Airship Formation Control

Estela Bicho, Andr

´

e Moreira, S

´

ergio Diegues,

Manuel Carvalheira and S

´

ergio Monteiro

Departamento de Electr

´

onica Industrial da Universidade do Minho,

4800-058 Guimaraes, Portugal

Abstract. This paper addresses the problem underlying the control and coordi-

nation of multiple autonomous airships that must travel maintaining a desired ge-

ometric formation and simultaneously avoid collisions with moving or stationary

obstacles. The control architecture is based on the attractor dynamics approach to

behaviour generation. The airship physical model is presented and the mathemat-

ical background for the control architecture is explained. Simulations (with per-

turbations) with formations of two and three autonomous airships are presented

in order to validate the architecture.

1 Introduction

In this paper we address the problem underlying the control and coordination of multi-

ple autonomous airships that must drive maintaining a desired geometric formation and

simultaneously avoid collisions with obstacles(e.g. another airship, a building, etc... )

see Fig.1. The problem of formation control on land mobile robots has received much

attention from researchers working on cooperative robotics (see e.g. [1], [2], [3],

[13], [5], [6] and [7] for some interesting works). Research on UAV formation control

as also been a subject of growing research(e.g. [8], [9], [10]). In respect to airships,

some work is being done on loose formations of stratospheric airships that will serve

as telecommunications relays and airborne radar stations(e.g. Lockheed Martin High

Altitude Airship).

This project is the next step after the group work on semi-autonomous airships

control [11] [12], where we presented a control architecure based on the dynamical

systems approach to behavior generation(see c.f [18][20][19]). In [4], [13] and [14]

the control of formations of land mobile robots was adressed and studied.

Here we show how a set of decentralized and distributed basic control architectures

for line, column and ”oblique” formations can be used for teams of two airships. These

dynamic control architectures can then be easily combined to generate more complex

geometric formations for larger teams of airships. As examples we show teams of 3 air-

ships ﬂying in line, column and V formation. We demonstrate the ﬂexibility of our dy-

namic control architectures, presenting the ability to avoid sensed obstacles integrated

with movement in formation. Although we present examples for formations for teams

of 3 airships, more complex general conﬁgurations (larger number of airships) can be

solved by our approach.

Bicho E., Moreira A., Diegues S., Carvalheira M. and Monteiro S. (2006).

Airship Formation Control.

In Proceedings of the 2nd International Workshop on Multi-Agent Robotic Systems, pages 22-33

DOI: 10.5220/0001223900220033

Copyright

c

SciTePress

We assume that the airships have no prior knowledge of the environment and we

follow a master-referenced strategy for each airship in the team. The control architec-

ture of each airship is structured in terms of elementary behaviors (i.e. obstacle avoid-

ance and “keep formation” behavior). The individual behaviors and their integration

are modelled by non-linear dynamical systems and bifurcations are used to make de-

sign decisions around points at which a system must switch from one type of solution to

another. The advantage is that the mathematical properties associated with the concepts

(c.f. section 3) enable system integration including stability of the overall behavior of

the autonomous systems. The dynamical systems that govern the behavior of each air-

ship are tuned so that the movement of each airship in time is generated as a time series

of attractor (i.e. asymptotically stable) states. The beneﬁt is that asymptotical stability

can be maintained and thus the systems are robust against environmental perturbations.

The rest of the paper is structered as follows; section 2 describes the background,

airship model and the system disturbances. In section 3, we explain how the forma-

tions are achieved and maintaned during ﬂight. the basic conﬁgurations in line, column

and oblique are explained and how they can be combined in larger and more complex

formations. Section 4 reports on the simulation results on various scenarios with the

airship formations avoiding obstacles and changing formations. The last section reports

on the conclusions and some facts pertinent to the ﬁnal results.

2 Aiship Model and Perturbations

The goal is to enable a team of lighter-than-air vehicles to autonomously navigate in

formation toward a target destination, avoiding obstacles and coping with environmen-

tal perturbations. We brieﬂy discuss the organization of the team of mobile airships and

we outline the basic assumptions behind this work.

A team of N airships has one designated Lead airship labelled A

1

(the notion of

a Lead airship is in analogy with the work of Desai, Ostrowsky and Kumar[2]). This

airship navigates from an initial position to a ﬁnal goal destination. Within the forma-

tion, each airship (except the Lead airship) depends on one of the others. Thus there are

many leaders and many followers but a unique Lead airship. We decompose the team

of N airships into N − 1 sub-teams of 2-airships each (Fig. 1). The control of each

sub-team follows a leader-follower decentralized motion control strategy (c.f. example

in Fig. 5).

Each follower airship takes its leader as a reference point and its motion must be

controlled in order to fulﬁll the following task requirements (see Fig. 2): i) To maintain a

desired relative angle between the leader and the follower, ∆γ

i,d

; ii) maintain a desired

distance to the leader, l

i,d

; iii) maintain a desired altitude to the leader, ∆h

i,d

; and iv)

simultaneously avoid collisions with obstacles that may appear.

2.1 Airship Kinematics

Each airship is a balloon in which the lift is independent of ﬂight speed, what is called

aerostatic lift. Its kinematics description is based on the reference frames presented in

ﬁgure 3.

23

A

A

A

2

3

1

Obstacle

Target

Fig.1. Example of a team with N(=3) airships in formation. A

1

is the Lead Airship, A

2

and A

3

follows A

1

in a oblique formation.

Fig.2. a) ∆γ

i,d

is the desired relative angle between the follower airship and its leader. l

i,d

is the

desired distance of the follower to the leader. γ

i

is the angle at which the follower sees the leader.

b) ∆h

i,d

is the desired altitude relative to the leader airship.

We use the following notation: The generalized coordinates for the airship are

η = (x, y, z, φ, θ, ψ)

T

∈ ℜ

6

(1)

where (x, y and z) denote the position of the centre of mass, relative to the earth-ﬁxed

reference frame, and (φ, θ, ψ) are the three Euler angles (i.e. roll, pitch and yaw angle)

and represent the orientation of the airship (see [15], [16] or [12]). Therefore, the model

partitions naturally into translational and rotational coordinates

η

T

1

= (x, y, z)

T

∈ ℜ

3

η

T

2

= (φ, θ, ψ)

T

∈ ℜ

3

(2)

The linear and angular velocity vector with coordinates in body-ﬁxed reference frame-

{w’} (see Fig.3) is

v = (v

x

, v

y

, v

z

, ω

x

, ω

y

, ω

z

)

T

∈ ℜ

6

(3)

which can be decomposed into:

υ

T

1

= (v

x

, v

y

, v

z

)

T

∈ ℜ

3

υ

T

2

= (ω

x

, ω

y

, ω

z

)

T

∈ ℜ

3

(4)

24

ay

obs

ay

tar

a

a

y

Target

Obstacle

y

xx

xx’

ay’

{w}

{w’}

e

obs

tar

a

e

Fig.3. a) Desired task for the lead airship is movement toward the target location while avoiding

obstacles. Constraints for the yaw control (i.e. ψ) are the directions at which obstacles and target

lie as seen from the current position of the airship, i.e. ψ

obs

and ψ

tar

. Obstacle challenges the

movement toward the target location. ǫ

obs

and ǫ

tar

are given by the vision system. b) We deﬁne

three coordinate frames: i) Earth-ﬁxed reference frame {ω}(X

ω

, Y

ω

, Z

ω

); ii) moving coordinate

frame b (X

b

,Y

b

,Z

b

) ﬁxed to the airship and origin coincident with the centre of gravity (CG) (i.e.

body-ﬁxed) and iii) w (X

ω

, Y

ω

, Z

ω

) is simply a translation of the earth-ﬁxed reference frame w

to the airships centre of the gravity.

The airships ﬂight path relative to the earth-ﬁxed coordinate system is given by a

velocity matrix transformation:

˙η

1

= J

1

(η

2

)v

1

(5)

The body-ﬁxed angular velocity vector υ

2

and the Euler rate vector ˙η

2

= (

˙

φ,

˙

θ,

˙

ψ)

are related through a transformation matrix according to:

˙η

2

= J

2

(η

2

)v

2

(6)

For further details on the Jacobeans J

1

(η

2

) and J

2

(η

2

) please refer to [16].

2.2 Airship Dynamics

The airship dynamics can be expressed by the following nonlinear dynamic equation of

motion:

M ˙v + C (v) v + D (v) v + g (η) = τ (7)

where variables are described in the reference frame of airship {w’} and τ is the vector

with the control inputs, i.e. forces (Fx, Fy, Fz) and torques (Nx , Ny, Nz):

τ = (τ

T

1

, τ

T

2

) (8)

τ

T

1

= [F

x

, F

y

, F

z

]

T

(9)

25

τ

T

2

= [N

x

, N

y

, N

z

]

T

(10)

The matrix M is the mass matrix (including added mass terms), C is the Coriolis

and centrifugal forces matrix, D is the aerodynamic damping matrix, g is the restoring

force (gravity and buoyancy)

In order to get a more realistic behaviour we use the non-linearized model of the

airship making the airship model time dependent. This means that C is:

C =

0 0 0 mz

G

ω

z

−m x

G

ω

y

− v

z

m x

G

ω

z

+ v

y

0 0 0 −mv

z

m z

G

ω

z

+ x

G

ω

x

mv

x

0 0 0 −m

z

G

ω

x

− v

y

−m z

G

ω

y

+ v

x

mx

G

ω

x

−mz

G

ω

z

mv

z

m

z

G

ω

x

− v

y

0 I

zz

ω

z

−I

yy

ω

y

m

x

G

ω

y

− v

z

−m z

G

ω

z

+ x

G

ω

x

m z

G

ω

y

+ v

x

−I

zz

ω

z

0 I

xx

ω

x

m

x

G

ω

z

+ v

y

−mv

x

−mx

G

ω

x

I

yy

ω

y

−I

xx

ω

x

0

(11)

It is not in the scope of this paper to go into further details on this (and also be-

cause space is limited) , but if you wish to further explore these items please check

[15][16][17]. The comprehensive non-linearized airship model is:

˙x

xy

=

−M

−1

xy

(D

xy

+ C

xy

) −M

−1

xy

g

xy

J

xy

[0]

3×3

x

xy

+

M

−1

xy

B

xy

[0]

3×2

u

xy

(12)

˙x

xz

=

−M

−1

xz

(D

xz

+ C

xz

) −M

−1

xz

g

xz

J

xz

[0]

3×3

x

xz

+

M

−1

xz

B

xz

[0]

3×2

u

xz

(13)

The general mass matrix is simpliﬁed for the xy and xz system, the same is true for the

entire matrix presented in the above systems.

The perturbed state variables for the heading direction (xy) matrix are x

xy

= (v

y

(t),

ω

x

(t), ω

z

(t), y(t), ψ(t), φ(t))

T

and the system input is u

xy

= F y; while for the xz

matrix they are x

xz

= (v

x

(t), v

z

(t), ω

x

(t), x(t), z(t), θ(t))

T

and u

xz

= (F x, F z)

T

.

2.3 Environmental Disturbances

In order to test the robustness of the controller design we simulate the behaviour of the

airship with perturbations acting along the ﬂight path, the perturbations are added to

the corresponding simulink models taking into account the Jacobean matrixes used (see

[12] for further details). Two types of environmental perturbations are simultaneously

considered and simulated: turbulence and wind. Both contributions were limited to a

maximum of one quarter of the airship thrusters maximum torque.

Turbulence. By deﬁnition turbulence is a state of ﬂuid ﬂow in which the instantaneous

velocities exhibit irregular and apparently random ﬂuctuations so that in practice only

statistical properties can be recognized and subjected to analysis. With this in mind a

perturbation model with stochastic white noise properties was used. This perturbation

affects the airship along the longitudinal and translational axis (see Fig. 4, panel A).

Wind. The second perturbation used is a gust wind. The wind model that we used

can be found in the Matlab Simulink aerospace block set. The mathematical form is as

follows:

26

u

w

= W

6

ln

z

z

0

ln

20

z

0

, 1 ≤ z ≤ 300m (14)

where u

w

is the mean wind speed, W

6

is the measured wind speed at an altitude of

6m, z is the airship altitude, and z

0

is a constant equal to 0.0045 for Category C ﬂight

phases and 0.18m for all other ﬂight phases. We considered phase C (terminal ﬂight)

due to the fact that this is the one that we feel is more adequate to a slow moving airship

(see Fig. 4, panel B). The effect of the wind on the airship frame is obtained through

the use of the corresponding Jacobean matrixes: J

1

for the heading system and J

2

for

the translation control system (see [11]).

A B

Fig.4. In panel A we can see a stochastic perturbation acting. In panel B we can see the wind.

These are the plots of the perturbations that acted on the airships during the simulation in Fig. 8.

3 Building Airship Formations

Our approach is based on the so called Dynamic Approach to Behaviour Generation

([18][19][20]). To model the airships ﬂight behaviour we use its heading direction ψ

(i.e. yaw), forward velocity, v

x,b

, and altitude, z. Behaviour is generated by continu-

ously providing values to these variables, which control then the airships motors. The

time course of each of these variables is obtained from (ﬁxed point) solutions of dy-

namical systems. The attractor solutions (asymptotically stable states) dominate these

solutions by design. In the present design the controller that governs the behavioural

dynamics of ψ(t), v

x,b

(t) and altitude z(t) is deﬁned as a set of differential equations:

˙

ψ

i

= f(ψ

i

, parameters)

˙v

x,b,i

= g(v

x,b,i

, parameters) i = follower1, f ollower2, ...

˙z

i

= h(z

i

, parameters)

(15)

Task constraints deﬁne contributions to the vectors ﬁelds f , g and h. The complete

control architecture for the trajectory generation of the lead airship is described in [12]

27

in detail and we will not enter into any particular details in this paper. Next we build the

vector ﬁelds which erects an attractor at z

leader

+ ∆h

i,d

with relaxation

˙z

i

= −λ

z

(z

i

− z

leader

+ ∆h

i,d

) with λ

z

> 0 (16)

where z

leader

is the targets altitude and deﬁnes the desired valuefor the airships

altitude and λ

z

is the relaxation rate.

Now, consider two airships that navigate in a world, keeping the distance between

them constant. Then, we state that they are either in a column formation, if one is exactly

behind the other (see ﬁgure 5.a)), or in a line formation, if they navigate side-by-side

(see ﬁgure 5.c)), or in an oblique formation, otherwise (see ﬁgure 5.b)).

From this set of basic two airship formations, more complex ones can be derived,

as we will see later in section 3.4. Next, in sections 3.1 to 3.3 we present the control

architecture for each of these two airship formations.

A)

B)

C)

Fig.5. Possible formation for teams with only two airships. Note that i will refer to the leader

and j to the follower(s). The airships can either be in a) column formation; b) oblique formation;

c) line formation. The heading direction of the leader and the follower are, respectively, ψ

i

and

ψ

j

. γ

i

is the direction at which the follower sees the leader. l

i,d

is the desired distance between

both airships. ∆γ

i,d

is the desired difference between the followers heading and the direction at

which sees the leader.

3.1 Two Airships in Column

A dynamical system that causes a follower airship to navigate in column formation,

maintaining a constant distance, with its leader is:

˙

ψ

i

= f

col,i

= −λ

col

sin (ψ

i

− γ

i

) (17)

This dynamical system ensures that the airship steers to the desired heading direc-

tion, ψ

i

(the direction at which the follower sees its leader), by making it an assimptot-

28

ically stable state of the system. Parameter λ

col

(> 0) is the strength of attraction to the

attractor and corresponds to the relaxation rate.

Path velocity is controlled to ensure that the follower adequates its velocity to the

leader’s one, while trying to maintain the desired distance to it. This is accomplished

by making the value of the desired velocity equal to

v

i,d

=

v

j

− (l

i,d

− l

i

)/T

2c

if l

i

≥ l

i,d

−v

j

− (l

i,d

− l

i

)/T

2c

else

(18)

T

2

C is a parameter that smoothes the airship movement, by controlling its accelerations

and decelerations.

3.2 Two Airships in Oblique

A dynamical system that causes a follower airship to navigate in an oblique formation,

maintaining a constant distance and relative orientation, with its leader is:

˙

ψ

i

= f

oblique

(ψ

i

)

= f

attract

(ψ

i

) + f

repel

(ψ

i

) (19)

where each term deﬁnes an attractive force (k = attract, repel)

f

k

(ψ

i

) = −λ

oblique

λ

k

(l

i

)sin(ψ

i

− γ

k

) (20)

where the ﬁrst contribution, f

attract

, erects an attractor at a direction

γ

attract

= γ

i

+ ∆γ

i,d

− π/4 (21)

The strength of this attractor (λ

oblique

λ

attract

(l

i

) with λ

oblique

ﬁxed), increases with

distance, l

i

, between the two airships:

λ

attract

(l

i

) = 1/(1 + exp (−(l

i

− l

i,d

)/µ)). (22)

The second contribution, f

repel

, sets an attractor at a direction pointing away from

the leader,

γ

repel

= γ

i

+ ∆γ

i,d

+ π/4 (23)

with a strength (λ

oblique

λ

repel

(l

i

)) that decreases with distance, l

i

, between the airships,

λ

repel

(l

i

) = 1 − λ

attract

(l

i

). (24)

The attractor location of the resultant vector ﬁeld, is thus dependent on the distance

between the two airships. When the distance between the two airships is larger than

the desired distance the attractive force erected at direction γ

attract

is stronger than the

attractive set at direction γ

repel

. Their superposition leads to an attractor at a direction

still pointing towards the movement direction of the leader airship. Conversely, when

the distance between the two airships is smaller than the desired distance, the reverse

holds, i.e. the attractive force set at direction γ

attract

is now stronger than the attractive

force at direction γ

repel

. The resulting oblique formation dynamics exhibits an attractor

29

at a direction pointing away from the leader’s direction of movement. When the airships

are at the desired distance the two attractive forces have the same strength which leads

to a resultant attractor at the direction γ

i,d

= γ

i

+ ∆γ

i,d

. Path velocity is controlled

exactly in the same way as for column formation.

3.3 Two Airships in Line

A dynamical system that causes a follower airship to navigate in a line formation, main-

taining a constant distance, with its leader is similar to the one of oblique formation.

The only difference lies in ∆γ

i,d

, which is ﬁxed and equal ±π/2 depending on the

follower driving on the right or left of the leader.

In line formation, the path velocity does not depend only on the distance and ve-

locity of the leader, but we also have to take into account the heading direction of the

leader and the direction at which it is seen by the follower. A set of heuristic rules have

been written that make the follower accelerate or decelerate depending on the leader’s

pose relative to the follower:

v

i,d,line

= DE

1

· v

j

(1 − |sin(γ

i

)|) +

+ DE

2

· v

j

(1 − |cos(γ

i

)|) +

+ AC

1

· v

j

(1 + K

v

|sin(γ

i

)|) +

+ AC

2

· v

j

(1 + K

v

|cos(γ

i

)|) (25)

where DE

1

, DE

2

, AC

1

and AC

2

are mutually exclusive activation variables that em-

bed the relative attitude of the follower airship regarding the leader. They are set and

reset by testing the direction at which the leader is seen by the follower and the heading

direction of the leader (see [13] for details).

3.4 N-airship Formations

Teams of airships with more than two airships are built by specifying pairs of leader-

follower teams and stating the particular conﬁguration to achieve. A complete team

speciﬁcation is accomplished by means of a formation matrix:

S =

L

1

∆γ

1,d

l

1,d

∆h

1,d

L

2

∆γ

2,d

l

2,d

∆h

2,d

... ... ...

L

N

∆γ

N,d

l

N,d

∆h

n,d

(26)

For a team of N airships, where each airship is identiﬁed by a speciﬁc identiﬁcation

number, the formation matrix has N rows and four columns. Row i relates to the airship

with identiﬁcation number i. The contents of the columns specify the values that char-

acterize a formation, l

i,d

(cm) and ∆γi, d(rad) in columns two and three, respectively,

the identiﬁcation number of this airships leader and the fourth column, ∆h

i,d

(cm), is

the altitude difference to the leader airship. The lead airship is identiﬁed by having its

row with l

i,d

= 0 and ∆γi, d = 0, while the third column is the distance it should stop

from the target and the fourth column is the desired altitude for the formation(see Fig. 6

for an example).

30

Fig.6. Example of a formation. airship A

1

is the Lead airship, airship A

2

follows A

1

on the

left side and maintaining an oblique formation, airship A

3

follows A

1

on the right side and

maintaining an oblique formation. airship A

4

follow airship A

2

in a column formation. airships

A

5

follow airship A

3

maintaining a column formation.

4 Results

In ﬁgure 7 we can see a simulation run where the possible conﬁgurations are depicted.

The airships change from line formation to column formation after reaching a waypoint.

At the next waypoint the formation changes to a V formation and at the ﬁnal waypoint

they change to an oblique formation where the airships are at different altitudes

Waypoint

Waypoint

Waypoint

Waypoint

Fig.7. Simulations of the possible conﬁguration with three airships. In panel A we can se the

airships in a line formation, in B the airships are in a column formation, in C in V formation and

in D in oblique formation with the airships at different altitudes.

In ﬁgure 8 we can observe the airships moving in a cluttered environment main-

taning formation (as close as possible) and avoiding obstacles. Depicted here is a case

31

simmilar to the one that is depicted in ﬁgure 1. The target is placed behind an obstacle

and the Leader airship travels to that position. The Followers airships follow the master

in a oblique formation avoiding obstacles and keeping formation.

Target

A

B

C

D

Fig.8. In panels A through D we can see the airship formation traveling to a target position while

avoiding an obstacle in its ﬂight path.

5 Conclusions

We demonstrated that the formation control architecture is capable of controling n-

airship formations using simple behaviours. These simple behaviours are the column,

line and oblique formations of two airships. All N-airship formations can be ”built”

from these simple behaviours. These behaviours enable the formation to avoid station-

ary obstacles.

The simulation efforts were conducted in the presence of perturbations on the main

axis of motion. Although the model of the airship is inertial, we must note that the

formation control is done at a kinematic level, where the behaviour based approach to

non-linear dynamical systems generates the next reference point for the airship control

variables. It is expected that the airship actuators are capable of ”following the orders”

given by the dynamical system. The actuator model is also taken into account during

the simulation, nevertheless, the dynamical system does not generate forces or torques

to be applied directly, it generates reference values to be followed by the robot. As with

everything this has advantages and disadvantages. The main advantage is the fact that,

with some parameters adjustmentes, the control arquitecture can be used to control any

kind of airhip - the control architecture is not model dependent. On the other hand, it

is expected that the physical platform be able to perform adequatly to the references

provided.

32

References

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Int. Conf. Robotics and Automation, 2000

2. J. Desai, J. Ostrowski and V. Kumar: Modeling and Control of Formations of Nonholo-

nomic Mobile Robots, in IEEE Transactions on Robotics and Automation, 905-908, Decem-

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