Necessary and Sufficient Graphical Conditions
1
for Formation Control of Unicycles
Zhiyun Lin*
Bruce Francis
Manfredi Maggiore
[email protected]
[email protected]
[email protected]
Tel: 416-978-6289
Tel: 416-978-6343
Tel: 416-946-5095
Department of Electrical and Computer Engineering
University of Toronto
10 King’s College Road
Toronto, Canada, M5S 3G4
Fax: 416-978-0804
Submitted for publication November 6, 2003
Abstract
The feasibility problem is studied of achieving a specified formation among a group of autonomous
unicycles by local distributed control. The directed graph defined by the information flow plays a key
role. It is proved that formation stabilization is feasible if and only if the sensor graph has a globally
reachable node. A similar result is given for formation stabilization to a line and to more general geometric
arrangements.
Keywords: Multiagent system, distributed control, time-varying feedback, nonholonomic mobile robots.
I. I NTRODUCTION
The problem of coordinated control of a group of autonomous wheeled vehicles is of recent interest
in control and robotics due to the broad range of applications of multi-vehicle systems in space missions,
operations in hazardous environments, and military missions. Distributed vehicles can readily exhibit the
attractive characteristics of structural flexibility, reliability through redundancy, and simple hardware as
compared to sophisticated individual vehicles.
* Corresponding author
2
Despite the many advantages inherent in distributed multi-vehicle systems, there are challenges in
coordination and control due to the absence of a centralized supervisor and global information. Individual
vehicles in a distributed system must be capable of collectively accomplishing tasks using only locally
sensed information and little or no direct communication. Coordinated control of multi-vehicle systems
includes many aspects, one of the most important and fundamental being formation control.
Over the past decade, many researchers have worked on formation control problems with differences
regarding the types of agent dynamics, the varieties of the control strategies, and the types of tasks
demanded. In 1990, Sugihara and Suzuki [24] proposed a simple algorithm for a group of point-mass
type robots to form approximations to circles and simple polygons. And in the years following, distributed
algorithms were presented in [3], [2], and [25] with the objective of getting a group of such robots to
congregate at a common location. Moving synchronously in discrete-time steps, the robots iteratively
observe neighbors within some visibility range and follow simple rules to update their positions. Also, in
[18], stability of asynchronous swarms with a fixed communication topology is studied, where stability
is used to characterize the cohesiveness of a swarm.
Convergence to a common point is an example of an agreement problem: Agents initially without
a common reference frame eventually come to agree upon a reference point by virtue of having arrived
at one. Besides being of relevance to the rendezvous problem, convergence to a common point is also
of interest because if it is feasible, then so is convergence to other formations, as shown for example
in [17]. Jadbabaie et al. [12] studied a different agreement problem: Getting autonomous agents in the
plane to move in a common direction.
In addition to the references mentioned so far, there have been many results on the mathematical
analysis of formation control. In [21], formation stabilization of a group of agents with linear dynamics
(double-integrator) is studied using structural potential functions. An alternative is to use artificial potential
functions and virtual leaders as in [16]. The approach known as leader-following has been widely used
in maintaining a desired formation while moving, e.g., [6], [7]. Similarly to [12], reference [17] focuses
on the problem of achieving a specified formation among a group of mobile autonomous agents by
distributed control when the communication topology is dynamic because agents come into and go out
of sensor range. Reference [13] studies achievable equilibrium formations of unicycles 1 each moving at
1
By a unicycle we mean a nonholonomic system moving in the plane and with one angle of orientation. It’s sometimes also
referred to as a shopping cart, the kinematics being identical.
3
unit speed and subject to steering control and presents stabilizing control laws, wherein each unicycle
senses all others. And in [19], a circular formation is achieved for a group of unicycles using the strategy
of cyclic pursuit.
Motivated by the fact that no continuous time-invariant feedback control law can stabilize a nonholonomic system to a point, in 1991 Samson and Ait-Abderrahim [23] showed that smooth time-varying
feedback can stabilize a unicycle. Since then, much work (for instance, [20], [22]) has focused on smooth
time-varying feedback control of nonholonomic systems, and averaging theory has been used to study
stability [20] and also motion planning [10]. On the question of formations, in [27], [28] a time-varying
feedback control law is proposed for multiple Hilare-type mobile robots and averaging theory is used to
analyze stability.
As a natural extension of our previous work [19], [17], and motivated by the proposed strategy in
[28], [27], in this paper the problem is studied of achieving a specified formation among a group of
unicycles by distributed control. Each unicycle is equipped with an onboard sensor, by which it can
measure relative displacements to certain neighbors; in particular, we do not assume that the unicycles
possess a common reference frame.
Central to a discussion of formation control is the nature of the information flow throughout the
formation. This information flow can be modeled by a sensor digraph (directed graph), where a link from
node i to node j indicates that vehicle i can sense the position of vehicle j —but only with respect to the
local coordinate frame of vehicle i. In this paper, we assume the sensor digraph is static—the dynamic
case, where ad hoc links can be established or dropped, is harder. Our analysis relies on several tools
from algebraic graph theory [9], non-negative matrix theory [5], [15], and averaging theory [1], [14],
[20]. We introduce a new concept for our analysis: H(α, m) stability of the Laplacian of the digraph.
Our first main result is that formation stabilization to a common point is feasible if and only if the
sensor digraph has a globally reachable node (a node to which there is a directed path from every other
node). That is, there exists at least one unicycle that is viewable, perhaps indirectly by hopping from one
unicycle to another, by all other unicycles. This is precisely the degree of connectedness required and is
much weaker than strong connectedness of the sensor digraph (as in cyclic pursuit [19], for example).
The key to this main result is a new result in graph theory: A digraph has a globally reachable node iff
0 is a simple eigenvalue of the Laplacian of the digraph. Our proof of sufficiency is constructive: We
present an explicit smooth periodic feedback controller, and prove convergence using averaging theory.
4
In addition, we show how formation stabilization to a common point can be adapted to any geometric
pattern.
Our second main result concerns formation stabilization to a line. This turns out to be feasible if
and only if there are at most two disjoint closed sets of nodes in the sensor digraph. Again, our proof
of sufficiency is constructive. Finally, we introduce a special sensor digraph which guarantees that all
vehicles converge to a line segment, equally spaced. This is an extension to unicycles of a line-formation
scheme of Wagner and Bruckstein [26].
II. M ATHEMATICAL P RELIMINARIES
In this section, we review some basic notions in graph theory [4], [8], algebraic graph theory [9], and
matrix theory [5], [15] that are useful in linking graph-theoretic and control-theoretic concepts. Finally,
an averaging result is given that will be the key tool in the stability analysis throughout the paper.
A. Graph Theory
A directed graph or digraph G = (V, E) consists of a finite nonempty set V of nodes and a (possibly
empty) set E of ordered pairs of nodes of G called edges or links, where e ij = (vi , vj ) ∈ E and vi , vj ∈ V .
If U is a nonempty subset of V , then the digraph (U, E ∩ (U × U)) is termed an induced subgraph. If
each edge is associated with a positive number, called the weight of the edge, then the digraph is named
a weighted digraph. An unweighted digraph is just a weighted digraph with each of the edges bearing
weight 1. The in-degree and out-degree of a node are the sum of the weights of the edges at its head
and tail, respectively.
A path on a digraph G = (V, E) of length N from vj0 to vjN is an ordered set of distinct nodes
{vj0 , vj1 , . . . , vjN } such that (vji−1 , vji ) ∈ E, ∀ i = 1, 2, . . . , N . If there is a path in G from one node v i
to another vj , then vj is said to be reachable from vi , written vi → vj . If not, then vj is said to be not
reachable from vi , written vi 9 vj . If a node vi is reachable from every other node in the digraph, then
we say it is globally reachable. A digraph is strongly connected if every two of its nodes, say u and v ,
are such that u is reachable from v and v is reachable from u. So a digraph with at least two nodes is
strongly connected if and only if each node is globally reachable. For a digraph with only one node, the
node is globally reachable, but the digraph is not strongly connected if the node has no self-loop. For a
digraph G = (V, E), if U is a nonempty subset of V and u 9 v for all u ∈ U and v ∈ V − U , then U is
said to be closed.
5
PSfrag replacements
As an example, in Fig. 1 the digraph (a) is strongly connected but the digraphs (b) and (c) are not.
The globally reachable node sets of digraphs (b) and (c) are {1, 2, 6} and {6} respectively.
1
1
6
2
6
2
6
2
5
3
5
3
5
3
4
(a)
Fig. 1.
1
4
(b)
4
(c)
Three digraphs.
B. Matrix Theory
We now turn to matrices associated with graphs. For this purpose, we first give some notions and
properties of a class of matrices. A matrix A is said to be nonnegative (positive) if all elements are
nonnegative (positive). An n × n matrix A is cogredient to a matrix E if for some permutation matrix
P , P AP T = E . The matrix A is reducible if it is cogredient to


B 0
,
E=
C D
(q)
where B and D are square matrices, or if n = 1 and A = 0. Otherwise, A is irreducible. Let a ij denote
the (i, j) element of Aq . A nonnegative matrix A is irreducible if and only if for every (i, j) there exists
(q)
q ≥ 1 such that aij > 0.
Suppose that the nodes of a weighted digraph G = (V, E) are enumerated and denoted {v 1 , . . . , vn }.
The adjacency matrix of G , denoted A, is a square nonnegative matrix with rows and columns indexed
by the nodes of G , such that the ij -entry of A is the weight associated with the edge if (v i , vj ) ∈ E , and
is zero otherwise. Let D be the diagonal matrix with the out-degree of each node along the diagonal and
call it the degree matrix of G . The Laplacian of the digraph is defined as L = D − A.
Taking the digraph (b) in Fig. 1 as an example, suppose each edge bears weight 1. Then we have,
6
respectively,

0 1


 0 0


 0 1
A=

 0 0


 0 0

1 0
0 0 0 0


1 0 0 0 0 0




 0 1 0
0 0 0 1 




 0 0 2
0 1 0 0 
, D = 


 0 0 0
0 0 1 0 




 0 0 0
1 0 0 1 


0 0 0
0 0 0 0


1 −1
0
0
0
0







 0
1
0
0
0 −1 
0 0 0 






 0 −1
2 −1
0
0 
0 0 0 
.
, L = 



 0
0
0
1 −1
0 
1 0 0 






 0
0 −1
0
2 −1 
0 2 0 



−1
0
0
0
0
1
0 0 1
On the other hand, given a nonnegative matrix, we can define a corresponding digraph. The associated
digraph G(A) of an n × n matrix A = (aij ) consists of n nodes v1 , v2 , . . . , vn , where an edge leads from
(q)
vi to vj if and only if aij 6= 0. Then aij > 0 if and only if there exists a sequence of q edges from v i
to vj . Furthermore, A is irreducible if and only if G(A) is strongly connected.
An important class of nonnegative matrices are row stochastic matrices. A nonnegative matrix is
called row stochastic, or stochastic for short, if all row sums are equal to one. In addition, a matrix A
of the form A = sI − B (s > 0, B a nonnegative matrix) for which s ≥ ρ(B) is called an M -matrix.
Next we will review some properties of the Kronecker product. If the orders of the matrices involved
are such that all the operations below are defined, then
(a) (A + B) ⊗ C = A ⊗ C + B ⊗ C ;
(b) (A ⊗ B)T = AT ⊗ B T ;
(c) If A, C ∈ Rm×m and B, D ∈ Rn×n , then (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD);
(d) If λ1 , . . . , λm are the eigenvalues of A ∈ Rm×m and µ1 , . . . , µn are the eigenvalues of B ∈ Rn×n ,
then the eigenvalues of A ⊗ B are the mn numbers λr µs , r = 1, . . . , m, s = 1, . . . n.
Finally a definition from [11] will be modified in order to better suit our application. Let α =
{α1 , α2 , . . . , αp } be a partition of {1, 2, . . . , n}. A block diagonal matrix with diagonal blocks indexed
by α1 , α2 , . . . , αp is said to be α-diagonal. A scalar multiple of the identity matrix is said to be a scalar
matrix. An α-diagonal matrix D is said to be an α-scalar matrix if each block matrix in D is a scalar
matrix. For α = {{1, 2}, 3}, simple examples of α-diagonal matrix



2
1 3 0






A1 =  2 2 0  and A2 =  0



0
0 0 5
and α-scalar matrix are, respectively,

0 0


2 0 .

0 5
7
Definition 1: Let α = {α1 , α2 , . . . , αp } be a partition of {1, 2, . . . , n} and m ≥ 0 an integer. An
n × n matrix A is said to be H(α, m)-stable if
(a) 0 is an eigenvalue of A of algebraic and geometric multiplicity m, while all other eigenvalues have
negative real part,
(b) for every α-diagonal positive definite symmetric matrix R, 0 is an eigenvalue of RA of algebraic
and geometric multiplicity m, while all other eigenvalues have negative real part.
Obviously, an H(α, 0)-stable matrix is asymptotically stable, but the converse does not hold in
general. A simple counterexample is

A=
−5 −4
4
3

,
and α = {{1}, {2}}.
Then A is asymptotically stable with eigenvalues{−1, −1}
 but RA is unstable with the choice of α1 0
.
diagonal positive definite symmetric matrix R = 
0 2
C. Averaging Theory
In this subsection we introduce an averaging result that is useful for analyzing slowly time-varying
systems. More details can be found in [1], [14], [20]. Consider the system
ẋ = f (t, x, )
(1)
where f (t, x, ) is T -periodic in t for some T > 0. We associate with (1) an autonomous averaged system
ẋ = fav (x),
where
1
fav (x) =
T
Z
(2)
T
f (τ, x, 0)dτ.
0
Then we arrive at the following theorem.
Theorem 1: [14] Let f (t, x, ) be continuous and bounded, and have continuous and bounded derivatives up to the second order with respect to (x, ) for (t, x, ) ∈ [0, ∞) × D × [0, 0 ], where D is a
neighborhood of the origin. Suppose f is T -periodic in t for some T > 0 and that f (t, 0, ) = 0 for all
t ≥ 0 and ∈ [0, 0 ]. If 0 is an exponentially stable equilibrium point of the averaged system (2), then
there is a positive constant ∗ ≤ 0 such that for all 0 < < ∗ , 0 is an exponentially stable equilibrium
point of (1).
8
III. P ROBLEM S TATEMENT
AND
M AIN R ESULTS
Before treating unicycles, it is perhaps illuminating to give a result for the much simpler case of point
masses. Consider n “point-mass robots” whose positions are modeled by complex numbers, z 1 , . . . , zn ,
in the plane. Assume a kinematic model of velocity control: ż i = ui . Assume each robot senses the
relative positions of a subgroup, Ni , of the other robots. Let yi denote the vector whose components are
the relative positions zm − zi , as m ranges over Ni . Thus yi , a vector of dimension the cardinality of Ni ,
represents the information available to ui . We allow controllers of the form ui = Fi yi , or ui = 0 if Ni
is empty. Thus, yi = 0 =⇒ ui = 0 =⇒ żi = 0; that is, robot i does not move if all robots it senses are
collocated with it (or if it does not sense any other robot). The problem of convergence to a common
point is this:
Problem 1: Find, if possible, F1 , . . . , Fn such that
(∀ z1 (0), . . . , zn (0))(∃ zss ) lim z1 (t) = · · · = lim zn (t) = zss .
t→∞
t→∞
Now define the sensor graph G for this setup: There is a directed edge from node i to node m if and
only if m ∈ Ni .
Theorem 2:
1) Problem 1 is solvable if and only if G has a globally reachable node.
2) When Problem 1 is solvable, one solution is
h
Fi = 1 · · ·
The proof is omitted.
1
i
.
Now we turn to the main topic of unicycles. We can identify the real plane, R 2 , and the complex
plane, C, by identifying a column vector, zi , and a complex number, zi . Now consider a wheeled vehicle
PSfrag replacements
with coordinates (xi , yi , θi ) with respect to PSfrag
a globalreplacements
frame g Σ (see Fig. 2).
vi
ri
ωi θi
yi
iΣ
gΣ
Fig. 2.
si
Wheeled vehicle.
iΣ
zi
gΣ
xi
Fig. 3.
Frenet-Serret frame.
trajectory
9
The location of the vehicle in the plane is


xi

zi = 
yi
or zi = xi + jyi .
The vehicle has the nonholonomic constraint of pure rolling and non-slipping and is described kinematically as



ẋ = vi cos(θi ),

 i
ẏi = vi sin(θi ),



 θ̇ = ω ,
i
or
i

 ż = v ejθi ,
i
i
 θ̇ = ω .
i
(3)
i
Following [13], we construct a moving frame i Σ, the Frenet-Serret frame, that is fixed on the vehicle
(see Fig. 3). Let ri be the unit vector tangent to the trajectory at the current location of the vehicle (r i
is the normalized velocity vector) and let si be ri rotated by π/2. Since the vehicle is moving at speed
vi , vi ri = żi , and so in complex form
ri = ejθi ,
Thus
ṙi =
d jθi = jejθi θ̇i = si ωi ,
e
dt
si = jri .
ṡi = j ṙi = jsi ωi = −ri ωi .
The kinematic equations using the Frenet-Serret frame are



ż = vi ri ,

 i
ṙi = ωi si ,



 ṡ = −ω r .
i
(4)
i i
Now consider n wheeled vehicles, indexed by i. We refer to the individual vehicles as nodes and
the information flows as links. Although the vehicles in the group are dynamically decoupled, meaning
the motion of one vehicle does not directly affect any of the other vehicles, they are coupled through the
sensor information flow. A natural way to model the interconnection topology is a digraph G = (V, E)
with V = {1, 2 . . . , n} in which the vehicles are the nodes and the links are directed edges. The directed
edge from node i to node m is one of the graph’s edges just in case vehicle i can sense vehicle m.
(Thus, information flows in the direction opposite to the orientation of the edges.) We refer to this as a
sensor digraph.
Let Ni denote the set of labels of those vehicles sensed by vehicle i. In this paper, we assume N i
is time-invariant, meaning the information flow topology is static. In the control law that we study, no
PSfrag replacements
mΣ
yim
xim
zm
zi
gΣ
Fig. 4.
10
iΣ
Local information.
vehicle can access the absolute positions of other vehicles or its own. Specifically, vehicle i can measure
only the relative positions of sensed vehicles with respect to its own Frenet-Serret frame (see Fig. 4),

 xim = (zm − zi ) · ri ,
m ∈ Ni ,
(5)
 y = (z − z ) · s ,
im
m
i
i
where dot denotes dot product. This leads to the following definition.
Definition 2: A controller (vi , wi ), i = 1, . . . n, is said to be a local information controller if

 v = g (t, x , y )|
i
i
im im m∈Ni ,
i = 1, . . . , n
 ω = h (t, x , y )|
i
i
im
im m∈Ni
where gi , hi are smooth functions of their arguments, and gi is such that {(∀ m ∈ Ni ) zm = zi } ⇒
{vi = 0}.
Notice that in our definition a vehicle does not translate (but it can rotate) when either it cannot
sense any other vehicle or its neighbors have all converged to its position.
In what follows, we present the two main problems investigated in this paper, together with necessary
and sufficient conditions for their solution.
Problem 2: (Formation Stabilization to a Point) Find, if possible, a local information controller
such that for all (xi (t0 ), yi (t0 ), θi (t0 )) ∈ R3 , i = 1, . . . n, and all t0 ∈ R, there exists zss ∈ R2 such that
lim zi (t) = zss , ∀ i.
t→∞
Theorem 3: (Section IV) Problem 2 is solvable if and only if the sensor digraph has a globally
reachable node.
Problem 3: (Formation Stabilization to a Line) Find, if possible, a local information controller such
that for all (xi (t0 ), yi (t0 ), θi (t0 )) ∈ R3 , i = 1, . . . n, and all t0 ∈ R, all vehicles converge to form a line.
11
Theorem 4: (Section V) Problem 3 is solvable if and only if there are at most two disjoint closed
sets of nodes in the sensor digraph.
In Section V we also introduce a special sensor digraph which guarantees that all vehicles converge
to a line segment, equally spaced. In Section VI we show how our solution to Problem 2 can be employed
to achieve formation stabilization to any geometric pattern.
IV. F ORMATION S TABILIZATION
TO A
P OINT
In this section we prove Theorem 3. The proof requires the following lemmas.
Lemma 1: A digraph G = (V, E) with |V| ≥ 2 has no globally reachable node if and only it has
two disjoint closed subsets of V .
Proof: (Sufficiency) The sufficient condition follows directly.
(Necessity) We prove the necessary condition by means of a constructive algorithm. Firstly, select
any node, say vi1 , in V and partition V as V = {vi1 } ∪ V1 ∪ V10 , where every node in V1 can reach vi1
and no node in V10 can reach vi1 . Then V10 is closed. Also, V10 6= φ, since vi1 is not globally reachable.
Secondly, select any node, say vi2 in V10 . Since V10 is not empty, we can always find one. Check if
the node vi2 is globally reachable in the induced subgraph G(V 10 ).
If so, then partition V as V = W1 ∪ W10 ∪ V10 , where every node in W1 can reach (some node in) V10
and no node in W10 can reach V10 . Then W10 is closed. Also, W10 6= φ, since vi2 is not globally reachable.
Thus V10 and W10 are two disjoint closed subsets for the digraph G .
If instead the condition above is false, partition V 10 as V10 = {vi2 } ∪ V2 ∪ V20 , where every node in
V2 can reach vi2 and no node in V20 can reach vi2 . Then V20 is closed and nonempty. Next, select any
node, say vi3 , in V20 and check if vi3 is globally reachable in the induced subgraph G(V 20 ). If so, then
the conclusion follows by the same argument as above. If it does not, repeat this procedure again until
this condition holds. Since the digraph has a finite number of nodes and V k0 is getting smaller each step,
eventually the condition must hold and two disjoint closed subsets will have been constructed.
Next, we present a set of useful lemmas regarding the graph Laplacian. In what follows, let L be
the Laplacian of a digraph G = (V, E) with n nodes.
Lemma 2: 0 is an eigenvalue of L, while all other eigenvalues have positive real part.
12
Proof: By the definition of Laplacian matrix and the Geršgorin disk theorem, the conclusion follows
immediately.
Lemma 3: If the digraph is strongly connected, then 0 is a simple eigenvalue of L. Furthermore, L
has a positive right-eigenvector 1 = [1 · · · 1]T and a positive left-eigenvector β = [ β1 β2 · · ·
β n ]T
corresponding to 0.
Proof: We write the Laplacian matrix as L = D−A, where D , A are the degree matrix and the adjacency
matrix, respectively. By definition, the Laplacian L has an eigenvalue at 0 and a right eigenvector 1 =
[1 · · · 1]T . Now we will show that 0 has algebraic multiplicity 1 and its corresponding left eigenvector
e := D −1 L = I − D −1 A =: I − A
e (D is invertible since each node has nonzero
is positive. Define L
e have the same distribution
out-degree, since the digraph is strongly connected). The matrices A and A
e is also strongly connected and
of zero entries and nonzero entries, and so the associated digraph G( A)
e is irreducible. In addition, A
e is row stochastic by its definition, so it follows that ρ( A)
e = 1.
therefore A
e = 1 is an eigenvalue of algebraic multiplicity 1 and its
Hence, by the Perron-Frobenius theorem ρ(A)
fn ]T . Hence, 0 is a simple
corresponding left eigenvector is positive, denoted βe = [ βe1 βe2 · · · β
e , and so of L. Furthermore, βe is the left eigenvector of L
e corresponding to 0. As a
eigenvalue of L
e −1 is a left eigenvector of L corresponding to the 0 eigenvalue, which is positive.
consequence, β = βD
Lemma 4: The digraph has a globally reachable node if and only if 0 is a simple eigenvalue of L.
Proof: (Necessity) Let V 0 be the set containing all the globally reachable nodes. Since the digraph has
a globally reachable node, V 0 contains either all n nodes or r (1 ≤ r < n) nodes.
In the first case that, V 0 = V , the digraph G is strongly connected. Then by Lemma 3, 0 is an
eigenvalue of L of algebraic multiplicity 1.
Next, for the case when V 0 contains r (1 ≤ r < n) nodes, we have that u → v and v 9 u for all
u ∈ V − V 0 , v ∈ V 0 . If necessary, we can renumber the nodes of the digraph G so that V 0 = {1, 2, . . . , r}.
So D, A, L have the form




A1 0
D1 0
,
, A = 
D=
A2 A3
0 D3

L=
L1
0
L2 L3


=
D1
0
0
D3


−
A1
0
A2 A3

,
13
where L1 ∈
Rr×r ,
L3 ∈
R(n−r)×(n−r) .
In particular, when r = 1, L1 = D1 = A1 = 0. Since D may be
singular, it’s convenient to define

 1,
if r = 1,
Ds1 =
 D , if 1 < r < n,
1
and As1

 1,
if r = 1,
=
 A , if 1 < r < n,
1
and rewrite L as (L doesn’t change)

 
 

A
0
D
0
L1 0
.
 −  s1
 =  s1
(6)
L=
A2 A3
0 D3
L2 L3


Ds1 0
 is invertible. Then by
The associated digraph G(As1 ) is strongly connected and Ds = 
0 D3
Lemma 3, L1 = Ds1 − As1 has a simple eigenvalue at 0. Define

 



e
e
A
0
A
0
L
0
e := D −1  s1
e := D −1 L =  1
e
= 1
 , and L
 = I − A.
A
s
s
e
e
e2 L
e3
A2 A3
A2 A3
L
e is row stochastic, and so is A
ek (the k th power of A
e). Furthermore, A
ek can be
Thus, the matrix A
expressed in the form

ek = 
A
ek
A
1
0
e(k) A
ek
A
3
2


e(k) depending upon the matrices A
e1 , A
e2 and A
e3 . Since every node in V − V 0 can reach
for some matrix A
2
e(n) must have a positive entry. Thus the
one of the nodes in V 0 after n steps, it follows that each row of A
2
en is less than one and therefore ρ(A
e3 ) < 1. Hence, L
e3 = I − A
e3 is nonsingular
maximal row sum for A
3
and so is L3 . In conclusion, 0 is an eigenvalue of L of algebraic multiplicity 1.
(Sufficiency) Suppose by way of contradiction that the digraph has no globally reachable node. By
Lemma 1, it follows that there exist two disjoint closed subsets of V . So if necessary, we can renumber
the nodes so that L has the form

L11
0
0


L =  0 L22 0

L31 L32 L33



.

By Lemma 2, L11 has a zero eigenvalue and L22 has a zero eigenvalue, too. Hence, 0 is not a simple
eigenvalue of L, a contradiction.
The sensors produce a kind of symmetry in the system equations, in that sensory capabilities with
respect to x and y coordinates are identical. For this reason, the Laplacian L leads to the matrix L ⊗ I 2 ,
which we now study.
14
Lemma 5: Let L(2) = L ⊗ I2 . If the digraph is strongly connected, then −L(2) is H(α, 2) stable,
where α = {{1, 2}, {3, 4}, . . . , {2n − 1, 2n}}.
Proof: By Lemma 3, L1 = 0 and β T L = 0, where β is positive. Then for P = diag(β1 , β2 , . . . , βn ), P
is a positive definite matrix. Let Q = LT P + P L. Then we have
Q1 = LT P 1 + P L1 = LT β = 0
and
Q = LT P + P L = (D − A)T P + P (D − A) = (D T P + P D) − (AT P + P A),
where D T P +P D is a positive diagonal matrix, and AT P +P A is a nonnegative irreducible matrix since
the sum of an irreducible matrix and a nonnegative matrix is still irreducible. If we treat A T P + P A as
the adjacency matrix of a weighted digraph G1 and D T P + P D as its degree matrix, then we know G1
is strongly connected and Q is its Laplacian matrix. By Lemma 2 and Lemma 3, 0 is an eigenvalue of Q
of algebraic multiplicity 1, while the others are in the open right half-plane, implying that Q is positive
semi-definite. Furthermore, 1 is its right eigenvector corresponding to 0.
Let α = {{1, 2}, {3, 4}, . . . , {2n − 1, 2n}}. Thus, for the matrix L (2) = L ⊗ I2 , we can define the
positive-definite α-scalar matrix
P(2) = diag(β1 I2 , β2 I2 , . . . , βn I2 ) = P ⊗ I2 ,
and then by properties (a), (b) and (c) of the Kronecker product we have
LT(2) P(2) + P(2) L(2) = (L ⊗ I2 )T (P ⊗ I2 ) + (P ⊗ I2 )(L ⊗ I2 )
= (LT P ) ⊗ I2 + (P L) ⊗ I2
= (LT P + P L) ⊗ I2
= Q ⊗ I2 =: Q(2) .
By property (d) of the Kronecker product, it follows that both L (2) and Q(2) have 2 zero eigenvalues
and the others are in the open right half-plane and therefore Q (2) is positive semi-definite. In addition,
ker(Q(2) ) = ker(L(2) ).
Given an α-diagonal positive definite symmetric matrix R, R −1 is also α-diagonal, positive definite
and symmetric. Since P(2) is a positive definite α-scalar matrix, P(2) R−1 = R−1 P(2) is an α-diagonal
positive definite matrix, too. It now follows that
(−RL(2) )T (R−1 P(2) ) + (P(2) R−1 )(−RL(2) ) = −LT(2) P(2) − P(2) L(2) = −Q(2) ,
15
which is negative semi-definite. Hence, by Lyapunov theory, the eigenvalues of −RL (2) are in the open
left half-plane or on the imaginary axis. Further, from the spectral properties of −L (2) we can infer that
0 is an eigenvalue of −RL(2) of algebraic and geometric multiplicity 2. Therefore, to show −L (2) is
H(α, 2) stable, we only need to show that no other eigenvalues of −RL (2) , except the 2 zero eigenvalues,
are on the imaginary axis. Suppose by way of contradiction that λ = jω (ω 6= 0) is one of the eigenvalues
with corresponding eigenvector x. Then
0 = x∗ (RL(2) )T (R−1 P(2) ) + (P(2) R−1 )(RL(2) ) − Q(2) x
= −jωx∗ (R−1 P(2) )x + jωx∗ (P(2) R−1 )x − x∗ Q(2) x
= −x∗ Q(2) x.
It follows that x ∈ ker(Q(2) ) = ker(L(2) ) = ker(RL(2) ) and therefore ω = 0, a contradiction.
Lemma 6: Let L(2) = L ⊗ I2 . If and only if the digraph has a globally reachable node, −L (2) is
H(α, 2) stable, where α = {{1, 2}, {3, 4}, . . . , {2n − 1, 2n}}.
Proof: (Sufficiency) Let V 0 be the set containing all the globally reachable nodes. It is not empty but
contains either all n nodes or r (1 ≤ r < n) nodes. In the former case, the digraph G is strongly connected
and therefore −L(2) is H(α, 2) stable by Lemma 5. In the latter case, from the proof of Lemma 4, we
have that L can be expressed as in (6), i.e.,

 
 

L1 0
Ds1 0
As1 0
=
−
,
L=
L2 L3
0 D3
A2 A3
where the associated digraph G(As1 ) is strongly connected and L3 is nonsingular. Hence, by Lemma 5,
−L1(2) is H(α, 2) stable. Furthermore, one can easily verify that L 3 is a nonsingular M -matrix. Then it
follows from the characterization of nonsingular M -matrices (page 136, [5]) that there exists a positive
diagonal matrix P = diag(p1 , p2 , . . . , p(n−r) ) such that
Q = LT3 P + P L3
is positive definite. Applying the Kronecker product with I 2 to both sides of the above equation and
using properties (a), (b) and (c) of the Kronecker product yields
Q(2) = LT3(2) P(2) + P(2) L3(2) ,
where P(2) is α-scalar positive definite and Q(2) is positive definite. Given an α-diagonal positive definite
symmetric matrix R2 , we let P̄ = P(2) R2−1 = R2−1 P(2) , which is positive definite. Then
(−R2 L3(2) )T P̄ + P̄ (−R2 L3(2) ) = −Q(2) < 0,
16
which meansthat −L3(2) is H(α, 0) stable. Notice that given
 an α-diagonal positive definite symmetric
R1 0
R L
0
, RL(2) =  1 1(2)
 . Therefore, −L(2) is H(α, 2) stable.
matrix R = 
0 R2
R2 L2(2) R2 L3(2)
(Necessity) Since −L(2) = −L ⊗ I2 is H(α, 2) stable, it follows that −L(2) has a 0 eigenvalue of
algebraic multiplicity 2 and then by property (d) of the Kronecker product, −L has a simple eigenvalue
at 0. Using Lemma 4, the digraph has a globally reachable node.
Proof of Theorem 3: (Sufficiency) We show it by presenting the explicit smooth time-periodic feedback
controller

P
P

 vi (t) = k
(zm (t) − zi (t)) · ri (t),
xim (t) = k
i = 1, 2, . . . , n.
m∈Ni
m∈Ni

 ω (t) = cos(t),
i
(7)
We begin by noticing that {(∀ m ∈ Ni ) zm = zi } ⇒ {vi = 0}. Further, for any t0 ∈ R, θi (t) =
θi (t0 ) + sin(t), i = 1, . . . , n, which is periodic with period 2π . Next, using the identity (z · r)r = (rr T )z ,
we get
żi = vi ri = k
X
[(zm − zi ) · ri ] ri = k ri riT

M (θi (t)) := ri riT = 
and




H(θ(t)) := 



cos2 (θi (t))
cos(θi (t)) sin(θi (t))
cos(θi (t)) sin(θi (t))
sin2 (θi (t))
M (θ1 (t))
0
0
0
0
M (θ2 (t))
0
0
0
0
..
0
0
0
Thus, the overall position dynamics read as
where z ∈ R2n
(zm − zi ) .
m∈Ni
m∈Ni
Define
X
.
0
M (θn (t))




.





(8)
(9)
ż = −kH(θ(t))L(2) z,
(10)
iT
h
and L(2) = L ⊗ I2 (L is the Laplacian of
is the position vector z = z1T z2T · · · znT
the sensor digraph).
The averaged system of (10) is
ż = −kHav L(2) z,
(11)
17
where Hav = diag(M̄1 , . . . , M̄n ) and

 
R 2π
1
2
1
2
mi mi
2π
0 cos (θi (τ ))dτ
=
M̄i := 
R
2π
1
m2i m3i
2π 0 cos(θi (τ )) sin(θi (τ ))dτ
1
2π
R 2π
0
1
2π
2

cos(θi (τ )) sin(θi (τ ))dτ
.
R 2π 2
0 sin (θi (τ ))dτ
Then by the Cauchy-Schwarz inequality, m1i m3i ≥ (m2i ) . Since θi (t) is not constant, the inequality holds
strictly. It follows that M̄i is positive definite and therefore Hav is positive definite. More exactly, we
say Hav is α-diagonal positive definite with α = {{1, 2}, {3, 4}, . . . , {2n − 1, 2n}}. Since the sensor
digraph has a globally reachable node, by Lemma 6 it follows that −L (2) is H(α, 2) stable. So there is
a similarity transformation F such that

−F −1 L(2) F = 
−L11
0
0
02×2

,
where −L11

is Hurwitz
 column vectors of F must be in the null space of L (2) , i.e.
 the last two
 and


a
2n


, a, b ∈ R . Without loss of generality, we can choose the last two column
ξ ∈R ξ =1⊗


b
vectors of F to be 1 ⊗ I2×2 .

Applying the transformation e = 


ė1
ė2
e1
e2


 = F −1 z to system (10) yields

 = −kF −1 H(θ(t))L(2) F e = −kF −1 H(θ(t))F 

=: k 
A11 (t)
0
A12 (t) 02×2


e1
e2

L11
0
0
02×2


e1
e2


.
And correspondingly, for the averaged system (11), we have





e
L
0
ė
 1 
 1  = −kF −1 Hav L(2) F e = −kF −1 Hav F  11
e2
0 02×2
ė2

=: k 
Ā11
0
Ā12 02×2


e1
e2

.
Thus, the reduced averaged system ė1 = k Ā11 e1 is exponentially stable since −L(2) is H(α, 2) stable.
Then, by Theorem 1, there exists a positive constant k ∗ such that, for all 0 < k < k ∗ , global exponential
stability of the reduced original system ė1 = kA11 (t)e1 is established. Also, since ė2 = kA12 (t)e1 and
18
A12 (t) is uniformly bounded, it follows that ė2 → 0 exponentially when t → ∞. This implies that e2
tends to some finite constant vector, say [xss yss ]T . In conclusion,




e1 (t)
xss
=1⊗
,
lim z(t) = lim F 
t→∞
t→∞
e2 (t)
yss
showing that the overall position dynamics globally exponentially converge to an invariant subspace, thus
solving Problem 2.
(Necessity) Assume formation stabilization to a point by local information controller is feasible. By
way of contradiction, suppose the sensor digraph has no globally reachable node. Then it follows from
Lemma 1 that there are two disjoint closed sets of nodes in the sensor digraph G = (V, E), say V 1
and V2 . Given the initial conditions satisfying zi (0) = zss1 , i ∈ V1 and zj (0) = zss2 , j ∈ V2 , then
for each vehicle i in V1 , (∀ m ∈ Ni ) zm = zi and so żi = 0. Meanwhile, for each vehicle j in V2 ,
(∀ m ∈ Nj ) zm = zj and so żj = 0. Hence, if zss1 6= zss2 , they can not gather at the same point, a
contradiction.
Remark 1: In [28], a general average of M (θi (t)) in (8)


R
R
1 T
1 T
2 (θ (τ ))dτ
cos
cos(θ
(τ
))
sin(θ
(τ
))dτ
i
i
i
T 0

Mav (T ) =  R T 0
R
1 T
1 T
2
cos(θ
(τ
))
sin(θ
(τ
))dτ
sin
(θ
(τ
))dτ
i
i
i
T 0
T 0
is considered and it is claimed that det(Mav ) > 0, where Mav = lim Mav (T ), if and only if θi (t) is
T →∞
not constant. This is only true if T is finite. Specifically, if T < ∞ and θ i (t) is not constant, then
Z T
2
Z T
Z T
1
1
1
2
2
cos (θi (τ ))dτ
sin (θi (τ ))dτ >
cos(θi (τ )) sin(θi (τ ))dτ .
T 0
T 0
T 0
When T → ∞ the above strict inequality may not hold. To see that, choose θ i (t) = sin−1 (e−t ). Then,
RT
lim T1 0 sin2 (θi (τ ))dτ = 0 and
T →∞
lim
T →∞
1
T
Z
T
2
cos (θi (τ ))dτ
0
1
T
Z
0
T
2
sin (θi (τ ))dτ
= lim
T →∞
1
T
Z
T
cos(θi (τ )) sin(θi (τ ))dτ
0
2
= 0,
and thus det(Mav ) = 0.
Remark 2: An alternative choice of controller for formation stabilization to a point is

P

 vi (t) =
xim (t),
m∈Ni
i = 1, 2, . . . , n.

 ω (t) = γ cos(γt),
i
(12)
By applying a time scaling, τ = γt , one can use Theorem 3 and conclude that if the sensor digraph has a
globally reachable node, there exists a positive constant γ ∗ such that, for all γ ∗ < γ < ∞, (12) achieves
formation stabilization to a point.
19
Remark 3: The above results show that all the wheeled vehicles exponentially converge to a point
formation through simple local actions, or we say they achieve an agreement about a common point.
Notice that the sensor digraph is required to have just one globally reachable node (as in Fig. 1(c)). So
if we treat a beacon placed at the proper location as one member of the group of vehicles and it is the
globally reachable node, the local actions of each individual vehicle result in the group gathering at the
beacon.
Fig. 5 depicts simulation results for formation stabilization to a point of ten wheeled vehicles using
the smooth time-varying feedback control law (7) with the choice of k = 1. The initial conditions are
randomly produced and the sensor digraph is given in Fig. 6. For example, node 5 is globally reachable,
while node 1 is not.
PSfrag replacements
Fig. 5.
Fig. 6.
Ten wheeled vehicles gather at a common position.
1
3
5
7
9
2
4
6
8
10
The sensor digraph of a group of ten wheeled vehicles.
20
V. F ORMATION S TABILIZATION
TO A
L INE
We begin this section with the proof of Theorem 4.
Proof of Theorem 4: (Sufficiency) By the condition that there are at most two disjoint closed sets of
nodes in the sensor digraph, by Lemma 1, either the sensor digraph has a globally reachable node, or there
are exactly two disjoint closed sets of nodes in it. In the first case, by Theorem 3, formation stabilization
to a point is feasible, which is a special instance of line formation. In the second case, we let V 1 , V2
be the two disjoint closed sets of nodes in the sensor digraph G = (V, E), and let V 3 = V − V1 − V2 .
Thus, the induced subgraphs G1 = (V1 , E ∩ (V1 × V1 )), G2 = (V2 , E ∩ (V2 × V2 )) both have a globally
reachable node. (To see this point, suppose one of these two induced subgraphs has no globally reachable
set. Then by Lemma 1 there are two disjoint closed sets in it and therefore there are three disjoint closed
sets in G , a contradiction.) If V3 is empty, then by Theorem 3 each group of vehicles whose indices are
in Vi , i = 1, 2 can converge to a point and therefore the whole group of wheeled vehicles form a line.
On the other hand, if V3 is not empty, then every node in V3 can reach either V1 or V2 . Without loss of
generality, we assume the graph Laplacian has the form

L
0
0
 11

L =  0 L22 0

L31 L32 L33



,

where L11 and L22 are the Laplacian matrices corresponding to the induced subgraph G 1 and G2 . One
can easily verify that L33 is a nonsingular M -matrix.
By using the controller (7), the overall position dynamics are given by

1
1
1


 ż = −kH(θ (t))L11(2) z ,

ż 2 = −kH(θ 2 (t))L22(2) z 2 ,



 ż 3 = −kH(θ 3 (t)) L
2 ,
1
3
33(2) z + L31(2) z + L32(2) z
where z i , θ i , i = 1, 2, 3 are the corresponding position vector and orientation vector respectively. From the
∗
∗
∗
proof of Theorem 3,
 that, for all 0 < k < k1 ,
 that there exist positive constants k 1 and k2 such
 we have
x
xss1
 and for all 0 < k < k2∗ , lim z 2 (t) = 1 ⊗  ss2  with exponential
lim z 1 (t) = 1 ⊗ 
t→∞
t→∞
yss2
yss1
convergence rate.
The change of variables ς = L33(2) z 3 + L31(2) z 1 + L32(2) z 2 yields
ς˙ = −kL33(2) H(θ 3 (t))ς − kL31(2) H(θ 1 (t))L11(2) z 1 − kL32(2) H(θ 2 (t))L22(2) z 2 .
(13)
21
Since L33 is a nonsingular M -matrix, by the same argument as in the proof of Lemma 6, −L 33(2) and
−LT33(2) are both H(α, 0) stable. By Theorem 1, there exists a positive constant k 3∗ such that, for all
0 < k < k3∗ , the origin of the nominal system
ς˙ = −kL33(2) H(θ 3 (t))ς
is globally exponentially stable. Furthermore, notice that the other two terms in (13) both exponentially
converge to zero. Hence (13) can be viewed as an exponentially stable system with an exponentially
vanishing input, and thus its origin is exponentially stable. Let k ∗ = min{k1∗ , k2∗ , k3∗ }. Hence, for all
0 < k < k∗ ,
lim z 3 (t) = − L33(2)
t→∞
= − L33(2)
h
−1
−1
L31(2) lim z 1 (t) − L33(2)
t→∞
L31(2) · 1 ⊗ 


= − L−1
33 · L31 · 1 ⊗
i


xss1
yss1

xss1
yss1
−1
L32(2) lim z 2 (t)
t→∞

 − L33(2)
−1


 − L−1
33 · L32 · 1 ⊗


L32(2) · 1 ⊗ 
xss2
yss2

xss2
yss2


.
−1
· 1 = 0 and so − L−1
33 · L31 · 1 − L33 · L32 · 1 = 1. Hence, all




xss1
xss2
 and 
, which means the wheeled vehicles
(zi (t))i∈V3 approach a convex combination of 
yss1
yss2
with indices in V3 eventually move to the line formed by two points (x ss1 , yss1 ) and (xss2 , yss2 ) in the
Notice that
L31 L32 L33
plane.
(Necessity) Suppose by way of contradiction that there are three disjoint closed sets of nodes in the
sensor digraph, say V1 , V2 and V3 . Then let the initial conditions of the vehicles in V j , j = 1, 2, 3 be
chosen such that zi (0) = zssj , i ∈ Vj . Hence, for each vehicle i in Vj , (∀ m ∈ Ni )zm = zi and so
żi = 0. Then three groups of vehicles form a geometric pattern specified by three points z ss1 , zss2 and
zss3 . These three points can be arbitrarily set that may not form a line, a contradiction.
Theorem 4 has an interesting special case when the two disjoint closed sets of nodes in the sensor
digraph both have only one member, say nodes 1 and n. Vehicles 1 and n are called edge leaders. The
edge leaders here are not necessarily wheeled vehicles. They can be virtual beacons or landmarks. But
the vehicles respond to these edge leaders much like they respond to real neighbor vehicles. The purpose
of the edge leaders is to introduce the mission: to direct the vehicle group behavior. We emphasize that
22
the edge leaders are not central coordinators. They do not broadcast instructions. They only play the role
of individual vehicles, but cannot sense other vehicles or communicate with them. As for the remaining
PSfrag replacements
vehicles, i, i = 2, . . . , n − 1, we assume that each agent can sense agents i − 1 and i + 1. This gives the
sensor digraph in Fig. 7. It is readily seen that the digraph in Fig. 7 has exactly 2 disjoint closed sets of
nodes. We now show that in this special case all vehicles converge to a uniform distribution on the line
segment specified by the two edge leaders.
2
1
Fig. 7.
n−2
3
n−1
n
The sensor digraph for a group of wheeled vehicles with two edge leaders.
Theorem 5: Consider a group of n wheeled vehicles with two stationary edge leaders labeled 1
and n. Then, there exists a positive constant k ∗ such that for all 0 < k < k ∗ , the following smooth
time-varying feedback control law

P

 vi (t) = k
xij (t),
Ni = {i − 1, i + 1},
i = 2, . . . , n − 1
j=Ni

 ω (t) = cos(t),
i
(14)
guarantees that all the vehicles converge to a uniform distribution on the line segment specified by the
two edge leaders.
Proof: First we observe that the Laplacian of the sensor digraph in Fig. 7 is given by


0
0
0
0 ··· 0




 −1 2 −1 0 · · · 0 




 0 −1 2 −1 · · · 0 


L= .
..  .
..
..
..
..
.
 .
.
.
.
.
. 




 0 · · · 0 −1 2 −1 


0
Let z =
h
z1 z2 · · ·
zn
iT
···
0
0
0
0
. It follows from Theorem 4 that L(2) z(∞) = 0. Consider the following
partition of {1, 2, . . . , n}, {m1 , m2 , . . . , m2n } = {1, 3, 5, . . . , 2n−1, 2, 4, 6, . . . , 2n}. Then the associated
permutation matrix P has the unit coordinate vectors e m1 , em2 , . . . , em2n for its columns. Now observe
23
that the matrix P performs the transformation
where x =
h

P T (L ⊗ I2 )P = I2 ⊗ L = 
x1
L
0
0
L
iT



and P T z = 
x
y

,
iT
h
and
y
=
. Thus,
x2 · · · x n
y1 y2 · · · y n





x(∞)
L
0
L 0


 P T z(∞) = P 
0 = L(2) z(∞) = P 
y(∞)
0 L
0 L
implies Lx(∞) = 0 and Ly(∞) = 0. Also note that



0









 1




Ker(L) = span ξ1 =  2



 ..


 .







n−1












 , ξ2 = 















n − 2 




n−3  ,


.. 


. 





0
n−1
so x(∞) = α1 ξ1 + α2 ξ2 and y(∞) = β1 ξ1 + β2 ξ2 . Since x1 (∞) = x1 (0), xn (∞) = xn (0) and
y1 (∞) = y1 (0), yn (∞) = yn (0), so we solve for α1 =
In conclusion, we have
 

x1 (∞)
 

 

 x2 (∞)  
 

..
 

=

.
 

 

 xn−1 (∞)  
 

xn (∞)
x1 (0)
(n−2)x1 (0)+xn (0)
n−1
..
.
x1 (0)+(n−2)xn (0)
n−1
xn (0)






,





xn (0)
n−1 ,
α2 =
x1 (0)
n−1


y1 (∞)


 y2 (∞)

..


.


 yn−1 (∞)

yn (∞)
 
 
 
 
 
=
 
 
 
 
and β1 =
yn (0)
n−1 ,
y1 (0)
(n−2)y1 (0)+yn (0)
n−1
..
.
y1 (0)+(n−2)yn (0)
n−1
yn (0)
This shows that all vehicles asymptotically approach a uniform distribution on the line.
β2 =
y1 (0)
n−1 .






.




Fig. 8 shows the simulation result of four wheeled vehicles and two stationary edge leaders to form
a uniform distribution on a line using the control law (14) with the choice of k = 1.
VI. F ORMATION S TABILIZATION
TO
A NY G EOMETRIC PATTERN
In this section, we turn our attention to the problem of formation stabilization to any geometric
pattern. Following [25], we let Π be a predicate describing a geometric pattern, such as a point, a regular
polygon, a line segment, etc. Such a predicate specifies a formation up to translation and rotation. By
formation stabilization of a group of n vehicles to Π, we mean that the vehicles (globally exponentially)
converge to a distribution satisfying Π.
24
Fig. 8.
Formation stabilization to a line of four wheeled vehicles and two edge leaders.
We suppose that a group of wheeled vehicles have a common sense of direction, represented by the
angle ψ in Fig. 9 . For instance, each vehicle carries a navigation device such as a compass. Alternatively,
all vehicles initially agree on their orientation and use it as the common direction. The common direction
may not coincide with the positive x-axis of the global frame. Let φ i = θi − ψ (see Fig. 9). We assume
PSfrag replacements
that vehicle i can measure its own φi .
φk
ψ
k
φj
ψ
φi
ψ
j
i
gΣ
Fig. 9.
A group of wheeled vehicles have a common sense of direction.
There are two ways to describe a geometric pattern in the plane. One way is by inter-node distances,
dij , as in the rigid formation framework of [21]. The other way is by specifying the position vector, c i , of
25
each node with respect to a common coordinate frame. The two equivalent representations are illustrated
in Figure 10(a) and (b) for a square formation.
d12
2
2
1
1
d41
d23
d24
3
c4
c3
3
4
d34
4
(a)
Fig. 10.
c1
c2
d13
(b)
An example for a square formation.
It is worth noting that, for all (R, b) ∈ SE(2), the vectors ĉ i = Rci + b describe the same geometric
formation as the one specified by ci . So given a desired geometric formation pictured by c i , i = 1, . . . , n,
our objective is to stabilize the position state z i of each vehicle to ĉi = Rci + b, i = 1, . . . , n for some
R and b. To achieve a desired geometric formation characterized by c = [c T1 · · · cTn ]T , we can simply
translate the formation vector c into a control offset d = L (2) c so that the forward control velocity is 0
PSfrag
whenreplacements
the group of vehicles has achieved a formation. We denote the offset for each vehicle by


d xi
 or di = dxi + jdyi .
di = 
d yi
As an example, consider the simple sensor digraph shown in Fig. 11(a) and a triangle formation described
by ci , i = 1, 2, 3. The corresponding control offsets di , i = 1, 2, 3 are shown in Fig. 11(b).
2
1
d1
c2
c1
d2
2
3
(a) Sensor digraph
Fig. 11.
An example of a simple triangle formation.
c3
1
d3
3
(b) Illustration for ci and di
26
Next, we show that the time-varying control law

)
(
h
i

P

 vi (t) = k
xim (t) ,
1 0 R(−φi (t))di +
i = 1, 2, . . . , n,
m∈Ni


 ω (t) = cos(t),
i
where R is a rotation matrix defined by

R(φ) = 
cos(φ) − sin(φ)
sin(φ)
cos(φ)
achieves formation stabilization to Π.
(15)

,
Theorem 6: Let Π be a desired geometric formation described by c. Suppose a group of n wheeled
vehicles have a common sense of direction and formation stabilization to a point is feasible. Then there
exists a positive constant k ∗ such that for all 0 < k < k ∗ , the smooth time-varying feedback control law
(15) with d = L(2) c guarantees global exponential formation stabilization to Π.
Proof: Notice that
vi = k
=k
(
(
h
1 0
i
R(−φi (t))di +
R(ψ)di +
xim
m∈Ni
P
(zm − zi )
m∈Ni
Thus, we have the following closed-loop system
(
P
żi = kM (θi (t)) R(ψ)di +
X
)
)
· ri .
)
(zm − zi ) ,
m∈Ni
or in vector form, we have
ż = kH(θ(t)) −L(2) z + (In ⊗ R(ψ)) d .
By the property of Kronecker product, it follows that
(In ⊗ R(ψ))(L ⊗ I2 ) = L ⊗ R(ψ) = (L ⊗ I2 )(In ⊗ R(ψ)).
Furthermore, since d = L(2) c, we obtain
ż = −kH(θ(t))L(2) {z − (In ⊗ R(ψ))c} .
Under the coordinate transformation ς = z − (In ⊗ R(ψ))c, we get
ς˙ = −kH(θ(t))L(2) ς.

By the proof of Theorem 3, lim ς(t) = 1 ⊗ 
t→∞
xss
yss

 for some constant position (xss , yss ). Hence,

lim zi (t) = R(ψ)ci + 
t→∞
27
xss
yss

,
i = 1, . . . , n,
which means that the group of vehicles form a geometric formation specified by c.
Remark 4: Notice that θi (t) = θi (t0 ) + sin(t), so if a group of n wheeled vehicles achieve an
agreement on their initial orientation θi (t0 ) and choose it as their common direction, the control law (15)
becomes

)
(

P

 vi (t) = k Re e−j sin(t) di +
xim (t) ,


 ω (t) = cos(t),
i
m∈Ni
i = 1, 2, . . . , n.
(16)
The agreement on their orientation can be implemented by an alignment strategy as shown in [12].
However, the control law (16) is not robust in practice.
Fig. 12.
Ten wheeled vehicles form a circle formation.
As an example, the simulation result for a circle formation of ten wheeled vehicles with the sensor
digraph in Fig. 6 is shown in Fig. 12. The circle formation is described by c i = 75e(j
The initial conditions are
x(0) = [10, 10, 10, 0, 0, 0, 0, −10, −10, −10] T ,
2(i−1)π
10
)
, i = 1, . . . , 10.
28
T
y(0) = [−5, 0, 5, 10, 3, −3, −10, −5, 0, 5] ,
h π π π π π π π π π π iT
θ(0) =
, , , , , , , , ,
,
6 6 6 6 6 6 6 6 6 6
and the control law (16) is used with the parameters k = 1, d 1 = 61.74 − j107.72, d2 = 78.26 − j25.43,
d3 = 91.63 + j66.57, d4 = 35 + j25.43, d5 = j82.29, d6 = −48.37 + j66.57, d7 = −78.26 + j25.43,
d8 = −78.26 − j25.43, d9 = −35 − j25.43, d10 = −j82.29.
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