Decentralized Cooperative Multi-Robot Localization with EKF
Ruihua Han, Shengduo Chen, Yasheng Bu, Zhijun Lyu and Qi Hao*
Abstract— Multi-robot localization has been a critical prob-
lem for robots performing complex tasks cooperatively. In
this paper, we propose a decentralized approach to localize
a group of robots in a large featureless environment. The
proposed approach only requires that at least one robot remains
stationary as a temporary landmark during a certain period of
time. The novelty of our approach is threefold: (1) developing
a decentralized scheme that each robot calculates their own
state and only stores the latest one to reduce storage and
computational cost; (2) developing an efficient localization
algorithm through the extended Kalman filter (EKF) that only
uses observations of relative pose to estimate the robot positions;
(3) developing a scheme has less requirements on landmarks
and more robustness against insufficient observations. Various
simulations and experiments using five robots equipped with
relative pose-measurement sensors are performed to validate
the superior performance of our approach.
I. INTRODUCTION
In the study of autonomous mobile robots, localization
is one of the most fundamental problems [1]. Mobile robot
localization, as known as position estimation, is the process
of determining the pose of a robot relative to a given map
of the environment. Compared to a single robot, multi-robot
systems are more robust and efficient to perform complex
tasks within complex environments, including rescue and
disaster management, surveillance and monitoring, and so
on [2]. For the autonomous multi-robot systems, the perfor-
mance of most functionalities heavily relies on the accuracy
of robot localization results, where individual localization
results easily interfere with each other.
The main multi-robot localization approaches can fall into
three categories: (1) external, (2) internal, and (3) coopera-
tive. The external methods help each robot to acquire their
position information independently through those external
active landmarks such as Global Position Systems (GPS)
or Ultra-Wideband (UWB) systems. The localization uncer-
tainty of the external methods can be bounded only when
those external signals are reliable and available. The internal
methods enable individual robots to localize their positions
by utilizing the sensors on each robot such as camera or laser
scanner to detect the features of the environment. However,
these methods can not work well in those environments
without rich feature information. The cooperative methods,
also known as cooperative localization (CL), allow a group
*This work was supported by the Science and Technology Innovation
Commission of Shenzhen Municipality (No. CKFW2016041415372174
and No. GJHZ20170314114424152), and the National Natural Science
Foundation of China (No. 61773197).
The authors are with Department of Computer Science and En-
gineering,
Southern
University
of
Science
and
Technology,
Shen-
zhen, Guangdong, 518055, China. {hanrh@mail.sustc.edu.cn
haoq@sustc.edu.cn} *Corresponding author
t 
1
2
3
4
5
(a)
t 
1
2
3
4
5
(b)
t 
1
2
3
4
5
(c)
t 
1
2
3
4
5
(d)
Fig. 1.
Illustration of the approach. The example during (a) denotes at the
time t, one robot is randomly chose to remain stationary, firstly, the nearest
moving robot 1 detect the stationary one to estimate its pose. During (b), the
next nearest robot to robot 1 estimate its pose by the two observation. During
(c) and (d) the robot estimate their pose by the observation of previous robot.
of communicating robots to estimate their states jointly
using relative observations of each other. Those methods
are advantageous in their low cost and high scalability,
receiving increasing attentions recently, especially for those
robot systems working in featureless environments.
However, many technical challenges need to be overcome
in using cooperative localization methods. Generally speak-
ing, the estimation accuracy in featureless environments is
inferior to that in environments with rich feature information.
Besides, as the group of robots becomes larger, the cost
of computation and communication grows much higher. In
practice, only using relative observations of each other may
not be enough to localize multiple robots given various
uncertainties and noises.
A number of approaches have been proposed to ad-
dress above challenges. Dead-reckoning(DR) is one common
method which estimates the robot pose from its previous
pose information by using proprioceptive sensors [3], but the
estimation errors will be accumulated over a long distance.
To improve the accuracy and consistency of estimated robot
states, various data fusion algorithms have been developed,
such as Markov localization (ML), extended Kalman filter
(EKF) localization and Monte Carlo localization (MCL) [4].
The main idea of these schemes is to optimally combine
the data derived from proprioceptive and exteroceptive sen-
sors and filter out the motion and measurement noises to
reduce the uncertainty of robot pose estimation. Usually, the
dependency on landmarks limits the applications of those
approaches.
arXiv:1811.07506v1  [cs.RO]  19 Nov 2018
Generally, the schemes to calculate the data fusion in CL
are either centralized, multi-centralized or decentralized [5].
In a centralized system, the computations for all robot data
are processed in one central robot. Central robot provides
feedback and required sensory information from all other
robots [6]. The major disadvantages of this scheme include
the high expense of computation and the high vulnerability
of the whole system. In a multi-centralized system, there
are more than one robot performing as the central robot to
process all the data of entire group [7]. Consequently, this
system needs higher computational power. The decentralized
system means each robot estimates its own pose respec-
tively through processing their own sensors data and updates
it locally. Nonetheless the problem of double-counting or
data incest may occur when two uncorrelated robots share
information [8]. The decentralized system yield the least
computational cost and highest system scalability.
In this paper, we present a decentralized approach for CL,
the approach is illustrated as Fig. 1. In this approach, at each
time instant only one random robot in the group remains
stationary, while the other moving robots estimate their state
by fusing the data of odometry and observations of relative
observation between two robots. The rearward moving robots
utilize the observation with the forward moving robots to
improve the accuracy and stability of the estimator. And the
order depends on the relative distance. The contributions of
this work include:
1) A decentralized scheme for CL is proposed, where
each robot estimates their own state that reduces the cost
of computation.
2) An EKF Based algorithm is proposed, where each robot
only maintains its own pose and corresponding covariance,
and the communication between two robots only occurs
when one robot detect another at each time instant, yielding
a computational complexity of O(N).
3) This approach only requires the odometry data and rela-
tive pose measurements so that it can perform in GPS-denied
featureless environments. Even though there are insufficient
relative pose measurements, the proposed approach can still
guarantee a moderate performance.
The rest of this paper is organized as follows. Section
II describes the related work. Section III presents the sys-
tem setup and the localization problem. Section IV gives
the details of the proposed algorithm. Section V provides
simulation and experiment results and discussions. Section
VI concludes the paper and outlines future works.
II. RELATED WORK
In the problem of multi-robot localization, the study of CL
receives considerable attention for a long period, because of
its scalability and superior performance. Many approaches
on CL have been proposed based on various schemes and
algorithms as mentioned above. In this section, we describe
several relevant approaches on CL to highlight the novelty
of this work.
To estimate the state of robots with high accuracy and
consistency, various filter algorithms have been tested for CL.
An EKF-based algorithm for heterogeneous outdoor multi-
robot localization is described in [9], where individual robot
equipped with encoders and a camera maintains an estimated
pose by using sensor data fusion through EKF. An improved
EKF on CL is studied in [10]. Additionally, a recent approach
that called recursive decentralized localization based on EKF
is presented in [8]. The proposed algorithm approximates
the inter-robot correlations and performs with asynchronous
pairwise communication. Other algorithms such as particle
filter (PF) [11] and maximum a posteriori (MAP) [12] are
also extensively studied for CL. The main limitation of these
approaches is that in certain case, none of the robots in
group has an access to the absolute state information that
will reduce the estimation accuracy and consistency. In our
approach, the stationary robot at each time can improve the
accuracy efficiently by an optimized EKF algorithm.
To reduce the cost of computation, various decentralized
schemes are proposed. Luis C et al. present a Covariance In-
tersection (CI)-based algorithm for reducing the complexity
of CL [13], where each robot locally maintains its state and
covariance. Amanda et al. provide a PF-based algorithm that
is completely decentralized and a low-cost particle clustering
method to reduce the computational complexity [14]. In a
decentralized system, it is important to avoid the problem
of double-counting or data incest. Masinjila et al. presents a
heuristically tuned EKF to address this problem [5], which
proposes an empirical methodology for improving the consis-
tency of EKF estimates by means of artificial inflation of the
landmark covariance. This problem is also discussed in [15]
and [8]. Our approach also employs the similar decentralized
scheme to reduce the system complexity.
The approach of CL with stationary robots was first
studied by Kurazume et al. [16] [17]. The author proposes a
method called Cooperative Positioning System(CPS). Within
the CPS, a team of robots are divided into a stationary group,
which acts as landmarks, and a mobile group. The roles of
groups would reverse at the next time instant until the team
reaches the destination. More related issues are discussed in
[18] and [19]. A leader-assistive approach is presented in
[20] and [21], where a team of robots are divided into two
groups, the performance-plus robots act as leader and the
rest as child. The leaders have high-resolution estimators and
help to localize the child robots. Such an approach is further
extended in [22]. The main limitation of these approaches is
that too many constrains(such as the order of movement) are
required on the robot motion. In our approach, each robot
moves with its own path.
III. SYSTEM SETUP AND PROBLEM STATEMENT
Consider a group of N robots navigating in 2D envi-
ronment, each robot is equipped with proprioceptive and
exteroceptive sensors. The state of individual robot at time
t is denoted as Xt = [xt, yt, θt]T , which contains the
information of position and orientation. In this approach,
we adopt the velocity motion model with the assumption
that a mobile robot can be controlled through a rotational
and translational velocity denoted by vt and ωt respectively.
Thus, the control vector is denoted as ut = [ vt
ωt ]T . It
derives from the proprioceptive sensors such as encoder. On
the basis of motion model, at time t, the state of robot can
be presented as following equation:
Xt = g (ut, Xt−1) + N (0, Rt)
g (ut, Xt−1) = Xt−1 +


−vt
ωt sin θt−1 + vt
ωt sin(θt−1 + ωt∆t)
vt
ωt cos θt−1 −vt
ωt cos(θt−1 + ωt∆t)
ωt∆t


(1)
where ∆t is the duration of a time step, N (0, Rt) is the
motion noise in an additive Gaussian with zero mean and
covariance matrix Rt .
Additionally, when one robot detects another by extero-
ceptive sensors at time t, the relative measurement zt =
 rt
φ t

T , rt and φt are relative distance and bearing
respectively between two robots. The measurement model
for robot i measuring robot j at time t is
zi,j
t
= h

Xi
t, Xj
t

+ N (0, Qt)
h

Xi
t, Xj
t

=
 
q
(xj
t −xi
t)
2+(yj
t −yi
t)
2
a tan 2(yj
t −yi
t, xj
t −xi
t) −θi
t
!
(2)
Here, Xi
t and Xj
t are the actual state of robot i and robot
j. N (0, Qt) is the additive Gaussian noise with zero mean
and covariance matrix Qt.
At each time step t, each robot is provided with an estimate
¯Xt of its own state and the corresponding covariance matrix
¯Σt which represents the uncertainty. The available data
is odometry of each robot and all relative measurements
between each two robots. For centralized scheme, center
robot stores and processes all data including the state and
measurement of each robot. It can be denoted as:
X = [ˆx1, ˆx2, ˆx3, ˆx4....ˆxi]
Z = [z1,2, z3,5......zi,j]
[¯x1, ¯x2, ¯x3, ¯x4....¯xi] = filter(X, Z)
(3)
For our decentralized scheme, each robot i only stores its last
state, odometry and measurement to estimate current state.
Depending on the odometry and motion model, each robot i
has a predicted state:
¯Xi
t = g
ui
t, ¯Xi
t−1


(4)
When the robot i have several relative measurements with
n robots. The objective of our approach is to find a proper
estimator which meets the following equation:
¯Xi
t = arg min
n
X
j=1
(zi,j
t
−¯zi,j
t )
= arg min
n
X
j=1

zi,j
t
−h

¯Xi
t, ¯Xj
t

(5)
IV. PROPOSED APPROACH
The purpose of our approach is to localize a group of
mobile robots navigating in featureless environment depend-
ing on the relative measurement between two robots. We
have the following assumptions: i) Each robot is equipped
with proprioceptive and exteroceptive sensors to obtain data
of odometry and relative measurement. ii) Each robot is
marked with tags in order to be detected. iii) Robot is able to
communicate with each other. In our localization algorithm,
at any given time, firstly, one robot is chose randomly to
remain stationary. Secondly, the moving robot which is clos-
est to the stationary robot estimates its state by localization
algorithm with available relative measurement. Sequentially,
the rest of moving robots can utilize the measurement with
former estimated robot. This process is usually broken into
two steps, namely prediction and correction. In the following
subsection, we would derive the equation of this process.
A. Prediction
Suppose that there are one stationary robot j and several
moving robots i at time t. Robot predicts its state mainly
by the odometry noted as ut. Thus, the predicted state of
stationary robot j is same as that at previous time step t−1:
¯Xj
t = ¯Xj
t−1
¯Σj
t = ¯Σj
t−1
(6)
Moving robots i change its state during each time step.
Therefore, depending on standard EKF, the predicted state
and the corresponding covariance matrix can be derived by
the following equation:
¯Xi
t = g
ut, ¯Xi
t−1


¯Σi
t = Gt ¯Σi
t−1GT
t + VtMtVT
t
(7)
where, ¯Xi
t−1 and ¯Σi
t−1 is the estimated state and cor-
responding covariance matrix at time t −1. Gt ¯Σi
t−1GT
t
represents the uncertainty of robot i and VtMtVT
t provides
an approximate mapping between the motion noise in control
space to the motion noise in state space. Mt is the covariance
matrix of the noise in control space. And Jacobian Gt is the
derivative of the function g with respect to the state vector,
Vt is the derivative of the function g with respect to control
vector.
The estimate is calculated based on the odometry in the
prediction step. However, this step is similar to DR that
would have a cumulative error over a long distance. Hence,
we need fuse the relative measurement to correct the estimate
and reduce the uncertainty.
B. Correction
The order of the robots to be corrected with the predicted
state depends on the relative distance to the stationary robot
j. We assume the closest one is robot k, its predicted state
and corresponding variance matrix at time t are ¯Xk
t and
¯Σk
t . Hence, we have the prediction of relative measurement
between k and j:
¯zk,j
t
= h

¯Xk
t , ¯Xj
t

(8)
We utilize the difference between the measurement zk,j
t
collected from exteroceptive sensors and the predicted mea-
surement ¯zk,j
t
calculated in (8) to correct the estimated state.
There are three factors that would influence the accuracy of
measurement, including the uncertainty of stationary robots,
the uncertainty of moving robot and the Gaussian noise in
exteroceptive sensors. Hence, we have:
Sk,j
t
= Hk
t ¯Σk
t

Hk
t
T + Hj
t ¯Σj
t
h
Hj
t
iT
+ Qt
(9)
Here, Hk
t is the Jacobian of h with respect to the state vector
of moving robot k. Hj
t is the Jacobian of h with respect to
the state vector of stationary robot. Qt is the covariance
matrix of measurement noise. This equation represents the
noise mapping into the measurement space.
Then, depending on standard EKF, we can calculate the
Kalman gain as following:
Kk,j
t
= ¯Σk
t

Hk
t
T Sk
t

−1
(10)
Fusing the previous formulas, the estimate ¯Xk
t and cor-
responding variance matrix ¯Σk
t of moving robot k can be
updated by the following equation:
¯Xk
t = ¯Xk
t + Kk,j
t

zk,j
t
−¯zk,j
t

¯Σk
t =

I −Kk,j
t Hk
t

¯Σk
t
(11)
Here I is the identity matrix of the same dimensions as
Kk
t Hk
t .
So far, we obtain the estimate state and corresponding
covariance matrix of moving robot k by fusing the odometry
and relative measurement with stationary robot j at time t.
Essentially, we have the assumption that all probabilities of
measurement between robots are independent that enable us
to incrementally add the information from multiple robots
into our filter [23]. Therefore, the rest of the robots can utilize
the measurement of estimated moving robot like robot k to
estimate its state efficiently. For the remainder moving robot
m and n estimated robot:
¯Xm
t =
n
X
j=1

Km,j
t

zm,j
t
−¯zm,j
t

¯Σm
t =
n
Y
j=1

I −Km,j
t
Hm
t

(12)
Accordingly, the belief derive from the stationary robot
propagates to all the robots that reduces the uncertainty of
moving robot and improves the stability.
C. Accuracy and Consistency
The estimated state of robot is represented by the approxi-
mation and corresponding covariance matrix. To evaluate the
accuracy of this estimator, we calculate the root mean square
error(RMSE) between two state vectors. The definition is as
following:
0
1
2
3
4
5
6
position x (unit:m)
-7
-6
-5
-4
-3
-2
-1
0
1
position y (unit:m)
DE-EKF
groundtruth
DR
Fig. 2.
The trajectory of one robot calculated from DR and De-EKF and
the groundtruth
RMSE := 1
n
n
X
i=1


Xt −¯Xt


2
(13)
Xt and ¯Xt is the actual and estimated state. RMSE indicates
how similar the actual and estimated state are over n time
steps.
V. SIMULATION AND EXPERIMENT
A. Simulation
The performance of our approach was tested in simulation
with N = 5 robots navigating in an area of approximately 10
m×10 m for 1000 time steps, each time step has a duration
of 0.05 sec. In order to validate the performance, we have
the following setting:
• The motion model and measurement model of all robots
are identical which conforms to (1) and (2).
• Each robot has individual translational velocity vt = 0.2
m/s and rotational ωt = 0.5 ∗randn rad/s, the randn
is a normally distributed random number.
• The initial pose error(m,m,rad)=[0.01, 0.01, 0.01], the
initial
actual
position(m,m,rad)=[5 ∗randn, 5 ∗
randn, 2
∗
randn]
and
the
measurement
error(m,rad)=[0.01, 0.01].
Under the above setting, the position of one robot calculated
from our approach, decentralized EKF(De-EKF) and the
dead-reckoning(DR) are shown in Fig. 2. We can see that, the
DR and De-EKF have similar performance in the beginning.
However, as the distance increases, estimator from DR have
a cumulative error, while that from De-EKF accompanies the
groundtruth constantly.
Then, we compare the performance of the following
approaches: i) DR which estimates the state depending on
odometry only. ii)L-EKF that one robot estimates its state
by four known landmarks. iii)our approach(DE-EKF) that
five robots estimates their states mainly utilize the relative
measurement. We employ RMSE to indicate the accuracy of
these approaches. The result of one robot is depicted in Fig.
3. As expected, the DR has the worst performance, while
the L-EKF has the best performance because the known
landmark with absolute position can reduce the uncertainty
0
100
200
300
400
500
600
700
800
900
1000
time step (0.05 s)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
RMSE position (unit:m)
DR
L-EKF
DE-EKF
0
100
200
300
400
500
600
700
800
900
1000
time step (0.05 s)
0
0.02
0.04
0.06
0.08
0.1
0.12
RMSE orientation (unit:rad)
DR
L-EKF
DE-EKF
(a)
(b)
Fig. 3.
Comparison of DR, DE-EKF, L-EKF: (a) RMSE of robot position;
(b) RMSE of robot orientation;
0
100
200
300
400
500
600
700
800
900
1000
time step (0.05 s)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
RMSE position (unit:m)
90% Zt
70% Zt
50% Zt
20% Zt
10% Zt
Fig. 4.
Comparison of the performance with insufficient measurements
of robot efficiently. Our approach is less accurate than the L-
EKF, since the portable landmarks which are stationary robot
or estimated robots have their own uncertainty. However, our
approach do not need the known landmark. And it’s capable
for featureless environment. Additionally, the accuracy of our
approach is close to that of L-EKF.
In order to test the stability and robustness of our ap-
proach, we simulate the condition with insufficient relative
measurement. The performance of our approach with 10%-
90% measurement in each time step is compared in Fig. 4.
It depicts that with over 50% available measurement, our
approach has normal performance as usual. Even though
there are 20% available measurement, our approach still
has a moderate performance that the RMSE in position is
approximately 0.12m.
B. Experiment
An experiment with five modified Turtlebot2 is performed
to validate the performance of our approach. The hardware
and software used in experiment are specified in table I and
II. Turtlebot2 is a low-cost personal wheeled robot kit with
open-source software. The structure of modified Turtlebot2
is illustrated in Fig. 5. Each Turtlebot2 is equipped with
encoder to acquire odometry. The exteroceptive sensors are
structure light camera kinect and full HD camera located in
front and back of this robot to acquire relative measurement.
The actual information of robot position and rotation are
provided by motion capture system(OptiTrack) with error
less than 0.3mm and 0.05◦respectively. Two features are
applied on Turtlebot2. One is the rigid body marker, used to
be detected by OptiTrack. Another is Apriltag, used to be
detected by camera. The data of each robot is processed in
mini PC which is a Inter NUC on our Turtlebot2.
The experiment environment is shown in Fig. 6. Five
Turtlebot2 navigate in this environment with the motion
strategy as we mentioned previously. The experiment result is
shown in Fig. 7 and Fig. 8. We can see the state of five robots
calculated by our approach approximates to the groundtruth.
The RMSE of position and orientation is less than 0.04m
and 0.04 rad respectively.
TABLE I
HARDWARE USED IN EXPERIMENT
Hardware
Type
Description
Turtlebot2 base
YMR-K01-W1
Provide Odometry
Full HD Camera
Microsoft LifeCam 1425
Relative pose detection
Kinect
Microsoft Kinect V1
Relative pose detection
rigid body marker
-
Unique feature
Apriltag
Tag36h11
Unique feature
Intel NUC Kit
J57645-309
Data processing
Optitrack Camera
Prime 13
Provide ground truth
TABLE II
SOFTWARE USED IN EXPERIMENT
Software
Version
Description
Optitrack Motive
2.0
System driver
usb cam
0.3.6
ROS camera driver
Openni camera driver
1.11.1
ROS Kinect driver
Apriltags2 ROS
0.9.8
Provide relative pose
ROS
Kinetic
Robot operating system
Mocap Optitrack
2.8.3
Optitrack driver
Fig. 5.
Structure of Turtlebot2
Fig. 6.
Experiment environment
0
2
4
6
8
10
12
14
position x (unit:m)
-4
-2
0
2
4
6
8
position y (unit:m)
DE-EKF
groundtruth
DR
Fig. 7.
Trajectory of five robots
0
100
200
300
400
500
600
700
800
900
1000
time step (0.05 s)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
RMSE position (unit:m)
Turtlebot2 position
0
100
200
300
400
500
600
700
800
900
1000
time step (0.05 s)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
RMSE orientation (unit:rad)
Turtlebot2 orientation
(a)
(b)
Fig. 8.
experiment result(a) RMSE of robot position; (b) RMSE of robot
orientation;
VI. CONCLUSION AND FUTURE WORK
In this paper, a decentralized approach with improved
EKF for cooperative localization is proposed. Compared
with other multi-robot localization methods, the proposed ap-
proach is advantageous in high accuracy, scalability and ro-
bustness of state estimation, as well as low hardware cost and
computational complexity. This approach provides a solution
for localizing a group of robots in featureless environments.
The experiment results show that the proposed approach can
yield similar localization accuracy as the landmark-based
method does, without using any pre-known landmarks. For
five robots, the RMSE of position and orientation is less than
0.04m and 0.04 rad respectively. When the measurements are
insufficient, the proposed approach still can yield a moderate
localization performance. However, the system performance
would decline when the robots move at high speeds. Future
work will be focused on addressing this problem through
using high sampling rates and faster estimation algorithms.
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