICan a fast deformation model be developed that could be used in a FSI module of a design tool?
Is formulating a geometric model possible?
Can the model predict both symmetric bending and asymmetric twist?
Can the model run real-time?
Does including empirical relations increase model accuracy?
II Can one perform a photogrammetry analysis of the V3 during flight using footage shot from the
KCU?
Can both straight and turning flight be analyzed?
Can one extract empirical ballooning relations?
3
Research approach
The literature was reviewed, scientific gaps were identified and research questions were formulated
(see Ch. 2). The main body of the rest of the research can be divided into three parts.
The first part is focused on data acquisition (see Ch. 4). Experimental data, empirical relations and the
best representation of the flown version during the analyzed measured campaign are extracted. The
latter comes in the form of geometry specifications of the V3 kite wing and its accompanying bridle line
system. The data acquisition efforts form the first building block of the research, by providing the input
data (see Fig. 3.1).
The second part covers the development of a representation of the kite wing (see Ch. 5). Using insights
from literature, kite designers, experimental experiences and common engineering knowledge a novel
wireframe representation of the V3 was developed in Python (see Fig. 3.1) [50]. Python is selected
as the programming language because it enables the possibility of open-sourcing the solution and it is
an efficient code enabling low computational costs [67]. The author has, furthermore, several years of
experience using Python, using it will therefore accelerate the research process.
The third part governs the bridle line system, due to the key role it plays in determining the shape de-
formations (see Ch. 6). Three different algorithms were developed and coupled to wing models (see
Fig. 3.1).
With all the components in place, the results can be analyzed and compared (see Ch. 7). The com-
parison serves to verify and validate the developed models, thereby indicating the accuracy of the
developed models. Based on the accuracy, modularity and computational cost the best models will be
selected. This step forms the last block and the research output. Based on these results, conclusions
will be made and recommendations for future research can be provided (see Sec. 8.1 and 8.2).
19
20
Figure 3.1: Research workflow.
4
Data acquisition
To develop the desired models, data is needed. Kitepower B.V. has provided experimental data contain-
ing both footage and measurements as well as several answers to specific geometric layout questions.
The experimental data can be used to verify whether deformation is indeed the key to closing the gap
between the experiments and simulations. The experimental data is from tests done in 2017, the period
in which the V3 was called the V3.A. Having both measurements and footage of the same flight would
allow one to assess relationships between measured variables and the deforming shape for each time
step. From analyzing the footage, it was observed that no measurement rack was present, the con-
clusion was therefore made that the measurement and footage were obtained during different flights.
Therefore, the deformation needed to be determined using only the footage. Achieved by using a one
camera-based photogrammetry analysis. One camera-based photogrammetry is possible because
distances to certain objects are known upfront [34]. The photogrammetry procedures are explained in
detail in section 4.1.
The V3 is Kitepower B.V.’s ‘workhorse’ kite and its bridle line system layout has been continuously
subject to change to improve its performance. The changes were mainly done during experimental
campaigns in the field, leaving one with many undefined parameters. For the validation to be useful,
it is key to obtain the closest approximation of the bridle line system that the V3.A kite was flying with
during the experiments in 2017. This is in part because the developed bridle line system models oper-
ate using geometrical input, i.e. bridle line length changes (see Ch. 6).
Besides uncertainties in the bridle line system, there are also uncertainties regarding the precise kite
wing shape. These are partly caused by the differences between drawings send to the kite manufac-
turer and the physical kite wing that one receives back [5]. Measuring the shape in hindsight is not trivial
either because the unloaded geometry is subject to change due to the high flexibility [5]. Therefore, one
would ideally measure the shape in flight to ensure the same design is analyzed as is flown [38]. Ex-
perimental efforts that use inertial measurements units for this purpose have recently been developed,
but not yet fully worked out nor tested on an AWES [34]. The specifications of the design geometry of
the V3 kite are relevant for the AWE field because the kite has the potential to serve as a benchmark
model. The V3 could serve as a benchmark model, because Kitepower B.V. no longer restricts data
access of the V3 to protect IP. Another reason is the relatively large, i.e. relative considering that the
AWE research field is a novel, body of research that has been done on the V3 which enables the use of
previously determined relationships and values. The steps and assumptions leading to a specification
of the design geometry of the V3 kite are presented in section 4.2.
4.1. Photogrammetry
On the 24th and the 30th of March 2017, Kitepower B.V. performed experiments with the V3.A kite at
Valkenburg Airport in the Netherlands. Several of these flights were filmed by attaching a camera to the
KCU (see Fig. 4.1). This footage will be used to track how the width and the amount of TE ballooning
change during flight. The models will predict the width change based on the bridle line attachment
21
4.1. Photogrammetry 22
points. Therefore, the distance between the bridle line attachment points at the tip will be determined
instead of the kite width. In the remainder of the report, width should be interpreted as the width be-
tween the bridle line attachment points at the tip. The width is chosen as a parameter to compare the
experiments and simulations, because it represents a global measure of the anhedral change. The
anhedral angle is important because it has a direct effect on the generated force by altering several
aerodynamic parameters, one of them being the projected surface area. Ballooning is studied experi-
mentally because it scales with aerodynamic force and the development of an accurate aerodynamic
model does not fall within the identified aim of this research. It will, however, be considered because
the expectation is that the combination of local changes in TE strut distance affects the global shape
of the kite.
Since no measurements of the same flight are present, it is difficult to couple the measured parameters
to certain states of the kite, e.g. the obtained deformation can’t be formulated as a function of the wind
speed. A solution was found by identifying extreme states and linearizing the behavior in between the
states.
Figure 4.1: Video still from the KCU of the V3 during a launch on the 24th of March 2017 [1].
For straight flight, the maximum and minimum powered state, also called depowered state, will be
identified. The powered state occurs in the reel-out phase, whereas the depowered state occurs in
the reel-in phase. In straight flight, a symmetrical deformation is assumed, with its symmetry line ly-
ing parallel to the struts crossing the middle canopy piece. Furthermore, the powered state is chosen
because it corresponds to the kite its reported computer-aided design (CAD) design (see Sec. 4.2).
The only difference is that the CAD displays no ballooning. B. van Ostheim, who is the kite designer
for Kitepower B.V., confirmed that the kite is designed such that with ballooning in powered flight the
tubular frame has the same orientation as the CAD shape, i.e. the tubes are aligned in the same di-
rection and under the same angle with respect to one another. In other words; the distance between
the TE tips of the struts is not that of a fully stretched canopy, but rather the distance designed for the
expected ballooning during powered flight.
For turning flight, three states are identified: the apex of a powered left turn, the apex of a powered
right turn and straight powered flight. All these states occur during the reel-out phase, as the ideal flight
path contains no turning maneuvers during the reel-in phase (see Fig. 1.4).
4.1.1. Analysis setup
Two videos, one from each measurement day were selected of sufficient quality to be used for the pho-
togrammetry. From each video, for each state, several time stamps were identified that corresponded
best to the respective description. The sampling type is non-probability purposive full sampling be-
4.1. Photogrammetry 23
cause each sample is selected based on knowledge of the desired sample. The sampling selection
is made using visual and audible input. The velocity is visually assessed by comparing the respective
surrounding velocity. Other input was obtained from estimating the position of the kite within the known
ideal flight path and for turning by studying the full movement and estimating the middle point within that
movement. The audible input was useful to identify the amount of loading, because the noise scales
with velocity, which in turn scales with the amount of loading.
From each selected time stamp a video still was extracted. Before one can extract the desired lengths
out of the resulting images one must deal with the distortion of these images. The camera uses a
wide-angle lens creating a so called ‘fisheye’ effect that distorts the image. A software called GIMP is
used since it allows one to manually remove the distortion by adding an opposite ‘lens distortion’ [30].
This procedure is done under the assumption that a straight horizon corresponds to an image without
camera-induced distortion. A main distortion setting of 25% and an edge distortion setting of 10%
was needed to straighten the horizon.
The video footage images appear in perspective view. Perspective view is how one normally sees
the world, i.e. objects further away appear smaller than they are. For taking three-dimensional (3D)
measurements out of a 2D collection of images this is a problem. To resolve this issue, one must
transform the image to an orthographic view or correct the extracted lengths based on their distance
to the camera. The latter approach is used by comparing the relative change in line-of-sight and angle
in comparison to the powered state in straight flight. The powered state in straight flight is used as the
baseline because it corresponds to the design state. The changes are caused by the KCU swing, kite
pitching motion and the deformation of the kite.
Figure 4.2: KCU swing demonstration of an 25 m2LEI kite during flight on the 24th of November 2010. It was filmed from a
camera attached to the KCU, which at the time was located lower down the tether [22]. The left image shows the KCU swinging
to the right during a turn. Upon comparison of the middle and right image, a ‘vertical’ difference is noticed, also indicating a
KCU swing.
KCU swing
With KCU swing, the change of position of the KCU with respect to the kite is meant (see Fig. 4.2).
This swing is caused partly by the relatively large mass of the KCU that carries momentum. The KCU
swing differs between turning and straight flight, its effect is so large, it even induces a roll effect on the
kite [53]. In the powered and depowered state, the distribution of forces between the LE and TE shifts,
due to a chord-wise tilting of the lift vector [47]. This is another contribution towards the KCU swing. A
comparable effect is found between turning and straight flight, but now in the span-wise direction. The
span-wise tilting causes the bridle line system to be asymmetrically loaded, which induces the KCU
swing effects. All of the discussed types of KCU swings cause changes in the line-of-sight and angle
at which the kite is observed, which due to the perspective view must be taken into account.
Interesting to notice is the relative inward folding movement of the tip when the kite is depowered (see
Fig. 4.3). This effect will be further discussed in chapter 6. A quick qualitative assessment between
4.1. Photogrammetry 24
powered and depowered reveals an inwards motion of the pulley points, an outward motion of the knots
and two slacking bridles on each side of the depowered wing. The knot and pulley positional changes
will be measured by drawing an imaginary line from the KCU through the points. This imaginary line
crosses the canopy of the kite wing at the TE and this location will be used as calculation input for a
particular bridle line system model (see Sec. 6.2.3).
Figure 4.3: Video stills indicating the height difference of the kite within the image frame [1]. The pulley point, knot point and
kite bridle tip movements from powered to the depowered state are identified. Furthermore, the red line indicates the present
slacking bridle in the depowered state.
Kite pitching
The pitching of the kite is controlled by changing the power setting (up). The power setting is found con-
stant during the reel-in and reel-out phases. The same observation was made by Oehler and Schmehl,
who studied other measurement data of the V3 [47]. The angle of the chord-wise direction of the kite
with respect to the KCU will change, as an effect of the pitching. For elements roughly perpendicular
to the change in angle, e.g. the TE of the middle plate, this mainly causes a general increase in the
length of the line-of-sight. For elements parallel to the angle, however, e.g. the struts, this causes both
a varying change in the line-of-sight length as well as in observed object size.
4.1. Photogrammetry 25
Figure 4.4: Effect of the kite pitching on the line-of-sight.
Kite deformation
Kite deformation is caused by a change in the loading conditions. An apparent velocity change, either
in direction or magnitude, affects the loading conditions and could be caused by a change in the wind
velocity, a different flight direction of the kite or a different kite velocity. Another factor contributing to a
changing loading condition, and hence deformation, is the shape deformation itself. This is the result of
the effect of the shape on the forces and the effect of the force on the shape, i.e. the FSI loop (see Sec.
2.1.2). The deformation of the kite changes the lengths of the line-of-sight to different kite elements
and can also affect the angle between the video still and the object (see Fig. 4.5).
4.1. Photogrammetry 26
Figure 4.5: Simplified 2D representation of how deformation can affect the length of the line-of-sight. The anhedral angle
change is taken here as example.
The changes in the length of line-of-sight and angle effects are taken into account by comparing the
strut lengths. By assuming that the strut lengths remain constant one can compare the measured
pixels between extreme states. Under the constant length assumption, the relative difference can only
be attributed to a change in line-of-sight and angle. Therefore, the difference is formulated as a ratio
and used to filter out the line-of-sight and angle-induced distortion. For measuring the TE lengths, it
is assumed that the inferred distortion will be the same as it is for the accompanying struts. For each
plate, the effect must be determined separately because a different angle and line-of-sight with respect
to the KCU could be present. For the width of the kite, the averaged relative distortion of all the struts
is used.
4.1.2. Results
In both videos, several stills were selected which represent the extreme states in straight and turning
flight. The distortions were measured and removed, leaving one with pixel lengths that can be com-
pared based on their relative differences expressed as percentages. The knot and pulley positions
were also measured in percentage change, based on the relative distance they are in between two
neighboring struts. An expression in percentages was chosen because the required output is the re-
spective change and not the number of meters. Another argument for leaving it as a percentage is that
the transformation step could cause additional uncertainty.
To remove random measurement errors, the resulting percentages are averaged for each video, i.e. the
mean of the found samples is used to reduce the variation induced by errors. Random measurements
errors could arise from an incorrect selection of the number of pixels by hand, the wrong timestamp
selection and due to vibrations of the system as a whole. The KCU, tether, bridle lines and kite wing
together are prone to vibrate causing a continuous change in distortion and shape. An indication of
the reliability of the measurements comes from the standard deviation (σ) (see Eq. 4.1). Where µ
represents the mean and Nthe number of samples. More samples would reduce the random error,
thereby increasing the accuracy, commonly represented using the standard error (SE ) (see Eq. 4.2).
σ=v
u
u
t
1
N
N
X
i=1
(xiµ)2(4.1)
4.1. Photogrammetry 27
SE =σ
N(4.2)
The results of both videos are averaged to decrease the existing systematic errors. Potential system-
atic errors could arise from how the camera distortion is removed, how a constant strut assumption
is used to deal with the distortion changes between states and due to a camera misalignment. The
latter can be reduced by averaging the measurements for the left and right parts of the straight flight.
For turning flight, one must average the results of both left and right turns, which only works if the
assumption of similarity in the turns holds. Regarding reliability, both test-retest and Intra-observer
reliability could be treated by doing multiple analyses. For the latter, the analysis must be done by
different researchers. It is decided to leave the quantitative assessment of the systematic errors for
another study due to the non-optimal experimental setup, a too-small sample size and time constraints.
The struts and panels have been numbered to enable reporting of the measurements (see Fig. 4.6).
The ‘strut’ length is indicated in quotation marks because the distance between the bridle line attach-
ment point was taken and not the true strut length. This is in order to be consistent with the developed
models, similarly as why the bridle line tip-to-tip width is measured and not the maximum width.
Figure 4.6: Video still indicating the numbered struts (St,i), plates (Pl,i ) and the measured lengths [1].
Straight flight
A5% decrease in width was found going from the powered to the depowered state (see Table 4.1).
Physically, this makes sense since, upon an increase in force, one would expect a flatter shape [46].
The TE lengths of the plates were found to decrease when powering the wing. The change is present
due to the ability of the canopy to balloon. Since ballooning scales with the force acting on the canopy, a
decrease in length is as expected. An effect that counteracts the decrease in length is a global one and
is caused by an increase in the sideways component of the local force vectors. These forces stretch
the kite out, i.e. they pull the struts outwards hence reducing the ballooning. Because the observed
behavior is a decrease in TE length, the force component that causes ballooning can be concluded
stronger. The underestimation of the angle change due to an increase in anhedral angle causes a
respective underestimation of the width of the TE of Pl,4. Therefore, the relatively large measured
ballooning for the TE of Pl,4is considered a faulty measurement. The averaged distortion effect used
for the total width is, therefore, based on St,1,St,2and St,3.
4.1. Photogrammetry 28
Table 4.1: Photogrammetry results of the relative length changes.
Width TE Pl,1TE Pl,2TE Pl,3TE Pl,4
Difference 5.0%1.9%2.0%3.4%14.2%
The pulley positional and knot positional values match the qualitatively observed trend (see Table 4.2).
The pulley points move inward when depowering, whereas the knot points move outwards.
Table 4.2: Photogrammetry results governing the changes of the pulley and knot line-of-sight canopy crossing position.
Pulley Knot
Powered 80% inwards from St,372% inwards from St,4
Depowered 3% inwards from St,217% inwards from St,4
The sampling distribution of the width measurements shows some degree of variation as expected
(see Fig. 4.7). The measured pixel values have been scaled such that the mean of the powered
measurements equals the design width. The behavior has been linearized in between the powered
and depowered state, to obtain estimates for the width at power settings (up) between zero and one.
The variation is represented by the standard deviation which is equal to 0.23 m. The standard error
(SE) was found equal to 0.73 m and is used to plot a confidence band around the linearized behavior.
This is done by adding half the SE above and below the mean.
Figure 4.7: Sampling distribution of the width. Where the grey band indicates the standard error (SE ) and each dot represents
an individual measurement.
Turning flight
The increased number of effects causing uncertainty has led to the conclusion that the overall inaccu-
racy of the resulting data coming out of the turning analysis is too low. One of the effects comes from
the difficulty of identifying the precise timestamp of the turning apex. Another effect is the increase in
4.2. Design geometry 29
distortion due to larger shape differences. The asymmetric deformation of the kite is one reason for the
increase in magnitude, another is the increased relevance of the KCU swing during turning maneuvers
(see Fig. 4.2).
A qualitative comparison of the experienced asymmetric deformation and the predicted deformation is,
however, still possible. From several video stills the expected asymmetric deformation was found (see
Fig. 4.8). Another relevant finding is the slacking of two bridle lines, different than those found in the
depowered state.
Figure 4.8: Video still of a left turn [1]. Where the white dotted lines indicate the slacking bridles and the full white lines
qualitatively show the expected asymmetric deformation.
4.2. Design geometry
The required inputs for the developed models are the 3D design shape specified through the bridle line
attachment coordinates, an estimate of the initial positions of the knots and pulleys in the bridle line
system, the extension of the depower and steering tape and the lengths of all the bridle lines.
A 3D CAD file of the V3 in 2012, i.e. one that is not equal to the V3.A used during the analyzed experi-
ments was provided by Kitepower B.V. (see Fig. 4.9). The 3D coordinates of the bridle line attachment
points were extracted (see App. A). The bridle fan connections attached near the LE are simplified
by representing them with one line. The struts are all parallel to each other, the measurement errors
between the LE and TE point were therefore removed by making its span-wise coordinate match. Here-
after, the coordinates are scaled such that the kite bridle line attachment tip width matches the design
width of 8.3m. Finally, a transformation of the coordinates is made to place the KCU at the origin.
(a) Front view (b) Side view
(c) Bottom view
Figure 4.9: CAD drawing of the initial design of the V3 [1].
In 2018 a document of the kite development at Kitepower B.V. was made which contained specifica-
tions of certain design aspects of the V3 [60]. Of particular interest are the provided specified bridle
4.2. Design geometry 30
line system lengths measured on 30 July 2017, which is five months after the measurements campaign.
The specification of the KCU bridle line system is still missing, which is a critical subset of the bridle line
system present at the TE (see Fig. 4.10). The KCU bridle line system is critical because it is used to
actuate the kite. Asymmetric actuation is done by changing the steering tape lengths and symmetric ac-
tuation by changing the depower tape length. The bridle line system starts from where the tether ends,
which is a point generally referred to as the bridle point. The model is, however, build starting from the
KCU. This is possible because J. Breuer, who is CTO at Kitepower B.V., identified that all bridles of the
LE and TE were attached to the KCU. This modeling decision was taken because it ensures that the
models have the same viewpoint as the photogrammetry. The only difference between modeling from
the KCU or the bridle point is a straight line connecting the two. For future research that would also
model the tether, one would only have to add a straight line. Furthermore, J. Breuer was so kind as to
provide the KCU bridle line lengths that were flown with in 2017 (see App. A).
Figure 4.10: Adjusted bridle line system representation, with the red square roughly outlying the KCU bridle line system [47].
The goal is to compare the model predictions with the photogrammetry results, which means the steer-
ing settings and the maximum line lengths must be known. To do so, the measurements made by
the potentiometers of the KCU, during the same day as the footage, were analyzed. The values of
multiple pumping cycles were averaged. The depower tape was found to differ with 628 mm between
minimal and maximal power settings. The minimal length of the depower tape (ld) is 1098 mm, which
is observed during the powered state. The power setting (up) scale has been adjusted such that the
maximum and minimum values correspond with those observed during flight. For the steering tape,
maximal steering of 40% of the total length was found. With a neutral state of 1600 mm, this leads to
a steering tape of 1040 mm on one side and 2160 mm. M. Schelbergen a Ph.D. researcher at the TU
4.2. Design geometry 31
Delft, found an average depower tape length difference (ld) of 8% between powered and depowered
state from investigating a larger data set of the V3 out of 2019. This is smaller than the ld= 13%
observed from analyzing just two samples, the difference could come due to the less extreme settings
used during the larger data set collection in 2019 [56]. To check the sensitivity of the models to ld,
the two most advanced models will be tested using both ld= 8% and ld= 13%.
Upon comparing the found bridle line system lengths with the footage, it was found that additional lines
were present. Two of those lines attach the tip to the line from the KCU to the pulley, somewhere on
the steering lines or the line above it. This results in a kink in this line and an additional knot (see Fig.
4.11). This is one reason why a reversed engineering process was used to find the ‘correct’ bridle line
lengths. Another reason was to ensure that the powered state result matches the design CAD state.
This required changing the line lengths of several bridles (see App. A). This approach has its short-
comings and ideally one would measure all bridle line lengths on the day of the test to ensure that the
proper input is used.
Figure 4.11: A video still indicating the positions of the knots and pulleys [1]. Where the red line indicates the ‘new’ line.
5
Wing models
For developing a model of the kite wing, a numerical representation is needed. To ensure the accuracy
of this representation, one must consider what the main contributing structural elements of the wing
are. From this analysis, a modelling hypothesis is formed (see Sec. 5.1). The hypothesis leads to a
wireframe representation of the kite wing (see Sec. 5.2). The goal is to develop a novel structural model
with an ideal balance between accuracy and computational cost. Several wing models with differing
fidelity levels have, therefore, been developed (see Sec. 5.3).
5.1. Modelling hypothesis
Airborne wind energy companies are aiming for a design that produces the most load because that
leads to more energy production. The parts of the canopy that produce lift parallel to the longitudinal
tether axis produce more energy compared to those that are less aligned. This is attributed partly to that
the latter parts of the canopy produce a local lift vector pointing outward, i.e. part of the load produced
is counteracted by the other side of the kite and is thus not as effective towards the energy production.
Another reason why the middle, more flatter parts, contribute more towards the energy production is
that they generate more force compared to the outer parts of the kite. The flatter parts have more
optimal flow angles and larger projected surface areas. The kites of Kitepower B.V. have likely been
becoming flatter with each evolution because of this increased useful load potential (see Fig. 5.1).
Figure 5.1: Kitepower B.V.’s kite evolution from left to right one sees the: V2, the Hydra and the V3 [60].
At the same time, the kites have also been getting more bridle lines, most likely to be able to handle
the increased wing loading. Adding bridles also has negative effects: it increases drag, adds weight
and it decreases the depowering ability of the kite. The depowering ability is reduced by restricting the
pitch movement, which depending on the design makes either the reel-in or reel-out phase less efficient.
If the kite wing were to be a ram-air kite, the contribution of the kite wing to the structural strength resist-
ing rotational deformation would be close to zero. With rotational deformation, the main deformation
modes, i.e. bending and twisting, are referred to. The LEI kite and ram-air are quite close in terms of
32
5.2. wireframe representation 33
structure. One of the differences is that the ram-air kite has open holes at the front making the pres-
sures depend on the amount of air coming into the kite. The LEI kite on the other hand has a closed
tubular frame that is pressurized upfront, which enables higher pressures. Therefore, the structural
strength contribution of the tubular frame of the LEI generally contributes more than the pressurized air
of the ram-air. The shape of any kite is determined by the force balance between lift and its counterpart,
which for the ram-air kite is carried mainly by the bridle line system. This is only possible because of
the elongation resisting properties of the membrane canopy. Without substantial elongation resistance,
large local ballooning effects would become possible resulting in a different less-optimal shape (see Fig.
5.2a and 5.2b). This illustrates the importance of bridles and membrane towards resisting deformation
of the ram-air kite.
(a) ‘Normal’ elongation resistance properties. (b) Low elongation resistance properties.
Figure 5.2: Schematic illustration of a kite with three segments.
Furthermore, ballooning is a relevant factor for the aerodynamics of the kite, because it changes the
aerodynamic force production by altering the chamber. Kitepower B.V. has put a wire at the TE of the
V3 to reduce these deformations, for LEI kites this generally leads to higher efficiency [5].
The V3 kite has the entire tubular frame supported, i.e. on all struts on both the LE and TE bridle
line attachment points are present. Assuming that the contribution of the tubular frame towards the
structural strength resisting rotational deformation is low compared to the bridle line system, the same
conclusion as drawn for the ram-air kite can be made. This leads to the main hypothesis under which
the deformation models operate: one can describe the shape of the V3 using a wireframe wing model
represented by the bridle line attachment points, whose coordinate changes are modeled using a bridle
line system model and ballooning relations.
Modeling the kite deformation based on this key hypothesis is a novel idea, making the developed mod-
els the first of their kind. The first point supporting the assumption is that the LEI kite wing structure
without aerodynamic loading is so flexible that it bends under its weight [5]. Another argument comes
from Kitepower B.V., who mentioned that during one of their experiments the tubular frame lost its air
pressure and thereby the structural properties arising from that pressure. The fully-bridled LEI kite that
they were flying did, however, not visually appear to change its shape as a result. In other words,
while the LEI kite structure was reduced to only bridle lines and a membrane canopy, the same shape
remained. The shape must thus have been dominated by these two components. Another qualitative
argument comes from several discussions had with kite designers, whom all identified that the main
role of the tubular frame is to resist compression and not bending.
5.2. wireframe representation
The kite wing can be represented by spanwise segments, where each segment consists of a LE tube,
two struts and the TE membrane (see Fig. 5.3). These segments are supported on the LE by a bridle
fan and at the TE by a single bridle, which together are modeled as four bridle line attachment points
placed in each corner of the respective kite segment.
5.2. wireframe representation 34
Figure 5.3: Two images of the V3.A kite, where the red quadrilateral indicates one of the spanwise segments and the red
dotted circles indicate the bridle fan [60]. The curved lines of the membrane canopy TE in between the strut tips demonstrate
the ballooning.
The kite wing is discretized into N panels, where each panel represents a kite segment. The corners
of the panels are the bridle line attachment points and the edges, representing the LE tube and struts,
are assumed of constant length. The TE represents the canopy membrane and can vary in length due
to ballooning, which is modeled using empirically obtained relations (see Sec. 4.1). Furthermore, it is
assumed that the panel itself does not bend, meaning that the lines describing the edges stay straight.
This leads to a wireframe representation of the kite (see Fig. 5.4) The deformation degrees of freedom
(DOF) of the plate representation of the kite are a possible twist deformation of the LE tube elements
and the rotations of the plates at their connections. The latter LE connections can be interpreted as
gimbal joints. An alternative way to describe this is that one approximates the continuous bending of
the tube by the rotational DOF of a multi-body model of the tubular frame. No structural properties are
modeled using this approach, meaning that when excluding ballooning effects the shape deformation
can be simulated based on pure geometrical inputs. These models become constraint-based, where
the shape deformation is induced by bridles and not by forces overcoming the structural resistance of
the kite wing and bridles.
Figure 5.4: wireframe representation of the kite wing, where each kite segment is represented by a plate. The red arrows
indicate the DOF of the plates.
5.3. Discretisation levels 35
5.3. Discretisation levels
The kite wing is represented by N panels, where each is supposed to represent one of the nine seg-
ments enclosed by the tubular frame. Therefore, the most accurate representation is to use nine pan-
els, since each panel edge than corresponds to the tubular frame edge, excluding the TE that is. This
makes the constant length assumption more likely to be an accurate representation considering the
compression resisting properties of the tubular frame.
In some modeling efforts, including more detail results in an enlargement of the errors. This is caused
by the additional assumptions that increase the discrepancy with reality, i.e. sometimes the accuracy
is higher when including fewer details. Models using less than nine panels will be developed to study
this particular effect and for studying the mechanistics of the model, for verification and because lower-
fidelity models generally have low computational cost. The following wing models are developed: a
triangular two plate model (see Sec. 5.3.1), a quadrilateral 2-plate model (see Sec. 5.3.2), a quadrilat-
eral 3-plate model (see Sec. 5.3.3), a quadrilateral 7-plate model (see Sec. 5.3.4) and a quadrilateral
9-plate model (see Sec. 5.3.5). Arguments for selecting some of these particular kite wing representa-
tions have arisen from how the bridle line system is modeled (see Ch. 6).
5.3.1. Triangular 2-plate model
The smallest functional substitute model of the kite model is a rigid rod. One can extend this 2D model
of a rigid rod to a 3D model with triangular faces. one can assume that the triangular face is of constant
shape, by excluding the ballooning of the trailing edge. For this model, two such triangular faces are
used, which can fold with respect to one another over the middle hinge line (see Fig. 5.5a).
Under the assumption that the edges remain of constant length, the angles of the triangular plates
remain constant. Because the plates must remain in contact over the middle line, this means that the
triangular 2-plate model can’t simulate asymmetric deformation. This would only be possible if the
plates collapse into one another or separate at the middle line. The model can simulate symmetrical
deformation, through the folding motion.
(a) Orthographic view of the wing model. (b) Side view, demonstrating the wind velocity vector at an angle of
attack.
Figure 5.5: Triangular 2-plate wing model.
A Cartesian reference frame (x, y, z) is defined with the KCU as the origin. The z-coordinate is defined
positively upwards, the x-coordinate positive in the downstream direction and the y-coordinate is de-
fined positive to the left. The triangular 2-plate representation in this reference frame has the middle
chord parallel to the (x, y) plane, which is how the design powered state will be represented throughout
5.3. Discretisation levels 36
this research. The apparent wind velocity vector (va) determines the angle of attack and is itself not
parallel to the (x, y). It is drawn to illustrate this point, from which it becomes clear that the angle of
attack of the middle chord is not zero in the design powered state (see Fig. 5.5b). The angle of attack
in the powered state is assumed equal to 10 ° [47, 53].
5.3.2. Quadrilateral 2-plate model
The quadrilateral 2-plate model is almost the same as the triangular 2-plate model. The difference
arises from the two additional points that turn the triangular faces into quadrilaterals (see Fig. 5.6). By
excluding ballooning, all the edges are of constant length. Asymmetric actuation is not possible be-
cause it would require the two plates to fold into one another or to separate the connection. Symmetric
actuation will be modeled and compared to the triangular 2-plate model, to see the effect of different
calculation methods and the effect of the additional two points.
Figure 5.6: Quadrilateral 2-plate model.
5.3.3. Quadrilateral 3-plate model
The quadrilateral 3-plate model is different from the quadrilateral 2-plate model since it has a center
plate, which represents the center canopy piece in between the two off-center center struts (see Fig.
5.7). Ballooning is excluded, the edges are therefore of constant length. Because there is an additional
plate there are now two hinge lines over which the kite can bend.
The middle plate TE points are free to move, which allows for twisting of the center plate. This twisting
enables asymmetric actuation, a similar process happens physically when the kite wing deforms. The
model allowing asymmetric actuation is important because it causes asymmetrical deformation which
enables the kite to turn. Symmetrical actuation is also possible and will be compared to the other
models, to study the effect of adding a plate.
5.3. Discretisation levels 37
Figure 5.7: Quadrilateral 3-plate model.
5.3.4. Quadrilateral 7-plate model
The quadrilateral 7-plate model is the same as the quadrilateral 3-plate model, only with more panels
(see Fig. 5.8). The outer two panels are modeled as one, due to slacking bridle lines observed during
the photogrammetry study (see Fig. 4.3). This model is developed with and without ballooning, mean-
ing that the plate TE edges can now vary as a function of the power setting. Lastly, both asymmetric
and symmetrical actuation can be simulated.
Figure 5.8: Quadrilateral 7-plate model
5.3.5. Quadrilateral 9-plate model
In the quadrilateral 9-plate, now all the plates represent a kite wing segment with at the LE and sides the
tubular frame and at the TE a ballooning canopy (see Fig. 5.9). This is the most detailed representation
of the kite wing that will be presented in this research and it follows the assumption that each kite wing
segment can be represented by a plate. The 9-plate model is developed with and without ballooning,
both are used to allow one to assess the effect including ballooning has. The bridle line system model
used to calculate the deformations of the 9-plate model can include the outer plates, because it can
deal with slacking lines (see Sec. 6.3).
5.3. Discretisation levels 38
Figure 5.9: Quadrilateral 9-plate model.
6
Bridle line system models
The bridle line system is relevant because its job is to distribute the load onto the kite wing, provide
stability and enable control. If control could be achieved through minimal actuation one could reduce
the KCU mass. If one could minimize the number of bridles, both mass and drag would be reduced. A
drag reduction causes an increase in aerodynamic performance and a mass reduction would increase
the operating wind range. The argument was made that bridles dominate the kite shape (see Ch. 5),
therefore, the bridle line system will be used to predict the deformation. Furthermore, because part
of the goal of this research is to develop insights that can increase aerodynamic performance, under-
standing the bridle line system mechanics is key.
The bridle line system models use symmetric and asymmetric actuation as input. The input values are
found using the empirically determined relationship between the power setting (up), the steering setting
(us) and the amount of line length change (see Sec. 4.2). Changes in the power setting cause the kite
to pitch. Previously, this has been modelled by Oehler and Schmehl using the extension of an imagi-
nary line (lim) connecting the KCU to the middle of the TE canopy of the V3 [47]. An imaginary line is
used in some bridle line models whereas the perpendicular assumption is used in all developed models.
For their model Oehler and Schmehl had to develop a relationship between depower tape extension
and imaginary line extension. Because the depower tape splits into two lines (see Fig. 4.10), the ex-
tension of the imaginary line was modelled as half of the depower tape extension [47]. This approach
is considered correct and is therefore incorporated in all discussed bridle line system models, that use
an imaginary line. The KCU is defined as a point, therefore, both steering tape and depower tape are
attached to the same point in space, i.e. the origin. In the models, the middle LE points are assumed
to remain in place. The assumption is necessary for some of the algorithms to work and to enable
consistent visualizations. The latter is achieved by not having a continuously changing image, which
otherwise occurs due to the rotation with respect to the wind of the whole KCU, bridle and kite wing ref-
erence frame. By using this reference frame, the frame rotations with respect to the wind do not appear.
Multiple models differing in their fidelity have been developed, which will enable verification procedures
and assessing if including more detail makes the prediction better or worse. Except for the particle sys-
tem model, all developed models operate under the assumption that all bridle lines are straight. For the
LE this is a valid assumption considering that the power lines of the V3 have been found to only change
by 0.1° to 0.2° during operation [47]. In the existing literature, there is no proof for the lines going to-
wards the TE. Straight TE bridle lines are, however, still assumed to allow a constraint-based geometric
approach. Qualitative observations made during the photogrammetry analysis in straight flight affirm
the validity of these assumptions, except for two slacking bridle lines. This problem is resolved by
not modelling these lines, possible by formulating a 7-plate model (see Sec. 6.2.3). For turning flight,
however, several lines did display slack. Because slack can’t be modelled using a constraint-based
geometric approach, most bridle line models are not able to accurately simulate asymmetric deforma-
tion.
39
6.1. Tetrahedon model 40
The tetrahedron-based model is the simplest of the developed model (see Sec. 6.1). The trilateration-
based model can deal with coordinate changes, hence able to predict more than only the change in
width (see Sec. 6.2). Finally, the particle system model is developed that operates without a straight
line assumption (see Sec. 6.3).
6.1. Tetrahedon model
The tetrahedron relations applied onto the triangular 2-plate wing make for the most simple represen-
tation of the relationship between change in anhedral and the power setting. The bridle line system is
simplified to four straight lines all coming from the KCU (see Fig. 6.1). Pitching is the only possible
motion and is controlled through a change in the imaginary line lim connecting the KCU to the middle
of the TE canopy.
Figure 6.1: Schematic representation of the triangular 2-plate model. The blue lines indicate the lines attached to the LE and
the red line the line attached to the TE (lim).
The shape change can be calculated using tetrahedron relationships, possible since lines: b,d,a,e,
cref are assumed of constant length. First the volume of the tetrahedron (VT) spanned by lines: b,d,
a,e,cref and lim is calculated (see Eq. 6.1).
X=d2+l2
im c2
ref
Y=b2+l2
im e2
Z=b2+d2a2
VT=1
12q4b2d2l2
im b2X2d2Y2l2
imZ2+X Y Z
(6.1)
The second step is to calculated the area of the triangle (At) spanned by lines: d,lim and cref . Lastly,
the length of the line from point 1 that is perpendicular to the face of this triangle is calculated and
multiplied by two, to obtain the width (W) of the kite (see Eq. 6.2). Due to symmetry multiplication by
two is possible, as the other side will give the same result.
At=1
4q(lim +d+cref )(lim +d+cref )(lim d+cref )(lim +dcr ef )
W= 23VT
At(6.2)
6.2. Trilateration 41
6.2. Trilateration
The next calculation technique is based on a mathematical algorithm called trilateration. Trilateration
in general is a geometric technique to find the position of a point based on distances to other points
[28]. In this context, trilateration is the technique of solving the intersection problem of three spheres.
Spheres and their respective centres and radii form the input and the result, if present, are their two
intersection points. Because line lengths can change, one can take into account the ballooning of the
TE canopy. The trilateration algorithm calculates coordinate changes and not width changes, therefore
it can describe the whole shape of the kite. Another advantage over the tetrahedron algorithm is that
besides triangular plates, this algorithm can calculate a kite representation of quadrilateral plates.
The problem trilateration solves is best illustrated in 2D, where it becomes the intersection problem of
three circles (see Fig. 6.2). The resulting intersection point in 2D is in 3D a line connecting the two
intersection points, one being in front of the paper and the other behind the paper.
Figure 6.2: 2D representation of the 3D sphere intersection problem, with P representing the intersection point in 2D.
6.2.1. Model workflow
The best way to explain how trilateration works is by walking through an example calculation. The
triangular 2-plate model is taken as a reference, where its change is caused by a symmetrical actua-
tion. Applying another technique on the same wing model is useful because it enables a verification
procedure.
Where the line length change of lim was sufficient input for the tetrahedon algorithm to calculate the
amount of pitching, trilateration also requires the coordinate change of point 4 (P4) (see Fig. 6.3). The
cos-law is used to calculate the depower angle (αd) based on the extension of line lim. Next step is to
calculate the new coordinate of P4using αd(see Eq. 6.3). The depowering of the kite is represented
by the rotation around the front suspension point, not around the actual LE because the bridle line
attachment points are located at about 0.5m behind the real LE [47].
αd=arccos d2+c2
ref l2
2dcref !π
2
P4x=cref cos (αd)
P4y= 0
P4z=P4z,old +cref sin (αd)
(6.3)
With the new location of P4known, point 1 (P1) and point 3 (P3) can be calculated using trilateration.
6.2. Trilateration 42
The trilateration algorithm will be illustrated for the calculation of P1. The calculation of P1is done sim-
ilarly, only mirrored over the symmetry (x, z) plane. It is verified that the output of the model is indeed
the same for P3and P1with only the y-value switched. The input into the trilateration algorithm for the
calculation of P3are the initial coordinates of the powered state (up= 1) and the new location of P4.
Calculating point 1
The first step is to define the spheres and radii. The radii are either known from lines of constant lengths
or calculated using the norm of the vector, i.e. Pythagoras theorem applied in 3D. Sphere 1 has its
center at the KCU (P0) and its radius equal to line b(see Fig. 6.3). Sphere 2 has its centre at P2and
its radius equal to line a. Sphere 3 has its centre at P4and its radius equal to line e. The intersection
points of these spheres lie at either side of the plane spanned by the three spheres, looking solely at
this plane the 2D representation of the 3D intersection is found again (see Fig. 6.2). By selecting the
intersection points in a systematic order, the negative zdirection (see Eq. 6.5) is present on the right
side. Meaning that when one wants to find the location of P3there can be only 1 solution.
Figure 6.3: Illustration of the 2D plane spanned by P0,P2and P4, shown in red.
The second step is to transform the reference frame (x, y, z) to (x, y, z) such that point P0remains
on the origin, point P2lies on the new x-axis and point P4on the (x, y) plane. The transformation is
defined by the unit vectors: ux,uyand uz(see Eq. 6.4) [28].
ux=P0P2
|P0P2|
uy=(P4P0)u2
x(P4P0)
|P4P0|
uz=ux×uy
(6.4)
Within this new reference frame the intersections points (P3x, P3y, P3z) are found using the trilatera-
tion algorithm (see Eq. 6.5) [28]. The unit vectors are used to transform the intersection points their
coordinates in the (x, y, z) frame back to the original (x, y, z) frame to find the coordinate of P3. As
mentioned previously, since P3is on the right side of the plane spanned by the three centres of the
spheres, the solution will have the addition of the negative z-term (P3zuz).
6.2. Trilateration 43
P3x=|P0P3|2− |P2P3|2+|P0P2|2
2|P0P2|
P3y=|P0P3|2− |P4P3|2+ux(P4P0)2+uy(P3P0)22ux(P4P0)P3x
2uy(P3P0)
P3z=q|P0P3|2P2
3x+P2
3y
P3=P0+P3xux+P3yuy±P3zuz
(6.5)
6.2.2. Uniform actuation
All bridle line system models that actuate the TE lines by the same amount, fall under the uniform
actuation category. For both the triangular 2-plate model and the quadrilateral 2-plate model, this holds.
For the quadrilateral 2-plate model the same cos-law (see Eq. 6.3) as was used for the triangular 2-
plate model is used. The cos-law is necessary because the coordinate of the point on the TE of the
middle chord is needed as input for the trilateration algorithm. The bridle system is modelled as six
straight lines coming from the KCU to each corner point (see Fig. 6.4). Only symmetric actuation can
be modelled and is done by actuating all TE bridle lines uniformly, i.e. all by the same line extension
equal to half of the depower tape extension.
Figure 6.4: Quadrilateral 2-plate model. The blue lines indicate the lines attached to the LE and the red lines those that are
actuated and attached to the TE.
The last uniformly actuated model is the quadrilateral 3-plate which does not require the cos-law. To
illustrate why it does not, a brief description of how one goes about calculating the location of a tip TE
point follows (see Fig. 6.5). Three points are needed as input for the trilateration algorithm, but now
two LE points and the KCU can be used as input and a known TE point is thus not required. This is
possible because both LE points were assumed to remain in place. The bridle line system is again
represented by straight lines that connect the KCU to the corner plate. Symmetric actuation is done
uniformly by all four TE bridles. Modelling asymmetric actuation is possible but will not be done due to
the existence of slacking bridles.
6.2. Trilateration 44
Figure 6.5: Quadrilateral 3-plate model. The blue lines indicate the lines attached to the LE and the red lines those that are
actuated and attached to the TE.
6.2.3. Non-uniform actuation
A change in depower tape length, resulting from symmetrical actuation, is likely to not result in a uniform
change in line length between the attachment points and the KCU. The first argument for why uniform
might not be correct, is that the actual bridle line system does not consist of uniform straight lines. In
reality, many other independent lines exist, which are connected by two pulleys and several knots. The
positions of the pulleys and two of the lowest knots have been identified to lie on a different line-of-sight
in the powered state compared to the depowered state (see Fig. 4.3). This horizontal shift is caused by
a change in loading conditions, supporting the non-uniformly loaded hypothesis. From an aerodynamic
perspective, the generated lifting force is known to be a function of the surface area and inflow angle.
The surface area of the middle plates is larger and the inflow angle of the outer plates is less optimal
with respect to the incoming flow, meaning more force is produced in the middle compared to the outer
parts. Tip vortices caused by the pressure difference between the top and the bottom surface further
decrease the lift force at the outer sections. In short, from an aerodynamic perspective, the loading
can be concluded as non-uniform. Because the loading determines the bridle line layout it, therefore,
provides another argument for why uniform actuation could not suffice. The last indication is that it
was found that a 7-plate or 9-plate quadrilateral model with uniform symmetric actuation can’t predict
the shape for low-powered settings. For these cases, errors were caused due to the inability of the
trilateration algorithm to find all sphere intersection points.
To achieve non-uniform actuation the level of detail of the bridle line system representation must be
increased (see Fig. 6.6). The lowest knots and the pulleys will be modelled to do so, thereby repre-
senting part of the KCU bridle line system. The knots and pulley positional change must be calculated,
using as input the empirically obtained relationships between the power setting and the line-of-sight of
the knots and pulleys (see Sec. 4.1). The rest of the bridle lines are modelled as straight lines in be-
tween the KCU, knots, pulleys and bridle line attachment points. A non-uniform actuation of the bridle
line attachment point is achieved because the actuation length and direction depends on the distance
each point has to the moving knot and pulley points. Only symmetric actuation is modelled because
for asymmetric actuation the lines attached to the tips slack and slack can’t be taken into account.
6.2. Trilateration 45
Figure 6.6: Quadrilateral 7-plate model. The blue lines indicate the lines attached to the LE and the red lines those that are
actuated and attached to the TE. On the right a front view is shown with only the TE bridles visible, the thick red lines indicate
the KCU bridle line system representation and the dots the knots and pulleys.
Once the knot and pulley position for a certain symmetrical actuation are known, all the coordinates of
the bridle line attachment points can be calculated using the trilateration algorithm. The main change
is that instead of the KCU, the knots and pulleys are now used as inputs forming the first sphere. Ob-
taining the knot and pulley position is, however, not trivial.
Calculating knot and pulley point positions
By thinking of the problem as a circle intersection problem, a solution can be found. The circles are
in a 2D plane spanned by a line from the KCU to the LE mid point (LEmp), a line from LEmp to the
line-of-sight its crossing point at the TE (Pcross), and from Pcross to the KCU (see Fig. 6.7). The first
circle centre is placed at the KCU and its radius is equal to the length from the KCU to Pcross (R). The
second circle is centered at the LEmp, which is a distance daway from the KCU and its radius is equal
to the length from LEmp to Pcross (r). Rewriting the two circle equations into one equation one can
solve for the z-coordinate of Pcross (zPcross ) (see Eq. 6.6).
zPcross =d2r2+ (R+ ∆L)2
2d(6.6)
6.2. Trilateration 46
Figure 6.7: 7-plate wing model, with 2D plane spanned by LEmp, Pcr oss and the KCU.
This described procedure is only possible by assuming that line rremains of constant length. The pa-
rameter dalso remains constant, because the LE bridles do not change length. The actuation input is
taken into account by adding a quarter of the depower tape extension (L) to R. A quarter of the value
is taken because at the pulley the lines have split twice, each time dividing the extension effects in half.
The parameters for the powered state can be obtained from the known geometry and the empirically
determined crossing point. Using the powered state as the initial state, the depowered state can be
calculated since all other required parameters are known.
Using similar triangles zPcross can be used to calculate the z-coordinate of the pulley point (zp) (see Eq.
6.7). The first triangle consists of: R,dand the distance along the new x-axis and the second triangle
of: the line from the KCU to the new pulley location (Lp+ ∆L) parallel to R,zpparallel to zPcross and
distance along the same x-axis (x
p) (see Fig. 6.8a). With zpknown one can calculate the pulley its
position along the x-axis (x
p).
Lp=q(Lp,x KC Ux)2+ (Lp,y K CUy)2+ (Lp,z KCUz)2
zp=Lp+ ∆L
R+ ∆LzPcross
x
p=q(Lp+ ∆L)2z2
p
(6.7)
6.2. Trilateration 47
(a) The (z, x) plane is shown. Where the blue triangle illustrates the
similar triangle to its larger brother, which is indicated by the red lines.
(b) The (y, z) plane is shown, indicating the definition of the angle θ.
Figure 6.8: Illustrations of the non-uniform calculation method.
The y-coordinate of the pulley (yp) can be found by looking at the (y, z)frame and calculating the
angle of the line R(θ) (see Fig. 6.8b). Using θand Lthe change in y-direction of the pulley (yp)
is calculated. This is only possible by assuming that θdoes not change. By adding ypto the old
y-position (yp,0) the y-coordinate of the pulley is found (yp).
θ=arccos yp,0
q(x2
p,0+y2
p,0+z2
p,0)!
yp= ∆Lcos(θ)
yp=yp,0+ ∆yp
(6.8)
With ypand x
pone can use Pythagoras theorem to calculate the x-coordinate of the pulley (xp) (see
Eq. 6.9).
xp=qx2
py2
p(6.9)
The same procedure as described above is applied to calculate the coordinate of the knot point.
Ballooning
Another version of the quadrilateral 7-plate model is developed incorporating ballooning, both with and
without is modelled to allow verification procedures through comparison. Ballooning is taken into ac-
count using the experimentally obtained relations between TE plate length and power setting (see Sec.
4.1). Since not all nine kite wing segments are represented by 7-plates, the outer plates their TE change
will be the percentage change of the outer segments. As a consequence of ballooning, the diagonal
length within each plate also changes. As the distance between points is a necessary input for the
6.3. Particle system model 48
trilateration algorithm, the diagonal length between the inner TE point and outer LE point is assumed
constant for the outer plates (see Fig. 6.9). The assumption is only needed one way because when
calculating the outer TE points of a plate three sphere’s can be selected that are connected through
the edges of the plate and thus do not lie at unknown distances.
An alternative to assuming that the diagonal line lengths are constant is to formulate a system of equa-
tions for each plate. With the two inner point locations known, one can write the location of both outer
points as a function of the edge length, the locations of the inner points, KCU and the outer points
themselves. One ends up with a system of six nonlinear equations with six unknowns, the latter are
the x,yand zlocation of each outer plate point. Solving this system of equations is not trivial and was
found to take too much computational time to be useful.
Figure 6.9: Top view of the quadrilateral 7-plate. The red arrows indicate the ballooning causing the TE length to differ, which
alters the diagonal distance.
The middle plate is the exception to the rule and does not require the constant diagonal length as-
sumptions. Because the struts of this plate are of equal length the middle plate is a special kind of
quadrilateral, called an ‘isosceles trapezoid’. For an isosceles trapezoid the Ptolemy’s theorem holds
which relates edge lengths (Ei,j ) to the diagonal length (Di,j ) (see Eq. 6.10) [12].
Ea,b =q(Pa,x Pb,x)2+ (Pa,y Pb,y )2+ (Pa,z Pb,z )2
Ec,d =q(Pc,x Pd,x)2+ (Pc,y Pd,y )2+ (Pc,z Pd,z )2
Ea,d =q(Pa,x Pd,x)2+ (Pa,y Pd,y )2+ (Pa,z Pd,z )2
Da,c =qEa,cEc,d +E2
a,d
(6.10)
6.3. Particle system model
The limitations of the trilateration based models arose in part from not being able to deal with slacking
bridles. Representing the bridles by straight lines of constant length does not allow one to accurately
model all nine kite wing segments nor asymmetric actuation. The latter is a must as it allows the kite
to turn, without which a kite can’t fly pumping cycles. Furthermore, to incorporate non-uniform steering
additional empirical relations were needed for the 7-plate model. These empirical relations stand in the
way of the potential of the model because needing upfront information makes applying it to other kites
or bridle line system configurations impossible without a test flight. Therefore, infeasible to use as the
basis for the desired design model.
The particle system model (PSM) overcomes these problems and is developed both with and without
ballooning (see Fig. 6.10). Because ballooning is a force-based phenomenon, there are extrapolation
problems, which are expected to be resolved when incorporating the PSM into a FSI framework.
6.3. Particle system model 49
Figure 6.10: Orthographic view of the PSM on the left. On the top right top the bridles attached to the LE are shown, whereas
on the bottom right those attached to the TE.
Solving the bridle line system with more detail to achieve the desired non-uniform actuation, was only
possible by using empirical relations. Without the relations, the free angles of the middle part of the
bridle line system cause infinitely many solutions, i.e. an under constraint problem. Thinking of 2D
geometric shapes explains best why the free angles exist. A triangle with fixed edge lengths has a
‘rigid’ shape and can not move its angles freely. A quadrilateral with fixed edge lengths on the other
hand, or any other shape with more than three lines, does not have a rigid shape. As the middle part of
the LE bridles forms a pentagon the problem, therefore, thus has free angles. By incorporating a force
input, the PSM can deal with the full bridle line system, allowing for an accurate representation of the
occurring non-uniform actuation.
AWES tether elements are often dynamically modelled as springs with a spring stiffness, this enables
dynamic models to reach a stable solution [23]. Modelling bridles as spring elements is generally not
done due to their short lengths in comparison to the tether elements [9, 66]. An exception is the work
of Geschiere, who extended the FSI model of Bosch et al. using the PSM of Fechner et al. to model an
extendible tether and more bridle lines (see Sec. 2.1.1) [5, 23, 27]. The model was developed using a
different approach that achieves less utility by not including the actuation system. To be more precise,
the described dynamic PSM differs from the one developed here in: how the kite wing is represented
(see Fig. 6.11a and 6.11b), which forces it takes into account, the numeric values, the addition of non-
physical elements (compliant element) necessary for stability and not modelling the KCU bridle line
system.
6.3. Particle system model 50
(a) Geschiere his model, where the horizontal line represents the
compliant element [27].
(b) The PSM model developed in this research.
Figure 6.11: Dynamic PSMs, depicted in the same colours for ease of comparison.
All particles are attributed a unit mass. Furthermore, a damping term is added because the dynamic
simulation must lose energy to reach a stable solution. The drag of the considered elements is ne-
glected, based on its relative minor size compared to the force generated by the kite wing [10].
The incorporated spring stiffness and damping terms are non-physical making the PSM pseudo-physical.
Non-physical parameters are needed for the dynamic simulation to remain stable during the transient
phase, e.g. a lower than real spring stiffness is selected. As only the steady-state is of interest, tran-
sient phase effects are neglected, e.g. the friction of a line rolling over the pulley. The actual spring
stiffness is of such magnitude that substantial stretching is unlikely. The solution criteria that the max-
imum stretching of the bridle must be below 0.25% in the powered CAD state is, therefore, adopted.
The non-relevance of the unphysical transient phase combined with strict solution criteria, make that
the pseudo-physical PSM its steady-state solution is considered physical.
The elements making up the particles (see Sec. 6.3.1), their new position is an outcome of a force
equilibrium (see Sec. 6.3.2). The forces that together reach an equilibrium are the damping force (see
Sec. 6.3.3), spring force (see Sec. 6.3.4) and lift force (see Sec. 6.3.5).
6.3.1. The particles
The vector x represents the location of all the particles in the PSM, which are the knots, pulleys and
bridle attachments points to the kite wing (see Fig. 6.12). The latter represents the kite shape, where
the kite wing segments edges are represented by elements with spring stiffness and a damping con-
stant. This makes for the discussed wireframe 9-plate representation of the kite wing (see Sec. 5.2).
Because the quadrilateral shape is not ‘stiff’, i.e. free to change its angles, additional diagonal elements
are added to prevent shear (see Fig. 6.13). As a side note, the trilateration based models that have
a wireframe without diagonals, hold their validity since one of the diagonals is used in the calculation
and is assumed constant.
6.3. Particle system model 51
Figure 6.12: Orthogonal view of the PSM, where each dot
represent a particle.
Figure 6.13: Top view of the PSM, where the diagonal kite
wing plate elements are clearly shown.
6.3.2. Equations of motion
The dynamic simulation solves a force equilibrium for each particle in the system. The force equilibrium
is based on Newton’s 2nd law and takes the general form of
Fr=m
¨x, where
Fris the resultant force, m
the mass and
¨xthe acceleration. Because the particles have unit mass the form is reduced to
Fr=
¨x.
Taken into account all the force components: spring stiffness (
Fs), damping force (
Fd) and lift force
vector (
L) the equation of motion (EOM) becomes a 2nd order non-homogeneous differential equation
(see Eq. 6.11).
¨x=
Fs+
Fd+
L(6.11)
The 2nd order non-homogeneous differential equation can be reduced to a system of 1st order coupled
non-homogeneous differential equations. This reduction is achieved by introducing a parameter q rep-
resenting the positions equal to x and u representing the velocities equal to ˙
x. One can represent the
acceleration term (¨
x) by the two equations formed by the derivative of q and u (see Eq. 6.12).
˙q=
˙x
˙u=
Fs+
Fd+
L(6.12)
The remaining system of 1st order coupled non-homogeneous differential equations are so called ‘stiff’,
i.e. difficult to numerically solve. An explicit solver known as the ‘leap-frog method’ was used but found
too unstable when incorporating all particles. It was decided that an implicit solver method is needed
for numerical stability reasons.
Implicit solvers generally require more computational cost per iteration compared to explicit solvers.
The benefit is however that they are stable at much larger time steps. Leaving one with a trade-off
decision, where for the PSM problem an implicit method was found best. The problem is solved using
a module of Python called ‘SciPy’, in which the PSM is formulated as an initial value problem [63]. The
selected solver is the ‘Radau’ solver, which implements an implicit Runge-Kutta method of the fifth-
order [31].
6.3.3. Damping force
The linear viscous damping force term (
Fd) is a function of the velocity (
˙x) and the damping constant
(C) (see Eq. 6.13). Because the transient phase is not relevant the damping constant is not selected
6.3. Particle system model 52
through experiments, rather through a trial and error procedure. Its magnitude was set high enough to
provide stability and low enough to allow movement of the points.
Fd=C
˙x(6.13)
6.3.4. Spring force
The spring force
Fsis equal to the sum of all attached bridle lines and kite elements that are extended.
The generated force is based on the spring stiffness (K) and the amount of extension (
l). The latter
is represented by vectors because all particles are connected by multiple elements, each in their own
direction (see Fig. 6.14). The amount of extension is calculated by taking the difference between two
particles and subtracting the restitution length (l0) (see Eq. 6.14). The spring stiffness is set equal for all
bridles and selected through trial and error. This process was done while ensuring that the maximum
line stretch in the design state (up= 1) is below 0.25% making its effect negligible.
To decrease numerical instability the spring force (
Fs) is modelled slightly different than merely a linear
decrease for positive extensions. The function is translated to reach a value of one at
l= 0 and when
l < 0the spring stiffness approaches zero asymptotically (see Eq. 6.14). This makes the function
smooth, hence decreasing numerical instability problems.
l= (lA,B lA,B0)
Fs=(K
l+ 1 if
l0
1/(
l1) if
l < 0
(6.14)
Figure 6.14: Spring force due to a line extension
between two knots.
Figure 6.15: Spring force due to a line extending between two knots, that
goes over a pulley.
Figure 6.16: Free body diagrams of the spring force (Fs).
The particles representing the pulleys are modelled separate from those that represent the knots be-
cause the line extension must be calculated differently. The bridle line is namely attached at knots and
not at the pulley itself (see Fig. 6.15). The line extension determining the spring force is, therefore, cal-
culated with respect to the total difference in distance, i.e. the sum of the current lengths ([lA,P +lB ,P ])
minus the sum of the restitution lengths ([lA,P 0+lB,P 0]) (see Eq. 6.15).
l=[lA,P +lB,P ][lA,P 0+lB,P 0]
[lA,P 0+lB,P 0](6.15)
6.3.5. Lift force
The vector
Lrepresents the lift force and is filled with zeros for all the entries representing the knots
and pulleys. Only the particles representing the bridle line attachment points, i.e. the corners of the
6.3. Particle system model 53
9-plate representation of the kite wing, are given a non-zero plate lift (
Lp).
Lpis calculated per plate
and is applied to each relevant particle in the outwards perpendicular direction (see Fig. 6.17). This
is achieved by multiplying
Lpwith the unit vector normal to the plate, calculated by taking the cross
product of the two diagonals of the respective plate. Updating the orientation of
Lpwith each iter-
ation ensures that the effect of panel rotations are taken into account.
Lpis assumed to act in the
middle of the plate. The distribution of
Lon each corner point is therefore assumed uniform locally in
both chord-wise and span-wise, hence each corner point is attributed 25% of
Lpof the respective panel.
Figure 6.17: 9-plate wing model, where the red arrows indicate each lift vector (
Lp) perpendicular to its respective panel.
Lpis multiplied by a factor to simulate a more accurate non-uniform spanwise loading condition. The
factor should represent the lifting conditions of the kite wing, which therefore should follow the general
lift-force equation (see Eq. 6.16). The general lift-force equation is based on the density (ρ), the surface
area (S), the apparent velocity ( va) and the lift coefficient (CL). From these parameters, the relevant
change per kite wing segment is governed by: S,CLand va.
L=1
2ρS ⃗va
2CL(6.16)
The change in vais dependent on the current flight direction of the kite with respect to the wind velocity
vector ( vw). Because the flight path is not simulated, the change can’t be taken into account. The
change in Scan be used as a factor and is calculated using Brahmagupta’s formula (see Eq. 6.17)
[43]. The edge lengths of the plate (ledge,i ) are used to calculate the semi-perimeter (s), which is used
to calculate S. To not alter the order of magnitude of
L,Sis non-dimensionalized.
s=1
2
4
X
i=1
ledge,i
S=q(sledge,1)(sledge,2)(sledge,3)(sledge,4)
(6.17)
The wind velocity vector vwis assumed to stay constant and parallel to the (x, z) plane hitting the middle
chord of the kite wing at an angle of 10 ° (see Fig. 5.5b). This enables the formulation of the change of
effective inflow angle for each plate, calculated with respect to a vector representing the middle chord
of each plate ( cm,i). By assuming that CLscales as a flat plate would do in its linear part of the lift-polar,
i.e. by 2πsin (α), the change of CLcan be taken into account. Because the kite wing is subject to
an anhedral angle, the effective inflow angle is calculated in both the (x, z)and (x, y )plane. The lift
scaling factor (L), therefore, becomes a function of both the angle of attack (α) and the side-slip angle
(αs). The respective effects of both angles are multiplied by the orientation of the LE of the plate (see
Eq. 6.18), expressed as unit vector (
LEi). To not change the order of magnitude of
L, the lift scaling
factor (L) is non-dimensionalized before multiplying.
L=|
LEi,y sin(α)|+|
LEi,z sin(αs)|(6.18)
6.3. Particle system model 54
Including the scaling of
Lpbased on the effective inflow angle has an additional benefit when actuating
the wing asymmetrically. The resultant lift vector namely tilts to the side upon experiencing the turning
input, both in flight and within the simulation environment. The result of the tilt is a change in panel
orientation, meaning that in the next time step the lift-vector will tilt even more. Without a counteracting
force, the result is a kite wing that keeps rotating around the origin. Leaving one with a simulation that
would never reach a stable solution. By including orientation based scaling the
Lpof the panels tilting
to the side reduces, causing a dynamic stabilisation of the kite wing.
7
Results
Several wing models and bridle line system representations have been developed. To assess the
accuracy the models will be compared based on the width change for different power settings. The
comparisons with other models serve as verification and the comparisons with the photogrammetry
results as validation. Besides accuracy, the computational cost of the models is compared based on
run times. The run times were obtained using code that is not optimised and a laptop with a single
processor, that has a processor base frequency of 2.7 GHz.
First, the triangular 2-plate model, used mainly for verification shall be discussed (see Sec. 7.1) followed
by the effect of adding TE bridles and an additional plate (see Sec. 7.2). Furthermore, the effect of
including more plates and using non-uniform actuation is discussed (see Sec. 7.3). The results for the
9-plate kite wing representation, modelled using the PSM, are discussed last (see Sec. 7.4).
7.1. Triangular 2-plate model
The 2-plate triangular wing model is simulated using the tetrahedron algorithm, trilateration algorithm
and the PSM. The output of the three models is plotted together with the photogrammetry results show-
ing the predicted kite width for different power settings (up) (see Fig. 7.1). The same behaviour of
increasing width for increasing power setting is observed, but the steepness of the slope and magni-
tude of the width differs. This could be caused by the extension of the imaginary line and the simplified
representation of the kite, which combined seem to overpredict the amount of deformation. Other fac-
tors contributing to the discrepancy could be caused by not including ballooning, incorrect geometric
input and incorrect actuation relations. In the powered state (up= 1) all lines end at the same point,
which they should because that is the design width upon which the models are built.
Comparing the models, one observes that both tetrahedron and trilateration predict the exact same out-
put. Considering their purely geometric nature this should be the case, thereby verifying both models.
The tetrahedron and trilateration algorithm both show a run time below 1ms, whereas the PSM runs in
2.5s. The large difference in run time is attributed to the non-geometric nature of the PSM. It should,
for small convergence criteria, still give roughly the same results as the geometric models do. That all
three models predict about the same width, therefore verifies the correct working of each code. The
only relevant difference is observed for up< 0.1, where the width prediction of the PSM is slightly higher.
This is attributed to several reasons, discussed along with the rest of the PSM its results (see Sec. 7.4).
55