A Novel FastSLAM Framework Based on 2D Lidar for Autonomous Mobile Robot

Abstract

The autonomous navigation and environment exploration of mobile robots are carried out on the premise of the ability of environment sensing. Simultaneous localisation and mapping (SLAM) is the key algorithm in perceiving and mapping an environment in real time. FastSLAM has played an increasingly significant role in the SLAM problem. In order to enhance the performance of FastSLAM, a novel framework called IFastSLAM is proposed, based on particle swarm optimisation (PSO). In this framework, an adaptive resampling strategy is proposed that uses the genetic algorithm to increase the diversity of particles, and the principles of fractional differential theory and chaotic optimisation are combined into the algorithm to improve the conventional PSO approach. We observe that the fractional differential approach speeds up the iteration of the algorithm and chaotic optimisation prevents premature convergence. A new idea of a virtual particle is put forward as the global optimisation target for the improved PSO scheme. This approach is more accurate in terms of determining the optimisation target based on the geometric position of the particle, compared to an approach based on the maximum weight value of the particle. The proposed IFastSLAM method is compared with conventional FastSLAM, PSO-FastSLAM, and an adaptive generic FastSLAM algorithm (AGA-FastSLAM). The superiority of IFastSLAM is verified by simulations, experiments with a real-world dataset, and field experiments.

Full text

electronics
Article
A Novel FastSLAM Framework Based on 2D Lidar for
Autonomous Mobile Robot
Xu Lei 1,2, Bin Feng 1,2 , Guiping Wang 1,2,*, Weiyu Liu 1,2 and Yalin Yang 1,2
1The School of Electronics and Control Engineering, Chang’an University, Xi’an 710064, China;
xulei@chd.edu.cn (X.L.); fengbin.dev@outlook.com (B.F.); liuweiyu@chd.edu.cn (W.L.);
yangyalin1994@outlook.com (Y.Y.)
2The National Traffic Information and Control Virtual Simulation Experimental Teaching Center,
Chang’an University, Xi’an 710064, China
*Correspondence: gpwang@chd.edu.cn
Received: 24 March 2020; Accepted: 21 April 2020; Published: 24 April 2020


Abstract:
The autonomous navigation and environment exploration of mobile robots are carried out
on the premise of the ability of environment sensing. Simultaneous localisation and mapping (SLAM)
is the key algorithm in perceiving and mapping an environment in real time. FastSLAM has played an
increasingly significant role in the SLAM problem. In order to enhance the performance of FastSLAM,
a novel framework called IFastSLAM is proposed, based on particle swarm optimisation (PSO). In this
framework, an adaptive resampling strategy is proposed that uses the genetic algorithm to increase
the diversity of particles, and the principles of fractional differential theory and chaotic optimisation
are combined into the algorithm to improve the conventional PSO approach. We observe that the
fractional differential approach speeds up the iteration of the algorithm and chaotic optimisation
prevents premature convergence. A new idea of a virtual particle is put forward as the global
optimisation target for the improved PSO scheme. This approach is more accurate in terms of
determining the optimisation target based on the geometric position of the particle, compared to an
approach based on the maximum weight value of the particle. The proposed IFastSLAM method
is compared with conventional FastSLAM, PSO-FastSLAM, and an adaptive generic FastSLAM
algorithm (AGA-FastSLAM). The superiority of IFastSLAM is verified by simulations, experiments
with a real-world dataset, and field experiments.
Keywords:
FastSLAM; genetic algorithm; resampling; particle swarm optimisation; fractional
differential; chaotic; robot
1. Introduction
Owing to the rapid development of autonomous mobile robots, simultaneous localisation and
mapping (SLAM) [
1
] has emerged as a crucial technology in a great many applications, such as
self-driving, exploration, and navigation. SLAM can help a robot acquire information on the surrounding
environment through a self-positioning process in an unknown environment, and gradually builds
an incremental map. The robot performs the SLAM process iteratively in order to eliminate uncertain
factors and to estimate the position accurately. Due to the high accuracy and high frequency of lidar,
it is very suitable for observation of complex environments. SLAM based on lidar is therefore widely
used in practical robot applications. Lidar-based SLAM technologies include 2D lidar SLAM and 3D
lidar SLAM. The 3D lidar can identify more details of the environmental space and is suitable for
high-precision application scenarios, such as autonomous vehicles, but its application is expensive. The
2D lidar has high measurement accuracy in the two-dimensional plane, which can meet the navigation
and positioning requirements of mobile robots. Compared with 3D Lidar, it has lower cost and has the
possibility of wide application.
Electronics 2020,9, 695; doi:10.3390/electronics9040695 www.mdpi.com/journal/electronics

Electronics 2020,9, 695 2 of 25
At present, the most widely used SLAM frameworks based on 2D lidar include the approach
based on extended Kalman filter (EKF-SLAM) [
2
,
3
] and that based on Rao-Blackwellised particle filter
(RBPF-SLAM, also known as FastSLAM) [
4
]. EKF-SLAM ignores the higher-order terms of the Taylor
expansion when performing state estimation, resulting in a large uncertainty in the state estimation of
the robot. As part of the flow of the algorithm, the covariance matrix must be continuously calculated,
which creates increasing computational complexity as the maps expand. In contrast, FastSLAM is
a SLAM framework based on a particle filter, which was inspired by early probabilistic experiments by
Murphy [5] and Thrun [6]. The particle filter in FastSLAM takes advantage of Monte Carlo sampling
technology, which can properly estimate non-linear systems with better state accuracy.
In this paper we focus on the FastSLAM based on 2D lidar. The main drawback of FastSLAM is the
particle degeneracy problem, which arises due to insufficient particle diversity during resampling [
7
].
To overcome the issue of particle degeneracy, researchers have proposed many optimisation algorithms
to improve the resampling process in FastSLAM. Cugliari [
8
] introduced an unscented Kalman filter
into FastSLAM and proposed a resampling strategy based on adaptive selection. Liu [
9
] used an
adaptive fading extended Kalman filter to compute the proposal distribution in FastSLAM, and showed
that the distribution was closer to the posterior position of the mobile robot and that the degree of
particle degradation was reduced. Zhang [
10
] used a nonlinear adaptive square-root unscented Kalman
filter to replace the particle filter in FastSLAM to solve the problem of particle degradation, and found
that the accuracy and reliability of the algorithm were improved. While these algorithms have made
some progress compared to conventional FastSLAM, they fail to make effective use of the information
carried by all particles, and do not use collaborative optimisation methods to solve the issue of particle
degeneracy [11].
The genetic algorithm (GA) [
12
] is a commonly used method of ensuring the diversity of particles
before the resampling step. Conventional resampling algorithms suffer from particle degeneracy
problems, since higher-weight particles are repeatedly selected and lower-weight particles are deleted
after resampling. As a result, the remaining particles cannot closely approximate the true state of the
robot [
13
]. Due to the crossover and mutation steps used, the GA can improve the particle degeneracy
problem to a certain extent. Tai-Zhi [
14
] proposed a new FastSLAM algorithm based on revised
genetic resampling and a square-root unscented particle filter, in which a fast Metropolis–Hastings
algorithm was used as the mutation operator and was combined with traditional crossover to form
a new resampling method. Khairuddin [
15
] integrated the GA and PSO into FastSLAM, and overcame
the particle depletion problem by improving the accuracy of FastSLAM in terms of robot and landmark
set position estimation. However, in this approach, the scalar parameters of GA are undetermined,
and need to be tuned empirically. To solve these problems, we propose an adaptive genetic resampling
algorithm in which the crossover and mutation coefficients are determined adaptively based on the
distribution of particle weights. The new resampling algorithm is called AGA-Resampling.
PSO [
16
] is a group algorithm that optimises a problem by iteratively updating the particle set using
local and global optimal values. Global optimal solution can be obtained with high probability. It has
been proved that the PSO algorithm can effectively improve positioning accuracy in the problem of
positioning of mobile robots [
17
]. Tdor [
18
] introduced the PSO algorithm into SLAM for the first time,
while Havangi [
19
] used PSO to optimise the proposal distribution. PSO makes the particles aggregate
at locations with high posterior probability before the weights are updated, and the impoverishment of
particles can be prevent. Ye [
20
] proposed a random weight PSO strategy to improve the FastSLAM
algorithm, to avoid particle degeneracy and maintain particle diversity. Seung [
21
] proposed a relational
FastSLAM approach that integrated the PSO algorithm into the calculation of the weights and formation
of the particles. Zuo [
22
] proposed a new FastSLAM method based on quantum-behaved PSO to
improve the proposal distribution of particles and optimise the estimated particles. Due to the
collaborative optimisation of PSO, the accuracy of SLAM was significantly improved. However,
there still are some problems in with FastSLAM based on the PSO algorithm: The most typical of
these is that the algorithm can easily fall into a local optimum, and another is the low calculation

Electronics 2020,9, 695 3 of 25
speed. To further enhance the algorithm, we propose an improved PSO algorithm FCPSO, in which
the fractional differential is employed to speed up the evolution of the PSO algorithm and chaotic
optimisation is utilised to address the problem of premature convergence.
After particle filtering process in FastSLAM is complete, the map and the robot’s path are stored
in separate particles, each of which stores different information. The robot autonomously chooses one
particle from the particle set to represent the results of FastSLAM. In most cases, the particle that has
the highest importance weight is selected to represent the results [
23
]; however, the particle with the
highest weight does not always produce the best result, and this cannot be guaranteed. Nosan [
24
] has
proved that representation by means of particles provides a better result with a higher probability than
when the particle with the highest weight is used alone. Hence, we define a virtual particle based on
the mean of the particles, and the global optimisation value of the FCPSO algorithm is calculated using
the position of this virtual particle. The improved FCPSO algorithm is then introduced into FastSLAM
based on AGA-Resampling, which not only maintains the diversity of the particles but also alleviates
the particle degeneracy problem in FastSLAM. We call this improved FastSLAM method IFastSLAM.
The remainder of this paper is organised as follows. In Section 2, we describe the standard
FastSLAM algorithm, and the proposed improved algorithm is introduced. In Section 3, we discuss
simulation experiments and field experiments. Finally, a summary and future prospects are given in
Section 4.
2. Methodology
2.1. FastSLAM Algorithm Based on AGA-Resampling
SLAM is the process by which the robot draws the environment map and estimates its position [
1
].
A schematic diagram of this approach is shown in Figure 1. The mobile robot uses LIDAR to observe
unknown landmarks by moving within the environment. The variables described below are defined at
time t:
xt: state vector that represents the location and orientation of the robot
ut: The control input that drives the robot from state xt−1to state xt
mt: A vector describing the position of the ith landmark
zt: The observation of the ith landmark from the robot
Besides, we define the following collections:
x1:t={x1,x2,...,xt}={x1:t−1,xt}: The trajectory of the robot
u1:t={u1,u2,...,ut}={u1:t,ut}: The control input of the robot
m={m1,m2,...,mn}: The collection of landmark
z1:t={z1,z2,...,zt}: The collection of landmarks’ observation
n1:t={n1,n2,...,nt}: The correspondence variable of landmarks
The core idea of the FastSLAM algorithm is to calculate the posterior probability of the robot
trajectory and map by factorisation, as shown in
p(x1:t,m|z1:t,u1:t,n1:t) = p(x1:t|z1:t,u1:t,n1:t)
N
∏
n=1
p(mn|x1:t,z1:t,n1:t). (1)
In FastSLAM, the Rao-Blackwellised particle filter is used to calculate the robot’s trajectory,
which is represented by
p(x1:t|z1:t
,
u1:t
,
n1:t)
. The possible trajectory of the robot is dictated by
particles. The map is represented by a series of landmarks that follow a Gaussian distribution,
which is updated using the observation model. Based on the premise that the position of the robot is
known, the extended Kalman filter is used to estimate the landmarks of the map, which are represented
by p(mn|x1:t,z1:t,n1:t).

Electronics 2020,9, 695 4 of 25
Figure 1. Schematic diagram of simultaneous localisation and mapping (SLAM).
2.1.1. Basic Flow of FastSLAM
Based on information from LIDAR and an odometer, the basic steps of the FastSLAM algorithm
are as follows:
1. Initialisation of the particles.
We initialize N particles, each of which has an initial weight ω(i)
0=0.
2. Retrieval.
The initial step involves retrieval of a particle representing the posterior at time
t−
1 and the
sampling of the robot’s position at time t. The position of the robot is predicted by sampling
from a proposal distribution
π
, and is denoted as
x(i)
t
for the
i
th particle. Normally, the proposal
distribution generally adopts the probabilistic motion model p(xt|xt−1,ut).
3. Importance weight.
The importance weight for the
i
th particle
ω(i)
t
represents the proportion between the target
distribution
p(x(i)
1:t|z1:t
,
u1:t−1
,
n1:t)
and the proposal distribution
π
mentioned above. It is defined
as follows:
ω(i)
t=p(x(i)
1:t|z1:t,u1:t−1,n1:t)
π(x(i)
1:t|z1:t,u1:t−1,n1:t),i=1, . . . , N(2)
4. Resampling.
We define the effective number of particles Ne f f as follows:
Ne f f =1
∑N
i=1(ω(i)
t)2. (3)
By comparing
Ne f f
with the threshold set in advance, it can be determined whether to perform
resampling. The threshold selected here is
Nth =N/
2, where
N
denotes the total number of
particles. When
Ne f f >Nth
, resampling is performed, in which higher weighted particles are
copied and lower weighted particles are discarded.
5. Measurement update.

Electronics 2020,9, 695 5 of 25
The posterior of the map feature estimation
p(mn|x1:t
,
z1:t)
is updated based on the robot trajectory
x1:tand the observation z1:tof each particle.
The process of the standard FastSLAM is graphically depicted in Figure 2.
(a) (b)
(c) (d)
Figure 2.
The process of the standard FastSLAM. (
a
) Sampling. (
b
) Measurement. (
c
) Importance
weight. (d) Resampling.
2.1.2. Particle Degeneracy Problem in FastSLAM
In the resampling phase, it is essential to remove those particles with large estimation errors.
However, the resampling process leads to insufficient particle diversity: Only the higher weighted
particles are replicated, while the lower weighted particles are rejected, and the diversity of the particles
is therefore reduced.

Electronics 2020,9, 695 6 of 25
The particle degeneracy problem can be effectively solved by adding those particles that can
accurately estimate the robot’s position. However, since the position of the robot is unknown, it is
difficult to maintain good particles. Only when the particle swarm is able to share the most likely
position of the robot can a sufficient number of good particles be maintained.
2.1.3. Adaptive GA Resampling
In this section, we propose an adaptive strategy by introducing an improved resampling method
based on the GA [
25
]. Before resampling is applied, higher weighted particles are used to correct for
lower weighted particles through the crossover and mutation operations of the GA. The adaptive
strategy is used to determine the crossover degree and mutation probability for the GA. Compared
with traditional particle filtering, this resampling method based on an adaptive GA can effectively
improve the diversity of the particle swarm, and this can to some extent solve the particle degradation
problem of FastSLAM.
We assume that the particle sets are represented as
x(i)
t,ω(i)
t
(i= 1, ..., N), where
ω(i)
t
represents
the normalised particle weights. We sort particle weights
ω(i)
t
(i= 1, ..., N) in descending order and
store them in W:
W=nω(1)
t,ω(2)
t,...,ω(N)
to. (4)
Then we define a threshold WTbased on Ne f f :
WT=W(Nef f ). (5)
where W(i)represents the ith weight in the set W.
The particles are divided into two groups by WTas follows:
x(i)
t∈XL,i f ω(i)
t≤WT
XH,i f ω(i)
t>WT
(6)
where
XL
and
XH
are particle sets that contain the lower and higher weight particles, respectively.
We assume that
x(i)
t,L
and
x(i)
t,H
represent the particles from
XL
and
XH
, respectively. If
x(i)
t,c
represents
the modified particles, the formulation of the crossover operator can be given by:
x(k)
t,c=µx(i)
t,L+ (1−µ)x(j)
t,H(7)
where
i=
1, . . .
NL
,
j=
1, . . .
NH
, and
k=i+j
.
NL
and
NH
denote the numbers of particles contained
in
XL
and
XH
, respectively. The parameter
µ∈[
0, 1
]
represents the crossover degree of the particles.
The greater the value of
µ
, the more information will be transferred from
x(i)
t,L
to
x(i)
t,c
and the less
information will be transferred from
x(j)
t,H
to
x(i)
t,c
. Especially when
µ
=1,
x(i)
t,c=x(i)
t,L
, no crossover occurs.
A mutation operator is applied to further increase the diversity of the particles. We assume that
x(i)
t,mrepresents the mutated particles. It may happen on the modified particles x(i)
t,caccording to:
x(i)
t,m=2x(j)
t,H−x(i)
t,c,i f rl≤PM
x(i)
t,c,i f rl>PM
(8)
where
rl
is a random variable drawn from a uniform distribution on [0,1].
PM
denotes the mutation
probability, which needs to be set in advance; if PM= 0, no mutation will occur.
An adaptive selection strategy is proposed here in order to determine the crossover degree
µ
and
the variation probability
PM
. It is well known that the smaller the variance in the particle weights,
the better the approximation to the true posterior distribution. We assume that
σ2
represents the
variance in the particle weights. Mathematical theorems have shown that for a sequence in the range

Electronics 2020,9, 695 7 of 25
[0,1], the variance is between zero and 1/4 [
26
], and the proof of this theorem is listed in Appendix A.
We can therefore draw the conclusion that when the weights of the particles are within the range [0,1],
the variance
σ2
is within the range [0,1/4]. The higher the value of
σ2
, the more crossover needs to be
performed. When
σ2
= 0, the quality of the particles is good enough that no crossover is necessary.
The relationship between µand σ2can therefore be represented as:
µ=1−4σ2(9)
When
σ2
= 0, the parameter
µ
= 1, meaning that no crossover occurs. Conversely, when
σ2
= 1/4,
the parameter
µ
= 0, meaning that a higher degree of crossover is needed to mitigate the degeneracy
problem of the particles.
Mutation is applied to the particles
x(i)
t,c
which are obtained from crossover with a certain
probability PM. In general, Pmin = 0.005 and Pmax = 0.01 [27].
When crossover is applied, the weights of the particle set also change. The variance in the new
particle weights is expressed as
σ2
∗
. Similarly, the probability of mutation can also be determined
based on
σ2
∗
; the greater the value of
σ2
∗
, the more mutation need to be performed, and the higher
the probability of mutation
PM
. In particular, when
σ2
∗
= 1/4, the mutation probability
PM=Pmax
,
and when
σ2
∗
= 0, the mutation probability
PM=Pmin
. We therefore propose an adaptive selection
strategy for PMas follows:
PM=Pmin + (Pmax −Pmin)·4σ2
∗(10)
In summary, we propose an improved resampling method based on the adaptive GA, which is
referred to here as AGA-Resampling. This approach is used to replace the classical resampling
of FastSLAM.
2.2. FastSLAM Optimised by FCPSO
2.2.1. PSO Improved by Fractional Differential
The PSO algorithm is derived from animal predation behavior. In this algorithm, the particle
swarm is randomly initialised using the fitness function to evaluate the solution, and the optimal
solution is found by iteration [28].
We assume that
Xi= (xi1
,
xi2
, . . . ,
xiN )
represents the position of the
i
th particle,
Pi=
(pi1
,
pi2
,...,
piN )
represents the local optimal of the particles, and
Pg= (pg1
,
pg2
,...,
pgN )
represents
the global optimisation of the particles. After t iterations, the velocities of the particles are updated by
Equation (11) and the positions are updated using Equation (12):
vt
id =ωivt−1
id +c1r1(pid −xt
id ) + c2r2(pgd −xt
id )(11)
xt+1
id =xt
id +vt
id. (12)
In Equation (11),
d
denotes the dimension, where
d=
1, ...,
N
;
vid
is the d-dimensional component
of the velocity of the
i
th particle;
xid
is the d-dimensional component of the position of the
i
th particle;
pid
is the d-dimensional component of
Pi
;
pgd
is the d-dimensional component of
Pg
;
t
is the number of
iterations;
ω
is the inertia factor that determines the global and local optimisation ability;
c1
represents
the individual learning factor of each particle;
c2
represents the collective learning factor for each
particle; and
r1
and
r2
are random numbers between zero and one. In Equation (12), because the
sample period between
t
and
(t+
1
)
is 1, the moving distance in unit time is
vt
id ×
1, so the position
xt+1
id
can be calculated with the sum of a position
xt
id
plus a velocity
vt
id
. The standard PSO algorithm
has the problems of slow convergence and easy to fall into local convergence. So it is necessary to
make improvements.
The convergence speed of PSO is limited by its algorithm structure. Since fractional-order model
has superior memory characteristics, fractional differential theory is of great assistance to the dynamic

Electronics 2020,9, 695 8 of 25
evolution of particles in a particle swarm [
29
]. In this part, we introduce the fractional differential into
the PSO algorithm to change the convergence speed.
The fractional differential equation [
30
] for the discrete-time domain defined by Grunwald–Letnikov
is as follows:
Dα[x(t)] = 1
Tα·
r
∑
k=0
(−1)kΓ(α+1)x(t−kT)
Γ(k+1)Γ(α−k+1)(13)
where
Γ
denotes the gamma function,
T
denotes the sampling period, and
r
denotes the cut-off order
of the equation.
The speed update formula for the PSO algorithm is rearranged as follows:
vt
id −ωivt−1
id =c1r1(pid −xt
id ) + c2r2(pgd −xt
id ). (14)
Assuming
ω
= 1, this equation is a discrete form of differentiation. Assuming
T=
1 and following
Pires et al. [30] , we transform Equation ( 14 ) as follows:
Dα[x(t)] = c1r1(pid −xt
id ) + c2r2(pgd −xt
id ). (15)
Non-integer calculus cannot be directly processed by existing simulation tools, so it is necessary
to use finite-dimensional ideas to approximate fractional calculus [
31
]. Only the first four items are
considered, and Equation ( 15 ) can be expressed as:
vt=αv(t−1)+1/2αvt−2+1/6α(1−α)vt−3+1/24αv(1−α)(2−α)t−4+c1r1(pid −xt
id ) + c2r2(pgd −xt
id ).(16)
By applying the idea of the fractional differential, the order of velocity differential
α
can be
generalised to a fraction between zero and one, which will result in a more stable change and
a longer memory effect. The value of
α
greatly affects the specific performance of the PSO algorithm:
The dynamic performance of the PSO algorithm is weakened when
α
is small, while if the value of
α
is
relatively large, the particle diversity will be increased and the global optimal value will be obtained
easier. Based on previous experiments [
32
], the fractional order
α
is set to 0.6 where the algorithm
converges at the fastest rate.
2.2.2. PSO Improved by Chaotic Optimisation
PSO is a relatively simple and fast-running algorithm. However, it cannot search for the global
optimal solution when all particles move toward a local optimum, and this gives rise to the premature
convergence phenomenon. The introduction of chaotic optimisation into the search algorithm can help
particles escape from local optima by generating many neighboring solutions for the local optimum [
33
].
Hence, we use a chaotic search to update a certain particle whenever it is judged that PSO is showing
premature convergence.
The judgment of premature convergence is based on the variance in the fitness of the particle
swarm [34]. The variance in the fitness of the particle swarm is denoted by σ2
f:
σ2
f=
N
∑
i=1
fi−favg
f, (17)
f=max1≤i≤m|fi−favg |,max1≤i≤m|fi−fav g |>1
1, others (18)
where
f
is the normalization factor,
fi
the fitness of the
i
th particle, and
favg
the average fitness of the
particle swarm.

Electronics 2020,9, 695 9 of 25
The fitness variance
σ2
f
reveals the aggregation degree of the particle swarm. The smaller the value
of
σ2
f
, the more aggregated the particle swarm. When
σ2
f
is less than a constant C, chaotic operation
is performed.
The chaotic search space in this work is determined by the optimal position
pg=
(pg1
,
pg2
, . . . ,
pgn)
, which is found when the PSO algorithm falls into local convergence. The chaotic
variables is generated by the Logistic function [35] as show in Equation (19).
yq0=4yq(1−yq),q=1, 2, . . . , n. (19)
N chaotic variables
yq0(
0
<yq0<
1
)
can be obtained by substituting n initial values
yq(
0
<yq<
1
)
with small differences into Equation (19). A solution vector to the objective function is then generated
according to Equation (20).
zq0=yq0−pgq,q=1, 2, . . . , n. (20)
In summary, when the particles fall into a local optimum, the following steps are performed:
(1) Use the Logistic function to generate the chaotic sequence with initial chaotic variable with values
in the range [0,1]. (2) Calculate the fitness of each chaotic sequence and record the sequence with
the best fitness value. (3) Replace the local optimal particle of the current particles with the chaotic
sequence with the best fitness.
The standard PSO is improved by both fractional differential theory and chaotic optimisation.
This improved algorithm is referred to here as fractional differential and chaotic particle swarm
optimisation (FCPSO). The particle update speed is accelerated, and the hereditary nature of PSO is
increased. The particle swarm is also updated when premature convergence occurs, which increases
the diversity of the particle swarm.
2.2.3. The Proposal of the Virtual Particle
In general particle filter algorithm based on a standard PSO algorithm, the particle fitness in the
PSO algorithm is calculated based on the weight of the particle, and the particle with the largest weight
is taken as the global optimisation target [
36
]. However, it is proved that the mean value of the particle
positions is closer to the real location than the particle with the maximum weight [
24
]. We therefore define
a virtual particle according the mean value of the particle positions in this part. The virtual particle is not
a real particle in PSO, and it is just used to determine a global optimum for PSO optimisation.
The position of the virtual particle is calculated as follows:
xvir =1
N
N
∑
i=1
xi, (21)
where xirepresents the initial position of the ith particle.
We introduce FCPSO optimisation into the FastSLAM algorithm. In particular,the fitness function
is calculated based on the position of the particle instead of the importance weight:
fi=kxi−xvir k, (22)
For each particle, we calculate the local optimum as follows:
Pi=xi. (23)
We select the virtual particle as the global optimum target rather than the largest weighted particle
as follows:
Pg=xvir. (24)
The main process of FCPSO is shown in Figure 3.

Electronics 2020,9, 695 10 of 25
We use the FCPSO algorithm to optimise the FastSLAM algorithm. First, we use the particle filter
based on AGA-Resampling to resample the possible robot position. We then define the virtual particle
and use it as the optimisation target of FCPSO, which attracts particles to move to the actual position
of the robot. The proposed method, which uses FastSLAM based on AGA-Resampling and is further
optimised by FCPSO, is called IFastSLAM. This approach is different from the standard FastSLAM
scheme because it uses an additional process to compensate for particle degeneracy. This compensation
process serves as a key strategy. The overall scheme of the algorithm is shown in Figure 4, and
pseudo-code is given in Algotithm 1.
IFastSLAM has several advantages: AGA-Resampling can effectively improve the diversity of
particles, and the iteration speed of PSO with fractional differential optimisation is increased, further
improving the speed of SLAM. In addition, PSO optimised by chaos can avoid local convergence and
can effectively improve the accuracy of SLAM.
(a) (b)
(c) (d)
Figure 3.
Particle update process of FCPSO. (
a
) Information sharing between particles. (
b
) Define the
virtual particle to get the global optimisation goal. (
c
) Deal with particle premature convergence using
chaotic optimisation. (d) Particles move toward global optimisation target.

Electronics 2020,9, 695 11 of 25
Figure 4. Scheme of IFastSLAM.

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Algorithm 1: IFastSLAM
1for m=1toNdo
2Retrieval. Retrieval mth particle from the particle set;
3Prediction. Sample a new pose x[k]
t∼p(xt|xt−1,µt,n1:t);
4Measurement update. For each observed feature zi
tidentify the correspondence j for the
measurement zi
t, and incorporate the measurement zi
tinto the corresponding EKF, by
updating the mean µ[k]
j,tand covariance ∑[k]
j,t;
5Importance weight. Calculate the importance weight ω(i)
tfor the new particle;
6AGA-Resampling. Use crossover and mutation operators [Equations (7) and (8)] where
the coefficients and mutation probability are given by [Equations (9) and (10)] to modify
the robot state;
7end
8Define the virtual particle: calculate the position of the virtual particle. [Equation (21)] ;
9for m=1toNdo
10 Initialize the velocity of the mth particle;
11 Initialize the velocity of the mth particle;
12 Initialize the fitness value of the mth particle;
13 end
14 for k=1toNdo
15 for m=1toNdo
16 Calculate the fitness value of the mth particle. [Equation (22)] ;
17 end
18 Determine the global optimal. [Equation (24)] ;
19 for m=1toNdo
20 Determine the local optimal of the mth particle. [Equation (23)] ;
21
Update the velocity of the m_th particle using Fractional Differential. [Equation (16)] ;
22 Update the position of the m_th particle. [Equation (12)] ;
23 end
24 Calculate the aggregation extent of the particle set. [Equation (17)] ;
25 if premature convergence occurred then
26 Implement Chaotic optimisation. [Equations (19) and (20)];
27 end
28 end
3. Experiments
In this section, we perform experiments with the simulation environment, the Victoria Park
dataset, and field experiments, in order to study the performance of the improved algorithm.
We performed simulation experiments using Matlab on Intel(R) Core (TM) i5-8250 CPU@1.6 GHz
RAM 8 GB PC. We used a robot equipped with LiDAR (SLAMTEC RPLIDAR) and an odometer for
field experiments.
3.1. Simulation
3.1.1. Simulation Setup
The simulation was performed using MATLAB to verify the accuracy and stability of IFastSLAM.
We developed the SLAM simulator based on the source code published by Bailey [37].

Electronics 2020,9, 695 13 of 25
In this work, we assumed that the robot moved at a constant speed. The model for the motion is
as follows: 


xt+1
yt+1
θt+1


=


xt
yt
θt


+


v∆tcos(G+θt)
v∆tsin(G+θt)
v∆tcos(G)/WB 

+Wt(25)
where
[xt
,
yt
,
θt]T
is the vector of the robot pose;
v
is the robot speed;
G
is the steering angle of the
robot;
WB
is the wheelbase of the robot;
∆t
is the sampling time interval;
Wt
is the system noise,
and Wt∼N(0, Q).
The observation model used in the simulation experiment is as follows:
zt="ρi
βi#="q(mx−xt)2+ (my−yt)2
atan(my−yt,mx−xt)−θt#+Vt(26)
where
[mx
,
my]T
is the coordinate of the
i
th landmark detected by the robot;
ρi
and
βi
represent
the distance and angle, respectively, of the
i
th landmark from the current robot position;
Vt
is the
observation noise; and Vt∼N(0, R).
The simulation environment is illustrated in Figure 5a. It is in effect a search area of size
250
×
200 m. In this simulation experiment, 17 waypoints and 35 landmarks were randomly set.
The red trajectory represents the designated simulation path, the red ‘o’ is the waypoint, and the blue
asterisk is the stationary landmark. The robot starts from (0,0) and moves counterclockwise along
the red trajectory shown in Figure 5a to obtain the estimated trajectory and the estimated landmark.
Figure 5b shows a screenshot of the process of simulation at a certain moment. In Figure 5b, the black
triangle represents the robot, and the green circle represents the observation range of the robot, which
is centered on the robot. The robot can observe landmarks within a certain measuring angle inside the
green circle. The estimated trajectory is represented by a black line, and the estimated landmark is
denoted by a red asterisk.
(a) (b)
Figure 5.
Simulation interface. (
a
) Simulation environment. (
b
) A screenshot of the process of simulation.
The relevant parameters in the simulation are as follows. The initial position of the robot is
[
0, 0, 0
]T
; its speed is
v=
3 m/s; the maximum steering angle is 20
◦
; The sampling time is 0.025 s; error
in the speed is
σv=
0.3 m/s; the sampling time of the laser radar is 0.2 s; the maximum measuring
distance is 30 m, the measurement error in the distance is
σρ=
0.1 m; and the error in the angle is
σβ= 1◦; The systematic noise Qand the observation noise Rare as follows:
Q="0.320
0(3π
180 )2#,R="0.120
0(π
180 )2#(27)
We use one hundred particles for simulation. Resampling is performed when Ne f f >50.

Electronics 2020,9, 695 14 of 25
3.1.2. Results and Analysis
We simulated FastSLAM, PSO-FastSLAM (FastSLAM based on PSO with particle weights),
AGA-FastSLAM (FastSLAM based on AGA-Resampling), and IFastSLAM with the same simulation
parameters in order to verify the superiority of the improved algorithm. The estimated robot path and
landmark location are shown in Figure 6.
(a)
(b)
Figure 6. Cont.

Electronics 2020,9, 695 15 of 25
(c)
(d)
Figure 6.
The simulation result of four SLAM algorithms. (
a
) FastSLAM. (
b
) Particle swarm
optimisation (PSO)-FastSLAM. (
c
) Adaptive generic FastSLAM algorithm (AGA-FastSLAM). (
d
)
IFastSLAM.
From the results, we can conclude that the estimated landmark location of IFastSLAM is essentially
consistent with the actual landmark location, and the estimated path substantially coincides with
the actual path. The proposed IFastSLAM algorithm is therefore more accurate than the other
three algorithms.

Electronics 2020,9, 695 16 of 25
In order to compare these four algorithms more intuitively, we compared the location estimation
error of the robot in X axis and Y axis, as shown in Figure 7. The positioning error is calculated
as follows:
error(t) = v
u
u
t(
N
∑
i=1
xi(t)/N−xtrue )T(
N
∑
i=1
xi(t)/N−xtrue )(28)
The positioning error reflects the absolute error between the actual position of the robot and the
average position of the particle swarm.
As shown in Figure 7a, the positioning error in X axis is less than 12 m in FastSLAM, less
than 7 m in the standard PSO-FastSLAM algorithm optimised by particle weight, less than 4 m in
AGA-FastSLAM, and within 3 m in IFastSLAM. The values of the estimation accuracy of the three
improved FastSLAM algorithms are higher than that of the standard FastSLAM. In PSO-FastSLAM,
the positioning accuracy is effectively improved due to the use of PSO optimisation. In AGA-FastSLAM,
adaptive genetic resampling is used to handle the particle degeneracy problem, and the positioning
accuracy is therefore also effectively improved. Furthermore, since the position of the virtual particle is
used as the optimisation target, and the premature convergence phenomenon is solved in the FCPSO
algorithm, the positioning error in IFastSLAM is smaller than in the other three algorithms. Similarly,
in the IFastSLAM algorithm, the positioning error in Y axis is also smaller than for the other three
algorithms, and the estimation accuracy is higher, as can be seen from Figure 7b.
As shown in Table 1, the IFastSLAM algorithm requires a shorter calculation time than
PSO-FastSLAM and IFastSLAM for the same number of particles; the reason for this is that the
use of fractional differential theory improves the evolution speed of the particle swarm. In addition,
the calculation time increases as the particle number increases.
(a) (b)
Figure 7. The estimation error. (a) Robot position error in X axis. (b) Robot position error in Y axis.
Table 1. Comparison of the RMSE and running time of SLAM algorithms.
Particle Number 20 50 100
FastSLAM Time (s) 30.49 76.13 153.57
RMSE (m) 8.43 7.90 6.84
PSO-FastSLAM Time (s) 34.02 81.35 159.05
RMSE (m) 6.31 5.10 4.79
AGA-FastSLAM Time (s) 33.95 80.96 158.84
RMSE (m) 5.12 3.78 3.25
IFastSLAM Time (s) 31.17 78.05 156.98
RMSE (m) 2.85 2.10 1.71

Electronics 2020,9, 695 17 of 25
As shown in Figure 8, the estimation accuracy of IFastSLAM is significantly higher than the other
three algorithms for the same number of particles. We can also clearly see that FCPSO-FastSLAM
maintains a low RMSE with fewer particles than the other three algorithms, since the use of the virtual
particle in the IFastSLAM algorithm, it can maintain more effective particles than the FastSLAM
algorithm and can therefore resolve the problem of particle degeneracy to some extent.
Compared with standard FastSLAM, IFastSLAM sacrifices a certain amount of calculation
time, but its estimation accuracy is significantly improved, and the running time is lower than for
PSO-FastSLAM and AGA-FastSLAM. In this sense, the IFastSLAM has better overall performance.
Figure 8. Simulation environment.
3.2. Verification Using the Victoria Park Dataset
In order to further compare the performance of IFastSLAM with PSO-FastSLAM and
AGA-FastSLAM, we applied several algorithms to the Victoria Park dataset [
38
]. This dataset is
provided by the Australian Center for Field Robotics (ACFR) and is widely applied by researchers to
verify the effectiveness and evaluate the performance of the SLAM algorithm in the real environment.
The mobile test platform used by the ACFR is shown as Figure 9. It is equipped with LiDAR,
an odometer, and GPS, and is driven a distance of 4 km over about half an hour. LiDAR is used to
provide observation information on the features of the road (trees in the park). The odometer is used
to provide information on the linear speeds and heading angles of vehicles. The GPS sensor is used to
provide reference vehicle location information. In this dataset, the errors caused by the inaccuracy of
GPS data is ignored, and the vehicle trajectory is approximately represented by GPS data [39].
Figure 9. The mobile test platform used by the ACFR.

Electronics 2020,9, 695 18 of 25
The validation results of several algorithms on the Victoria Park dataset are shown in Figure 10a–c,
where the blue line represents the real GPS track of the vehicle, the red line represents the vehicle track
estimated by the algorithm, and the yellow point represents the position of the landmark feature point.
From the white box in Figure 10, it can be seen that the IFastSLAM algorithm obtains an estimation
result that is closer to the real GPS track.
The RMSE of the robot trajectory and CPU running times of three algorithms on the Victoria Park
dataset are shown in Table 2. We can draw the conclusion that the RMSE of IFastSLAM is lower than
the other two algorithms, and it also requires less running time. Overall, we have verified the superior
performance of IFastSLAM on the Victoria Park dataset.
(a)
(b)
Figure 10. Cont.

Electronics 2020,9, 695 19 of 25
(c)
Figure 10.
The Victoria Park dataset validation results. (
a
) PSO-FastSLAM. (
b
) AGA-FastSLAM.
(c) IFastSLAM.
Table 2. Comparison of RMSE and running time with Voctoria Park dataset.
Algorithms Running Time (s) RMSE (m)
PSO-FastSLAM 1359.22 16.37
AGA-FastSLAM 1491.30 11.49
IFastSLAM 1406.57 9.15
3.3. Field Experiment
We used the mobile robot shown in Figure 11a to perform field experiments to test the
practicability of IFastSLAM. The robot was equipped with an internal odometer and a SLAMTEC
RPLIDAR A2 LiDAR, and moved at a speed of 0.3 m/s to create a grid map in real-time using LiDAR
and odometer data. The National Key Laboratory of Virtual Simulation of Chang’an University was
selected as the experimental site, as shown in Figure 11b.
(a) (b)
Figure 11.
Field experiment condition. (
a
) Mobile robot used in the field experiment. (
b
) Field
experimental site.

Electronics 2020,9, 695 20 of 25
The robot was driven along the specified path within the experimental site to build an
environmental map using radar scanning. Three algorithms were compared in this field experiment.
Figure 12a shows the map built by the PSO-FastSLAM algorithm, while Figure 12b,c show the
maps built by AGA-FastSLAM and IFastSLAM in the same environment, respectively. There are
obvious defects in the grid map built by PSO-FastSLAM as can be seen from Figure 12a, and a large
deviation appears at the boundary of the map. Similarly, blurring occurs at the edge of the grid map
built by the AGA-FastSLAM, as shown in Figure 12b. In contrast, the grid map built by the IFastSLAM
is more precise, as shown in Figure 12c.
(a)
(b)
(c)
Figure 12. Grid map by different algorithm. (a) PSO-FastSLAM. (b) AGA-FastSLAM. (c) IFastSLAM.
The results reveal that the map generated by IFastSLAM has fewer defects and higher precision,
while the maps generated by the PSO-FastSLAM and AGA-FastSLAM algorithms are of low quality.
Figure 13 shows the estimated trajectory for the different algorithms. We can see that the
trajectory estimated by the IFastSLAM algorithm correctly follows the ground truth, while the trajectory
estimated by PSO-FastSLAM and AGA-FastSLAM obviously deviates from the ground truth of the

Electronics 2020,9, 695 21 of 25
robot in some places. IFastSLAM has better localisation accuracy than the other two algorithms.
In addition, the running times of PSO-FastSLAM, AGA-FastSLAM, and IFastSLAM are 592.53 s,
647.09 s, and 629.74 s respectively, meaning that the computational efficiency of IFastSLAM is clearly
higher. Table 3show the RMSE of different algorithms in field experiment. It can be seen that the
RMSE of IFastSLAM algorithm is the smallest. It can be therefore concluded that IFastSLAM has better
overall performance.
Table 3. Comparison of RMSE for different algorithms in field experiment.
Algorithms PSO-FastSLAM AGA-FastSLAM IFastSLAM
RMSE (m) 0.27 0.16 0.09
Figure 13. The estimated trajectory for different algorithms in the field experiment.
4. Conclusions
In the resampling process of the conventional FastSLAM process, the accuracy decreases with the
number of iterations, due to particle degeneracy. We propose a framework named IFastSLAM that can
enhance the performance of FastSLAM based on PSO. In this framework, an adaptive GA is introduced
at the resampling stage, and a novel algorithm called FCPSO is proposed based on fractional differential
and chaotic optimisation to improve PSO-FastSLAM. Compared with conventional frameworks,
IFastSLAM has four significant benefits: (i) The diversity of particles is enriched by an adaptive GA;
(ii) the evolutionary speed of the PSO algorithm is increase; (iii) the problem of premature convergence
in PSO is alleviated; and (iv) the global optimisation accuracy is improved by using a virtual particle
as the optimisation target, and the errors in the trajectory and landmark estimations are reduced.
We compare our proposed IFastSLAM approach with FastSLAM, PSO-FastSLAM,
and AGA-FastSLAM on a series of SLAM issues. This comparison was made based on simulations, an
experiment with the Victoria Park dataset provided by ACFR, and a field experiment with a mobile
robot, showing promising results. In future work, we plan to integrate IFastSLAM into 3D maps and
carry out further study on path planning and collision detection for mobile robots. In addition, the
parameters used in FCPSO are taken directly from other studies, and we will readjust the parameters
in the next study to make them more suitable for robot systems.

Electronics 2020,9, 695 22 of 25
Author Contributions:
Conceptualization, X.L. and B.F.; methodology, X.L. and B.F.; software, B.F.; validation,
X.L., B.F. and W.L.; formal analysis, X.L., B.F. and W.L.; investigation, X.L., B.F., G.W. and Y.Y.; resources, X.L. and
G.W.; data curation, B.F. and Y.Y.; writing—original draft preparation, X.L. and B.F.; writing—review and editing,
G.W. and W.L.; visualization, B.F. and Y.Y.; supervision, G.W.; project administration, X.L. and G.W.; funding
acquisition, X.L. and G.W. All authors have read and agreed to the published version of the manuscript.
Funding:
This work was funded by the Key Research and Development Programs of Shaanxi Province (grants
2019ZDLGY15-04-02, 2018ZDCXL-GY-05-04, 2018ZDCXL-GY-05-07-02) and the Fundamental Research Funds for
the Central Universities (grants 300102329502, 300102320502).
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this paper:
Symbols Implication
SLAM Simultaneous localization and mapping
FastSLAM Rao-Blackwellized particle filter SLA
EKF-SLAM extended Kalman filter SLAM
AGA-Resampling adaptive genetic algorithm resampling
PSO Particle Swarm Optimisation
FCPSO fractional differential and chaotic PSO
IFastSLAM FastSLAM based on AGA-Resampling and optimized by FCPSO
xtthe state vector describing the position and orientation of the robot
utthe control vector, drive the robot from state xt−1to state xt
mithe vector describing the position of the ith landmark
ztthe observation of the ith landmark from the vehicle at time t
ω(i)
tthe importance weight of ith particle
πthe proposal distribution of particles
Ne f f the effective number of particles
µthe crossover degree of the particles
PMthe mutation probability
Pithe local optimal position of the particles of PSO
Pgthe global optimal position of the swarm of PSO
vt
id the velocity of ith particle at time t
xt
id the position of ith particle at time t
σ2
fthe variance of particles’ population fitness
fithe fitness of the ith particle
favg the average fitness of the particle swarm
fthe normalization factor
mxthe mean of the particle position
vthe moving speed of the robot
Gthe steering angle of the robot
WB the wheelbase of the robot
W(t)the system noise at time t
V(t)observation noise at time t
Qsystematic noise covariance
Rthe observation noise covariance
Appendix A. Mathematical Proofs in the Paper
Theorem A1. For a sequence in the range [0,1], the variance is in the range [0,1/4].
Proof of Theorem A1.
Assume that there is a sequence consists of n random variables in the range
[0,1], namely
X1
,
X2
, ...,
Xn
. The average value of this sequence is denoted by Y. The variance is denoted
by Z. According to the calculation formula of variance,
Z= [(X1−Y)2+ (X2−Y)2+... + (Xn−Y)2]/n

Electronics 2020,9, 695 23 of 25
= [(X1)2+ (X2)2+... + (Xn)2−2Y(X1+X2+... +Xn) + nY2]/n
= [(X1)2+ (X2)2+... + (Xn)2−2Y(nY) + nY2]/n
= [(X1)2+ (X2)2+... + (Xn)2−nY2]/n
= [n(X1)2+n(X2)2+... +n(Xn)2−(X1+X2+... +Xn)2]/n2
When
X2
,
X3
, ...,
Xn
are regarded as fixed values, the above formula is a quadratic function of
opening upward for
X1
. So when
X1
= 0 or
X1
= 1 (one of the two endpoints of the domain), the
variance Z takes the maximum value.
Similarly, when X2= 0 or X2= 1, the variance Z takes the maximum value.
...
Similarly, when Xn= 0 or Xn= 1, the variance Z takes the maximum value.
Therefore, only when the values of all random variables take values in 0 or 1, the variance Z takes
the maximum value. We will seek the maximum variance next. Assuming that in the sequence, the
number of random variables with a value of 0 is a, and the number of random variables with a value
of 1 is (n−a), then the average Y= (n−a)/n.
Z=[a(0−Y)2+ (n−a)(1−Y)2]/n
=a[0−(n−a)/n]2/n+ (n−a)[1−(n−a)/n]2/n
=[a(n−a)2+ (n−a)a2]/n3
=a(n−a)/n2
=(−a2+na)/n2
=[−(a−n/2)2+n2/4]/n2
≤1/4
Therefore, when
a=n/
2, in other words, the number of random variables with a value of 0 is
n/2, and the number of random variables with a value of 1 is n/2. Under these circumstances, the
variance Z takes the maximum value of 1/4. Because Z is a positive number, the variance Z is therefore
in the range [0,1/4].
In conclusion, for a sequence in the range [0,1], the variance is in the range [0,1/4].
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