Preprints of the 4th IFAC Workshop on Distributed Estimation
and Control in Networked Systems
September 25-26, 2013. Rhine-Moselle-Hall, Koblenz, Germany
Controlling Formations of Autonomous
Agents with Distance Disagreements
Hector Garcia de Marina ∗ Ming Cao ∗ Bayu Jayawardhana ∗
∗
Research Institute for Technology, Engineering and Management
University of Groningen (e-mail: {h.j.de.marina, m.cao,
b.jayawardhana}@rug.nl)
Abstract: We address the robustness issue for controlling, using only local information, the
shapes of undirected rigid formations of autonomous agents when the agents disagree with
their neighboring peers on the prescribed or measured distances between them. We propose
to make use of simple local estimators as part of the distributed controllers. It is proved then
that for infinitesimally rigid undirected formations satisfying a specific condition determined
by the geometric shape of the desired formation and which agents are chosen to estimate the
disagreements, our controller locally stabilizes exponentially the formations when the distance
disagreements are small. In addition, the formation under control stops moving in the end and
does not exhibit any undesirable motion caused by the distance disagreements. The final actual
distances between the neighboring agents can be calculated directly from the steady-state values
of the estimators. The simulation results for a six-agent formation validates the performance of
the proposed controller.
Keywords: Formation Control, Distributed Control, Distributed Estimation, Distance
Disagreement
1. INTRODUCTION
Teams of mobile robots with sensing, communication, and
computing capabilities are used more and more widely for
a range of robotic tasks (Kumar et al. (2005); Bullo et al.
(2009)). More and more often, robots have been deployed
in formations with different shapes in order to facilitate the
adaptive sampling of an unknown environment (Leonard
et al. (2007)) or to achieve better cooperation efficiency
(Stipanovic et al. (2008)). As a result, considerable research efforts have been made in the past few years on
designing distributed control laws to stabilize the shapes of
formations of autonomous agents (Suzuki and Yamashita
(1999); Marshall et al. (2004); Dimarogonas and Johansson
(2010)). In particular, designing formation control laws
using the notion of graph rigidity has proven to be a
promising approach (Anderson et al. (2008); Krick et al.
(2009); Yu et al. (2009); Cao et al. (2011)) and such control laws are usually based on potential functions closely
related to the graphs describing the neighbor relationships
between the agents.
However, it has been recently reported in (Belabbas et al.
(2012)) that for such potential-based control laws, if agents
disagree with their neighboring peers on the prescribed or
measured distances between them, undesirable formation
motion might appear. The robustness issue reported has
critical impact since in real practice, distance disagreements may arise for several reasons. First, in a distributed
setting, agents may have access only to part of the prescribed shape of the overall formation and a pair of neighboring agents may disagree on the prescribed distance between them. Second, agents may have their own guidance
systems, which may have different setting points. Third,
Copyright © 2013 IFAC
411
sensors equipped on the mobile agents may not return the
same value even if they are measuring the same distance
due to heterogeneity in manufacturing processes. Fourth,
the same sensor can produce different readings in face of
random measurement noises.
In this paper, we address the robustness issue just described by proposing simple local estimators. Using our
proposed estimators, we are able to both locally exponentialy stabilize a large class formations and to eliminate
the undesirable periodic asymptotic motions reported in
(Belabbas et al. (2012)). The main contribution of the paper is the new distributed control law incorporating local
estimators that can handle small distance disagreements
for a large class of rigid formations. The new control law
still takes full advantage of the existing potential-based
control laws for undirected rigid formations and thus can
be combined with other controllers designed so far for rigid
formations using potential functions.
The rest of the paper is organized as follows. In Section
2 we introduce some basic notions about rigid formations
in the plane and formulate the formation control problem for rigid formations. In Section 3 we emphasize the
robustness issues in maintaining infinitesimally rigid undirected formations due to distance disagreements between
neighboring agents. In Section 4 we propose and discuss
the performances of a distributed controller incorporating
a simple local estimator. Simulation results are shown in
Section 5 where we apply our distributed control on a
formation of six agents.
NecSys 2013
September 25-26, 2013. Koblenz, Germany
2. CONTROLLING RIGID FORMATIONS
2.1 Rigid formations
We consider a system consisting of n autonomous agents
whose neighbor relationships are described by an undirected graph G with the vertex set V = {1, . . . , n} and
the edge set E ⊆ V × V. We use |E| to denote the number
of edges of G. For i ∈ V, we use Ei to denote the edges
containing vertex i. Following (Krick et al. (2009)), to
define formations in IR2 , we embed G in IR2 by assigning to
each vertex i a Cartesian coordinate xi ∈ IR2 . A framework
is a pair (G, x), where x = (xT1 , . . . , xTn )T is a multi-point
in IR2n . For every framework (G, x), we define the edge
function fG : IR2n → IR|E| by
fG (x) = col (||zkj ||2 ),
∀(k,j)∈E
where col(·) defines the column vector by collecting all its
arguments as the vector’s components, zkj = xk − xj is
the relative position vector between vertices k and j in
the framework and || · || denotes the Euclidean norm in
IR2 .
We first review some basic notions on rigidity.
Definition 1. (Asimow and Roth (1979)) A framework
(G, x) is locally rigid if for every x ∈ IR2n there exists
a neighborhood X of x such that fG−1 (fG (x)) ∩ X =
fK−1 (fK (x)) ∩ X , where K is the complete graph with the
same vertex set V of G.
Definition 2. (Asimow and Roth (1979)) A framework
(G, x) is globally rigid if fG−1 (fG (x)) = fK−1 (fK (x)).
Roughly speaking a framework (G, x) is rigid if it is not
possible to smoothly move some vertices of the framework without moving the rest while maintaining the edge
lengths specified by fG (x). Most of the existing literature
has focused on a special class of rigid frameworks. We need
some more definition to introduce such frameworks.
Let us take the following approximation of fG (x)
fG (x + δx) = fG (x) + dfG (x)δx + O(δx2 ),
where dfG (x) denotes the Jacobian matrix of fG (x) and δx
is an infinitesimal displacement of x. The matrix dfG (x) is
then called the rigidity matrix of the framework (G, x).
Definition 3. (Asimow and Roth (1979)) A framework
(G, x) is infinitesimally rigid in IR2 if the dimension of
the kernel of dfG (x) is three, or equivalently rank dfG (x) =
2n − 3.
An infinitesimally rigid framework allows its vertices to
move infinitesimally, while keeping the edge function constant up to the first order approximation. Roughly speaking, an infinitesimally rigid framework (G, x) only admits
rotations and translations of the whole framework in order
to satisfy fG (x + δx) = fG (x). The edge function remains
constant up to the first order when δx belongs to the kernel
of dfG (x).
An example of a rigid framework and an infinitesimally
rigid framework is shown in Figure 1. While the framework
in Figure 1a does not admit any other possible configuration which shares the same fG (x), the framework in
Figure 1b shows that if the interior vertices lie at the same
412
(a)
(b)
Fig. 1. An example of an infinitesimally rigid framework
is shown in (a), while the framework shown in (b) is
rigid but not infinitesimally rigid.
distance from the exterior vertices on the line from the
baricentre to the exterior vertices, then there is a nontrivial infinitesimal movement of the internal vertices as
shown by the arrows, which also preserves its fG (x).
Note that an infinitesimally rigid framework is also rigid,
but in general the converse is not true.
2.2 Controling infinitesimally rigid formations
Now we consider how to control the shape of the n-agent
formation of autonomous agents. The geometric features
of the formation can be conveniently described by the
concept of frameworks that we have discussed in the
previous subsection. Assume that the dynamics of every
agent i are described by
ẋi = ui ,
(1)
where xi is the Cartesian coordinate of agent i with respect
to some fixed global coordinate system and ui ∈ IR2 is
the control input. In (Krick et al. (2009)), an elegant
distributed control law has been presented to stabilize a
large class of rigid formations in the plane. This control law
uses only the agents’ local information about the measured
relative positions of their neighbors specified by G
X
ui = −
zk` ek` ,
(2)
(k,`)∈Ei
where ek` is the error between the squared actual and
prescribed distances between agents k and `
ek` = ||zk` ||2 − d2k` ,
(3)
and here dk` denotes the prescribed distance between
agents k and `. The prescribed distance between agents
are given by the edge function fG (x∗ ) where x∗ ∈ IR2n is a
non-unique multi-point that satisfies all the prescribed distance constraints between the agents. Obviously, when the
framework is rigid, any such x∗ satisfies the control goal
of achieving the desired shape of the n-agent formation.
To study the overall dynamics of the n-agent formation,
one can analyze the dynamics of the error system. Using
(1)-(2), we describe the dynamics of zij by
X
X
żij = −
zk` ek` +
zk` ek` .
(4)
(k,`)∈Ei
(k,`)∈Ej
It thus follows that the dynamics of the error system can
be given by
NecSys 2013
September 25-26, 2013. Koblenz, Germany
T
ėij = 2zij
żij
T
= −2zij
X
T
zk` ek` + 2zij
(k,`)∈Ei
X
zk` ek` .
(5)
(k,`)∈Ej
This error system can be written in a compact form as
ė = f (e, z)
(6)
where e = col (eij ) ∈ IR|E| and z ∈ IR2|E| . In (Belabbas
(i,j)∈E
et al. (2012)) it has been shown that f (e, z) in (6) can be
expressed in terms of a smooth function depending only
on e, and not on z. This smooth function exists for any
given rigid framework, but only if this function is locally
defined. For some rigid frameworks like the triangles, this
function is globally defined. In (Belabbas et al. (2012)) it
has been shown that the error system has an exponentially
stable equilibrium at the origin, which corresponds to
the prescribed shape given by the framework (G, fG (x∗ )).
Following (Belabbas et al. (2012)), we rewrite the error
system (5) as
ė = −Q(e)e,
(7)
T
where Q(e) = dfG (z)dfG (z) is determined by the Jacobian matrix of the edge function given by the framework
(G, x). The exponential stability of (7) can be shown by
using V = 12 eT e as the Lyapunov function which satisfies
V̇ = −eT Q(e)e, where Q(e) is a positive definite matrix
if dfG (z) is derived for an infinitesimally rigid framework,
since for such a case dfG (z) is a full row-rank matrix.
Appealing as the exponential stability is, there are subtle
but critical robustness issues associated with such formation control systems as discussed in (Belabbas et al.
(2012)) when a pair of neighboring agents disagree on
their prescribed or measured distance. In the next section,
we focus on the robustness problem of the potential-based
controllers in the form of (2).
Fig. 2. The disagreement in edge (1, 2) can be described
in two different ways depending on which agent is set
to be the estimating agent. In the left figure, agent 1
is chosen to be the estimating agent and in this case
∆12 = 0. On the other hand, in the right figure, agent
2 is the estimating agent and thus ∆21 = 0.
(resp. j) to be the estimating agent for this edge if the
disagreement ∆ij (resp. ∆ji ) is decided in the controller
design stage to be estimated. For simplicity of notation,
we define dh = dji as the common prescribed distance for
the hth edge when agent i is the estimating agent, and
dh = dij otherwise. The common error distance (w.r.t. the
dh ) is denoted by ēij = ēji = kzij k2 − d2h 1 . Accordingly,
for this hth edge, we denote the vertex of the estimating
agent by vh and the corresponding other vertex by vh0 and
the disagreement between vh0 and vh is denoted by ∆h , i.e.,
dvh vh0 = ∆h + dh .
Figure 2 illustrates this notion of an estimating agent
and the reference prescribed distance for the edge that
links vertex 1 and 2. In this particular example, the
edges (3, 1) and (2, 3) are not associated with any distance
disagreement and so ∆13 = ∆23 = 0.
3. ROBUSTNESS ISSUES WITH DISTANCE
DISAGREEMENT
3.2 Robustness issues due to the presence of disagreements
3.1 Distance disagreements
So far we have assumed that the same information on the
desired infinitesimally rigid framework is shared among
all the agents involved in maintaining the formation. In
(Belabbas et al. (2012)) it has been shown that robustness
issues may arise when these agents share the same graph
G given by the infinitesimally rigid framework, but the
edge function fG,i (x∗ ) of agent i is biased compared to the
edge functions fG,j (x∗ ) of agent j, where j is a neighbor of
agent i. Therefore the prescribed distance considered by
agent i to maintain its distance with respect to agent j is
different from dji to be maintained by agent j. When such
“disagreement” exists, the design and analysis of formation
controllers become tricky. For any edge in G with vertices i
and j, we define the disagreement ∆ij to be ∆ij = dij −dji .
Obviously ∆ij = −∆ji .
In order to deal with the possible disagreement in the
formation, we need to distinguish the two associated
agents for any given edge in G and thus introduce the
notion of estimating agent for every edge whose task is to
estimate the disagreement ∆ij or ∆ji . For an hth edge
in G with the pair of vertices (i, j), we define vertex i
413
Let us now review the influence of disagreement on the
formation. From (3), it is immediate to see that if there
is a non-zero disagreement between agents i and j, then
eij 6= eji . By denoting µh := d2h − (dh + ∆h )2 for every
edge 2 h, the errors satisfy evh vh0 = evh0 vh +µh . Using ēij , ēji
as defined before, the dynamics (1)-(2) can be rewritten
as
X
X
ẋi = −
zk` ēk` −
zk` µh ,
(8)
(k,`)∈Ei
h∈{1,...,|E|}
(k,`)∈E h
i
where Eih denotes the hth edge for which the associated
agent i is not chosen to be the estimating agent.
Using the above equation, the error dynamics can be
written as a function of ē = col(ēij ) and µ = col(µh )
1
2
Here, we have kzij k = kzji k
When there is no disagreement at hth edge then trivially µh = 0
NecSys 2013
September 25-26, 2013. Koblenz, Germany
T
ē˙ ij = 2zij
żij

X
T 
= 2zij
−
X
zk` ēk` +
(k,`)∈Ei
zk` ēk`
(k,`)∈Ej

X
−
zk` µh +
h∈{1,...,|E|}
(k,`)∈E h
i
X
h∈{1,...,|E|}
(k,`)∈E h
j

zk` µh 

or equivallently
ē˙ = − Q(ē)ē + A(ē)µ,
(9)
(10)
where Q(ē) has been defined after equation (7) and A(ē)
is obtained by comparing (9) and (10).
Here, we call the matrix A(ē) the disagreement matrix.
We remark that the matrix A(ē) is unique only after
one picks a particular choice of the estimating agent for
every edge. For example, consider the formation of three
agents in Figure 2, where if we set agents 2 and 3 to
be the estimating agents for the edges (2, 3) and (3, 1),
respectively, the disagreement matrix for the formation in
the figure on the left is given by

T
||z12 ||2
0
−z12
z31
T
A(ē) = −2 −z23
z12 ||z23 ||2
0 ,
T
0
−z31
z23 ||z31 ||2

(11)
where, following (Belabbas et al. (2012)), the z elements
of an infinitesimally rigid framework can be rewritten in
terms of ē. In fact for the particular case of the triangle
formation, we can write it down just using (3) and the law
of cosines.
In (Belabbas et al. (2012)), the robustness issues of (1)-(2)
due to the presence of µ has been studied and the results
can be summarized as follows:
(1) Unknown final shape of the formation. Because exponential stability is a robust property with respect to
parametric perturbations such as µ, the perturbed
system (10) for µ being sufficiently small has an
exponentially stable equilibrium ē∗ (µ) which is close
to ē = 0. Since the disagreements µ are unknown, the
final shape of the formation determined by ē∗ (µ) is
also unknown.
(2) Undesirable steady-state formation motion induced
by distance disagreements. An equilibrium in ē only
implies an equilibrium in the norm of z, but not
necessarily an equilibrium of z. The equilibrium ē∗ (µ)
for ē can lead to a formation that follows a closed orbit
with a constant angular velocity for a large set of µ’s.
For some particular µ, the z system converges to an
equilibrium point but this does not necessarily imply
that the x system reaches an equilibrium.
We give the following example to illustrate the last point
above for the triangular case.
Example 4. Consider an infinitesimally rigid framework
(G, x) with n = 3 which is embedded in IR2 . In (Belabbas
P3
et al. (2012)) it has considered the case when i µi = 0,
z and ē converge to an equilibrium point z ∗ and ē∗ ,
respectively. In this case,
414
∗ ∗
∗ ∗
∗
ẋ1 = −z12
ē12 + z31
ē31 + z31
µ3
(12)
∗ ∗
∗ ∗
∗
ẋ2 = −z23 ē23 + z12 ē12 + z12 µ1
(13)
∗ ∗
∗ ∗
∗
ẋ3 = −z31
ē31 + z23
ē23 + z23
µ2 .
(14)
The velocity ẋ∗cm of the center of mass of the system
defined by (12) - (14) is
1 ∗
∗
∗
ẋ∗cm = (z31
µ3 + z12
µ1 + z23
µ2 ) 6= 0,
(15)
3
which is a constant vector different from zero since two of
the three vectors in z ∗ are independent. If the center of
mass in the plane is moving with a constant velocity and
the z system is at an equilibrium, then all the agents have
the same constant velocity ẋ∗cm .
This example and the results from (Belabbas et al. (2012))
T
show that for almost all µ = [µ1 µ2 µ3 ] , there is a
steady-state movement of the formation in the plane. Note
that in the absence of disagreements, the center of mass
formed by all the agents remains constant, which has
already been shown in (Krick et al. (2009)). This result
for the triangular case can also be extended to other
infinitesimally rigid formations.
In view of such robustness issues, in the next section, we
propose new controllers incorporating a simple estimator
to stabilize rigid formations with distance disagreements.
4. MAIN RESULT
In this section, we design a distributed controller that
estimates µ. The motivation is to compensate distributively the distance disagreements, and thus to ensure that
in steady states, there is no undesirable collective motion
induced by distance agreements. Although µ is unknown,
we consider the following assumption on µ for the wellposedness of the formation control problem.
Assumption 5. The unknown µ is such that the formation
is still infinitesimally rigid at the equilibrium e∗ (µ).
For instance, in the particular example of triangular formations, Assumption 5 means that the following triangular
inequality must be held by µ, which is a function of ∆:
dm + ∆m < dn + ∆n + dl + ∆l ,
(16)
where m, n, l are the labels of the three edges of the
triangle.
We now present our main result in the following theorem.
Theorem 6. Consider an infinitesimally rigid formation
with the disagreement vector µ satisfying Assumption 5.
For 1 ≤ i ≤ n, consider the dynamic distributed controller
for each agent i given by
X
X
ui = −
zk` ek` +
zk` µ̂h
(17)
(k,`)∈Ei
h∈{1,...,|E|}
(k,`)∈W h
i
where Wih denotes the hth edge for which agent i is the
estimating agent and the dynamics of the estimator µ̂h ,
for which agent i is the estimating agent, are given by
µ̂˙ h = κ(ek` − µ̂h ) where (k, `) = W h .
(18)
i
Define the vector µ̂ ∈ R|E| to be the collection of all the
µ̂h for 1 ≤ h ≤ |E|. Then if A(−µ) is Hurwitz, it holds
that the equilibrium set Q := {(x, µ̂)|ē = −µ, µ̂ = −µ} is
NecSys 2013
September 25-26, 2013. Koblenz, Germany
locally exponentially stable, i.e, for some neighborhood C
of Q, there exists c > 0 such that for every κ > c > 0 and
every (x(0), µ̂(0)) ∈ C, the trajectory (x, µ̂) converges to a
point in Q exponentially fast.
Proof: To prove Theorem 6, we first prove that the error
system with the control law (17)-(18) has an exponentially
stable equilibrium at ē∗ (µ) = µ̂∗ (µ) = −µ. Following the
same derivation as in (10), the error system can be written
in the following compact form
ē˙ = −(B(ē) − A(ē))ē + A(ē)µ + B(ē)µ̂
(19)
µ̂˙ = K(ē − µ̂),
(20)
where B(ē) = Q(ē) + A(ē), K = κI. Using a coordinate
change α = ē − µ̂ and ẽ = ē + µ, we obtain
ẽ˙ = Aẽ − B α̃ = f (ẽ, α)
(21)
α̇ = Aẽ − B α̃ − Kα = f (ẽ, α) − Kα,
(22)
where for conciseness of presentation, we denote A =
A(ẽ, µ) and B = B(ẽ, µ). The new-coordinate system (21)(22) has an equilibrium at ẽ = 0 and α = 0, which implies
that ē = −µ and µ̂ = −µ.
Since A(−µ) is Hurwitz, it follows that
∂f (ẽ, 0)
F0 :=
|ẽ=0 = A(−µ),
(23)
∂ẽ
is also Hurwitz. Therefore, there exists P = P T 0 such
that
F0T P + P F0 = −2I.
(24)
Consider the next candidate Lyapunov function for the
system (21)-(22)
1
V (ẽ, α) = ẽT P ẽ + αT α,
(25)
2
which satisfies
V̇ (ẽ, α) = 2ẽT P f (ẽ, α) + αT f (ẽ, α) − κ||α||2
= 2ẽT P f (ẽ, 0) + 2ẽT P (f (ẽ, α) − f (ẽ, 0)) +
+ αT f (ẽ, α) − κ||α||2
= 2ẽT P (F0 ẽ + g(ẽ) + f¯(ẽ)α) + αT f (ẽ, α) − κ||α||2 ,
(26)
where g(ẽ) is the Taylor-series residue that satisfies
||g(ẽ)||
lim
= 0,
(27)
||ẽ||→0 ||ẽ||
and f¯(ẽ) = −B(ẽ). In particular, (27) implies that for all
> 0, there exists a δ such that ||ẽ|| ≤ δ =⇒ ||g(ẽ)|| ≤
||ẽ||.
1
Taking = 2||P
|| with the corresponding δ, using (24) and
(26), and since f¯(ẽ) is locally Lipschitz, we know that for
all ||ẽ|| ≤ δ, ||α|| ≤ δ, there exist M, N > 0 such that
1
V̇ (ẽ, α) ≤ −2||ẽ||2 + 2||ẽ||||P ||
||ẽ|| + M ||ẽ||||α||+
2||P ||
+ N ||α||2 − κ||α||2 ,
(28)
where the third and fourth terms on the right-hand side
are due to the boundeness of f¯(ẽ) in an open ball. Hence,
by choosing κ > 0 such that
1 M2
(N − κ) ≤ − −
,
(29)
2
2
which implies, in view of Young’s inequality, that
1
V̇ (ẽ, α) ≤ − (||ẽ||2 + ||α||2 ),
(30)
2
415
and thus the system (21)-(22) converges locally exponentially fast to the origin. Substituting ē∗ (µ) = −µ and
µ̂∗ (µ) = −µ into (17), we get ẋi = ui = 0 for all 1 ≤ i ≤ n,
which implies the trajectory (x, µ̂) converges to a point in
Q exponentially fast as the error system converges.
Remark 7. For the triangular formation with A(ē) defined
as in (11), we have the particular case of B(ē) = −A(ē)T .
Since Q(ē) = B(ē) − A(ē) = −(A(ē)T + A(ē)) is a positive
definite matrix at ē = −µ under assumption (5), we know
A(−µ) is Hurwitz.
Remark 8. In view of the definition of µh , we know that
at the equilibrium when the agents stop moving, ē∗kl =
||z ∗ ||2kl −d2h = (dh +∆h )2 , where (k, l) = Whi . Therefore, we
know further that the final distances between the agents
satisfy ||z ∗ || = dh + ∆h , which are determined by the
prescribed distances for the non-estimating agents.
Remark 9. When there are no disagreement, i.e. µ = 0,
the equilibrium point of (19)-(20) is ē∗ (0) = µ̂∗ (0) = 0,
i.e., it corresponds to the desired formation as in (1)-(2).
Remark 10. The condition of A(−µ) being Hurwitz is a
sufficient condition for the local exponential stability of
the system (19)-(20). This sufficient condition requires
the checking of the eigenvalues of an |E| × |E| square
matrix, which in general is impossible to be done since
µ is unknown. However, one can check whether A(0) is
Hurwitz. Since the elements of A(ē) depends smoothly on
ē, there exists a positive number ρ sufficiently small so that
A(−µ) is Hurwitz in the compact set H = {µ : ||µ|| ≤ ρ}.
5. NUMERICAL EXAMPLE
In the numerical setup, we consider a formation of six
agents with disagreements. We implement the distributed
controller given in Theorem 6. The six agents are prescribed to maintain a regular hexagon with edge length
d = 1. The agents and the edges are labeled, and the
estimating agents are chosen as shown in Figure 3. For
this formation, one can check that A(0) is Hurwitz. We
consider the following randomly generated disagreement
vector
T
µ = [0.044 0.065 0.019 0.15 0.15 0.109 0.68 0.168 0.111] ,
(31)
which satisfies Assumption 5.
Fig. 3. Six-agent formation in a regular hexagon shape.
The tail of the arrows indicate the corresponding
estimating agents.
The six agents are placed randomly in the plane within a
bounded area of 50 square units. We choose K = I. For
simplicity, the values of the estimators µ̂ at time zero are
set to zero.
NecSys 2013
September 25-26, 2013. Koblenz, Germany
This work was supported by the EU INTERREG program
under the auspices of the SMARTBOT project. The work
of Cao was also supported by the European Research
Council (ERC-StG-307207).
Fixed rigid formation
4
speed1
speed2
0.5
speed3
3
speed4
speed5
0.4
2
speed6
1
0.3
0
0.2
REFERENCES
−1
0.1
−2
0
−5
−4
−3
−2
−1
0
1
2
3
1
2
3
4
5
6
7
8
Fig. 4. Evolution of the formation converging to a sightly
deformed regular hexagon. The speed of the six agents
converge to zero.
0.01
−0.012
0.005
0
−0.014
−0.005
−0.016
−0.01
e4
−0.018
e1
−0.015
−mu4
e5
−mu1
−0.02
−0.02
e2
−0.025
−mu2
−mu5
e6
−0.022
e3
−0.03
−mu3
−0.035
10
15
20
25
30
35
40
−mu6
−0.024
45
0
10
20
30
40
50
Fig. 5. Evolution of some of the components of the error
vector for the formation given in Figure 3.
0.14
muhat4
0.12
0.02
−mu4
0.1
0
−0.02
−0.04
−mu5
0.06
muhat6
muhat1
0.04
−mu1
0.02
muhat2
muhat5
0.08
−mu6
0
−mu2
−0.06
muhat3
−mu3
−0.08
−0.02
−0.04
−0.06
5
10
15
20
25
30
35
40
45
50
55
0
10
20
30
40
50
60
Fig. 6. Evolution of some of the states of the estimators µ̂i
for the formation given in Figure 3.
By using the proposed controller, the agents’ velocities
converge to zero and the shape of the overall formation
also converges to a shape close to a regular hexagon as
it is shown in Figure 4. Moreover, the variables ē and µ̂
converge to the disagreement vector −µ as it is shown in
Figures 5-6. Therefore the final distances between agents
can be computed.
6. CONCLUDING REMARKS
In this paper, we have proposed a dynamic distributed
controller incorporating simple estimators for infinitesimally rigid formation in order to deal with the robustness issues discussed in (Belabbas et al. (2012)). The
proposed distributed control law also removes the steadystate movement of the formation induced by the distance
disagreements.
Currently we are working on the implementation of the
proposed control laws on a robotic testbed consisting of
10 mobile robots. Some preliminary data have validated
the effectiveness of the formation control law. We are
also looking into the case when the neighbor relationships
between the agents are asymmetric and thus directed
frameworks have to be used to study the rigidity of
formations.
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