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4598 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 69, NO. 11, NOVEMBER 2022
q-R´enyi Kernel Functioned Kalman Filter for Land
Vehicle Navigation
Jiangnan Xing , Tao Jiang ,Member, IEEE, and Yingsong Li ,Senior Member, IEEE
Abstract—In this brief, a q-R´enyi kernel functioned Kalman
filter (qRKFKF) is proposed based on the q-R´enyi kernel func-
tion which is to provide better flexibility and performance
in non-Gaussian environment. The concrete realization of the
proposed qRKFKF are created and analyzed in detail, and
its performance is presented and discussed. Three examples
are carried out to verify the performance of the proposed
qRKFKF for land vehicle navigation via numerical simula-
tions: large outliers in non-Gaussian noise, alpha-stable noise
and diverse noise distributions that combines Gaussian and
Laplace noises. Compared with Kalman filter (KF), Maximum
Correntropy Kalman Filter (MCKF) and minimum error entropy
KF (MEE-KF), the qRKFKF is superior to others for combating
non-Gaussian noises.
Index Terms—Kalman filter, q-R´enyi kernel function, non-
Gaussian measurement noise, large outlier.
I. INTRODUCTION
STATE parameter estimation plays a crucial role in the
fields of target tracking, simultaneous localization and
signal processing in the past decades [1], [2], [3]. To meet the
requirements, the well-known Kalman filtering (KF) theory is
one of the optimal filtering methods to solve the estimation
of dynamic linear system [4], whose criterion is minimum
mean square error (MMSE) and so does its related extended
algorithm [5]. However, it has two drawbacks. It can only
capture the second order statistical characteristic of the signal
interested [6] and is usually sensitive to large outliers [7]. The
performance of those algorithms are not quite satisfactory when
non-Gaussian environment is happened, especially when lager
outliers or diverse noise distributions occur in the system [8].
To expand the applications of the KF, several extended algo-
rithms are proposed. Kalman and Bucy devoted to the study
of extended Kalman filter (EKF) in nonlinear systems [9].
Julier et al. proposed unscented Kalman filter (UKF) [10]
Manuscript received 15 May 2022; revised 5 June 2022; accepted
11 June 2022. Date of publication 16 June 2022; date of current version
28 October 2022. This work was supported in part by the Shanghai Aerospace
Science and Technology Innovation Fund under Grant SAST2019-002, and
in part by the Key Laboratory of Advanced Marine Communication and
Information Technology, Ministry of Industry and Information Technology,
Harbin Engineering University, Harbin, China. This brief was recommended
by Associate Editor L. Gan. (Corresponding author: Yingsong Li.)
Jiangnan Xing and Tao Jiang are with the College of Information and
Communication Engineering, Harbin Engineering University, Harbin 150001,
China.
Yingsong Li is with the Key Laboratory of Intelligent Computing and Signal
Processing, Ministry of Education, Anhui University, Hefei 230601, China
(e-mail: liyingsong@ieee.org).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TCSII.2022.3183617.
Digital Object Identifier 10.1109/TCSII.2022.3183617
which is based on the unscented transformation instead of
the first-order linearization for nonlinear system. Arasaratnam
and Haykin constructed cubature Kalman filter (CKF) to solve
the problem of precision degradation of the UKF in high-
dimensional system filtering [11]. However, these extended
algorithms still use MMSE as their error criterion, so that they
are still sensitive to non-Gaussian noises [12]. Fortunately,
Chen et al. proposed Maximum Correntropy Kalman Filter
(MCKF) algorithm which uses the maximum correntropy cri-
terion (MCC) [13] as the error criterion and can deal with the
estimation problem under non-Gaussian noises [14]. Then, he
proposed minimum entropy error Kalman filter (MEE-KF) [15]
which is derived from minimum entropy error (MEE) crite-
rion [16]. There are some other class of algorithms proposed
to tackle the outliers from the perspective of adaptive filter,
such as Exponential Hyperbolic Cosine Robust Adaptive Filter
and Champernowne adaptive filter (CMAF) [17], [18], [19].
However, it is still impossible to solve the problem of diverse
noise distributions in the measurement.
In practical application, we should consider the background
of Gaussian noise, non-Gaussian noise and diverse noise dis-
tributions [20], [21]. To solve the above problems, q-R´enyi
kernel function is considered as the error criterion of the KF to
propose a q-R´enyi kernel function Kalman Filter (qRKFKF),
which can capture the second and higher order statistical char-
acteristic of the signal and combat large outliers. As a result,
the qRKFKF has advantages in dealing with the estimation
under non-Gaussian noise and diverse noise distributions obvi-
ously. Its performance is proved via three examples for land
vehicle navigation under different noise conditions which are
non-Gaussian noise, alpha-stable noise and diverse noise dis-
tributions, respectively. The results show that the performance
of the qRKFKF is superior to the other popular KFs for
combating non-Gaussian noises.
II. THE PROPOSED qRKFKF ALGORITHM
The proposed qRKFKF is different from the classical
MMSE promoted KF algorithm, which uses q-R´enyi kernel
function to model a new error criterion. Then, considerating a
dynamic linear system, the state and measurement equations
are modeled respectively as
x(l)=(l−1)x(l−1)+w(l−1), (1)
z(l)=F(l)x(l)+v(l), (2)
where x(l)is a Ndimensions state vector, z(l)is a measure-
ment vector with Mdimensions. and Fare state transition
1549-7747 c
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XING et al.: qRKFKF FOR LAND VEHICLE NAVIGATION 4599
matrix and observation matrix, respectively. In the Eqs. (1)
and (2), process noise w(l−1)and measurement noise v(l)
have zero mean, which are statistically independent. The
covariance matrices of w(l−1)and v(l)are defined in Eqs.
(3) and (4), respectively.
E[w(l−1)wT(l−1)]=W(l−1), (3)
E[v(l)vT(l)]=V(l). (4)
From Eqs. (1) and (2), we have
ˆ
x(l|l−1)
z(l)=I
F(l)x(l)+u(l), (5)
where Iis an identity matrix with Ndimensions, and u(l)is
defined as
u(l)=ˆ
x(l|l−1)−x(l)
v(l),(6)
and u(l)has the property which is shown below
E[u(l)uT(l)]=P(l|l−1)0
0V(l)
=BP(l|l−1)BPT(l|l−1)0
0Bv(l)BvT(l)
=B(l)BT(l), (7)
where
P(l|l−1)=(l−1)P(l−1|l−1)T(l−1)+W(l−1), (8)
where P(l|l−1)is the covariance matrix of system state
equation. B(l)is obtained by Cholesky decomposition of
E[u(l)uT(l)]. Both sides of Eq. (5) are premultiplied by
B−1(l),wehave
D(l)=Y(l)x(l)+e(l), (9)
where
D(l)=B−1(l)ˆ
x(l|l−1)
z(l),(10)
Y(l)=B−1(l)I
F(l),(11)
e(l)=B−1(l)u(l), (12)
E[e(l)eT(l)]=E[(B−1(l)u(l))(B−1(l)u(l))T]
=E[B−1(l)u(l)(u(l))T(B−1(l))T]
=E[B−1(l)B(l)(B(l))T(B−1(l))T]
=I.(13)
where Eis the expectation. According to Eq. (13), the residual
error e(l)are white noise process which satisfied the demand
of the KF.
After getting the residual error e(l), the optimal solution is
obtained via the q-R´enyi kernel functioned error criterion
ˆ
JqR(x)=1
L
L
i=1
(κq,σ (ei(l)), (14)
with
ei(l)=di(l)−yi(l)x(l). (15)
where di(l)is the ith element of D(l),yi(l)is the ith row
of Y(l), and L=N+M.Theq-R´enyi kernel function is
defined by
κq,σ (x(l), z(l))=1−q−1
3q−1x(l)−z(l)2
σ21
q−1
+
.(16)
where σis the kernel width, +represents select the maximum
value between the number in [1 −(q−1
3q−1)x(l)−z(l)2
σ2] and 0.
When q →1/3, q-R´enyi kernel function closes to impulse.
When q >1, q-R´enyi kernel function reduces to uniform dis-
tribution. When 1/3 <q<1, q-R´enyi kernel function is shaper
than Gaussian distribution, so that it has better performance
to combat non-Gaussian noises. Therefore, the value of qis
only considered in the region of (1/3,1)in this brief.
Then, the optimal solution to x(l)is achieved via maximiz-
ing the error criterion function. The optimum estimation of
x(l)is
ˆ
x(l)=arg max
y
1
L
L
i=1
(κq,σ (ei(l)), (17)
Then, the solution is obtained by solving
∂ˆ
JqR(x)
∂x(l)=0.(18)
Without any difficulty, we have
x(l)=(
L
i=1
(κq,σ (ei(l))2−qyT
i(l)yi(l))−1
×(
L
i=1
(κq,σ (ei(l))2−qyT
i(l)di(l)). (19)
In fact, Eq. (19) is a fixed-point equation of x(l)and usually
is written as
x(l)=f(x(l)). (20)
where
f(x(l)) =L
i=1kq,σ di(l)−yi(l)x(l)2−qyT
i(l)yi(l)−1
×L
i=1kq,σ di(l)−yi(l)x(l)2−qyT
i(l)di(l)(21)
Then, an iterative equation is achieved
ˆ
x(l)t+1=f(ˆ
x(l)t). (22)
The iterative Eq. (21) can also be rewritten as
x(l)=(YT(l)C(l)Y(l))−1(YT(l)C(l)D(l)), (23)
where
C(l)=Cx(l)0
0Cy(l),(24)
Cx(l)=diag(fc(e1(l)), . . . , fc(en(l))), (25)
Cy(l)=diag(fc(en+1(l)), . . . , fc(en+m(l))), (26)
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4600 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 69, NO. 11, NOVEMBER 2022
where
fc(ei(l)) =κq,σ (ei(l))2−q.(27)
Using the Eqs. (7) and (11), we have
Y(l)=B−1(l)I
F(l)
=Bp−1(l|l−1)0
0Bv−1(l) I
F(l)
=Bp−1(l|l−1)
Bv−1(l)F(l).(28)
Combining Eqs. (24) and (28), we obtain
(YT(l)C(l)Y(l))−1
=[(B−1
p)TCxB−1
p+(F)T(B−1
v)TCyB−1
vF]−1,(29)
where Bp,Bv,Cx, and Cyrepresent BP(l|l−1),Bv(l),Cx(l),
and Cy(l), respectively. Thus, the variables in Eq (29) are
(B−1
p)TCxB−1
p=A,FT=B,
(B−1
v)TCyB−1
v=C,F=D.(30)
Using the transformation of matrix inversion
(A+BCD)−1
=(A−1−A−1B(C−1+DA−1B)−1DA−1). (31)
We obtain
(YT(l)C(l)Y(l))−1=(A+BCD)−1
=(A−1−A−1B(C−1+DA−1B)−1DA−1)
=BpC−1
xBT
p−BpC−1
xBT
pFT
×(BT
vC−1
yBT
v+FBpC−1
xBT
pFT)−1×(FBpC−1
xBT
p). (32)
By the similar argument, we obtain
D(l)=B−1(l)ˆ
x(l|l−1)
z(l)
=B−1(l)ˆ
x(l|l−1)
B−1(l)z(l).(33)
Further, we have
YT(l)C(l)D(l)
=(B−1
p)TCxB−1
pˆ
x(l|l−1)+FT(B−1
v)TCyB−1
vz(l). (34)
Combining Eqs. (23), (32) and (34), we obtain the posterior
state estimate equation
x(l)=ˆ
x(l|l−1)+K(l)(z(l)−F(l)ˆ
x(l|l−1)), (35)
where K(l)is the gain of qRKFKF obtained from
K(l)=P(l|l−1)FT(l)(F(l)P(l|l−1)FT(l)+V(l))−1,
(36)
P(l|l−1)=BP(l|l−1)C−1
x(l)BPT(l|l−1), (37)
V(l)=Bv(l)C−1
y(l)BvT(l). (38)
A summary of the proposed qRKFKF is listed as follows:
1) Select a suitable σand a tiny positive value ε, respec-
tively; initialize the estimation ˆ
x(0|0)and the covariance
matrix P(0|0);letl=1.
TAB LE I
COMPUTATIONAL COMPLEXITY OF THE QRKFKF
2) Calculate the prior estimation ˆ
x(l|l−1)using
ˆ
x(l|l−1)=(l−1)ˆ
x(l−1|l−1), (39)
Meanwhile, calculate P(l|l−1)using Eq. (8), and derive
BP(l|l−1)using Cholesky decomposition.
3) Set t=1 and initialize ˆ
x(l|l)0at the iteration t=0,
where ˆ
x(l|l)0=ˆ
x(l|l−1), and ˆ
x(l|l)trepresents the state at
the iteration t.
4) Calculate ˆ
x(l|l)tusing the following equation:
ˆ
x(l|l)t=ˆ
x(l|l−1)+˜
K(l)(z(l)−F(l)ˆ
x(l|l−1)), (40)
where
˜
K(l)=˜
P(l|l−1)F(l)T(F(l)˜
P(l|l−1)F(l)T+˜
V(l))−1,(41)
˜
P(l|l−1)=BP(l|l−1)˜
C−1
x(l)BPT(l|l−1), (42)
˜
V(l)=Bv(l)˜
C−1
y(l)BvT(l), (43)
˜
Cx(l)=diag(fc(˜e1(l)), . . . , fc(˜en(l))), (44)
˜
Cy(l)=diag(fc(˜en+1(l)), . . . , fc(˜en+m(l))), (45)
˜ei(l)=di(l)−yi(l)ˆ
x(l|l)t−1.(46)
5) Compare the estimation between the current step and
previous step, using
ˆ
x(l|l)t−ˆ
x(l|l)t−1
ˆ
x(l|l)t−1
≤ε. (47)
If Eq. (47) is satisfied, the estimation of the current step is
very close to the true value. Then, we set ˆ
x(l|l)=ˆ
x(l|l)tand
move to next step. Otherwise, let t +1→t, and go back to
step 4.
6) Updating the posterior covariance matrix using
P(l|l)=(I−˜
K(l)F(l))P(l|l−1)(I−˜
K(l)F(l))T
+˜
K(l)V(l)˜
KT(l)(48)
Then, let l+1→land return to 2) to get the next time
estimation.
And the computational complexity of qRKFKF is given in
Table I.
The average fixed-point iteration number is set at t, then the
total computational complexity of the qRKFKF is
CqRKFKF =(2t+8)N3+(6+4t)N2M+(2t−1)N2
+(4t+2)NM2+(3t−1)NM +(4t−1)N+2tM
+2tM3+tON3+2tOM3(49)
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XING et al.: qRKFKF FOR LAND VEHICLE NAVIGATION 4601
The computational complexity of the KF, MCKF and MEE-
KF are given by Eqs. (49), (50) and (51), respectively.
CKF =8N3+10N2M−N2+6NM2−N+OM3(50)
CMCKF =(2t+8)N3+(6+4t)N2M+(2t−1)N2
+(4t+2)NM2+(3t−1)NM +(4t−1)N+2tM
+2tM3+tON3+2tOM3(51)
CMEE−KF =(7t+8)N3+(19 +6t)N2M−N2
+(15t+2)NM2+(6t−1)NM +(5t−1)N+tM
+7tM3+tO(MN)+tON3+2tOM3.(52)
III. ILLUSTRATIVE EXAMPLES
To verify the effectiveness of the proposed qRKFKF algo-
rithm for land vehicle navigation, three test scenarios includ-
ing non-Gaussian noise with outliers, alpha-stable noise and
diverse noise distributions in measurements are considered in
the following simulations. The state equation and measurement
equation of the linear land vehicle system are given by
⎡
⎢
⎢
⎣
x1(l)
x2(l)
x3(l)
x4(l)
⎤
⎥
⎥
⎦=⎡
⎢
⎢
⎣
x1(l−1)
x2(l−1)
x3(l−1)
x4(l−1)
⎤
⎥
⎥
⎦+⎡
⎢
⎢
⎣
w1(l)
w2(l)
w3(l)
w4(l)
⎤
⎥
⎥
⎦
,(53)
z1(l)
z2(l)=F⎡
⎢
⎢
⎣
x1(l−1)
x2(l−1)
x3(l−1)
x4(l−1)
⎤
⎥
⎥
⎦+v1(l)
v2(l),(54)
with the state transition matrix
=⎡
⎢
⎢
⎣
10T0
010T
00 1 0
00 0 1
⎤
⎥
⎥
⎦
,(55)
and the observation matrix
F=1010
0101
.(56)
where T=0.3 is the interval of vehicles.
The x1,x2,x3and x4are the north location, east location,
north velocity, and east velocity of the system, respectively.
The output z1and z2are the north location and the east
location next time. It should be noted that process noise
wi(l),i=1,2,3,4 and measurement noise vi(l), i=1,2are
uncorrelated.
A. Example 1: Non-Gaussian Noise With Outliers in
Measurements
First, a test is created, where the observation noise vi(l)is
mixed-Gaussian and the process noises wi(l)are Gaussian. In
the simulation,
wi(l)∼N(0,0.01), and
vi(l)∼0.9N(0,0.01)+0.1N(0,100),
Clearly, Gaussian distribution with a larger variance can create
stronger impulsive noises (outliers).
TAB LE II
THE PERFORMANCE OF FOUR ALGORITHM UNDER DIFFERENT σ
The KF, MCKF, MEE-KF and qRKFKF are used to esti-
mate the state variables x1,x2,x3and x4in the above
linear system, respectively. We calculate the mean square error
(MSE) between the estimation and the true values to obtain
the estimation errors. Meanwhile, we test the performance of
the MCKF, MEE-KF, qRKFKF algorithm with different kernel
widths σwhich is shown in Table II. In this simulation, we
choose the kernel width σof 5 and qof 0.4 for qRKFKF,
the kernel width 2 and 1 for the MCKF and the MEE-KF,
respectively, which can make their performance reach the best
according to the Table II. The simulation shows the estima-
tion accuracy of the qRKFKF is the best among the four
algorithms.
B. Example 2: Alpha-Stable Noise in Measurements
In order to test the performance of the qRKFKF algorithm
in the alpha-stable noise, the kernel widths of the MCKF,
MEE-KF, qRKFKF algorithms are set as 2, 1 and 7, respec-
tively, which can optimize their performance in this noise. The
characteristic function of the alpha-stable noise is given by
v(t)=exp{jδt−α|t|β1+jζsign(t)h(t,β)},(57)
with
h(t,β)=tan πβ
2β= 1
2
πlog|t|β=1.(58)
where β∈(0,2] is the characteristic index, ζ∈[−1,1] is
the symmetry, δis the location factor and α∈(0,+∞]isthe
dispersion factor. We set (δ,α,ζ,β) =(0,1,0,1.4)in this
simulation.
Then, we plot probability density function (PDF) curves in
Fig. 1(POEE-Probability of estimation error). The higher the
kurtosis of the PDF is, the smaller the error variance estimated
is, which proves that the qRKFKF algorithm estimates the
state variables x1,x2,x3and x4more accurately. The kurtosis
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4602 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 69, NO. 11, NOVEMBER 2022
Fig. 1. Probability densities of x1,2,3,4estimation errors.
TABLE III
MSESUNDER DIVERSE NOISE DISTRIBUTIONS
of the PDF of the qRKFKF in Fig. 1are higher than the KF,
MCKF and MEE-KF. As we expected, qRKFKF can maintain
its excellent performance when the measurement data contains
alpha-stable noise.
C. Example 3: Diverse Noise Distribution in Measurements
The measurement noise in this simulation is the combination
of Gaussian and Laplace distributions. We choose the same
kernel width for qRKFKF, MCKF and MEE-KF as them in
example 1, which can make their performance reach the best.
The noise is described as
vi(l)∼0.9N(0,100)+0.1L0,1/√2,(59)
where L(0,1/√2)denotes the Laplace distribution with zero
mean and scale parameter 1/√2.
Table III give the simulation results of the KF, MCKF, MEE-
KF and qRKFKF for land vehicle navigation under diverse
noise distributions in measurement. Without any difficulty, we
find that the qRKFKF has the minimum MSEs among the four
algorithms under discussion.
IV. CONCLUSION
In this brief, a q-R´enyi kernel functioned Kalman filtering
algorithm (qRKFKF) is proposed, which is realized by using
q-R´enyi kernel function to deal with the measurement uncer-
tainty caused by large outliers, alpha-stable noise and diverse
noise distributions. Simulation results illustrate that the esti-
mation accuracy of the qRKFKF is highly improved compared
with the KF, MCKF and MEE-KF for combating non-Gaussian
noises.
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