Maximum Correntropy Criterion-based UKF for Tightly Coupling

INS and UWB with non-Gaussian Uncertainty Noise

Seong Yun Cho

, Jae Hong Lee

and Chan Gook Park

Division of Robotics and Mobility, Kyungil University, Gyeongsan, Republic of Korea

Department of Aerospace Engineering, Seoul National University, Seoul, Republic of Korea

Keywords: Maximum Correntropy Criterion, UKF, INS/UWB.

Abstract: In this paper, unscented Kalman filter (UKF) based on maximum correntropy criterion (MCC) instead of

minimum mean square error (MMSE) criterion, and it is applied to tightly coupled integration of inertial

navigation system (INS) and ultra wide-band (UWB). UWB can measure distance with an accuracy of less

than 30cm in line-of-sight environment, but provides distance measurement with various types of non-

Gaussian uncertainty noise in non-line-of-sight environment. In this case, if the INS/UWB system is

configured with the existing MMSE-based filter, a large error occurs. To solve this problem, in this paper,

UKF is designed based on MCC. Through simulation analysis, it is confirmed that the proposed filter has

robust characteristics against UWB uncertainty and enables stable INS/UWB integration.

1 INTRODUCTION

Consider a system integrating inertial navigation

system (INS) and ultra wide-band (UWB) for indoor

navigation. Although INS can accurately calculate 6-

degree of freedom motion, it has a problem of

accumulation of errors over time. UWB enables

accurate distance measurement and location

estimation in a line-of-sight (LoS) environment, but

in non-line-of-sight (NLoS) environment such as

indoors, accurate becomes impossible as various

types of uncertainty noise are included in distance

measurements (Cho, 2019). To integrate these two

systems, nonlinear filters such as extended Kalman

filter (EKF) (Brown and Hwang, 2012) and unscented

Kalman filter (UKF) (Julier et al., 2000) are generally

used in consideration of their nonlinear

characteristics. However, since these filters designed

based on minimum mean square error (MMSE) do

not properly respond to UWB uncertainty noise, a

large error may occur. In this paper, considering this

problem, maximum correntropy criterion (MCC)-

based UKF (MCUKF) is introduced.

The MCC-based filter is a filter that maximizes

the similarity between the measurements and the

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estiamtes, and is designed based on the kernel

function reflecting the error characteristics of the

measurement. The kernel function is to indicate the

similarity of two random variables, and the total sum

of kernel function values including innovation and

residuals calculated in the measurement-update

process is determined as a cost function. Then, the

state variables are estimated so that this cost function

is maximized. In this process, when uncertainty noise

occurs in the UWB measurement, the MCC-based

filter adjusts the P matrix and R matrix, thereby

reducing the effect of the measurement uncertainity

noise on the system (Chen et al., 2017).

Recent research on MCC-based filters has

focused on kernel function design considering

various error probability distributions (Li et al., 2022,

Huang and Zhang, 2022). There are not many studies

to apply MCC to nonlinear systems. Ther purpose of

this study is to apply MCC to UKF so that is can be

used in nonlinear systems. And the designed MCUKF

is applied to tightly coupled INS/UWB integration

system.

The performane of the MCUKF-based INS/UWB

integrated navigation system is verified based on

simulation. Through simulation analysis, if non-

Cho, S., Lee, J. and Park, C.

Maximum Correntropy Criterion-based UKF for Tightly Coupling INS and UWB with non-Gaussian Uncertainty Noise.

DOI: 10.5220/0011286000003271

In Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2022), pages 209-213

ISBN: 978-989-758-585-2; ISSN: 2184-2809

209

Gaussian uncertainty noise is included in the UWB

measurement, it is confirmed that MCUKF provides

a stable navigation solution by largely tuning the

value of the R matrix corresponding to the erroneous

measurement.

2 MCUKF-BASED INS/UWB

INTEGRATION

Considering the following discrete-time nonlinear

system model:

() , ~(0,)

() , ~ (0,)

kkk

fx w w N Q

yhx vvNR

−−

(1)

where subscript k is the discrete time,

x ∈  is the

state, and

y ∈  is the output.

and

are

uncorrelated zero-mean white Gaussian noise

processes, and the covariance matrices of the

processes are denoted by Q and R, respectively. The

state variables considering INS are set as

xPosVelAtt



=∇



(2)

where

Pos and

Vel are the position and velocity

in the local level coordinate system,

is the

attitude expressed in Euler angles, and

∇

and

are

the accelerometer bias and gyro bias, respectively.

In UKF, these state variables are converted into

sigma points. The number of sigma points is 2

N+1,

which is 31 because the system dimension

N is 15

(Julier et al., 2000).

The system function can be expressed as an INS

equations as follows (Farrell and Barth, 1999):

{

}

(

{

)

}

1,111

,1 ,1 1 1

11,11

,1 ,1 ,1

()

(2 )

()

LL L

kkk

LL Lb

kk bkkk

LL LL

ie k eL k k k

kk kibkk

bL L

Lk iek eLk

Pos Pos Vel dt

Vel Vel C f

Vel g dt

Qtn Qtn Qtn

Cdt

ωω

ωε

ωω

−−

−−−−

−− −

=+⋅

=+ −∇−

+×+⋅

=+ ∗ −

−+⋅

(3)

where

dt is the IMU output period, Qtn is the

quaternion,

and

are the accelerometer output

and gyro output, respectively,

∇

and

are the

estimated accelerometer bias and gyro bias,

respectively,

is the Earth’s angular velocity

vector, and

is the rotational angular velocity

vector of the local level coordinate system caused by

the velocity.

In the time-propagation process, the sigma points

are propagated in synchronization with the IMU

output period based on (3).

When the distance measurement is obtained in

UWB, the measurement-update is processed. The

measurement function in (1) is as follows (Cho, 2019):

() ( () ) ( () ) ()

kkkk

ri xi Pos yi Pos wi=− +− +

(4)

where

[() ()]

iyi is the location of the anchor

node i,

os is the j-axis location of the mobile node

in the local level coordinate system, and

()wi is the

noise contained in channel i. And

{1, 2, , }iM∈  .

In case of using calibrated UWB, w in (4) is

modelled as zero-mean noise in the LoS environment.

However, w may appear in the form of heavy-tailed

impulsive noise in the probability distribution in the

indoor environment. This error can be filtered out in

an MCC-based filter. In MCUKF, the cost function is

set as follows (Chen et al., 2017):

() (())

xGei



(5)

where G() is a kernel function, and the Gaussian

kernel function can be expressed as follows:

()

Ge e

−

(6)

where

is the kernel bandwidth and is an important

design parameter in MCC.

The state variables are estimated to maximize the

cost function.

arg max ( )

Jx= (7)

The convergence performance of the filter can be

improved by iteratively using one measurement based

on the fixed-point iteration algorithm in the

measurement-update process (Chen et al., 2017).

Using the innovation and residual calculated in this

process, the P matrix and R matrix are adjusted as

follows:

() ()

()

ka P ka P

ka R ka R

BC B

RBCB

−−

−

(8)

where a is the iteration order,

BB P

−

= and

BB R= .

−

should be calculated before the

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

210

measurement-update using the time-propagated

sigma points as follows:

()

ˆˆ

(() )(() )

kkkkk

PixixQ

χχ

−−

−−−−−

=−−+



(9)

where

21nN=+

, Q is the process noise covariance

matrix, and

()

−

is the set of time-propagated i

sigma points.

In (8),

()

and

()

are calculated as follows:

()

(1) 0

0()

(1) 0

0()

Ge N

Ge M















 



 



(10)

where

() ( 1)

() ()

()

ka P k ka

ka R k ka

eBxx

eByy

−− −

−

−−

=−



(11)

Through this, the P matrix and R matrix are

adjusted based on the values calculated by inputting

the normalized innovation and the normalized

residual as input to the kernel function, respectively.

When the fixed-point iteration algorithm is

finished, the state variables and error covariance

matrix are updated as follows:

,() ,() ()

() ,() ,() ,()

ˆˆ

()

k k xy k a yy k a k k a

k ka xyk a yyka xyk a

xxP P yy

PP P P P

−− −

−−

=+ −

=−



(12)

where

,() () ( 1) ()

,() () () ()

ˆˆ

()()

ˆˆ

()()

xyka ka ka k ka

yyka k ka k ka ka

Pxyy

PyyyyR

−− −

−

−−

=−−

=−−+







(13)

In (12),

()

−

is constructed by calculating for

each channel based on (4).

3 SIMULATION RESULTS

To verify the performance of the proposed MCUKF-

based INS/UWB integrated navigation, a simulation

was carried out to apply this filter to navigation for

robots and pedestrians in an indoor environment. The

IMU used in the simulation is OEM-IMU-EG-320N,

and the specifications are as follows:

• gyro bias repeatability is 0.5 deg/sec

• angular random walk is 0.1 deg/√hr

• accelerometer bias repeatability is 15mg, and

• velocity random walk is 0.05 m/s/√hr.

The output frequency of the IMU was set to 50Hz,

and that of the UWB was set to 1Hz. Four anchor

nodes are located as shown in Figure 1, and the

mobile node moves along the trajectory shown in this

figure for 60 seconds.

Figure 1: Simulation trajectory.

(a) measurement error

Figure 2: Simulation result in the first case.

Ch. 1 (m)Ch. 2 (m)Ch. 3 (m)Ch. 4 (m)

Maximum Correntropy Criterion-based UKF for Tightly Coupling INS and UWB with non-Gaussian Uncertainty Noise

211

(b) square root of adjusted R matrix

(d) positioning error

Figure 2: Simulation result in the first case (cont.).

In MCUKF, the kernel bandwidth was set to 1.3.

First, it is the case where w in (4) is white

Gaussian noise with zero-mean. The result is shown

in Figure 2. As can be seen in this figure, the

measurements have only Gaussian noise, and the R

matrix adjusted in MCUKF is similar to that set as the

initial value. And it can be confirmed that the

positioning performance of MCUKF is almost similar

to that of UKF in general case.

(a) measurement error

(b) square root of adjusted R matrix

(d) positioning error

Figure 3: Simulation result in the second case.

Ch. 1 (m)Ch. 2 (m)Ch. 3 (m)Ch. 4 (m)

y (m)

Ch. 1 (m)Ch. 2 (m)Ch. 3 (m)Ch. 4 (m)

N (m)

(m)

ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics

212

Table 1: Summary of the simulation results.

positioning errors (first case)

filter mean standard deviation

UKF 0.16127 0.21896

MCUKF 0.16602 0.22280

positioning errors (second case)

filter mean standard deviation

UKF 0.51414 0.44778

MCUKF 0.26019 0.25141

In the second case, a total 10 impulse errors were

randomly generated in w, and the result of MCUKF

in this case is shown in Figure 3. It can be seen that

large and small impulse errors are randomly entered

into each channel of the distance measurements. And

it can be seen that the adjusted R in MCUKF increases

largely according to the impulse error generated for

each channel. Due to this, the channel with the

impulse error momentarily loses its function, and the

INS error is corrected using the measurements

obtained from the remaining channels. Therefore,

MUCKF is hardly affected by the impulse error.

It can be seen that the UKF positioning result is

greatly affected by the impulse error and the error

increases. The reason is that UKF is a filter designed

based on MMSE and cannot cope with non-Gaussian

noise. On the other hand, it is confirmed that the

positioning result of MCUKF is not affected by

impulse error. Therefore, MCUKF is evaluated to be

able to provide stable navigation information

regardless of positioning error.

The number of the estimated position information

for 60 seconds is 3000, and the mean and standard

deviation of the positioning errors are calculated for

each filter. And the result is summarized in Table 1.

Based on this table, the excellent performance of the

proposed MCUKF can be confirmed.

4 CONCLUSIONS

In this paper, MCUKF-based INS/UWB integrated

navigation system was introduced. To use MCC in

nonlinear system, MCUKF was designed by

combining MCC with UKF. And this filter was used

to integrated INS and UWB. UWB has non-Gaussian

uncertainty noise in an indoor environment. While

this causes a large estimation error in the existing

UKF, it is proven based on simulation that MCUKF

provides a stable navigation solution by tuning the R

matrix for each channel in which this error occurs.

Based on this paper, it is expected that stable

variables can be reliably estimated in a nonlinear

system including heavy-tailed non-Gaussian impulse

noise.

ACKNOWLEDGEMENTS

This work was supported by Institute of Information

& Communications Technology Planning &

Evaluation (IITP) grant funded by the Korea

government (NFA)

(No. 2019-0-01325, Development

of wireless communication tracking-based location

information system in disaster scene for fire-fighters

and person who requested rescue).

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