International Journal of Control, Automation, and Systems (2015) 13(4):978-985
DOI 10.1007/s12555-014-0149-6
ISSN:1598-6446 eISSN:2005-4092
http://www.springer.com/12555
A Grey Probability Measure Set based Mobile Robot Position
Estimation Algorithm
Peng Wang, Qi-Bin Zhang, and Zong-Hai Chen*
Abstract: A novel algorithm for the position estimation of a mobile robot is proposed, which is based
on probability statistics and grey system theory. For the proposed algorithm, the grey probability
measure set is established, which is composed of a base set and the corresponding grey probability
measure. The base set is used to represent the uncertain information and the grey probability measure
distributes probability on the base set. Moreover, the integrating rules are formulated using the grey
probability measure set and the q-satisfied rule to estimate the position of a mobile robot. In addition to
providing a new way of representing the uncertain information, results of the proposed algorithm are
also more reliable in the presence of error and outliers. The algorithm is applied in the position estimation of a Pioneer 3-DX robot in a corridor-office environment. Experimental results have shown that
the estimation accuracy of the algorithm is as good as that of the particle filter and better than that of
the extended Kalman filter.
Keywords: Grey probability measure set, mobile robot, position estimation, probability fusion.
1. INTRODUCTION
Position estimation, which is one of the most important issues in autonomous navigation, aims to update the
robot position by integrating the uncertain information
[1,2]. Two categories of approaches have been proposed
for this purpose: the probability statistics based approach
and the set-membership based approach [3,4].
The extended Kalman filter (EKF) and the particle filter (PF) are widely used algorithms based on probability
statistics. The EKF linearizes the state equation around
working points and then approximates the state posterior
probability distribution as a Gaussian distribution [3]. It
has been successfully applied to position estimation [5-8].
However, if the state equation is highly nonlinear, linearization may reduce the estimation accuracy [3].
Meanwhile, the performance of the EKF can be adversely affected if state perturbations and measurement
errors are not Gaussian. Recently, the PF has attracted
considerable attention in position estimation because it
performs well even when the posterior probability distribution is not unimodal. It has been widely used by mobile robots for position estimation [9-14]. However, the
__________
Manuscript received April 9, 2014; revised September 22,
2014; accepted October 16, 2014. Recommended by Associate
Editor Hyun Myung under the direction of Editor Myotaeg Lim.
This paper was supported by the National Natural Science
Foundation of China (Grant No. 61375079).
Peng Wang and Qi-Bin Zhang are with the Department of
Automation, University of Science and Technology of China,
West Campus, Hefei, Anhui 230027, P. R. China (e-mails:
{pwang, zqb101}@mail.ustc.edu.cn).
Zong-Hai Chen is with the Department of Automation, University of Science and Technology of China, West Campus, Hefei,
Anhui 230027, P. R. China (e-mail: [email protected]).
* Corresponding author.
© ICROS, KIEE and Springer 2015
performance of the PF is determined by the number of
particles and the weight reallocation strategy. A large
number of particles have to be used when the measurement is uncertain, which affects the real-time implementation of the PF; meanwhile, the accuracy of the PF can
be reduced due to particle degradation [15]. Set membership, however, has been considered as another approach
to solve the position estimation problem. In such an approach, the bounded interval is used to represent the uncertain information, and the position estimation is computed by solving the constraint satisfaction problem
[4,16-18]. However, it is usually difficult to determine
bounds of the interval owing to the lack of useful information, which in turn limits the application of setmembership based approach in situations that require
high precision [19]. As the interval bounds are fixed,
measurement error and outliers may not be considered
[17,18].
In this paper, we estimate the position by combining
probability statistics and grey system theory. Grey system theory [20], proposed by Deng, simulates the cognition process of human beings using the uncertain information. In grey system theory, the uncertain information
is represented by an interval grey number. A whitening
function is also constructed in case a representative value
of the interval grey number is needed. A method of representing the uncertain information employing grey system theory has been proposed [21] and applied in localization and path planning [22]. Huang et al. [23] presented the idea of combining grey system theory and
probability statistics, but the interval grey number and
whitening function were both defined as being fixed. If
there is measurement error or outliers, it may not be possible to use the method. In this paper, we represent and
process the uncertain information using the grey probability measure set (GPMS), which contains a base set
A Grey Probability Measure Set based Mobile Robot Position Estimation Algorithm
and the corresponding grey probability measure. The
base set is used to represent the uncertain information,
while the grey probability measure distributes probability
on the base set. While the robot is moving, the grey
probability measure set and the q-satisfied rule are used
to represent and integrate the uncertain information. A
finer base set can then be generated and a better position
thus estimated.
The remainder of the paper is organized as follows:
Section 2 describes the problem to be solved. Section 3
gives a brief introduction to grey system theory, defines
basic concepts and arithmetic operations of the GPMS
and presents the GPMS-based algorithm. Section 4 presents the experimental results and analysis. Section 5
concludes our work.
2. PROBLEM DESCRIPTIONS
With the assumption that the initial pose is known, position estimation extracts the robot position from sensor
measurements [2]. The robot state equation and observation equation are
xk +1 = f (x k ,u k , m k ),
y k = g(xk , n k ),
(1)
(2)
where xk +1 is the state vector at time step k + 1, u k is
the input vector, y k is the observation vector , and m k
and n k are the state noise and observation noise.
In this paper, the robot is assumed to move without
slippage in a horizontal planar environment. Let x k =
( xk , yk ,θ k ) be the state vector at time step k. The state
equation can then be written as
⎧ xk +1 = xk + T ⋅ vk ⋅ cos θ k
⎪
⎨ yk +1 = yk + T ⋅ vk ⋅ sin θ k
⎪θ = θ + T ⋅ ω ,
k
k
⎩ k +1
(3)
in which T = tk +1 − tk is the sampling time and vk and
ωk are the velocity and rotation velocity, respectively.
Odometer and sonar devices are typical sensors that
are widely used, but their measurement errors are greater
than those of laser or visual sensors. Thus, it is essential
to establish a robust method of representing and integrating the uncertain information and estimating the robot
position. On the basis of the assumption and state equations given above, we propose a position estimation algorithm that performs well even with measurement error
and outliers.
3. GREY PROBABILITY MEASURE SET
3.1. Interval grey number
The interval grey number is used to describe a grey
quantity with a true value in an interval, but we do not
know exactly what it is. Usually, given a grey quantity g,
the corresponding interval grey number is defined as
g ∈ a = [ a, a ],
(4)
where a and a are the lower and upper bounds. A
979
whitening function is usually defined on a to indicate the
preference of each value in the interval. In grey system
theory, the whitening function of a is usually defined as
µ a ( g ) → [0,1], g ∈ [ a, a ].
(5)
3.2. GPMS and arithmetic operations
3.2.1 Basic concepts and definitions
Let Ω be a given domain with interval grey number
x ⊆ Ω, and Ux be the power set of x. We then have the
following definition.
Definition 1: Let X = { A1 , A2 , , Am }, Ai ∈ U x , m ∈
N. X is a base set deduced from x if 1) Px ( Ai ∩ A j ) = 0,
m
i ≠ j; 2) x = ∪ i =1 Ai ; 3) Given any M ∈ F , there
exists a set of Ai ∈ X such that M = ∪ Ai .
We conclude that F is a sigma algebra generated from
X, which is denoted σ ( X ). For any Ai ∈ X , if
Px ( Ai ) ≥ 0, then Px and ( X , Px ) are defined as the
grey probability measure and a GPMS on x, respectively.
Remark 1: Px acts as the whitening function of x. Different GPMSs of x can be generated according to different X.
3.2.2 Arithmetic operations
Let (Oi , Poi ) be the GPMS defined on oi with
i
O = {S1i , S2i , , Smi i }, Smi i ∈ U oi , i = 1, , n, mi ∈ N. We
then have the following definition.
Definition 2: o and O are the interval grey number
and finer base set generated from oi and Oi, i = 1,..., n, if
n
1) o = ∩ i =1 oi ;
{
n
n
}
2) O = S j S j = ∩ i =1 Ski i ,1 ≤ ki ≤ mi , j ≤ ∏ i =1 mi .
1
n
Theorem 1: Given o = = o = o and the finer
base set O generated from O1 , , O n , we have σ (Oi )
⊆ σ (O), i = 1. , n.
Proof: We only need to prove that for any Oi,
{S1i , S2i , , Smi i } ⊆ σ (O).
Given Ski i ∈ Oi , 1 ≤ ki ≤ mi , we find that Ski i =
∪ (Skii ∩ (i ∩ j ≠i Smj j )),j j = 1. , n.
As Ski ∩ (∩ j ≠i Sm ) ∈ O, we have Ski i ∈ σ (O);
j
therefore, σ (Oi ) ⊆ σ (O), i = 1, , n.
Definition 3: Given o and O, we define Po as the integrated grey probability measure on σ(O) if
1) ∀Si ∈ O, Po ( Si ) ≥ 0;
2) Po (o) = 1;
3) let Q be the finer base set generated from
O1 , , O n-1 , which are arbitrarily selected from O1 ,
, O n . Suppose that Li , Ll ∈ Q, Si , Sl ∈ O, and Smn k
∈ O n , where Si = Li ∩ Smn k , Sl = Ll ∩ Smn k . If Pq ( Li )
≥ Pq ( Ll ), then Po ( Si ) ≥ Po ( Sl ).
Let Poi be the grey probability measure on σ (Oi ),
i = 1, , n. The integrated grey probability measure Po
on σ (O) is then computed according to the following
rules.
1) Let O i = {Shii = Ski i ∩ o | 1 ≤ ki ≤ mi , hi ≤ ji ≤ mi }, in
which we suppose that Shii = Shii ∩ o if hi ≤ ji , and
Peng Wang, Qi-Bin Zhang, and Zong-Hai Chen
980
Shii = S hii ∩ o = ∅ if hi > ji , i = 1, , n.
2) The grey probability measure on σ (O i ) is defined as
( )
Poi Shi i
( )
Po i Shii =
( )
1 − ∑ h > j Poi Shii
i
i
.
(6)
Proof: We need only prove that for any Si(,qp+1) ⊆
O(pq +1) , there exist Sh( q,m) ⊆ Om( q ) and Sk( q,n) ⊆ On( q ) such
that Si(,qp+1) = Sh( q,m) ∩ Sk( q,n) .
q +1 i ,( q +1)
q +1
Si(, p ) = ∩ j =1 Sh j , p
ij
( )
Po S j =
∏ i =1 Po i ( )
,
n
1 − ∑ n S i =∅ ∏ i =1 Po i ( Shii )
∩ i =1 hi
where S j = ∩
n
Si
i =1 hi
=∩
Shii
i
is the finer base set generated from Oi1 , , O q and
Si( q,)i is the ik-th element of Oi( q ) . P q is the grey probk
oi
ability measure on σ (Oi( q ) ). According to (7), we have
( )
o
q
in which
k
∑i
k
1− ∑
Pq
oi
∩ j =1 Smji j =∅ ∏
q
i
i
P ⎛S j
j =1 oi j ⎜ mi j
⎝
q
q +1
⎞
⎟
⎠
,
( ) = 1.
Theorem 2: Let O ( q ) and O( q +1) be the finer base
sets generated from {Oi( q ) | 1 ≤ i ≤ Cnq } and {O (jq +1) | 1 ≤
j ≤ Cnq +1}, and σ (O ( q ) ) and σ (O ( q +1) ) be the corresponding sigma algebras. We then have σ (O ( q +1) ) ⊆
σ (O ( q ) ).
k
i j ,( q )
)]
ij ,p
q +1
∩ j = 2 Sh
q
q +1
∩ [(∩ k =1 oipk ) ∩ (∩ k = 2 oipk )]
i ,( q )
)∩(
ij , p
q
= [(∩ j =1 Shj
i ,( q )
)∩(
ij ,p
q +1
= [∩ j =1 ( Shj1
1
1
q
∩ [∩ j
2 =1
k
q +1
∩ k =2 oip )]
i ,( q )
i j ,m
q
q
∩ k =1 oip )]
∩ [(∩ j = 2 Shj
i
( Shj2
k
∩ om( ) )]
q
,( q )
i j2 ,n
q
∩ on( ) )]
= Sh( ,m) ∩ Sk( , n) .
q
q
It is obvious that Sh( q,m) ⊆ Om( q ) and S k( q,n) ⊆ On( q ) . We
thus have σ (O ( q +1) ) ⊆ σ (O ( q ) ).
Remark 2: Theorem 2 intends to illustrate that with
more information accumulated (q increasing), the estimation accuracy improves.
3.4. Processing uncertainty information
3.4.1 Generation of the GPMS
Suppose that f1 , , f nk are nk sets of measurements
of the robot position at time step k. From the measurements, we can estimate the position using the mean values f1 , , f nk and the corresponding root-mean-square
errors (RMSEs) σ1 , , σ nk . We also know that f1 ,
, f nk can be replaced by y ik = [ fi − cσ i , f i + cσ i ],
i = 1, , nk , such that the position belongs to y ik with
probability Pri (c). Therefore, we define the base set of
y ik as
{
}
Yki = S ij , j = 1, , 2c
= {[ fi − cσ i , fi − (c − 1)σ i ],
(8)
Si( q,)i
k
q +1
∩ k =1 oip )
i ,( q )
)∩(
ij , p
q
q
i
q
⎧ q
⎫
q
q
Oi( ) = ⎨ Si( ,i) Si( ,i) = ∩ j =1 Smj , ik ≤ ∏ j =1 mi j ⎬
k
k
ij
⎩
⎭
oi
∩ (∩ k =1 oipk )]
= [(∩ j =1 Shj
3.3. q-satisfied GPMS
Suppose that o1 , …, o n are interval grey numbers
and O1 , , O n are the base sets. For 1 ≤ q ≤ n, we can
obtain Cnq subsets of o1 , …, o n denoted as oi1 ,
i
i
i
…, o q with base sets Oi1 , , O q , where O j =
i
i
{S1 j , , Smj }, 1 ≤ j ≤ q, 1 ≤ i ≤ Cnq .
ij
i
Definition 4: oi1 , …, o q are q-satisfied if they satq
ij
(q)
isfy oi = ∩ j =1 o ≠ ∅, 1 ≤ q ≤ n. oi( q ) is called a qsatisfied interval grey number.
{oi( q ) | i ≤ M ≤ Cnq } are M q-satisfied interval grey
numbers.
P q Si( ,i) =
i ,( q +1)
ij ,p
= (∩ j =1 Shj
According to the above discussion and probability statistics, we have the following corollaries.
Corollary 1: With O and O i defined as above, we
have σ (O i ) ⊆ σ (O
), i = 1, , n.
i
Corollary 2: o and (Oi , Poi ) are defined as above.
Suppose that there exists a one-to-one mapping
b = h(a) such that bi = h(oi ). Let Bi = { A1i , , Ami i }
and (Bi , Pbi ) be the base set and the GPMS. Aki and
S ki are in one-to-one correspondence with each other
through b = h(a), and we then have Pbi ( Aki ) =
Poi ( Ski ), k = 1, , mi .
∏ j =1 P i j
q +1
q +1 i ,( q +1)
)∩(
ij , p
(7)
Si .
i =1 ki
⎛ Sij ⎞
⎜ mi j ⎟
⎝
⎠
q +1
∩ o(p ) ]
= ∩ j =1[ Shj
n
q
i ,( q +1)
ij ,p
= ∩ j =1[ Shj
3) The grey probability measure on σ(O) is then defined as
n
q +1
(9)
,[ fi + (c − 1)σ i , f i + cσ i ]},
where i = 1, , nk .
Without loss of generality, we set Pri (0) = 0. The
grey probability measure of Yki is then defined as
( )
(
Pyi S ri = Pyi S 2i c −( r −1)
k
k
)
= ( Pri (c − (r − 1)) − Pri (c − r ) ) / 2,
where r = 1, , c.
(10)
A Grey Probability Measure Set based Mobile Robot Position Estimation Algorithm
981
Table 1. The GPMS-based position estimation algorithm.
Algorithm 1: GPMS-based position estimation algorithm
n
Input: y1k , y 2k , , y kk , a time widow with length w.
Output: state x k and the corresponding GPMS.
( )
(q)
j
Step 1: compute y k ,hk = ∩ j =1 y k and the GPMSs of y k , hk , hk = 1, , mk .
−1 ( q )
Step 2: compute x k , h = g k (y k , h ) and the GPMSs of x k ,hk , hk = 1, , mk .
i
q
q
k
k
Step 3: for q = mk → 1
q
(q)
if xk ,lk = ∩ j=1 x k ,hk j ≠ ∅, break.
endfor
(q)
(q)
store N ( k , q ) non-empty x k ,lk and compute the GPMSs of x k ,lk , lk = 1, , N ( k , q ) .
Step 4: for lk − w = 1 to N ( k − w, q )
…
for lk −1 = 1 to
N ( k − 1, q )
( )
(q)
q
= f k − w:k ( x k − w,l ) ∩ ∩1≤i ≤ w f k −i:k ( x(k −)i ,l
if k < w, xk ,lk = f0:k x0,l0
)
else xk ,lk
k -w
k −i
endfor
…
endfor
w
Step 5: compute the GPMSs of x k ,lk , lk = 1, , N k , N k = ∏ i =1 N ( k − i, q ).
Step 6: for q = N k → 1
q
(q)
if xk , pk = ∩ j=1 xk ,lk j ≠ ∅, break.
endfor
(q)
store Pk non-empty x k , p .
(q)
(q)
Step 7: x k = x k , pk , x k , pk
i
i
k
is determined according to the constraints in 3.4.3. Compute the GPMS of x k .
Yki and Pyi constitute the GPMS ⎛⎜ Yki , Pyi
k
k
⎝
⎞.
⎟
⎠
(
3.4.2 At time step k
q
i
q
Let y (k ,h) = ∩ j=1 y kj , hk = 1, , mk , be mk q-satisfied
k
interval grey numbers generated from y1k , y 2k , , y nkk . If
fk (⋅) and g k (⋅) are one-to-one mappings, then the
q
state x k ,hk corresponding to y (k ,h) is defined as
x k , hk =
g k−1
(y ),
(q)
k , hk
k
hk = 1, , mk .
(11)
Let Yki be the base set of y ik , and (Yki , Pyi ) be the
k
GPMS with i = 1, , nk . The GPMS on y (kq,h) can then
k
be computed using Definition 3 and Theorem 2, and the
grey probability measure on x k , hk can be computed
according to Corollary 2.
To reduce the effect of measurement error and outliers
on position estimation, we can further compute all x(kq,l)
k
and the GPMSs. q is the maximum value that ensures
q
q
x(k ,l) = ∩ j=1 x k ,hk ≠ ∅, lk = 1, , N ( k , q ) .
k
)
xk ,lk = f0:k x0,l0 , if k < w,
j
3.4.3 Time window
We define a time window with length w at time step k.
x(kq,l) , lk = 1, , N (k , q) are defined as above. Then,
k
according to [24], we have
(
)
(q)
(12)
(
x k ,lk = f k − w:k x k − w,lk -t ∩ ∩1≤i ≤ w f k −i:k x(k −)i ,l
if k ≥ w,
q
k −i
),
(13)
where fk1:k2 +1 (xk1 ) = fk2 (fk1:k2 (xk1 )), k1 ≤ k2 , and fk :k (x k )
= xk .
To obtain the final estimation of the state, we first
q
compute x(kq, )p = ∩ j=1 xk ,lk ≠ ∅, pk = 1, , Pk , with
k
j
maximum q. We then decide the final result x k = x(kq, )p
ki
according to the two constraints on x(kq, )p :
ki
1) the maximum interval length;
2) a centralized probability distribution.
xk and x(kq, )p share the same GPMS, which can be
ki
computed according to Corollary 2, Definition 4, and
Theorem 2.
3.4.4 Position estimation algorithm
On the basis of the above discussion, we propose the
GPMS-based position estimation algorithm in Table 1.
4. EXPERIMENT AND ANALYSIS
4.1. Experimental settings
Experiments were conducted in a corridor-office envi-
Peng Wang, Qi-Bin Zhang, and Zong-Hai Chen
982
40
x/m
30
21
20
20
19
10
(a)
18
0
40
0
45
x
EKF
95% CI
50
10 20 30 40 50 60 70 80 90
Time/s
(a) x.
(c)
0.4
(b)
Fig. 1. Experimental settings: (a) environment, (b) robot, and (c) environmental layout.
95% CI
y/m
0
-0.2
-0.4
0
10 20 30 40 50 60 70 80 90
Time/s
(b) y.
Fig. 2. Position estimates of the robot obtained with the
EKF.
40
30
x/m
4.2. Experimental results and analysis
The EKF, PF (with 100 particles), and GPMS-based
algorithm were applied in the position estimation of the
mobile robot. Figs. 2 through 4 show the estimation results obtained with the three algorithms. The 95% confidence interval (CI) is also given. For the PF, the CI is
computed for the effective particles obtained at each recursive step.
Figs. 2 through 4 show that the EKF, PF, and GPMSbased algorithm all perform well. However, owing to
measurement error and outliers, the estimates of the EKF
and PF may exceed the 95% CI. In contrast, the position
estimates of the GPMS-based algorithm are all within the
95% CI, and the results are therefore more reliable than
those of the EKF and PF.
Figs. 5 and 6 present the estimation results and errors
of the three algorithms, respectively. It is clear that the
results obtained with the proposed algorithm are relatively stable. According to the layout of the environment,
we know that features along the x-axis are sparser than
those along the y-axis. Thus, errors along the x-axis are
greater, as errors along the y-axis can be well compensated. From the results presented in the figures, we can
conclude that the performance of our algorithm along the
x-axis is as good as that of the PF, and better than that of
the EKF; however, our algorithm performs better than
the EKF and PF along the y-axis.
To further verify the effectiveness of the new algo-
EKF
0.2
20
21
20
19
10
0
0
18
x
PF
95% CI
10 20 30 40 50 60 70 80 90
40
45
50
Time/s
(a) x.
0.6
y
PF
95% CI
0.4
y/m
ronment (Fig. 1(a)) with a Pioneer 3-DX robot (Fig. 1(b)).
The robot was expected to move along the center line of
the corridor (the x-axis in Fig. 1(c)). The robot was
equipped with a sonar ring of 16 sonar sensors (with
measurement range 10 cm–5 m), an odometer, and a gyroscope. In addition, a laptop with 2GB random access
memory and a 2.13-GHz processor was used to control
the robot. The floor was planar and slippage was therefore not considered. The state equation is given by equation (3). During the experiments, the robot moved along
the x-axis with mean speed of 0.4 m/s; i.e., v(k) = 0.4 m/s
and ω(k) = 0 rad/s. The sampling rate was 24 Hz.
In the experiments, we supposed that the robot could
extract and match available features.
y
0.2
0
-0.2
-0.4
0
10 20 30 40 50 60 70 80 90
Time/s
(b) y.
Fig. 3. Position estimates of the robot obtained with the
PF.
rithm, six experiments are conducted under the same
conditions. The RMSEs are listed in Table 2. From all
the experimental results given, we conclude that the
GPMS-based algorithm provides a new way of represent-
A Grey Probability Measure Set based Mobile Robot Position Estimation Algorithm
40
x
Bounds
0.7
95% CI
0.6
20
20
18
10
0.3
0.1
40
45
50
10 20 30 40 50 60 70 80 90
0
0
Time/s
y
10 20 30 40 50 60 70 80 90
Time/s
(a) x.
GPMS
Bounds
0.12
95% CI
EKF
PF
GPMS
0.1
0.2
0.08
y/m
y/m
GPMS
0.4
(a) x.
0.4
PF
0.2
16
0
0
EKF
0.5
22
x/m
x/m
30
GPMS
983
0
0.06
0.04
-0.2
-0.4
0
0.02
0
0
10 20 30 40 50 60 70 80 90
10 20 30 40 50 60 70 80 90
Time/s
Time/s
(b) y.
(b) y.
Fig. 4. Position estimates of the robot obtained with the
GPMS-based algorithm.
40
x/m
30
21
20
20
19
18
10
0
0
17
40
45
50
x
EKF
PF
GPMS
10 20 30 40 50 60 70 80 90
Time/s
(a) x.
0.4
y/m
0
-0.4
0
Table 2. RMSEs of the three algorithms.
State
x
y
x
y
x
y
x
y
x
y
x
y
EKF(m)
0.480
0.087
0.451
0.087
0.442
0.080
0.439
0.086
0.410
0.083
0.438
0.092
PF(m)
0.437
0.095
0.447
0.085
0.417
0.084
0.425
0.093
0.394
0.092
0.419
0.093
GPMS(m)
0.463
0.029
0.449
0.028
0.433
0.024
0.438
0.026
0.408
0.028
0.433
0.030
5. CONCLUSION
0.2
-0.2
Fig. 6. Errors of the three algorithms.
y
EKF
PF
GPMS
10 20 30 40 50 60 70 80 90
Time/s
(b) y.
Fig. 5. Position estimates comparison of the three algorithms.
ing the uncertain information while having almost the
same estimation accuracy as the PF and better accuracy
than the EKF.
On the basis of probability statistics and grey system
theory, a GPMS-based position estimation algorithm is
presented. The algorithm uses a base set to represent the
uncertain information, and employs a grey probability
measure to distribute probability on the base set. By
combining the GPMS and q-satisfied rule, measurements
are integrated to estimate mobile robot position more
accurately. It turns out that the new algorithm is robust
with respect to error and outliers. We also demonstrate
that the new algorithm is more reliable compared to the
EKF and PF. The new algorithm was applied to position
estimation of a mobile robot in a corridor-office environment, in comparison with the EKF and PF. The new
algorithm was shown to provide a more accurate estimation of the position of the mobile robot than the EKF,
and performed as well as the PF.
The base set is the most important component of the
984
Peng Wang, Qi-Bin Zhang, and Zong-Hai Chen
GPMS. In this paper, the base set was built with a limited
number of measurements. When the measurement error
and outliers become significant, the performance of the
algorithm may be affected. We will therefore search for a
more robust method of establishing a base set in the future.
REFERENCES
S. Thrun, D. Fox, W. Burgard, and F. Dellaert,
“Robust Monte Carlo localization for mobile robots,” Artificial Intelligence, vol. 128, no. 1, pp. 99141, May 2001.
[2] I. J. Cox, “Blanche-an experiment in guidance and
navigation of an autonomous robot vehicle,” IEEE
Trans. on Robotics and Automation, vol. 7, no. 2,
pp. 193-204, April 1991.
[3] G. Nassreddine, F. Abdallah, and T. Denoux, “State
estimation using interval analysis and belieffunction theory: application to dynamic vehicle localization,” IEEE Trans. on Systems, Man, and Cybernetics, vol. 40, no. 5, pp. 1205-1218, October
2010.
[4] M. Delafosse, A. Clerentin, L. Delahoche, and E.
Brassart, “Uncertainty and imprecision modeling
for the mobile robot localization problem,” Proc. of
the 20th IEEE International Conference on Robotics and Automation, pp. 4550-4555, 2005.
[5] M. Choi, J. Choi, and W. K. Chung, “Correlationbased scan matching using ultrasonic sensors for
EKF localization,” Advanced Robotics, vol. 26, no.
13, pp. 1495-1519, July 2012.
[6] L. Teslic, I. Skrjanc, and G. Klancar, “EKF-based
localization of a wheeled mobile robot in structured
environments,” Journal of Intelligent and Robotic
Systems, vol. 62, no. 2, pp. 187-203, May 2011.
[7] W. S. Moon, B. S. Cho, J. W. Jang, and K. R. Baek,
“A multi-robot positioning system using a multicode ultrasonic sensor network and a Kalman filter,” International Journal of Control, Automation
and Systems, vol. 8, no. 6, pp. 1349-1355, December 2010.
[8] H. Myung, H. K. Lee, K. Choi, and S. Bang, “Mobile robot localization with gyroscope and constrained Kalman filter,” International Journal of
Control, Automation and Systems, vol. 8, no. 3, pp.
667-676, June 2010.
[9] T. B. Kwon, J. H. Yang, and J. B. Song, “Efficient
and reliable Monte Carlo localization with thinning
edges,” International Journal of Control, Automation and Systems, vol. 8, no. 2, pp. 328-338, April
2010.
[10] S. H. Park, Y. J. Kim, and M. T. Lim, “Novel adaptive particle filter using adjusted variance and its
application,” International Journal of Control,
Automation and Systems, vol. 8, no. 4, pp. 801-807,
August 2010.
[11] F. Dellaert, D. Fox, W. Burgard, and S. Thrun,
“Monte Carlo localization for mobile robots,” Proc.
of the 14th IEEE International Conference on Robotics and Automation, pp. 1322-1328, 1999.
[12] D. Fox, “Adapting the sample size in particle filters
[1]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
through KLD-sampling,” International Journal of
Robotics Research, vol. 22, no. 12, pp. 985-1003,
December 2003.
R. Havangi, M. Teshnehlab, M. A. Nekoui, and H.
D. Taghirad, “A novel particle filter based SLAM,”
International Journal of Humanoid Robotics, vol.
10, no. 3, pp. 1-23, September 2013.
M. Pfingsthorn and A. Birk, “Simultaneous localization and mapping with multimodal probability
distributions,” International Journal of Robotics
Research, vol. 32, no. 2, pp. 143-171, February
2013.
Z. Khan, T. Balch, and F. Dellaert, “MCMC-based
particle filtering for tracking a variable number of
interacting targets,” IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 27, no. 11, pp.
1805-1819, November 2005.
A. Garulli and A. Vicino, “Set membership localization of mobile robots via angle measurements,”
IEEE Trans. on Robotics and Automation, vol. 17,
no. 4, pp. 450-463, August 2001.
L. Jaulin, “Robust set-membership state estimation;
application to underwater robotics,” Automatica,
vol. 45, no. 1, pp. 202-206, January 2009.
V. Drevelle and P. Bonnifait, “Localization confidence domains via set inversion on short-term trajectory,” IEEE Trans. on Robotics, vol. 29, no. 5,
pp. 1244-1256, October 2013.
U. D. Hanebeck, J. Horn, and G. Schmidt, “On
combining statistical and set-theoretic estimation,”
Automatica, vol. 35, no. 6, pp. 1101-1109, June
1999.
J. L. Deng, “Introduction to grey system theory,”
The Journal of Grey System, vol. 1, no. 1, pp. 1-24,
March 1989.
C. L. Chen, D. Y. Dong, Z. H. Chen, and H. B.
Wang, “Grey systems for intelligent sensors and information processing,” Journal of Systems Engineering and Electronics, vol. 19, no. 4, pp. 659-665,
August 2008.
S. J. Li, P. Wang, and Z. H. Chen, “Artificial potential field without traps based on gray qualitative
theory,” Control and Decision, vol. 28, no. 6, pp.
879-883, June 2013.
Y. L. Huang, Z. H. Chen, and W. S. Gui, “Gray
qualitative simulation,” The Journal of Grey System,
vol. 16, no. 1, pp. 5-20, March 2004.
L. Jaulin, “Set-membership localization with probabilistic errors,” Robotics and Autonomous Systems,
vol. 59, no. 6, pp. 489-495, June 2011.
Peng Wang received his B.S. degree
from the University of Science and
Technology of China (USTC) in 2010.
He is now a Ph.D. candidate at the Laboratory of Simulation and Intelligent
Control in the Department of Automation,
USTC. His research interests include
system modeling and simulation, mobile
robot localization, uncertain information
processing and knowledge representation.
A Grey Probability Measure Set based Mobile Robot Position Estimation Algorithm
Qi-Bin Zhang received his B.S. degree
from the University of Science and
Technology of China (USTC) in 2012.
He is now a Ph.D. candidate at the Laboratory of Simulation and Intelligent
Control in the Department of Automation,
USTC. His research interests include
mobile robot localization and landmark
extraction, knowledge representation and
interval analysis.
985
Zong-Hai Chen received his B.S. degree
from the University of Science and Technology of China (USTC) in 1988. He is
currently a professor at the Laboratory of
Simulation and Intelligent Control in the
Department of Automation, USTC. His
research interests include simulation and
optimization control of complex system,
theory and technology of intelligent system, and quantum control theory. Prof. Chen is now a member
of the Robotics Technical Committee of the International Federation of Automatic Control (IFAC).