1
Semi-centralized control for multi-robot formation
and theoretical lower bound
Shuo Wan, Jiaxun Lu, Pingyi Fan, Senior Member, IEEE and Khaled B. Letaief*, Fellow, IEEE
Tsinghua National Laboratory for Information Science and Technology(TNList),
Department of Electronic Engineering, Tsinghua University, Beijing, P.R. China
E-mail: wan-s17@mails.tsinghua.edu.cn, lujx14@mails.tsinghua.edu.cn, fpy@tsinghua.edu.cn
*Department of Electronic Engineering, Hong Kong University of Science and Technology, Hong Kong
Email: eekhaled@ece.ust.hk
Abstract—Multi-robot formation control enables robots to
cooperate as a working group in completing complex tasks,
which has been widely used in both civilian and military
scenarios. Before moving to reach a given formation, each robot
should choose a position from the formation so that the whole
system cost is minimized. To solve the problem, we formulate
an optimization problem in terms of the total moving distance
and give a solution by the Hungarian method. To analyze
the deviation of the achieved formation from the ideal one,
we obtain the lower bound of formation bias with respect to
system’s parameters based on notions in information theory. As
an extension, we discuss methods of transformation between
different formations. Some theoretical results are obtained to
give a guidance of the system design.
Index Terms—position arrangement, formation bias, leader-
follower, formation conversion
I. INTRODUCTION
R
ECENT years, multi-robot moving in a formation are
replacing complex single robots for both civilian and
military applications. The multi-robot system has many advan-
tages. By simply replacing broken robots, the overall system
performance will not be degraded largely. In this way, it can
reduce the complexity by decomposing complex tasks into
small ones [1]. Furthermore, multi-robot system can complete
more complex tasks with a lower cost. In military applica-
tions, robots can cooperate as a working group to complete
surveillance tasks [2] or do spying work in adversarial areas
based on techniques in [3][4][5]. In civilian applications, auto-
matically driving car system may be helpful in the intelligent
transportation systems.
Formation control is a critical problem for robots to coordi-
nate. To complete a certain task, we often need robots to move
by maintaining a sequential formations. For example, when
cooperatively transporting a large item, certain formations are
required, which enable them to complete the task [6]. In this
process, the formation should be reached as fast as possible
to improve the working efficiency. Furthermore, the deviation
of achieved formation from the exactly required one should
be constrained in a reasonable range to ensure the task’s
completion. In practice, we also need to change formation
according to different tasks and the process should be fast as
well.
Multi-robot formation usually requires several basic condi-
tions or assumptions. Here, it assumes that: 1) Robots are able
to set relative positioning of others around them by using the
assistant essential information to guide themselves [7][8]; 2)
Robots are able to control themselves to reach their desired
places; 3) The basic communication capability is necessary,
so that robots are able to cooperate with each other [9][10].
The
multi-robot
formation
controlling
approaches
are
mainly divided into three classes: decentralized control, cen-
tralized control and the combination of centralization and
decentralization, refer to as semi-centralized control. In cen-
tralized control, there is a central agent monitoring the whole
environment and controlling all robots’ motion to reach a
formation. The agent can be either a computer or a robot. In
decentralized control system, there is no central agent. Each
robot makes their own decision. They monitor the environment
by themselves and exchange their observations with others by
communication. In the semi-centralized control method, there
exists a central agent distributing tasks and sending commands
to robots and each robot reaches its goal by decisions of their
own.
To cope with different application scenarios, researchers
presented several different methods based on different assump-
tions and requirements. In [11], it discussed the problem of
planning collision-free paths for permutation-invariant multi-
robot formations. In [12], it presented a formation controller
for a scalable team of robots where communications are
unavailable and sensor ranges are limited. Robots can reach a
locally desired formation and change gradually to a globally
desired one. In [13], it proposed a method of distributed esti-
mation and control for preserving formation rigidity of mobile
robot teams, which is a kind of virtual structure approach. In
[14][15], they proposed a leader-follower controller for multi-
robot teams. In [16], it developed a suboptimal controller for
a leader-follower formation problem of quadrotors with the
consideration of external disturbances. In [17], it combined
centralization and decentralization methods and proposed a
coordination architecture for spacecraft formation control.
In [18], it presented an approach for formation control of
autonomous vehicles traversing along a multi-lane road with
obstacles and a certain degree of traffic.
However, to the best knowledge of us, the previous works
didn’t discuss how to arrange each robot with a position
in the formation. Instead, this is usually requirement from
applications. Furthermore, they also didn’t present theoretical
analysis of the achieved formation compared to the exactly
required one.
arXiv:1709.03641v1  [cs.RO]  12 Sep 2017
2
To solve the problem of arranging robots in a formation, we
transform the problem to a task distribution problem and solve
it with the Hungarian method. Then we define the formation
bias. To estimate the bias, we apply notions in information
theory. We do an estimation of the Bayes risk by calculating
the mutual information between measured value and true value
to solve the problem.
In this paper, we focus on leader-follower method which
is more similar to human’s action mode when standing in
a formation. Each robot has a reference and follows it until
the robot arrives at the right position. Based on this, we first
propose a method to obtain an optimal solution for arranging
robots to positions in a given formation by minimizing the
predicted total moving distance. Then we also consider the
achieved formation’s bias to the ideal one. Applying methods
in decentralized estimation, we obtain a lower bound of the
formation bias with respect to system parameters. This can
guide us to design the system according to user’s requirements.
Finally, we discuss the transformation between different for-
mations. We optimize the choice of new formation center and
confirm it by simulation.
Our discussed system belongs to the third class. It has a
central agent to send commands to robots. Each robot receives
the commands and reaches its goal by itself. In the process,
they detect possible collisions and avoid them. Each robot can
measure the relative position of its reference and follow it to
reach the formation.
The rest paper is organized as follows: A brief problem and
notation statement are given in Section II. In Section III, we
give method to arrange robots to positions in the formations
by transforming it into a distribution optimization problem. In
Section IV, we give a depiction of the leader-follower method
in our considered system. A lower bound of the formation
bias is derived in Section V. The proof is given in Section VI.
Afterwards, we discuss formation conversion in Section VII.
Some simulation results are presented in Section VIII. Finally,
we give the conclusion.
II. PROBLEM STATEMENTS AND RELEVANT NOTATIONS
Consider a group of robots, each has a serial number ranging
from 1 to n. Their initial positions {xi(0)|i = 1, 2, ......, n}
are randomly distributed. Their motion can be described as
follows:
xi(k + 1) = xi(k) + vi(k)
(1)
where xi(k) ∈R2 is the position of robot i at time slot k
with i ∈{1, 2, ......, n} and ∥vi(k)∥≤Umax. Umax is the
maximum of velocities. vi(k) is the speed of robot i. Given a
formation, the robots need to move to reach it, during which
there should not have any collision.
In the real world, there are many different demands for robot
formation configuration. As for the square formation, we tend
to find a leading robot and let other robots follow it to reach a
formation. Sometimes, we want the center of the formation to
be at a certain position. Thus, we may let robots keep up with
the center to move. The center can be given previously by the
planner or it can be decided by the system automatically.
The formation is represented by F = {f1, f2, ......, fn},
where fi is the position of the i-th node in the formation.
Considering the formation on 2D plane, we have fi ∈R2.
Besides, we also let origin to be the center of the formation,
which means Pn
i=1 fi = 0. The formation F is predefined by
users, which is used by the system as reference to produce
controlling forces for reaching the formation.
Having those positions in the formation, the robots have to
be arranged according to them. The arrangement is denoted
by D = {d1, d2, ......, dn}, where di ∈{1, 2, ......, n}. di
represents the serial number of the i-th robot’s position in
the formation.
When configuring a square formation, we assume there
exists a leading robot which is chosen by the controlling
center. Each robot keeps up with another robot according to the
arrangement mentioned above, while some robots will follow
the leading robot directly. The leading robot’s serial number is
denoted as ilead. Besides, we define H = {h1, h2, ......, hn},
where hi ∈{1, 2, ......, n} represents the serial number of
the robot followed by robot i. In addition, we also define
P = {p1, p2, ......, pn}, where pi ∈R2 represents the re-
quired relative position in the formation between robot i and
the robot it’s following. pi can obviously be calculated by
pi = fdhi −fdi. Assuming that robot i can measure the
relative position of robot hi to itself, the robot can move
to adjust the relative position. If for all i ̸= ilead, there is
xi(k) −xhi(k) = pi, the formation is reached at time slot k.
Besides, if the robots collide with another, they may break
down. So our controlling method should avoid such event
happen.
When requiring the formation to be around a center, we may
not require to find a leader. Assuming robots can know the
relative position between the center and themselves, they can
move to adjust the relative position to achieve the formation.
Denoting the position of the center to be Cx, if at time slot
k, there are xi(k) −Cx = fi for all i ∈{1, 2, ......, n}, the
formation is reached at time slot k.
The movement of the robots should be synchronized so that
equation (1) can be meaningful to describe the robots’ motion
[19]. Assuming that the center or the leading robot can act
as the synchronizer, we can achieve this. Besides, the speed
should be adjusted from vi(k −1) to vi(k) at time slot k. We
assume this can be achieved by a motion controller on the
robot. As mentioned before, the controlling should be based
on the ability of measuring the relative position of other robots
[20]. We assume the on board sensor and communication
system will help to achieve this.
III. OPTIMIZATION OF FORMATION MAPPING
As mentioned, the formation can be described by F =
{f1, f2, ......, fn}. Besides, the initial positions of robots are
randomly deployed as X(0) = {x1(0), x2(0), ......, xn(0)}.
Given X(0), we can optimise the arrangement of the
robots to the position in the formation described by D =
{d1, d2, ......, dn}. The aim of the optimization is to minimize
the robots’ total moving distance.
Assuming there is a center controller, it will collect the
robots’ initial position X(0) and calculate the arrangement D
3
according to X(0) and formation F. Then it will broadcast
the arrangement to robots.
A. Cost Function
As previously stated, searching for the optimal arrangement
D requires minimizing the total moving distance. Moreover, if
all the robots choose a position ensuring the shortest path, the
total consuming time will be smaller and the moving process
seems more reasonable. Before the optimization, we should
set up a cost function to describe the moving distance.
To get the cost function, one can estimate the distance
between the initial position and the final position in the
formation. However, in real system, the moving distance is
actually very complex to describe, as the robots may get away
the calculated path in the planning stage to avoid other robots
or obstacles. For example, when it comes to square formation,
each robot follows another one. If it is not following the leader,
its destination will keep moving. So the actual path may not
be straight, which will result in an extra length compared to
the estimated distance. In this way, the real moving distance
may be very difficult to be represented. As an alternative, we
use the estimated distance to get a near optimal result. We
assume that the robots are not very dense, and the collision
does not frequently happen. Besides, considering the actual
curve path will make the problem much more complex, which
is not worth to do so. Therefore using the estimated distance
may be a good selection in the initial planning stage.
Choosing the leader, others will move to the right position
relative to it. Besides, fdilead should also be the leading
position in the square formation. Given the leading robot’s
position xilead and assuming that the formation is reached at
time slot k, the i-th robot’s current position is
xi(k) = xilead −fdilead + fdi
Therefore the estimated moving distance of robot i should be
Ci = ||xi(0) −xi(k)|| = ||(xi(0) −fdi) −(xilead −fdilead )||
So we can get the cost function for robot i.
Ct(i) = C2
i = ||(xi(0) −fdi) −(xilead −fdilead )||2
(2)
When it requires to reach a formation around a center, it
is not need to choose a leading robot. Given a center with
position Cx, each robot may move according to Cx. Assuming
that collision does not happen frequently, the extra path length
caused by avoiding possible collisions can be ignored first.
Besides, each robot can move according to the fixed center,
which means that without considering the few collisions, the
path is straight with high possibility. In this way, if the
formation is reached at time slot k, the right position for robot
i will be
xi(k) = Cx + fdi
Therefore the estimated moving distance of robot i is
Ci = ||xi(0) −xi(k)|| = ||(xi(0) −fdi) −Cx||
The corresponding cost function for robot i is defined as
Ct(i) = C2
i = ||(xi(0) −fdi) −Cx||2
(3)
B. Optimization problem and Hungarian method
1) Problem Statement: Considering the square formation
with a leader, the optimization problem can be formulated as
follows by using cost function (2).
D = argmin
D
n
X
i=0,i̸=ilead
||(xi(0) −fdi) −(xilead −fdilead )||2
(4)
When it comes to reaching a formation around a center, the
cost function is updated by equation (3) and the optimization
problem is formulated as
D = argmin
D
n
X
i=0
||(xi(0) −fdi) −Cx||2
(5)
In fact, with semi-centralized control mode, the formation
can be divided into the two scenarios above along the time. In
the first phase, all robots are far from its required positions.
So selecting a leader may be a good way to move close to the
required positions. Then the second phase is triggered, one can
select the mode where the center of formation is known. Now,
we first introduce the Hungarian Method by Kuhn in [21][22].
2) Review of Hungarian Method: Considering a general
assignment problem, there is an n by n cost matrix A = [aij],
where ai,j satisfies aij
≥0. The aim is to find a set
j1, j2, ......, jn, which is a permutation of 1, 2, ......, n, so that
the sum Ct = r1,j1 + r2,j2 + ...... + rn,jn is minimized.
As for matrix A, if there are zero elements existing both in
different rows and columns, one can get the assignments easily,
and the Ct mentioned above is obviously zero. However, A
doesn’t necessarily contain enough such zero elements. So the
matrix have to be transformed by the Hungarian Method. Two
basic theorems are introduced here, refer to Lemma 1 and 2,
respectively.
Lemma 1. Koning’s theorem [23]: In any bipartite graph, the
number of edges in a maximum matching equals the number
of vertices in a minimum vertex cover.
Remark 1. In Lemma 1, bipartite graph refers to those whose
vertices can be partitioned into two sets such that each edge
has one endpoint in each set. A vertex cover in a graph is a set
of vertices that includes at least one endpoint of every edge,
and a vertex cover is minimum if no other vertex cover has
fewer vertices. A matching is a set of edges none of which
share an endpoint. If no other matching has more edges,
a matching is maximum [24]. Based on Lemma 1, there is
a conclusion that the minimum number of rows or columns
needed to contain matrix A’s all zero elements is equal to the
maximum number of zeros that can be chosen, in which none
of two zero elements is on the same line.
Lemma 2. the distribution problem is unchanged if the matrix
A is replaced by A
′ = (a
′
ij), with a
′
ij = aij −ui −vj for
constants ui and vj, and i, j = 1, 2, ......n [22].
With Lemma 2, one can transform matrix A to obtain some
zero elements. With Lemma 1, one can find the independent
zero elements and minimize Ct. If there are not enough zero
elements, one can continue to transform A with Lemma 2 until
success. The detailed procedure can refer to [21][22].
4
3) Problem Transformation and Solution: The formation
mapping problem can be transformed to the assignment prob-
lem and solved by the Hungarian Method.
As for the optimization problem (4), given robot i ̸= ilead,
its assigned position’s serial number is di. Let di = j, we can
have the following equation:
aij = ||(xi(0) −fdi) −(xilead −fdilead )||2
(6)
where aij is an element in the cost matrix A. It represents the
cost of arranging robot i to the position with serial number j.
In the considered system, the leading robot’s position should
be the explicit reference marker in the square formation.
Therefore dilead can be fixed. However, to calculate aij, we
should select the leading robot ilead. The value of aij is then
relevant to ilead, which means we can calculate a cost matrix
A given an ilead. In this case, A is (n−1) by (n−1), for the
leader is excluded. With A, we can use the Hungarian Method
to get an optimal arrangement conditionally for a given ilead.
If we let each robot to be the leader and separately calculate
the result, an optimal arrangement can be finally acquired by
comparison of the conditional minimum cost results.
Note that in the optimization problem (5), given di = j
for robot i, the cost matrix A is calculated by the following
equation:
aij = ||(xi(0) −fdi) −Cx||2
(7)
That is, given Cx, the cost matrix A can be acquired. Then
the optimal arrangement can be directly achieved by the
Hungarian Method. The result is correlated with Cx. In the
considered system, Cx is usually predetermined.
IV. MOVING STRATEGIES
Given n robots’ initial position x(0), formation F and
calculated formation mapping D, the robots can move to reach
the formation. The moving strategy should not only achieve
the goal, but also prevent possible collisions.
A good approach to solve the problem is to imitate human.
When people need to stand in a formation, they tend to find a
reference and move to the exactly desired position relative to it.
During the process, if two or more people approach the same
position, they may slow down or deviate to avoid possible
collisions. The reference can be selected as the leader or other
people beside them in the formation. When people want to
reach a formation around a center, they can also choose the
center as the reference. To conclude, this is actually a leader-
follower mode. As for robots, they can also follow such mode.
Therefore a polar partitioning model [14][15] can be applied
and extended to our scenario.
Fig. 1 represents the coordinate system. The follower can
measure the leader’s position. Together with the relative po-
sition in the formation, it can calculate a currently desired
destination, which is the origin of the polar coordinates. The
follower can measure its destination’s position and locate itself
in the polar coordinates.
Considering a follower located in circle Rm with the radius
R, the center of Rm is the follower’s current destination. Fig.
2 shows that Rm is divided by curves

r = rh|rh =
R
nr −1(h −1), h = 1, 2, ......, nr

(8)
Fig. 1.
leader follower model and Circle coordinate system
Fig. 2.
A partitioned circle whose center is the present destination

θ = θj|θj =
2π
nθ −1(j −1), j = 1, 2, ......, nθ

(9)
A region is denoted by
Rhj = {x = (r, θ)|rh ≤r ≤rh+1, θ(j) ≤θ ≤θ(j + 1)}
(10)
whose boundaries are given in (8) and (9).
Supposing a follower is denoted by robot i, at time slot k,
let xi(k) ∈Rhj. The motion of robot i is described by (1).
Then the moving strategies should be
xi(k + 1) ∈R(h−1)j
(h ̸= 1)
(11)
vi(k) = 0
(h = 1)
(12)
However, if it detects a possible collision in R(h−1)j, it will
choose to move to Rh(j+1) at time slot (k + 1). If Rh(j+1)
is also possible for collision, it will choose Rh(j−1). If no
possible moving is safe, it will stop for a time slot. When robot
i detects possible collision with robot j and i > j, only robot
i will deviate from the path and robot j will not be affected.
In this way, it can improve the system’s efficiency intuitively.
Therefore the formation can be reached if each robot follows
this procedure to find its destination.
V. THEORETICAL ANALYSIS OF FORMATION BIAS
Based on the procedure described previously, the formation
can be finally reached. However, the measurement of position
may not be accurate. Besides, the moving strategies need to
partition the space, which will cause inaccuracy. So robots’
final position may have a bias compared to the exactly required
one, which causes a formation bias.
5
In this section, we provide a theoretical analysis of the
formation’s bias. We first introduce the notion of the formation
bias and analyze possible issues causing it. Then we give a
lower bound of it, which is related to the system’s parameters.
Finally we use simulations to check the theoretical results.
A. Formation bias definition
In the system, each robot should measure the position of
its destination, including relative distance r and relative angle
θ. Then the robot need to quantify r and θ to complete the
space partition. The measurement of r and θ may bring some
bias. Besides, the quantification will also bring bias to the final
formation.
Assuming that robot can move from one region to another
accurately, the bias mainly results from measurement and
quantification. Besides, the angle θ represents the direction
from which the robot approaches its destination. This does not
contribute to the final formation bias. So we mainly consider
bias caused by r here.
Supposing
the
formation
is
reached
at
time
slot
k,
the
robots’
destinations
are
denoted
by
DS
=
{ds1, ds2, ......, dsn}. For robot i, its position
bias Er(i) is
Er(i) = |xi(k) −dsi|
(13)
Therefore the formation bias Er can be defined as
Er = 1
n
n
X
i=1
Er(i)
(14)
which represents the mean deviation of robots. The bias is
mainly from the measurement of distance r. We will analyze
the bias of r to give an estimation of Er.
B. Problem Modeling
A model of decentralized estimation with single processor
is first considered. In this process, the estimator can’t gain
direct access to the parameter of interest. It can receive
the measurement from a local sensor. The sensor can make
several measurements about the parameter. After acquiring
the samples, it will quantize them and send them to the
estimator for analysis. Finally, it makes an estimation about
the parameter.
The estimation performance can be evaluated by the distor-
tion, which is represented by a function of the parameter value
and its estimated value. The minimum possible distortion is
defined as Bayes risk. In [25], it gives the lower bounds of
the Bayes risk for the estimator problems.
Fig. 3.
The model of decentralized estimation with single processor
Fig. 3 shows the decentralized estimation system. W is
the parameter of interest, which is measured by local pro-
cessor. W is derived from a prior distribution PW . Given
W = w, the sensor can get a sample X generated from the
distribution PX|W =w. Typically, the sensor gets n samples
independently according to the distribution PXn|W =w, where
Xn = {X1, X2, ......, Xn}. After that, the system uses quan-
tification function ϕQ to map the observed message Xn into
a b-bit message
Y = ϕQ(Xn)
(15)
Then the encoder uses the coding function ϕE to transform Y
into codeword
U = ϕE(Y )
(16)
U is transmitted through a noisy channel and received as V .
Finally, the estimator calculates
ˆW = ψ(V )
(17)
as the estimates of W. Acquiring this estimation model, we
can use it to estimate our formation bias. Before this, we
should fit our problem to this model.
* W is defined as the robot’s distance variable from its present
destination and w is its value. We consider robots in the
circle Rm. Assuming w satisfies w ∈[0, l0] in practical
system, we have l0 < R.
* Xn are n independently drawn samples of W. The sensor
on the robot measures the distance r for n times. The n
independent measures are denoted as {x1, x2, ......, xn}.
* After acquiring the measurement, the system will use curves
(8) to divide R and locate the measured distance into one of
the regions. This is a process of quantification. The number
of divisions nr represents the quantization precision. The
result is b-bit length Y in the system.
* In our system, the controller can gain direct access to Y .
There is not process through the noisy channel. So we don’t
need to make the channel coding and transmit it to the
receiver. Therefore we have V = U = Y .
Before the analysis, the relative distributions and equations
should also be defined.
PW
is a prior distribution of W. In this system, W
represents the distance r. As mentioned above, robot i is in
circle Rm, whose center is the robot’s present destination. As
mentioned above, we assume W ∈[0, l0]. l0 is an empirical
constant of the system. To ensure the quantification being
effective, there is R > l0. To calculate PW , we assume that
robot i has an equal possibility to be located at each point in
the circle with the radius l0. Therefore
P(W < r) = Sr
Sl0
= πr2
πl2
0
= r2
l2
0
(18)
where P(W < r) represents the probability that the robot’s
distance to its destination is smaller than r. Sr and Sl0
respectively represents the acreage of the circle with the radius
r and l0.
Acquiring the distribution function, we need to calculate the
density function.
P(W = r) = dP(W < r)
dr
= 2r
l2
0
(19)
PX|W represents the distribution of measurement X con-
ditioned on W. Given W = w, the measurement random
6
variable X is generated according to the distribution PX|W =w.
In this system we choose Gaussian distribution as PX|W =w.
PX|W =w =
1
√
2πσ2 × e
(X−w)2
−2σ2
(20)
where the real value of the parameter is w, which is also the
mean value of the Gaussian distribution. σ is the measurement
variance.
In the considered system, it assumes that the sensor mea-
sures the distance for n times independently. The mean and
variance value are with the same w and σ. In this case, a joint
Gaussian distribution is applied here to represent PXn|W =w.
PXn|W =w =
1
(
√
2πσ2)n × e
Pn
1 (xi−w)2
−2σ2
(21)
The measured distance is not directly used by the controller.
It has to be quantified first. The distance r satisfies r ∈[0, l0]
and there is l0 < R, then [0, R] is the quantitative range.
The range is divided into nr regions, so the quantization bit b
satisfies
b = log2(nr)
(22)
In this way, the distributed estimation system is applied
to the formation bias estimation problem. Then we can use
theories in distributed estimation to give a lower bound on the
Bayes risk of the estimation. This will guide us to choose the
system’s parameter.
C. Lower Bounds Estimation
As shown in Fig. 3, the parameter of interest is W, which
is derived from a prior distribution PW . After sampling and
quantification, the parameter is estimated as ˆW = ψ(V ). By
using a non-negative distortion function l : W × W →R+,
we can define the Bayes risk of estimating ˆW as
RB = inf
ψ
E(l(W, ψ(V )))
(23)
where the distortion function l is of the form l(w, ˆw) = ||w −
ˆw||q and q ≥1. In this paper, W represents the distance r.
Therefore function l is defined on R1. In [25], it provides
lower bounds on the Bayes risk for estimation. Then based
on the above model and distributions, we can obtain a lower
bound of RB and derive the formation’s bias.
Theorem 1. In a multi-robot formation control system, we
measure the distance n times with a measuring variance σ.
The radius of controlling circle Rm is R and the quantification
rate is set as b. If the measured distance satisfies r ∈[0, l0],
we can derive the lower bound of RB:
RB ≥l0
2emax







2−(log nl0
2σ + 1
2 −1
2 log(2πe)), 2
−

1−e
−nl2
0
2σ2

b







(24)
where l(w, ˆw) = ||w −ˆw||.
Remark 2. The above RB describes measuring bias of the
distance. This bias describes the robot’s deviation from its
right position in the formation. The whole formation bias is
the mean value of each robot’s deviation as defined in (14).
Therefore the lower bound of RB can be viewed as the lower
bound of formation bias.
VI. PROOF OF LOWER BOUND
In this section, we give proof of the lower bound in Theorem
1. In [25], the author gives theorems to estimate lower bound
of Bayes risk. We introduce the theorems as lemmas to analyze
the formation bias.
Lemma 3. For any arbitrary norm ∥·∥and any q ≥1, when
the parameter of interest W ∈Rd and W is distributed in
[0, 1]. The Bayes for estimating the parameter based on the
sample X with respect to the distortion function l(w, ˆw) =
∥w −ˆw∥r satisfies
RB ≥
sup
PU|W,X
d
qe

VdΓ(1 + d
q )
−q
d
2−(I(W ;X|U)−h(W |U)) q
d
(25)
To make problems simple, when we are estimating a real-
valued W with respect to l(w, ˆw) = ∥w −ˆw∥,
RB ≥
sup
PU|W,X
1
2e 2−(I(W ;X|U)−h(W |U))
(26)
Considering the unconditional version, there’s a simpler form
RB ≥1
2e 2−I(W ;X)
(27)
In fact, (27) is an unconditional version of lemma 3, which
can give us a lower bound of RB. This lower bound is for the
case W ∈[0, 1]. If the distribution range is changed, we can
multiply a parameter to correct it.
In the single processor estimating system displayed by Fig.
3, the estimator can’t gain direct access to the measurement
Xn. Xn is quantified and transmitted to the estimator. In fact,
the estimation is based on V . So we can use unconditional
version of lemma 3 to obtain a lower bound of Bayes risk
RB by replacing I(W; X) with I(W; V ). Now, we need to
calculate an upper bound of I(W; V ).
Lemma 4. In decentralized estimation with a single processor,
for any choice of φQ and φE, there is
I(W, V ) ≤min{I(W, Xn)ηT , η(PXn, PW |Xn)(H(Xn) ∧b)ηT ,
η(PXn, PW |Xn)CT}
(28)
where C is the Shannon Capacity of the noisy channel PV |U,
and
ηT =



1 −(1 −η
PV |U)

T
with feedback
η

P
N T
V |U

without feedback
(29)
In (29), T is the time spent transmitting a message Y
through the channel PV |U, and η is a constant.
On the calculation of η, it can be described as follows. Given
a channel K whose input alphabet is X and output alphabet
is Y . There is a reference input distribution µ on X. If for a
constant c ∈[0, 1) and any other input distribution ν on X,
there is D(νK||µK) ≤cD(ν||µ), we say K satisfies an SDPI
at µ.
7
The SDPI constant of K at input distribution µ is defined
as
η (µ, K) = sup
ν:ν̸=µ
D (νK||µK)
D (ν||µ)
(30)
The SDPI constant of K is defined as
η (K) = sup
µ η (µ, K)
(31)
With Lemma 3, one can obtain a lower bound of Bayes
risk RB relative to the mutual information I(W, V ). With
Lemma 4, one can calculate an upper bound of I(W, V ) in
our formation bias estimating system based on the model in
section V.B, and then derive the result of Lemma 3.
In our considered formation bias estimation model, the
quantified result Y is directly used for estimation. So the
transmitting channel can be viewed as lossless. From the
definition of η (30) (31), we can obtain
η(PV |U) = 1
(32)
Then according to the definition (29), there is
ηT = 1
(33)
Besides, our model considers no channel loss. That is, the
channel capacity C is infinite. In (28), the upper bound of
I(W, V ) is the minimum value of three parts. The last part
η(PXn, PW |Xn)CT is related with C. Therefore it can be
ignored. Then the upper bound (28) can be simplified to the
following
I(W, V ) ≤min{I(W, Xn), η(Pxn, PW |Xn)(H(Xn) ∧b)}
(34)
In this considered system, Xn is the measurement of the
distance between the robot and its present destination. It
contains n samples taken from the sensor. Its element Xi
satisfies Xi ∈[0, l0]. This is a continuous variable. When
quantifying it with b bit, it will certainly result in some bias.
So H(Xn) is larger than b. That is,
H(Xn) ∧b = b
(35)
Therefore the upper bound (34) can be simplified further as
I(W, V ) ≤min{I(W, Xn), η(PXn, PW |Xn)b}
(36)
Next
we
should
separately
estimate
I(W, Xn)
and
η(Pxn, PW |Xn)b to finally obtain the upper bound.
Clarke [26] shows that
I (W, Xn) = d
2log
 n
2πe

+ h (W) + 1
2E

log det JX|W (W)

+o (1)
(37)
where h(W) is the differential entropy of W, and JX|W (W)
is the Fisher information matrix about w contained in X.
From (19), we get the density function of W
p(W) = 2W
l2
0
(38)
The differential entropy of W is
h(W) = E(−log(p(W)))
=
Z l0
0
−2W
l2
0
log(2W
l2
0
) dW
= 1
2 −log 2
l0
(39)
Now, we need to calculate the fisher information. The fisher
information can be written as
det JX|W (W) = −E[ ∂2
∂W 2 log
PX|W


]
(40)
From (21), P(X|W) is a joint Gaussian distribution of n
independent samples. Then we have
log
PX|W


= −n
2 log
2πσ2

−
1
2σ2
n
X
i=1
(xi −W)2
(41)
Then
det JX|W (W) = −E[ ∂2
∂W 2 (−1
2σ2
n
X
i=1
(xi −W)2)]
=
1
2σ2 E[ ∂
∂W (−2
n
X
i=1
(xi −W))]
=
1
2σ2 E[2n]
= n
σ2
(42)
In this system, since W ∈R1, we have d = 1. By using
(39) and (42), one can obtain an estimation of I(W, Xn).
I (W, Xn) = 1
2log
 n
2πe

+ 1
2 −log
 2
R

+ 1
2log
 n
σ2

+ o(1)
= log nR
2σ + 1
2 −log 2πe + o(1)
(43)
The first part I (W, Xn) is estimated. We shall estimate the
second part of (36). The critical part of this is to estimate
η(PXn, PW |Xn). Here, a lemma in [25] can help us achieve
it.
Lemma 5. For a joint distribution PW,X, suppose there is
a constant α ∈(0, 1] such that the forward channel PX|W
satisfies
dPX|W =w
dPX|W =w′ (x) ≥α
(44)
for all x ∈X and w, w
′ ∈W.
Then the SDPI constants of the forward channel PX|W and
the backward channel PW |X satisfy
η
PX|W


≤1 −α
(45)
and
η
PW |X


≤1 −α
(46)
In the considered system, we measure the distance indepen-
dently for n times and get Xn. So we replace η
PW |X


with
η
PW |Xn

. From (31), we have
η
PW |Xn

≥η(PXn, PW |Xn)
(47)
8
Then if (46) holds, we have
η(PXn, PW |Xn) ≤η
PW |Xn

≤1 −α
(48)
Therefore the critical step of this part is to estimate α.
From (21), PX|W =w is a joint Gaussian distribution, with
w as its mean. Then we can use the density function (21) to
calculate
dPX|W =w
dPX|W =w′ (x) =
1
(
√
2πσ2)n e
Pn
i=1 −(xi−w)2
2σ2
1
(
√
2πσ2)n e
Pn
i=1 −(xi−w′)
2
2σ2
= e
Pn
i=1

xi−w
′ 2
2σ2
−Pn
i=1
(xi−w)2
2σ2
= e
Pn
i=1

w−w
′ 
2xi−w−w
′ 
2σ2
(49)
As each measurement xi is obtained independently. There-
fore finding the minimum value of

w −w
′ 
2xi −w −w
′
is equivalent to the calculation of minimum value of (49).
h = min
n
w −w
′ 
2x −w −w
′o
(50)
where x ∈[0, l0] and w, w
′ ∈[0, l0].
It is easy to see that

w −w
′ 
2x −w −w
′
= 2(w−w
′)x−(w2 −w
′2) (51)
When w > w
′, (51) increases as x increases. So the minimum
value is obtained as x = 0. Then we have
h = −(w2 −w
′2)
(w > w
′)
(52)
For w, w
′ ∈[0, l0], we have
h = −l2
0
(w = 0, w
′ = l0)
(53)
Similarly, when w < w
′, (51) decreases as x increases. So
the minimum value is obtained as x = l0. Then we have
h = 2(w −w
′)l0 −(w2 −w
′2)
(54)
To calculate the minimum value, we transform (54) as the
following.
h = w
′2 −2w
′l0 −w2 + 2wl0
(55)
(55) can be viewed as a quadratic function of w
′. The
minimum value can be reached as w
′ = l0, for w < w
′.
That is, we have
h = −l2
0 −w2 + 2wl0
(w
′ = l0)
(56)
(56) can be further viewed as a quadratic function of w. The
minimum value can be reached as w = 0. Therefore
h = −l2
0
(w = 0)
(57)
Together with (53) and (57), we have
min
n
w −w
′ 
2x −w −w
′o
= −l2
0
(58)
Then there is
Pn
i=1

w −w
′ 
2xi −w −w
′
2σ2
≥−nl2
0
2σ2
(59)
Together with (49), we have
dPX|W =w
dPX|W =w′ (x) ≥e−
nl2
0
2σ2
(60)
That is, (see (44))
α = e−
nl2
0
2σ2
(61)
Finally, from Lemma 5 and (48), we have
η(PXn, PW |Xn) ≤1 −e−
nl2
0
2σ2
(62)
Now we have finished estimating the second part of (36). We
can get a conclusion
I (W, V ) ≤min

lognl0
2σ + 1
2 −1
2log (2πe) ,

1 −e−
nl2
0
2σ2

b

(63)
In (27), we replace I(W, X) with I(W, V ). The lower bound
in (27) should be corrected by multiplying l0. According to
the upper bound given in (63), the final lower bound of the
Bayes risk can be obtained
RB ≥l0
2emax







2−(log nl0
2σ + 1
2 −1
2 log(2πe)), 2
−

1−e
−nl2
0
2σ2

b







(64)
Therefore the proof of theorem 1 is completed.
VII. FORMATION CONVERSION CONTROL
In the practical use, robots should not only form a formation,
but also change to another one to accomplish a different task.
The formations are usually planned according to the routine
requirements. This includes several different cases.
* Robots reach a formation with a leader chosen by the user.
* Robots reach a formation with a leader at the leading place.
The leader is chosen by the system. The leading place is
determined by the formation.
* Robots reach a formation around a center chosen by the
planner according to its requirement.
* Robots reach a formation around a center chosen by the
system to optimize the process.
In the following discussion, we assume the new formation’s
area should be the same as the former one.
In this part, we mainly discuss the transformation between
square formation, triangle formation and circle formation. The
method can also be applied to other transformations. Besides,
we also give a moving strategy for a given center by the
planner and a method to adaptively choose a center. We also
confirm that the chosen center can minimize the total moving
distance.
A. Triangle Formation
Fig. 4 shows the structure of a triangle formation with its
center on the origin. This is a isosceles triangle, with equal
numbers of robots on its two waist edges. From geometry, we
know AO = 2OD. Excluding robots on the three vertices, we
assume there are x robots on one waist edge and y robots on
9
Fig. 4.
a triangle formation with robots stand on the black points along the
edges
the bottom edge. The total number of robots is n, then we
have
3 + 2x + y = n
(65)
Therefore given y, we can have the arrangement of robots.
Given the area of the formation as a × b, we can set
AD = a BC = 2b
or
AD = b BC = 2a
Then the position of A,B and C can be determined to locate
every point in the formation.
B. Circle Formation
Fig. 5.
a circle formation with robots stand on the arc evenly
Fig. 5 shows a circle formation with its center at the origin.
Given the formation’s area S, we can calculate the radius by
r =
q
S
π . Then the positions of all points can be determined
by partitioning the whole circle equally into n sections.
C. Transformation Strategies
Given a new formation, the robots need to move from the
present one to it.
In Section III and Section IV, we have discussed methods
to arrange robots in a given formation for arbitrary initial
positions for robots and move to reach the formation.
1) Center given by planner: If the center is given and is
not far, the robots can move to the proper positions around
the center. But if the center is too far, this may be very costly.
We should first reduce the distance between the desired center
and those robots.
Assuming robots’ position at time slot k is
X(k) = {x1(k), x2(k), ......, xn(k)}
Their present center is
Cxp = 1
n
n
X
i=1
xi(k)
(66)
while the desired center given by planner is denoted as Cx.
Then we let all robots move towards the direction of Cx−Cxp
at a given speed. Given a threshold d0, if there is
||Cx −Cxp|| ≤d0
(67)
the robots are close enough to the center. Then They follow
method in section III and section IV to reach the formation
around the center.
2) Center determined by the system: If the planner don’t
give the position of the center, the system need first find a
center. From (3), the expected moving cost for reaching the
formation is
C =
n
X
i=1
||(xi(0) −fdi) −Cx||2
(68)
We need to choose the center to minimize C.
Theorem
2.
For
n
robots
with
the
initial
position
X(0) = {x1(0), x2(0), ......, xn(0)} and a formation F =
{f1, f2, ......, fn} which satisfies Pn
i=1 fi = 0, when choosing
the center as
Cx = 1
n
n
X
i=1
xi(0)
(69)
the moving cost C is minimized
Proof: Defining a function F of Cx as
F(Cx) = C =
n
X
i=1
||(xi(0) −fdi) −Cx||2
(70)
then
dF
dCx
=
n
X
i=1
−2(xi(0) −fdi −Cx)
= −2[
n
X
i=1
xi(0) −
n
X
i=1
fdi −nCx]
(71)
For Pn
i=1 fi = 0, we have
dF
dCx
= −2[
n
X
i=1
xi(0) −nCx]
(72)
Let
dF
dCx = 0, we have
Cx = 1
n
n
X
i=1
xi(0)
(73)
and it’s obviously that this is the minimum point.
10
According to Theorem 2, one can choose the optimal center
and follow the methods in Section III and Section IV to reach
the formation.
VIII. SIMULATIONS
In this section, we shall present some simulations for the
multi-robot formation control. At the beginning, we present
a group of robots reaching a square formation with a leader.
Afterwards, we provide a display of robots in square formation
changing to circle formation, then to triangle formation, and
finally to a circle formation with a far away center. Then the
moving cost of our position arrangement strategy in Section
III is compared with other position arrangement strategies
by simulations. At last, we discuss the relations between
formation bias and system parameter. All simulations are run
on MATLAB.
A. Formation Illustration
In this simulation, all robots are in a 1100 × 1100 square
area. The number of robots n is set to be 15, and their initial
positions are generated randomly in a 300 × 300 area.
We set the radius of Rm as R = 300, the quantification
rate as b = 7, the area of the formation as S = 28800 and the
variance of measuring the distance as σ = 1. The estimated
total moving cost is calculated by definitions in (2) and (3),
while the practical moving cost is calculated with simulated
data.
(a)
(b)
(c)
(d)
Fig. 6.
Sub-figures (a) shows a group of robots in random initial position.
(b)-(c) show the robots begin to move according to the calculated mapping
arrangement. (d) shows robots having reached a square formation.
1) Reaching a square formation from random initial po-
sitions: In Fig. 6(a), robots’ initial positions are randomly
distributed. By applying methods in section III, the system
selects a leading robot and arranges each robot in a position
of the formation. Then robots move and reach the square
formation. Finally, the square formation is reached as shown
in Fig. 6(d).
(a)
(b)
(c)
(d)
Fig. 7. Sub-figures (a) shows a group of robots in a square formation. (b)-(c)
show the robots begin to move according to new mapping in circle formation.
(d) shows robots having reached a circle formation.
(a)
(b)
(c)
(d)
Fig. 8.
Sub-figures (a) shows a group of robots in a circle formation. (b)-
(c) show the robots begin to move according to new mapping in triangle
formation. (d) shows robots having reached a triangle formation.
In this example, the estimated cost is 72000 and the practical
cost is 150860. In fact, when reaching the square formation,
each robot chooses another one in the nearby position as the
reference. It adjusts to stand at the right position relative to
the reference. For the robot referred to is also moving, the
destination keeps moving as well. Then the robot’s practical
path is far from straight. That is why the practical path causes
an extra distance compared to the estimated one resulting from
calculating by using straight line distance.
2) Conversion from square formation to circle formation:
The process of transformation from square formation to circle
formation is shown in Fig. 7. The user doesn’t give the center.
The system determines the center’s position according to
11
(a)
(b)
(c)
(d)
Fig. 9.
Sub-figures (a) shows a group of robots in a triangle formation
preparing to reach a formation around a center given by the user. (b) shows
the robots is moving together near to the given center. (c) shows robots are
moving to reach a formation around the given center. (d) shows robots having
reached a circle formation around the center.
Theorem 2. Then the system arranges each robot to a position
in the circle formation. Finally, robots move and reach the
circle formation as shown in Fig. 7(d).
In the example, the estimated cost is 27211, while the
practical moving cost is 27470. In the process, each robot
moves referring to the fixed center. Regardless of the unfre-
quent collision avoidance, the practical path is nearly straight.
Therefore the estimated cost is a little larger than the practical
one. The collision avoidance also results in an extra moving
cost.
3) Conversion from circle formation to triangle formation:
In Fig. 8, robots transform gradually from circle formation
to triangle formation. There is no center given by the user.
The system first determines the center of the formation and
obtains the triangle formation according to section VII.A. Then
the system completes arrangement of robots in the formation.
Finally, robots move and reach the triangle formation as shown
in Fig. 8(d).
In this example, the estimated cost is 15642, while the
practical cost is 15652. During this transformation, there is
nearly no collision avoidance. Therefore the practical cost is
nearly the same as the estimated one.
4) Reaching a circle formation around a faraway center:
In Fig. 9(a), robots are in the triangle formation. User gives
a center (-100,0). Robots first move together towards the
center as shown in Fig. 9(b). When they are near the desired
positions, the system calculates the arrangement of robots.
Then robots move to reach the circle formation as shown in
Fig. 9(c). Finally they reach the circle formation in Fig. 9(d).
In this example, the estimated cost is 2.71 × 106, while the
practical moving cost is 2.76×106. This includes the extra cost
caused by collision avoidance. Furthermore, robots move near
the center rather than move directly towards the right position
relative to the center. This causes an extra cost compared with
the straight distance.
B. Moving Cost comparison
To confirm the advantage of our proposed position arrange-
ment strategy in reducing the moving cost, we compare it with
other strategies. One strategy is called fixed position moving
mode, where each robot is arranged in a fixed position in
the formation. Another one is referred to as random position
moving mode, where robots are arranged randomly in the
formation. We do simulations and obtain the moving cost for
each method. The comparison is done for two different cases.
We first do experiments of reaching a square formation with
a leader from random initial positions. Then we simulate the
case of reaching a circle formation with a center. For each
case, we do 200 experiments and compare the moving cost of
the three strategies. In this simulation, our system parameter
is the same as the one in Section VIII.A.
1) moving cost for square formation: As shown in Fig. 10,
the moving cost of our proposed position arrangement is lower
than the other two methods. Furthermore, its moving cost
is more stable. Therefore the proposed position arrangement
based on the Hungarian method can improve the system
performance both in moving cost and in stability.
Fig. 10.
Comparison of the moving cost of different position arrangement
strategies for reaching the square formation. The graph represents moving
cost of the three methods in different experiments.
2) moving cost for circle formation: As shown in Fig. 11,
in the case of reaching the circle formation with a center, the
result is similar as that presented in Fig. 10. Our arrangement
strategy based on the Hungarian method can improve the
system performance compared to other two methods.
C. Formation bias analysis
To investigate the offsets of the theoretical lower bound on
the formation bias for the circle formation process, we carry
out simulations to investigate the change of formation with
respect to three different system parameters. The experimental
data is fitted with the least squares method. The calculated
lower bound is also depicted.
In each simulation, we adjust the conditioning parameter in
a previously set range. In each parameter value, we carry out
12
Fig. 11.
Comparison of the moving cost for different position arrangement
strategies for reaching a formation around a center. The graph represents
moving cost of the three methods in different experiments.
20 experiments. We let robots be in random initial positions
and then complete reaching the circle formation to obtain the
formation bias for each experiment.
1) Measuring times n: In this simulation, the formation area
is S = 28800 and the measuring variance of angle is 0.05.
There are 15 robots whose measuring variance of distance is
2. The radius of the system’s controlling circle Rm is 200.
The distance’s partition number is 200. Converting it to the
quantification rate, we have b = 7.64. The measuring number
ranges from 1 to 60. In theoretical calculation, the distance’s
empirical ranging threshold is l0 = 120.
Fig. 12(a) shows the change of formation bias with respect
to measuring times n. All curves decrease as n increases.
From the point of common sense, when n is small, increasing
the measuring times can reduce the formation bias apparently.
When n is large enough, increasing n can’t obtain obvious
evolution. The tendency of curves is consistent with common
sense. The red curve represents experimental data and the
black curve is the fitting result with least square method.
The blue curve is the calculated lower bound according to
Theorem 1. In the beginning, the theoretical result is a little
larger than experimental data. This is due to less statistics be
done. However, in the rest majority part, the lower bound is
below the experimental data and the tendency is the same.
2) Distance measuring variance σ: In this simulation, the
formation area is S = 28800 and the measuring variance of
angle is 0.05. There are 15 robots. Each robot measures the
distance for 10 times. The radius of the system’s controlling
circle Rm is 200. The distance’s partition number is 200.
Converting it to the quantification rate, we have b = 7.64.
The distance’s measuring variance ranges from 0.1 to 1.3.
In theoretical calculation, the distance’s empirical ranging
threshold is l0 = 120.
Fig. 12(b) shows the change of formation bias with respect
to distance measuring variance σ. In general, all curves are
increasing with σ. From the point of common sense, when the
measuring variance increases, the measuring bias increases and
causes the formation bias to increase. The tendency of curves
is consistent with common sense. The theoretical lower bound
(a)
(b)
(c)
Fig. 12. Sub-figures (a) shows the formation bias decreases as the measuring
number n increases. It depicts the experimental data, the fitted curve and
the calculated lower bound. (b) shows the formation bias increases as
the distance’s measuring variance increases. (c) shows the formation bias
decreases as the quantification rate b increases.
13
is below the experimental data and the tendency is the same.
3) Quantification rate b: In this simulation, the formation
area is S = 28800 and the measuring variance of angle is 0.05.
There are 15 robots. Each robot measures the distance for 10
times. The radius of the system’s controlling circle Rm is 200.
The measuring variance of distance is 0.01. The distance’s
partition number ranges from 50 to 200. Converting it to the
quantification rate, we have b ∈[5.64, 7.64]. In theoretical
calculation, the distance’s empirical ranging threshold is l0 =
120.
Fig. 12(c) shows the change of formation bias with respect
to the quantification rate b. All curves increase with b. From
the point of common sense, when b increases, the measuring
bias decreases, causing the formation bias to decrease. When
b is large enough, the decreasing tendency slows down. The
tendency of curves is consistent with common sense. The
lower bound is below the experimental data. Though there
is a gap, the tendency is the same.
IX. CONCLUSION
In this paper, we discussed the control of multi-robot forma-
tion and analyzed the system theoretically. First, we presented
a method to arrange robots to positions in a formation by
optimizing the total moving distance. We formulated the
problem into a task distribution optimization problem and
solved it in Hungarian method. Then we employed a leader-
follower method to achieve the formation. Afterwards, we
defined the formation bias and presented a lower bound of
it, which can help us determine the parameter of the system.
Besides, we also discussed control in formation changing. We
also carried out various simulations to investigate the effects
of system parameters on the formation bias.
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