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A Lyapunov-based approach forTime-Coordinated 3D Path-Following of Multiple Quadrotors
Venanzio Cichella, Isaac Kaminer, Enric Xargay, Vladimir Dobrokhodov,Naira Hovakimyan, A. Pedro Aguiar, and Ant´onio M. Pascoal
Abstract
—This paper focuses on the problem of developingcontrol laws to solve the
Time-Coordinated 3D Path-Following
task for multiple quadrotor UAVs in the presence of time-varying communication networks and spatial and temporalconstraints. The objective is to enable a ﬂeet of quadrotorsto track predeﬁned spatial paths while coordinating to achievesynchronization in both time and heading. One scenario is asymmetric exchange of position by four quadrotors initiallypositioned in four corners of a square room. When the missionstarts, every quadrotor is required to execute collision freemaneuvers and arrive at the opposite corner at the same desiredinstant of time. In this paper, the time-coordination task issolved by adjusting the second derivative of the coordinationvariable along the desired paths. Conditions are derived underwhich the coordination and path-following errors converge toa neighborhood of zero. Flight test results are presented tovalidate the theoretical ﬁndings.
I. I
NTRODUCTION
Avoiding harm’s ways requires the employment of in-telligent autonomous vehicles. This, along with recent ad-vances in miniature technology, brings a global spotlighton the development of Unmanned Aerial Vehicles (UAVs).Currently, the use of UAVs plays a crucial role in pre-venting exposure of human beings to uncertain and hostileenvironments, therefore avoiding any danger to the lives of operators. For instance, after being struck by the biggestrecorded earthquake and a devastating tsunami, Japan hasbeen ﬁghting a potential nuclear catastrophe by deployingUAVs in situations where the presence of human operatorswas hazardous.From a design point of view, and with a slight abuse of terminology, UAVs can be classiﬁed in two main categories:ﬁxed-wings and rotatory-wings.Compared to the ﬁxed-wings–which cannot freely move in any direction (rotate) orhold a constant position–, rotorcrafts can be deployed ina much wider variety of scenarios. Among rotatory-wingsaircraft, quadrotors play an important role in research areasas prototypes for real-life missions, including monitoring andexploration of small areas.A quadrotor consists of four blades, whose motion controlis achieved by adjusting the angular rate of one or morerotor discs. Control of quadrotors is quite challenging and
Research is supported in part by USSOCOM, ONR, AFOSR, ARO, andCO3AUVs of the EU.V. Cichella, E. Xargay, and N. Hovakimyan are with UIUC, Urbana,IL 61801, e-mail:
{
cichell2,xargay,nhovakim
}
@illinois.edu. I. Kaminerand V. Dobrokhodov are with NPS, Monterey, CA 93943, email:
{
kaminer,vldobr
}
@nps.edu. P. Aguiar and A. Pascoal are with IST, Lisbon,1049 Portugal, email: pedro,antonio@isr.ist.utl.pt.
has been addressed in many recent publications. To mentiona few, in [1] and [2] a stabilization and control algorithmis developed using Lyapunov stability theory. In [3] and [4]PD
2
and PID architectures are compared with LQR basedcontrol theory. Backstepping control is proposed in [5],while in [6] and [7] a visual-based feedback control law ispresented using camera measurements for pose estimation.Fuzzy-logic control techniques are proposed in [8]. Intelli-gent control, based on neural networks, is introduced in [9]to achieve vertical take-off and landing. Integral sliding modeand reinforcement learning control are presented in [10]as solutions for accommodating the nonlinear disturbancesfor outdoor altitude control. Finally, in [11] a trajectory-tracking control algorithm is formulated using the SpecialOrthogonal group
SO
(
3
)
for attitude representation, leadingto a simple and singularity-free solution for the trajectorytracking problem.Cooperation between multiple unmanned vehicles has alsoreceived signiﬁcant attention in the control community inrecent years. Relevant work includes spacecraft formationﬂying [12], UAV control [13], [14], coordinated controlof land robots [15], and control of multiple autonomousunderwater vehicles [16], [17]. However, much work remainsto be done to overcome numerous critical constraints. Forexample, one of the crucial problems is the presence of time-varying communication networks that arise due to temporaryloss of communication links and switching communicationtopologies [18], [19].Motivated by these challenges, we address the problemof
Time-Coordinated 3D Path-Following
(TCPF), where aset of quadrotor UAVs are requested to
converge to and follow desired prespeciﬁed paths under stringent temporalconstraints
. In the solution adopted, the path-following (PF)and time-coordination (TC) problems are almost decoupled.At the PF level, we assume there exists a control law capableof steering a quadrotor along its assigned path. At the TClevel, the synchronization problem is solved by adjustingthe commanded position and velocity of the quadrotorsinvolved in the mission, thus obtaining –indirectly– vehiclecoordination. Figure 1 captures the key concept describedabove.This paper is organized as follows. In Section II, wedeﬁne the PF and TC control problems, and present stability-related properties that the PF closed-loop system must sat-isfy. Then, a formal deﬁnition of the TCPF problem isgiven. In Section III we propose a solution for the TC prob-lem. Section IV formulates a PF algorithm that enables an
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A Lyapunov-based approach for Time-Coordinated 3D Path-Following of Multiple Quadrotors
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Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
PathGeneration
PF
i
TimeCoordinationNetworkExchangeQuadrotorsPFerrors
x
d,i
(
γ
i
)
x
i
u
i
γ
i
γ
i
γ
i
γ
Fig. 1. TCPF Control Scheme.
AR.Drone quadrotor to follow a desired path, and shows thatthe convergence properties of the TCPF system hold for thisparticular vehicle. Section V presents and discusses ﬂighttests results that illustrate the effectiveness of the proposedPF and TC algorithms. Finally, Section VI contains the mainconclusions.II. P
ROBLEM
F
ORMULATION
A. 3D Path-Following for a single quadrotor
Let
I
denote an inertial reference frame, andlet
x
i
(
t
)
∈
R
3
be the position of the center of massof the
i
th quadrotor in this inertial frame, resolved in
I
.Also, let
B
i
=
{
b
1
, b
2
, b
3
}
denote the body frame with itssrcin located at the center of mass of the
i
th quadrotor;vector
b
3
is normal to the plane deﬁned by the centers of thefour rotors –pointing upwards in non-inverted ﬂight–, whilevectors
b
1
and
b
2
lie in this plane, with
b
1
pointing out thenose and
b
2
completing the right-hand system. Further, let
x
d,i
(
γ
i
)
∈
R
3
be a desired path parameterized by
γ
i
, andassume that
∂x
d,i
∂γ
i
≤
v
d
max
,i
,
(1)for some
0
< v
d
max
,i
< v
max
, where
v
max
is the maximumoperational speed of the quadrotors. The choice of theparameterizing variable
γ
i
will be discussed later.Then, we can deﬁne the position error vector as
e
x,i
=
x
d,i
(
γ
i
)
−
x
i
∈
R
3
(2)and the velocity error vector as
e
v,i
=
∂x
d,i
(
γ
i
)
∂γ
i
˙
γ
i
−
˙
x
i
= ˙
x
d,i
(
γ
i
)
−
˙
x
∈
R
3
.
(3)Additionally, similar to [11], deﬁne the errors
e
˜
R,i
= 12
R
⊤
d,i
R
i
−
R
⊤
i
R
d,i
∨
,
(4)
e
Ω
,i
= Ω
i
−
R
⊤
i
R
d,i
Ω
d,i
,
(5)where
R
i
∈
SO
(
3
)
and
Ω
i
∈
R
3
are, respectively, the ro-tation matrix from the body-ﬁxed frame
B
i
to the inertialframe
I
and the angular velocity of the
i
th quadrotor in thebody-ﬁxed frame
B
i
;
R
d,i
∈
SO
(
3
)
represents the desiredattitude of the
i
th quadrotor with respect to the inertial frameand is generally expressed as a function of the position andvelocity errors,
e
x,i
and
e
v,i
, as well as the desired head-ing
ψ
d,i
;
Ω
d,i
satisﬁes
S
(Ω
d
) =
R
⊤
d
˙
R
d
; while the operators
(
·
)
∨
and
S
(
·
)
denote the
vee
and
hat
maps [11].With the above notation, we deﬁne the path-followinggeneralized error vector
x
PF,i
=
e
⊤
x,i
, e
⊤
v,i
, e
⊤
˜
R,i
, e
⊤
Ω
,i
⊤
∈
R
12
.
(6)The dynamics of the
i
th vehicle’s PF error vector can bemodeled as
˙
x
PF,i
=
f
i
(
x
PF,i
,u
i
)
,
(7)where
f
i
(
·
)
is a general nonlinear vector map and
u
i
is thecontrol signal vector. Then, the PF control problem can bedeﬁned as:
Problem 1 (Path-Following Problem):
Consider the
i
th quadrotor UAV and a given trajectory
x
d,i
(
γ
i
)
satisfying (1). We say that a controller
u
i
(
t
)
solves thePF control problem if the generalized PF error vector
x
PF,i
with the dynamic described in (7) satisﬁes
x
PF,i
(
t
)
≤
k
x
PF,i
(0)
e
−
λ
PF
t
,
for some parameter
k >
0
, rate of convergence
λ
PF
>
0
, anddomain of attraction
D
=
{
x
PF,i
∈
R
12
:
x
PF,i
≤
r
}
, r >
0
.
Assumption 1:
We assume that there exists a control law
u
i
(
t
)
that solves the PF problem deﬁned in Problem 1.
B. Time-Coordination
We now address the TC problem of a ﬂeet of
n
quadrotorUAVs. As will become clear, this problem will be solvedby adjusting –for each vehicle– the second derivative of theparameterizing variable
γ
i
(
t
)
.As described earlier, the desired path assigned to eachvehicle is parameterized by a variable
γ
i
,
i
= 1
,...,n
. Thechoice of the parameter
γ
i
is such that, if
γ
i
(
t
)
−
γ
j
(
t
) = 0
,
∀
i,j
∈ {
1
,...,n
}
,
i
=
j
and
˙
γ
i
(
t
) = 1
,
∀
i
∈ {
1
,...,n
}
,at some time
t
, then all the vehicles are synchronized andevolve at the desired speed.To achieve synchronization, the coordination variables
γ
i
have to be exchanged among the quadrotors over a sup-porting communications network. Using tools from graphtheory, we can model the information ﬂow as well as theconstraints imposed by the communicationtopology.We startby assuming that the
i
th UAV communicates only with aneighboring set of vehicles, denoted by
N
i
. We also assumethat the communication between two UAVs is bidirectionalwith no delays. The reader is referred to [20] for key conceptsand details on algebraic graph theory.Following the notation used in [21], we now let
L
(
t
)
∈
R
n
×
n
be the Laplacian of the graph
Γ(
t
)
.Let
Q
∈
R
(
n
−
1)
×
n
be a matrix such that
Q
1
n
= 0
,
QQ
⊤
=
I
n
−
1
, and deﬁne
¯
L
(
t
) =
QL
(
t
)
Q
⊤
; it can be shown
that
¯
L
∈
R
(
n
−
1)
×
(
n
−
1)
has the same spectrum as the Lapla-cian
L
(
t
)
without the eigenvalue
λ
1
= 0
. Finally, we let
¯
L
(
t
)
satisfy the persistency of excitation (PE) assumption:
t
+
T t
¯
L
(
τ
)
dτ
≥
µI
n
−
1
.
(8)Next, letting
γ
(
t
) = [
γ
1
(
t
)
,...,γ
n
(
t
)]
⊤
and
˙
γ
(
t
) = [˙
γ
1
(
t
)
,...,
˙
γ
n
(
t
)]
⊤
, we deﬁne the coordinationerror vectors
ξ
(
t
) =
Qγ
(
t
)
∈
R
n
−
1
,
(9)
z
(
t
) = ˙
γ
(
t
)
−
1
n
∈
R
n
.
(10)From the deﬁnition of
Q
it follows that, if
ξ
(
t
) = 0
n
,then
γ
i
−
γ
j
= 0
,
∀
i,j
∈ {
1
,...,n
}
. Note that convergenceof
z
(
t
)
to zero implies that the individual parameterizingvariables
γ
i
(
t
)
evolve at the desired rate 1.With the above notation, the coordination problem cannow be deﬁned as:
Problem 2 (Time-Coordination Problem):
Given a set of
n
3D desired trajectories
x
d,i
(
γ
i
)
, design feedback controllaws for
¨
γ
i
for all vehicles such that the coordinationerror vectors
ξ
and
z
, deﬁned in (9) and (10) respectively,
converge exponentially to a neighborhood of zero with rateof convergence
λ
TC
>
0
.
C. Time-Coordinated 3D Path-Following
Considering the PF and TC problems described above, wecan now deﬁne the combined TCPF control problem for aﬂeet of quadrotor UAVs.
Problem 3 (Time-Coordinated Path-Following Problem):
Consider a set of
n
quadrotor UAVs and a set of
n
3D desired trajectories
x
d,i
(
γ
i
)
. Assume the quadrotors cancommunicate over a communications network satisfying (8).Design feedback control laws
u
i
(
t
)
and
¨
γ
i
(
t
)
such that1) for each vehicle, the generalized PF error vec-tor
x
PF,i
(
t
)
deﬁned in (6) converges to a neighborhoodof zero;2) the coordination error vectors deﬁned in (9) and (10)
converge exponentially to zero.III. T
IME
-C
OORDINATED
3D P
ATH
-F
OLLOWING
:M
AIN
R
ESULT
To solve the TCPF problem, we let the evolution of
γ
i
(
t
)
be given by
¨
γ
i
=
−
b
(˙
γ
i
−
1)
−
a
j
∈N
i
(
γ
i
−
γ
j
)
−
d
¯
α
i
(
x
PF,i
)
,γ
i
(0) = 0
,
˙
γ
i
(0) = 1
,
where
a
,
b
,
d
are positive coordination control gains, while
¯
α
i
(
x
PF,i
)
is deﬁned as
¯
α
i
(
x
PF,i
) =˙
x
⊤
d,i
e
x,i
˙
x
d,i
+
δ
+˙
x
⊤
d,i
e
v,i
˙
x
d,i
+
δ ,
with
δ
being a positive design parameter. The dynamics of
γ
(
t
)
can be written in compact form as
¨
γ
=
−
bz
−
aLγ
−
d
¯
α
(
x
PF
)
, γ
(0) = 0
n
,
˙
γ
(0) = 1
n
,
(11)where
x
PF
= [
x
⊤
PF,
1
,...,x
⊤
PF,n
]
⊤
∈
R
12
n
,
¯
α
(
x
PF
) = [¯
α
1
(
x
PF,
1
)
,...,
¯
α
n
(
x
PF,n
)]
⊤
∈
R
n
.
Then, the Lemma below states the main result of thispaper:
Lemma 1:
Consider a set of
n
quadrotor UAVs and a setof
n
3D desired trajectories
x
d,i
(
γ
i
)
. Given
n
PF algorithmssatisfying Assumption 1 and the coordination control lawdescribed in (11), then there exist control gains
a
,
b
,
d
, and
δ
that solve the TCPF control problem 3. In particular, it can beshown that the vector
x
TCPF
= [
x
⊤
PF
,ξ
⊤
,z
⊤
]
⊤
convergesexponentially fast to a neighborhood of zero with rate of convergence
λ
= min(
λ
PF
, λ
TC
)
,
(12)where
λ
TC
< µ
2
T
(1 +
n
2
T
)
2
,
(13)and with domain of attraction
D
c
x
TCPF
∈
R
14
n
−
1
:
x
PF,i
≤
r
.
(14)
Proof.
An outline of the proof is given in the Appendix.
Remark 1:
The rate of convergence
λ
PF
depends on theproperties of the adopted PF control law. If the PF controllaw has a rate of convergence greater than
λ
TC
, then the rateof convergence of the TCPF system is equal to the rate of convergence of the TC algorithm.
Remark 2:
Note that the rate of convergence of the TC al-gorithm strictly depends on the quality of the communicationnetwork (parameters
µ
and
T
).IV. I
LLUSTRATIVE EXAMPLE
: TCPF
WITH
AR.D
RONES
To test the performance of the algorithm presented in theprevious section, we adopted the ﬂying robot architecturerealized by Parrot AR.Drone company. To this end, we ﬁrstdeveloped a PF algorithm that satisﬁes the conditions de-scribed in Section II and that uses the control input providedby the AR.Drone autopilot, which accepts control commandsfor linear velocity along the inertial vertical channel
˙
z
, Eulerangles
θ
and
φ
for the horizontal motion, and yaw rate
˙
ψ
.Next, we reformulate the PF problem presented in Section IIfor this particular platform, and derive a PF algorithm basedon simple linear control.
A. PF Error Dynamics
For simplicity, we write separately the horizontal andvertical motions:
x
= [(Π
x
)
⊤
, e
⊤
3
x
]
⊤
,v
= [(Π
v
)
⊤
, e
⊤
3
v
]
⊤
,

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