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Cascaded Pumping Cycle Control for

Rigid Wing Airborne Wind Energy Systems

Sebastian Rapp∗and 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,τ

φ

ey,τ

ex,τ

ez,τ

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,τ

ey,τ

ex,B

ey,B

ex,τ

vk,τ

χτey,τ

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)B∈R3x1

is the kinematic aircraft velocity in the

B

frame with components

uk,vk

and

wk

,

mk

is the mass of the

aircraft,

ωOBB∈R3x1

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)B∈R3x1

is the aerodynamic force,

FgB∈R3x1

is the gravity force and

(Ft)B∈R3x1

is the

Table 2 Actuator Parameters.

Parameters Values Units

Natural frequency ω035 rad s−1

Relative damping ζ1 -

Max./Min. aileron deﬂection ±15 °

Max./Min. elevator deﬂection ±15 °

Max./Min. rudder deﬂection ±20 °

Rate limits (all actuators) ±300 °s−1

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 m−3

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

=J−1−ωOBB×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

J∈R3x3

is the inertia tensor, and

(Ma)B∈R3x1

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

=

−q2−q3−q4q1

q1−q4q3q2

q4q1−q2q3

−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

1−q2

2−q2

3−q2

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=J−1

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 s−1

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

lt≥lt,max

¬f V T OL

λ≥λwp1

λ≤0

lt≤lt,min∨

kpGk ≤ lt,min∨

φ≥φmax

Va≥Va,min

φ≤φ0+∆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°≤∆φ0≤10°

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°≤∆t≤10°

) 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,τ

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,τ

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,τ

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

+

2∂2Γ

∂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 Newton’s method. With

d

ds pG

⊥·t(s)=pG

⊥·dt(s)

ds (20)

The update equation for Newton’s method is then

s+=s−−pG

⊥·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=s∗yields

Û

δ=−1

√1−cos2δÛ

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=−vk,τ cos θ(37)

where

vk,τ

is the magnitude of

vk,τ =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/2−∆χτ+eχτ,for σ < 0

π/2+∆χτ−eχτ,for σ > 0

(38)

This yields for Û

δ

Û

δ=

−vk,τ sin ∆χτ−eχτ,for σ < 0

−vk,τ sin −∆χτ+eχτ,for σ > 0

(39)

with Eq. (30) it follows

Û

δ=−σvk,τ sin −∆χτ+eχτ

=−σvk,τ −sin ∆χτcos eχτ

+cos ∆χτsin eχτ

=−σvk,τ σδ/δ0cos eχτ

1+(δ/δ0)2

+

sin eχτ

1+(δ/δ0)2

=−σvk,τ

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. eχτ→0then

Û

δ=−vk,τ

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,τ

ey,τ

vk,τ

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=

vk,τ σ/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,τ ·tG

ex,τ ·tG(46)

Û

χτ, k=

cos χτ, k−sin χτ, k

∂ey,τ

∂λ Û

λ+∂ey,τ

∂φ Û

φ·tG+ey,τ ·dtG

ds Û

s

∂ex,τ

∂λ Û

λ+∂ex,τ

∂φ Û

φ·tG+ex,τ ·dtG

ds Û

s

ktGk(47)

with

ex,τ =

−sin φcos λ

−sin φsin λ

cos φ

,ey,τ =

−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,τ

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

µk−Û

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 λ−Û

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 µk−ωO¯

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=γf−γi

φmax −φ0(φ−φ0)+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+kp,χ χk,c−χk+ki,χ t

0χk,c−χkdτ

νγ=Û

γk,c+kp,γ γk,c−γk+ki,γ t

0γk,c−γkdτ

(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)K∈R3x1

, gravitational force

FgK∈R3x1

as well as the tether force

(Ft)K∈R3x1

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,¯

K−sin 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,B−sin 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,B−sin 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 γkvk−ft,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,c−tan 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 ξ−cos 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,χ =kbc νχ ,k,c−ˆ

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+kp,χ χk,c−χk

+ki,χ t

0χk,c−χk−νh, 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,χ =−2ζ ω0νr, χ +ω2

0χk,c−χk,r

Û

χk,r=νr,χ −νh

(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

µk→Φ

Kinematic

Transforma-

tion

Φ→µa

µa

- Reference

Filter

α

- Reference

Filter

Kµa,p

Kµa,i

νχ, νγ

µk,c

αc

Φcµa,c

µa,r

+−+

+

+νµa

µa

νr,µa

+

αr

νr,α

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,ω

+

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=νr,µa+Kµ,pµa,r−µa+Kµ ,it

0µa,r−µadτ

να=νr,α +Kα,p(αr−α)+Kα, it

0(αr−α)dτ

νβ=Kβ,p(βr−β)+Kβ ,it

0(βr−β)dτ

(79)

where

νr,µa

and

νr,α

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 βνµ−cos 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+β−arcsin 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 s−1

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,τ =−ωxvy,τ +ωyvx,τ +

Fg,z,τ +Fa,z,τ +Ft

mk(87)

This can be written more compactly as

Û

vz,τ =Faircraft +Ft

mk(88)

with

Faircraft =mk−ωxvy,τ +ωyvx,τ +Fg,z,τ +Fa,z,τ (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,τ , hence

Û

vr=Ft,m−Ft

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

rwmk−rwFt,m−Jw

rwmkFt,c(91)

32

Substituting this expression back into the winch model yields the closed loop winch model

Û

ωw=1

rwmkFt,m−Ft,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

0Fm−Ft,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

rwmkFm−Ft,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,τ

yτ(m)

ex,τ

-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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