INT J COMPUT COMMUN, ISSN 1841-9836
Vol.7 (2012), No. 4 (November), pp. 632-644

Formation Control of Multiple Agents with Preserving
Connectivity and its Application to Gradient Climbing

K.D. Do

K.D. Do
Department of Mechanical Engineering,
Curtin University of Technology,
Perth, WA 6845, Australia
E-mail: duc@curtin.edu.au

Abstract: A design of cooperative controllers that force a group of N mobile agents
with limited communication ranges to perform a desired formation is presented. The
proposed formation control system also preserves initial communication connectivity
and guarantees no collisions between the agents. The formation control design is based
on smooth step functions, potential functions, and the Lyapunov direct method. The
proposed formation control system is applied to solve a gradient climbing problem
where the gradient average of a distributed field is estimated over a bounded region
using the field measurement by the agents.
Keywords: Formation control, collision avoidance, gradient climbing.

1 Introduction

Formation control involves controlling positions of a group of agents such that they perform
desired tasks such as optimizing objective functions from measurements taken by each agent,
and stabilization/tracking desired locations relative to reference point(s). Various methods have
been proposed for formation control of multiple agents.

Here, three popular methods are briefly mentioned. The leader-follower method (e.g., [1], [2])
uses several agents as leaders and others as followers. This method is easy to understand and
ensures formation maintenance if the leaders are disturbed. However, the desired formation
cannot be maintained if followers are perturbed unless a formation feedback is implemented, [3].
The behavioral method (e.g., [4], [5]), where each agent locally reacts to actions of its neighbors,
is suitable for decentralized control but is difficult in control design and stability analysis since
group behavior cannot explicitly be defined. The virtual structure method (e.g., [6], [7]) treats all
agents as a single entity. This method is amenable to mathematical analysis but is difficult to deal
with time-varying formation structure. Research works on formation control usually utilize one
or more of the above methods in a centralized or a decentralized manner. Centralized strategies
(e.g., [8], [3]) use a single controller that generates collision free trajectories in the workspace.
These strategies guarantee a complete solution but require high computational power and are
not robust. Decentralized schemes (e.g., [9], [10], [7]) require less computational effort but have
difficulties in controlling critical points, especially when collision avoidance between the agents
is a must.

The control design in the above works did not put hard constraints on the controlled outputs
except for those papers considered the problem of collision avoidance. Without hard constraints
on controlled outputs, overshoot might result in loss of initial communication between agents
due to limited communication between the agents. Hard constraints on the controlled outputs
were applied to design cooperative controllers for mobile agents to preserve initial communication.
These constraints on the controlled outputs were obtained through barrier Lyapunov or potential
functions using non-trivial bump functions or switching control strategies in [11] for the agreement
problem, [12] for the centralized approach, and [13] for the swarm aggregation.

Copyright c⃝ 2006-2012 by CCC Publications



Formation Control of Multiple Agents with Preserving
Connectivity and its Application to Gradient Climbing 633

This paper contributes two main folds. The first one is a design of smooth and bounded
cooperative controllers for a group of mobile agents to perform a desired formation task. The
desired formation task includes collision avoidance and communication connectivity preserva-
tion between the agents, time-varying desired formation shape, and stabilization of the desired
formation shape at any reference trajectories with bounded time derivatives. The second contri-
bution is an algorithm for estimating gradient average of a distributed field over a region in two
dimensional space. This algorithm uses only the field measurement on the boundary of a region,
over which the gradient average is to be estimated. The two contributions are then combined
to provide an effective gradient climbing system for a group of mobile agents by allowing the
reference trajectory for each agent generated based on the gradient average.

2 Preliminaries and Formation Control Objective

2.1 Smooth step function

This section presents a construction of a smooth step function. The smooth step function is to
be embedded into a potential function to avoid discontinuities in the control law due to the agents’
communication limitation in solving collision avoidance and connectivity preserving problems.

−1 0 1 2 3 4
−6

−4

−2

0

2

x

h
(x

,0
,3

,2
) 

a
n

d
 it

s 
d

e
ri
va

tiv
e

s

 

 

h
h’
h’’

Figure 1: A smooth step function and
its first and second derivatives.

Definition 1. A scalar function h(x,a,b,c) is said to be a
smooth step function if it possesses the following properties

where x ∈ R, h′(x,a,b,c) = ∂h(x,a,b,c)
∂x

, h′′(x,a,b,c) =
∂2h(x,a,b,c)

∂x2
, a and b are constants such that a < b, and c is

a positive constant.

Lemma 2. Let the scalar function h(x,a,b,c) be defined
as

h(x,a,b,c) =
f(τ)

f(τ) + cf(1− τ)
with τ =

x−a
b−a

, (1)

where

f(τ) = 0 if τ ≤ 0 and f(τ) = e−
1
τ if τ > 0, (2)

with a and b being constants such that a < b, and c being a positive constant. Then the function
h(x,a,b,c) is a smooth step function.

Proof. Proof of this lemma follows the same lines as the proof of Lemma 1 in [7]. An
illustration of a smooth step function (a = 0,b = 3,c = 2) is given in Figure 1.



634 K.D. Do

2.2 Problem statement

Agent dynamics
Y

(
1)

i

d

q

X
O

(
1)

i
d

q

Desired shape

od
q

id
l

d
O

idq

1i
O

Agent i -1
Agent i

1i
O

Agent i+1

i
O

i
R

i
R

Figure 2: Formation setup.

We assume that the agent i has the dy-
namics:

q̇i = ui, i ∈ N, (3)

where N is the set of all agents in the group,
ui ∈ Rn is the control input vector and qi ∈
Rn the position vector of the agent i.

Formation control objective

Each agent in the group needs its refer-
ence trajectory to track. The reference trajec-
tories can be predefined or determined from
measurement data. Furthermore, each agent
needs to communicate with other agents in the
group to perform its cooperative mission. Therefore, before stating formation control objective
we impose the following assumption on the reference trajectories, communication and initial
conditions between the agents in the group:

Assumption 3.
1) The agent i has a physical safety ball, which is centered at the point Oi and has a radius

Ri, and has a communication ball, which is centered at the point Oi and has a radius Ri, see
Figure 2. The radius Ri is such that

Ri ≥ Ri + Rj + ε1ij, (4)

for all j ∈ N,j ̸= i, where ε1ij is a strictly positive constant.
2) The reference trajectory qid for the agent i is generated by

qid = qod(sod) + lid, (5)

where qod(sod) is referred to as the common reference trajectory with sod being the common
trajectory parameter, and lid is to specify a desired formation shape. The trajectory qod has its
bounded derivatives. The vectors lid, i ∈ N have bounded derivatives, and satisfy

(Ri + Rj + ε2ij) ≤∥lid − ljd∥≤ min(Ri,Rj)−ε2ij, (6)

for all (i,j) ∈ N, i ̸= j, where ε2ij is a strictly positive constant, and is strictly less than
ε1ij
2

.
3)The agent i broadcasts its trajectory, qi, and its reference trajectory qid in its communi-

cation ball. Moreover, the agent i can receive the trajectory, qj, broadcasted by other agents j,
j ∈ N, j ̸= i in the group if the points Oj of these agents are in the communication ball of the
agent i.

4) At the initial time t0 ≥ 0, all the agents in the group are sufficiently far but not too far
away from each other in the sense that the following condition holds:

(Ri + Rj + ε3ij) ≤∥qi(t0)−qj(t0)∥≤ (min(Ri,Rj)−ε3ij), (7)

for all (i,j) ∈ N, i ̸= j, where ε3ij is a strictly positive constant and is strictly less than
ε1ij
2

.



Formation Control of Multiple Agents with Preserving
Connectivity and its Application to Gradient Climbing 635

Remark 4. Item 2) in Assumption 3 defines a desired formation (by vectors lid) and how this
desired formation moves (by the common reference trajectory qod). Item 3) specifies the way
each agent communicates with other agents in the group within its communication range. In
Figure 2, the agents i and i − 1 are communicating with each other since the points Oi−1 and
Oi are in the communication areas of the agents i and i− 1, respectively. Item 4) implies that
at the initial time t0 there are no collision between all the agents, and that all the agents are
communicating with each other. The conditions (4), (6) and (7) are imposed to avoid conflict
when solving collision avoidance and connectivity preserving problems. This is because we will
design a formation control system so that qi to track its reference trajectory qid.

Under Assumption 3, for each agent i design the control input vector ui to achieve a desired
formation consisting of 1) no switchings in the controllers; 2) no collisions between any agents; 3)
asymptotic convergence of each agent’s trajectory qi to its reference trajectory qid; and 4) initial
connectivity preservation. Mathematically, the objective is to design a smooth ui to achieve:

∥qi(t)−qj(t)∥ > (Ri + Rj), lim
t→∞

(qi(t)−qid(t)) = 0, ∥qi(t)−qj(t)∥ < min(Ri,Rj), (8)

for all ∀t ≥ t0 ≥ 0 and (i,j) ∈ N and j ̸= i.

3 Formation Control Design

Consider the following potential function

φ =

N∑
i=1

(
γi +

1

2
βi

)
. (9)

The aim of the goal function γi is to achieve asymptotic convergence of each agent’s trajectory
qi to its reference trajectory qid. As such the function γi puts penalty on the tracking errors
between the trajectory qi of the agent i and its reference trajectory qid = qod + lid. We choose
the function γi as:

γi =
1

2
∥qi −qid∥2. (10)

The purpose of the collision avoidance and connectivity preserving function βi is to force the
agent i to move away from other agents, and to maintain communication connectivity between
the agent i and other agents in the group. This function is chosen as follows:

βi =
∑
j∈Ni

βij, (11)

where Ni is the set of all the agents in the group except for the agent i. The function βij = βji
is a function of ∥qij∥2/2 with qij = qi −qj , and possesses the following properties:

1) βij = 0, βij′ = 0, βij′′ = 0, ∀ ∥qij∥∈
(
(Ri + Rj + δij),(min(Ri,Rj)− δij)

)
,

2) βij > 0, ∀ ∥qij∥∈
((

(Ri + Rj),(Ri + Rj + δij)
)
∪
(
(min(Ri,Rj)− δij),min(Ri,Rj)

))
,

3) lim
∥qij∥→(Ri+Rj)

βij = ∞, lim
∥qij∥→min(Ri,Rj)

βij = ∞,

4) βij is smooth for all ∥qij∥∈
(
(Ri + Rj),(min(Ri,Rj))

)
,

(12)



636 K.D. Do

where δij is a strictly positive constant and is strictly less than ε2ij specified in Assumption 3.
The terms βij′ and βij′′ are defined as follows:

βij′ = ∞, βij′′ = ∞, if ∥qij∥ = Ri + Rj, or ∥qij∥ = min(Ri,Rj),

βij′ =
∂βij

∂(∥qij∥2/2)
, βij′′ =

∂2βij
∂(∥qij∥2/2)2

, elsewhere.
(13)

Based on the smooth step function in Section 2.1, we can find many functions that satisfy
all properties listed in (12). As an example, we will use the following function βij:

βij =κij

[
1−h

(
∥qij∥2

2
,
(Ri+Rj)

2

2
,
(Ri+Rj+δij)

2

2
,cij

)
(
∥qij∥2

2
−

(Ri+Rj)
2

2

)2 + h
(
∥qij∥2

2
,
(min(Ri,Rj)−δij)2

2
,
min(Ri,Rj)

2

2
,cij)(

min(Ri,Rj)2

2
− ∥qij∥

2

2

)2
]
,

(14)

where κij and cij are positive constants, and the function h(•) is a smooth step function defined
in Definition 1. An illustration of βij defined in (14) is given in Figure 3 with Ri + Rj = 1,
min(Ri,Rj) = 11, δij = 2, cij = 1, κij = 1.

0 5 10
0

2

4

6

8

10

||q
ij
||

β i
j

Figure 3: An illustration of βij.

The derivative of φ along the solutions of (3)
satisfies

φ̇ =

N∑
i=1

ΩTi (ui − q̇id) +
N∑
i=1

( ∑
j∈Ni

βij′qTij
)
l̇id, (15)

where
Ωi = qi −qid +

∑
j∈Ni

βij′qij. (16)

From (15), we design the control input ui to make
the sum

∑N
i=1 Ω

T
i (ui − q̇id) negative definite as

ui = −kΨ(Ωi) + q̇od + l̇id, (17)

where k is a positive constant, and Ψ(Ωi) denotes a vector of bounded functions of elements of
Ωi in the sense that Ψ(Ωi) =

[
ψ(Ω1i ) ...,ψ(Ω

l
i), ...,ψ(Ω

n
i )
]T

with Ωli the l
th element of Ωi, i.e.,

Ωi = [Ω
1
i ...,Ω

l
i...Ω

n
i ]

T . The function ψ(x) satisfies

1) |ψ(x)| ≤ M1, 2)ψ(x) = 0 if x = 0, xψ(x) > 0 if x ̸= 0,

3)ψ(−x) = −ψ(x),(x−y)[ψ(x)−ψ(y)] ≥ 0, 4)
∣∣∣ψ(x)
x

∣∣∣ ≤ M2, ∣∣∣∂ψ(x)
∂x

∣∣∣ ≤ M3, ∂ψ(x)
∂x

∣∣∣
x=0

= 1,

(18)

for all x ∈ R,y ∈ R, where M1,M2,M3 are strictly positive constants. Some functions that
satisfy the above properties are arctan(x) and tanh(x). The above bounds mean that the large
control effort problem is avoided when the distance ∥qij∥ between the agent i and an agent j in
the group reaches a collision limit Ri + Rj or a connectivity preserving limit min(Ri,Rj).

To deal with the sum
∑N

i=1

( ∑
j∈Ni βij′q

T
ij

)
l̇id in (15), we observe that βij′ = 0 for all

∥qij∥ ∈
(
(Ri + Rj + δij),(min(Ri,Rj) − δij)

)
, see Property 1) of the function βij in (12). This

observation motivates us to design an update law for lid so that
∑N

i=1

( ∑
j∈Ni βij′q

T
ij

)
l̇id = 0 for

all time and l̇id tends to its desired value vid asymptotically. As such, we choose:

l̇id = Hivid, (19)



Formation Control of Multiple Agents with Preserving
Connectivity and its Application to Gradient Climbing 637

where

Hi =
∏
j∈Ni

h(∥qij∥2/2,(Ri + Rj + δij)
2/2,(Ri + Rj + δ

v
ij)

2/2,cij)×

(
1−h(∥qij∥2/2,min(Ri + Rj − δvij)

2/2,(Ri + Rj − δij)2/2,cij)
)
,

(20)

with δvij being a positive constant such that δij < δ
v
ij < ϵ2ij, and h(•) being a smooth step

function defined in Definition 1. With the choice of δij < δvij < ϵ2ij, we can see that

Hi = 1, ∀ ∥qij∥∈
(
(Ri + Rj + δ

v
ij),(min(Ri,Rj)− δ

v
ij)

)
,

Hi = 0, ∀ ∥qij∥∈
(
(0,(Ri + Rj + δij))∪ (min(Ri,Rj)− δij),∞)

)
,

0 < Hi < 1, elsewhere.

(21)

Obviously, the choice of the update law for lid in (19) with Hi being satisfied (21) gives:∑
j∈Ni

βij′qTij l̇id = 0, ∀ ∥qij∥∈ ((Ri + Rj),min(Ri,Rj)). (22)

Remark 5. 1) A careful look at the control law ui in (17) with Ωi in (16) shows that the
argument of the bounded Ψ (with the negative sign moved in) consists of two parts. The first
part is −(qi −qid), and the second part is −

∑
j∈Ni βij′qij. The first part together with q̇od + l̇id

is referred to as the attractive force plays the role of forcing the agent i to track its reference
trajectory. The second part is referred to as the repulsive force takes care of collision avoidance
and connectivity preserving for the agent i with the other agents in the group. Moreover, the
control ui is a smooth function of and depend on only its own state and reference trajectory,
and the states of other neighbor agents j if the agents j are sufficiently close to the agent i for
collision avoidance, or are sufficiently far away from the agent i for connectivity preserving.

2) The choice of the update law in (19) ensures that when the collision avoidance or con-
nectivity preserving is active, i.e., when the sum

∑
j∈Ni βij′qij is non-zero, the vector lid is not

updated, i.e., the desired formation shape is not changed. This implies that the control law ui
gives priority to the collision avoidance and/or connectivity preserving mission or the desired
formation shape updating mission whenever which mission is more important.

Substituting the control law ui in (17) and the update law l̇id in (19) into (15) gives

φ̇ = −k
N∑
i=1

ΩTi Ψ(Ωi), (23)

where we have used (22). On the other hand, substituting the control law the control law ui in
(17) into (3) including the update law l̇id in (19) results in the closed loop system:

q̇i = −kΨ(Ωi) + q̇od + l̇id,
l̇id = Hivid,

(24)

for all i ∈ N. We now present the main result of our paper in the following theorem.
Theorem 6. Under Assumption 3, the smooth control input ui = given in (17) and the update
law l̇id in (19) for the agent i solve the formation control objective. In particular:

1) There are no collisions between any agents, connectivity between the agents is maintained,
and the closed loop system (24) is forward complete. The first and last inequalities in (8) hold.

2) The reference velocity l̇id approaches its desired reference velocity vid asymptotically.
3) The trajectory qi of each agent i tracks its reference trajectory qid asymptotically, i.e., the

limit in the second equation of (8) holds.

Proof. See Appendix 1.



638 K.D. Do

4 Gradient climbing

4.1 Approach

In this section, we present an application of our proposed formation control to solve a gradient
climbing mission in a distributed environment Φ(t,η). To do so, we consider each agent in the
group as a mobile sensor and the network as a reconfigurable sensor array. As such, at each time
t the agent i with i ∈ N in the group of N agents is equipped with a sensor that can measure
Φ(t,qi) at the location qi. With Φ(t,qi), we estimate/calculate an approximation of the gradient
average, ∇Φ, of the distributed environment over a region A bounded by a contour C, on which
the agents in the group are positioned. After ∇Φ is estimated/calculated, we let the gradient
of the common reference trajectory qod equal to ∇Φ. This means that the common reference
trajectory qod is simply generated by

q̇od =
∂qod
∂sod

ṡod = ∇Φṡod, (25)

with some initial condition qod(t0), where ṡod specifies how fast the desired formation moves along
the common reference trajectory qod. For the case of gradient descent, we can use q̇od = −∇Φṡod
instead of (25). Moreover, we can specify the desired formation shape velocity vid to change
(expand/shrink/rotate) the formation shape, i.e., change the shape define vector lid, see (19),
to improve the gradient average approximation. We propose the the desired formation shape
velocity vid as follows:

vid = −K1v(lid − l∗id) + K2vΨ(∇Φ), (26)

where K1v and K2v are diagonal positive definite matrices. The constant vectors l∗id, i ∈ N
are chosen so that they specify the minimum desired formation shape, which is such that the
condition (6) holds with lid replaced by l∗id for all i ∈ N. The vector function Ψ(∇Φ) is a
bounded vector function of ∇Φ, see the paragraph just below (17). Once the common reference
trajectory qod and the desired formation shape lid are available, the formation control design
proposed in Section 3 can be used directly to drive the agents in the group. The following section
gives a method to estimate an approximation of the gradient average, ∇Φ, of the distributed
environment Φ(t,η) from measurements Φ(t,qi) on the boundary, i.e., the contour or surface C,
carried out by the agents in the group. Therefore, we will present a method to calculate the
gradient average of a distributed field in the following subsection.

4.2 Average gradient estimate of a distributed field

A

1
C

2
C

1
( )f x

2
( )f x

O

Y

Xa b

r
n t

P

Figure 4: Coordinates for a gradient
computation

We consider a region A, see Figure 4, bounded by a
contour C, such that any line through A parallel to either
one of the coordinate axes intersects C in only two points.
The curve C is divided by its leftmost and rightmost points
(x = a and x = b) into a lower segment C1, described by
y = f1(x), and an upper segment C2 described by y =
f2(x). With the position vector to a point P on C given
by r = xex + yey, where ex and ey are the unit vector
on the OX and OY axes, respectively. The unit tangent
vector at P is t = dr

ds
= dx

ds
ex +

dy
ds
ey, where ds is the

differential length along C, and the unit normal vector is
n = t×ez = dydsex −

dx
ds
ey = nxex +nyey. For the function



Formation Control of Multiple Agents with Preserving
Connectivity and its Application to Gradient Climbing 639

Φ(t,x,y) defined in A, consider the area integral∫
A

∂Φ

∂y
dA =

∫ b
a

(
Φ(t,x,f2(x))−Φ(t,x,f1(x))

)
dx =

∫ b
a

(
[Φ]C2 − [Φ]C1

)
dx. (27)

As shown in Fig. 4, a positive contour integration corresponds to a counter-clockwise traversal
of C. To make the first integral in (27) consistent with this connection, we write∫

A

∂Φ

∂y
dA = −

∫ a
b
[Φ]C2dx−

∫ b
a
[Φ]C1dx = −

∫
C
Φdx = −

∫
C
Φ
dx

ds
ds, (28)

which combines with n = nxex + nyey to yield
∫
A

∂Φ
∂y
dA =

∫
C
Φnyds. A similar computation

gives
∫
A

∂Φ
∂x
dA =

∫
C
Φnxds. Therefore, we have∫

A
∇ΦdA =

∫
C
nΦds (29)

where ∇Φ =
[
∂Φ
∂x
, ∂Φ
∂y

]T
. It is of interest to note that the total gradient

∫
A
∇ΦdA of the

distributed field Φ(t,η) over the region A is completely determined from the integral
∫
C
nΦds

carried out on the boundary C only. From (29), we can calculate the gradient average of Φ(t,η)
over the region A as

∇Φ =
∫
C
nΦds

ΩA
(30)

where ΩA is the area of the region A. Usually, it is not possible to obtain an explicit result of the
integral

∫
C
nΦds because the distributed field Φ is unknown. Hence, we approximate this integral

from measurement Φ(t,qi) at the time t and the location (qi) by each agent i, and approximate
the area ΩA. We assume that the formation shape is a convex polygon whose vertices are at qi.
The steps to calculate an approximate value of the integral

∫
C
nΦds and the region area ΩA are

as follows:
1) Using a curve fitting method such as Spline or least square to find a best fitted and smooth

contour C(θ), where θ is the curve parameter, that goes through all vertices at the time t;
2) Calculating an approximate value of

∫
C
nΦds and ΩA as follows:∫

C
nΦds ≈

N∑
i=1

Φ(t,qi)n(θi)∆Ci, ΩA ≈
1

2

N∑
i=1

(
xiyi+1 −xi+1yi

)
, (31)

where qN+1 = q1, n(θi) is the unit vector normal to C(θ) at θi corresponding to the position of
the vertex qi, and ∆Ci is the arc length from the middle point Mi−1 between qi−1 and qi an the
middle point Mi+1 between qi and qi+1, see Fig.5.

A

i
xO

Y

X

( )
i

n

1i
x

1i
x

i
y

1i
y

1i
y

( )C

1i
M

1i
M

1i
q

i
q

1i
q

Ci

Figure 5: Coordinates for gradient
average calculation.

For a special case where the formation shape is a reg-
ular simple polygon, which has the center at qod and the
vertices at qi, i ∈ N, and that the contour C goes through
all the vertices at the time t. Moreover, the unit vector
n normal to the contour C at qi is in the direction from
qod to qi at the time t. The the integral

∫
C
nΦds and the

region area ΩA can be approximated as∫
C
nΦds ≈

N∑
i=1

Φ(t,qi)
qi −qod
∥qi −qod∥

∥∥∥∥qi+1 −qi−12
∥∥∥∥,

ΩA ≈
1

2

N∑
i=1

∣∣∣det([qi,qi+1])∣∣∣,
(32)

with qN+1 = q1 and q−1 = qN , and det([qi,qi+1]) is the determinant of the matrix [qi+1,qi].



640 K.D. Do

5 Simulation results

In this section, we a problem of gradient climbing by our proposed formation controller using
a group of N = 6 identical agents. Each agent i has a physical safety radius Ri = 0.5 and a
communication radius Ri = 10. The control design parameters are taken as k = 4, δij = 0.5,
δvij = 0.75, cij = 1, and the bounded function ψ(.) taken as arctan(.).

The desired formation shape specification vectors l∗id are chosen as l
∗
id = Rf[cos(

2(i−1)π
N

),
sin(

2(i−1)π
N

)]T with Rf = 3, and the gain K1v = diag(2.5,2.5). This choice of l∗id means that
the desired formation configuration is a polygon whose vertices uniformly distribute on a circle
centered on the common reference trajectory and with a radius Rf . The initial conditions are
lid(0) = l

∗
id, qod(0) = [0 0]

T , and qi(0) = Rf[cos(
2(i−1)π

N
+π), sin(2(i−1)π

N
+π)]T . These particular

initial qi(0) were chosen to illustrate the collision avoidance capability of our proposed formation
control system as all the agents have to across the center of the desired formation shape to
track their desired reference trajectories. The distributed environment Φ(t,x,y) is taken as

Φ(t,x,y) = e−
(x−15)2+(y−15)2

150 , which has a global maximum value at (x = 15,y = 15).
We set K2v = diag(1.5,1.5) to improve the gradient climbing, i.e., the desired formation

shape is adapted to the distributed field. Simulation results are plotted in Figure 6. From these
figures, it is seen that our proposed formation is able to achieve the objective of both formation
control and gradient climbing. The control inputs ui, see sub-figure 6D, force the agents to move
in such a way that collision between the agents is avoided and that communication between
the agents is preserved, see sub-figure 6A where trajectories of the agents are plotted in XY-
plane. These sub-figures also show that our proposed formation control performs the gradient
climbing mission very well in the sense that the center of the formation shape, see the polygon of
which vertices are the agents, converges to the global maximum location of the function Φ(t,x,y).
Collision avoidance and communication preserving are also confirmed in sub-figure 6C, where the
distances ∥q∥1i between the agent 1 and other agents in the group are plotted. These distances
are within the range of (1,10) since Ri + Rj = 1 and min(Ri,Rj) = 10.

−5 0 5 10 15 20
−5

0

5

10

15

20

x

y

A

0 20 40 60 80
0

0.5

1

1.5

2

2.5

3

3.5

4

Time [s]

M
ea

n 
G

ra
d.

B

0 10 20 30 40 50 60 70 80
0

5

10

Time [s]

||q
1i

||

C

0 10 20 30 40 50 60 70 80
−5

0

5

Time [s]

u 1
x, 

u 1
y D

Figure 6: Simulation results with formation shape adaptation.



Formation Control of Multiple Agents with Preserving
Connectivity and its Application to Gradient Climbing 641

6 Conclusions

A constructive method has been proposed to design smooth and bounded cooperative con-
trollers for a group of N mobile agents with limited communication to perform a desired for-
mation. Novel potential functions encoding desired formation mission tasks with smooth step
functions embedded in were constructed to design the controllers that guaranteed all equilibrium
(critical) sets, except for the desired set in formation, are unstable. The proposed formation
control system is applied to solve a gradient climbing problem. An extension of the proposed
formation control design in this paper and those controllers designed for single underactuated
ships in [17] to provide a formation control system for a group of underactuated ships is under
consideration.

1 Proof of Theorem 6

Proof of no collisions, connectivity preserving, and forward completeness of the
closed loop system: It is seen from (23) that φ̇ ≤ 0. Integrating φ̇ ≤ 0 from t0 to t and using
the definition of φ in (9) with its components defined in (10) and (11) results in

φ(t) ≤ φ(t0), (33)

where φ(t0) =
∑N

i=1(γi(t0) +
1
2

∑
j∈Ni βij(t0)) and φ(t) =

∑N
i=1(γi(t) +

1
2

∑
j∈Ni βij(t)), for all

t ≥ t0 ≥ 0. From the condition specified in item 4) of Assumption 3, and properties of βij, we
have the right hand side of (33) is bounded by a positive finite constant depending on the initial
conditions. Boundedness of the right hand side of (33) implies that the left hand side of (33)
must be also bounded. As a result, βij(∥qij∥2/2) must be smaller than some positive constant
depending on the initial conditions for all t ≥ t0 ≥ 0. From properties of βij, see (12), ∥qij∥,
for all (i,j) ∈ N and i ̸= j, must be in the interval

(
(Ri + Rj),min(Ri,Rj)

)
. Hence, there are

no collisions between any agents and connectivity between agents is preserved for all t ≥ t0 ≥ 0.
Boundedness of the left hand side of (33) also implies that of (qi(t) − qid(t)) also bounded for
all t ≥ t0 ≥ 0. Moreover, from (21) we can see that |Hi| ≤ 1. Therefore, ∥l̇id(t)∥ ≤ ∥vid(t)∥ for
all t ≥ t0 ≥ 0. Therefore, the closed loop system (24) is forward complete.

Equilibrium set: We will use Lemma 2 in [7] to find the equilibrium set, which the trajec-
tories of the closed loop system (24) converge to. Integrating both sides of (23) yields∫ ∞

0
ω(t)dt = φ(t0)−φ(∞) ≤ φ(t0), (34)

with ω(t) :=
∑N

i=1 Ω
T
i (t)Ψ(Ωi(t)), where Ωi(t) is given in (16), and the function Ψ(Ωi(t))

is the bounded vector function of Ωi(t) with properties listed in (18). Indeed, the function
ω(t) is scalar, nonnegative and differentiable. Now differentiating ω(t) along the solutions of
the closed loop system (24) and using properties of the function βij given in (12) readily show
that

∣∣dω(t)
dt

∣∣ ≤ Mω(t) with M being a positive constant. Therefore Lemma 2 in [7] results in
limt→∞ ω(t) = 0, which implies from the expression of ω(t) and properties of the bounded vector
function Ψ(Ωi(t)) in (18) that limt→∞ Ωi(t) = 0. Therefore, from the expression of Ωi(t) the
limit limt→∞ Ωi(t) = 0 given in (16) implies that

lim
t→∞

N∑
i=1

(
qi(t)−qid(t) +

∑
j∈Ni

βij′(t)qij(t)
)

= 0. (35)

The limit in (35) implies that q(t) = [qT1 (t) q
T
2 (t), . . . ,q

T
N(t)]

T can tend to qd = [qT1d q
T
2d, . . . ,q

T
Nd]

T

denoted by the set Ξd, since βij′(t) = 0 at qi = qid and qj = qjd, for all (i,j) ∈ N and i ̸= j or



642 K.D. Do

tend to the set qc = [qT1c q
T
2c, . . . ,q

T
Nc]

T denoted by the set Ξc as the time goes to infinity, i.e.,
the equilibrium sets can be Ξd or Ξc. The equilibrium set Ξc is such that

Ω
∣∣∣
q∈Ξc

=

(
qi −qid +

∑
j∈Ni

βij′qij
)∣∣∣∣

q∈Ξc
= 0, (36)

for all i ∈ N. Thus, we have already proved that the trajectory q can approach either the desired
equilibrium set denoted by Ξd or the undesired equilibrium set denoted by Ξc ’almost globally’.
The term ’almost globally’ refers to the fact that the agents start from a set that includes
the condition (7) and that does not coincide at any point in the undesired equilibrium set Ξc.
Therefore, we now need to prove that Ξd is a locally asymptotically stable set and that Ξc is a
locally unstable set. Once this is proved, we can conclude that the trajectory q approaches qd
from almost everywhere except for from the set denoted by the condition (7) and the undesired
equilibrium set Ξc, which is unstable (to be proved below). To prepare for showing that Ξd is
asymptotically stable and that Ξc is unstable. We write the first equation of the closed loop
system (24) for all i ∈ N in a vector form as

q̇ = −kΦ(q,qd) + q̇d (37)

where Φ(q,qd) = [ΨT (Ω1), ...,ΨT (ΩN)]. Linearizing (37) around qo = [qT1o, . . . ,q
T
No]

T , and
letting the set Ξo contain qo results in

q̇ = −k
∂Φ(q,qd)

∂q

∣∣∣
q∈Ξo

+ q̇d, (38)

where ∂Φ(q,qd)
∂q

= [∆ij] with ∆ij =
∂Ψ(Ωi)
∂Ωi

∂Ωi
∂qj

and

∂Ωi
∂qi

=

(
1 +

∑
i∈Ni

βij′
)
In +

∑
j∈Ni

βij′′qijqTij,
∂Ωi
∂qj

= −βij′In×n −βij′′qijqTij, (39)

for all (i,j) ∈ N. Let N∗ be the set of the agents such that if the agents i and j belong to the
set N∗ then ∥qij∥ ∈ ((Ri + Rj),min(Ri,Rj). Next we will show that the equilibrium set Ξd is
asymptotically stable and that the equilibrium set Ξc is unstable.

Proof of Ξd being asymptotically stable: As mentioned above, to prove that the equi-
librium set Ξd is asymptotically stable, we just need to show that Ξd is locally asymptotically
stable. Letting Ξo be Ξd in (38), we obtain

q̇ = −k(q −qd) + q̇d, (40)

where we have used the fact that βij′
∣∣
q∈Ξd

= 0 and βij′′
∣∣
q∈Ξd

= 0, see Property 1) of the function
βij in (12). Local asymptotic stability of the equilibrium set Ξd follows from (40) since the first
time derivative of the function Vd = 12∥q−qd∥

2 along the solutions of (40) satisfies V̇d = −2kVd.
Proof of Ξc being asymptotically stable: Let us define

q̄ = [qT12, ...,q
T
1Nq

T
23, ...,q

T
2N, ...,q

T
N−1,N]

T , q̄c = [q
T
12c, ...,q

T
1Ncq

T
23c, ...,q

T
2Nc, ...,q

T
N−1,Nc]

T ,

βijc′ = βij′
∣∣
q∈Ξc

, βijc′′ = βij′′
∣∣
q∈Ξc

, qijc = qic −qjc.

With the above definitions, we can see that stability of Ξc is equivalent to that of Ξ̄c = q̄c. Define
Ωijc = Ωic − Ωjc, ∀(i,j) ∈ N, i ̸= j where Ωic = Ωi|q∈Ξc = 0, see (36). Therefore Ωijc = 0.
Hence

∑
(i,j)∈N∗ q

T
ijcΩijc = 0, i ̸= j, which by using (36) is expanded to∑

(i,j)∈N∗

(
qTijc(qijc−qijd)+Nβijc′q

T
ijcqijc

)
= 0 ⇒

∑
(i,j)∈N∗

(1+Nβijc′)qTijcqijc =
∑

(i,j)∈N∗
qTijcqijd (41)



Formation Control of Multiple Agents with Preserving
Connectivity and its Application to Gradient Climbing 643

where i ̸= j. The sum
∑

(i,j)∈N∗ q
T
ijcqijd is strictly negative since at the point F where qij =

qijd, ∀(i,j) ∈ N∗, i ̸= j all attractive and repulsive forces are equal to zero while at the point C
where qij = qijc ∀(i,j) ∈ N∗, i ̸= j the sum of attractive and repulsive forces are equal to zero
(but attractive and repulsive forces are nonzero). Therefore the point O where qij = 0, ∀(i,j) ∈
N∗, i ̸= j must locate between the points F and C for all (i,j) ∈ N∗, i ̸= j. That is the points
F , O, and C must be co-linear. Hence, there exists a strictly positive constant b such that∑

(i,j)∈N∗ q
T
ijcqijd < −b, which is substituted into (41) to yield∑

(i,j)∈N∗
(1 + Nβijc′)qTijcqijc < −b,i ̸= j. (42)

Since qTijcqijc > 0,∀(i,j) ∈ N
∗, i ̸= j, there exists a nonempty set N∗∗ ⊂ N∗ such that for all

(i,j) ∈ N∗∗, i ̸= j, (1 + Nβijc′) is strictly negative, i.e., there exists a strictly positive constant
b∗∗ such that (1 + Nβijc′) < −b∗∗, ∀(i,j) ∈ N∗∗, i ̸= j.

We now define a subspace Υ as Υ :=
(
qij −qijc = 0, ∀(i,j) ∈ N\N∗∗

)
∩
(
qTijc(qij −qijc) =

0, ∀(i,j) ∈ N∗, i ̸= j
)
. In the subspace Υ, we have

V̄c =
1

2

∑
(i,j)∈N∗∗

∥qij −qijc∥2, ˙̄Vc = −k
∑

(i,j)∈N∗∗
(1 + Nβijc′)∥qij −qijc∥2 ≥ 2kb∗∗V̄c (43)

where we have used (1 + Nβijc′) < −b∗∗, ∀(i,j) ∈ N∗∗, i ̸= j. Since the set N∗∗ is nonempty,
(43) implies that the equilibrium set Ξ̄c is unstable by Chetaev’s Theorem (Theorem 4.3 in [15]).
This implies the desired result that the equilibrium set Ξc is unstable. We can further explore
instability of the equilibrium set Ξc based on (43) as follows. From (43), we have∑

(i,j)∈N∗∗
∥qij(t)−qijc∥≥

∑
(i,j)∈N∗∗

∥qij(t0)−qijc∥ekb
∗∗(t−t0), i ̸= j, t ≥ t0 ≥ 0. (44)

Now assume that the equilibrium set Ξc is stable, i.e., limt→∞ ∥qi(t) − qic∥ = di,∀i ∈ N with
di a nonnegative constant. Note that N∗∗ ⊂ N, we have limt→∞ ∥qi(t) − qic∥ = di,∀i ∈ N∗∗,
which implies that limt→∞

∑
(i,j)∈N∗∗ ∥qij(t) − qijc∥ = d

∗∗,∀(i,j) ∈ N∗∗, i ̸= j with d∗∗ a non-
negative constant, since qij = qi − qj and qijc = qic − qjc. This contradicts (44) for the case∑

(i,j)∈N∗∗ ∥qij(t0)−qijc∥ ̸= 0, since the right hand side of (44) is divergent (so does the left hand
side). For the case

∑
(i,j)∈N∗∗ ∥qij(t0)−qijc∥ = 0, there would be no contradiction. However this

case is never observed in practice since the ever-present physical noise would cause ∥qij(t∗)−qijc∥
for some (i,j) ∈ N∗∗, i ̸= j to be different from 0 at the time t∗ ≥ t0. Proof of Theorem 6 is
completed.

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