1. Determination of kite forces using three-dimensional flight trajectories
for ship propulsion
George M. Dadd*, Dominic A. Hudson, R.A. Shenoi
University of Southampton, Southampton, England SO17 1BJ, United Kingdom
a r t i c l e i n f o
Article history:
Received 15 November 2010
Accepted 23 January 2011
Available online 21 April 2011
Keywords:
Kite
Dynamics
Trajectories
Ship propulsion
Optimisaton
Experiment
a b s t r a c t
For application of kites to ships for power and propulsion, a scheme for predicting time averaged kite
forces is required. This paper presents a method for parameterizing figure of eight shape kite trajec-
tories and for predicting kite velocity, force and other performance characteristics. Results are
presented for a variety of maneuver shapes, assuming realistic performance characteristics from an
experimental test kite. Using a 300 m2
kite, with 300 m long flying lines in 6.18 msÀ1
wind, a time
averaged propulsive force of 16.7 tonne is achievable. A typical kite force polar is presented and
a sensitivity study is carried out to identify the importance of various parameters in the ship kite
propulsion system. Small horizontally orientated figure of eights shape kite trajectories centred on an
elevation of 15
is preferred for maximizing propulsive benefit. Propulsive force is found to be highly
sensitive to aspect ratio. Increasing aspect ratio from 4 to 5 is estimated to yield up to 15% more
drive force.
Ó 2011 Published by Elsevier Ltd.
1. Introduction
Kite propulsion is an attractive means to reduce fuel
consumption on ships by assisting the main engine using the power
of the wind. Recent developments, such as in autopilot kite control
and in launch and recovery systems1
have enabled them to be used
commercially for trans-oceanic voyages, yielding financial savings
through reduced fuel costs as well as minimizing emissions that are
harmful to the environment.
The determination of drive forces using a kite performance
model is required for ship velocity prediction, for enabling design,
for synthesising fuel savings and for optimizing kite systems for the
best propulsive effect. In addition, a kite performance model can be
used to implement carefully considered kite trajectories for
a desired force output.
Kite performance prediction models have been previously
established by Lloyd, [1], Wellicome [2], Naaijen [3], Williams [4]
and Argatov [5,6] although only Wellicome’s zero mass theory
has received published experimental validation. Dadd et al. (2010)
previously used the zero mass kite manoeuvring theory [2] to
predict kite line tension and other performance parameters. These
results were compared with real kite trajectories that had been
recorded using a purpose-specific kite dynamometer. The results
were shown to agree favourably; that work focused on the
validation of performance prediction based on kite position only.
The onset velocity and resulting line tension were calculated
without directly knowing the kite velocity itself. This paper focuses
on the additional modelling required in order to determine kite
velocity theoretically, an essential feature to enable the kite
performance to be established as a function of time.
Section 2 in this paper discusses the assumptions made in the
kite performance model. Section 3 presents a method for creating
kite trajectory shapes theoretically [2] and extends previous
developments by allowing the parameterized kite trajectories to be
transformed to simulate different mean angles to the wind. Section
4 defines the mathematical model. Section 5 describes the imple-
mentation and presents results using a case study for a typical ship
kite propulsion system. A new kite force polar diagram is developed
showing the propulsive drive for different wind angles. The
investigations are carried out considering the influence of the
Earth’s natural boundary layer. Section 6 presents an optimization
and sensitivity study that shows how various parameters effect
system performance including elevation, kite aspect ratio, angle of
attack, maneuver pole separation and pole circle size. Section 7
provides validation by way of comparison between theoretical
and experimental results [7].
* Corresponding author. Tel.: þ44 7815 044873.
E-mail address: georgedadd@hotmail.com (G.M. Dadd).
1
Pamphlet “Skysails Technological Information” available from www.skysails.
com.
Contents lists available at ScienceDirect
Renewable Energy
journal homepage: www.elsevier.com/locate/renene
0960-1481/$ e see front matter Ó 2011 Published by Elsevier Ltd.
doi:10.1016/j.renene.2011.01.027
Renewable Energy 36 (2011) 2667e2678

2. 2. Assumptions in the kite force model
1. The zero mass theory [2] assumes that the kite and the lines are
weightless. This is reasonable provided that the real weight is
very small compared to the aerodynamic forces, as shown by
Dadd et al. [7].
2. The kite is assumed to maneuver on the surface of a sphere of
radius defined by the flying line.
3. The kite lift and drag coefficients are assumed to remain
constant. The aerodynamic lift and drag coefficients are given
by expressions of the form
CL ¼ f ðae; RnÞ (1)
and
CD ¼ gðae; RnÞ: (2)
This implies firstly that the dependence of the force coefficients on
Rn is negligible and secondly that the angle of attack is unchanging.
To explain the third assumption, the kite is in a condition of
force equilibrium during static flight, where the line tension is
equally opposed to the aerodynamic force (neglecting weight). The
kite assumes its position in the flight envelope where this condition
is met and the effective angle of attack (ae) is dependent on the
relative wind velocity and the angle of mount to the flying lines.
During dynamic flight, the kite seeks the same force equilibrium
condition. When an imbalance of force arises, the kite accelerates
almost instantaneously to achieve the apparent kite onset velocity
at which this equilibrium is again achieved. The angle of attack
remains the same as the static flight case where CL and CD are
constant.
The Reynolds number effects which can also influence CL and CD
are not expressly included in the zero mass model, although it is
noted that from Dadd et al. [7] that Rn was seen to vary between
7 Â 105
for static flight and 4.3 Â 106
for dynamic flight using a small
3 m2
kite in light winds. These are above the critical Rn number
(w5 Â105
) at which transition between laminar to turbulent flow
tends to occur and thus it can be expected that the flow will remain
substantially turbulent during dynamic flight and expectedly more
so for larger kites or for stronger winds. Thus with transition
between laminar and turbulent flow being unlikely during normal
flying conditions, the Rn effects are very minor and safe to neglect
whilst maintaining good predictions for kite performance.
Based on the above principles, Wellicome showed that the onset
wind velocity at the kite can be established in terms of its azimuth
and elevation spherical position angles, using the fundamental zero
mass equation [2].
U ¼ VA
cos qcos f
sin 3
: (3)
Here, q ¼ 0 f ¼ 0 defines the downwind direction.
Lloyd [1] had found that where the kite passes directly through
the downwind position, the onset velocity can be approximated by
Nomenclature
AK projected kite area, m2
AR aspect ratio
e aerodynamic planform efficiency factor (lifting line
theory)
f,g generic functions
F aerodynamic force magnitude, N
CL lift coefficient
CLa
lift coefficient at a
CL0
lift coefficient at a ¼ 0
CD drag coefficient
CD0
drag coefficient at a ¼ 0
D drag force magnitude, N
E rotation matrix
H pole of trajectory sphere
l aerodynamic lift force unit vector, N
L lift force magnitude, N
n exponent dependant on atmospheric and surface
conditions
n vector normal to great circle (right to left sweeps), m
n1,n2,n3 components of vector n, m
m vector normal to great circle (left to right sweeps), m
m1,m2,m3 components of vector m, m
O origin of trajectory sphere
P pole of small circle sweep
Q pole of small circle sweep
r kite position unit vector, m
ro small circle pole position vector, m
R kite position vector magnitude, m
R kite position vector, m
Re Reynolds number (Uck/n)
T time taken to traverse between two maneuver points A
and B, s
u onset velocity unit vector, msÀ1
U intersection node on trajectory
U onset velocity vector, msÀ1
U onset velocity magnitude, msÀ1
v apparent wind velocity unit vector, msÀ1
V intersection node on trajectory
V apparent wind velocity magnitude at the kite when
static, msÀ1
VT true wind speed, msÀ1
VTref
true wind speed at reference altitude, msÀ1
W intersection node on trajectory
V apparent wind at the kite, as though it were static,
msÀ1
x,y,z Cartesian position coordinates, m
X,Y,Z global Cartesian position coordinates, m
dt time step, s
f, g generic functions
ae effective angle of attack,
a semi-vertex cone angle,
a1 semi-vertex cone angle at P,
a2 semi-vertex cone angle at Q,
b azimuth angle of air onset velocity,
d variable,
3 aerodynamic drag angle,
g elevation angle of air onset velocity,
f azimuth angle,
h1,2,3 transformation rotation angles about axis X, Y and Z,
q elevation angle,
ra density of air (1.19 kg mÀ3
at 20, 1 bar)
s variable,
s variable,
m substitution variable, (m ¼ 1/2raAKCLsec3)
z variable,
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e26782668

3. the apparent wind speed multiplied by the lift to drag ratio of the
kite. Thus, with present day kites, such as the Flexifoil blade II para-
foil type test kite, used by Dadd [7], having lift to drag ratios of the
order of six, onset velocities six times wind speed are typical for
downwind dynamic kite flight and since aerodynamic force
increases with the square of the onset velocity, line tensions thirty
six times that of static kite flight are feasible.
Practical kite trajectories must remain downwind of the kite
tether. The preferred maneuver shapes are those that do not result in
undesirable twisting of the flying lines or collision with the ground.
Consequently, vertical or horizontal figure of eight type motions are
commonplace for substantially downwind sailing courses, whilst
sine-wave (up and down) motions and static kite flight are
commonplace for upwind sailing courses. The manner in which
propulsive forces develop is largely dependent on the shape of
the kite trajectory adopted. Section 3 describes the way in which the
trajectories in the present paper were parameterized mathematically.
3. Parameterization of kite trajectory shapes
The trajectory parameterization follows the work of Wellicome
[2] and is further modified to allow vertically and diagonally
orientated trajectory shapes. The figure of eight flying maneuver
shape can be mathematically defined as comprising two small
circle sweeps for the ends of the maneuver and two great circle
sweeps which connect the small circular ends (see Figs. 1 and 2).
The great circle sweeps are along the intersections of a plane
passing through the origin with the sphere surface. The small circle
ends are defined using a semi-vertex angle for the cone swept out
by the radius and the position of the circle pole at its centre.
The spherical position angles, azimuth and elevation, are as
shown in Fig. 1. The position vector of the kite is given by
r ¼ (cos q cos f, cos qsin f, sin q). The unit vector n ¼ (1,2,3) is
chosen normal to the plane of the great circle sweep such that
nr ¼ 0.
rn1cos q cos f þ n2cos q sin f þ n3sin q ¼ 0 (4)
rtan q ¼ À
n1
n2
cos f þ
n2
n3
sin f
(5)
Hence Eq. (5) can be used to establish q, 4 ordinates on any great
circle defined by n.
The direction of the kite is defined by the value of f
=q
which can be determined by differentiation of Eq. (4) for
great circle sweeps as
ð À n1cos q sin f þ n2cos q cos fÞf
þ ð À n1sin q cos f À n2sin q sin f þ n3cos qÞq
¼ 0:
(6)
r
f
q
¼
sin qðn1cos f þ n2sin fÞ À n3cos q
cos qðn2cos f À n1sin fÞ
: (7)
For the small circle ends in Fig. 2, the pole P with position vector ro
defines the centre of the small circle sweep that is inscribed by the
kite with semi-vertex angle a. Points which lie on the small circular
arc conform to
r ro ¼ cos a; (8)
where
r ¼ ðcos q cos f; cos q sin f; sin qÞ (9)
and
ro ¼ ðcos qocos fo; cos qosin fo; sin qoÞ: (10)
rcos a ¼ cos q cos qocosðf À foÞ þ sin q sin qo; (11)
rf ¼ fo Æ cosÀ1
cos a À sin q sin qo
cos q cos qo
: (12)
Hence Eq. (12) can be used to establish q, f ordinates on any small
circle sweep. The direction of flight for the small circle ends is
determined through differentiation of Eq. (11) as
f
q
¼
cos q sin qo À sin q cos qocosðf À foÞ
cos q cos qosinðf À foÞ
: (13)
Thus, to fully define the maneuver geometry, two poles (P(q1, f1)
and Q(q2, f2)) are required with corresponding semi-vertex angles
(a1 and a2) for each of the small circle ends, as well as calculation of
the two normal vectors n and m to define the great circle sweeps,
right to left and left to right respectively. It is necessary to establish
these maneuver segments as forming a continuous path by
ensuring tangency at the respective joins. For this, consideration is
given to the spherical triangle shown in Fig. 3, with a view to
finding the ratios n1/n3 and n2/n3 which can later be used to ensure
that tangency and continuity are correctly obtained. PH and QH in
Fig. 3 are great circles passing through the Z axis and PQ is another
great circle passing through O. The angle subtended between OP
and OH is 90 À q1, the angle subtended between OQ and OH is
90 À q2, the angle subtended between OP and OQ is z and the angle
subtended between HP and HQ is f1 À f2.
Applying the standard spherical cosine formula to triangle HPQ,
z can be found.
Fig. 1. Great circle sweep geometry.
Fig. 2. Small circle sweep geometry.
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e2678 2669

4. cos z ¼ cosð90 À q1Þcosð90 À q2Þ
þ sinð90 À q1Þsinð90 À q2Þcosðf1 À f2Þ (14)
or
cosz ¼ sinq1sinq2 þ cosq1cosq2cosðf1 À f2Þ (15)
Fig. 4 shows the intersection node points for consecutive great
circle and small circle sweeps at R, T, U and V. z1 and z2 are the
angles subtended between OP and OS and between OQ and OS
respectively. a1, a2, s1 and s2 equivalently represent their included
angles with the origin. The angle formed by the intersection of PQ
and RT is d. The angle RPQ is s1, the angle HOR is 90 À qR, the angle
HOP is 90 À q1 and the angle RHP is f1 À 4R. HR, HP, RP, PQ, RT and
QT are all great circles. DPRS and DSTQ are right triangles.
Applying the spherical sine formula to DPRS,
sin a1
sin d
¼
sin z1
sin 90
(16)
rsin d ¼
sin a1
sin z1
(17)
Similarly for DSQT,
sin d ¼
sin a2
sin z2
(18)
Through equating Eqs. (17) and (18) and using z2 ¼ z À z1, the
value of z1 can be found,
tan z1 ¼
sin z sin a1
sin a2 þ cos z sin a1
(19)
Applying the cosine formula to DPRS, s1 can be obtained with
cos z1 ¼ cos a1cos s1 þsin a1sin s1cos 90 ¼ cos a1cos s1 (20)
rcos s1 ¼
cos z1
cos a1
(21)
Applying sine formula to DPRS, s1 is obtained using
sin s1 ¼
sin s1
sin z1
(22)
Employing the sine formula with DHPQ in Fig. 3 gives
sin
HbPQ
¼
cos q2sinðf1 À f2Þ
sin z
(23)
then
HbPQ ¼ HbPQ À s1: (24)
Applying cosine formula to DHPR enables calculation of qR,
cosð90 À qRÞ ¼ cos a1cosð90 À q1Þ
þ sin a1sinð90 À q1Þcos HbPQ (25)
rsinqR ¼ cosa1sinq1 þ sina1cosq1cos
HbPQ
(26)
Then, applying the sine formula to DHPR allows fR to be solved
using
sinðf1 À fRÞ ¼
sin a1sin
HbPQ
cos qR
(27)
The use of Eqs. (15)e(27) allows the locations the nodes qR and fR to
be solved for a given input set [q1, f1, a1, q2, f2, a2] that define the
maneuver shape. A similar set of equations can be developed to
establish the values of qT and fT. These values for points R and T can
be input in Eq. (4) and solved to establish the ratios n1/n2 and n3/n2.
Thus Eq. (5) may be used to define a series of data points along the
great circle sweep, RT. A similar process is repeated to establish
a series of positions along the other great circle sweep, UV. Points
which lie on the small circular end about the pole P are established
using Eq. (12) and similarly for points which lie on the small
circular end about pole Q. Data points on the surface of the sphere
which form a continuous and tangent curve around a figure of eight
shape have now been defined.
The parameterized trajectory may be transformed using Euler
rotations to achieve a vertically or diagonally orientated figure of
eight maneuver shape positioned at different angles relative to the
onset wind. Rotations are applied to the position vector, r, first about
the X axis, then about Y and about Z respectively in that order as
r0
¼ E3E2E1r: (28)
The rotation angles h1, h2 and h3 about X, Y and Z respectively
are user selected and the rotation matrices are E1, E2 and E3
given byFig. 4. Spherical triangles PRS, QST and HPR.
Fig. 3. Spherical triangle HPQ.
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e26782670

5. E1 ¼
2
4
1 0 0
0 cosh1 sinh1
0 Àsinh1 cosh1
3
5;E2 ¼
2
4
cosh2 0 Àsinh2
0 1 0
sinh2 0 cosh2
3
5andE3
¼
2
4
cosh3 sinh3 0
Àsinh3 cosh3 0
0 0 1
3
5 (29)
f
=q
is then determined numerically in place of Eqs. (7) and (13).
4. Determination of kite forces, kite velocity and time
averaged quantities
The remainder of this section essentially follows the develop-
ment of Wellicome [2]. It is intended here to provide the necessary
background to Section 5.
The onset velocity can be expressed as
U ¼ Uu ¼ VAv À Rr
(30)
The aerodynamic force magnitude is given by
F ¼ L sec 3 ¼
1
2
raAKU2
CLsec 3 ¼ mU2
(31)
The Lift force magnitude is given by
L ¼
1
2
raAKU2
CL ¼ mU2
cos 3 (32)
The drag force magnitude is given by
D ¼
1
2
raAKU2
CD ¼ mU2
sin 3 (33)
By vector addition
F ¼ Fr ¼ Ll þ Du (34)
where l is a unit vector perpendicular to u so that lu ¼ 0.
Substituting for FL and D,
mU2
r ¼ mU2
cos 3l þ mU2
sin 3u (35)
or
r ¼ l cos 3 þ u sin 3 (36)
Taking the scalar product with onset unit direction u,
r u ¼ l u cos e þ u u sin 3
rr u ¼ sin 3
(37)
By taking scalar products with the position unit vector, Eq. (30)
becomes:
Uu r ¼ U sin 3 ¼ Vv$r À R_r r (38)
Since kite motion is confined to the surface of a sphere, its
velocity is tangential to the spheres’ surface such that ðr$r
¼ 0Þ. It
follows that
U ¼
Vv$r
sin 3
(39)
A Cartesian right handed system is chosen with the wind parallel to
the X axis. The position angles of the vectors R and U are as shown
in Fig. 5. The position angles of U are b and g. The unit vectors v, r
and u are expressed in terms of their position angles as
v ¼ f1; 0; 0g (40)
r ¼ fcos q cos f; cos q sin f; sin qg (41)
u ¼ fcos g cos b; cos g sin b; sin gg (42)
r
is then obtained by differentiation of r as
r
¼
n
À cos q sin ff
À sin q cos fq
; cos q cos ff
À sin q sin f_q; cos q_q
o
(43)
Substituting the expression for r
into 15 yields
Ufcos g cos b; cos g sin b; sin gg
¼
È
V þ R cos q sin ff
þ R sin q cos f q
; R sin q sin fq
À R cos q cos ff
; ÀR cos q q
É
(44)
Since r v ¼ cos q cos f, Eq. (39) can be expressed as
U ¼ VA
cos q cos f
sin 3
(45)
which is the same as Eq. (3), thus the onset velocity of air onto the
kite can be calculated according to any given kite position in terms
of azimuth and elevation, such as those positions which may be
defined using the parameterization of kite trajectory shapes
developed in Section 3 or alternatively, by using experimentally
recorded trajectories.
In real conditions, the wind speed can be expected to increase
with altitude due to viscous effects that slow the air in the Earth’s
atmospheric boundary layer. Calculation of Eq. (45) can therefore
be improved by employing [8]
VT ¼ VTref
Z
Zref
!n
(46)
to account for the difference in altitude relative to the wind refer-
ence height. The value of the exponent n may be expected to vary
depending on atmospheric conditions, however a value of 1/7 is
suitable for typical conditions at sea [8]. The lift force is calculated
using
L ¼
1
2
rAKU2
CL: (47)
The drag force is calculated using
Fig. 5. Flight envelope showing position angles for the kite and the onset velocity, U.
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e2678 2671

6. D ¼
1
2
rAKU2
CD: (48)
The total aerodynamic kite force, assumed here to be equal to
the line tension is obtained as
F ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L2 þ D2
p
: (49)
or
F ¼
1
2
rAU2
CF (50)
The established onset velocity in Eq. (45) does not require or allow
the calculation of the kite velocity. This is obtained through the
implementation of the following formulae, Eqs. (51)e(58). Then, it
becomes possible to integrate the kite force around the maneuver
in order to determine time averaged force, magnitude and direc-
tion, using Eqs. (59)e(63).
The X, Yand Z components of Eq. (44) can be written separately as
X : U cos g cos b ¼ V þ cos q sin f Rf
þ sin q cos f Rq
(51)
Y : U cos g sin b ¼ Àcos q cos fRf
þ sin q sin fRq
(52)
Z : U sin g ¼ Àcos qR_q (53)
Combining Eqs. (53) with (45) yields an expression for R_q, for the
velocity of the kite in the q direction.
rRq
¼ ÀV
cos f sin g
sin 3
(54)
Substituting Eqs. (45) and (54) in Eq. (52) yields an expression
for R _f, for the velocity of the kite in the f direction,
Rf
¼ ÀV
tan q sin f sin g þ cos g sin b
sin 3
!
: (55)
Eqs. (45) and (54) can be combined to yield an expression for the
direction of the kite from kite flight dynamic considerations,
f
q
¼
tan q sin fsin g þ cos gsin b
cos fsin g
(56)
Eq. (51) combined with Eqs. (45), (54) and (55) and after manipu-
lation becomes
sin 3 ¼ sin q sin g þ cos q cos g cosðf À bÞ; (57)
or
b ¼ f Æ cosÀ1 sin 3 À sin q sin g
cos q cos g
!
(58)
For a stationary kite, g ¼ b ¼ 0, so that substitution of these values
into Eq. (58) enables determination of azimuth and elevation values
along the line of static equilibrium that limits the motion of the kite.
The time taken for the kite to maneuver from a point qA to qB can
be obtained from
T ¼
ZqB
qA
dt
dq
dq ¼
ZqB
qA
1
q
dq (59)
and the time averaged aerodynamic force F is given by
F ¼
1
T
Z
Fdt ¼
1
T
ZqB
qA
F
dt
dq
dq ¼
1
T
ZqB
qA
F
q
dq (60)
The angle of elevation for the time averaged force F is deter-
mined as
q ¼ tanÀ1
0
B
@
Fy
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F2
x þ F2
y
r
1
C
A (61)
The azimuth angle for the time averaged force is determined as
f ¼ tanÀ1
Fy
Fx
(62)
The magnitude of force acting in the horizontal plane for useful
ship propulsion is thus obtained as
Fxy ¼ F cos q (63)
5. Implementation and results
Using the procedure outlined in Section 3, a horizontally
orientated figure of eight maneuver was defined as the default
maneuver shape (q1 ¼ À25, f1 ¼7, a1 ¼8, q2 ¼ 25, f2 ¼ 7, a2 ¼ 8).
This was manipulated using Euler rotation angles h1, h2 and h3 in Eq.
(29) to produce vertically orientated trajectories with 15 different
mean wind angles across the flight envelope. The Euler angles for
transformations are shown in Table 1 and the resulting maneuver
shapes are shown graphically in Fig. 6.
The theory of Sections 3 and 4 was implemented using Matlab
[9] to obtain performance results. The flow diagram in Fig. 7 shows
the order in which calculations were done.
The kite performance input parameters were previously
measured for an experimental test kite (Model: Flexifoil Blade III
3 m2
) similar to the type of kite that is typically used for ship
propulsion. These data are shown in Table 2 from Dadd et al. [7]. For
this purely theoretical study, the kite area used is 300 m2
and
default line length used is 300 m. These values are in keeping with
typical kite sizes and line lengths that are presently used on
commercially available ship kite propulsion systems.
During the implementation, it was found that for some input
sets, more than one solution was feasible for the parameters b and
g. The correct solution selected was that which ensured continuous
Table 1
Euler rotation angles for vertical trajectory manipulation.
Trajectory h1 h2 h3
1 105 35 67
2 102 35 60
3 100 35 50
4 98 35 40
5 96 35 30
6 94 35 20
7 92 35 10
8 90 35 0
9 88 35 À10
10 86 35 À20
11 84 35 À30
12 82 35 À40
13 80 35 À50
14 78 35 À60
15 75 35 À67
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e26782672

7. results throughout the maneuver trajectory and satisfied the
requirement that the proper sense of the force vector was achieved.
Selected results arising for trajectory number 8 (highlighted in
Fig. 6) are shown in are in Fig. 8. These show the azimuth and
elevation angles that define kite position as an input, the kite
velocity together with the air onset velocity and the aerodynamic
force generated. These results show how the kite trajectory shape is
related to force output and provides a basis with which shapes can
be tailored to attain the desired output. This may be useful for
producing a control algorithm to regulate the force output. In
Fig. 8B it can be noted that the kite velocity closely follows the onset
velocity. This shows that the majority of the onset velocity is due to
the motion of the kite itself and explains why large force amplifi-
cation arises as a result of traversing the kite across the sky. Fig. 8C
shows that the line tension reduces significantly from 2 Â 105
to
0.3 Â 105
N as it is elevated during the maneuver. This is expected
since the kite is moved away from its location of maximum force
output which occurs directly downwind.
For predicting ship performance, it is useful to provide a single
time averaged force vector for a particular maneuver shape. The
time averaged aerodynamic forces were obtained using Eqs. (59)
and (60) for each of the trajectories in Table 1. To demonstrate
the effect of the wind increase with altitude due to the Earths
atmospheric boundary layer, kite performance was investigated
first using a wind gradient according to Eq. (46), and secondly with
no velocity gradient (see Fig. 9). The two cases have equal wind
speed at 10 m altitude (6.18 msÀ1
).
Fig.10A, shows the time averaged force for trajectories in Table 1,
not including the effect of the wind gradient, whilst Fig. 10B shows
the time averaged force in the presence of a wind gradient
(6.18 msÀ1
at Zref ¼ 10 m). The difference between the two
demonstrates the improved performance that arises as a result of
increasing wind speed with altitude. This is a significant added
benefit which is not realized by other wind propulsion systems,
such as conventional sails.
Fig. 10C shows the aerodynamic force which is realized
instantaneously as the kite is traversed along a straight path from
left to right at constant elevation, q ¼ 15.0 (inc. Wind gradient).
This result can be viewed alternatively as the force realised from
a figure of eight trajectory centred at q ¼ 15 with infinitely long
lines, such that the full extent of the trajectory may be considered
to remain at a single discreet value of azimuth and elevation. By
comparing Fig. 10B with Fig. 10C it is possible to see the extent by
which the propulsive drive force is reduced by traversing the kite
around the trajectories of Fig. 6. Fig. 10D shows the time averaged
aerodynamic forces from horizontally orientated trajectories that
have resulted from the optimization study in Section 6. These are
the maneuver shapes which are recommended for practical ship
propulsion.
6. Optimization and sensitivity study
Having defined a generic kite force polar diagram using a time
averaged result in both an even wind and in a wind gradient,
investigations have been undertaken to establish factors which
influence optimal trajectories. These factors include
1. The ideal elevation for maximizing line tension,
2. The ideal elevation to maximize the useful propulsive drive
force,
3. Kite aspect ratio,
Fig. 6. Vertical trajectory parameterization.
Fig. 7. Flow diagram for implementation.
Table 2
statically measured performance characteristics for the test kite (Model: Flexifoil
Blade III 3 m).
3 L=D CF CL CD
N samples 21 21 21 21 21
Mean 9.55 6.07 0.786 0.776 0.128
Standard deviation 1.48 0.88 0.115 0.116 0.012
95% Confidence limit 0.63 0.38 0.049 0.050 0.005
99% Confidence limit 0.83 0.50 0.065 0.065 0.006
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e2678 2673

8. 4. Angle of attack,
5. Maneuver pole separation and Pole circle size.
6.1. Elevation for maximizing line tension
The wind speed increases with altitude in the Earth’s natural
boundary layer. Hence the onset velocity in Eq. (45) and the
resulting lift through Eq. (47) is also increased. However, the
elevation angle in Eq. (45) reduces the onset velocity due to its
influence on kite motions. There is an optimum elevation at which
the line tension is maximized. In order to find this Eqs. (45) and (46)
are combined with Z ¼ R sin q to obtain onset velocity (U) as
a function of its kite position angles,
U ¼ VTref
R sin q
Zref
!n
cos q cos f
sin 3
: (64)
The maximum onset velocity (and thus lift) with respect to
elevation is the solution to DU/Dq ¼ 0, subject to D2
U=Dq2
0,
hence
n
R sin q
Zref
!nÀ1
R cos2q
Zref
À sin q
R sin q
Zref
!n
¼ 0 (65)
Taking n ¼ 1/7 for typical conditions at sea [8], Zref ¼ 10 m,
R ¼ 300 m and 3 ¼ 9.55, the solution to Eq. (65) is numerically
obtained as q ¼ 20.7. Therefore in the presence of a wind gradient
caused by the Earths boundary layer, the onset velocity and hence
line tension can be maximized by maintaining a trajectory that is
Fig. 8. Parameters arising through trajectory number 8. (A) Position angles azimuth and
elevation; (B) kite onset flow velocity and actual kite velocity; (C) aerodynamic force.
Fig. 9. Wind profiles.
Fig. 10. Horizontal component of kite force, FAxy
[N/104
] for different cases. (A) Time
averaged vertical trajectories, wind speed ¼ 6.18 msÀ1
, wind gradient off. (B) Time
averaged vertical trajectories, wind speed ¼ 6.18 msÀ1
at Zref ¼ 10 m, wind gradient on.
(C) Instantaneous kite force along straight horizontal trajectory q ¼ 15, wind
speed ¼ 6.18 msÀ1
at Zref ¼ 10 m, wind gradient on. (D) Time averaged horizontal
trajectories, wind speed 6.18 msÀ1
at Zref ¼ 10 m, wind gradient on.
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e26782674

9. centred closely to this value of elevation. This result is useful where
the absolute value of the line tension magnitude is important, such
as for producing electricity using kite power.
6.2. Elevation for maximizing propulsive drive force
For ship propulsion, only the horizontal component of the line
tension leads to useful propulsive force. The horizontal component
reduces with the cosine of elevation, so that the optimum elevation
is obtained by maximizing the value of U cos q.
This is given by
U cos q ¼ VTref
R sin q
Zref
!n
cos2q cos f
sin 3
(66)
Setting D(U cos q)/Dq ¼ 0, subject to D2
U cos q=Dq2
0 to find
the maximum gives,
n
R sin q
Zref
!nÀ1
R
Zref
!
$cos2
q À 2 sin q
R sin q
Zref
!n
¼ 0 (67)
When solved numerically, Eq. (67) yields the maximum horizontal
component of drive force where q ¼ 15.0. Hence for ship pro-
pulsion in the presence of a wind gradient, an optimal trajectory
should centre closely to this elevation. In this respect, horizontally
orientated figure of eight trajectories are preferential to vertical
ones for downwind trajectories, since elevation angles during the
maneuver remain closer to the optimum value. However, for
courses sailed close to the wind a vertically orientated trajectory
must be used in order to maintain a close winded time averaged
force, such as for trajectories 1 and 15 in Table 1.
Trajectories which reflect this optimization have been investi-
gated using the default parameters (q1 ¼ À25, f1 ¼7, a1 ¼8, q2 ¼ 25,
f2 ¼ 7, a2 ¼ 8). These have been transformed using Eq. (28) and
Table 3. The resulting trajectory shapes are shown in Fig. 11. The
time averaged horizontal force output for each trajectory is shown
in Fig. 10D. By comparing Fig. 10B with Fig. 10D, horizontally
orientated maneuvers centred proximally to the optimum eleva-
tion are seen to make a significant improvement to the drive force
compared to vertical ones.
6.3. Kite aspect ratio
To investigate the effects of kite aspect ratio on aerodynamic
force, the kite drag angle is expressed as a function of the lift to drag
ratio using
3 ¼ tanÀ1
1
ðL=DÞ
(68)
Eqs. (45) and (50) are combined to yield an expression that relates
the drag angle to aerodynamic force as
FA ¼
1
2
rA
VAcos q cos f
sin 3
2
CF (69)
The drag coefficient of a kite with a dissimilar aspect ratio to that
used in the present study can be calculated using
C0
D ¼ CD þ
C2
L
p
1
AR0 À
1
AR
; (70)
where C0
D corresponds tothe drag coefficientof awingof a newaspect
ratio AR0 [10]. It is assumed here that the kite is set so that the lift
coefficient remains unchanged by modified aspect ratio. The lift to
drag ratio of a wing with modified aspect ratio is therefore given by
L
D
¼
CL
CD þ
C2
L
p
1
AR0 À
1
AR
: (71)
To find the sensitivity of the kite system to aspect ratio, the
derivative dFA=dðARÞ is sought. This is given by
dFA
dðARÞ
¼
dFA
d3
$
d3
dðL=DÞ
$
dðL=DÞ
dðARÞ
; (72)
for which Eqs. (68)e(71) are differentiated to obtain
dFA
d3
¼ ÀrACFðVAcos q cos fÞ
2cos 3
sin3
3
; (73)
d3
dðL=DÞ
¼ À
1
ðL=DÞ2
þ1
; (74)
and
dðL=DÞ
dðARÞ
¼
C3
L
pAR2
h
CD þ
C2
L
p
1
AR0 À
1
AR
#2
(75)
where 3 is in radians. Using r ¼ 1.19, A ¼ 300, VA ¼ 6.18, q ¼ 15,
f ¼ 0, CL ¼ 0.776, CD ¼ 0.128, CF ¼ 0.786 and AR ¼ 4.86, Table 4 was
formulated using Eq. (68)e(75), for a range of values of aspect ratio
modified from the experimentally measured performance data in
Table 2.
Table 3
Euler rotation angles for horizontal trajectory manipulation.
Trajectory h1 h2 h3
1 0 15 À48
2 0 15 À40
3 0 15 À32
4 0 15 À24
5 0 15 À16
6 0 15 À8
7 0 15 0
8 0 15 8
9 0 15 16
10 0 15 24
11 0 15 32
12 0 15 40
13 0 15 48
Fig. 11. Horizontal trajectory parameterization.
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e2678 2675

10. For the example input set, Table 4 shows that increasing aspect
ratio from 4 to 5 for example, results in a dramatically improved
instantaneousaerodynamicforce.19.21 Â104
N is achieved insteadof
16.67 Â 104
N yielding 15% more aerodynamic force per unit increase
in aspect ratio. There are other factors that require consideration in
selectionofkiteaspect ratio,suchashandling characteristics,stability
and response ingusts.Anecdotal evidence[11]suggeststhat theseare
factors which would tend to favour lower aspect ratios.
6.4. Kite angle of attack
The lift coefficient at which the kite operates has a significant
influence on aerodynamic force; firstly through its direct influence
on the lift generated, but also through a secondary effect due to the
change in the operational L/D that accompanies a change in angle of
attack. The optimum angle of attack for maximizing aerodynamic
force may be determined by considering Prandtl lifting line theory to
establish a functional relationship that can be assessed analytically
when combined with the zero mass theory. If an increase in lift
coefficientoccurstogetherwith anincrease in liftto drag,thepositive
effect of each will be compounded. If an increase in lift coefficient
accompanies a reduction in lift to drag, the effect will be negated. The
optimum condition for maximized propulsive force occurs when
dFA
da
¼
dFA
dCL
$
dCL
da
þ
dFA
dðL=DÞ
$
dðL=DÞ
da
¼ 0: (76)
From Eqs. (50), (45) and (68)
dFA
dCL
¼
1
2
rAU2
¼
1
2
rA
VA
cos q cos f
sin 3
2
; (77)
and through combining (50), (68) and (71),
dFA
dðL=DÞ
¼
À rACFðVAcos q cos fÞ
2cos 3
sin3
3
'
$
(
À
1
ðL=DÞ2
þ1
)
:
(78)
The 3D lift coefficient from Prandtl lifting line theory is
CLa
¼ CL0
þ 2p
AR
AR þ 2
a: (79)
The induced drag coefficient from Prandtl lifting line theory is
CD ¼ CD0
þ
C2
L
pARe
: (80)
From Eq. (79)
dCL
da
¼ 2p
AR
AR þ 2
: (81)
Combining Eqs. (79) and (80) gives
L=D ¼
CL0
þ 2p
À AR
AR þ 2
Á
a
CD0
þ
À
CL0
þ 2p
À AR
AR þ 2
Á
a
Á2
pARe
(82)
Differentiating Eq. (82) gives
Table 4
Effect of aspect ratio on aerodynamic force.
AR’ 3 [
] L/D FA [104
N] d(FA)/d(AR) [104
N]
1 19.86 2.77 4.34 5.25
2 13.37 4.21 9.36 4.60
3 11.11 5.09 13.46 3.62
4 9.97 5.69 16.67 2.84
5 9.29 6.12 19.21 2.26
6 8.83 6.44 21.24 1.83
7 8.50 6.69 22.91 1.51
8 8.25 6.90 24.29 1.27
9 8.06 7.06 25.46 1.08
10 7.90 7.20 26.46 0.92
dðL=DÞ
da
¼
0
@CD0
þ
À
CL0
þ 2p
À AR
AR þ 2
Á
a
Á2
pARe
1
A$2p
AR
AR þ 2
À
CL0
þ 2p
AR
AR þ 2
a
$
0
@
2CL0
$2p
À AR
AR þ 2
Á
pARe
þ
8p2
À AR
AR þ 2
Á2
a
pARe
1
A
0
@CD0
þ
À
CL0
þ 2p
À AR
AR þ 2
Á
a
Á2
pARe
1
A
2
(83)
Fig. 12. Trajectories shapes with different pole circle sizes.
Fig. 13. Trajectory shapes with different pole circle separation.
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e26782676

11. Using AR ¼ 4.86, CD0
¼ 0:05, CL0
¼ 0:337, ra ¼ 1.19, A ¼ 300,
V ¼ 6:18, q ¼ 15, f ¼ 0, and e ¼ 0.9 Eq. (76) was found yield
a ¼ 15.28. This is the theoretical optimum as determined by
Prandtl lifting line theory, though there are other factors which
might influence the best angle of attack for practical use. In
particular, the way that the kite recovers from off-design flight
conditions, such as stall, is likely to be significantly improved by
mounting the kite to operate at a smaller angle of attack.
6.5. Pole separation and circle size
The effects of maneuver pole separation and pole circle size
were investigated by comparing a series of different kite trajecto-
ries (See Figs. 12 and 13). Results were obtained through imple-
mentation of the flow chart in Fig. 7, including the effect of the wind
gradient as determined by Eq. (46). The effect of increasing pole
circle size and pole circle separation on time averaged force output
is shown in Figs. 14 and 15 respectively. These figures show that
increasing pole circle size and separation both lead to a marked
reduction of time averaged propulsive drive force since the
maneuver is moved away from its optimal position for maximized
propulsive effect. The limitation of these investigations is that no
account is taken of the potential reduction in lift coefficient asso-
ciated with canopy deformation and angle of roll during a tight
turn.
7. Experimental validation
Previously Eq. (45), for obtaining onset velocity as a function of
kite position, was validated by comparing predicted aerodynamic
force with measured kite line tension [7]. The kite forces, force
direction and wind speed had been recorded by flying a kite from
a dynamometer in natural winds. In the present paper, the
implementation of the theory has been developed to allow the kite
velocity and motions to be computationally predicted, as well as
the instantaneous force based on position. For experimental
validation of the present work, two trajectories are shown in
Fig. 16.
The theoretically parameterized trajectory is defined according
to Section 3, whilst the experimentally determined trajectory was
recorded using a purpose built kite dynamometer described by
Dadd [7]. The theoretically determined shape was selected to
closely match that of the experimental one (see Fig. 16). Small
differences are apparent since the manually flown trajectory
exhibits irregularities. The aerodynamic force was determined in
three ways (see Fig. 17). First, the zero mass theoretical aero-
dynamic force was determined using the experimentally deter-
mined position coordinates as an input. Second, the actual
aerodynamic kite force was obtained using the dynamometer.
Comparing these two results serves to validate Eq. (45), which
determines kite onset velocity as a function of position, as
described by Dadd et al [7]. The third result in Fig. 16 is also based
on Eq. (45), but uses Eqs. (51)e(59) to establish the kite velocity
and hence to theoretically determine the results to a base of time,
by integrating around the maneuver. It can be seen that the
theoretical time for the kite to progress around the maneuver
closely matches that of the experimental kite trajectory. This
serves to validate Eqs. (51)e(59). The errors arising through
determination of kite force through each of these methods is
Fig. 14. Force reduction with circle size.
Fig. 15. Force reduction with pole separation.
Fig. 16. Experimentally and theoretically defined trajectories for comparison.
Fig. 17. Comparison between theoretical predictions and experimental measurements.
G.M. Dadd et al. / Renewable Energy 36 (2011) 2667e2678 2677

12. discussed in detail by Dadd et al [7]. The most significant error for
experimentally determined kite trajectories was shown to be due
to spatial and temporal differences in wind speed across the flight
envelope which was not properly accounted for.
8. Conclusions
This paper has presented a scheme for predicting time averaged
kite forces together with a method for prameterizing figure of eight
shape kite trajectories. The theoretical model was modified to allow
a high degree of control over input maneuver paths in order to
assess the effects of maneuver orientation, shape and true wind
angle. The influence of the changing wind strength in the Earth’s
natural boundary layer is included. The scheme has been imple-
mented to show variation of onset velocity, kite velocity, onset
angles and aerodynamic force for an example maneuver. In a case
study with a 300 m2
kite, 300 m long flying lines and 6.18 msÀ1
wind speed (at 10 m), peak drive force up to 30 Â 104
N was pre-
dicted. A typical vertical figure of eight maneuver in the same
conditions was shown to produce a time averaged horizontal drive
force component 16.7 Â 104
N. A complete force polar showing the
horizontal component of aerodynamic force for different true wind
angles has thus been obtained.
The calculation of kite velocity and the propagation of the kite
around a theoretically defined maneuver have been validated by
comparing a real recorded trajectory to a simulated one. The time
scales for a theoretical and practical trajectory are shown to agree
favourably, as do the magnitudes of minimum and maximum kite
force. The increase in wind with altitude is found to double the aero-
dynamic force compared to a hypotheticalcase with nowind gradient.
Through considering the wind gradient in combination with kite
motions, the optimum mean kite elevation angle for ship pro-
pulsion during dynamic flight was found to be 15. Horizontally
orientated trajectories are shown to improve downwind perfor-
mance by a factor of 1.4 compared to vertical ones, since the
maneuver can remain closer to the optimum elevation angle for
propulsion. Vertical maneuvers are, however, preferred when
maintaining close winded performance is necessary.
By combining Prandtl lifting line theory with the zero mass
model, the derivative dFA=dðARÞ was calculated at a variety of
different aspect ratios. For the downwind dynamic flight condition,
increasing the kite aspect ratio from 4 to 5, for example, was found
to yield a 15% increase in drive force showing that force is highly
sensitive to aspect ratio through its effect on kite L/D. Low aspect
ratios may however be favourable to kite stability characteristics.
The angle of attack at which the kite operates modifies the
aerodynamic force generated through its influence on lift and drag
coefficients, but also through its effect on L/D. Optimizing the angle
of attack with respect to both of these yields a ¼ 15.
Optimal trajectories were found to be those that are centred
most closely to the best region of the flight envelope. Increasing
maneuver pole circle size and pole separation both diminished the
time averaged propulsive drive force arising. If the potential effect
of reduced CL during tight turns is negligible, the best maneuver is
found to be a figure of eight shape with small circular ends and
small pole separation, centred closely to an elevation of 15.
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