International Journal of Computers, Communications & Control
Vol. I (2006), No. 3, pp. 41-59

Formation Control of Mobile Robots

Dang Binh Nguyen, Khac Duc Do

Abstract: A constructive method is presented to design cooperative controllers that
force a group of N mobile robots to achieve a particular formation in terms of shape
and orientation while avoiding collisions between themselves. The control devel-
opment is based on new local potential functions, which attain the minimum value
when the desired formation is achieved, and are equal to infinity when a collision
occurs. The proposed controller development is also extended to formation control
of nonholonomic mobile robots.
Keywords: Formation control, mobile robot, local potential function, nonholonomic
mobile robot.

1 Introduction

Over the last few years, formation control of multiple vehicles has received a lot of attention from
the control community. Applications of vehicle formation control include the coordination of multiple
robots, unmanned air/ocean vehicles, satellites, aircraft and spacecraft [1]-[28]. For example, a coopera-
tive mobile sensor network, where each mobile robot serves as a mobile sensor, is expected to outperform
a single large vehicle with multiple sensors or a collection of independent vehicles when the objective is
to climb the gradient of an environmental field. The single, heavily equipped vehicle may require con-
siderable power to operate its sensor payload, it lacks robustness to vehicle failure and it cannot adapt
the configuration or resolution of the sensor array. An independent vehicle with a single sensor may
need to perform costly maneuvers to effectively climb a gradient, for instance, wandering significantly
to collect rich enough data much like the "run and tumble" behavior of flagellated bacteria. In military
missions, a group of autonomous vehicles are required to keep in a specified formation for area coverage
and reconnaissance. In automated highway system, the throughput of the transportation network can be
greatly increased if vehicles can form to platoons at a desired velocity while keeping a specified distance
between vehicles. Research on formation control also helps people to better understand some biological
social behaviors, such as swarm of insects and flocking of birds.

In the literature, there have been roughly three methods to formation control of multiple vehicles:
leader-following, behavioral and virtual structure. Each method has its own advantages and disadvan-
tages. In the leader-following approach, some vehicles are considered as leaders, whist the rest of robots
in the group act as followers [1], [2], [3], [4]. The leaders track predefined reference trajectories, and the
followers track transformed versions of the states of their nearest neighbors according to given schemes.
An advantage of the leader-following approach is that it is easy to understand and implement. In addi-
tion, the formation can still be maintained even if the leader is perturbed by some disturbances. However,
a disadvantage is that there is no explicit feedback to the formation, that is, no explicit feedback from
the followers to the leader in this case. If the follower is perturbed, the formation cannot be maintained.
Furthermore, the leader is a single point of failure for the formation. In the behavioral approach [5],
[6], [7], [8], [9], [10], [11], [12], [13], [14], few desired behaviors such as collision/obstacle avoidance
and goal/target seeking are prescribed for each vehicle and the formation control is calculated from a
weighting of the relative importance of each behavior. The advantages of this approach are: it is natural
to derive control strategies when vehicles have multiple competing objectives, and an explicit feedback is
included through communication between neighbors. The disadvantages are: the group behavior cannot
be explicitly defined, and it is difficult to analyze the approach mathematically and guarantee the group
stability. In the virtual structure approach, the entire formation is treated as a single entity [15], [16],
[17], [18]. When the structure moves, it traces out desired trajectories for each robot in the group to

Copyright c© 2006 by CCC Publications



42 Dang Binh Nguyen, Khac Duc Do

track. Some similar ideas based on the perceptive reference frame, the virtual leader, and the formation
reference point are given in [14], [17], [19]. The advantages of the virtual structure approach are: it is
fairly easy to prescribe the coordinated behavior for the group, and the formation can be maintained very
well during the manoeuvres, i.e. the virtual structure can evolve as a whole in a given direction with
some given orientation and maintain a rigid geometric relationship among multiple vehicles. However
requiring the formation to act as a virtual structure limits the class of potential applications such as when
the formation shape is time-varying or needs to be frequently reconfigured, this approach may not be the
optimal choice. The virtual structure and leader-following approaches require that the full state of the
leader or virtual structure be communicated to each member of the formation. In contrast, behavior-based
approach is decentralized and may be implemented with significantly less communication. Formation
feedback has been recently introduced in the literature [18], [20], [21], [22]. In [18], a coordination
architecture for spacecraft formation control is introduced to incorporate the leader-following, behav-
ioral, and virtual structure approaches to the multi-robot coordination problem. This architecture can
be extended to include formation feedback. In [20], a Lyapunov formation function is used to define a
formation error for a class of robots (double integrator dynamics) so that a constrained motion control
problem of multiple systems is converted into a stabilization problem for one single system. The error
feedback is incorporated to the virtual leader through parameterized trajectories. In terms of information
from the robots in the group used for feedback in the control design for each robot, there are two main
approaches to solve the problem of motion planning/control of a group of mobile robots: centralization
and decentralization. In the centralized approach, see for example [18], a single controller and collision
free trajectories are constructed in a workspace. The centralized approach has a drawback of computa-
tion complexity but guarantees a complete solution. The decentralized approach, see for example [23],
requires less computational effort, and offers an easy way to scale the size of the robot group. This
approach usually involves a combination of robot based local potential fields [14], [24], [25].

The main problem with the decentralized approach is that it is unable or extremely difficult to pre-
dict and control the critical points. Basically, the closed loop system under a controller designed by the
decentralized approach has multiple equilibrium points. It is rather difficult to design a controller such
that all the equilibrium points except for the desired equilibrium one are unstable/saddle points for a
group of many robots. Moreover even the formation control system is designed in a centralized manner,
the tuning constants in several aforementioned papers (e.g. [26], [27], [28], [29]), which are crucial to
guarantee that the only desired equilibrium points are asymptotic stable and that the other critical points
are unstable, are extremely difficult to obtain for practical implementation. In most of the above papers,
point-robots with simple (single or double integrator) dynamics (e.g. [14], [24], [29]) or fully actuated
vehicles [19] (which can be converted to a double integrator dynamics via a feedback linearization) were
investigated. Vehicles with nonholonomic constraints were also considered (e.g. [5]). However, the non-
holonomic kinematics are transformed to a double integrator dynamics by controlling the hand position
instead of the inertial position of the vehicles. Consequently, the vehicle heading is not controlled. In
addition, in the tracking control of single nonholonomic mobile robots (e.g. [30], [31], [32]) the track-
ing errors are often converted into special forms to deal with nonholonomic constraints using several
non-trivial coordinate transformations. If these techniques are migrated to formation control of a group
of nonholonomic mobile robots, it is extremely difficult to incorporate collision avoidance between the
robots. The above problems motivate the contribution of this paper.

In this paper, we propose a constructive method to design a cooperative formation control system
for a group of N mobile robots. The simple point-mass robots are first considered to clarify the de-
sign philosophy. The proposed technique is extended to mobile robots with nonholonomic constraints
(nonholonomic mobile robots). New local potential functions are constructed to design gradient based
cooperative controllers to achieve almost global asymptotic convergence of a group of mobile robots
to a particular formation in term of both shape and orientation, and guarantee no collisions between
themselves. Formal proof of the results is given.



Formation Control of Mobile Robots 43

2 Problem statement

We consider a group of N simple point-mass mobile robots, of which each has the following dynam-
ics

q̇i = ui, i = 1, ..., N (1)

where qi ∈ Rn and ui ∈ Rn are the state and control input of the robot i. We assume that n > 1 and
N > 1. The assumption that each robot is represented as a point is not as restrictive as it may seem since
various shapes can be mapped to single points through a series of transformations [26], [27], [28]. Our
task is to design the control input ui for each robot i that forces the group of N robots to stabilize with
respect to their group members in configurations that make a particular formation specified by a desired
vector l(η) = [lT12(η), l

T
23(η), ..., l

T
N−1,N (η)]

T , where η ∈ Rm is the formation parameter vector to specify
the formation change, while avoiding collisions between themselves. The parameter vector η is used to
specify rotation, expansion and contraction of the formation such that when η converges to its desired
value η f , the desired shape of the formation is achieved. In addition, it requires all the robots align their
velocity vectors to a desired bounded one ud ∈ Rn, and move toward specified directions specified by the
desired formation velocity vector. The control objective is formally stated as follows:

Control objective: Assume that at the initial time t0 each robot initializes at a different location, and
that each robot has a different desired location, i.e. there exist strictly positive constants ε1, ε2 and ε3
such that

||qi(t0)−q j(t0)|| ≥ ε1,
||li j(η)|| ≥ ε2,
||∂ li j(η)/∂ η|| ≤ ε3, ∀i, j ∈ {1, 2, ...N}, ∀η ∈ Rm.

(2)

Design the control input ui for each robot i , and an update law for the formation parameter vector η
such that each robot (almost) globally asymptotically approaches its desired location to form a desired
formation, and that the robots’ velocity converges to the desired (bounded) velocity ud while avoiding
collisions with all other robots in the group, i.e.

limt→∞(qi(t)−q j(t)−li j(η(t))) = 0,
limt→∞(η(t)−η f ) = 0,
limt→∞(ui(t)−ud ) = 0,
||qi(t)−q j(t)|| > ε4, ∀i, j ∈ {1, 2, ...N}, ∀t ≥ t0 ≥ 0

(3)

where ε4 is a strictly positive constant, and η f is a vector of constants that determine the desired forma-
tion. The desired formation can be represented by a labeled directed graph ([29], [34]) in the following
definition.

Definition 1. The formation graph, G = {V, E, L} is a directed labeled graph consisting of:
-a set of vertices (nodes), V = {ϑ1,··· , ϑN} indexed by the mobile robots in the group,
-a set of edges, E = {(ϑi, ϑ j) ∈V ×V}, containing ordered pairs of vertices that represent inter-robot

position constraints, and
-a set of labels, L = {γi j|γi j = ||qi − q j − li j||2, ∀(ϑi, ϑ j) ∈ E}, li j = qi f − q j f ∈ Rn indexed by the

edges in E.
Indeed, when the control objective is achieved, the edge labels become ||qi−q j −li j||2 = 0, ∀(ϑi, ϑ j) ∈

E, i.e. the relative distance between the robots i and j is li j.

3 Control design

We consider the following local potential function

ϕi = γi + δ βi (4)



44 Dang Binh Nguyen, Khac Duc Do

where δ are positive tuning constants, the functions γi and βi are the goal and related collision avoidance
functions for the robot i specified as follows:

-The goal function γi is essentially the sum of all distances from the robot i to its adjacent group
members, Ni. A simple choice of this function is

γi = ∑
j∈Ni

γi j, γi j =
1
2
||qi −q j −li j||2. (5)

-The related collision function βi should be chosen such that it is equal to infinity whenever any
robots come in contact with the robot i, i.e. a collision occurs, and attains the minimum value when the
robot i is at its desired location with respect to other group members belong to Ni, which are adjacent to
the robot i. This function is chosen as follows:

βi = ∑
j∈Ni

(
β ki j
β 2ki jl

+
1

β ki j

)
(6)

where k is a positive constant to be chosen later, βi j and βi jl are collision and desired collision
functions chosen as

βi j =
1
2
||qi −q j||2, βi jl =

1
2
||li j||2. (7)

It is noted from (7) that βi j = β ji and βi jl = β jil .

Remark 1.
1. The above choice of the potential function ϕi given in (4) with its components specified in

(5)-(6), has the following properties: 1) it attains the minimum value when the robot i is at the de-
sired location with respect to other group member belong to Ni, which are adjacent to the robot i, i.e.
qi − q j − li j = 0, j ∈ Ni, and 2) it is equal to infinity whenever one or more robots come in contact with
the robot i, i.e. when a collision occurs.

2. The potential function (4) is different from the ones proposed in [14] and [33] in the sense that
the ones in [14] and [33] are centralized and do not put penalty on the relative distance between the
robots, i.e. do not include the goal function γi. Therefore, the controllers developed in [14] and [33] do
not guarantee the formation converge to a specified configuration but to any configurations that locally
minimize the potential functions (these potential functions in [14] and [33] are nonconvex).

3. Our potential function (4) is also different from the navigation functions proposed in [26] and [29]
in the sense that our potential function is of the form of sum of collision avoidance functions while those
navigation functions in [26] and [29] are of the form of product of collision avoidance functions . This
feature makes our potential function "more decentralized". Furthermore, our potential function is equal
to infinity while those in [14], [26] and [29] is equal to a finite constant when a collision occurs. However,
those in [26] and [29] also cover obstacle and work space boundary avoidance. Although these issues are
not included in this paper for clarity, considering these issues is possible and is the subject of future work.

4. Our potential function does not have problems like local minima and non-reachable goal as listed
in [24].

To design the control input ui, we differentiate both sides of (4) along the solutions of (1) to obtain

ϕ̇i = ∑
j∈Ni

[ΩTi j(ui −u j)−ΨTi jη̇]
= ∑

j∈Ni
[ΩTi j(ui −ud −(u j −ud ))−ΨTi j η̇]

= ∑
j∈Ni

ΩTi j(ui −ud )− ∑
j∈Ni

ΩTi j(u j −ud )− ∑
j∈Ni

ΨTi jη̇
(8)



Formation Control of Mobile Robots 45

where

Ωi j = qi −q j −li j + δ k
(

1
β 2ki jl
− 1

β 2ki j

)
β k−1i j (qi −q j)

Ψi j =

[(
qi −q j −li j +

2δ kβ ki j
β 2k+1i jl

li j

)T
∂ li j
∂ η

]T
.

(9)

From (8), we simply choose the control ui for the robot i and the update law for η as follows:

ui = −C ∑
j∈Ni

Ωi j + ud

η̇ = −Γ(η −η f ) (10)

where C ∈ Rn×n+ and Γ ∈ Rm×m+ are symmetric positive definite matrices. Substituting (10) into (8)
yields

ϕ̇i = − ∑
j∈Ni

ΩTi jC ∑
j∈Ni

Ωi j − ∑
j∈Ni

ΩTi j(u j −ud ) + ∑
j∈Ni

ΨTi jΓ(η −η f ). (11)

Substituting (10) into (1) results in the closed loop system

q̇i = −C ∑
j∈Ni

Ωi j + ud , i = 1, ..., N. (12)

Since the desired formation is specified in terms on relative distances between the robots, we write
the closed loop system of the inter-robot dynamics from the closed loop system (12) as

q̇i j = −C
(

∑
a∈Ni

Ωia − ∑
b∈N j

Ω jb

)
, (i, j) ∈ {1, ..., N}, i 6= j (13)

where qi j = qi −q j. We now state the main result in the following theorem.

Theorem 1. Under the assumptions stated in the control objective, the control for each robot i given in
(10) with an appropriate choice of the tuning constants δ and k, solves the control objective.

Proof. See Appendix.

4 Simulations

We carry out a simulation example in two-dimensional space to illustrate the results. The number
of robots is N = 4. The initial positions of robots are chosen randomly in the circle with a radius of 0.5
centered at the origin. The design constants are chosen as C = diag(0.4, 0.4), k = 0.5, δ = 0.1. It is
noted that this choice satisfies the conditions in the proof of Theorem 1. We run two simulations with
ud = [1 0.2]T (linear formation motion meaning that each robot will move on a rectilinear line to form the
desired formation) and ud = [sin(0.5t) cos(0.5t)] (circular formation motion meaning that each robot will
move on a circle to form the desired formation). For clarity, we take the formation parameter η as a scalar
to implement formation expansion. The desired formation is depicted in Figure 1. These simulations are
motivated by gradient climbing missions in which the mobile sensor network (each mobile robot serves
as a mobile sensor) seeks out local maxima or minima in the environmental field. The network can
adapt its configuration in response to the sensed environment in order to optimize its gradient climb. For
example, gradients in temperature fields (among others) can be estimated from the data collected by the



46 Dang Binh Nguyen, Khac Duc Do

mobile robots; these are of interest for enabling gradient climbing to locate and track features such as
fronts and eddies. These gradients can be used to compute the desired reference velocity vector ud in our
simulations in this section. In the first 4.5 seconds (for the linear formation motion case) and 15 seconds
(for the circular formation case), η is set to zero then is updated to η f = 3 for the rest of simulation time.
The update gain is chosen as Γ = 2 (scalar).

Robot 1

Robot 4

Robot 3

Robot 2

2(
1

)

2
(1

) 2(1

)

2(
1

)2(1

)

2 (1 )

Figure 1: Desired formation for simulation.

Figures 2 and 3 plot simulation results for the linear formation motion and circular formation cases,
respectively. For clarity, we only plot the control u1 = [ux1 uy1]T and distances from the robot 1 to other
members in the group, i.e. ‖q12‖,‖q13‖ and ‖q14‖. It is seen from these figures that the desired formation
shapes are nicely achieved and there are no collisions between any robots, see the bottom right figures in
Figures 2 and 3, where the distances from the robot 1 to other members in the groups are plotted. Clearly,
these distances are always larger than zero. It is also seen from Figures 2 and 3 that at the beginning
all the robots rapidly move away from each other to avoid collisions since they start pretty close to each
other.

5 Extension to formation control of nonholonomic mobile robots

Control of single nonholonomic mobile robots receives considerable attention, and is complicated
due to the fact that they have less controls than the outputs to be controlled, see for example [30],
[31], [32] and references therein. Indeed, control of a group of nonholonomic mobile robots is more
complicated due to some nonholonomic (non-integral) constraint. However, in this section we show that
the control method developed in Section 3 can be readily extended to force a group of N nonholonomic
mobile robots of unicycle type to move in such a way that a desired formation is achieved. For clarity, we
consider only the kinematic model of the nonholonomic mobile robots. Designing the control system at
the dynamic level even without requiring robot velocities be measured can be carried out using one more
"backstepping" step [35] and our proposed exponential observer in [31]. Consider the kinematic model
of the unicycle mobile robot i, whose only two wheels are actuated and the third wheel is not actuated



Formation Control of Mobile Robots 47

0 2 4 6 8 10 12
−2

0

2

4

6

8

x

y

t=9 s

0 2 4 6 8 10
−1

0

1

2

3

4

5

Time [s]

C
o

n
tr

o
l u

1

0 2 4 6 8 10
0

1

2

3

4

Time [s]

D
is

ta
n

ce
s 

−2 0 2 4 6 8
−2

0

2

4

6

8

x

y

t=4.5 s

||q
12

||

||q
13

||

||q
14

||

u
x1

u
y1

Robot 1

Robot 2

Robot 3

Robot 4

Figure 2: Linear formation motion: simulation results.

2 0 2 4 6
4

2

0

2

4

x

y

t=30 s

0 10 20 30
2

0

2

4

6

Time [s]

C
o
n
tr

o
l 
u

1

0 10 20 30
0

1

2

3

4

Time [s]

D
is

ta
n
c
e
s

2 0 2 4 6
4

2

0

2

4

x

y

t=15 s

u
x1

u
y1

||q
12

||

||q
13

||

||q
14

||

Robot 3
Robot 2

Robot 1

Robot 4

Figure 3: Circular formation motion: simulation results.



48 Dang Binh Nguyen, Khac Duc Do

(see Figure 4), given by

ẋi =
Ri
2

(cos(θi)ω1i + cos(θi)ω2i)

ẏi =
Ri
2

(sin(θi)ω1i + sin(θi)ω2i)

θ̇i =
Ri
2bi

(ω1i −ω2i) (14)

where (xi, yi) denote the coordinates of the middle point, P0i, between the left and right driving wheels,
and θi denotes the heading of the robot i coordinated in the earth-fixed frame OXY , see Figure 4, ω1i
and ω2i denote the angular velocities of the wheels of the robot i. Moreover Ri and bi are defined in
Figure 4. The task now is to design the control inputs ω1i and ω2i to achieve the control objective
stated in Section 3. We require an additional assumption on the desired formation velocity vector ud that
limt→∞ ||ud (t)|| 6= 0, i.e. we do not consider the stabilization/regulation problem. For convenience, we
convert the angular velocities of the wheels to the linear and angular velocities (vi and ri) of the robot i
by the following relationship:

[
vi
ri

]
=

(
1
Ri

[
1 bi
1 −bi

])−1 [ ω1i
ω2i

]
. (15)

With (15), we can write (14) as

ẋi = vi cos(θi)
ẏi = vi sin(θi)
θ̇i = ri

(16)

Y

X
O

Surg
e ax

is

i
y

i
x

2
i

b

0i
P

i

Sway axis

Actuated wheel

Passive wheel

2
i

R

Robot i

Figure 4: Geometric description of a nonholonomic mobile robot.

Indeed, the kinematic model (14) or (16) possesses the following nonholonomic constraint:

ẋi sin(θi)− ẏi cos(θi) = 0. (17)

Moreover, we will consider the the linear and angular velocities (vi and ri) of the robot i as the control
inputs. After these inputs are designed, ω1i and ω2i are calculated from (15).



Formation Control of Mobile Robots 49

5.1 Control design

The control design consists of two steps. At the first step, we consider the control vi and the yaw
angle θi as a virtual control to steer the robot position (xi, yi) to its desired location. At the second step,
the control ri will be desired to force the virtual yaw angle to converge to its actual yaw angle.

Step 1. Define
θei = θi −αθi (18)

where αθi is a virtual control of θi. With (18), we can write (16) as

q̇i = ui + Λθei (19)

where

qi =
∣∣∣∣

xi
yi

∣∣∣∣ , ui =
∣∣∣∣

cos(αθi )
sin(αθi )

∣∣∣∣ vi, Λθei =
∣∣∣∣

(cos(θei)−1) cos(αθi )−sin(θei) sin(αθi )
sin(θei) cos(αθi ) + (cos(θei)−1) sin(αθi )

∣∣∣∣ vi. (20)

It is seen that (19) is almost of the same form as (1). However, the problem is that the controls vi and αθi
are not solvable directly from the control ui if ui is not designed properly. We therefore present briefly
how ui is designed to tackle that problem. Consider the following potential function (the same form as
(4))

ϕi = γi + δ βi (21)
where δ , γi and βi are defined in Section 3, see (6) and (7). Differentiating both sides of (21) along

the solutions of (19) gives

ϕ̇i = ∑
j∈Ni

[ΩTi j(ui + Λθei −u∗d −(u j + Λθe j −u∗d ))−ΨTi jη̇] (22)

where Ωi j and Ψi j are defined in (9), and u∗d =

√
1 +

N
∑

i=1
|| ∑

j∈Ni
Ωi j||2 ud . It is noted that we use u∗d instead

of ud in (22) to overcome the nonholonomic problem of the mobile robot under investigation. Indeed,
limt→∞ ∑

j∈Ni
Ωi j(t) = 0 implies that limt→∞ u∗d (t) = ud . From (22), we choose the control ui and the update

law for η as

ui = −C||ud|| ∑
j∈Ni

Ωi j + u∗d

η̇ = −Γ(η −η f )
(23)

where C and Γ are diagonal positive definite matrices. Again, ||ud|| is included in the control ui to
overcome the nonholonomic problem. Defining θd = arctan(udy/udx), then from the first equations of
(23) and (20), we have

cos(αθi )vi = −c1||ud|| ∑
j∈Ni

Ωxi j +

√
1 +

N
∑

i=1
|| ∑

j∈Ni
Ωi j||2||ud||cos(θd )

sin(αθi )vi = −c2||ud|| ∑
j∈Ni

Ωyi j +

√
1 +

N
∑

i=1
|| ∑

j∈Ni
Ωi j||2||ud||sin(θd )

(24)

where Ωxi j and Ωyi j are defined as Ωi j = [ Ωxi j Ωyi j]T , c1 and c2 are defined as C = diag(c1, c2). We now
need to solve (24) for vi and αθi . To do this, multiplying both sides of the first and second equations of
(24) with cos(θd ) and sin(θd ), respectively, then adding them together result in

cos(αθi −θd )vi = −c1||ud|| ∑
j∈Ni

Ωxi j cos(θd )−

c2||ud|| ∑
j∈Ni

Ωyi j sin(θd ) +

√√√√1 +
N

∑
i=1
|| ∑

j∈Ni
Ωi j||2||ud||. (25)



50 Dang Binh Nguyen, Khac Duc Do

On the other hand, multiplying both sides of the first and second equations of (24) with sin(θd ) and
cos(θd ), respectively, then subtracting from each other result in

sin(αθi −θd )vi = c1||ud|| ∑
j∈Ni

Ωxi j sin(θd )−c2||ud|| ∑
j∈Ni

Ωyi j cos(θd ). (26)

From (25) and (26), we have

αθi = θd + arctan




c1 ∑
j∈Ni

Ωxi j sin(θd )−c2 ∑
j∈Ni

Ωyi j cos(θd )

−c1 ∑
j∈Ni

Ωxi j cos(θd )−c2 ∑
j∈Ni

Ωyi j sin(θd ) +

√
1 +

N
∑

i=1
|| ∑

j∈Ni
Ωi j||2




. (27)

It is seen that (27) is well-defined if the positive constants c1 and c2 are chosen such that

c1 + c2 < 1. (28)

The control vi is found by solving (24) as

vi = cos(αθi )||ud||
(
−c1 ∑

j∈Ni
Ωxi j +

√
1 +

N
∑

i=1
|| ∑

j∈Ni
Ωi j||2 cos(θd )

)
+

sin(αθi )||ud||
(
−c2 ∑

j∈Ni
Ωyi j +

√
1 +

N
∑

i=1
|| ∑

j∈Ni
Ωi j||2 sin(θd )

)
.

(29)

Substituting (23) into (22) results in

ϕ̇i = −||ud|| ∑
j∈Ni

ΩTi jC ∑
j∈Ni

Ωi j + ∑
j∈Ni

[ΩTi j(Λθei −(u j + Λθe j −u∗d )) + ΨTi jΓ(η −η f )]. (30)

Step 2. To design the control ri, differentiating both sides of (18) along the solutions of the third
equation of (16) and choosing the control ri as

ri = −diθei −α̇θi − ∑
j∈Ni

ΩTi jΛθei /θei (31)

where di is a positive constant, and the term ∑
j∈Ni

ΩTi jΛθei /θei is to cancel the cross term ∑
j∈Ni

ΩTi jΛθei in

(30), result in
θ̇ei = −diθei − ∑

j∈Ni
ΩTi jΛθei /θei. (32)

Note that Λθei /θei is well defined since sin(θie)/θie =
1∫
0

cos(θieλ )dλ and (cos(θie)−1)/θie =
1∫
0

sin(θieλ )dλ

are smooth functions.

4.2 Stability analysis
We consider the following function

ϕtot = log(1 +
N

∑
i=1

(ϕi + θ 2ei)) +
1
2
(η −η f )T Γ(η −η f ) (33)

whose derivative along the solutions of (30), (32) and the second equation of (23) satisfies

ϕ̇tot = −2
||ud||

N
∑

i=1
∑

j∈Ni
ΩTi jC ∑

j∈Ni
Ωi j +

N
∑

i=1
diθ 2ei

1 +
N
∑

i=1
(ϕi + θ 2ei)

+

N
∑

i=1
∑

j∈Ni
ΨTi jΓ(η −η f )

1 +
N
∑

i=1
(ϕi + θ 2ei)

−(η −η f )T Γ(η −η f ) (34)



Formation Control of Mobile Robots 51

where we have used

−
N

∑
i=1

∑
j∈Ni

[ΩTi j(u j + Λθe j −u∗d ) =
N

∑
i=1

∑
j∈Ni

[ΩTi j(ui −u∗d + Λθei )

= −||ud||
N

∑
i=1

∑
j∈Ni

ΩTi jC ∑
j∈Ni

Ωi j +
N

∑
i=1

∑
j∈Ni

ΩTi jΛθei . (35)

The rest of stability analysis can be carried out in the same lines as in Proof of Theorem 1 since (34)
is of the same form as (40) and limt→∞ ||ud (t)|| 6= 0 by assumption. Finally, note that limt→∞ θei(t) = 0
and limt→∞ ∑

j∈Ni
Ωi j(t) = 0 implies that limt→∞(θi(t)−θd ) = 0 , i.e. the yaw angle of all robots converge

to the desired angle θd = arctan(udy/udx).

5.2 Simulation results

We now perform a simulation to illustrate the results in the previous subsection. The number of
robots, initial conditions of the robot positions, control gains, desired formation velocity and desired
formation shape are the same as in Section 4. The robot heading angles are initialized randomly in the
circle with a radius of 0.5 centered at the origin. For clarity, we only simulate the circular formation
motion, and we do not include simulation results on the formation expansion as in Section 4, i.e. the
formation parameter η is set to zero in all the simulation time. The other design constants are chosen as
di = 5. Simulation results are plotted in Figure 5. Again, it is seen that the robots are forced to move
to nicely achieve the desired formation and no collisions between the robots occur. Moreover, the yaw
angle of all robots converges to the desired value θd , see the top-right figure in Figure 5, where the
yaw angle errors are plotted. A close look at Figure 5 shows that the main different between simulation
results in this subsection and those in Section 4 is that the robots take a longer time to approach the
desired formation. This is because we use the heading angles θi as the virtual controls to steer the robots
to overcome the noholonomic constraint.

6 Summary and Conclusions

This paper has contributed the method to construct local potential functions, based on which gradient-
like cooperative controllers were designed for a group of mobile robots both with and without nonholo-
nomic constraints to perform certain formation missions. Formal analysis of the convergence and fea-
sibility of the control solutions have also been provided. In the near future, it is of interest to apply the
proposed control design method combined with the control design scheme for other single underactuated
robots with second order nonholonomic constraints such as underactuated ships [37] to achieve a desired
formation for a group of underactuated systems.

7 Appendix: Proof of Theorem 1

We prove Theorem 1 in two steps. At the first step, we show that there are no collisions between any
robots and the solutions of the closed loop system exist. At the second step, we prove that the equilibrium
point of the inter-robot dynamics closed loop system (13), at which qi − q j − li j = 0, is asymptotically
stable. Finally, we show that all other equilibrium(s) of (13) are either unstable or saddle.

Step 1. Proof of no collision and existence of solutions:
We consider the following common potential function ϕ given by

ϕ =
N

∑
i=1

ϕi (36)



52 Dang Binh Nguyen, Khac Duc Do

2 0 2 4 6
3

2

1

0

1

2

3

4

x

y

0 2 4 6 8 10 12

1

0.5

0

0.5

1

Time [s]

H
e
a
d
in

g
 a

n
g
le

 e
rr

o
rs

0 5 10 15 20
5

0

5

10

Time [s]

C
o
n
tr

o
ls

 v
1
, 

r 1

0 5 10 15 20
0

0.5

1

1.5

2

Time [s]

D
is

ta
n
c
e
s

1 d

2 d

3 d

4 d

v
1

r
1

||q
12

||

||q
13

||

||q
14

||

Robot 1
Robot 2

Robot 3

Robot 4

Figure 5: Mobile robot circular formation motion: simulation result.



Formation Control of Mobile Robots 53

whose derivative along the solutions of (11) is

ϕ̇ = −
N

∑
i=1

∑
j∈Ni

ΩTi jC ∑
j∈Ni

Ωi j −
N

∑
i=1

∑
j∈Ni

ΩTi j(u j −ud ) +
N

∑
i=1

∑
j∈Ni

ΨTi jΓ(η −η f ). (37)

Since li j = −l ji and Ωi j = −Ω ji, we have
N

∑
i=1

∑
j∈Ni

ΩTi j(u j −ud ) = −
N

∑
i=1

∑
j∈Ni

ΩTi j(ui −ud ). (38)

Substituting (38) into (37) gives

ϕ̇ = −2
N

∑
i=1

∑
j∈Ni

ΩTi jC ∑
j∈Ni

Ωi j +
N

∑
i=1

∑
j∈Ni

ΨTi jΓ(η −η f ). (39)

We now consider the following total function ϕtot = log(1 + ϕ) + 0.5||η − η f ||2 whose derivative
along the solutions of (39) the second equation of (10) satisfies

ϕ̇tot = −
2

1 + ϕ

N

∑
i=1

∑
j∈Ni

ΩTi jC ∑
j∈Ni

Ωi j +
1

1 + ϕ

N

∑
i=1

∑
j∈Ni

ΨTi jΓ(η −η f )−(η −η f )T Γ(η −η f ) (40)

which implies that

ϕ̇tot ≤ −
2

1 + ϕ

N

∑
i=1

∑
j∈Ni

ΩTi jC ∑
j∈Ni

Ωi j +
λmax(Γ)

4ε(1 + ϕ)2

∥∥∥∥∥
N

∑
i=1

∑
j∈Ni

ΨTi j

∥∥∥∥∥
2

−

(λmin(Γ)−ε λmax(Γ))||η −η f ||2 (41)

where ε is a positive constant, λmin(Γ) and λmax(Γ) denote the minimum and maximum eigenvalues of
Γ respectively. From (9), and definition of the function ϕ , it can be readily shown that there exists a
positive constant ωmax such that

1
(1 + ϕ)2

∥∥∥∥∥
N

∑
i=1

∑
j∈Ni

ΨTi j

∥∥∥∥∥
2

≤ ωmax (42)

With (42) in mind, picking ε = λmin(Γ)/λmax(Γ) we can write (41) as

ϕ̇tot ≤
λ 2max(Γ)
4λmin(Γ)

ωmax
∆
= ϖmax. (43)

Integrating both sides of (43) results in

ϕtot (t) ≤ ϕtot (t0) + ϖmax(t −t0). (44)

where ϕtot (t) and ϕtot (t0) are (from the definition of ϕtot )

ϕtot (t) = log
[

1 +
N
∑

i=1

(
γi(t) + δ ∑

j∈Ni

(
β ki j (t)
β 2ki jl

+ 1
β ki j (t)

))]
+ 12||η(t)−η f ||2

ϕtot (t0) = log
[

1 +
N
∑

i=1

(
γi(t0) + δ ∑

j∈Ni

(
β ki j (t0)

β 2ki jl
+ 1

β ki j (t0)

))]
+ 12||η(t0)−η f ||2.

(45)

The right hand side of (44) cannot escape to infinity unless when t = ∞ since βi jl > 0 and βi j(t0) > 0 (see
definition of βi jl and βi j given in (7)). Therefore the left hand side of (44) cannot escape to infinity for



54 Dang Binh Nguyen, Khac Duc Do

all t ∈ [t0, ∞). This implies that βi j(t) cannot be zero for all t ∈ [t0, ∞), i.e. no collisions can occur for all
t ∈ [t0, ∞). On the other hand, it is true from the second equation of (10) that

||η(t)−η f || ≤ ||η(t0)−η f ||e−λmin(Γ)(t−t0) (46)
which means that the desired formation shape is achieved exponentially. Using (42) and (46), we can
write (41) as

ϕ̇tot≤ λmax(Γ)
√

ωmax||η(t0)−η f ||e−λmin(Γ)(t−t0). (47)
Integrating both sides of (47) from t0 to t results in

ϕtot (t) ≤ ϕtot (t0) + λmax(Γ)
√

ωmax||η(t0)−η f ||/λmin(Γ). (48)
It is seen that the right hand side of (48) is bounded. Therefore the left hand side of (48) must also
be bounded. This implies that βi j(t) must be larger than a strictly positive constant for all t ∈ [t0, ∞),
which in turn means that there exists a strictly positive constant ε4 such that the last inequality of (3)
holds. To prove that the solutions of the closed loop system (12) exist, we consider the function W =

0.5
N
∑

i=1
||qi||2 whose derivative along the solutions of (12), after some simple manipulation, satisfies Ẇ ≤

ρ1(1 + 1/ min(βi j))W + ρ2 , where ρ1 and ρ2 are some positive constants, which implies that the solutions
of (12) exist since βi j(t) is larger than a strictly positive constant for all t ∈ [t0, ∞). Furthermore, applying
Barbalat’s lemma found in [36] to (41) gives

lim
t→∞

1
1 + ϕ(t)

N

∑
i=1

∑
j∈Ni

ΩTi j(t)C ∑
j∈Ni

Ωi j(t) = 0 (49)

which implies that
{

limt→∞ ∑
j∈Ni

Ωi j(t) = 0

limt→∞ ϕ(t) = χ1
or

{
limt→∞ ∑

j∈Ni
Ωi j(t) = χ2

limt→∞ ϕ(t) = ∞
(50)

where χ1 and χ2 are some constants. From definitions of Ωi j and ϕ , the second limit set in (50) cannot
be true. Therefore, the first limit set in (50) implies that limt→∞ ∑

j∈Ni
Ωi j(t) = 0.

Step 2. Behavior near equilibrium points.
At the steady state, the equilibrium points are found by solving the following equations

∑
j∈Ni

Ωi j = ∑
j∈Ni

(
qi j −li j + δ k

(
1

β 2ki jl
− 1

β 2ki j

)
β k−1i j qi j

)
= 0, i = 1, ..., N. (51)

It is directly verified that q̄ = l̄ where q̄ and are stack vectors of qi j and li j, respectively, i.e. q̄ =
[qT12, q

T
13..., q

T
N−1,N ]

T and l̄ = [lT12, l
T
13..., l

T
N−1,N ]

T , is one root of (51). In addition there is (are) another
root(s) denoted by q̄c = [qT12c, q

T
13c..., q

T
N−1,Nc]

T of (51) different from l̄ satisfying

∑
j∈Ni

Ωi j

∣∣∣∣∣
q̄=q̄c

= ∑
j∈Ni

(
qi jc −li j + δ k

(
1

β 2ki jl
− 1

β 2ki jc

)
β k−1i jc qi jc

)
= 0, i = 1, ..., N (52)

where βi jc = 0.5||qic − q jc||2. In the following, we will show that the equilibrium point q̄ = l̄ is asymp-
totically stable, and the equilibrium point(s) q̄ = q̄c is (are) unstable or saddle. We now write the closed
loop system of the inter-robot dynamics (13) as

˙̄q = −C̄F(q̄, l̄). (53)



Formation Control of Mobile Robots 55

where C̄ = diag(C,··· ,C︸ ︷︷ ︸
E

) with E the number of edges of the formation graph, and

F(q̄, l̄) = [ ∑
a∈N1

ΩT1a − ∑
b∈N2

ΩT2b, ∑
a∈N1

ΩT1a − ∑
b∈N3

ΩT3b, ...., ∑
a∈Ni

ΩTia − ∑
b∈N j

ΩTjb, ....,

∑
a∈NN−1

ΩTN−1,a − ∑
b∈NN

ΩTNb]
T . (54)

Since (52) holds for all i = 1, ..., N, at the steady state we have ∑
a∈Ni

Ωia− ∑
b∈N j

Ω jb = 0, ∀(i, j) ∈{1, ..., N}, i 6=
j. Therefore the equilibrium points q̄ = l̄ and q̄ = q̄c are also the equilibrium points of (53). The general
gradient of F(q̄, l̄) with respect to q̄ is given by

∂ F(q̄, l̄)
∂ q̄

=




∂ Ξ12
∂ q12

∂ Ξ12
∂ q13

··· ∂ Ξ12∂ qN−1,N
...

...
...

...
∂ Ξi j
∂ q12

··· ∂ Ξi j∂ qi j
∂ Ξi j

∂ qN−1,N
...

...
...

...
∂ ΞN−1,N

∂ q12
··· ··· ∂ ΞN−1,N∂ qN−1,N




,

Ξi j = ∑
a∈Ni

Ωia − ∑
b∈N j

Ω jb, (i, j) ∈ {1, ..., N}, i 6= j. (55)

It can be checked that

∂ Ξi j
∂ qi j

= NIn×n + 2δ k
(

1
β 2ki jl
− 1

β 2ki j

)
β k−1i j In×n + 2δ k

(
(k −1)

(
1

β 2ki jl
− 1

β 2ki j

)
β k−2i j +

2k
β k+2i j

)
qi j q

T
i j

∆
= Hi j

∂ Ξi j
∂ qcd

= sδ k
(

1
β 2kcdl

− 1
β 2kcd

)
β k−1cd In×n + sδ k

(
(k −1)

(
1

β 2kcdl
− 1

β 2kcd

)
β k−2cd +

2k
β k+2cd

)
qcd q

T
cd ,

(56)

where (c, d) ∈{1, ..., N}, (c, d) 6= (i, j), c 6= d , and s = 1 or s = −1 depending on value of c, d, i and j.
However, we do not need to specify the sign of s for our next task. We now investigate properties of the
equilibrium points q̄ = l̄ and q̄ = q̄c based on the general gradient ∂ F(q̄, l̄)/∂ q̄ evaluated at those points.

Step 2.1 Proof of q̄ = l̄ being the asymptotic stable equilibrium point:
At the equilibrium point q̄ = l̄ , we have

∂ Ξi j
∂ qi j

∣∣∣∣
q̄=l̄

= NIn×n +
4δ k2

β k+2i jl
li jl

T
i j ,

∂ Ξi j
∂ qcd

∣∣∣∣
q̄=l̄

= s
2δ k2

β k+2cdl
lcd l

T
cd , (57)

where βcdl = 0.5||lcd||2 . With (57), let ξ ∈ RnE we have

ξ T
∂ F(q̄, l̄)

∂ q̄

∣∣∣∣
q̄=l̄

ξ ≥
(

N −
4δ k2nE max(l2i ja)

min(β k+2i jl )

)
ξ T ξ , (i, j) ∈ {1, ..., N}, i 6= j (58)

where li ja is the a
th element of li j. Therefore, for any given constant k if we choose the tuning

constant δ such that

N −
4δ k2nE max(l2i ja)

min(β k+2i jl )
> 0 → δ <

N min(β k+2i jl )
4k2nE max(l2i ja)

, (i, j) ∈ {1, ..., N}, i 6= j (59)

then the matrix ∂ F(q̄, l̄)/∂ q̄
∣∣
q̄=l̄ is positive definite, which in turn implies that the equilibrium point

q̄ = l̄ is asymptotically stable.



56 Dang Binh Nguyen, Khac Duc Do

Step 2.2. Proof of q̄ = q̄c being the unstable/saddle equilibrium point(s): The idea is to consider
block matrices on the main diagonal of the matrix ∂ F(q̄, l̄)/∂ q̄

∣∣
q̄=q̄c

and show that there exists at least

one block matrix whose determinant is negative. Define Hi jc = ∂ Ξi j/∂ qi j
∣∣
q̄=q̄c

and let φa and φb be the
ath and bth elements of qi jc, (a, b) ∈ {1, ..., n}, a 6= b. We form the matrices H abi jc from the matrix Hi jc as
follows

H abi jc =
[

h11 h12
h21 h22

]

h11 = N + 2δ kΠi jcβ k−1i jc + 2δ k[(k −1)Πi jcβ k−2i jc + 2k/β k+2i jc ]φ 2a
h12 = 2δ k[(k −1)Πi jcβ k−2i jc + 2k/β k+2i jc ]φaφb
h21 = 2δ k[(k −1)Πi jcβ k−2i jc + 2k/β k+2i jc ]φaφb
h22 = N + 2δ kΠi jcβ k−1i jc + 2δ k[(k −1)Πi jcβ k−2i jc + 2k/β k+2i jc ]φ 2b

(60)

where Πi jc = 1/β 2ki jl −1/β 2ki jc . The determinant of H abi jc is given by

det(H abi jc) = (N + 2δ kΠi jcβ
k−1
i jc )∆

ab
i jc (61)

where
∆abi jc = N + 2δ kΠi jcβ

k−1
i jc + 2δ k[(k −1)Πi jcβ k−2i jc + 2k/β k+2i jc ](φ 2a + φ 2b ) (62)

Let us consider the sum:

n−1
∑
a=1

n

∑
b=a+1

∆abi jc = n(n−1)N + 2δ k(n−1)(2(k−1) + n)β k−1i jc /β 2ki jl + 2δ k(n−1)(2(k + 1)−n)/β k+1i jc . (63)

Since n > 1, picking k > n/2 − 1 ensures that
n−1
∑

a=1

n
∑

b=a+1
∆abi jc > 0. Therefore, there exists at least one

pair (a, b) ∈ {1, ..., n} denoted by (a∗, b∗) such that ∆a∗b∗i jc > 0. Now for all (i, j) ∈ {1, ..., N}, i 6= j let us
consider the sum:

N−1
∑
i=1

N

∑
j=i+1

det(H a
∗b∗

i jc )

∆a∗b∗i jc
βi jc =

N−1
∑
i=1

N

∑
j=i+1

(Nβi jc + 2δ kΠi jcβ ki jc). (64)

On the other hand, multiplying both sides of F(q̄c, l̄) = 0 with q̄Tc results in q̄
T
c F(q̄c, l̄) = 0, which is

expanded to
N−1
∑
i=1

N

∑
j=i+1

(NqTi jc(qi jc −li j) + 2δ kNΠi jcβ ki jc) = 0. (65)

Substituting (65) into (64) results in

N−1
∑
i=1

N

∑
j=i+1

det(H a
∗b∗

i jc )

∆a∗b∗i jc
βi jc =

N−1
∑
i=1

N

∑
j=i+1

(N −2)βi jc +
N−1
∑
i=1

N

∑
j=i+1

qTi jcli j. (66)

The term
N−1
∑

i=1

N
∑

j=i+1

(
qTi jcli j

)
is strictly negative since at the point where qi j = li j (the point F in Figure

6) all attractive and repulsive forces are equal to zero while at the point where qi j = qi jc (the point C in
Figure 6) the sum of attractive and repulsive forces is equal to zero (see Section 2 for discussion of a
simple case). Therefore the point qi j = 0 (the point O in Figure 6) must locate between the points qi j = li j
and qi j = qi jc , see Figure 6. Furthermore if we write (65) as

2
N−1
∑
i=1

N

∑
j=i+1

βi jc + δ k(β ki jc/β
2k
i jl −1/β ki jc) =

N−1
∑
i=1

N

∑
j=i+1

qTi jcli j (67)



Formation Control of Mobile Robots 57

12
q

13
q

1N
q

21
q

2 N
q

1,1N
q

1,N N
q

O

F

C

The point where all attractive
and repulsive forces are zero

The point where sum of attractive and repulsive forces are zero.

Figure 6: Illustration of location of critical points.

we can see that deceasing δ results in decrease in βi jc since βi jl is a bounded constant and the right
hand side of (67) is negative. Therefore, choosing a sufficiently small δ ensures that the right hand of
(64) is strictly negative since βi jc = 0.5||qi jc||2 . That is

N−1
∑
i=1

N

∑
j=i+1

det(H a
∗b∗

i jc )

∆a∗b∗i jc
βi jc < 0 (68)

which implies that there exists at least one pair (i, j) ∈ {1, ..., N} denoted by (i∗, j∗) such that

det(H a
∗b∗

i∗ j∗c) < 0. (69)

The inequality implies that at least one eigenvalue of the matrix ∂ F(q̄, l̄)/∂ q̄
∣∣
q̄=q̄c

is negative. This in
turn guarantees that q̄c is an unstable/saddle equilibrium point of (53). Proof of Theorem 1 is completed.

References

[1] P. K. C. Wang, Navigation Strategies for Multiple Autonomous Mobile Robots Moving in Formation,
J. Robot. Syst., Vol. 8, No. 2, pp. 177-195, 1991.

[2] P. K. C. Wang and F. Y. Hadaegh, Coordination and Control of Multiple Microspacecraft Moving in
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Dang Binh Nguyen
Thai Nguyen University of Technology

Rector’s Office
3-2 Street, Thai Nguyen City, Vietnam

E-mail: nguyen.d.binh.tnut@gmail.com
Khac Duc Do

School of Mechanical Engineering
The University of Western Australia

35 Stirling Highway, Crawley, WA 6009, Australia
E-mail: duc@mech.uwa.edu.au