International Journal of Computers, Communications & Control
Vol. II (2007), No. 4, pp. 340-354

Bounded Controllers for Decentralized Formation
Control of Mobile Robots with Limited Sensing

K.D. Do

Abstract: This paper presents a constructive method to design bounded cooperative
controllers that force a group of N mobile robots with limited sensing ranges to stabi-
lize at a desired location, and guarantee no collisions between the robots. The control
development is based on new general potential functions, which attain the minimum
value when the desired formation is achieved, and are equal to infinity when a colli-
sion between any robots occurs. Smooth and p times differential jump functions are
introduced and embedded into the potential functions to deal with the robot limited
sensing ranges. Formation tracking is also considered.
Keywords: Formation control, mobile robot, local potential function, nonholonomic
mobile robot.

1 Introduction

Formation is an extremely useful tool mimicking from biological systems to man-made teams of
vehicles, mobile sensors and embedded robotic systems to perform tasks such as jointly moving in a
synchronized manner or deploying over a given region with applications to search, rescue, coverage,
surveillance, reconnaissance and cooperative transportation. Formation control can be roughly under-
stood as controlling positions of a group of the robots such that they stabilize/track desired locations
relative to reference point(s), which can be another robot(s) within the team, and can either be stationary
or moving. Three popular approaches to formation control are leader-following (e.g. [2], [3]), behavioral
(e.g. [4], [5]), and use of virtual structures (e.g. [6], [7]. Most research works investigating formation
control utilize one or more of these approaches in either a centralized or decentralized manner. Central-
ized control schemes, see e.g. [3], [11], and [10] where actual dynamics of nonholonomic robots were
considered in the formation control design, use a single controller that generates collision free trajecto-
ries in the workspace. Although these guarantee a complete solution, centralized schemes require high
computational power and are not robust due to the heavy dependence on a single controller. A nice ap-
plication of formation control based on potential field method [3] and Lyapunov’s direct method [17] to
gradient climbing is recently addressed in [18]. However, the final configuration of formation cannot be
foretold. On the other hand, decentralized schemes, see e.g. [8], [12], require less computational effort,
and are relatively more scalable to the team size. The decentralized approach usually involves a combi-
nation of robot based local potential fields (e.g. [3], [15], [16]). The main problem with the decentralized
approach, when collision avoidance is taken into account, is that it is extremely difficult to predict and
control the critical points of the controlled systems. Recently, a method based on a different navigation
function from [13] provided a centralized formation stabilization control design strategy is proposed in
[11]. This work is extended to a decentralized version in [12]. However, the potential function, which
possesses all properties of a navigation function (see [13]), is finite (but its gradient with respect to the
system states can be unbounded) when a collision occurs. This complicates analysis of collision avoid-
ance. Moreover, the formation is stabilized to any point in workspace instead of being "tied" to a fixed
coordinate frame. In [13], [11] and [12], the tuning constants, 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. This problem has been removed in [9] where new potential functions were
introduced. It is however noted that in [9] each robot requires knowledge of position of all other robots
in the group. Moreover, the control design methods (e.g. [3], [14], [15], [18]) based on the potential
functions that are equal to infinity when a collision occurs exhibit very large control efforts if the robots

Copyright © 2006-2007 by CCC Publications



Bounded Controllers for Decentralized Formation
Control of Mobile Robots with Limited Sensing 341

are close to each other. Hence, a bounded control is called for. In addition, switching control theory [21]
is often used to design a decentralized formation control system (e.g. [2], where a case by case basis
is proposed), especially when the vehicles have limited sensing ranges and collision avoidance between
vehicles must be considered. Clearly, it is more desirable if we are able to design a non-switching for-
mation control system that can handle the above decentralized and collision avoidance requirements.

In this paper, bounded cooperative controllers are designed for formation stabilization of a group of
mobile robots with limited sensing ranges. New general potential functions are constructed to design the
controllers that yield (almost) global asymptotic convergence of a group of mobile robots to a desired
formation, and guarantee no collisions among the robots. Smooth and p times differential jump functions
are introduced and embedded into the potential functions to deal with the robot limited sensing ranges.
Moreover, the controlled system exhibits multiple equilibria due to collision avoidance taken into ac-
count. We therefore investigate the behavior of equilibrium points by linearizing the closed loop system
around those points, and show that critical points, other than the desired point for an robot, are unstable.
The proposed formation stabilization solution is then extended to solve a formation tracking problem.

2 Problem statement

We consider a group of N mobile robots, of which each has the following dynamics

q̇i = ui, i = 1, ..., N (1)

where qi ∈ Rn and ui ∈ D ⊂ Rn are the state and control input of the robot i. We assume that n > 1 and
N > 1. Here, we treat each robot as an autonomous point.

Control objective: Assume that at the initial time t0 ≥ 0 each robot starts at a different location, and
that each robot has a different desired location, i.e. there exist strictly positive constants ε1 and ε2 such
that for all (i, j) ∈ {1, 2, ..., N}, i 6= j

‖qi(t0)−q j(t0)‖ ≥ ε1,‖qi f −q j f ‖ ≥ ε2 (2)

where qi f , i = 1, .., N, is the desired location of the robot i. Moreover, the robot i can only measure its own
state and can only detect the other group members if these members are in a sphere, which is centered
at the robot and has a radius of Ri larger than a strictly positive constant. Design the bounded control
input ui for each robot i such that each robot asymptotically approaches its desired location while avoids
collisions with all other robots in the group, i.e. for all (i, j) ∈ {1, 2, ..., N}, i 6= j,t ≥ t0 ≥ 0

‖ui(t)‖ ≤ δ , lim
t→∞

(qi(t)−qi f ) = 0,‖qi(t)−q j(t)‖ ≥ ε3 (3)

where δ is a strictly positive constant and ε3 is a positive constant.

3 Preliminaries

We present one definition and one lemma to be used in the control design and stability analysis in
the next section.



342 K.D. Do

Definition 1. A scalar function h(x, a, b) is called a p times differential jump function if it enjoys the
properties:

1) h(x, a, b) = 0 if 0 ≤ x ≤ a,
2) h(x, a, b) = 1 if x ≥ b,
3) 0 < h(x, a, b) < 1 if a < x < b, (4)
4) h(x, a, b) is p times differentiable with respect to x

where p is a positive integer, x ∈ R+, and a and b are constants such that 0 ≤ a < b. Moreover, if p = ∞
then the function h(x, a, b) is called a smooth jump function.

Lemma 1. Let the scalar function h(x, a, b) be defined as

h(x, a, b) =
∫ x

a f (τ −a) f (b−τ)dτ∫ b
a f (τ −a) f (b−τ)dτ

(5)

with the function f (y) being defined as follows

f (y) = 0 if y ≤ 0, f (y) = g(y) if y > 0 (6)

where the function g(y) enjoys the following properties

a) g(τ −a)g(b−τ) > 0 a < τ < b,
b) g(y) is p times differentiable with respect to y,

and lim
y→0+

∂ kg(y)
∂ yk

= 0, k = 1, 2, ..., p−1. (7)

Then the function h(x, a, b) is a p times differentiable jump function.

Proof. See Appendix A.

Remark 1. Several examples of the function g(y) are g(y) = yp, g(y) = tanh(y)p, g(y) = arctan(yp)
for any positive integer p, and g(y) = sin(y)p for any even positive integer p.

Corollary 1. If the function g(y) in (7) is taken as g(y) = exp(−1y ) then the function h(x, a, b) defined
in (5) is a smooth jump function.

Proof. See Appendix B.

4 Control design

To achieve the control objective, we design the control ui for the robot i based on the new potential
function ϕ . This potential function must attain its unique minimum value when all robots are at their
desired positions, and must equal infinity when there is a collision between any robots. The potential
function ϕ should also be chosen such that the gradient based control ui for the robot i can handle the
limited sensing range of the robot. As such, we propose the following potential function

ϕ =
eγ

β κ
−1 (8)



Bounded Controllers for Decentralized Formation
Control of Mobile Robots with Limited Sensing 343

where κ is a positive constant, γ and β are the goal and collision avoidance functions. These functions
are specified as follows:
-The goal function is designed such that it puts penalty on stabilization errors for all robots, and is equal
to zero when the robots are at their final positions. A simple choice of this function is

γ =
1
2

N

∑
i=1
‖qi −qi f ‖2. (9)

-The collision function β is chosen such that it equals zero when there is a collision between any robots,
and equals 1 when the robots are at their desired positions. We choose this function as follows:

β = ∏
i, j

βi j, i = 1...N −1, j = i + 1, ..., N. (10)

The function βi j = β ji is designed as

βi j = h(0, b2i j/2,‖qi j‖2/2) (11)

where h(0, b2i j/2,‖qi j‖2/2) is a smooth or p > 2 times differentiable jump function presented in the pre-
vious section, qi j = qi −q j, bi j is a strictly positive constant such that bi j ≤ min (Ri, R j, ε2) with ε2 given
in (2).

Remark 2. Thanks to properties of the smooth or p > 2 differentiable jump function (see Definition
1), the collision function β equals zero when a collision between any robots occurs, i.e. ‖qi j‖ = 0 for any
i 6= j. The function β equals 1 when all robots are at their desired locations, i.e. qi = qi f for i = 1, ..., N.
The function β is at least twice differentiable with respect to qi j. Hence, the choice of the goal function
γ in (9) and the collision function β in (10) with its components given in (11) ensures that the potential
function ϕ in (8) attains the (unique) minimum value of zero when all the robots are at their desired
positions, and equals infinity whenever a collision between any robots occurs. Moreover, the potential
function ϕ is at least twice differentiable.

The derivative of ϕ along the solutions of (1) satisfies

ϕ̇ =
eγ γ̇ β κ −eγ κ β κ−1β̇

β 2κ
=

eγ

β κ

(
γ̇ −κ

N−1
∑
i=1

N

∑
j=i+1

β̇i j
βi j

)

=
eγ

β κ
N

∑
i=1

ΩTi ui (12)

where we have used β̇ = β ∑N−1i=1 ∑
N
j=i+1

β̇i j
βi j

, and

Ωi = qi −qi f −κ ∑
j∈Ni

β̄ ′i jqi j (13)

with β̄ ′i j =
β ′i j
βi j

, β ′i j =
∂ βi j

∂ (‖qi j‖2/2) , and Ni the set of all robots, denoted by N, in the group except for the
robot i. From (12), a bounded control ui for the robot i is simply designed as follows:

ui = −cΨ(Ωi) (14)

where c 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

2
i ), ..., ψ(Ω

l
i ), ...., ψ(Ω

n
i )

]T
with Ωli the l

th element of Ωi, i.e.



344 K.D. Do

Ωi = [Ω1i Ω
2
i ...Ω

l
i ...Ω

n
i ]

T . The function ψ(x) is a scalar, differentiable and bounded function, and satisfies

1) |ψ(x)| ≤ M1,
2) ψ(x) = 0 if x = 0, xψ(x) > 0 if x 6= 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

(15)

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). Indeed, the control ui is bounded, i.e. ‖ui(t)‖ ≤ c

√
nM1 :=

δ ,∀t ≥ t0 ≥ 0.

Remark 3. When Ωi defined in (13) is substituted into (14), the lth element of the control ui can be
written as uli = cψ

(
−(qli − qli f )− ∑ j∈Ni β̄ ′i jqli j

)
with qli , q

l
i f and q

l
i j being the l

th elements of qi, qi f , and
qi j. The argument of ψ consists of two parts: −(qli −qli f ) and −∑ j∈Ni β̄ ′i jqli j. The first part, −(qli −qli f ),
referred to as the attractive force plays the role of forcing the robot to its desired location. The second
part, −∑ j∈Ni β̄ ′i jqli j, referred to as the repulsive force, takes care of collision avoidance for the robot i
with the other robots. Moreover, the control ui of the robot i given in (14) depends on only its own state,
and the states of other neighbor robots j if these robots are in a sphere, which is centered at the robot and
has a radius no greater than Ri because outside this sphere β̄ ′i j = 0.

Now substituting (14) into (12) results in

ϕ̇ = −c e
γ

β κ
N

∑
i=1

ΩTi Ψ(Ωi). (16)

Substituting (14) into (1) results in the closed loop system

q̇i = −cΨ(Ωi), i = 1, ..., N. (17)

Theorem 1. Assume that at the initial time t0 ≥ 0 each robot starts at a different location, and that
each robot has a different desired location, i.e. the conditions given in (2) hold, the bounded controls
given in (14) guarantee that no collisions between any robots can occur, the solutions of the closed loop
system (17) exist and the robots asymptotically approach their desired positions (a set of equilibria) de-
fined by qi f , i = 1, ..., N.

Proof. See Appendix C.

5 Simulations

We carry out a simulation with n = 2, N = 10. The robots are initialized randomly in a circle, which
is centered at the origin and has a radius of 1. The desired formation is specified in shape, location and
orientation as qi f = R f [sin((i − 1)2π/N); cos((i − 1)2π/N)], i = 1, ..., N with R f = 8, i.e. the desired
formation is a polygon whose vertices are uniformly distributed on a circle, which is centered at the
origin and has a radius of R f . All robots have the same sensing range: Ri = 2. The parameters of the p
times differential jump functions are p = 2, bi j = 1, g(y) = yp. The function ψ is taken as arctan. The
control gains are chosen as κ = 1, c = 2. Simulation results are plotted in Fig. 1. It is seen that all
robots nicely approach their desired locations. Since the robots initialize pretty close to each other, they
quickly move away from each other then approach their desired locations, see Sub-figure A) of Fig. 1,
where the trajectory of the robot 1 is plotted in the thick line, and robots 1 and 2 are indicated by 1 and
2. For clarity, only the control input u1 = [u1x u1y]T is plotted in Sub-figure B) of Fig. 1. Noticing that



Bounded Controllers for Decentralized Formation
Control of Mobile Robots with Limited Sensing 345

the values of the continuous controls u1x and u1y are in the range ±π . Sub-figure C) of Fig.1 plot the
functions β1 j, j = 2, ..., N. It is seen that these functions are always greater than zero and approach 1.
Sub-figure D) of Fig. 1 plots a ’mean-product’ distance, distall =

(
∏(i, j)∈N‖qi j‖

)N(N−1)/2
, see the thick

line, and the distances between the robot 1 and other robots in the group. Clearly, no collisions between
any robots occurred since distall is larger than zero for all simulation time. The bottom figure in Fig. 1
plots the functions β1 j, j = 2, ..., N. It is noted that all β1 j equal 1 when ‖q1 j‖ are larger or equal to 1.

10 5 0 5 10

10

5

0

5

10

X

Y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
6

4

2

0

2

4

6

Time [s]

u
1
x
, 

u
1
y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0

5

10

15

20

Time[s]

d
is

a
ll,

 |
|q

1
j||

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

||q
1j

||

1
j

u
1x

u
1y

dis
all

1

2

A)

B)

C)

D)

Figure 1: Simulation results.

6 Extension to formation tracking

This section extends the results developed in the previous sections to solve the problem of designing
a control input ui for each robot i that forces the group of N mobile robots whose dynamics are given in
(1) to track a moving Desired Formation Graph (DFG), in the sense that the DFG is allowed to move on a
common desired trajectory qod (s) with s being the common reference trajectory parameter defined in the
fixed coordinate system ΠF , see Fig. 2 and Fig. 3. We consider the DFG whose center moves along the
common reference trajectory qod (s). We assume that qod (s) is regular in the sense that it is single valued
and its first derivative exists and is bounded. Since the DFG under consideration is only representative,
the center does not have to be the "true" center of the DFG but can be any convenient point. When the
DFG moves along the trajectory qod (s), the vertex i of DFG generates the reference trajectory qid (s) for
the robot i to track. We limit our consideration to two- and three-dimensional (2D and 3D) spaces, which
are most common in practice. The control objective is now stated as follows.
Control objective. Assume that at the initial time t0 ≥ 0, for all (i, j) ∈ {1, 2, ...N}, i 6= j, s ∈ R there



346 K.D. Do

exist strictly positive constants ε1, ε2 and ε3 such that

‖qi(t0)−q j(t0)‖ ≥ ε1, ‖qid (s)−q jd (s)‖ ≥ ε2,
‖∂ qid (s)

∂ s
‖ ≤ ε3 (18)

Moreover, the robot i can only measure its own state and can only detect the other group members if
these members are in a sphere with a radius of Ri larger than a strictly positive constant, and centered at
the robot i. Design the control input ui for each robot i such that

lim
t→∞

(qi(t)−qid (s)) = 0, ‖qi(t)−q j(t)‖ ≥ ε4 (19)

for all (i, j) ∈ {1, 2, ...N}, i 6= j, s ∈R, where ε4 is a positive constant.
Control design. Let us construct qid (s). Let the moving coordinate system ΠM , of which the origin

_
O

coincides with the center of the desired formation graph, move along qod (s). Let
_
q id be the coordinate

vector of the vertex i of the DFG in the moving coordinate system. We then have

_
q id = J(•)(qid (s)−qod (s)) (20)

where J(•) whose elements depend on ∂ qod (s)/∂ s and is the rotational (invertible) matrix, which de-
scribes the rotation of ΠM with respect to ΠF , and is such that δ1 ≤ ‖J(•)‖ ≤ δ2, δ3 ≤ ‖J(•)−1‖ ≤ δ4
with δi, i = 1, .., 4 strictly positive constants. Therefore, by specifying

_
q id in ΠM , qid (s) in ΠF for the

robot i can be calculated from (20). Similarly, in ΠM the coordinate vector of each robot i satisfies

_
q i= J(•)(qi −qod (s)). (21)

From (20) and (21), we can see that the first two conditions in (18) imply the following condition

‖
_
q i (t0)−

_
q j (t0)‖ ≥

_
ε 1, ‖

_
q id (s)−

_
q jd (s)‖ ≥

_
ε 2 (22)

where
_
ε 1 and

_
ε 2 are some strictly positive constants. Moreover, the tracking control goal specified in

(19) is achieved by designing the control ui for each robot i such that

lim
t→∞

(
_
q i (t)−

_
q id (s)) = 0, ‖

_
q i (t)−

_
q j (t)‖ ≥

_
ε 4 (23)

where
_
ε 4 is a positive constant, and by letting the DFG move along the common reference trajectory via

giving ṡ some desired value.
Now differentiating both sides of (21) gives

_̇
q i =

_
ui (24)

where
_
ui is the new control, and we have chosen the control ui as

ui = q̇od (s) + J(•)−1(
_
ui −J̇(•)(qi −qod (s))). (25)

The problem of designing
_
ui for (24) to achieve (23) under (22) is exactly the same as the control objective

in Section 2. Therefore, the control design in Section 4 can be used directly to design a bounded control
_
ui to achieve the goal (23). After

_
ui is designed, the actual tracking control ui is calculated from (25). Let

us give the expression of the rotational matrix J(•) in 2D and 3D spaces.



Bounded Controllers for Decentralized Formation
Control of Mobile Robots with Limited Sensing 347

Figure 2: Formation coordinates in 2D.

( )Y j

( )X i

( )Z k

i
y

i
x

i
z

1
Y

1
X

1
Z

( )Y n

ix

iy

i
z

( )Z b

( )X t

od
x

od
y

od
z

O

O

i
v

11

13

12

11
12

13, ,

21
22

23

,
,

31

32

33

,
,

odq

Agent i

Moving frame coordinate system

Formation graph

M

Figure 3: Formation coordinates in 3D.



348 K.D. Do

Two-dimensional space. Consider the moving coordinate frame,
_
O

_
X

_
Y attached to the DFG, as shown

in Fig. 2. The origin
_
O coincides with the center of the graph, and is on the common reference trajec-

tory qod (s) = [xod (s) yod (s)]T . The
_
O

_
X and

_
O

_
Y axes are tangential and perpendicular to the reference

trajectory qod (s). Therefore the angle θ between
_
O

_
X and OX is calculated as θ = arctan(y

′
d /x

′
d ), where

•′ ∆= ∂ •/∂ s. Hence the rotational matrix J(•) is given by J(•) =
[

cos(θ ) sin(θ )
−sin(θ ) cos(θ )

]
, which is indeed

invertible for all θ ∈R, and ‖J(•)‖ = 1.
Three-dimensional space. Consider the moving coordinate frame,

_
O

_
X

_
Y

_
Z , attached to the DFG as

shown in Fig. 3. The coordinate frame
_
O X1Y1Z1 is parallel to OXY Z . The origin

_
O coincides with

the center of the graph, and is on the reference trajectory qod (s) = [xod (s) yod (s) zod (s)]T . The
_
O

_
X ,

_
O

_
Y

and
_
O

_
Z axes coincide with the unit tangent vector~t , the unit principal vector ~n , and the unit binormal

vector ~b of the trajectory qod (s) at the point
_
O. These unit vectors form a positively oriented triple of

vectors called the moving triad, and are given by~t = ~q′ od /‖~q′ od‖, ~n =~t′/‖~t′‖, ~b =~t ×~n, where × stands
for the vector cross product operation, (~i, ~j, ~k) are the unit vectors of the OXY Z coordinate frame. Let
(ξi1, ξi2, ξi3), i = 1, 2, 3 be the directional cosines of

_
O

_
X ,

_
O

_
Y and

_
O

_
Z with respect to the fixed axes

OX , OY and OZ, respectively. This notation means that if we denote (θi1, θi2, θi3), i = 1, 2, 3 as the
angles between the axes

_
O

_
X ,

_
O

_
Y and

_
O

_
Z , and the axes OX , OY and OZ (see Fig. 3 for representation

of θ11, θ12 and θ13), we have ξi j = cos(θi j), ∀i, j ∈ {1, 2, 3}. The rotational matrix J(•) is given by

J(•) =



ξ11 ξ12 ξ13
ξ21 ξ22 ξ23
ξ31 ξ32 ξ33


. It is shown in [20] that the determinant of J(•) is equal to 1, i.e. J(•) is

globally invertible, and ‖J(•)‖ = 1.

7 Summary and Conclusions

This paper has presented a method to design bounded and continuous (even smooth) controllers to
force a group of mobile robots with limited sensing ranges to achieve a desired formation while avoiding
collisions among themselves. The control development was based on construction of new potential
functions, and guaranteed that all critical points, except for the desired points in formation, are unstable
points. These potential functions with embedded smooth or p times differential jump functions are
attractive parts of the paper since they do not require the use of switching control theory despite the robot
limited sensing ranges. These functions can certainly be modified to solve other cooperative control
problems such as flocking and consensus of mobile robots. The problem of formation tracking was also
addressed. Future work is to extend the proposed techniques in this paper and those for centralized
formation control of nonholonomic mobile robots in [10] to design a decentralized formation control
system for a group of nonholonomic mobile robots.

8 Appendix A: Proof of Lemma 1

We need to verify that the function h(x, a, b) given in (5) satisfy all properties defined in (4). Property
1) holds because by (6) for all 0 ≤ x ≤ a, we have ∫ xa f (τ −a) f (b−τ)dτ = 0. Property 2) holds since by
(6) we have

∫ x
a f (τ −a) f (b−τ)dτ =

∫ b
a f (τ −a) f (b−τ)dτ for all x ≥ b. To prove Property 3), we first

note from Property a) of the function g(y) given in (7) that
∫ x

a f (τ −a) f (b−τ)dτ > 0 for all a < x < b.
Therefore, 0 <

∫ x
a f (τ−a) f (b−τ)dτ∫ b
a f (τ−a) f (b−τ)dτ

< 1, which means that Property 3) of the function h(x, a, b) holds. To

prove Property 4), we just need to show that f (y) is p−1 times differentiable. We first note that f (y) is



Bounded Controllers for Decentralized Formation
Control of Mobile Robots with Limited Sensing 349

p times differentiable except at y = 0. Hence, we only need to verify that f (k)(0) = ∂
k f (y)
∂ yk

∣∣∣
y=0

= 0 for

any positive integer k < p. Clearly, limy→0− f (k)(y) = 0 since f (y) = 0, ∀y ≤ 0. On the other hand, since
f (y) = g(y), y > 0, from Property b) of the function g(y) we have limy→0+ f (k)(y) = limy→0+ g(k)(y) = 0,

where g(k) = ∂
k g(y)
∂ yk . Since both left- and right-hand limits are equal to 0, we have f

(k)(0) = 0. Hence
Property 4) holds. 2

9 Appendix B: Proof of Corollary 1

We first note that Property a) of the function g(y) in (7) can be proven without a difficulty. We focus
on proof of Property b). We note that g(k)(y) = ∂

k g(y)
∂ yk = Qk(

1
y )e

− 1y where Qk( 1y ) is a polynomial function
of 1y , and k is any positive integer. We will prove Property b) of the function g(y) in (7) by induction.

It is clear that limy→0+ g(1)(y) = limy→0+
g(y)−g(0)

y−0 = limy→0+
e−

1
y

y = limξ→∞
1
eξ

= 0 where ξ = 1y and we
have used l’Hopital’s rule. This means that Property b) of the function g(y) holds for k = 1. Assuming
that limy→0+ g(k)(y) = 0, we now compute limy→0+ g(k+1)(y) as follows:

lim
y→0+

g(k+1)(y) = lim
y→0+

g(k)(y)−g(k)(0)
y−0 =

lim
y→0+

Q̃k(
1
y
)e−

1
y = lim

ξ→∞
Q̃k(ξ )

eξ
= 0 (26)

where l’Hopital’s rule has been used, ξ = 1y , and Q̃k(ξ ) = ξ Qk(ξ ) is another polynomial of ξ . Therefore
we have proved that limy→0+ g(k)(y) = 0 for any k, which means Property b) of the function g(y) holds
for any positive integer p, i.e. p can be equal to infinity. By Definition 1, the function h(x, a, b) is a
smooth jump function.2

10 Appendix C: Proof of Theorem 1

We prove Theorem 1 in two steps. In the first step, we prove that no collisions between the robots
can occur and that the robots asymptotically approach their target points or some critical points. In the
second step, we investigate stability of the closed loop system (17) at these points, we linearize the closed
loop system at these points. We then prove that only desired target points are unique asymptotic stable
and that other critical points are unstable.

Step 1. Proof of no collision and existence of solutions. From (16) and properties of the function ψ
, see (15), we have ϕ̇ ≤ 0, which implies that ϕ(t) ≤ ϕ(t0),∀t ≥ t0. With definition of the function ϕ in
(8) and its components in (9) and (10), we have

e0.5 ∑
N
i=1 ‖qi(t)−qi f ‖2

∏ h(0, b2i j/2,‖qi j(t)‖2/2)
≤ e

0.5 ∑Ni=1 ‖qi(t0)−qi f ‖2

∏ h(0, b2i j/2,‖qi j(t0)‖2/2)
(27)

for all t ≥ t0 ≥ 0, i = 1, ..., N − 1, j = i + 1, ...N. From the first condition in (2) and properties of the
jump function h(0, b2i j/2,‖qi j‖2/2), we have ∏ h(0, b2i j/2,‖qi j(t0)‖2/2) is larger than a strictly positive
constant. Therefore the right hand side of (27) is bounded by a positive constant depending on the initial
conditions. Boundedness of the right hand side of (27) implies that the left hand side of (27) must be also
bounded. As a result, ∏ h(0, b2i j/2,‖qi j(t)‖2/2) must be larger than some positive constant depending



350 K.D. Do

on the initial conditions for all t ≥ t0 ≥ 0. From properties of h(0, b2i j/2,‖qi j‖2/2), ‖qi j(t)‖ must be
larger than some positive constant depending on the initial conditions denoted by ε3, i.e. there are no
collisions for all t ≥ t0 ≥ 0, i = 1, ..., N −1, j = i + 1, ...N. Boundedness of the left hand side of (27) also
implies that of ‖qi j(t)‖ and ‖qi(t)‖ for all t ≥ t0 ≥ 0, i.e. the solutions of the closed loop system (17) exist.

+Equilibrium points. Since we have already proved that there are no collisions between any robots
and that the solutions of the closed loop system (17) exist, an application of Theorem 8.4 in [19] to (16)
yields

lim
t→∞

ΩTi (t)Ψ(Ωi(t)) = 0,∀i = 1, 2, ..., N. (28)
Thanks to Property 2) of the function ψ , see (15), the limit equation (28) implies that

lim
t→∞

Ωi(t) = lim
t→∞

[
qi(t)−qi f −κ ∑

j∈Ni
β̄ ′i j(t)qi j(t)

]
= 0 (29)

for all i = 1, 2, ..., N. The limit equation (29) implies that the state q(t) = [qT1 (t) q
T
2 (t), ..., q

T
N (t)]

T

converges to the manifold M of (17) contained in E = {q ∈ Rn×N|Ω=0} with Ω = [ΩT1 ΩT2 , ..., ΩTN ]T ,
i.e. on the surface where ϕ̇ = 0. This surface is continuous because we have already proved that
‖qi j‖ > 0,∀(i, j) ∈ {1, 2, ..., N}, i 6= j, i.e. β̄ ′i j is continuous. As the time t goes to infinity, it can be veri-
fied that one solution of (29) is q f = [qT1 f q

T
2 f , ..., q

T
N f ]

T since β ′i j|‖qi j‖=‖qi j f ‖ = 0 ⇒ β̄ ′i j|‖qi j‖=‖qi j f ‖ = 0 (we
have chosen bi j ≤ min (Ri, R j, ε2), see (11)), and other solutions are denoted by qc = [qT1c qT2c, ..., qTNc]T .
It is noted that some elements of qc can be equal to that of q f . However, for simplicity we abuse the
notation, i.e. we still denote that vector as qc. Indeed, the vector qc is such that

Ωi|q=qc =
[
qi −qi f −κ ∑

j∈Ni
β̄ ′i jqi j

]∣∣∣∣
q=qc

= 0 (30)

for all i = 1, ..., N. Next, we will show that q f is stable and qc is unstable, by linearizing (17) at these
points.

+Properties of equilibrium points. The closed loop system (17) can be written in a vector form as
q̇ = −cΨq(q, q f ), and Ψq(q, q f ) = [ΨT (Ω1), ..., ΨT (Ωi), ..., ΨT (ΩN )]T . Therefore, near an equilibrium
point qo, which can be either q f or qc, we have

q̇ = −c ∂ Ψq(q, q f )/∂ q
∣∣
q=qo

(q−qo) (31)

where

∂ Ψq(q, q f )
∂ q

=




∆11 ∆12 ··· ··· ∆1N
...

. . .
...

...
...

∆i1 ··· ∆ii ··· ∆iN
...

...
...

. . .
...

∆N1 ··· ··· ··· ∆NN




(32)

with ∆i j =
∂ Ψ(Ωi)

∂ Ωi
∂ Ωi
∂ qi

, (i, j) ∈N, where N denotes the set of all robots in the group. A simple calculation
shows that for all i = 1, ..., N, j ∈Ni, j 6= i

∂ Ωi
∂ qi

=
(

1−κ ∑
i∈Ni

β̄ ′i j

)
In −κ ∑

j∈Ni
β̄ ′′i jqi jq

T
i j,

∂ Ωi
∂ q j

= κ β̄ ′i jIn×n + κ β̄
′′
i jqi jq

T
i j (33)



Bounded Controllers for Decentralized Formation
Control of Mobile Robots with Limited Sensing 351

where β̄ ′′i j =
∂ β̄ ′i j

∂ (‖qi j‖2/2) . Let N
∗ be the set of the robots such that if the robots i and j belong to the set N∗

then ‖qi j‖ < bi j. Next we will show that q f is asymptotically stable and that qc is unstable.

Step 2. Behavior of equilibrium points. Evaluating (33) at q = q f gives

∂ Ψ(Ωi)
∂ Ωi

∣∣∣∣
q=q f

= In,
∂ Ωi
∂ qi

∣∣∣∣
q=q f

= In,
∂ Ωi
∂ q j

∣∣∣∣
q=q f

= 0 (34)

where we have used β̄ ′i j
∣∣
qi j =qi j f

= 0 and β̄ ′′i j
∣∣
qi j =qi j f

= 0 since β ′i j
∣∣
qi j =qi j f

= 0 and β ′′i j
∣∣
qi j =qi j f

= 0 (we have

chosen bi j ≤ min (Ri, R j, ε2), see (11)). We consider the Lyapunov function candidate Vf = 0.5‖q−q f ‖2
whose derivative along the solutions of the linearized closed loop system (31) with qo replaced by q f ,
and using (34) satisfies V̇f = −c ∑Ni=1 ‖qi −qi f ‖2 = −2cVf , which implies that q f is asymptotically stable.

- Proof of qc being unstable. Now evaluating (33) at q = qc give that for all i = 1, ..., N, i 6= j :
∂ Ψ(Ωi)

∂ Ωi

∣∣∣∣
q=qc

= In,
∂ Ωi
∂ qi

∣∣∣∣
q=qc

=
(
1−κ ∑

j∈Ni
β̄ ′i jc

)
In −

κ ∑
j∈Ni

β̄ ′′i jcqi jcq
T
i jc,

∂ Ωi
∂ q j

∣∣∣∣
q=qc

= κ β̄ ′i jc + κ β̄
′′
i jcqi jcq

T
i jc (35)

where qi jc = qic − q jc, β̄ ′i jc = β̄ ′i j
∣∣
qi j =qi jc

and β̄ ′′i jc = β̄
′′
i j

∣∣
qi j =qi jc

. Since the related collision avoidance

functions βi j, (hence β̄ ′i j and β̄
′′
i j), are specified in terms of relative distances between robots and it is

extremely difficult to obtain qc explicitly by solving (30), it is very difficult to use the Lyapunov function
candidate Vc = 0.5‖q − qc‖ to investigate stability of (31) at qc. Therefore, we consider the following
Lyapunov function candidate

V̄c = 0.5‖q̄− q̄c‖2 (36)
where q̄ = [qT12, q

T
13, ...q

T
1N , q

T
23, ..., q

T
2N , ..., q

T
N−1,N ]

T and q̄c = [qT12c, q
T
13c, ...q

T
1Nc, q

T
23c, ..., q

T
2Nc, ..., q

T
N−1,Nc]

T .
Differentiating both sides of (36) along the solution of (31) with qo replaced by qc gives

˙̄Vc = −c ∑
(i, j)∈N\N∗

‖qi j −qi jc‖2 −c ∑
(i, j)∈N∗

(1−κ Nβ̄ ′i jc)×

‖qi j −qi jc‖2 + κ cN ∑
(i, j)∈N∗

β̄ ′′i jc
(
qTi jc(qi j −qi jc)

)2
(37)

where i 6= j and (35) has been used. To investigate stability properties of q̄c based on (37), we will use
(30). Define Ωi jc = Ωic − Ω jc, ∀(i, j) ∈ {1, ..., N}, i 6= j where Ωic = Ωi|q=qc = 0, see (30). Therefore
Ωi jc = 0. Hence ∑(i, j)∈N∗ qTi jcΩi jc = 0, i 6= j, which by using (30) is expanded to

∑
(i, j)∈N∗

(
qTi jc(qi jc −qi j f )−κ Nβ̄ ′i jcqTi jcqi jc

)
= 0

⇒ ∑
(i, j)∈N∗

(1−κ Nβ̄ ′i jc)qTi jcqi jc = ∑
(i, j)∈N∗

qTi jcqi j f (38)

where i 6= j. The sum ∑(i, j)∈N∗ qTi jcqi j f is strictly negative since at the point where qi j = qi j f , ∀(i, j) ∈
N∗, i 6= j (the point F in Fig. 4) all attractive and repulsive forces are equal to zero while at the point
where qi j = qi jc ∀(i, j) ∈ N∗, i 6= j (the point C in Fig. 4) the sum of attractive and repulsive forces are
equal to zero (but attractive and repulsive forces are nonzero). Therefore the point where qi j = 0, ∀(i, j) ∈
N∗, i 6= j (the point O in Fig. 4) must locate between the points F and C for all (i, j) ∈ N∗, i 6= j. That is



352 K.D. Do

12
q

13
q

*
1| |

q

23
q

*
2| |

q

* *
| | 1,| |

q

O

F

C

Figure 4: Illustration of equilibrium points.

there exists a strictly positive constant b such that ∑(i, j)∈N∗ qTi jcqi j f < −b, which is substituted into (38)
to yield

∑
(i, j)∈N∗

(1−κ Nβ̄ ′i jc)qTi jcqi jc < −b, i 6= j. (39)

Since qTi jcqi jc > 0,∀(i, j) ∈ N∗, i 6= j, there exists a nonempty set N∗∗ ⊂ N∗ such that for all (i, j) ∈
N∗∗, i 6= j, (1 − κ Nβ̄ ′i jc) is strictly negative, i.e. there exists a strictly positive constant b∗∗ such that
(1−κ Nβ̄ ′i jc) < −b∗∗, ∀(i, j) ∈N∗∗, i 6= j. We now write (37) as

˙̄Vc = −c
[

∑
(i, j)∈N\N∗

‖qi j −qi jc‖2 + ∑
(i, j)∈N∗\N∗∗

(1−κ Nβ̄ ′i jc)×

‖qi j −qi jc‖2 −κ N ∑
(i, j)∈N∗

β̄ ′′i jc
(
qTi jc(qi j −qi jc)

)2
]
−

c ∑
(i, j)∈N∗∗

(1−κ Nβ̄ ′i jc)‖qi j −qi jc‖2 (40)

where i 6= j. We now define a subspace such that qi j − qi jc = 0, ∀(i, j) ∈ N\N∗∗ and qTi jc(qi j − qi jc) =
0, ∀(i, j) ∈N∗, i 6= j. In this subspace, we have

V̄c = 0.5 ∑
(i, j)∈N∗∗

‖qi j −qi jc‖2, (41)

˙̄Vc = −c ∑
(i, j)∈N∗∗

(1−κ Nβ̄ ′i jc)‖qi j −qi jc‖2 ≥ 2cb∗∗V̄c

where we have used (1−κ Nβ̄ ′i jc) < −b∗∗, ∀(i, j) ∈N∗∗, i 6= j. Clearly (41) implies that

∑
(i, j)∈N∗∗

‖qi j(t)−qi jc‖ ≥ ∑
(i, j)∈N∗∗

‖qi j(t0)−qi jc‖ecb
∗∗(t−t0) (42)

for all i 6= j, t ≥ t0 ≥ 0. Now assume that qc is a stable equilibrium point of the closed loop system
(17), 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∗∗ ‖qi j(t) − qi jc‖ = d∗∗,∀(i, j) ∈
N∗∗, i 6= j with d∗∗ a nonnegative constant, since qi j = qi −q j and qi jc = qic −q jc. This contradicts (42)
for the case ∑(i, j)∈N∗∗ ‖qi j(t0)− qi jc‖ 6= 0, since the right hand side of (42) is divergent (so does the left



Bounded Controllers for Decentralized Formation
Control of Mobile Robots with Limited Sensing 353

hand side). For the case ∑(i, j)∈N∗∗ ‖qi j(t0) − qi jc‖ = 0, there would be no contradiction. However this
case is never observed in practice since the ever-present physical noise would cause ‖qi j(t∗)− qi jc‖ for
some (i, j) ∈N∗∗, i 6= j to be different from 0 at the time t∗ ≥ t0. We now write (42) as

∑
(i, j)∈N∗∗

‖qi j(t)−qi jc‖ ≥ ∑
(i, j)∈N∗∗

‖qi j(t∗)−qi jc‖ecb
∗∗(t−t∗) (43)

for all i 6= j, t ≥ t∗ ≥ t0 ≥ 0. Since ∑(i, j)∈N∗∗ ‖qi j(t∗)−qi jc‖ 6= 0, the right hand side of (43) is divergent
(so does the left hand side). This contradicts limt→∞ ∑(i, j)∈N∗∗ ‖qi j(t)− qi jc‖ = d∗∗,∀(i, j) ∈ N∗∗, i 6= j.
Therefore qc must be an unstable equilibrium point of the closed loop system (17). Proof of Theorem 1
is completed.

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K.D. Do
School of Mechanical Engineering

The University of Western Australia
35 Stirling Highway, Crawley, WA 6009, Australia

E-mail: duc@mech.uwa.edu.au

Received: December 26, 2006