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Airborne wind energy is an emerging technology that uses tethered unmanned aerial vehicles for harvesting wind energy at altitudes higher than conventional towered wind turbines. To make the technology competitive to other renewable energy technologies a reliable control system is required that allows autonomously operating the system throughout all phases of flight. In the present work a cascaded nonlinear control scheme for reliable pumping cycle control of a rigid wing airborne wind energy system is proposed. The high-level control strategy in the form of a state machine as well as the flight controller consisting of path-following guidance and control, attitude, and rate loop is presented along with a winch controller for tether force tracking. A mathematical model for an existing prototype will be derived, and results from a simulation study will be used to demonstrate the robustness of the proposed concept in the presence of turbulence and wind gusts.
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Cascaded Pumping Cycle Control for
Rigid Wing Airborne Wind Energy Systems
Sebastian Rappand Roland Schmehl
Delft University of Technology, Faculty of Aerospace Engineering
Espen Oland
Kitemill AS
Thomas Haas§
KU Leuven, Department of Mechanical Engineering
Airborne wind energy is an emerging technology that uses tethered unmanned aerial
vehicles for harvesting wind energy at altitudes higher than conventional towered wind turbines.
To make the technology competitive to other renewable energy technologies a reliable control
system is required that allows autonomously operating the system throughout all phases of
flight. In the present work a cascaded nonlinear control scheme for reliable pumping-cycle
control of a rigid-wing airborne wind energy system is proposed. The high level control strategy
in form of a state machine as well as the flight controller consisting of path-following guidance
and control, attitude and rate loop is presented along with a winch controller for tether force
tracking. A mathematical model for an existing prototype will be derived and results from a
simulation study will be used to demonstrate the robustness of the proposed concept in presence
of turbulence and wind gusts.
I. Introduction
Airborne wind energy
(AWE) is an emerging branch within the sustainable energy systems portfolio that aims to
exploit wind energy resources at altitudes higher than conventional towered wind turbines by means of kites and
tethered aircraft. In general, AWE systems can be subdivided into two main categories. On the one hand, AWE systems
with on-board generators can fly crosswind patterns with constant tether length. The kinetic energy of the relative flow
is in this case directly converted into electrical power and the electricity is transmitted via a conductive tether to the
ground. On the other hand, AWE systems with a ground-based generator operate in a so-called pumping-cycle mode and
use the aerodynamic force of the kite or aircraft to uncoil the tether from a drum, which turns a generator that converts
PhD Researcher, Delft University of Technology, Kluyverweg 1, 2629HS, Delft, The Netherlands, s.rapp@tudelft.nl. Member AIAA.
Associate Professor, Delft University of Technology, Kluyverweg 1, 2629HS, Delft, The Netherlands, r.schmehl@tudelft.nl.
Control System Manager, Kitemill AS, Miltzowsgate 2, 5700 Voss, Norway, eo@kitemill.no
§PhD Researcher, KU Leuven, Celestijnenlaan 300, 3001 Leuven, Belgium, thomas.haas@kuleuven.be
the mechanical torque into electrical power on the ground. When the maximum tether length is reached, the aircraft will
fly back towards the ground station, while the tether is reeled in. Since the generator acts as a motor during this phase a
fraction of the produced power is consumed. Once the minimum tether length is reached, the cycle starts all over again
[
1
,
2
]. For a more detailed comparison of the different concepts it is referred to [
3
]. In the present work the focus lies on
the controller development for AWE systems operated in pumping cycle mode, although the controller can partially also
be implemented for AWE systems which fly on a constant tether length.
Historically, most researchers in this field started to study the potential of flexible kite power systems, which is
also reflected by the fact that most of the published papers are dedicated to the design of control systems applicable
to flexible wing kite power systems [
4
8
]. However, due to better scaleability and efficiency the trend goes towards
rigid wing AWE systems reflected by the fact that almost all companies in the field operate rigid wing prototypes.
Nevertheless, available publications on rigid wing kite control are rare. Although the reliability of the control system
plays a paramount role that decides upon the success of this new technology most of the available literature focuses
on flight path optimization instead of the development of more robust control solutions. One recent control approach
that is not dedicated to flight path optimization is presented in [
9
]. In the paper, the authors focus on take-off and
landing control, including a transition to a loiter-like figure of eight flight pattern on a constant tether length using linear
controllers.
To the best authors knowledge no modular control architecture for the full operational envelope for rigid wing AWE
systems has been published yet. The term modularity is used to clearly distinguish the control approach from more
integral approaches, usually based on nonlinear model predictive control such as in [
10
]. The present work tries to fill
this gap where a modular control architecture similar to the one presented in [
9
], but eventually applicable to the whole
range of operational modes including take-off, transition, pumping cycle mode and landing is presented. Moreover,
instead of using linear control techniques a model-based nonlinear flight controller is developed that eventually increases
the operational envelope and the performance of the AWE system in situations where linear control techniques might
fail. In the future, the presented control approach could be augmented with adaptive control techniques to increase the
robustness towards failures or unforseen environmental conditions. The modularity of the control architecture aims to
achieve a high degree of reusability especially of the outer-loop module, such that it can be implemented conveniently
on different platforms. The modules have defined interfaces that allow to exchange, modify and test different parts
of the entire controller conveniently. This enables operators with existing prototypes to only use specific modules
without the need to re-implement the entire control system. Especially the guidance module might be of interest for
AWE companies, since it is entirely model independent, and can be implemented for AWE systems either operated in
pumping-cycle mode or on a fixed tether length with airborne generators. Furthermore, applying systematically the
concept of pseudo control hedging [
11
] a flight envelope protection system is implemented ensuring that no unfeasible
commands are passed to the next loop. Constraining states is of particular importance in this application since the
2
aircraft is usually operated at near stall conditions while following a three dimensional curved path which requires
to constrain commands from the outer loops in a systematic manner. Such an envelope protection for airborne wind
energy systems has not been presented yet apart from model predictive control approaches where constraints are directly
embedded in the optimal control problem formulation [10].
The performance of the control system is demonstrated by means of a simulation study. To create a realistic simulation
framework a detailed aerodynamic analysis using computational fluid dynamics (CFD) and XFLR5 calculations of the
5 kW
prototype of Kitemill AS have been carried out. The robustness of the control system towards wind gusts and
atmospheric turbulence is assessed using three-dimensional transient wind field data generated by large-eddy simulations
(LES) of a pressure-driven boundary layer.
The contributions of the present paper to the research community can be summarized as follows. First, an extension
of the path-following controller which has been previously developed by one of the authors for flexible kite power
systems is presented such that it can also be implemented for rigid-wing AWE systems. Furthermore, we present an
intuitive way to calculate the required tangential plane course rate according to the three-dimensional path curvature to
keep the aircraft on the path. Moreover, an approach for radial direction control using tether force tracking is presented
and it will be demonstrated that this approach can be used at the same time for gust load alleviation. For a complete
pumping cycle control we additionally propose a retraction phase controller which has not been presented for rigid wing
AWE systems in the literature yet. Finally, we present a detailed description of the Kitemill
5 kW
prototype, which can
be used in the future as a reference model for other researchers in this field.
The paper is structured as follows. In section II the simulation models for aircraft, tether, ground station as well as
the wind field are presented. In section III a detailed derivation of the different controllers is presented. Simulation
results are presented in section IV followed by a conclusion in section V.
II. Reference Frames and Simulation Models
A. Reference Frames
Fig. 1displays the wind frame
W
where the
xW
axis is pointing in downwind direction, the
zW
axis is the local earth
surface normal vector, and the
yW
forms a right-hand coordinate system together with
xW
and
zW
. The origin of the
W
frame is at the ground station. Note, this definition of the wind frame differs from the conventional definition found in
the aerospace literature where the wind frame is a local body fixed frame [
12
, p. 76]. Furthermore, Fig. 2displays the
tangential plane frame
τ
which will be used as a reference frame for the guidance loop. The
zτ
axis is pointing towards
the origin of the wind frame
W
, the
xτ
axis points towards the zenith position which is located above the ground station.
Note that the
τ
-frame is defined equivalently to the North-East-Down frame (
O
) (see [
13
, p. 12]) for a small earth with
radius one and center at the origin of the
W
frame. The position of the aircraft with respect to the
W
frame will be given
3
either in Cartesian coordinates
xW,yW
and
zW
or in spherical coordinates using longitude
λ
and latitude
φ
as well as the
Euclidean distance of the aircraft to the origin of
W
. The body-fixed frame
B
[
14
, p. 57], the kinematic frame
K
[
14
, p.
ex,B
ey,Bez,Bvk
φ
ey
ex
ez
xwλ
O
Small Earth
r=1
yw
zw
Zenith
vw
vk
vk,r
h
ex,B
τ
Fig. 1 Visualization of wind frame W, body-fixed frame Band tangential plane frame τ.
ψτ
ex
ey
ex,B
ey,B
ex
vk
χτey
ex,B
ey,B
Fig. 2 Definition of the tangential plane heading Ψτand tangential plane course χτ.
58] as well as the aerodynamic frame A[14, p. 61] are defined according to aerospace convention.
B. Tethered Aircraft Model
The control strategy in this work will be tested within a simulation environment. The aircraft simulation model
represents the
5 kW
prototype which has been developed by Kitemill AS. Relevant aircraft parameters are summarized in
Table 1and a visualization of the aircraft is shown in Fig. 3. The actuators of the aircraft are modeled as second order
systems with natural frequency
ω0
and relative damping
ζ
, including limits on deflections and deflection rates. The
Table 1 Aircraft Parameters.
Parameters Values Units
Aircraft mass mk4.778 kg
Inertia Jxx,yy,zz,xz 1.74,0.28,1.83,-0.02 kg m2
Wing area SW0.76 m2
Wingspan b3.7 m
Mean chord ¯c0.22 m
4
numerical values are summarized in Table 2. The aircraft is modeled as a standard six degrees of freedom rigid body
Fig. 3 5 kW prototype of Kitemill AS with vertical takeoff- and landing capabilities.
with an additional term in the translational equations of motion representing the tether force. No additional term in the
rotational dynamics appears since it is assumed that the tether is attached to the center of gravity of the aircraft. A
detailed derivation of the governing equations of motion can be found for instance in [
12
]. The translational dynamics
in the bodyfixed frame Bare defined as
(Û
vk)B=
Û
uk
Û
vk
Û
wk
B
=ωOBB×(vk)B
+1
mk(Fa)B+FgB+(Ft)B
(1)
where
(vk)BR3x1
is the kinematic aircraft velocity in the
B
frame with components
uk,vk
and
wk
,
mk
is the mass of the
aircraft,
ωOBBR3x1
is the angular velocity vector between the
B
and
O
frame containing the roll rate
p
, pitch rate
q
as well as yaw rate
r
,
(Fa)BR3x1
is the aerodynamic force,
FgBR3x1
is the gravity force and
(Ft)BR3x1
is the
Table 2 Actuator Parameters.
Parameters Values Units
Natural frequency ω035 rad s1
Relative damping ζ1 -
Max./Min. aileron deflection ±15 °
Max./Min. elevator deflection ±15 °
Max./Min. rudder deflection ±20 °
Rate limits (all actuators) ±300 °s1
5
-10 0 10 20 30 40
α()
0
0.2
0.4
0.6
0.8
1
1.2
CD(α) (−)
Fig. 4 Drag coefficient as a function of angle of attack.
tether force. All forces are defined with respect to the center of gravity. The aerodynamic force is modeled according to
(Fa)B=1
2ρV2
aSWMBA
CD
CY
CL
A
(2)
where
ρ=1.225 kg m3
is the air density and
MBA
is the transformation matrix from the aerodynamic frame
A
to the
bodyfixed frame
B
[
12
, p. 77]. The coefficients
CD,CY
, and
CL
are nonlinear functions of the aircraft states and surface
deflections. For the purpose of this paper CFD and XFLR5 was used to create lookup tables that capture the main
dependencies of the coefficients on states and surface deflections. The modeled dependencies on angle of attack, sidelip
angle and the control surface deflections are displayed in Fig. 4-6. Note, the contributions of the surface deflections to
the drag coefficient where negligible and are therefore not displayed. Additionally, damping coefficients (see Table 3)
are added which in total yields
CD=CD(α)
CY=CY(β,δr)+CYp pb
2Va
+CYr rb
2Va
CL=CL(α, δe)+CLq q¯c
2Va
(3)
where CYp,CYr and CLq are defined in Table 3.
6
-16 -12 -8 -4 0 4 8 12 16
β()
-0.08
-0.04
0
0.04
0.08
CY(β, δr) (−)
δr=20
δr=0
δr=20
Fig. 5 Side force coefficient as a function of sideslip angle and and rudder deflection.
-10 0 10 20 30 40
α()
0
0.5
1
1.5
CL(α, δe) (−)
δe=8
δe=0
δe=8
Fig. 6 Lift coefficient as a function of angle of attack and elevator deflection.
The rotational dynamics are defined as
Û
ωOBB
=
Û
p
Û
q
Û
r
B
=J1ωOBB×JωOB B
+(Ma)B
(4)
Table 3 Rate dependencies of the force coefficients.
Coefficients Values
CYp -0.133
CYr 0.172
CLq 7.267
7
-10 0 10
β()
-0.02
-0.01
0
0.01
0.02
Cl(β, δr) (−)
δr=20
δr=0
δr=20
Fig. 7 Rollmoment coefficient as a function of sideslip angle and rudder deflection.
0 10 20 30
α()
-0.05
0
0.05
Cl(α, δa) (−)
δa=15
δa=0
δa=15
Fig. 8 Roll moment coefficient as a function of angle of attack and aileron deflection.
where
JR3x3
is the inertia tensor, and
(Ma)BR3x1
is the resulting aerodynamic moment around the center of gravity
of the aircraft. Similar to the aerodynamic force the aerodynamic moment is defined using moment coefficients:
(Ma)B=1
2ρV2
aSW
bCl
¯cCm
bCn
(5)
The relevant dependencies of the moment coefficients on states and surface deflections are depicted in Fig. 7-11. The
damping terms are summarized in Table 4, which in total yields for the moment coefficients
8
0 10 20 30 40
α()
-0.8
-0.4
0
0.4
Cm(α, δe) (−)
δe=8
δe=0
δe=8
Fig. 9 Pitch moment coefficient as a function of angle of attack and elevator deflection.
-15 -10 -5 0 5 10 15
β()
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Cn(β, δr) (−)
δr=20
δr=0
δr=20
Fig. 10 Yaw moment coefficient as a function of sideslip angle and rudder deflection.
Cl=Cl(α, δa)+Cl(β, δr)+Clp pb
2Va
+Clr rb
2Va
Cm=Cm(α, δe)+Cmq q¯c
2Va
Cn=Cn(α, δa)+Cn(β, δr)+Cnp pb
2Va
+Cnr rb
2Va
(6)
Table 4 Damping coefficients.
Coefficients Values
Clp -0.6450
Clr 0.2190
Cmq -16.3740
Cnp -0.1310
Cnr -0.0335
9
0 10 20 30
α()
-0.01
-0.005
0
0.005
0.01
Cn(α, δa) (−)
δa=15
δa=0
δa=15
Fig. 11 Yaw moment coefficient as a function of angle of attack and aileron deflection.
The attitude is parameterized using quaternions, hence the equation for the attitude propagation is given by
Û
q=
Û
q1
Û
q2
Û
q3
Û
q4
=
q2q3q4q1
q1q4q3q2
q4q1q2q3
q3q2q1q4
p
q
r
2kκ
(7)
The quaternion attitude propagation equation Eq. (7) is implemented with gradient feedback as described in [
15
, p.64]
with
κ=
1
q2
1q2
2q2
3q2
4
otherwise numerical inaccuracies can lead to a violation of the unity norm condition of the
quaternion vector. The position of the aircraft’s center of gravity
pGO
in the
O
frame will be propagated according to
Û
pGO
=
Û
pG
x
Û
pG
y
Û
pG
z
O
=MOB
uk
vk
wk
B
(8)
where MOB is the transformation matrix from the Bto the Oframe (see [12, p. 12]).
The states of the tethered aircraft are the three kinematic velocity components in the bodyfixed frame
uk,vk
and
wk
, the body rates
p,q,r
, the quaternions
q1,q2,q3
and
q4
, as well as the position in the
O
frame with components
pG
x,pG
y
and
pG
z
. At the moment full state feedback is assumed, and the controller requires measurements for mean wind
direction on the ground
ξ
, position, velocity, orientation, angle of attack
α
, sideslip angle
β
, airspeed
Va
, rotational rates
as well as the total tether force
Ft
measured on the ground and at the aircraft. The reason for measuring the tether force
10
on the aircraft as well as on the ground is that due to the tether drag and weight the force measured on the ground differs
from the tether force acting on the aircraft.
C. Tether Model
The tether is modeled as a particle system where the individual particles are connected via spring-damper elements.
For each particle the point mass dynamics are formulated incorporating tether drag and tether weight. During reel-out
or reel-in the unstretched length of each spring-damper as well as the mass of each particle is adapted proportionally to
the current change in tether length. A detailed explanation of the implemented tether model can be found in a previous
work of the second author [16].
D. Ground Station
In general, the ground station consists of the generator and the winch. In this work the only relevant component for
the controller development is represented by the winch which can be modeled as a scalar first order system given by
Û
ωw=J1
w(κwωw+rwFt+Mc)(9)
where
ωW
represents the rotational speed of the winch,
rW
is the radius of the winch, which is assumed to be constant
despite the reeling-in or -out of the tether,
κW>t
is a viscous friction coefficient,
Ft
is the tether force and
Mc
is the
motor/generator torque which represents the control input. The electrical drive system of the ground station is not
modeled in this work. The utilized values for the winch are summarized in Table 5.
E. Wind Field Model
In order to test the controller in a realistic wind field, a four-dimensional velocity field is integrated into the simulation
framework. The wind field data was generated by means of large-eddy simulations of a pressure-driven boundary
layer. The computations were carried out using SPWind, a pseudo-spectral simulation code developed at KU Leuven.
Information on the specification and the implementation of the flow solver can be found in [
17
19
]. The wind field data
is available at a spatial resolution of approximately
20 m ×15 m ×7 m
in
xW
,
yW
and
zW
direction, respectively, for a
time series of several minutes and stored in form of lookup tables. During the simulation the wind velocity vector at the
Table 5 Winch Parameters.
Parameters Values Units
Winch radius rW0.1 m
Inertia JW0.08 kg m2
Viscous friction κW0.6 kg m s1
11
Guidance
Module
Traction
Guidance
Module
Retraction
Path Loop Attitute Loop Rate Loop Control
Allocation
Set Point
Generator
Traction
Set Point
Generator
Retraction
Speed Control
Flight Controller:
Winch Controller:
Γ(s)
¯γk,cχk,c
γk,c
χk,c, γk,c
Ûχk,c,Ûγk,cµa,c
αc
βc
pc
qc
rc
Lc
Mc
Nc
δa,c
δe,c
δr,c
Ft,c
τm/g,c
vr,c
τm/g,c
Fig. 12 Cascaded control structure of flight and winch control system for traction and retraction mode.
location of the aircraft is obtained through linear interpolation of the adjacent vertex velocity vectors.
III. Controller Development
A. Control Architecture and State Machine
The high level control architecture is displayed in Fig. 12. On the highest level the controller can be decomposed
into the flight and the winch control system, represented by the upper and lower cascade in Fig. 12. The task of the flight
control system is to control the tangential motion on the sphere while the radial direction is controlled by the winch.
The blocks correspond to modules that will be discussed in more detail in the following sections. In general, each block
has one input and one output signal corresponding to the set point that has to be tracked by the module as well as the
commanded set point for the next module. Blocks with two inputs are subdivided into two submodules (not displayed),
one module for the traction and one for the retraction phase. All remaining modules are the same for both traction and
retraction, although different gains and filter bandwidths are used for increased performance. Based on the current state
πi
, as defined in Table 6, the output from either the traction or retraction module is passed on to the next module. The
flight control guidance module input of the traction phase is the path parameterization
Γ(s) ∈ R3x1
with
s∈ (
0
,
2
π)
.
Within this module the required kinematic (subscript
k
) course
χk,c
and kinematic path angle
γk,c
as well as the required
course rate
Û
χk,c
and path angle rate
Û
γk,c
are calculated based on the current position. The guidance module input of the
retraction phase is the desired path angle
¯γk,c
and the output signal is the kinematic course
χk,c
and kinematic path
angle
γk,c
. Note,
¯γk,c
and
γk,c
differ from each other only in the final part of the retraction phase where the path angle
¯γk,c
is linearly increased to a fixed value before the transition back into the traction phase is triggered. This maneuver is
used to dissipate kinetic energy of the aircraft before the turn. The path loop will track the commands from the guidance
12
π0
start π1
π2q5
π3π4
ltlt,max
¬f V T OL
λλwp1
λ0
ltlt,min
kpGk lt,min
φφmax
VaVa,min
φφ0+t
Fig. 13 State-machine for the pumping cycle mode.
module and calculates attitude commands for aerodynamic (subscript
a
) bank angle
µa,c
and angle of attack
αc
. Note,
α
and
β
always refer to the aerodynamic and not kinematic angles if not indicated otherwise. The attitude loop tracks the
attitude commands and transforms them into roll-, pitch-, and yaw rate commands
pc,qc
and
rc
, respectively. Finally,
the rate loop calculates the control moments which are then distributed among the actuators in the control allocation
block, which results in an aileron command
δa
, elevator command
δe
and rudder command
δr
. The winch controller
requires only a set point generator for traction and retraction phase as well as a speed controller. During the traction
phase, a reference torque
τm/g,c
is directly calculated based on the tether force set point
Ft,c
. During the retraction phase
a fixed reeling in speed
vr,c
is commanded that will be tracked by a speed controller, which outputs a corresponding
torque command τm/g,c. In both cases, the torque commands will be tracked by the electrical drive control system.
Fig. 13 shows the state-machine that is used to switch between the different control modules. The individual states
are defined in Table 6. The modeled prototype of Kitemill AS allows vertical takeoff and landing (VTOL). A VTOL
controller including the transition into pumping cycle mode is implemented in the simulation framework, however a
detailed description of the VTOL controller is out off the scope of this paper and will be part of a future publication.
Essentially, a similar control approach for the winch and the flight controller is adapted from [
20
], where a VTOL
controller for a flexible kite system is presented. The interface to the pumping cycle mode is given by a transition into
π0
. In this work it will be assumed that the aircraft was guided in downwind direction to the operational altitude that
Table 6 State definitions.
State Description
π0Transition from take-off to aircraft mode.
π1Capture crosswind pattern.
π2Traction phase.
π3Intermediate state.
π4Transition to retraction.
π5Retraction phase.
13
fulfills the latitude condition
φ > φ0+φ0
, where the VTOL controller keeps the aircraft in a hover state (not displayed)
until
π0
is triggered.
φ0
is the mean latitude angle of the path and
0°φ010°
is a small offset. The transition
from the launching state to the crosswind flight state is initiated by fast reeling in of the tether. As soon as the airspeed
exceeds the minimum airspeed, here denoted with
Va,min
, the transition to
π1
is triggered. In this state the path-following
controller is activated and the guidance law is initialized with a first guess of the closest point on the path relative to
the current aircraft position. Flying towards the path decreases the elevation angle, which triggers the transition into
the traction phase state
π2
if it reaches a value below
φm+t
(
0°t10°
) and the winch starts reeling out the
tether. The intermediate state
π1
was added to start reeling out after the aircraft is sufficiently steered into the downwind
direction. If the tether is reeled out immediately this could lead to a drop in tether tension during the initial turn. As
long as no landing is issued by the supervisory layer (
f VT O L =
1) the kite remains in state
π2
. The transition into
π3
is triggered as soon as the specified tether length is reached. This state can be interpreted as an intermediate state
which is left as soon as the aircraft flies into the negative half plane of the wind window defined by a negative longitude
λ <
0. This triggers the transition to
π4
. The retraction phase is initiated as soon as the aircraft flies past
wp1
which is
defined as the outermost point on the path. This procedure ensures that before the reeling-in of the tether is triggered the
aircraft always has to fly downwards through the center and flies towards the ground station on the same side of the
wind window. Before the aircraft transitions back into the traction mode, one out of three conditions has to be satisfied:
Either the tether length or the Euclidean distance
kpGk
of the aircraft relatively to the ground station is below a specified
value, or the elevation angle of the aircraft exceeds a maximum value. The latter can be regarded as a safety mechanism
that prevents the aircraft from overshooting the ground station.
B. Guidance Modules
In the existing AWE literature [
6
,
7
,
21
,
22
] the kite is steered according to the tangential plane course set point
χτ,c
.
It is defined as the angle between the
ex
axis of the tangential plane frame
τ
and the x-axis of the kinematic frame
K
as depicted in Fig. 2. This strategy is mainly motivated by the fact that a direct relationship between the steering
input of a flexible kite and the tangential plane course rate can be derived [
8
,
23
] which allows to directly calculate the
steering input based on the course rate. In this work the guidance problem will be solved as well by first calculating the
desired
χτ
course set point, which will be then however transformed into a corresponding set point for the course
χk
and path angle
γk
, which specify the orientation of the
K
frame relatively to the
O
frame. This approach provides an
additional control degree of freedom to track the desired flight direction. Moreover, controlling course and path angle in
the traction phase allows to use the same medium loop control structure for the retraction phase in which the kite is not
steered on a tangential plane anymore. Furthermore, providing set points for course and path angle allows to integrate
the guidance module easier into existing autopilot architectures for conventional aircraft. Hence this approach also fits
better into the modular control philosophy proposed in this work.
14
1. Traction Phase Guidance
Separating the radial and the tangential motion of the aircraft the control objectives for the traction phase can be
stated as follows: On the one hand, the radial direction needs to be controlled by the winch such that the tether force
set point is tracked. Moreover, the radial direction controller needs to ensure that the maximum tether tension is not
exceeded to avoid tether rupture or aircraft damage. On the other hand, for the tangential motion control the aircraft
position will be projected onto the unit sphere. In that case, the flight controller needs to follow a predefined flight path
on a sphere with a constant radius of one. The path on the unit sphere is adapted proportionally to the distance of the
aircraft to the ground station such that the real path the aircraft traces has a constant shape during the reel-out phase.
Fig. 14 depicts an example flight path, including a visualization of the aircraft and the flexible tether. Note that the
depicted vectors and the aircraft model are scaled, and the physical flight path and not the path on the unit sphere that is
used for the guidance is shown for visualization purposes.
-300 -200 -100 0 100 200 300
yW(m)
0
100
200
300
zW(m)
vk
t(s)
Γ
Γ
Γ(s)
pG
Fig. 14 Reference flight path on a sphere.
Parts of the guidance module are based on a previous work of the second author [
8
] where it is used to steer a
flexible kite along a prescribed path. In this work some modifications are introduced such as a novel predictive part that
takes the instantaneous path curvature into account in order to calculate the reference course rate. Furthermore, the
interface to a rigid-wing aircraft path-following controller will be presented taking into account a generalization of the
rotational rate vector
ωτ¯
K¯
K
which describes the relative rotation between the rotated kinematic and the tangential
plane frame. Since the terminology slightly deviates from [
8
] the main steps of the derivation will be presented again in
addition to the novel extensions for completeness.
The objective of the guidance module is twofold. First, it needs to calculate the flight direction that leads to a reduction
of the distance
δ
(i.e. the cross track error) as defined by the arc length between the projected aircraft position on
the unit sphere
pG
and the path
Γ
. Second, for zero cross-track error the kinematic velocity vector projected onto the
tangential plane
vk
needs to be aligned with the path direction as defined by the tangent vector
t
. For clarification, all
relevant vectors are depicted in Fig. 14. The path is defined in spherical coordinates on the unit sphere, hence a point on
the path is fully defined by its longitude
λΓ
and latitude
φΓ
. Note, all vectors are given in the
W
reference frame, if not
15
indicated otherwise. In Cartesian coordinates the path is given as
Γ(s)=
cos λΓ(s)cos φΓ(s)
sin λΓ(s)cos φΓ(s)
sin φΓ(s)
(10)
For subsequent calculations the tangent and its derivative need to be known. The tangent can be calculated according to
t(s)=dΓ
ds
=Γ
∂λΓ
dλΓ
ds
+Γ
φΓ
dφΓ
ds (11)
and its derivative is given by
t0(s)=2Γ
∂λ2
ΓdλΓ
ds 2
+
22Γ
φΓ∂λΓ
dφΓ
ds
dλΓ
ds
+2Γ
φ2
ΓdφΓ
ds 2
+t
s
(12)
The last partial derivative is given by
t
s
=Γ
∂λΓ
d2λΓ
ds2+Γ
φΓ
d2φΓ
ds2(13)
Furthermore, the speed of the path parameter
s
is denoted with
ds/dt =Û
s
and is given by the projection of the velocity
vector onto the path tangent:
ds
dt
=Û
s=
t>vG
kW
ktk2kpGWk2
(14)
The flight path can be fully described as a planar curve using scalar functions of
s
for longitude and latitude. The flight
path in this work will be defined as a Lemniscate of Booth, given by
λΓ(s)=aBooth sin s
1+aBooth
bBooth 2cos2s
φΓ(s)=
a2
Booth
bBooth sin scos s
1+aBooth
bBooth 2cos2s
(15)
which can be derived from the equation of a hyperbolic lemniscate as defined for instance in [
24
, p.164] with
y=xa
bcos s
.
aBooth
and
bBooth
are parameters that define height and width of the curve. A detailed comparison with other curve
parameterzations is out of the scope of this paper. Note, however, that the Lemniscate of Booth offers for a large range
of width and height parameters smaller curvature peaks compared to the Lissajous figure parameterization utilized in
16
[
8
] which is why it is chosen in this work. Ultimately, the planar curve can be transformed into a three dimensional
curve using Eq. (10).
The distance between a point on the curve and the kite position can be calculated using the definition of the arc length.
δ(s)=arccos pG
·Γ(s)(16)
In order to determine the closest point (defined by s) requires to solve
dδ
ds s=s
=0(17)
where the derivative is given by
dδ
ds
=1
sin δ
dpG
·Γ(s)
ds
=pG
·t(s)
sin δ(18)
Eventually, the following root-finding problem needs to be solved:
pG
·t(s)=0(19)
the solution can be determined using for instance Newtons method. With
d
ds pG
·t(s)=pG
·dt(s)
ds (20)
The update equation for Newton’s method is then
s+=spG
·t(s)
pG
·t0(s)(21)
In the simulations, the method converged usually quickly after two to three iterations if the previous solution is selected
as a starting point.
Knowing the closest point on the curve relative to the current aircraft position enables to calculate the desired flight
direction. The vector at the current aircraft position pointing towards
Γ(s)
perpendicularly along a great circle can be
expressed as
bG=
Γ(s) − cos δpG
sin δ(22)
This can be derived simply by looking at the normal projection of Γ(s)onto pG
(see Fig. 15) given by
Γproj(s)=cos δpG
(23)
17
δ
Γ(s)
bG
pG
OW
Γproj(s)
Γ(s)
Fig. 15 A slice of the unit sphere containing a segment of the great circle that connects pG
with Γ(s).
and
Γ(s)=Γ(s) − Γproj(s)(24)
where
Γ(s)
denotes the vector of the projection direction, which is by definition perpendicular to
pG
. Normalizing
Γ(s)yields:
bG=
Γ(s) − Γproj(s)
kΓ(s) − Γproj(s)k2
=
Γ(s) − cos δpG
sin δ(25)
Equation (19) can be rewritten using Eq. (25):
Γ(s) · t(s) − sin δbG·t(s)
cos δ=0(26)
The first scalar product is zero, since Γ(s)is perpendicular to the tangent vector, which yields
tan δbG·t(s)=0(27)
If this equation is divided by
tan δ
and bearing in mind that the only relevant singularity is located at
δ=
0this yields
for δ,0
bG·t(s)=0(28)
which proves that the direction vector pointing towards the path is indeed orthogonal to the tangent at
Γ(s)
. Hence,
if the kite would fly in
bG
direction it would intercept the path perpendicularly. From a practical point of view it is
however not desired that the aircraft intercepts the path perpendicularly. Instead, it is desirable that the commanded
flight direction smoothly transitions from an orthogonal interception if the aircraft is farther away from the curve to a
tangential, hence curve aligned, flight direction. If the aircraft is on the path it is desired that the path controller tracks
the directional angle of the tangent vector on the curve. This behavior can be achieved as follows: If
δ,
0the course
18
angle
χτ, k
, which can be obtained from the tangent on the path, has to be adapted such that the distance to the curve
δ
decreases over time. In [8] the following set point definition is proposed, which is utilized in this work as well:
χτ,c=χτ, k+χτ(29)
with
χτ=arctan2(σ(ι)δ,δ0)(30)
and
ι=(t(s) × Γ(s))·pG
Γ(s)(31)
where
σ
denotes the sign of
ι
. Depending if the aircraft is on the left or right hand side of the path, as depicted in
Fig. 16, the sign of
χτ
is adapted accordingly. If the course as defined in Eq. (29) is tracked by the flight control
system, the relative distance
δ
between aircraft and path decreases over time, i.e.
Û
δ <
0with can be proven as follows.
Taking the derivative of Eq. (16) with respect to time at s=syields
Û
δ=1
1cos2δÛ
pG
·Γ(s)+pG
·Û
Γ(s)(32)
with
Û
Γ(s)=t(s)Û
s(33)
pG
·t(s)is zero, therefore,
Û
δ=1
sin δÛ
pG
·Γ(s)(34)
With Eq. (25) the dot product can be written as
Û
pG
·Γ(s)=Û
pG
·bGsin δ+Û
pG
·pG
cos δ(35)
Per definition, the second scalar product on the right hand side is zero. Inserting the result into Eq. (34) yields
Û
δ=Û
pG
·bG(36)
This can be further simplified to
Û
δ=vkcos θ(37)
where
vk
is the magnitude of
vk=Û
pG
and
θ
denotes the angle between the vector pointing perpendicularly to
Γ(s)
19
and the projected aircraft velocity on the tangential plane. To calculate θtwo cases have to be distinguished:
θ=
π/2χτ+eχτ,for σ < 0
π/2+χτeχτ,for σ > 0
(38)
This yields for Û
δ
Û
δ=
vksin χτeχτ,for σ < 0
vksin χτ+eχτ,for σ > 0
(39)
with Eq. (30) it follows
Û
δ=σvksin χτ+eχτ
=σvksin χτcos eχτ
+cos χτsin eχτ
=σvkσδ/δ0cos eχτ
1+(δ/δ0)2
+
sin eχτ
1+(δ/δ0)2
=σvk
1+(δ/δ0)2σδ/δ0cos eχτ
+sin eχτ
(40)
where the identities
sin(arctan(x)) =x/1+x2
cos(arctan(x)) =1/1+x2
sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
(41)
have been utilized. If the course error dynamics are asymptotically stable i.e. eχτ0then
Û
δ=vk
δ/δ0
1+(δ/δ0)2(42)
where the fact that
σ2=
1has been exploited. Equation (42) shows that if the commanded course according to Eq. (29)
is tracked, the distance δstrictly decreases over time, which concludes the proof.
The input signal to the path-following controller will be the desired course and flight path angle rates. In an inversion
based control approach these rates are usually obtained by filtering the corresponding course and flight path angles.
From a geometrical point of view, the reference course rate contains information about the future course angle and
hence is linked to the curvature of the path that needs to be followed. If the rate of a reference filter is used only an
approximation is obtained if the to be followed path is not a straight line, or a combination thereof, which results in
step commands in the course reference angle that only require a course rate in the transients. If the path curvature is
20
t(s)
bG
ex
ey
vk
Γ(s)
σ < 0
σ > 0
θ
χτ
eχτ
pG
χτ, c
Fig. 16 Visualization of the angles utilized in Eq. (38).
not zero the approximated rate by the filter will not keep the system on the path since in general the rate of the filter
does not correspond to the rate imposed by the geometry of the path. Hence, although the path-following controller
would steer the aircraft towards the path, once the aircraft is on the path it would leave the path again, which can lead to
unnecessary control effort and oscillations of the aircraft around the path. Theoretically, this effect can be minimized
with high gain tracking error feedback, which however can lead to an unstable closed loop system. To avoid this behavior
a different approach is pursued where the exact required course rate based on the path geometry will be calculated
analytically instead of numerically using a filter. The commanded tangential plane course rate is given by
χτ,c
, hence
the commanded rate can be calculated by taking the derivative of the terms in Eq. (29) which yields
Û
χτ,c=Û
χτ, k+Û
χτ(43)
with
Û
χτ=σ/δ0
1+(δ/δ0)2Û
δ(44)
with Eq. (42) this leads to
Û
χτ=
vkσ/δ2
0
1+(δ/δ0)23/2δ(45)
It can be seen that for decreasing
δ
, hence small
δ/δ0
, the contribution of
Û
χτ
converges linearly to zero. Note,
Û
χτ
is
not linked to the path geometry directly. It improves however the path-following performance if
δ,
0. If
Û
χτ
would be
neglected only the course error feedback part would adapt
Û
χτ, k
such that the commanded course rate does not only
contain a component that would keep the aircraft parallel to the path. Since this contribution is mainly required for
21
δ,
0, a too high gain for the course tracking feedback would probably dominate also
Û
χτ, k
for
δ=
0. Hence, using
a small gain for the course error feedback in combination with the additional feed-forward part
Û
χτ
increases the
performance of the path-following controller. The derivative of χτ, kis given by
Û
χτ, k=d
dt arctan ey·tG
ex·tG(46)
Û
χτ, k=
cos χτ, ksin χτ, k
ey
∂λ Û
λ+ey
∂φ Û
φ·tG+ey·dtG
ds Û
s
ex
∂λ Û
λ+ex
∂φ Û
φ·tG+ex·dtG
ds Û
s
ktGk(47)
with
ex=
sin φcos λ
sin φsin λ
cos φ
,ey=
sin λ
cos λ
0
(48)
and
Û
λ=
vG
k
kpGWkcos φ,Û
φ=
uG
k
kpGWk(49)
where
uG
k
and
vG
k
are the
x
and
y
components of the kinematic velocity vector of the aircraft in the
τ
reference frame.
Equation (47) defines the rate with which the angle between the tangent vector
tG
at the aircraft and the basis vector of
the tangent plane frame
ex
changes as a function of path geometry and aircraft velocity. It hence corresponds to the
required course rate imposed by the path curvature.
Using kinematic manipulations the desired tangential plane course rate
Û
χτ,c
can be converted into the corresponding
rates for the course and flight path angle
Û
χk,c
and
Û
γk,c
, respectively. The tangential plane course rate occurs in the
angular velocity vector between the τand the ¯
Kframe for instance given in the rotated kinematic frame ¯
K:
ωτ¯
K>
¯
K
=Û
χτsin γτÛ
γτÛ
χτcos γτ¯
K
(50)
The
¯
K
frame is obatained by rotating the kinematic frame around the
xK
axis by
µk
such that the
y¯
K
axis is in the
tangential plane. Note, in [
8
] it is assumed that
γτ
0which is only justified if the reeling-out speed is small compared
to the onto the tangential plane projected kinematic velocity vector. Hence, Eq. (50) generalizes the result in [
8
].
Furthermore, Eq. (50) offers through
Û
γτ
another control degree of freedom that can be used to assist the winch controller
in the radial direction motion control. In this work this has not been further investigated, hence Û
γτ,cis set to zero.
ωτ¯
K¯
K
can be converted into the angular velocity vector between the
O
and
¯
K
frame, denoted with
ωO¯
K¯
K
22
according to
ωO¯
K¯
K
=
=M¯
KO ωOWO
+MOW ωWτW+ωτ¯
K¯
K
(51)
It is reasonable to assume that the mean wind direction changes much slower than the transport rate
ωWτW
and the
course rate vector ωτ¯
K¯
Khence ωOWKcan be set to zero. This yields
ωO¯
K¯
K
=M¯
KOMOW ωWτW
+ωτ¯
K¯
K
=
Û
µkÛ
χk,csin γk
Û
γk,ccos µk+Û
χk,csin µkcos γk
Û
γk,csin µk+Û
χk,ccos µkcos γk
¯
K
(52)
with ωWτ>
W
=Û
φsin λÛ
φcos λÛ
λW
(53)
Note, the second equality in Eq. (52) is a generic expression which can be obtained from the literature for instance from
[
14
, p. 75]. The transformation matrix
M¯
KO
can be calculated using the knowledge of course and path angle as well as
the position of the aircraft in the Wframe. With
ex,¯
K,O=
cos χkcos γk
sin χkcos γk
sin γk
ey,¯
K,O=MOW pGW×ex,¯
K,O
k − MOW pGW×ex,¯
K,Ok
ez,¯
K,O=ex,¯
K,O×ey,¯
K,O
(54)
this yields
M¯
KO =
e>
x,¯
K,O
e>
y,¯
K,O
e>
z,¯
K,O
(55)
23
Ultimately, the desired course and path angle rates can be calculated according to
Û
χk,c=
ωO¯
K
y,¯
Ksin µk+ωO¯
K
z,¯
Kcos µk
cos γk
Û
γk,c=ωO¯
K
y,¯
Kcos µkωO¯
K
z,¯
Ksin µk
(56)
with
µk=arctan M¯
KO,23
M¯
KO,33 (57)
2. Retraction Phase Guidance
The retraction phase guidance module is separated from the traction phase module. The supervisory logic switches
to the retraction phase according to the high-level state machine status. The outputs of the retraction guidance module
are again course and path angle commands. In contrast to the traction phase the aircraft will not follow a prescribed
path but directly flies towards the zenith position with a predefined path angle. The path angle set point is given by a
fixed descend angle which is chosen manually. The course angle is calculated based on the relative position of the
aircraft and the waypoint which is located at the zenith position of the small earth. The choice of this waypoint seems
naturally because reeling in the tether will automatically pull the aircraft towards the zenith position. Additionally, in
order to achieve a smoother transition back into the traction phase a flare-like maneuver is commanded that increases
the descent rate linearly leading to a slight pull-up maneuver before the aircraft goes back into cross wind flight. The
flare is initiated as a function of the aircraft latitude:
γk,c=γfγi
φmax φ0(φφ0)+γi
¯γk,c=max(min(γk,c, γf), γi)
(58)
with
φ0=φmax φ
. The parameters
φ, γf, γi
are chosen manually by the operator and characterize the length of the
flare, in terms of elevation angle, as well as the final and initial descent angle. The desired course angle is calculated
based on the relative position of the aircraft and the origin of the wind frame:
b>
O=pO,xpO,y0(59)
The course set point is then given by
χk,c=arctan2bO,y,bO,x(60)
24
Navigator
Kχ,p
Kχ,i
Kinematic
Transfor-
mation
pG
,Γ(s)χk,c
+
Û
χτ,c
Û
χk,c
+
Û
γk,c
+
+
+νχ
χk
γk,c
Fig. 17 Course controller block diagram.
C. Path Loop
1. Traction Phase
In the path loop the commanded course and path angle as well as their corresponding rates (output of guidance
module) are used to calculate the set points for the attitude loop. The overall pseudo control inputs are given by
νχ=Û
χk,c+kpχk,cχk+kit
0χk,cχkdτ
νγ=Û
γk,c+kpγk,cγk+kit
0γk,cγkdτ
(61)
The set points of the attitude controller will be derived using a model for the path dynamics. The total acceleration of
the aircraft in the kinematic frame is given by:
(Û
vk)O
K=
Û
vk
0
0
K
+ωOKK×
vk
0
0
K
=
Û
vk
Û
χkcos γkvk
Û
γkvk
K
=
ax,K
ay,K
az,K
K
(62)
The path dynamic are then defined according to
m
ax,K
ay,K
az,K
K
=(Fa)K+FgK+(Ft)K(63)
25
involving the aerodynamic force
(Fa)KR3x1
, gravitational force
FgKR3x1
as well as the tether force
(Ft)KR3x1
in the Kframe, where gravity and tether force are calculated with
Fg>
K=sin γkmkg0 cos γkmkg(64)
and
(Ft)K=MKO (p)O
(p)O2
Ft(65)
Solving for the aerodynamic force yields
fx,a,K
fy,a,K
fz,a,K
K
=m
ax,K
ay,K
az,K
K
FgK(Ft)K=(Fa)K(66)
The last two rows can be written as
fy,a,K=cos µkfa,y,¯
Ksin µkfa,z,¯
K
fz,a,K=sin µkfa,y,¯
K+cos µkfa,z,¯
K
(67)
where µkis the kinematic bank angle, i.e. the rollangle around the kinematic velocity vector and
fa,y,¯
K=cos αksin βkfa,x,B
+cos βkfa,y,Bsin αksin βkfa,z,B
fa,z,¯
K=sin αkfa,x,B+cos αkfa,z,B
(68)
Note that
αk
and
βk
are the kinematic angle of attack and kinematics sideslip angle. Since the inner loop controller
actively controls the sideslip angle
β
, i.e. the aerodynamic sideslip angle, the aerodynamic side force
fa,y,B
is
approximately zero. Contrarily, the kinematic sideslip angle βkis in presence of wind not zero. Hence,
fa,y,¯
K=cos αksin βkfa,x,Bsin αksin βkfa,z,B
fa,z,¯
K=sin αkfa,x,B+cos αkfa,z,B
(69)
26
The set point for the kinematic bank angle based on the required course and path angle rate is calculated by solving Eq.
(67) for µkand inserting the pseudo control signals for the course and path angle rates:
µk,c=arctan2mkνχcos γkvkft,y,K
mkνγvk+mkgcos γk+ft,z,K
+arctan fa,y,¯
K
fa,z,¯
K(70)
which requires estimates for the aerodynamic forces fa,y,¯
Kand fa,z,¯
K.
Based on the set point for the kinematic banking angle the corresponding Euler roll angle can be calculated according
to
Φc=arcsin cos γkcos βksin µk,ctan γktan βk
cos Θ(71)
Equation (71) can be obtained by comparing the relevant coefficients of
MBτ=MBOMOWMWτ
. The matrix
MBO
is
obtained for instance from [
12
, p. 12]. The matrix
MWτ
is equivalent to the transformation from the Earth-Centered-
Earth-Fixed (
E
) frame into the
O
frame (see [
12
, p. 31]) where the
E
frame corresponds to the
W
frame and the
O
frame corresponds to the τframe. MOW is given by
MOW =
cos ξsin ξ0
sin ξcos ξ0
0 0 1
(72)
where ξdenotes the wind direction measured from the north direction. Note, the structure of MBτis equivalent to the
structure of MBO.Φccan then be transformed into an aerodynamic banking angle command µa,cusing Eq. (73).
µa,c=arcsin cos Θsin Φc
cos γacos β+tan γatan β(73)
The required aerodynamic path angle can be calculated using Eq. (74), which has been derived in [13, p. 20-23].
γa=arcsin vksin γk+vw,O,z
va
arcsin vksin γk
va(74)
Notice that, the calculation of
γa
requires the knowledge of the wind component in
zO
direction
vw,O,z
which is however
usually negligibly small compared to the horizontal components. The angle of attack set point can be calculated similarly
to the approach presented in [25] with
Lreq ¯
f2
y,K+¯
f2
z,K(75)
27
Note, due to the wind influence this is only an approximation which is neglected in [
25
]. However, since the available
traction force needs to be maximized it makes sense to choose a fixed set point during the traction phase close to the
maximum angle of attack. Setting the angle of attack to a fixed value is similar to the case where the angle of attack
saturates. This can lead to a windup of the integrators in the path loop. One approach so mitigate the windup is to adapt
the reference model by the control deficit that results from the saturation (i.e. pseudo control hedging, PCH). However,
for the traction phase controller the reference course rate is directly calculated based on the path geometry, as discussed
in the previous section. This prevents a standard implementation of PCH, since no reference filter is used. Instead, an
anti-windup scheme based on back-calculation is used, where the feedback part corresponds to the deficit between for
instance the commanded course rate
νχ,k,c
and the expected course rate
Û
ˆχk,c
. The hedge signal is in this case defined by
νh=kbc νχ ,k,cˆ
Û
χk,c(76)
The gain
kbc
is chosen to be smaller than the integrator gain, as recommended in [
26
, pp. 79-80]. The feedback law for
the pseudo control input is then adapted according to
νχ=Û
χk,c+kpχk,cχk
+kit
0χk,cχkνh, χ dτ
(77)
The adaption of the flight path rate channel follows analogously.
2. Retraction Phase
For the retraction phase the course and path angle controller are designed similarly to the traction phase controller,
the only difference consists of the calculation of the course and path angle rate commands. Since in the retraction phase
no defined path needs to be followed, the rate commands are generated with second order reference filters. Although
first order filters would be sufficient second order filters lead to an additional smoothing of the derivatives [
27
]. Instead
of using a back-calculation anti-windup scheme a conventional PCH approach is chosen using estimates for the feasible
course and path angle rates. With the hedging signal
νh
the equations of the second order filter, here displayed for the
course filter, are defined by
Û
νr=2ζ ω0νr, χ +ω2
0χk,cχk,r
Û
χk,r=νrνh
(78)
and an equivalent pseudo control law with PI controller as for the traction phase is used (see Eq. (61)). Note, in
contrast to a fixed value for the angle of attack set point, the approximate expression of the required lift in Eq. (75) is
28
Inversion Path
Dynamics
Kinematic
Transforma-
tion
µkΦ
Kinematic
Transforma-
tion
Φµa
µa
- Reference
Filter
α
- Reference
Filter
Kµa,p
Kµa,i
νχ, νγ
µk,c
αc
Φcµa,c
µa,r
++
+
+νµa
µa
νra
+
αr
νr
Fig. 18 Path and attitude loop block diagram.
Kinematic
Transformation
Reference
Filter
Kω, p
Kω, i
Inversion Rate
Dynamics
ω
ω
ω¯
AB
cBω
ω
ωOB
cB
ω
ω
ωOB
rB
++
+
+ν
ν
νω
ω
ω
ωOBB
ν
ν
νr
+
Mc
Fig. 19 Rate loop block diagram.
used to determine the corresponding lift coefficient and by inversion of the lift coefficient the angle of attack set point
αc
for the attitude loop is determined.
D. Attitude Loop
The pseudo-control inputs for the attitude to rate inversion are given by
νµa=νra+Kµ,pµa,rµa+Kµ ,it
0µa,rµadτ
να=νr+Kα,p(αrα)+Kα, it
0(αrα)dτ
νβ=Kβ,p(βrβ)+Kβ ,it
0(βrβ)dτ
(79)
where
νra
and
νr
are calculated with an equivalent reference filter as defined for the course angle in Eq. (78). The
inversion of the attitude to rate dynamics is purely kinematic and given by
ω
ω
ωOB
cB
=MB¯
A
Û
χasin γa
Û
γa
Û
χacos γa
¯
A
+ω¯
ABB(80)
with
ω
ω
ω¯
ABB
=
cos αcos βνµ+νβsin α
sin βνµ+να
sin αcos βνµcos ανβ
B
(81)
29
The matrix
M¯
AB
is defined for instance in [
14
, p. 62].
Û
χa
and
Û
γa
are estimated by filtering Eq. (74) and Eq. (82), as
derived in [13, p. 23]
χa=χk+βarcsin 1
Vacos γavw,O,ycos χk,c
vw,O,xsin χk,c (82)
using a washout-filter, as proposed in [28]:
G(s)=
sω2
f
s2+2ωfs+ω2
f
(83)
where
ωf=90 rad s1
is the chosen filter bandwidth. Note, a better accuracy could be achieved by calculating
Û
χk
and
Û
γk
analytically using the model of the course and path rate dynamics as defined in Eq. (63) and only filter the remaining
terms. Alternatively, a model can be used to estimate
Û
χa
and
Û
γa
which requires to write down the path dynamics with
respect to the aerodynamic frame assuming a stationary wind field.
E. Rate Loop
Note, since it is assumed that the tether is connected close to the center of gravity of the aircraft the rate loop of the
tethered aircraft can be implemented analogously to the rate loop of a conventional aircraft. In the literature there exists
an ample amount of different approaches to control the rate dynamics of aircraft, in this work a conventional first order
dynamic inversion controller with second order reference filters and an incremental control allocation as presented
in [
25
] is used. Note, the incremental approach is necessary since in general the relationship between actuator inputs
and aerodynamic moments is nonlinear and not globally invertible. Since up to now and in the future extensive effort
is and will be put into the modeling and identification of the AWE system, a model-based inversion is chosen over a
sensor-based inversion as for instance presented in [29].
The commanded attitude rates as calculated by Eq. (80) are filtered and the resulting rate accelerations are added to
a PI control part analogously to Eq. (79) yielding the pseudo-control input
ν
ν
νω
for the inversion of the rate dynamics as
defined in Eq. (4). From the resulting moment the current acting moment on the aircraft, estimated using a model, is
30
subtracted yielding the required moment increment to track the commanded rates:
L
M
N
=
Lc
Mc
Nc
L0
M0
N0
=Jν
ν
νω+ωOBB×JωOB B
L0
M0
N0
(84)
F. Control Allocation
Eventually, the moment increments are mapped to a surface deflection increment that is added to the current surface
deflection resulting in the final actuator command:
δa,c
δe,c
δr,c
=
δa,0
δe,0
δr,0
+
δa
δe
δr
=
δa,0
δe,0
δr,0
+
Clδa0Clδr
0Cmδe0
Cnδa0Cnδr
1
L
M
N
(85)
where the
Ci,j
coefficients represent roll- (
L
), pitch- (
M
) or yaw-moment (
N
) control derivatives that are obtained by
linearizing the aerodynamic moment model with respect to the control surface deflections.
G. Winch Controller
The winch controller is derived based on the model defined in Eq. (9) without explicitly taking into account the
aircraft dynamics as presented for instance in [
30
]. The reason is that if the aircraft dynamics are taken into account, the
full state vector of the aircraft needs to be available to the winch controller including a tether model with measurable
states. So far no reliable information about the communication between the aircraft and the ground station is available
and feedback of tether states is not practical. Hence, it is decided to control the winch only based on the measured tether
force on the ground. In AWE, two high level control objectives for the winch controller can be formulated. First, the net
power output has to be maximized by controlling the radial motion of the aircraft in an optimal way, second, the winch
31
controller needs to prevent too high tension in the tether, for instance as a result of sudden wind speed changes, which
would lead to a tether rupture or damage of the aircraft. In this work, the focus is on the second control objective, since
it is more critical for the reliable operation of the AWE system.
Note, from the perspective of the winch, the dynamics of the aircraft and the tether represent a disturbance that the
winch controller needs to regulate in order to track a force set point. If a tether force measurement on the ground is
available, which is usually the case in this application, a complex disturbance model is not necessary because all relevant
information is condensed in the force measurement. Note that this approach assumes implicitly that the difference
between the tether force measured on the ground and the tether force measured at the aircraft is negligible. Simulation
results show that this assumption is valid during the traction phase. The set point for the reeling speed can be derived as
follows. The aircraft dynamics in the tangential plane, or spherical coordinates, are given by
Û
vGτ
+(ω)Wτ
τ×vGτ
=Fgτ+(Fa)τ+(Ft)τ
mk(86)
Assuming a straight tether only the third row is relevant which is given by
Û
vz=ωxvy+ωyvx+
Fg,z+Fa,z+Ft
mk(87)
This can be written more compactly as
Û
vz=Faircraft +Ft
mk(88)
with
Faircraft =mkωxvy+ωyvx+Fg,z+Fa,z(89)
Note,
Faircraft
requires the knowledge of the full aerodynamic model of the aircraft as well as the relevant measured states
if used for the set point calculation. However, instead of an estimation of
Faircraft
the measured tether force on the ground
can be used, if it is assumed that Faircraft ≈ −Ft,m. If the tether is straight, the reeling speed vris equal to vz, hence
Û
vr=Ft,mFt
mk(90)
If
Ft
is replaced by the desired traction force
Ft,c
the resulting acceleration can be interpreted as a reference acceleration
proportional to the tether force tracking error. With
Û
ωw=Û
vr/rw
this expression can be substituted into the winch model
in Eq. (9) and solved for the reference torque:
Mc=Jw
rwmkrwFt,mJw
rwmkFt,c(91)
32
Substituting this expression back into the winch model yields the closed loop winch model
Û
ωw=1
rwmkFt,mFt,c+w(92)
where
w
is the model mismatch as a result of an imperfect inversion of the plant dynamics. Note, if the measured
tether force deviates from the set point the winch will reel out faster or slower. Although simple, this approach proved to
be highly effective in dealing with varying wind conditions and wind gusts as will be shown in section IV, while being
independent of any aircraft state. In order to get rid of steady state errors an integrator term
kit
0FmFt,sdτ
can be
added to Eq. (91). For the stability of Eq. (92) only a qualitative but intuitive stability proof is given. If the tether force
becomes larger than the set point force the winch will start to accelerate according to Eq. (92). Of course this is strictly
only true if
1
rwmkFmFt,s>w
. However, in the opposite case the acceleration will only be delayed, since if the
winch further decelerates the tension in the tether would increase steadily until the tracking error contribution will be
larger than
w
. If the winch accelerates the kinematic radial speed of the aircraft will increase which decreases the
apparent wind speed. As a consequence the lift force will drop, which decreases the tension in the tether and therefore
decreases the tether force tracking error. The causal chain holds of course for the opposite case as well, where the tether
force is smaller than the force set point.
During the retraction phase the reeling-in speed is set to a fixed value, usually the maximum reeling-in speed that
the winch can achieve is chosen in order to minimize the retraction time. For the tracking task of the speed controller
a dynamic model based feed-forward controller (see [
31
, pp. 324-328]) for fast tracking is combined with a linear
quadratic feedback regulator with servomechanism [
32
, pp. 51-62]. The prefilter is used to create smooth transitions
between set point changes. Additionally, a feed-forward disturbance compensation is added since from the perspective
of the speed controller the tether force represents a measurable disturbance.
IV. Results
In this section two different simulation campaigns are used to investigate the robustness of the control system. First,
the robustness with respect to modest changes in the wind speed due to turbulence and wind shear is assessed. In the
second part, the effect on the control performance due to sudden and significant wind speed changes caused by gusts is
analyzed.
A. Consecutive Pumping Cycles in a Turbulent Wind Field
Fig. 20 and Fig. 21 show the resulting flight paths projected into the
xWzW
and
xWyW
plane, respectively. Fig. 22
depicts the path projected into the tangential plane at
λ=0°
and
φ=φ0
(center of the figure of eight). Despite the
turbulent wind field, shown in Fig. 23 and Fig. 24, the control system is able to guide the aircraft along the defined flight
33
path reliably. The visible deviations between the reference path (light grey curve in Fig. 22) and the real flight path are
acceptable and are caused by the limited bandwidth of the control system. This limitation results in a repetitive non-zero
cross track error during the turns. The results display roughly three consecutive pumping cycles. The reoccurring flight
pattern demonstrates the robustness of the closed loop system towards modest changes in wind speed caused by wind
shear and turbulence.
0 100 200 300 400 500
xW(m)
0
100
200
300
400
zW(m)
Fig. 20 Flight path in xWzW-plane).
0 100 200 300 400 500
xW(m)
-200
-100
0
100
200
yW(m)
Fig. 21 Flight path in xWyW-plane.
As described in section III, reference filters are used to generate the course and path angle rates during the retraction
phase. This allows to implement PCH to adapt the reference filters in case of saturation of the control signal. From
the point of view of the path loop, the control signals are the bank angle command
µa,c
as well as the angle of attack
command
αc
. In Fig. 25 it can be observed that during a significant part of the retraction phase, e.g. for instance between
34
-100 0 100
ey
yτ(m)
ex
-50
0
50
100
150
200
xτ(m)
Fig. 22 Figure of eight flight path projected into the tangential plane.
0 100 200 300 400
Time (s)
8
10
12
14
16
vw,x(m/s)
Fig. 23 x component of the wind velocity vector in the Wframe.
0 100 200 300 400
Time (s)
-3
-2
-1
0
1
2
3
vw,i(m/s)
vw,y
vw,z
Fig. 24 y and z components of the wind velocity vector in the Wframe.
35
226 s
and
234 s
, the angle of attack is saturating. In this case the commanded pseudo-control inputs
νγ
and
νχ
will
225 230 235 240 245
Time (s)
-10
-5
0
5
10
15
α()
αc
αr
αis
bounds
Fig. 25 Angle of attack tracking.
deviate from the actual plant responses. The adaptation of the course and path angle reference filters can be observed in
Fig. 26 and Fig. 27. The effect is especially visible for the path angle whose primary control variable is the angle of
attack. As the angle of attack is saturating the reference path angle increases (e.g. at
226 s
) as a result of the hedge
signal before it decreases again and eventually converges towards the negative commanded set point γk,c.
225 230 235 240
Time (s)
-30
-20
-10
0
10
χk()
χk,c
χk,r
χk,is
Fig. 26 Course angle tracking during retraction.
During the pumping cycles the sideslip angle varies most of the time between
2°
to
+2°
. Larger sideslip angles
occur during the transition phases from traction into retraction and vice verse as can be seen in Fig. 28. The evolution
of the aircraft control surface deflections is depicted in Fig. 29. It can be observed that the highest control effort is
required in the transition phases where the control surfaces partially saturate. During the traction phases the aileron
δa
and rudder
δr
inputs vary in a repetitive manner between
5°
to
+5°
while the elevator deflection
δe
remains almost
36
225 230 235 240
Time (s)
-20
0
20
40
60
γk()
γk,c
γk,r
γk,is
Fig. 27 Flight path angle tracking during retraction.
0 100 200 300 400
Time (s)
-10
-5
0
5
β()
Fig. 28 Sideslip angle regulation.
37
constant at around 9°as a results of the fixed angle of attack set point during the traction phase.
0 100 200 300 400
-20
0
20
δa()
0 100 200 300 400
-20
0
20
δe()
0 100 200 300 400
Time (s)
-20
0
20
δr()
Fig. 29 Control surface deflections with limits (dashed lines).
Besides the analysis of the flight control performance the winch control performance needs to be assessed. Fig. 30
shows the evolution of the tether force as measured on the ground. During the conducted simulations a tether force
set point of
1000 N
is chosen, which is well beyond the structural limitations of around
1500 N
. The tether force
oscillates around the set point with an amplitude of around
50 N
to
100 N
. The oscillations are a result of the continues
acceleration and deceleration of the aircraft while flying down and upwards during the figure of eight flight patterns.
To further reduce these oscillations an improved feed-forward winch controller could be implemented in the future
that systematically reels out slower during upward and faster during downward flight. At the moment this is partially
achieved via feedback control of the tether force. Furthermore, the resulting variations in the reeling speed depicted in
Fig. 31 should be reduced in the future since variations in reeling speed would lead to large oscillations in the mechanical
power output in combination with a constant tether force. One option to tackle this problem would be to use the pitch
channel of the aircraft to control the airspeed, which is out of the scope of this paper.
B. Robustness towards Wind Gusts
In this section the robustness of the control system towards rapid changes in the mean wind speed will be analyzed.
For that purpose a mexican hat gust as defined in [
33
] is implemented and activated during the simulation at a specified
instant in time. In this work only the response of the aircraft towards gusts in up- and downwind direction as depicted
in Fig. 32 and Fig. 33 is analyzed. In both cases the gust leads to a significant increase or decrease in airspeed and
therefore tether force (see Fig. 34 and Fig. 35). In order to keep the tether force around the set point the winch controller
has to adapt the reeling out speed according to Eq. (91) (see Fig. 36 and Fig. 37). It can be observed that the reeling
speed change follows the shape of the gust proportionally. The adaptation of the reeling speed has a direct effect on the
38
0 100 200 300 400
Time (s)
0
0.5
1
1.5
Ft(kN )
Ft,s
Ft,m
Fig. 30 Tether force tracking.
0 100 200 300 400
Time (s)
-20
-15
-10
-5
0
5
10
vr(m/s)
Fig. 31 Reeling speed.
0 20 40 60
Time (s)
-5
0
5
10
15
20
vw,x(m/s)
Fig. 32 Gust in upwind direction.
39
0 20 40 60
Time (s)
5
10
15
20
25
30
vw,x(m/s)
Fig. 33 Gust in downwind direction
0 20 40 60
Time (s)
0
0.5
1
1.5
Ft(kN )
Ft,s
Ft,m
Fig. 34 Tether force with gust in upwind direction.
0 20 40 60
Time (s)
0
0.2
0.4
0.6
0.8
1
1.2
Ft(kN )
Ft,s
Ft,m
Fig. 35 Tether force with gust in downwind direction.
40
0 20 40 60
Time (s)
-15
-10
-5
0
5
10
vr(m/s)
Fig. 36 vrwith gust in upwind direction.
0 20 40 60
Time (s)
-20
-10
0
10
20
vr(m/s)
Fig. 37 vrwith gust in downwind direction.
41
flight path in radial direction. The flight path gets either compressed (Fig. 38) or stretched (Fig. 39) depending on the
gust direction as a result of the increasing or decreasing reeling out velocity. Contrarily, Fig. 40 and Fig. 41 show that
100 200 300 400
xW(m)
50
100
150
200
250
300
zW(m)
Fig. 38 Flight path with gust in upwind direction.
100 200 300 400
xW(m)
50
100
150
200
250
300
zW(m)
Fig. 39 Fight path with gust in downwind direction.
the adaptation of the reeling speed has only a small effect on the path-following performance in the tangential plane.
V. Conclusion
In this paper a novel cascaded model based control architecture for rigid-wing airborne wind energy systems operated
in pumping-cycle mode has been presented. The proposed control approach leads to a robust control performance while
flying in a realistic turbulent wind field. The extended geometric path-following approach guided the aircraft along a
three dimensional curve reliably. State and input constraints are systematically handled using pseudo control hedging,
which turns out to be beneficial especially during the retraction phase where the commanded flight path is adapted
42
-100 0 100
yτ(m)
-50
0
50
100
150
200
xτ(m)
Fig. 40 Flight path with gust in upwind direction.
-100 0 100
yτ(m)
-50
0
50
100
150
200
xτ(m)
Fig. 41 Fight path with gust in downwind direction.
43
automatically in case of angle of attack saturation. Challenging phases during the pumping cycle are the transitions
from the traction to the retraction phase and vice versa. Due to the rapid tether force changes in these phases, overshoots
in sideslip angle and angle of attack are present although these peaks occurred only for a short period of time and the
resulting tracking errors could be regulated back to the set point by the respective feedback controller. Moreover, the
results show that the tether force set point can be tracked effectively by directly calculating a torque command as a
function of the force tracking error. However, the excellent tether force tracking performance leads to a high variance in
the reeling speed and therefore to oscillations in the mechanical power output. This effect could be reduced in the future
by additionally using the pitch angle of the aircraft to control the airspeed. In return, this would lead to a less aggressive
reeling speed adaption and hence a reduced variance of the mechanical power. In addition to the ability of tracking a
constant tether force the proposed winch controller can react to sudden wind speed changes, such as gusts, through
adaption of the reeling-out speed, effectively, ensuring the structural integrity of the aircraft.
Acknowledgments
This research has been supported financially by the project AWESCO (H2020-ITN-642682), funded by the European
Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 642682.
The authors would like to thank the team of Kitemill AS for supporting this work, in particular Sture Smidt for carrying
out the CFD and XFLR5 analysis of the aircraft.
44
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... Licitra et al. [12,13] present a dynamic model of the Ampyx Power AP-3 (12 m 2 wing area) aircraft. Rapp et al. [14,15] present the dynamic model and flight control framework underlying also the present work. Eijkelhof [16] shows the detailed design process of a large-scale reference kite for a Ground-Gen system. ...
... In the following, a concise description of the utilised control system is given. For a detailed derivation please refer to [14,25]. respect to the retraction path is translated into a desired course and path angle which are then both passed through a second order reference filter (illustrated in Figure 3) to generate, similar to the figure of eight guidance, smooth reference course and path angle rates. ...
... The winch controller is a simplified version of the controller introduced by Rapp et al. [14]. Instead of calculating a reference torque with a feed forward control structure, the torque command is generated by a simple PI controller where the input is given by the difference between the measured and the commanded tether force. ...
... The tracking control needs a reference point on the target trajectory which can be used by the guidance. Therefore, the nearest point on the lemniscate to the actual kite position (the geodetic distance) is calculated as proposed in [20] using Newton's method. In the following, the nearest point * is assumed to be known. ...
... The transfer of the tangent at the target trajectory Γ ( * ) to the point ⊥ is a parallel shift along the vector ( * ). Combining the previous with the geometric relations in Fig. 11 the flight path azimuth angle and its derivative can be calculated as follows: For more information, see [20]. ...
Conference Paper
Full-text available
Airborne Wind Energy has a wide range of applications and can help to cover the energy demand in the future. Using tethered kites is one possibility to harvest this green energy while having low material costs. The most important part of such a system is the control system, ensuring energy optimal trajectories and keeping the kite aloft, in the first place. The understanding and modeling of the system dynamics is an important part of the successful design of a control system. In this paper, an investigation of the eigenmotions of a 4-line kite is presented. Additionally, a cascaded nonlinear controller is designed. The nonlinear guidance law calculates the azimuth angle on a unit sphere to lead the kite on a predefined figure-eight trajectory. This is followed by a stabilization and tracking controller which is based on the eigenmotions of the kite as well as verified relationships between the yaw rate and the steering input of the kite. The deduced overall control is shown to precisely track the kite on a Bernoulli lemniscate.
... They find that, under some prescribed constraints, 60 circular and figure of eight trajectories produce similar mean power and that closed-loop control enhance robustness but decreases power production of about 10 %. Control in all operation phases is studied by Rapp et al. (2019) and Todeschini et al. (2021): the present work can be understood as a study of the guidance (or the reference trajectory) used during the power generation phase of their study. Pasquinelli (2021) investigates the power losses in a circular trajectory with a dynamical quasi-analytical model. ...
Preprint
Full-text available
The optimal control problem for flight trajectories for Fly-Gen Airborne Wind Energy Systems (AWES) is a crucial research topic for the field, as suboptimal paths can lead to a drastic reduction in power production. One of the novelties of the present work is the expression of the optimal control problem in the frequency domain through a Harmonic Balance formulation. This allows to reduce the problem size by solving only for the main harmonics and to implicitly impose periodicity of the solution. The trajectory is described by the Fourier coefficients of the dynamics (elevation and azimuth angles) and of the control inputs (on-board wind turbines thrust and AWES roll angle). To isolate the effects of each physical phenomenon, optimal trajectories are presented with an increasing level of physical representation from the most idealized case: i) If the mean thrust power (mechanical power linked to the dynamics) is considered as the objective function, optimal trajectories are characterized by a constant AWES velocity over the loop and a circular shape. This is done by converting all the gravitational potential energy into electrical energy. At low wind speed, on-board wind turbines are then used as propellers in the ascendant part of the loop; ii) If the mean shaft power (mechanical power after momentum losses) is the objective function, a part of the potential energy is converted into kinetic and the rest into electrical energy. Therefore, the AWES velocity fluctuates over the loop; iii) If the mean electrical power is considered as the objective function, the on-board wind turbines are never used as propellers because of the power conversion efficiency. Optimal trajectories for case ii) and iii) have a circular shape squashed along the vertical direction. The optimal control inputs can be generally modelled with one harmonic for the on-board wind turbines thrust and two for AWES roll angle without a significant loss of power, demonstrating that the absence of high-frequency control is not detrimental to the power generated by Fly-Gen AWES.
... When modelling AWE systems, the wind environment is often approximated as a height-dependent power law or logarithmic distribution (Archer, 2013;Zanon et al., 2013b;Horn et al., 2013;Bauer et al., 2018). A limited number of studies have used more realistic virtual wind conditions from large-eddy simulations to assess the robustness of their control strategies (Sternberg et al., 2012;Rapp et al., 2019). However, none of the aforementioned studies have measured how the atmospheric boundary layer flow is altered by the operation of airborne wind energy systems. ...
Article
Full-text available
The future utility-scale deployment of airborne wind energy technologies requires the development of large-scale multi-megawatt systems. This study aims at quantifying the interaction between the atmospheric boundary layer (ABL) and large-scale airborne wind energy systems operating in a farm. To that end, we present a virtual flight simulator combining large-eddy simulations to simulate turbulent flow conditions and optimal control techniques for flight path generation and tracking. The two-way coupling between flow and system dynamics is achieved by implementing an actuator sector method that we pair to a model predictive controller. In this study, we consider ground-based power generation pumping-mode AWE systems (lift-mode AWES) and on-board power generation AWE systems (drag-mode AWES). The aircraft have wingspans of approximately 60 m and fly large loops of approximately 200 m diameter centred at 200 m altitude. For the lift-mode AWES, we additionally investigate different reel-out strategies to reduce the interaction between the tethered wing and its own wake. Further, we investigate AWE parks consisting of 25 systems organised in five rows of five systems. For both lift- and drag-mode archetypes, we consider a moderate park layout with a power density of 10 MW km−2 achieved at a rated wind speed of 12 m s−1. For the drag-mode AWES, an additional park with denser layout and power density of 28 MW km−2 is also considered. The model predictive controller achieves very satisfactory flight path tracking despite the AWE systems operating in fully waked, turbulent flow conditions. Furthermore, we observe significant wake effects for the utility-scale AWE systems considered in the study. Wake-induced performance losses increase gradually through the downstream rows of systems and reach up to 17 % in the last row of the lift-mode AWE park and up to 25 % and 45 % in the last rows of the moderate and dense-drag-mode AWE parks respectively. For an operation period of 60 min at a below-rated reference wind speed of 10 m s−1, the lift-mode AWE park generates about 84.4 MW of power, corresponding to 82.5 % of the power yield expected when AWE systems operate ideally and interaction with the ABL is negligible. For the drag-mode AWE parks, the moderate and dense layouts generate about 86.0 and 72.9 MW of power respectively corresponding to 89.2 % and 75.6 % of the ideal power yield.
... Similarly, the focus in [27] is on the pumping mode and correct power output computation, and the model is confirmed using flight test data. In a recent research [28], the authors created nonlinear flight dynamics models and used a cascaded nonlinear control strategy to manage the pumping cycle. The authors investigated the influence of vertical wind gusts on the kite's capacity to follow a figure-8 path, but the kite's figure-8 path was not updated. ...
Conference Paper
View Video Presentation: https://doi.org/10.2514/6.2022-0136.vid This work describes the crosswind (figure-8 pattern) flight dynamics modeling of a tethered tailless kite, as well as the controller architecture for path following and path planning using the pure pursuit algorithm. It is based on a previously developed nonlinear flight dynamic Lagrange formulations-based model. The primary goal of this article is to model and investigate the impact of incoming wind direction fluctuations on the kite's ability to fly in a figure-8 pattern. Then, based on the wind velocity, utilize the path planning algorithm to adjust the orientation and placement of the figure-8 pattern. There were two types of simulation cases studied: 1) path following only, 2) path following, and path planning. In the first scenario, modeling findings reveal that a significant shift in wind direction can impair the kite's ability to fly in a fixed figure-8 pattern. Variations in wind speed direction can lower the lift and apparent wind speed of the kite, potentially leading to a crash. The second simulation example scenario investigated a similar wind profile as the first case, but the orientation and position of the figure-8 flight path were varied in real time depending on the direction of wind speed in the y and z axes. The results reveal that the kite can continue the figure-8 path following pattern flight if the figure-8 pattern flight is updated in the presence of changes in wind speed and direction.