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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1
Taming mismatches in inter-agent distances for the
formation-motion control of second-order agents.
Hector Garcia de Marina, Member, IEEE, Bayu Jayawardhana, Senior Member, IEEE, and Ming Cao, Senior
Member, IEEE
Abstract—This paper presents the analysis on the influence
of distance mismatches on the standard gradient-based rigid
formation control for second-order agents. It is shown that,
similar to the first-order case as recently discussed in the
literature, these mismatches introduce two undesired group
behaviors: a distorted final shape and a steady-state motion of the
group formation. We show that such undesired behaviors can be
eliminated by combining the standard formation control law with
distributed estimators. Finally, we show how the mismatches can
be effectively employed as design parameters in order to control
a combined translational and rotational motion of the formation.
Index Terms—Formation Control, Rigid Formation, Motion
Control, Second-Order dynamics.
I. INTRODUCTION
RECENT years have witnessed a growing interest in
coordinated robot tasks, such as, area exploration and
surveillance [1], [2], robot formation movement for energy
efficiency [3], and tracking and enclosing a target [4], [5].
In many of these team-work scenarios, one of the key tasks
for the agents is to form and maintain a prescribed formation
shape. Gradient-based control has been widely used for this
purpose [6]. In particular, distance-based control [7]–[11] has
gained popularity since the agents can work with their own
local coordinates and the desired shape of the formation under
control is exponentially stable [12], [13]. However, exponen-
tial stability cannot prevent undesired steady-state collective
motions, e.g., constant drift, if disturbances, such as biases
in the range sensors or equivalently mismatches between the
prescribed distances for neighboring agents, are present. This
misbehavior has been carefully studied for agents governed by
single integrator dynamics in [14], [15] and an effective solu-
tion to get rid of such misbehavior using estimators has been
reported in [16]. The proposed estimator-based tool works
without requiring any communication among agents. This is
a desired feature since the above mentioned issue with the
mismatches cannot usually be solved in a straightforward way
by sending back and forth more communication information
between the agents for several reasons: the sensing bias can
be time-varying due to factors such as the environment’s
H. Garcia de Marina is with the Ecole Nationale de l’Aviation Civile,
University of Toulouse, Toulouse, France. (e-mail: hgdemarina@ieee.org).
B. Jayawardhana and M. Cao are with the Engineering and Technol-
ogy Institute of Groningen, University of Groningen, 9747 AG Groningen,
The Netherlands. (e-mail: {b.jayawardhana, m.cao}@rug.nl). This work was
supported by the the EU INTERREG program under the auspices of the
SMARTBOT project and the work of Cao was also supported by the European
Research Council (ERC-StG-307207).
temperature; the same range sensor can produce different
readings for the same physical distance in face of random
measurement noises, making communication-based correction
costly; a continuous (or regular) communication among agents
may not even be possible or desired; and the agents may have
different clocks complicating the data comparison in real-time
computation.
In this paper for second-order agents, i.e., agents modeled
by double integrator dynamics, we extend the recent findings
in [14]–[17] on mismatched formation control, the cancellation
of the undesired effects via distributed estimation, and the
formation motion control by turning the mismatches into
distributed parameters. There are practical benefits justifying
such extension. For example, a formation controller employing
undirected sensing topologies has inherent stability properties
that are not present if directed topologies1are employed, espe-
cially for agents with higher order dynamics [6]. Another key
aspect of employing second-order dynamics is that the control
actions can be used as the desired acceleration in a guidance
system feeding the tracking controller of a mechanical system,
such as the ones proposed for quadrotors in [18], [19] or
marine vessels in [20]. This clearly simplifies such tracking
controllers compared to the case of only providing desired
velocities derived from first-order agent dynamics.
The extensions presented in this paper require new technical
constructions that go much beyond what is needed for first-
order agents. For example, a key step in the logic presented
in [14]–[17] is based on the fact that the dynamics of the
error signal, which measures the distortion with respect to
the desired shape, is an autonomous system for first-order
agents. This does not hold anymore in the second-order case.
Consequently, additional technical steps are developed for
extending the results of a mismatched formation control to
second-order agents.
The problem of motion and formation control can be solved
simultaneously if the mismatches are well understood and
not treated as disturbances but as design parameters. Indeed,
this is the strategy followed in [17], where we have turned
mismatches in the prescribed distances into design parameters
in order to manipulate the way how the collective motion is
realized. The act of turning mismatches to distributed motion
parameters allows one to address more complicated problems
such as moving a rigid formation (including rotational motion)
without leaders [6], or tracking and enclosing a free target.
1For directed topologies, only one agent per edge controls the inter-agent
distance. Hence, it is free of mismatches.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2
The desired motion is designed with respect to a frame of
coordinates fixed at the desired rigid-body shape. This design
allows to preserve the ordering of the agents during the motion,
e.g., which agent is leading at the front of the group. An impor-
tant aspect of this approach is that for low motion speeds the
agents do not need to measure any relative velocities as what
is commonly required in swarms of second-order agents [21].
Consequently, the sensing requirements are reduced. It will
be shown that the motion parameters for second-order agents
can be directly computed from the first-order case as in [17].
Unfortunately, the Lyapunov function for proving the stability
of the desired steady-state motion is not as straightforward
as in [17], since the standard quadratic function, involving
the norms of the errors regarding the distortion of the desired
shape and the velocity of the agents, can only be used to
prove asymptotic [6] but not exponential stability, which is the
key property for determining the necessary gains and region
of attraction for the formation and motion controller in [17].
Nevertheless, we will be able to show that the stability of the
closed-loop formation motion system is indeed exponential for
second-order agents too.
The rest of the paper is organized as follows. After some
definitions and clarification of notation in Section II, we
show in Section III the analysis about the undesired effects
on a formation of second-order agents in the presence of
small mismatches. It turns out that in addition to a distortion
with respect to the desired shape, we have an undesired
steady-state collective motion which is consistent with the
one described in [22] for first-order agents. In Section IV,
we propose two designs for a distributed estimator in order
to prevent the mentioned robustness issues. The first design
eliminates the undesired steady-state motion and is able to
bound the steady-state distortion with respect to a generic
rigid desired shape. In particular, this first design completely
removes the distortion for the special cases of triangles and
tetrahedrons. The second design is less straightforward and
requires the calculations of some lower bounds for certain
gains. However, it eliminates the undesired steady-state motion
and distortion for any desired shape. In Section V, we turn the
mismatches into distributed motion parameters in a similar
way as in [17] in order to design the stationary motion of
the formation without distorting the desired shape. Finally, in
order to validate the proposed algorithms from Sections IV and
V, simulation results with second-order agents are provided in
Section VI.
II. PRELIMINARIES
In this section, we introduce some notations and concepts
related to graphs and rigid formations. For a given matrix
A∈IRn×p, define A∆
=A⊗Im∈IRnm×pm, where the
symbol ⊗denotes the Kronecker product, m= 2 for the
2D formation case or m= 3 for the 3D one, and Im
is the m-dimensional identity matrix. For a stacked vector
x∆
=xT
1xT
2. . . xT
kTwith xi∈IRn, i ∈ {1, . . . , k},
we define the diagonal matrix Dx
∆
= diag{xi}i∈{1,...,k}∈
IRkn×k. We denote by |X| the cardinality of the set Xand by
||x|| the Euclidean norm of a vector x. We use 1n×mand
0n×mto denote the all-one and all-zero matrix in IRn×m
respectively and we will omit the subscript if the dimensions
are clear from the context.
A. Graphs and Minimally Rigid Formations
We consider a formation of n≥2autonomous agents whose
positions are denoted by pi∈IRm. The agents can measure
their relative positions with respect to its neighbors. This
sensing topology is given by an undirected graph G= (V,E)
with the vertex set V={1, . . . , n}and the ordered edge set
E ⊆ V ×V. The set Niof the neighbors of agent iis defined
by Ni
∆
={j∈ V : (i, j)∈ E}. We define the elements of the
incidence matrix B∈IR|V|×|E| for Gby
bik
∆
=
+1 if i=Etail
k
−1if i=Ehead
k
0otherwise,
where Etail
kand Ehead
kdenote the tail and head nodes, respec-
tively, of the edge Ek, i.e. Ek= (Etail
k,Ehead
k). A framework
is defined by the pair (G, p), where p= col{p1, . . . , pn}is
the stacked vector of the agents’ positions pi, i ∈ {1, . . . , n}.
With this last definition at hand, we define the stacked vector
of the measured relative positions by
z=BTp,
where each vector zk=pi−pjin zcorresponds to the relative
position associated with the edge Ek= (i, j).
For a given stacked vector of desired relative positions
z∗= [ z∗
1
Tz∗
2
T... z∗
|E|
T]T, the resulting set Zof the possible
formations with the same shape is defined by
Z∆
=I|E| ⊗ Rz∗,(1)
where Ris the set of all rotational matrices in 2D or 3D.
Roughly speaking, Zconsists of all those formation positions
that are obtained by rotating z∗.
Let us now briefly recall the notions of infinitesimally rigid
framework and minimally rigid framework from [23]. Define
the edge function fGby fG(p) = col
kkzkk2and we denote
its Jacobian by
2R(z)=2DT
zBT,(2)
where R(z)is called the rigidity matrix. A framework (G, p)
is infinitesimally rigid if rank{R(z)}= 2n−3when it is
embedded in R2or if rank{R(z)}= 3n−6when it is
embedded in R3. Additionally, if |E| = 2n−3in the 2D
case or |E| = 3n−6in the 3D case, then the framework is
called minimally rigid. Roughly speaking, the only motions
that one can perform over the agents in an infinitesimally
and minimally rigid framework, while they are already in
the desired shape, are the ones defining translations and
rotations of the whole shape. Some graphical examples of
infinitesimally and minimally rigid frameworks are shown
in Figure 1. If (G, p)is infinitesimally and minimally rigid,
then, similar to the above, we can define the set of resulting
formations Dby
D∆
=nz|||zk|| =dk, k ∈ {1,...,|E|}o,
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 3
(a) (b) (c)
Fig. 1: a) The square without an inner diagonal is not rigid
since we can smoothly move the top two nodes while keeping
the other two fixed without breaking the distance constraints;
b) The square can be done locally minimally rigid in IR2
if we add an inner diagonal; c) The tetrahedron in IR3is
infinitesimally and minimally rigid.
where dk=||z∗
k||, k ∈ {1,...,|E|}.
Note that in general it holds that Z ⊆ D. For a desired
shape, one can always design Gto make the formation
infinitesimally and minimally rigid. In fact, an infinitesimally
and minimally rigid framework with two or more vertices in
IR2can always be constructed through the Henneberg con-
struction [24]. In IR3one can construct a set of infinitesimally
and minimally rigid frameworks via insertion starting from a
tetrahedron, if each newly added vertex with three newly links
forms another tetrahedron as well.
B. Frames of coordinates
It will be useful for describing the motions of the infinites-
imally and minimally rigid formation to define a frame of
coordinates fixed to the desired formation itself. We denote
by Ogthe global frame of coordinates fixed at some point of
IRmwith some arbitrary fixed orientation. In a similar way,
we denote by Obthe body frame fixed at the centroid pcof
the desired rigid formation. Furthermore, if we rotate the rigid
formation with respect to Og, then Obis also rotated in the
same manner. Let bpjdenote the position of agent jwith
respect to Ob. To simplify the notation whenever we represent
an agents’ variable with respect to Og, the superscript is
omitted, i.e., pj
∆
=gpj.
III. ROBU ST NE SS I SS UE S DU E TO MISMATCHES IN
FORMATION GRADIENT-BAS ED C ON TRO L
A. Gradient Control
Consider a formation of nagents with the sensing topology
Gfor measuring the relative positions among the agents. The
agents are modelled by a second-order system given by
(˙p=v
˙v=u, (3)
where uand vare the stacked vector of control inputs
ui∈IRmand vector of agents’ velocity vi∈IRmfor
i={1, . . . , n}respectively.
In order to control the shape, for each edge Ek= (i, j)in
the infinitesimally and minimally rigid framework we assign
the following potential function Vk
Vk(||zk||) = 1
4(||zk||2−d2
k)2,
with the gradient along pior pjgiven by
∇piVk=−∇pjVk=zk(||zk||2−d2
k).
In order to control the agents’ velocities, for each agent iin
the infinitesimally and minimally rigid framework we assign
the following (kinetic) energy function Ti
Ti(vi) = 1
2||vi||2,
with the gradient along vibe given by
∇viTi=vi.
One can check that for the potential function
φ(p, v) =
|V|
X
i=1
Ti+
|E|
X
k=1
Vk,(4)
the closed-loop system (3) with the control input
u=−∇vφ− ∇pφ, (5)
becomes the following dissipative Hamiltonian system [25]
(˙p=∇vφ
˙v=−∇vφ− ∇pφ. (6)
Considering (4) as the storage energy function of the Hamil-
tonian system (6), one can show the local asymptotic conver-
gence of the formation to the shape given by Dand all the
agents’ velocities to zero [6], [26].
Consider the following one-parameter family of dynamical
systems Hλgiven by
˙p
˙v=−λIm|V | −(1 −λ)Im|V|
(1 −λ)Im|V| Im|V | ∇pφ
∇vφ,(7)
where λ∈[0,1], which defines all convex combinations of
the Hamiltonian system (6) and a gradient system. The family
Hλhas two important properties summarized in the following
lemma.
Lemma 3.1: [26]
•For all λ∈[0,1], the equilibrium set of Hλis given by
the set of the critical points of the potential function φ,
i.e. Ep,v =npTvTT:∇φ=0o.
•For any equilibrium pTvTT∈Ep,v and for all
λ∈[0,1], the numbers of the stable, neutral, and unstable
eigenvalues of the Jacobian of Hλare the same and
independent of λ.
This result has been exploited in [13] in order to show the
local exponential convergence of z(t)and v(t)to Dand 0
respectively. In the following brief exposition we revisit such
exponential stability via a combination of Lyapunov argument
and Lemma 3.1, which will play an important role in Section
III-B.
Define the distance error corresponding to the edge Ekby
ek=||zk||2−d2
k,(8)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 4
whose time derivative is given by ˙ek= 2zT
k˙zk. Consider the
following autonomous system derived from (7) with λ= 0.5
˙p=−1
2BDze+1
2v
˙z=−1
2BTBDze+1
2BTv(9)
˙e=−DT
zBTBDze+DT
zBTv(10)
˙v=−1
2BDze−v, (11)
where eis the stacked vector of ek’s for all k∈ {1,...,|E|}.
Define the speed of the agent iby
si
∆
=||vi||,
whose time derivative is given by ˙si=vT
i˙vi
si. The compact
form involving all the agents’ speed can be written as
˙s=D˜sDT
v˙v=−1
2D˜sDT
vBDze−D˜sDT
vv, (12)
where sand ˜sare the stacked vectors of si’s and 1
si’s for all
i∈ {1,...,|V|} respectively. Now we are ready to show the
local exponential convergence to the origin of the speed of the
agents and the error distances in the edges.
Lemma 3.2: The origins e=0and s=0of the error and
speed systems derived from (6) are locally exponentially stable
if the given desired shape Dis infinitesimally and minimally
rigid.
Proof:Consider eλand sλas the stacked vectors of
the error signals ekand speeds skderived from (7) for any
λ∈[0,1], which includes the system (6) for λ= 1. From
the definition of ek, we know that all the eλshare the same
stability properties by invoking Lemma 3.1, so do sλas well.
Consider the following candidate Lyapunov function for the
autonomous system (9)-(12) derived from (7) with λ= 0.5
V=1
2||e||2+||s||2,
whose time derivative satisfies
dV
dt=eT˙e+ 2sT˙s
=−eTDT
zBTBDze+eTDT
zBTv−sTD˜sDT
v
| {z }
vT
BDze
−2sTD˜s
|{z}
11×|V|
DT
vv
=−eTDT
zBTBDze−2||s||2.(13)
We first note that the elements of the matrix DT
zBTBDzare
of the form zT
izjwith i, j ∈ {1,...,|E|}. It has been shown
in [14] that for minimally rigid shapes these dot products
can be expressed as a (local) smooth functions of e, i.e.
zT
izj=gij (e), allowing us to write DT
zBTBDz=Q(e). For
infinitesimally minimally rigid frameworks R(z)is full rank,
so we have that Q(0) = R(z∗)RT(z∗)is positive definite
with z∗∈ D. Moreover, since the eigenvalues of a matrix
are continous functions of their entries, we have that Q(e)is
positive definite in the compact set Q∆
={e:||e||2≤ρ}for
some ρ > 0. Therefore, if the initial conditions for the error
signal and the speed satisfy ||e(0)||2+||s(0)||2≤ρ, since V
is not increasing we have that
dV
dt≤ −σmin||e||2−2||s||2,(14)
where σmin >0is the smallest (and always positive) eigen-
value of Q(e)in the compact set Q. Hence we arrive at the
local exponential convergence of e(t)and s(t)to the origin.
Remark 3.3: It is worth noting that the region of attraction
determined by ρin the proof of Lemma 3.2 for λ= 0.5might
be different from the one for λ= 1, since Lemma 3.1 only
refers to the Jacobian of (7), i.e., the linearization of the system
about the equilibrium.
It can be concluded from the exponential convergence to zero
of the speeds of the agents s(t)that the formation will even-
tually stop. This implies that p(t)will converge exponentially
to a finite point in IRmas z(t)converges exponentially to D.
B. Robustness issues caused by mismatches
Consider a distance-based formation control problem with
n= 2. It is not difficult to conclude that if the two agents
do not share the same prescribed distance to maintain, then
an eventual steady-state motion will happen regardless of
the dynamics of the agents since the agent with a smaller
prescribed distance will chase the other one. Therefore, for
n > 2it would not be surprising to observe some collective
motion in the steady-state of the formation if the neighboring
agents do not share the same prescribed distance to maintain.
Consider that two neighboring agents disagree on the de-
sired squared distance d2
kin between, namely
d2tail
k=d2head
k−µk,(15)
where dtail
kand dhead
kare the different desired distances that the
agents iand jrespectively in Ek= (i, j)want to maintain for
the same edge, so µk∈IR is a constant mismatch. It can be
checked that this disagreement leads to mismatched potential
functions, therefore agents iand jdo not share anymore the
same Vkfor Ek= (i, j), namely
Vi
k=1
4(||zk||2−d2
k+µk)2, V j
k=1
4(||zk||2−d2
k)2,
under which the control laws for agents iand juse the
gradients of Vi
kand Vj
krespectively for the edge Ek= (i, j).
In the presence of one mismatch in every edge, the control
signal (5) can be rewritten as
u=−v−BDze−S1Dzµ, (16)
where S1is constructed from the incidence matrix by setting
its −1elements to 0, and µ∈IR|E| is the stacked column
vector of µk’s for all k∈ {1,...,|E|}. Note that (16) can be
also written as
u=−v−BDze−A1(µ)z, (17)
where the elements of A1are
aik
∆
=(µkif i=Etail
k
0otherwise. (18)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 5
Inspired by [14], we will show how µcan be seen as a
parametric disturbance in an autonomous system whose origin
is exponentially stable. Consider the dynamics of the error
signal eand the speed of the agents sderived from system
(3) with the control input (5)
˙e= 2DT
zBTv(19)
˙s=−s−D˜sDT
vBDze, (20)
and define
αki =zT
kvi, k ∈ {1,...,|E|}, i ∈ {1,...,|V|} (21)
βij =vT
ivj, i, j ∈ {1,...,|V|}, i 6=j. (22)
We stack all the αki’s and βij ’s in the column vectors
α∈IR|E||V| and β∈IR |V|(|V |−1)
2respectively and define
γ∆
=eTsTαTβTT. We recall the result from Lemma
3.2 that for any infinitesimally and minimally rigid framework
there exists a neighborhood Uzabout this framework such that
for all zi, zj∈ Uzwith i, j ∈ {1,...,|E|} we can write zT
izj
as a smooth function gij (e). Then using (19)-(22) we get
˙γ=f(γ),(23)
which is an autonomous system whose origin is locally expo-
nentially stable using the results from Lemmas 3.1 and 3.2.
Obviously, in such a case, the following Jacobian evaluated at
γ=0
J=∂f (γ)
∂γ γ=0
,
has all its eigenvalues in the left half complex plane. From the
system (3) with control law (16) we can extend (23) but with
a parametric disturbance µbecause of the third term in (16),
namely
˙γ=f(γ , µ),(24)
where f(γ, 0)is the same as in (23) derived from the gradient
controller. Therefore, for a sufficiently small ||µ||, the Jacobian
∂f (γ,µ)
∂γ γ=0 is still a stable matrix since the eigenvalues of
a matrix are continuous functions of its entries. Although
system (24) is still stable under the presence of a small
disturbance µ, the equilibrium point is not the origin in
general anymore but γ(t)→ˆγ(µ)as tgoes to infinity, where
γµ
∆
= ˆγ(µ)is a smooth function of µwith zero value if
µ=0[27]. This implies that in general each component
of e, s, α and βconverges to a non-zero constant with the
following two immediate consequences: the formation shape
will be distorted, i.e., e6=0; and the agents will not remain
stationary, i.e., s6=0. The meaning of having in general non-
zero components in αis that the vector velocities of the agents
have a fixed relation with the steady-state shape, while the
fixed components in βdenote a constant relation between the
vectors of the agents’ velocities. If the disturbance ||µ|| is
sufficiently small, then ||γµ|| < ρ for some small ρ∈IR+
implying that ||eµ|| < ρ, and if further ρis sufficiently small,
then the stationary distorted shape is also infinitesimally and
minimally rigid. In addition, since the speeds of the agents
converge to a constant (in general non-zero constant), then
only translations and/or rotations of the stationary distorted
shape can happen.
Theorem 3.4: Consider system (3) with the mismatched
control input (16) where the desired shape for the formation
is infinitesimally and minimally rigid. There exist sufficiently
small 1, 2∈IR+such that if the mismatches satisfy ||µ|| ≤
1, then the error signal e(t)→e∗∈IR|E| as t→ ∞ with
||e∗|| ≤ 2such that the steady-state shape of the formation
is still infinitesimally and minimally rigid but distorted with
respect to the desired one. Moreover, the velocity of the agents
vi(t)→v∗
i(t)as t→ ∞, where all the v∗
i(t),∀i∈ {1, . . . , n}
define a steady-state collective motion that can be captured
by constants angular and translational velocities bω∗and bv∗
c,
respectively, where Obhas been placed in the centroid of the
resultant distorted infinitesimally and minimally rigid shape.
Proof:We have that system (24), derived from (3) and
(16), is self-contained and its origin is locally exponentially
stable with µ=0|E|×1. Then, a small parametric perturbation
µsuch that ||µ|| ≤ 1, for some positive small 1, does not
change the exponential stability property of (24). However,
the equilibrium point of (24) at the origin can be shifted. In
particular, the new shifted equilibrium is a continuous function
of µ, therefore e(t)→eµ∈IR|E| as tgoes to infinity, where if
1is sufficiently small, then ||eµ|| ≤ 2such that the stationary
shape is infinitesimally rigid. We also have that the elements
of s(t)→sµas tgoes to infinity with sµ∈IR|V|. Note that
the elements of sµare non-negative and in general non-zero.
Hence, the agents will not stop moving in the steady-state.
Since the steady-state shape of the formation locally converges
to an infinitesimally and minimally rigid one, from the error
dynamics (19) we have that
DT
z(t)BTv(t) = R(z(t)) v(t)→0m|V|×1, t → ∞,
therefore v(t)→vµ(t)as tgoes to infinity, where the
non-constant vµ(t)∈IR|V| belongs to the null space of
R(zµ(t)),zµ(t)∈ Zµand the set Zµis defined as in (1)
but corresponding to the inter-agent distances of the distorted
infinitesimally and minimally rigid shape with e=eµ. Note
that obviously, the evolution of z(t)is a consequence of the
evolution of agents’ velocities in v(t). The null space of R(zµ)
corresponds to the infinitesimal motions δpifor all isuch
that all the inter-agent distances of the distorted formation are
constant, namely
R(zµ)δp =R(zµ)vµδt =0m|V|×1,
or in order words
vi(t)→vµi(t), t → ∞,(25)
where the velocities vµi(t)’s for all the agents are the result of
rotating and translating the steady-state distorted shape defined
by Zµ. This steady-state collective motion of the distorted
formation can be represented by the rotational and translational
velocities bω∗(t)∈IRmand bvc(t)∗∈IRmat the centroid of
the distorted rigid shape.
Now we are going to show that the velocities bω∗(t)
and bvc(t)∗are indeed constant. By definition we have that
||vµi(t)|| =sµi. Since the speed sµifor agent iis constant
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 6
OgOb
bω∗
bv∗
c
Fig. 2: The velocities bω∗and bv∗
cat the centroid of the tetra-
hedron rotates and translates the infinitesimally and minimally
rigid formation respectively. If the velocitiy vectors bω∗and
bv∗
care constant, then the formation describes a closed orbit
in the plane where the projection of bv∗
cover such a plane
and bω∗is perpendicular (always the case in 2D formations)
plus a constant drift along the projection of bv∗
cover bω∗.
Note that for the case where bv∗
cand bω∗are only parallel or
perpendicular, then only the drift or the closed orbit motion,
respectively, will occur.
but not its velocity vµi(t), the acceleration aµi(t) = dvµi(t)
dt
is perpendicular to vµi(t). The expression of aµi(t)can be
derived from (17) and is given by
aµi(t) = −vµi(t)−
|E|
X
k=1
bikzµk(t)eµk+
|E|
X
k=1
aikzµk(t),(26)
where bik are the elements of the incidence matrix, and aik
are the elements of the perturbation matrix A1as defined in
(18). From (26) it is clear that the norm ||aµi(t)|| = Γi(γµ)
is constant. In addition, since aµi(t)is a continuous function,
i.e., the acceleration vector cannot switch its direction, and
it is perpendicular to vµi(t), the only possibility for the
distorted formation is to follow a motion described by constant
velocities bω∗and bv∗
cat its centroid.
Remark 3.5: In particular, in 2D the distorted formation will
follow a closed orbit if Γi(γµ)6= 0 for all i, or a constant drift
if Γi(γµ)=0for all i. This is due to the fact that in 2D, bω∗
and bv∗are always perpendicular or equivalently aµi(t)and
vµi(t)lie in the same plane. The resultant motion in 3D is the
composition of a drift plus a closed orbit, since bω∗and bv∗
c
are constant and they do not need to be perpendicular to each
other as it can be noted in Figure 2.
Remark 3.6: Although the disturbance µacts on the acceler-
ation of the second-order agents, it turns out that the resultant
collective motion has the same behavior as for having the
disturbance µacting on the velocity for first-order agents. A
detailed description of such a motion related to the disturbance
in first-order agents can be found in [17], [22].
IV. EST IM ATOR -BASED GRADIENT CONTRO L
It is obvious that if for the edge Ek= (i, j)only one of
the agents controls the desired inter-agent distance, then a
mismatch µkcannot be present. However, this solution leads
to a directed graph in the sensing topology and the stability
of the formation can be compromised [6]. It is desirable to
maintain the undirected nature of the sensing topology since
it comes with intrinsic stability properties. Then, the control
law (16) must be augmented in order to remove the undesired
effects described in Section III. A solution was proposed in
[16] for first-order agents consisting of estimators based on the
internal model principle. For each edge Ek= (i, j), there is
only one agent that is assigned to be the estimating agent
which is responsible for running an estimator to calculate
and compensate the associated µk. The estimator proposed
in [16] is conservative since the estimator gain has to satisfy
a lower-bound (which can be explicitly computed based on
the initial conditions) in order to guarantee the exponential
stability of the system. Using such distributed estimators, all
the undesirable effects are removed at the same time as the
estimating agent calibrates its measurements with respect to
the non-estimating agent. Another minor issue in the solution
of [16] is that the estimating agents cannot be chosen arbi-
trarily. Here we are going to present an estimator for second-
order agents where the estimating agents and the estimator
gain can be chosen arbitrarily (thus, removing the restrictive
conditions in [16]). The solution removes the effect of the
undesired collective motion but at the cost of not achieving
accurately the desired shape Z, where a bound on the norm of
the signal error e(t)for all time t, however, can be provided.
Furthermore, we will show that for the particular cases of
the triangle and tetrahedron, the proposed estimator achieves
precisely the desired shapes.
Let us consider the following distributed control law with
estimator
(˙
ˆµ= ˆu
u=−v−BDze−S1Dz(µ−ˆµ),
where ˆµ∈IR|E | is the estimator state and ˆuis the estimator
input to be designed. Substituting the above control law to (3)
gives us the following autonomous system
˙p=v(27)
˙v=−v−BDze−S1Dz(µ−ˆµ)(28)
˙z=BT˙p=BTv(29)
˙e= 2DT
z˙z= 2DT
zBTv(30)
˙
ˆµ= ˆu. (31)
Note that the estimating agents are encoded in S1, in other
words, for the edge Ekthe estimating agent is Etail
k.
Theorem 4.1: Consider the autonomous system (27)-(31)
with non-zero mismatches and a desired infinitesimally and
minimally rigid formation shape Z. Consider also the follow-
ing distributed control action for the estimator ˆµ
ˆu=−DT
zST
1v, (32)
where the estimating agents are chosen arbitrarily. Then the
equilibrium points (p∗, v∗, z∗, e∗,ˆµ∗)of (27)-(31) are asymp-
totically stable where v∗=0and the steady-state deformation
of the shape satisfies ||e∗||2≤2||µ−ˆµ(0)||2+ 2||v(0)||2+
||e(0)||2. Moreover, for the particular cases of triangles and
tetrahedrons, the equilibrium e∗=0, i.e. ˆµ∗=µand z∗∈ Z.
Proof:First we start proving that (32) is a distributed
control law. This is clear since the dynamics of ˆµk(the k’th
element of ˆµ) are given by
˙
ˆµk=zT
kvEtail
k,(33)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 7
which implies that the estimating agent Etail
kfor the edge Ekis
only using the dot product of the associated relative position
zkand its own velocity. Note that using the notation in (21),
the above estimator input is given by αkEtail
k. Consider the
following Lyapunov function candidate
V=1
2||ξ||2+1
2||v||2+1
4||e||2,
with ξ=µ−ˆµ, which satisfies
dV
dt=ξT˙
ξ+vT˙v+1
2eT˙e
=ξTDT
zST
1v− ||v||2−vTBDze−vTS1Dzξ
+eTDT
zBTv
=−||v||2.(34)
From this equality we can conclude that ξ, v and eare
bounded. Moreover, from the definition of e,zis also bounded.
Thus all the states of the autonomous system (28)-(31) are
bounded, so one can conclude the convergence of v(t)to
zero in view of (34). Furthermore, since the right-hand side
of (28) is uniformly continuous, ˙v(t)converges also to zero
by Barbalat’s lemma. By invoking the LaSalle’s invariance
principle, looking at (28) the states e, ξ and zconverge
asymptotically to the largest invariance set given by
T∆
={e, z, ξ :S1Dzξ+BDze=0m|V |×1},(35)
in the compact set
Q∆
={ξ, v, e :||ξ||2+||v||2+1
2||e||2≤ρ},(36)
with 0< ρ ≤2V(0). Since v=0for all points in this invari-
ant set, it follows from (29)-(31) that z, e and ˆµare constant
in this invariant set. In other words, z(t)→z∗, e(t)→e∗
and ξ(t)→ξ∗as tgoes to infinity, where z∗, e∗and ˆµ∗
are fixed points satisfying (35). Note that by comparing (27)
and (29) one can also conclude that p(t)→p∗as tgoes to
infinity. In general we have that e∗and ξ∗are not zero vectors,
therefore z∗/∈ Z. It is also clear that ||e∗||2≤2ρ, therefore
for a sufficiently small ρ, the resultant (distorted) formation
will also be infinitesimally and minimally rigid.
Now we are going to show that e∗, ξ∗=0for triangles and
tetrahedrons. Since triangles and tetrahedrons are derived from
complete graphs, the distorted shape when ρis sufficiently
small will also be a triangle or a tetrahedron, i.e., we are
excluding non-generic situations, e.g., collinear or coplanar
alignments of the agents in IR2or IR3. In the triangular
case we have two possibilities after choosing the estimating
agents: their associated directed graph is cyclic (each agent
estimates one mismatch) or acyclic (one agent estimates two
mismatches and one of the other two agents estimate the
remaining mismatch).
The cyclic case for the estimating agents in the triangle
corresponds to the following matrices
B=
1 0 −1
−1 1 0
0−1 1
, S1=
100
010
001
,
and by substituting them into the equilibrium condition in T
we have that
z∗
1e∗
1−z∗
3e∗
3+z∗
1ξ∗
1= 0
z∗
2e∗
2−z∗
1e∗
1+z∗
2ξ∗
2= 0
z∗
3e∗
3−z∗
2e∗
2+z∗
3ξ∗
3= 0
.(37)
Since the stationary formation is also a triangle for a suffi-
ciently small ρ, then z∗
1, z∗
2and z∗
3are linearly independent.
Therefore from (37) we have that e∗
3, e∗
1, e∗
2= 0 respectively
and consequently we have that ξ∗
1, ξ∗
2, ξ∗
3= 0.
Without loss of generality the acyclic case for the estimating
agents in the triangle corresponds to the following matrices
B=
−1 0 −1
110
0−1 1
, S1=
000
110
001
,(38)
and by substituting them into the equilibrium condition in T
we have that
−z∗
1e∗
1−z∗
3e∗
3= 0
z∗
2e∗
2+z∗
1e∗
1+z∗
2ξ∗
2+z∗
1ξ∗
1= 0
z∗
3e∗
3−z∗
2e∗
2+z∗
3ξ∗
3= 0
.(39)
It is immediate from the first equation in (39) that e∗
1, e∗
3= 0
and then from the third equation in (39) we derive that e∗
2, ξ∗
3=
0, and hence ξ∗
1, ξ∗
2= 0 from the second equation in (39).
For the sake of brevity we omit the proof for the tetra-
hedrons, but analogous to the analysis for the triangles, the
key idea behind the proof is that the three relative vectors
associated to an agent are linearly independent in 3D.
Remark 4.2: The solution proposed in Theorem 4.1 is
distributed in the sense that each agent runs its own estimator
based on only local information in order to address or solve
the global problem of a distorted and moving formation.
Remark 4.3: Since the estimating agents are defined by S1,
by arbitrarily chosen we mean that for a given edge in the
incidence matrix B, the order of the agents (tail and head)
can be chosen arbitrarily.
Remark 4.4: A wide range of shapes can be constructed
through piecing together triangles or tetrahedrons. For exam-
ple, a multi-agent deployment of an arbitrary geometric shape
in 2D can be realized by employing a mesh of triangles. If we
consider the case when the interaction between different sub-
groups of triangular formations defines a directed graph (so
without mismatches), but each groups’ graphs are undirected
triangles or tetrahedrons, then one can employ the results of
Theorem 4.1.
The use of distributed estimators in Theorem 4.1, except for
triangles and tetrahedrons, does not prevent the undesirable
effect of having a distortion in the steady-state shape with
respect to the desired one. Nevertheless, the error norm ||e|| is
bounded by a constant √2ρ > 0. Since ||e|| is a combination
of all errors in every edge, we cannot use ρto prescribe a
zero asymptotic error for some focus edges or to concentrate
the bound only on some edges. This property is relevant if
we want to reach a prescribed distance for a high-degree of
accuracy for some edges. By exploiting the result in Theorem
4.1 for triangles and tetrahedrons, we can construct a network
topology (based on a star topology) that enables us to impose
the error bound only on one edge while guaranteeing that the
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 8
other errors converge to zero. We show this in the following
proposition.
Proposition 4.5: Consider the same mismatched formation
control system as in Theorem 4.1 with the equilibrium set
given by v=0m|V|×1and Tas in (35). Consider the
triangular formation defined by Band S1as in (38) for the
incidence matrix and the estimating agents respectively. For
any new agent i, i ≥4added to the formation, if we only link
it to the agents 2and 3, and at the same time we let the agents
2and 3to be the estimating agents for the mismatches in the
new added links, then for a sufficiently small ρas in (36)
lim
t→∞ ek(t)=0,∀k6= 2,(40)
and |e2(t)| ≤ √2ρfor all t.
Proof:Clearly a star topology has been used for the
newly added agents, where the center is the triangle formed by
agents 1,2and 3. Note that the newly added agent i, i ≥4is
forming a triangle with agents 2and 3. Therefore, as explained
in Theorem 4.1, if ρis sufficiently small, then the resultant
distorted formation is also formed by triangles. We prove the
claim by induction. First we derive the equations from Tas
in (55) for the proposed star topology with four agents
−z∗
1e∗
1−z∗
3e∗
3= 0
z∗
2e∗
2+z∗
1e∗
1−z∗
4e4+z∗
2ξ∗
2+z∗
1ξ∗
1+z∗
4ξ4= 0
z∗
3e∗
3−z∗
2e∗
2−z∗
5e5+z∗
3ξ∗
3+z∗
5ξ5= 0
z∗
4e∗
4+z∗
5e∗
5= 0
.(41)
As explained in the last part of the proof of Theorem 4.1, it is
clear that the errors e∗
1, e∗
3, e∗
4and e∗
5must be zero and e2
2≤ρ.
For any newly added agent i≥5, we add a new equation to
(41) of the form
z∗
le∗
l+z∗
l+1e∗
l+1 = 0,(42)
where land l+1 are the labels of the two newly added edges.
Thus for a sufficiently small ρwe have that z∗
land z∗
l+1 are
linearly independent so e∗
l, e∗
l+1 = 0.
It is possible to be more accurate in the estimation of the
mismatches under mild conditions. We in [16] have proposed
the following control law for the estimators in order to remove
effectively both, the distortion and the steady-state collective
motion
ˆuk=κ(ek+µk−ˆµk), k ∈ {1,...,|E|},(43)
where κ∈IR+is a sufficiently high gain to be determined.
Consider the following change of coordinates hk=ek+µk−
ˆµkand let h∈IR|E | be the stacked vector of hk’s for all
k∈ {1,...,|E|}. By defining S2
∆
=B−S1it can be checked
that the following autonomous system derived from (28)-(31)
˙v=−v−S2Dze−S1Dzh(44)
˙e= 2DzBTv(45)
˙
h= 2DzBTv−κh (46)
˙z=BTv, (47)
has an equilibrium at e=h=0,v=0and z∗∈ Z. The
linearization of the autonomous system (44)-(47) about such
an equilibrium point leads to
˙v
˙e
˙
h
˙z
=
−I|V| −S2Dz∗−S1Dz∗0
2Dz∗BT0 0 0
2Dz∗BT0−κI|E| 0
BT0 0 0
v
e
h
z
.(48)
From the Jacobian in (48) we know that the stability of the
system only depends on v, e and αand not on z. We consider
the following assumption as in [16].
Assumption 4.6: The matrix F∆
=h−I|V| −S2Dz∗
2Dz∗BT0iis
Hurwitz.
Theorem 4.7: There exists a positive constant κ∗such that
the equilibrium corresponding to ˜µ=µ, v =0and e=0
(with z∗∈ Z) of the autonomous system (28)-(31) with
the estimator law (43) is locally exponentially stable under
Assumption 4.6 for all κ > κ∗>0.
Proof:The resulting Jacobian by the linearization of (44)-
(47), derived from (28)-(31), evaluated at the desired shape
z∗∈ Z is given by (48). Note that the stability of the linear
system (48) does not depend on z, therefore the (marginal)
stability of (48) is given by analyzing the eigenvalues of the
first 3×3blocks, i.e., the dynamics of v, e and h. Also note that
by Assumption 4.6 the first 2×2blocks, i.e., the matrix F, of
the matrix in (48) is Hurwitz. In addition, one can also easily
check that the third block, i.e., −κI|E| , of the main diagonal of
(48) is negative definite. We consider the following Lyapunov
function candidate
V(v, e, h) = vTeTPv
e+1
2||h||2,(49)
where Pis a positive definite matrix such that P F T+F P =
−2I. For brevity, following the same arguments used in the
proof of the main theorem in [16], the time derivate of Vin
a neighborhood of the equilibrium h=0, v =0and e=0
(with z∗∈ Z) can be given by
dV
dt ≤ −||v||2− ||e||2+ (m−κ)||h||2,(50)
where m∈IR comes from the cross-terms that eventually are
going to be dominated by κ>κ∗=m. We also refer to
the book [28] for more details about the employed technique
of requiring Fto be Hurwitz and employing the Lyapunov
function (49). This completes the proof.
Remark 4.8: Since v(t)converges exponentially to zero, it
follows immediately that p(t)converges exponentially to a
fixed point p∗.
Assumption 4.6 is also related to the stability of formation
control systems whose graph Gdefines a directed sensing
topology. In fact, it is straightforward to check that the matrix
in Assumption 4.6 is the Jacobian matrix for vand ein a
distance-based formation control system (without mismatches)
with only directed edges in G, i.e., at the equilibrium (desired
shape) the linearized non-linear system is given by
˙v
˙e=Fv
e.(51)
Therefore, in order to satisfy Assumption 4.6, one has to
choose the estimating agents with the same topology as in
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 9
stable directed rigid formations, e.g., a spanning tree. This
selection can be checked in more detail in [16], [29].
These two presented strategies for the estimators in Theo-
rems 4.1 and 4.7 have advantages and drawbacks: the main
advantage of (32) over (43) is that we do not need to compute
any gain and that there is a free choice of the estimating agents;
on the other hand, (43) guarantees exponential convergence to
Zand a distributed estimation of µ. It is also worth noting that
a minor modification in (43) also compensates time varying
mismatches as it was studied in [16].
V. MOT IO N CO NT ROL OF SECOND-ORDER RIGID
FORMATIONS
In this section we are going to extend the findings in [17]
on the formation-motion control from the single-integrator to
the double integrator case. In particular, we consider how to
design the desired constant velocities bω∗and bv∗
cas in Figure
2 for an infinitesimally and minimally rigid formation, i.e., for
e= 0. The main feature of this approach is that the desired
motion of the shape is designed with respect to Obbut not Og.
Note that the latter is the common approach in the literature
[6], [11].
Following a similar strategy as in [17], [30], we solve
the motion control of rigid formation problem for double-
integrator agents by employing mismatches as design parame-
ters. More precisely, we assign two motion parameters µkand
˜µkto agents iand jin the edge Ek= (i, j )resulting in the
following control law2
u=−c1v−c2BDze+A(µ, ˜µ)z, (52)
where c1, c2∈IR+are gains, µ∈IR|E | and ˜µ∈IR|E | are the
stacked vectors for all µkand ˜µkand Ais defined by
aik
∆
=
µkif i=Etail
k
˜µkif i=Ehead
k
0otherwise.
(53)
The design of bω∗and bv∗
cis done via choosing appropriately
the motion parameters µand ˜µ, in the sense that we allow a
desired steady-state collective motion but remove any distor-
tion of the final shape. The design of the motion parameters
in Amust take into account not only the desired acceleration
but also the damping component in (52) (which is different
from the single-integrator case considered in [17]).
Let the velocity error
ev=v−Av(µ, ˜µ)z, (54)
where Av(µ, ˜µ)is designed employing the motion parameters
described in [17] directly related to the desired steady-state
collective velocity. For the sake of completeness, we briefly
describe how to compute µand ˜µin Avfor the prescribed
bv∗
cand bω∗as in Figure 2. Since
Av(µ, ˜µ)bz∗=¯
S1Dbz∗¯
S2Dbz∗µ
˜µ=T(bz∗)µ
˜µ,
(55)
2As a comparison, in [17] the proposed controller is of the form of u=
−c2BDze+A(µ, ˜µ)z.
defines the steady-state velocity in the body frame, one can
derive the following two conditions
BTT(bz∗)µ
˜µ= 0 (56)
DbzBTT(bz∗)µ
˜µ= 0,(57)
where (56) stands for translations and (57) for rotations and
translations, i.e., we set dz∗
k(t)
dt = 0 and d||z∗
k(t)||
dt = 0 for all
k∈ {1,...,|E|} in (56) and (57) respectively. Let us split µ=
µv+µωand ˜µ= ˜µv+ ˜µω. In order to compute the distributed
motion parameters µv,˜µvfor the translational velocity bv∗
cwe
eliminate the components of µand ˜µthat are not responsible
for any motion by projecting the kernel of BTT(bz∗)onto the
orthogonal space of the kernel of T(bz∗), namely
µv
˜µv∈ˆ
U∆
=PKer{T(bz∗)}⊥nKer{BTT(bz∗)}o,(58)
where the operator PXstands for the projection over the space
X. In a similar way, we need to remove the space responsible
for the translational motion in the null space of the matrix
in (57). Therefore, the computation of the distributed motion
parameters µω,˜µωfor the rotational motion bω∗of the desired
shape is obtained from (57) and (58) as
µω
˜µω∈ˆ
W∆
=Pˆ
U⊥nKer{DT
bz∗BTT(bz∗)}o.(59)
When the velocity error evis zero and we are at the desired
shape z∗(t)∈ Z with the desired velocities in v∗(t), then
from (54) we have that
v∗(t) = Avz∗(t)(60)
˙v∗(t) = Av˙z∗(t) = AvBTv∗(t) = AvBTAvz∗(t)
=Aaz∗(t).(61)
Note that the desired parameters in Aa(µ, ˜µ)correspond to
the desired acceleration of the agents at the desired shape Z.
With this knowledge at hand, we can design the needed motion
parameters for Ain the control law in (52) as
A(µ, ˜µ) = c1Av(µ, ˜µ) + Aa(µ, ˜µ),(62)
since for ev=0m|V|×1and e=0|E |×1the control law (52)
becomes
u=Aa(µ, ˜µ)z∗(t).(63)
Note that A(µ, ˜µ)can be computed directly from the motion
parameters for the desired velocity as in the first-order case.
Therefore, there is no need of designing desired accelerations,
which can be a more tedious task. We show in the following
theorem that the desired collective-motion for the desired
formation is stable for at least sufficiently small speeds.
Theorem 5.1: There exist constants ρ, ρµ, , c1, c2>0for
system (3) with control law (52), A(µ, ˜µ)as in (62) and with
a given desired infinitesimally and minimally rigid shape Z,
such that if [µ
˜µ]∈ M ∆
={µ, ˜µ:||[µ
˜µ]|| ≤ ρµ}, then the origin
of the error dynamical system ev=0and e=0corresponding
to z∗(t)∈ Z is exponentially stable in the compact set Q∆
=
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 10
{e, ev:c1+c2
4||e||2+1
2||ev||2≤ρ}. In particular, the steady-
state shape is the same as the desired one and the steady-state
collective motion of the formation corresponds to v∗(t) =
Avz∗(t).
Proof:First we rewrite the control law (52) employing
(54) and (62) as
u=−c1ev−c2BDze +Aaz. (64)
Consider the following candidate Lyapunov function
V=c1+c2
4||e||2+1
2||ev||2+eT
vBDze, (65)
where Vis positive definite in a neighborhood about e=0and
ev=0for some sufficiently small ∈IR+in the compact
set Qwith e=0corresponding to z∈ Z. Note that even
without knowing c1and c2yet, one can safely compute by
assuming c1= 0 and c2=c∗
2for an arbitrary c∗
2∈IR+. Then,
later we restrict the choosing of c1and c2to be bigger than 0
and c∗
2respectively, since by this choice one does not change
the positive semi-definite nature of (65) for the calculated
for c1= 0 and c2=c∗
2.
The time derivative of (65) is given by
dV
dt=1
2(c1+c2)eT˙e+eT
v˙ev+eT
vBDz˙e+eTDT
zBT˙ev
+eT
vBD(BTv)e
= (c1+c2)eTDT
zBT(ev+Avz)−c1||ev||2−c2eT
vBDze
+eT
vAvBTAvz−eT
vAvBTAvz−eT
vAvBTev
+ 2eT
vBDzDT
zBTv−c1eTDT
zBTev−c2eTDT
zBTBDze
+eTDT
zBTAvBTAvz−eTDT
zBTAvBTAvz
−eTDT
zBTAvBTev+eT
vBD(BTv)e
= (c1+c2)eTDT
zBTAvz
| {z }
f1(e,µ,˜µ)
−c1||ev||2−eT
vAvBT
| {z }
f2(µ,˜µ)
ev
+ 2eT
vBDzDT
zBT
| {z }
f3(z)
ev+ 2eT
vBDzDT
zBTAvz
| {z }
f4(µ,˜µ,z,e)
−c2eTDT
zBTBDz
| {z }
f5(e)
e−eTDT
zBTAvBT
| {z }
f6(µ,˜µ,z)
ev
+eT
vBD(BTv)
| {z }
f7(v)
e. (66)
Since all the fi, i ∈ {1,...,7}are locally Lipschitz functions
in the compact sets Qand Mand by using Young’s inequality
to every cross-term in (66), we can bound ˙
Vas follows
dV
dt≤c2M1(µ, ˜µ)−λ5+c1M1(µ, ˜µ) + 3
2||e||2
+−c1+M2(µ, ˜µ)
+22λ3
+M4(µ, ˜µ, z) + M6(µ, ˜µ, z ) + M7||ev||2,
(67)
where M1and M4are related to the Lipschitz constant of f1
and f4in the compact set Qgiven µand ˜µ,M2is the induced
2-norm of f2given µand ˜µ,M6is the squared induced 2-
norm of f4in the compact set Qgiven µand ˜µ, and finally M7
is the maximum squared induced 2-norm for f7in Qas well.
We also have that λ3is the maximum eigenvalue of f3and λ5
is the minimum eigenvalue of f5in the compact set Q. First
we note that for a sufficiently small ρ,λ5>0by the same
argument of having a desired infinitesimally and minimally
rigid formation as in Lemma 3.2. The time derivative (67) can
be made negative as a result of the following steps:
•Choose a sufficiently small ρµin Msuch that M1(µ, ˜µ)−
λ5<0, i.e., downscale if necessary µand ˜µby the same
factor.
•Compute M2for the given µ, ˜µ.
•Compute M4and M6for the given µ, ˜µin the compact
set Q.
•Compute M7considering all the vsuch that ||ev||2≤ρ.
•Choose c1such that the second bracket in (67) is negative.
•Given c1choose c2> c∗
2(employed for the calculation
of ) such that the first bracket in (67) is negative.
This guarantees the local exponential convergence of e(t)
and ev(t)to their origins, hence z(t)→z∗(t)∈ Z,
v(t)→Avz∗(t)and the stacked acceleration of the agents
a(t)→Aaz∗(t)as tgoes to infinity.
Remark 5.2: The limitation given by ρµis exclusively
related to the desired speed of the agents [17] and it does
not restrict in any other way the desired collective motion
for the formation. Therefore, once the motion parameters are
given, for asserting the exponential stability of the system, one
only has to downscale them if necessary. This (conservative)
downscale has an intuitive physical explanation related to the
condition of requiring λ5>0in Theorem 5.1, e.g., the agents
cannot be in a collinear configuration at any moment. In order
to avoid such configurations where λ5is zero we need e∈ Q,
i.e., for arbitrary velocities and positions of the agents such
that e, ev∈ Q, the third acceleration term in (64) given by
the motion parameters µand ˜µmust be small enough in
order to avoid the possibility of pushing the agents to become
collinear. Note that when the agents start close to the desired
shape and have low velocity, the gradient terms in (64) will
not push the agents to such a configuration but to the desired
shape. Nevertheless, imposing λ5>0for all time is a very
conservative condition and we have verified in simulation that
indeed it is not a necessary condition.
Remark 5.3: For µ, ˜µ=0we have that
M1, M2, M3, M4, M6= 0. Then employing (65) and
applying differently Young’s inequalities in (66) one can
prove that for c1, c2= 1 the dissipative Hamiltonian system
(6) is exponentially stable for a sufficiently small .
Remark 5.4: For desired constant drifts in triangular and
tetrahedron formations, it can be checked from [17] that
M1= 0. Therefore there is no restriction in the speed for
such particular cases. In particular, one can use the Lyapunov
function (65) with = 0. It turns out that the formation with
the proposed motion-shape controller is asymptotically stable
for any c2>0for c1>||Av(µ, ˜µ)BT||2.
Remark 5.5: For a desired rotation about the centroid of
an equilateral triangle, it can be checked from [17] that in
addition to M1= 0 we have that eT
vf2(µ, ˜µ)ev= 0 since f2
is skew symmetric. Therefore by using (65) with = 0 one
can prove the asymptotic stability of the origin of evand e
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 11
for any c1, c2>0.
Remark 5.6: We recall the mismatched case in Section
III-B. So far, we can conclude that the upper bound for the
distortion and speed of the resulting formation depends on
the steady-state of the formed geometrical shape. This can
be seen by looking at the third term on the right hand side
of (52) by just taking ˜µ=0. On the other hand, there is
a direct relation between a formation containing mismatches
with a formation containing motion parameters by checking
the two cases in (16) and (64). In the latter case we are able
to calculate the steady-state of both shape and velocity. In the
former (mismatched) case, we refer to the results in [22] in
order to check the approximated resultant motion. As a first
step, the distortion in the mismatched case can be approached
by comparing how far the mismatches are from the ones
calculated employing (58), (59) and (62), where no distortion
occurs.
The result in Theorem 5.1 allows one to design the desired
velocity for the given formation with respect to Obas in Figure
2. This result extends to the applications proposed in [17]
by using distributed motion parameters, such as steering an
infinitesimally and minimally rigid formation by controlling
the heading of the formation with respect to Og, and for the
tracking and enclosing of a target.
VI. SIMULATIONS
In this section we validate the results in Theorems 4.1, 4.7
and 5.1 with numerical simulations.
We first start validating Theorem 4.1 for a regular tetra-
hedron formation, with side length equal to 70 units, whose
associated incidence matrix is given by
B=
−1 0 −1 1 0 0
110001
0−1 1 0 1 0
000−1−1−1
.(68)
We then randomly generate the following vector of mis-
matches µfor each edge Ek
µ=12.14 −41.12 −16.64 −5.91 0.45 18.41T.
(69)
We randomly spread the four agents within an area of 100
cubic units and with random initial velocities but with speeds
smaller than 2units per second. We apply the control law as
in (28) with the estimator dynamics (32). We remark that with
this setup, one can choose arbitrarily the estimating agents, i.e.
how one chooses Bfor defining the (mismatched) tetrahedron
formation does not matter. The results are shown in Figure 3.
We now have a team of six agents whose prescribed shape
is a regular hexagon, with side length equal to 50 units, whose
incidence matrix is given by
B=
101000000
−110001000
0−1−1110000
000−1 0 0 1 1 0
0 0 0 0 −1−1−1 0 1
0000000−1−1
,
(70)
(a) (b)
(c) (d)
Fig. 3: Numerical simulation of a team of four agents with
mismatches in their prescribed distances. We employ the
estimator proposed in Theorem 4.1. The final positions of the
agents are marked with dots respectively and the red, green,
blue and black colors correspond to the agents 1,2,3and 4
respectively. The effectiveness of the estimator is shown in (b)
and (c), where all the agents’ speed and errors (as defined in
(8)) converge effectively to zero. The plot (d) shows how the
six elements of the estimator state ˆµconverge to the actual µ
(shown in black dashed lines).
and we add the following arbitrary vector of mismatches
µ= 0.1
−0.43
7.09
0.08
−1.19
−5.55
−0.574
7.33
1.85
−1.05
.(71)
We use the results in Theorem 4.7 in order to eliminate the
undesired steady-state motion and distortion in the desired
shape. We first notice that the estimating agents defined by S1
derived from Bare not defining any cycles as it can be checked
in Figure 4. Therefore, the topology for the estimating agents
makes Assumption 4.6 to be satisfied. Since the equilibrium
set for the shape is given by Dand not by Z, in order to have
a hexagon as a final shape, one restricts the initial positions
of the agents to be close to the desired shape and with small
initial velocities. We consider the gain κ= 1 for (43) noting
that this value is smaller than the conservative one that can be
derived from Theorem 4.7. The results are shown in Figure 5.
In order to validate the results of Theorem 5.1 we consider
the previous four agents from the first experiment but with a
regular tetrahedron of side length equal to 25 units as a desired
shape. The chosen incidence matrix is the same as (68). The
desired motion of the tetrahedron is given in Figure 2 but
with bv∗
cbeing parallel to bω∗, i.e., the agent on the top of the
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 12
2 5
3 4
1 6
Fig. 4: Regular hexagon formation where the tails of the
arrows indicate the corresponding estimating agents. Note that
this configuration does not contain any cycles, and therefore
Assumption 4.6 is satisfied.
(a) (b)
(c) (d)
Fig. 5: Numerical simulation of a team of six agents for
forming a hexagon but with mismatches in their prescribed
distances. We employ the estimator proposed in Theorem
4.7. The final positions of the agents are marked with dots
respectively and the red, green, blue, black, magenta and cyan
colors correspond to the agents 1,2,3,4,5and 6respectively.
The effectiveness of the estimator is shown in (b) and (c),
where all the agents’ speed and errors (as defined in (8))
converge effectively to zero. The plot (d) shows how the nine
elements of the estimator state ˆµconverge to the actual µ
(shown in black dashed lines).
tetrahedron is following a linear velocity perpendicular to the
base, the other three agents follow the same linear velocity,
and in addition these three agents also make spinning about
the centroid of the base. In order to have such a motion, the
distributed motion parameters for Avas in (54) are given by
µv=svh111−3−3−3iT
˜µv=svh−1−1−1−1−1−1iT
µω=sωh111000iT
˜µω=sωh111000iT
,
(72)
(a) (b)
Fig. 6: Numerical simulation of a team of four agents travelling
in a tetrahedron formation. We employ the results from The-
orem 5.1. In the first plot the agents 1,2,3and 4are marked
with the red, green, blue and black colors respectively. The
crosses indicate the initial positions. The black agent on top
of the tetrahedron follows a linear velocity while the other
tree agents follow the same velocity and in addition they are
spinning about the centroid of the base of the tetrahedron.
We show in plots (a) and (b) the evolution of the inter-agent
distances and the speeds of the agents, converging to the
desired ones (dashed-lines).
where we set sv= 0.15 and sω= 0.25, i.e., we regulate
the speeds of bv∗
cand bω∗such that the speed of the agent
4is 0.15||z∗
4+z∗
5+z∗
6|| = 9.184 units per second. Similar
calculations can be done for the other three agents in order
to derive that their stationary speed will be 11.113 units per
second. We randomly spread the four agents within an area
of 50 cubic units and with random initial velocities with the
initial speeds smaller than 2units per second. We apply the
control law (52) to system (3), constructing Awith (72) as
in (62) and with control gains c1= 1 and c2= 1, which are
smaller than the conservative ones derived from Theorem 5.1,
showing the conservative nature of the result of the theorem.
The numerical results are shown in Figure 6.
VII. CONCLUSIONS
In this paper we have analyzed the effects of having a
distance-based controller for rigid formations and second-
order agents with the presence of mismatches in their desired
inter-agent distances. These effects are a stationary distorted
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 13
shape with respect to the desired one and an undesired collec-
tive motion of the formation. It turns out that both first-order
and second-order agents share precisely the same behaviour
for the undesired collective motion. We have extended the
estimator based solution proposed in [16] to remove the effects
of the mismatches to second-order agents. We have also
proposed another estimator with fewer requirements although
it only eliminates all the undesired effects for both triangles
and tetrahedrons. For the rest of shapes, the new estimator is
only effective for removing the undesired state-state collective
motion. Nevertheless, a bound on the distortion of the steady-
state shape with respect to the desired is given. We have
further extended the results from [17] of employing distributed
motion parameters in order to control the motion of a desired
rigid shape to the second-order agents case. Consequently,
it opens possibilities to apply this method directly to actual
systems governed by Newtonian dynamics such as quadrotors
or marine vessels as it has been shown in [31]. We are
currently working on extending the recent results for flexible
formations with mismatches [32], [33] to second-order agents
as well.
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Hector G. de Marina received the M.Sc. degree in
electronics engineering from Complutense Univer-
sity of Madrid, Madrid, Spain in 2008 and the M.Sc.
degree in control engineering from the University
of Alcala de Henares, Alcala de Henares, Spain in
2011. He is a postdoctoral research associate with
the Ecole Nationale de l’Aviation Civile (ENAC)
in Toulouse, France. His research interests include
formation control and navigation for autonomous
robots, and drones in particular.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL 14
Bayu Jayawardhana (SM13) received the B.Sc.
degree in electrical and electronics engineering from
the Institut Teknologi Bandung, Bandung, Indonesia,
in 2000, the M.Eng. degree in electrical and elec-
tronics engineering from the Nanyang Technological
University, Singapore, in 2003, and the Ph.D. de-
gree in electrical and electronics engineering from
Imperial College London, London, U.K., in 2006.
Currently, he is an associate professor in the Faculty
of Mathematics and Natural Sciences, University of
Groningen, Groningen, The Netherlands. He was
with Bath University, Bath, U.K., and with Manchester Interdisciplinary Bio-
centre, University of Manchester, Manchester, U.K. His research interests are
on the analysis of nonlinear systems, systems with hysteresis, mechatronics,
systems and synthetic biology. Prof. Jayawardhana is a Subject Editor of the
International Journal of Robust and Nonlinear Control, an associate editor of
the European Journal of Control, and a member of the Conference Editorial
Board of the IEEE Control Systems Society.
Ming Cao is currently professor of systems and
control with the Engineering and Technology In-
stitute (ENTEG) at the University of Groningen,
the Netherlands, where he started as a tenure-track
assistant professor in 2008. He received the Bachelor
degree in 1999 and the Master degree in 2002
from Tsinghua University, Beijing, China, and the
PhD degree in 2007 from Yale University, New
Haven, CT, USA, all in electrical engineering. From
September 2007 to August 2008, he was a post-
doctoral research associate with the Department of
Mechanical and Aerospace Engineering at Princeton University, Princeton, NJ,
USA. He worked as a research intern during the summer of 2006 with the
Mathematical Sciences Department at the IBM T. J. Watson Research Center,
NY, USA. He is the 2017 recipient of the Manfred Thoma medal from the
International Federation of Automatic Control (IFAC) and the 2016 recipient
of the European Control Award from the European Control Association
(EUCA). He is an associate editor for IEEE Transactions on Automatic
Control, IEEE Transactions on Circuits and Systems and Systems and Control
Letters, and for the Conference Editorial Board of the IEEE Control Systems
Society. He is also a member of the IFAC Technical Committee on Networked
Systems. His main research interest is in autonomous agents and multi-agent
systems, mobile sensor networks and complex networks.
