JOURNAL OF COMPUTING, VOLUME 2, ISSUE 3, MARCH 2010, ISSN 2151-9617 
HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ 
 
83 
 
A Neuro-Fuzzy Multi Swarm 
 FastSLAM Framework  
 
R. Havangi, M.Teshnehlab and M.A. Nekoui  
 
Abstract— FastSLAM is a framework for simultaneous localization using a Rao-Blackwellized particle filter. In FastSLAM, 
particle filter is used for the mobile robot pose (position and orientation) estimation, and an Extended Kalman Filter (EKF) is 
used for the feature location’s estimation. However, FastSLAM degenerates over time. This degeneracy is due to the fact that a 
particle set estimating the pose of the robot loses its diversity. One of the main reasons for loosing particle diversity in 
FastSLAM is sample impoverishment. It occurs when likelihood lies in the tail of the proposal distribution. In this case, most of 
particle weights are insignificant. Another problem of FastSLAM relates to the design of an extended Kalman filter for landmark 
position’s estimation. The performance of the EKF and the quality of the estimation depends heavily on correct a priori 
knowledge of the process and measurement noise covariance matrices (Q and R ) that are in most applications unknown. On 
the other hand, an incorrect a priori knowledge of Q and R may seriously degrade the performance of the Kalman filter. This 
paper presents a Neuro-Fuzzy Multi Swarm FastSLAM Framework. In our proposed method, a Neuro-Fuzzy extended kalman 
filter for landmark feature estimation, and a particle filter based on particle swarm optimization are presented to overcome the 
impoverishment of FastSLAM. Experimental results demonstrate the effectiveness of the proposed algorithm. 
Index Terms— SLAM, Mobil robot, Particle filter, Particle Swarm Optimization, Neuro-Fuzzy and Extended Kalman Filter 
——————————      —————————— 
1 INTRODUCTION
he simultaneous localization and mapping (SLAM) is 
a fundamental problem of robots to perform auto-
nomous takes such as exploration in an unknown envi-
ronment. It represents an important role in the autonomy 
of a mobile robot. The two key computational solution to 
the SLAM are extended kalman filter (EKF-SLAM) and 
Rao-Blackwellized particle filter (FastSLAM). The EKF-
SLAM approach is the oldest and the most popular ap-
proach to solve the SLAM. Until now extensive research 
works have been reported employing EKF to the SLAM 
problem [1], [2], [3]. Several applications of EKF-SLAM 
have been developed for indoor applications [2], [4], out-
door applications [5], underwater applications [6] and 
underground applications [7]. However, EKF-SLAM suf-
fers from two major problems: the computational com-
plexity and data association [12]. Recently, FastSLAM 
algorithm approach has been proposed as an alternative 
approach to solve the SLAM problem. FastSLAM is an 
instance of Rao-Blackwellized particle filter, which parti-
tions the SLAM posterior into a localization problem and 
an independent landmark position estimation problem. In 
FastSLAM, particle filter is used for the mobile robot posi-
tion estimation and EKF is used for the feature location’s 
estimation. The key feature of FastSLAM, unlike EKF-
SLAM, is the fact that data association decisions can be 
determined on a per-particle basis, and hence different 
particles can be associated with different landmarks. Each 
particle in FastSLAM may even have a different number 
of landmarks in its respective map. This characteristic 
gives the FastSLAM the possibility of dealing with multi-
hypothesis association problem. The ability to simulta-
neously 
pursue 
multiple 
data 
associations 
makes 
FastSLAM significantly more robust to data association 
problems than algorithms based on incremental maxi-
mum likelihood data association such as EKF-SLAM. The 
other advantage of FastSLAM over EKF-SLAM arises from 
the fact that particle filters can cope with nonlinear and 
non-Gaussian robot motion models, whereas EKF ap-
proaches approximate such models via linear functions. 
However, FastSLAM also has some drawbacks. In refer-
ences [13], [14], [22], [27], it has been noted that FastSLAM 
degenerates over time. This degeneracy is due to the fact 
that a particle set estimating the pose of the robot loses its 
diversity. One of main reasons for losing particle diversi-
ty in FastSLAM is sample impoverishment. 
It occurs when likelihood lies in the tail of the proposal 
distribution [25]. On the other hand, FastSLAM highly 
relies on the number of particles to approximate the dis-
tribution density. Researchers have been trying to solve 
those problems in [16], [18],[25]. Another problem of 
FastSLAM relates to the design of EKF for landmark posi-
———————————————— 
 R. Havangi is with the Faculty of Electrical Engineering, K.N. Toosi Uni-
versity of Technology,Tehran, Iran 
 M.Teshnehlab S is with the Faculty of Electrical Engineering, K.N. Toosi 
University of Technology,Tehran , Iran 
 M.A.Nekoui is with the Faculty of Electrical Engineering, K.N. Toosi Uni-
versity of Technology, Tehran, Iran 
 
 
T
© 2010 Journal of Computing
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JOURNAL OF COMPUTING, VOLUME 2, ISSUE 3, MARCH 2010, ISSN 2151-9617 
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84
 
tion estimation. A significant difficulty in designing EKF 
can often be traced to incomplete a priori knowledge of 
the process covariance matrix Q and measurement noise 
covariance matrix R . In most application these matrixes 
are unknown. On the other hand, an incorrect a prior 
knowledge of Q and R may seriously degrade the Kal-
man filter performance. In this paper to solve these prob-
lems, particle filter based on Particle Swarm Optimization 
(PSO) is proposed to overcome the impoverishment of 
FastSLAM. Also, a Neuro-Fuzzy extended kalman filter is 
used for landmark feature estimation. 
 
2  THE SLAM PROBLEM 
The goal of SLAM is to simultaneously localize a robot 
and determine an accurate map of the environment. To 
describe SLAM, let us denote the map by  and the pose 
of the robot at time by
ts . The map consists of a collection 
of features, each of which will be denoted by
n
and the 
total number of stationary features will be denoted by N . 
In this situation, the SLAM problem can be formulized in 
a Bayesian probabilistic framework by representing each 
of the robot’s position and map location as a probabilistic 
density function as: 
( ,
|
,
,
)
t
t
t
t
p s
z
u
n

                                                                (1) 
In essence, it is necessary to estimate the posterior density 
of maps and poses 
ts given that we know the observa-
tion
1
{
,...,
}
t
t
z
z
z

, the control input 
1
{ ,...,
}
t
t
u
u
u

and 
the data association
tn . Here, data association represents 
the mapping between map points in and observation in 
tz .The SLAM problem is then achieved by applying 
Bayes filtering as follows: 
1
( ,
|
,
,
)
(
|
,
,
)
( ,
|
,
,
)
t
t
t
t
t
t
t
t
t
t
t
p s
z
u
n
p z
s
n
p s
z
u
n





                                (2) 
with 
1
1
1
1
1
( ,
|
,
,
)
(
|
,
)
(
,
|
,
,
)
t
t
t
t
t
t
t
t
t
t
t
t
p s
z
u
n
p s
s
u
p s
z
u
n ds









                              (3) 
Where
1
(
|
,
)
t
t
t
p s
s
u

is the dynamics motion model and 
(
|
,
,
)
t
t
t
p z
s
n

is the measurement model. The standard 
Bayesian solution (2) and (3) can be extremely expensive 
for high dimensional map. However, equation (3) can 
not be computed in closed form. FastSLAM is an efficient 
algorithm for the SLAM problem that is based on a 
straightforward factorization as follows: 
1
(
,
|
,
,
)
(
|
,
,
)
(
|
,
,
,
)
t
t
t
t
t
t
t
t
N
t
t
t
t
n
n
p s
z
u
n
p s
z
u
n
p
s
z
u
n





                               (4)                                                                                                                             
where
1
{ ,...,
}
t
t
s
s
s

 is a robot path or trajectory. This 
factorization states that the SLAM problem can be de-
composed into estimating the product of a posterior over 
robot path and N landmark posteriors given the know-
ledge of the robot’s path. The FastSLAM algorithm im-
plements the path estimator 
(
|
,
,
)
t
t
t
t
p s
z
u
n
 using a par-
ticle filter and the landmarks pose 
(
|
,
,
,
)
t
t
t
t
n
p
s
z
u
n

are 
realized by EKF, using separate filters for different land-
marks. Fig.1 shows the structure of the M particles in 
FastSLAM. Each particle forms the following [22], [26]: 
[
]
,[
]
[
]
[
]
[
]
[
]
,
,
1,
1,
,
,
,...,
,
m
t m
m
m
m
m
t
N t
N t
t
t
s
s





                            (5) 
Where [
]
m indicates the index of the particle, and 
,[
]
t
m
s
 
is the m th particle's path estimate, and  
[
]
[
]
,
,
,
m
m
N t
N t


 are 
the mean and the covariance of the Gaussian distribution 
representing the n th feature location conditioned on the 
path 
,[
]
t
m
s
. 
 
 
Fig.1. the structure of particles in FastSLAM 
 
The algorithm update of posterior of FastSLAM can be 
described as: 
1. Sample a new robot pose. 
2. Update estimation of observed landmark with EKF. 
3. Calculate importance weight and resample.  
 
2.1 SAMPLING A NEW OF POSE 
As mentioned in the previous section, FastSLAM employs 
a particle filter for estimating the path posterior 
(
|
,
,
)
t
t
t
t
p s
z
u
n
by particle filter. The particle set 
tS is 
calculated from the set 
1
tS at time 
1
t , the control 
tu , 
and the observation
tz . In general, it is not possible to 
draw samples directly from the SLAM posterior. Instead, 
the samples are drawn from a simpler distribution called 
the proposal distribution
,[
]
(
|
,
,
)
t
m
t
t
t
q s
z
u
n
.The mis-
match between of SLAM posterior and proposal distribu-
tion correct using a technique called importance sam-
pling. Therefore, in regions where the target distribution 
is larger than the proposal distribution, the samples are 
assigned a larger weight and in regions where the target 
distribution is smaller than the proposal distribution the 
samples will be given lower weights. For FastSLAM, the 
important weights of each particle can be calculated as: 
JOURNAL OF COMPUTING, VOLUME 2, ISSUE 3, MARCH 2010, ISSN 2151-9617 
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85 
 
  
,[
]
,[
]
target distribution
(
|
,
,
)
proposal distribution
(
|
,
,
)
t
m
t
t
t
i
k
t
m
t
t
t
p s
z
u
n
w
q s
z
u
n


          (6) 
As a result, the important weight of each particle is equal 
to the ratio of the slam posterior and the proposal distri-
bution. 
The 
proposal 
,[
]
(
|
,
,
)
t
m
t
t
t
q s
z
u
n
can 
be 
represented by a recursive form as: 
  
,[
]
[
]
1,[
]
1,[
]
[
]
1,[
]
1,[
]
1
1
1
(
|
,
,
)
(
|
,
,
,
)
(
|
,
,
)
(
|
,
,
,
)
(
|
,
,
)
t
m
t
t
t
m
t
m
t
t
t
t
Markov
t
m
t
t
t
m
t
m
t
t
t
t
t
m
t
t
t
q s
z
u
n
q s
s
z
u
n
q s
z
u
n
q s
s
z
u
n
q s
z
u
n









    (7) 
Similarity, the posterior can also be given by a recursive 
form using Baye’s rule as follows: 
,[
]
,[
]
1
[
]
1
1,[
]
1
1
1
(
|
,
,
)
(
|
,
,
,
)
(
|
,
,
) (
|
,
,
)
t
m
t
t
t
t
m
t
t
t
t
m
t
t
t
t
m
t
t
t
t
p s
z
u
n
p z
s
z
u
n
p s
z
u
n
p s
z
u
n








               (8) 
 
Where  is a normalizing constant. By (7) and (8), a se-
quential importance weight of m
th

particle can be ob-
tained as following: 
,[
]
1
[
]
1
1
[
]
1,[
]
(
|
,
,
,
) (
|
,
,
)
(
|
,
,
,
)
t
m
t
t
t
m
t
t
t
i
i
t
t
k
k
m
t
m
t
t
t
t
p z
s
z
u
n
p s
z
u
n
w
w
q s
s
z
u
n






 
                                                                                                  (9) 
The 
choice 
of 
the 
proposal distribution  
[
]
1,[
]
(
|
,
,
,
)
m
t
m
t
t
t
t
q s
s
z
u
n

is one of the most critical issues 
in the design of a FastSLAM. Two of those critical reasons 
are as follows: samples are drawn from the proposal dis-
tribution, and the proposal distribution is used to valuate 
important weights. The optimal importance density func-
tion that minimizes the variance of the importance 
weights is the following equation (FastSLAM 2.0 [9]) 
,[
]
1
1
,[
]
1
1
(
|
,
,
)
(
|
,
,
)
t
m
t
t
t
t
m
t
t
t
q s
z
u
n
p s
z
u
n





            (10) 
However, there are some special cases where the use of 
the optimal importance density is possible. The most 
popular suboptimal choice is the transitional prior 
(FastSLAM1.0 [11]) 
,[
]
1
1
[
]
1
(
|
,
,
)
(
|
,
)
t
m
t
t
t
m
t
t
t
q s
z
u
n
p s
u
s




                        (11) 
In this paper, FastSLAM1.0 is used due to its easy calcula-
tion. Hence, by substitution of (11) into (8), the weight’s 
update equation is: 
,[
]
1
1 (
|
,
,
,
)
i
i
t
m
t
t
t
k
k
t
w
w
p z
s
z
u
n




                                   (12) 
Not that the 
i
k
w
is updated incrementally while consider-
ing a full trajectory of the robot state 
,[
]
t
m
s
.  
 
2.2 UPDATING THE OBSERVED LANDMARK ESTIMATE 
The FastSLAM represents the posterior landmark esti-
mates 
(
|
,
,
,
)
t
t
t
t
n
p
s
z
u
n

 using low-dimensional EKFs. 
In fact FastSLAM 1.0 updates the posterior over the 
landmark estimates, respected by the mean 
[
]
,
1
m
n t

and the 
covariance 
[
]
,
1
m
n t 

.the updated values 
[
]
,
m
n t

and 
[
]
,
m
n t

 are 
then added to the temporary particle set
tS , along with 
the new pose. The update depends on whether or not a 
landmark n was observed at time t . For 
t
n
n

, the 
posterior over the landmark remains unchanged as fol-
lowing [26]: 
 
[
]
[
]
[
]
[
]
,
,
,
1
,
1
          
m
m
m
m
n t
n t
n t
n t







                                       (13) 
For the observed feature
t
n
n

, the update is specified 
through the following equation [26]: 
( )
[ ]
[ ]
,
1
,
1
,( )
,( )
1
,( )
1
,( )
1
,( )
1
,( )
1
(
,
,
)
(
,
(
,
,
)
(
|
,
,
)
(
|
,
,
,
) (
|
,
,
)
(
|
,
,
)
(
|
,
,
,
)
(
|
,
,
)
t
t
t
t
t
i
i
i
n
t
n
t
t
n t
n t
t
t
t
i
t
t
n
t
i
t
t
t
i
t
t
t
n
n
t
i
t
t
t
t
i
t
t
t
i
t
t
t
n
n
N
N z
g
s
R
p
s
n
z
p z
s
n
z
p
s
n
z
p z
s
n
z
p z
s
n
z
p
s
n
z




















             (14) 
The probability 
,( )
1
(
|
,
,
)
t
t
i
t
t
n
p
s
n
z


 at time 
1
t is 
represented by a Gaussian distribution with mean 
[ ]
,
1
i
n t

 
and covariance 
[ ]
,
1
i
n t 

 . For the new estimate at time t to 
also be Gaussian, FastSLAM linearizes the perceptual 
model 
,( )
1
(
|
,
,
,
)
t
t
i
t
t
t
n
p z
s
n
z


 
by 
EKF. 
Especially, 
FastSLAM approximates the measurement function g by 
the following first degree Taylor expansion [26]: 
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
,
1
,
1
,
1
ˆ
[ ]
[ ]
[ ]
,
1
(
,
)
(
,
)
(
,
)(
)
ˆ
=
(
)
t
t
i
i
t
t
t
i
i
i
i
i
i
n
t
n t
t
t
n t
n
n t
z
G
i
i
i
t
t
n
n t
g
s
g
s
g s
z
G




















 
 
  
 
  
     (15) 
Under this approximation, the posterior of landmark 
tn is 
indeed Gaussian. The mean and covariance are obtained 
using the following measurement update:  
[
]
,
1
ˆ
(
,
)
t
m
t
t
n t
z
g s



                                                            (16) 
[
]
[
]
,
1
;
(
,
) |
m
m
n
n
t
t
t
t
t
n
n
t
t t
t
n
s
s
G
g s









                                 (17) 
[
]
,
,
1
n
n
t
t
t
m
T
n t
t
n t
Z
G
G
R






                                                 (18) 
[
]
1
,
,
1
n
t
t
m
T
t
n t
n t
K
G
Z




 
                                                  (19) 
,
,
1
[
]
[
]
ˆ
(
)
t t
t t
m
m
t
t
t
n
n
K
z
z






                                                 (20) 
[
]
[
]
,
,
1
(
)
n
t
t
t
m
m
t
n t
n t
I
K G





                                                  (21) 
 
2.3 RESAMPLING  
Sine the variance of the importance weights increase over 
time [23], [24], resampling plays a vital role in FastSLAM. 
In the resampling process, particles with low importance 
weight are eliminated and particles with high weights are 
multiplied. After, the resampling, all particle weights are 
then reset to  
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86
 
1
i
t
w
N

                                                                                (22) 
This enables the FastSLAM to estimate increasing envi-
ronmental sates defiantly without growing a number of 
particles. However, resampling can delete good samples 
from the sample set, in the worst case, the filter diverges. 
The decision on how to determine the point of time of the 
resampling is a fundamental issue. Liu introduced the so-
called effective number of particles 
eff
N
to estimate how 
well the current particle set represents the true posterior. 
This quality is computed as 
( )
1
1
eff
N
i
i
N
w



                                                               (23) 
Where 
( )
i
w
refers to the normalized weight of par-
ticle i .The resampling process is operated whenever 
eff
N
is bellow a pre-defined threshold, 
tf
N
. Here 
tf
N
 is 
usually a constant value as following  
3
4
tf
N
M

                                                                           (24) 
Where M is number of particles. 
 
3 A MODIFIED FASTSLAM FRAMEWORK 
In this section, a modified FastSLAM framework is pre-
sented. For this purpose, a particle filter based on PSO is 
presented to overcome the impoverishment of the particle 
filter. Also a Neuro-Fuzzy extended kalman filter (EKF) 
for landmark feature estimation is proposed.  
 
3.1 A MODIFIED SAMPLING OF POSE 
Because the proposal function is suboptimal, there are 
two serious problems in FastSLAM. One problem is sam-
ple impoverishment, which occurs when the likelihood 
,[
]
1
(
|
,
,
,
)
t m
t
t
t
t
p z
s
z
u
n

 is very narrow or likelihood lies 
in the tail of the proposal distribution
[
]
1
(
|
,
)
m
t
t
t
p s
u
s 
. The 
prior distribution is effective when the observation accu-
racy is low. But not when prior distribution is a much 
broader distribution than the likelihood (such as Fig.2.). 
Hence, in the updating step, only a few particles will have 
significant importance weights.  
 
Fig.2. Prior and Likelihood 
 
This problem implies that a large computational effort is 
devoted to update the particles with negligible weight. 
Thus, the sample set only contains few dissimilar par-
ticles and sometimes they will drop to a single sample 
after several iterations. As a result, important samples 
may be lost. Another problem of FastSLAM is the number 
of particles dependency that estimates the pose of the 
robot. If the number of particles is small, then there might 
not have been particles distributed around the true pose 
of the robot. So after several iterations, it is very difficult 
for particles to converge to the true pose of the robot. For 
standard particle filter, there is one method to solve the 
problem. This is to augment the number of the particles. 
But this would make the computational complexity unac-
ceptable. To solve these problems of FastSLAM, particle 
swarm optimization is considered to optimize the sam-
pling process of FastSLAM. In fact, PSO encourages par-
ticles to high likelihood area before the sampling process. 
 
3.1.1 PARTICLE SWARM OPTIMIZATION 
James Kennedy and Russell C.Eberhart [17] originally 
proposed the PSO algorithm for optimization. PSO is a 
population-based search algorithm based on the simula-
tion of the social behavior of birds with a flock. PSO is 
initialized with a group of random particles and then 
computes the fitness of each one. Finally, it can find the 
best solution in the problem space via many repeating 
iterations. In each iteration, each particle keeps track of its 
coordinates which are associated with the best solution it 
has achieved so far (pbest) and the coordinates which are 
associated with the best solution achieved by any particle 
in the neighboring of the particle (gbest). Supposing that 
the search space dimension is D and number particles is 
N, the position and velocity of the i-th particle are 
represented 
as
1
(
,...,
)
i
i
iD
x
x
x

and 
1
(
,...,
)
i
i
iD
v
v
v

 
respectively. 
 Let 
1
[
,...
]
bi
i
iD
P
p
p

denote the best position which the 
particle i has achieved so far, and 
g
P  the best of 
bi
P
for 
any 
1,...,
i
N

. The PSO algorithm could be performed 
by the following equations:  
( )
(
1)
( )
i
i
i
x
t
x
t
v
t






                                              (25)  
1 1
2 2
( )
(
1)
(
(
1))
(
(
1))
i
i
bi
i
g
i
v
t
wv
t
c r P
x
t
c r P
x
t














                           (26) 
Where t represents the iteration number and
1c , 
2c  are 
the learning factors. Usually 
1
2
2
c
c


. 
1r , 2r  are ran-
dom numbers in the interval (0,1) . w is the inertial factor, 
and the bigger the value of w , the wider is the search 
range. 
 
3.1.2  A PARTICLE FILTER BASED ON PSO 
As discussed in the previous section, Impoverishment 
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87 
 
occurs when the number of particles in the high likelih-
ood area is low. In recent years, several researchers have 
studied this problem [16], [17], [18], [19]. The Main disad-
vantage of these approaches is that they are more difficult 
to implement than FastSLAM. Also, they are more ma-
thematically involved. Here, we use an easy idea to solve 
this problem. The idea is that the sampling process par-
ticles are encouraged to be at the right place (in the region 
of high likelihood) by incorporating the current observa-
tion. This implies that a Simple and effective method for 
this purpose is the using of PSO. In fact, by using PSO, we 
can move all the particles towards the region of high like-
lihood before the sampling process. For this purpose, first 
we must define a fitness function. The fitness function 
must take the newest observations into account and also 
fitness function’ particle with high likelihood have small 
value. For this purpose, we consider a fitness function as 
following: 
1
0
,
,
,
ˆ
ˆ
(
)
(
)
(
)
t
t
t
T
k
t
n t
n t
t
n t
f
x
z
z
Z
z
z









                         (27) 
Here, 
,
tn t
Z
is the residual covariance matrix defined in 
(12), 
,
ˆn t
z
is the predicted measurement and 
tz is the ac-
tual measurement. The particles should be moved such 
that the fitness function is optimal. This is done by tuning 
the position and velocity of the PSO algorithm. The stan-
dard PSO algorithm has some parameters that need to be 
specified before using it. Most approaches use uniform 
probability distribution to generate random numbers. 
However it is difficult to obtain fine tuning of the solution 
and escape from local minima using a uniform distribu-
tion. Hence, we use velocity updates based on the Gaus-
sian distribution. In this situation, there is no more need 
to specify the parameter learning factors 
1c and
2c . Fur-
thermore, using the Gaussian PSO the inertial factor 
was set to zero and an upper bound for the maximum 
velocity 
max
v
is not necessary anymore [21]. So, the only 
parameter to be specified by the user is the number of 
particles. Initial values of particle filter are selected as the 
initial population of PSO. Initial velocities of PSO are set 
equal to zero. The PSO algorithm updates the velocity 
and position of each particle by following equations [21]: 
 
( )
(
1)
( )
i
i
i
x
t
x
t
v
t






                                                     (28) 
( )
(
(
1))
(
(
1))
i
pbest
i
gbest
i
v
t
randn P
x
t
randn P
x
t









                                    (29) 
PSO moves all particles towards the high likelihood re-
gions which is the global best of PSO. When the best fit-
ness value reaches a certain threshold, the optimized 
sampling process is stopped. 
With this set of particles the sampling process will be 
done on the basis of proposal distribution. The Corres-
ponding weights will be as follows: 
,[
]
1
(
|
,
,
,
)
k
t
m
t
t
t
i
t
w
p z
s
z
u
n


                                            (30) 
Where 
,[
]
1
,
1
,
,
,
1
(
|
,
,
,
)
exp
(2 ) |
|
1
ˆ
ˆ
{
(
)
(
)}
2
t
t
t
t
t
m
t
t
t
t
n t
T
t
n t
n t
t
n t
p z
s
z
u
n
Z
z
z
Z
z
z











                    (31) 
The flowchart of the proposed algorithm to pose estima-
tion is shown in fig.3. 
 
 
Fig3. Modified particle filter for FastSLAM (pose estimation) 
 
As observed, this algorithm can be described with the 
following steps 
Step1. General initial population  
Initialization-randomly generation initial pose of mobile 
robot 
 
Initialized  particle velocity  
 
Initial particle position 
 
Initialized particle fitness value 
 
Initialized pbest and gbest 
Step2. Improve Sampling by PSO 
Step5. Sampling 
 Sampling is obtaining samples from the prior 
Step3. Update 
On receipt of the measurement saw
k
z
, evaluate the like-
lihood of each prior sample and obtain a normalized-
weight for each sample. 
Step4. Resampling 
Multiply Suppress samples with high /low importance 
weights. 
Step5. Prediction 
 Each pose is passed through the system model to obtain 
samples from the prior 
Setp6. Increase time t and iterate from step 2 to step 4. 
 
3.2 FEATURE UPDATE BY ADAPTIVE NEURO-FUZZY INFE-
RENCE SYSTEM (ANFIS) 
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88
 
As stated earlier, FastSLAM represents the conditional 
landmark estimates 
(
|
,
,
,
)
t
t
t
t
n
p
s
z
u
n

 by EKF. The tra-
ditional EKF assumes complete a priori knowledge of the 
process and measurement noise statistics; matrices Q and 
R. However, in most applications these matrixes are un-
known. An incorrect a prior knowledge of Q and R may 
lead to performance degradation [15] and it can even lead 
to practical divergence [16]. One of the efficient ways to 
overcome the above weakness is to use an adaptive algo-
rithm .Two major approaches that have proposed for 
adaptive EKF are multiple model adaptive estimation 
(MMAE) and innovation adaptive estimation (IAE) [15]. 
Although the implementation of these approaches is dif-
ferent, they both share the same concept of utilizing new 
statistical information obtained from the residual (innova-
tion) sequence. As landmarks are static, we assume that 
the noise covariance Q is completely known. Hence, the 
algorithm to estimate the measurement noise covariance 
R can be derived. The adaptation is adaptively adjusting 
the measurement noise covariance matrix R by using 
Neuro-fuzzy System. In this case, an innovation based 
adaptive estimation (IAE) algorithm to adapt the mea-
surement noise covariance matrix R is derived. The tech-
nique known as covariance-matching is used. The basic 
idea behind this technique is to make the actual value of 
the covariance of the residual to be consistent with its 
theoretical 
value. 
The 
innovation 
sequence 
,
ˆ
(
)
t
k
t
n t
r
z
z


has a theoretical covariance that is ob-
tained from the EKF algorithm 
[
]
,
1
n
n
t
t
t
m
T
k
t
n t
S
G
G
R






                                                    (32)                                                                                                                            
The actual residual covariance ˆ
k
C can be approximated by its 
sample covariance, through averaging inside a moving window 
of size N as following 
1
1
ˆ
(
)
k
T
k
i
i
k
N
i
C
r r
N





                     (33)                                                                                                                           
Where 
0i is first sample inside the estimation window. If the 
actual value of covariance ˆ
k
C  has discrepancies with its theo-
retical value, then the diagonal elements of 
k
R based on the 
size of this discrepancy can be adjusted. The objective of these 
adjustments is to correct this mismatch as far as possible. The 
size of the mentioned discrepancy is given by a variable called 
the degree of mismatch (
k
DOM
), defined as 
ˆ
k
k
k
DOM
S
C


                                                                     (34)                                                                                                                           
The basic idea used to adapt the matrix 
k
R is as follows: from 
equation (24) an increment in 
k
R will increase 
k
S
and vice 
versa. Thus, 
k
R can be used to vary 
k
S in accordance with the 
value of 
k
DOM
 in order to reduce the discrepancies between 
k
S
and ˆ
k
C .The adaptation of the ( , )
i i th element of 
k
R  is 
made in accordance with the ( , )
i i th element of
k
DOM
.The 
general rules of adaptation are as follows: 
If 
( , )
0
k
DOM
i i

 then maintain 
k
S
unchanged 
If 
( , )
0
k
DOM
i i

 then decrease 
k
S
 
If 
( , )
0
k
DOM
i i

 then increase 
k
S
 
In this paper the IAE adaptive scheme of the EKF coupled 
with adaptive Neuro-fuzzy inference system (ANFIS) is 
presented to adjust R . As the size 
k
DOM
and R  is two, 
two system ANFIS to adjust EKF is used. The structure of 
one of these systems is described in the next section. 
 
3.2.1 THE ANFIS ARCHITECTURE 
The ANFIS model has been considered as a two-input-
single-output 
system. 
The 
inputs 
of 
ANFIS 
are 
k
DOM
and   
k
DeltaDOM
. Here, 
k
DeltaDOM
is defined 
as following 
1
k
k
k
DeltaDOM
DOM
DOM



                                 (35) 
Fig.4 
and 
Fig.5 
presents 
membership 
functions 
for  
( , )
k
DOM
i i  and
k
DeltaDOM
.  
 
 
Fig.4. Membership function 
k
DOM
 
  
 
Fig .5.Membership function 
k
DeltaDOM
 
 
 
Fig .6.Membership function 
k
AjdR  
 
Finally, adjustments of 
k
R
is performed using the following 
relation 
k
k
k
R
R
R


                                                                        (29) 
where 
k
R

is the ANFIS output and membership function of 
k
R

 is shown in Fig.6. This ANFIS is a five layers network as 
shown in Fig.7. 
Let 
l
iu and 
l
io denote the input to output from the i th 
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89 
 
node of the 
 th
l
 layer, respectively. To provide a clear 
understanding of an ANFIS, the function of layer 1 to 
layer 5 are defined as follows: 
 
 
Fig .7.The Neuro-fuzzy system for feature update 
 
Table 1 
Rule Table 
 
 
Layer 1: The node in this layer only transmits input val-
ues to the next layer directly, i.e. 
1
1
i
i
o
u

                                                                                  (36) 
Layer2: In this layer, each node only performs a member-
ship function. Here, the input variable is fuzzified em-
ploying five membership functions (MFs). The output of 
the  th
i
 MF is given as: 
2
2
2
2
2
(
)
(
)
exp
(
)
ij
ij
ij
ij
ij
u
m
o
u
















                                  (37) 
Where 
ij
m
 and
ij

, respectively, are the mean and with 
of the Gaussian membership function. The subscript 
ij indicates the jth term of the ith input. Each node in 
this layer has two adjustable parameters: 
ij
m
 and 
ij

 
Layer3: The nodes in this layer are rule nodes. The rule 
node performs a fuzzy and operation (or product infe-
rence) for calculating the firing strength. 
3
3
l
i
i
o
u

                                                                            (38) 
Layer4: The node in this layer performs the normalization 
of firing strengths from layer 3, 
4
4
9
4
1
l
l
l
l
u
o
u


                                                                        (39) 
Layer5: This layer is the output layer. The link weights in 
this layer represent the singleton constituents (
i
W ) of the 
output variable. The output node integrates all the nor-
malization firing strength from layer 4 with the corres-
ponding singleton constituents and acts as defuzzfier, 
  
25
5
1
i
l
l
l
R
u w



                                                                    (40) 
The fuzzy rules which complete the ANFIS rule base as table1. 
 
 
3.2.2 LEARNING ALGORITHM 
The aim of the training algorithm is to adjust the network 
weights through the minimization of the following cast 
function: 
2
1
2
k
E
e

                                                                              (41) 
Where 
ˆ
k
k
k
e
S
C


                                                                        (42) 
By using the bake propagation (BP) learning algorithm, 
the weighting vector of ANFIS is adjusted such that the 
error defined in (41) is less of than a desired threshold 
value after a given number of training cycles. The well-
known BP algorithm may be written as: 
 
( )
(
1)
( )
(
)
( )
E k
W k
W k
W k






                                      (43) 
Here  and W represent the learning rate and tuning 
parameter of ANFIS respectively. Let 
[
, ,
]T
W
m
w


be 
the weighting vector of ANFIS. The gradient of E  with 
respect to an arbitrary weighting vector W as following: 
  
( )
( )
( )
( )
( )
E k
R k
e k
W k
W k











                                            (44) 
By the recursive application of the chain rule, the error 
term for each layer is first calculated, and then the para-
meters in the corresponding layers are adjusted. 
 
4  IMPLEMENTATION AND RESULTS 
Simulation experiments have been carried out to evaluate 
the performance of the proposed approach in comparison 
with the classical method. The proposed solution for the 
SLAM problem has been tested for the benchmark envi-
ronment, with varied number and position of the land-
marks, available in [20] .Fig.8 shows the robot trajectory 
and landmark location. The star points (*) depict location 
of the landmarks that are known and stationary in the 
environment. The state of the robot can be modeled as 
(
,
,
x y ) that (
,
x y ) are the Cartesian coordinates and  
is the orientation respectively to the global environment. 
The kinematics equations for the mobile robot are in the 
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90
 
following form 
 
-150
-100
-50
0
50
100
-100
-80
-60
-40
-20
0
20
40
60
80
100
X(m)
Y(m)
 
Fig.8. The experiment environment. The star point “*” denote the 
landmark positions and blue line is the path of robot. 
 
(
)cos(
[
])
(
)sin(
[
])
(
) sin(
)
v
v
v
x
V
v
v
y
V
v
v
V
v
v
B












































                                              (45)                                                                                                     
Where B is the base line of the vehicle and 


T
u
V


is 
the control input at time t consisting of a velocity input 
V and a steer input. The process noise 
T
v
v
v
v 



is 
assumed to be Gaussian. The vehicle is assumed to be 
equipped with a range-bearing sensor that provides a 
measurement of range 
ir and bearing 
ito an observed 
feature 
irelative to the vehicle. The observation z of 
feature 
iin the map can be expressed as: 
2
2
1
(
)
(
)
tan
i
i
r
i
i
i
i
x
x
y
y
r
y
y
x
x































                                       (46) 
Where (
,
)
i
i
x
y
is the landmark position in map and 


T
r
W




related to observation noise.  
The initial position of the robot is assumed to be
0
0
x 
. 
The robot moves at a speed 3m/s and with a maximum 
steering angle 30 deg. Also, the robot has 4 meters wheel 
base and is equipped with a range-bearing sensor with a 
maximum range of 20 meters and a 180 degrees frontal 
field-of-view. 
The 
control 
noise 
is
0.3 m/s
v

 
and
3o



. A control frequency is 40 HZ  and observa-
tion scans are obtained at 5 HZ . The measurement noise 
is 0.2 m in range and 1o in bearing. Data association is 
assumed known. At first, the performance of the two al-
gorithms can be compared by keeping the noises level 
(process noise and measurement noise) and varying the 
number of particles. Fig.9 to Fig.16 shows the perfor-
mance of the two algorithms for number of particles 
equivalent to 20 and 5. As observed, modified FastSLAM 
is more accurate than the FastSLAM. Also, performance 
of modified FastSLAM does not depend on the number of 
particles while the performance of FastSLAM highly de-
pends on the number of particles. For very low numbers 
of 
particles 
FastSLAM 
diverges 
while 
modified 
FastSLAM is completely robust. This is because PSO in 
the modified FastSLAM places the particles in the high 
likelihood region. In addition, we observed that the mod-
ified FastSLAM requires fewer particles than FastSLAM 
in order to achieve a given level of accuracy for state es-
timates. 
 
 
Fig.9. Estimated robot path and estimated landmark using modified 
FastSLAM with true robot path and true landmark. The “…” is path 
estimated, the “+” are the estimated landmark positions. In this expe-
riment the control noise is
0.3 m/s
v

,
3o



. The measure-
ment noise is
0.2 m
r

, 
1deg



 and number of particles is 
20 (n=20).Also, the results obtain over 50 Monte Carlo runs. 
 
 
Fig.10.Estimated pose error with 2


 bound using modified 
FastSLAM 
.In 
this 
experiment 
the 
control 
noise 
is
0.3 m/s
v

,
3o


. The measurement noise is
0.2 m
r

, 
1deg



 and number of particles is 20 (n=20).Also, the results 
obtain over 50 Monte Carlo runs. 
 
 
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91 
 
 
Fig.11. Estimated robot path and estimated landmark using 
FastSLAM with true robot path and true landmark. The “…” is path 
estimated, the “+” are the estimated landmark positions. In this expe-
riment the control noise is
0.3 m/s
v

,
3o


. The measure-
ment noise is
0.2 m
r

, 
1deg



 and number of particles is 
20 (n=20).Also, the results obtain over 50 Monte Carlo runs. 
 
 
Fig.12. Estimated pose error with 2


 bound using modified 
FastSLAM. 
In 
this 
experiment 
the 
control 
noise 
is
0.3 m/s
v

,
3o


. The measurement noise is
0.2 m
r

, 
1deg



 and number of particles is 20 (n=20).Also, the results 
obtain over 50 Monte Carlo runs. 
 
 
Fig.13. Estimated robot path and estimated landmark using 
FastSLAM with true robot path and true landmark. The “…” is path 
estimated, the “+” are the estimated landmark positions. In this expe-
riment the control noise is  
0.3 m/s
v

, 
3o


. The measurement 
noise is
0.2 m
r

, 
1deg



 and number of particles is 5 
(n=5).Also, the results obtain over 50 Monte Carlo runs. 
 
 
 
Fig.14. Estimated pose error with 2


 bound using modified 
FastSLAM. In this experiment the control noise is  
0.3 m/s
v

, 
3o


. The measurement noise is
0.2 m
r

, 
1deg


 and num-
ber of particles is 5 (n=5).Also, the results obtain over 50 Monte 
Carlo runs. 
 
 
 
Fig.15. Estimated robot path and estimated landmark using 
FastSLAM with true robot path and true landmark. The “…” is path 
estimated, the “+” are the estimated landmark positions. In this expe-
riment the control noise is
0.3 m/s
v

,
3o


. The measurement 
noise is
0.2 m
r

, 
1deg


 and number of particles is 5 
(n=5).Also, the results obtain over 50 Monte Carlo runs. 
 
 
Fig.16. Estimated pose error with 2


 bound using modified 
FastSLAM. 
In 
this 
experiment 
the 
control 
noise 
is
0.3 m/s
v

,
3o


. The 
measurement noise is
0.2 m
r

, 
1deg


 and number of particles is 5 (n=5).Also, the results obtain 
over 50 Monte Carlo runs. 
 
 
Now, we compare the performance of the two algorithms 
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92
 
while varying the level of measurement noise as 
0.01
r

 a, 
0.1



 and fixing the control noise i.e. 
0.3 m/s
v

 ,
3o


.  All experiments run with 5 par-
ticles in this case. From Fig.17 to Fig.20 we observe that 
the performance of our proposed method will outperform 
classical FastSLAM. 
This is because sample impoverishment occurs in 
FastSLAM. In fact, because the proposal distribution is 
broader than likelihood in FastSLAM, the weight of most 
of the participles are insignificant when their likelihoods 
are evaluated. As a result, the estimation accuracy of 
FastSLAM will decrease whenever the robot's motion is 
noisier and the range-bearing sensor is more accurate. In 
this situation, the performance of FastSLAM is reduced 
mostly while the performance of the modified FastSLAM 
is almost fixed. For very low values of measurement error 
FastSLAM clearly begins to diverge while the proposed 
algorithm is robust. 
 
 
Fig.17. Estimated robot path and estimated landmark using modified 
FastSLAM with true robot path and true landmark. The “…” is path 
estimated, the “+” are the estimated landmark positions In this expe-
riment the control noise is
0.3 m/s
v

,
3o


.The measurement 
noise is
0.01 m
r

, 
0.1deg


 and number of particles is 5 
(n=5).Also, the results obtain over 50 Monte Carlo runs. 
 
 
Fig.18. Estimated pose error with 2


 bound using modified 
FastSLAM. 
In 
this 
experiment 
the 
control 
noise 
is
0.3 m/s
v

,
3o


.The 
measurement 
noise 
is
0.01 m
r

, 
0.1deg


 and number of particles is 5 (n=5).Also, the results ob-
tain over 50 Monte Carlo runs. 
 
 
Fig.19. Estimated robot path and estimated landmark using 
FastSLAM with true robot path and true landmark. The “…” is path 
estimated, the “+” are the estimated landmark positions. In this expe-
riment the control noise is
0.3 m/s
v

,
3o


. The measurement 
noise is
0.01 m
r

, 
0.1deg


 and number of particles is 5 
(n=5).Also, the results obtain over 50 Monte Carlo runs. 
 
Fig.20. Estimated pose error with 2


 bound using modified 
FastSLAM. 
In 
this 
experiment 
the 
control 
noise 
is
0.3 m/s
v

,
3o


. 
The 
measurement 
noise 
is
0.01 m
r

, 
0.1deg



 and number of particles is 5 
(n=5).Also, the results obtain over 50 Monte Carlo runs. 
 
Finally,  we  study  affect  of  the  designed  ANFIS  on  the 
performance of FastSLAM. For this purpose, we consider 
the situation where measurement noise is wrongly consi‐
dered  as
0.5
r

  , 
3.0



  and  control  noise  is  rightly 
considered  as
0.3 m/s
v

, 
3o


.The  performance  of 
the  modified  FastSLAM  is  compared  with  classical 
FastSLAM  that  its  measurement  covariance  matrix  R  is 
kept static throughout the experiment. The proposed al‐
gorithm starts with a wrongly known statistics and then 
adapts  the 
k
R
  matrix  through  ANFIS  and  attempts  to 
minimize  the  mismatch  between  the  theoretical  and  ac‐
tual values of the innovation sequence. The free parame‐
ters  of  ANFIS  are  automatically  learned  by  SD  during 
training.  Fig.21‐22  and  Fig.24‐25  show  the  comparison 
between the proposed algorithm and the FastSLAM when 
the particle number is 20. It can be seen clearly that the 
results of the proposed algorithm are better than that of 
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93 
 
classical FastSLAM. This is because  modified FastSLAM 
adaptively  tuned  the  measurement  covariance  matrix  R 
while  covariance  matrix  measurement  R  in  standard 
FastSLAM is kept fixed over time as shown in Fig.23 and 
Fig.26.  Also  Fig.23  and  Fig.26  show  that  measurement 
covariance matrix measurement R converges to the actual 
covariance matrix R in our proposed method. 
 
Fig.21. Estimated robot path and estimated landmark using modified 
FastSLAM with true robot path and true landmark. The “…” is path 
estimated, the “+” are the estimated landmark positions. In this expe-
riment, measurement noise is wrongly considered as
0.5
r

 , 
3.0



 and control noise is truly considered as
0.3 m/s
v

, 
3o


 . Also, number of particles is 5 (n=5) and  the results obtain 
over 50 Monte Carlo runs.  
 
 
Fig.22. Estimated pose error with 2


 bound using modified 
FastSLAM. In this experiment, measurement noise is wrongly consi-
dered as
0.5
r

 , 
3.0



 and control noise is truly considered 
as
0.3 m/s
v

, 
3o


 . Also, number of particles is 5 (n=5) 
and  the results obtain over 50 Monte Carlo runs.  
 
Fig.23. A modified FastSLAM (n=5) 
 
 
Fig.24. Estimated robot path and estimated landmark using 
FastSLAM with true robot path and true landmark. The “…” is path 
estimated, the “+” are the estimated landmark positions. In this expe-
riment, measurement noise is wrongly considered as
0.5
r

 , 
3.0



 and control noise is truly considered as
0.3 m/s
v

, 
3o


 . Also, number of particles is 5 (n=5) and the results obtain 
over 50 Monte Carlo runs. 
 
 
Fig.25. Estimated pose error with 2


 bound using FastSLAM.  In 
this experiment, measurement noise is wrongly considered 
as
0.5
r

 , 
3.0



 and control noise is truly considered 
as
0.3 m/s
v

, 
3o


 . Also, number of particles is 5 (n=5) 
and the results obtain over 50 Monte Carlo runs. 
 
 
Fig.26. A FastSLAM (n=5) 
 
 
5 CONCLUSION 
In this Paper, two problems of FastSLAM are im-
proved. The first problem is that FastSLAM degene-
rates over time due to the loss of particle diversity. 
One of the main reasons for losing particle diversity is 
sample impoverishment. It occurs when likelihood lies 
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94
 
in the tail of the proposal distribution. In this case, 
most of participles weights are insignificant. Another 
problem of FastSLAM relates to the design of EKF for 
the landmark position’s estimation. A significant diffi-
culty in designing the extended kalman filter can often 
be traced to incomplete a priori knowledge of the 
process covariance matrix Q and the measurement 
noise covariance matrix R . On the other hand, an in-
correct a prior knowledge of Q and R may seriously 
degrade the Kalman filter performance. This paper 
presents a modified FastSLAM framework by soft 
computing. In the proposed method, a Neuro-Fuzzy 
extended kalman filter for landmark feature estima-
tion, and a particle filter based on particle swarm op-
timization is presented to overcome the impoverish-
ment of FastSLAM. Finally, Experimental results con-
firm the effective of the proposed algorithm. The main 
advantage of our proposed method is its more consis-
tency than classical FastSLAM. This is because in our 
proposed method, the theoretical value of the innova-
tion sequence matches with its real value. Also, when 
motion model is noisier than measurement, the per-
formance of the proposed method outperforms the 
standard method. 
 
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