Reliability Analysis of the Kalman Filter for INS/GPS Integrated

Navigation System Applied to Train

Seong Yun Cho

, Chang Ho Kang

and Kyung Ho Shin

Department of Robotics Engineering, Kyungil University, Gyeongsan, South Korea

Research Institute of Engineering and Technology, Korea University, Seoul, South Korea

Railroad Control & Communication Research Team, Korea Railroad Research Institute, Uiwang, South Korea

Keywords: INS/GPS, Train Application, Kalman Filter, Observability, RMSE.

Abstract: This paper aims to analyse the navigation performance that can be provided by the navigation system when

applying the INS/GPS integrated navigation system to the train. The performance of the Kalman filter

integrating INS and GPS can be summarized by the integrity of the measurement and the observability of the

filter. Assuming the integrity of the GPS information used as a measurement is always satisfied, the

performance of the filter can eventually be analysed by the observability. The observability of the filter

depends on the dynamic trajectory of the train. Because the train has a non-holonomic constraint and one-

dimensional motion, the filter design and the performance analysis are carried out considering this. We

analyse the observability of the filter through simulation and explain the limit of the filter and the flaw of the

observability. We also analyse the reliability of the navigation system and present additional research

directions.

1 INTRODUCTION

Accurate navigation information of a train must be

provided to control the train. Generally, trains are

detected on the basis of a fixed infrastructure installed

along the railway to track the position of trains. If the

train detection fails, the location information of the

train will not be provided. When new railways are

built, therefore, new infrastructure must be installed

at a high cost, infrastructure faults must be detected,

and infrastructure must be maintained periodically.

To overcome this realistic problem, in this paper, the

application of the inertial navigation system (INS)

/global positioning system (GPS) integrated

navigation system to trains is considered. That is, a

system that estimates the position of a train with only

INS and GPS receiver mounted on a train without any

additional infrastructure is dealt (Presti and Sabina,

2018).

The INS uses an inertial measurement unit (IMU)

consisting of 3-axis accelerometers and gyros. If the

sensor outputs are processed by the INS algorithm,

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the position, velocity, and attitude information is

provided at a frequency faster than 100Hz. However,

INS errors gradually increas over time due to the

errors included in the sensor outputs, the initial

attitude errors, the non-commutative errors occurring

in the digital computer, and so on. This is a bigger

problem if the low-level IMU is used (Titterton and

Weston, 1997; Cho, 2014). This error can be

estimated and compensated using the information

provided by the GPS receiver. This system is called

the INS/GPS integrated navigation system (Liu et al.,

2010; Miller and Campbell, 2012; Cho, 2014). INS

and GPS are integrated using a Kalman filter.

Integration types can be divided into loosely coupling,

tightly coupling, and ultra-tightly coupling depending

on the type of measurement. In this paper, we discuss

loosely coupling method considering simplicity of

implementation.

The performance of INS/GPS integrated

navigation system depends on the specifications of

the IMU and GPS receiver and the performance of the

integration filter. In this paper, it is assumed that 2.0

Cho, S., Kang, C. and Shin, K.

Reliability Analysis of the Kalman Filter for INS/GPS Integrated Navigation System Applied to Train.

DOI: 10.5220/0007832602370242

In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 237-242

ISBN: 978-989-758-380-3

237

deg/hr grade MEMS type IMU and integrity

guaranteed GPS receiver are used. Therefore, the

performance of INS/ GPS integrated navigation can

be predicted through the performance analysis of the

integration filter.

The filter used in INS/GPS integration can be

selected from nonlinear Kalman filters such as

extended Kallman filter (EKF), unscented Kalman

filter (UKF), cubature Kalman filter (CKF), etc.

considering the nonlinearity of INS (Reif et al., 1990;

Julier et al., 2000; Cho et al., 2017). In this paper,

EKF is used in consideration of small initial attitude

error. The performance of the filter can be explained

by the observability. The observability of the INS/

GPS integration filter is determined by the trajectory

of the trains. In this paper, considering the trajectory

of trains, we analyze the observability of the filter

through the estimation error covariance of the filter

state variables (Cho et al., 2007). Since the train has

a non-holonomic constraint and moves in one

dimension, the filter is designed considering this. In

this case, the observability is analyzed through

covariance analysis. Also, the limitation of the filter

and the loopholes in the analysis of the observability

are discussed by analyzing the covariance of the filter

and the root mean squared errors (RMSE) based on

Monte-Carlo simulation results.

All this analysis is done through simulation. The

information provided in this paper can be used to

analyze the reliability of the location information

provided by the INS/GPS integrated navigation

system installed on trains without infrastructure.

2 INS/GPS INTEGRATION

FILTER AND ANALYSIS

In this section, we design a loosely coupled INS/GPS

integration filter based on EKF using the position and

velocity information of GPS as a measurement, and

analyse the navigation performance when it is applied

to the train.

2.1 INS Algorithm

For INS, initial alignment is performed in the stopped

state first. In the case of train application, the final

navigation information can be stored in memory

when the train operation is finished. This information

can be used at the next operation. Therefore, INS can

be performed without initial alignment.

INS calculates the navigation information in

synchronization with the output period of the sensor

data output from the IMU. First, the attitude is

updated based on the following quaternion

differential equation using the gyro output (Titterton

and Weston, 1997).



()

bbnn

ib n ie en

qq C



  



(1)

where

is the quaternion,



is the quaternion

product,



is the gyro output,



is the Earth

rotation angular velocity, and



is the rotation

angular velocity of the navigation frame. And

the direction cosine matrix (DCM) from the

navigation frame to the body frame, and the

quaternion, DCM, and Euler angles are mutually

convertible.

The velocity is updated using the updated attitude

information and accelerometer output, and the

position is update by integrating the velocity.

(2 )

nnb nn nn

bieen

VCf Vg









(2)

/( )

/( )cos

LvRh

lvRhL







































(3)

where

[]

NED

Vvvv

is the velocity on the

navigation frame,

is the accelerometer output,

is the gravitational acceleration vector,

[]

Llh

is the position (latitude, longitude,

and altitude), and

R and

R are the Earth radii

calculated in the latitude and longitude directions,

respectively.

2.2 Filter Design

The integration filter is driven in synchronization

with the GPS signal output. First, the error state

variables are set as follows for EKF-based filter

design.

[]

nn bbT

XPV



  



(4)

where



is the position error,



is the velocity

error,



is the attitude error shown in the navigation

frame, and



and



are the accelerometer bias and

gyro bias, respectively.

The discrete-time system and measurement

equations of the filter are expressed as follows:

,~(0,)

kkkk

k kkk

XwwNQ

HX vvN R







 



(5)

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

238

The systems matrix is derived by applying the

linear perturbation method to (1) to (3) (Titterton and

Weston, 1997). In case of using the position and

velocity information of GPS, the measurement matrix

can be denoted as follows:

66 69

[0]





(6)

Table 1: Specification of the sensors used in simulation.

Sensor Error Spec.

Accelerometer

Bias Repeatability 3.0 mg

Noise 0.05 mg/√hr

Gyro

Bias Repeatability 2.0 deg/hr

Noise 0.07 deg/√hr

GPS

Position Noise 5.0 m (1σ)

Velocity Noise 0.1 m/s (1σ)

2.3 Simulation Analysis

Simulation is performed to analyse the navigation

performance considering the application of the INS/

GPS integrated navigation system to the train. The

specifications of the IMU and GPS used in the

simulation are set as shown in Table 1. The model of

IMU is Northrop Grumman’s MEMS type μIMU-I,

and the model of GPS receiver is u-blox.

The maximum speed of the train is 200

km/hr

and the moving trajectory is set as shown in Fig. 1.

The train accelerates first and then runs at constant

Figure 1: Moving trajectory.

speed and the total driving time is 2 hours. After 10

minutes of straight running, rotate 30.0 degrees at a

rotational angular velocity of 1.0

deg/s for 30 seconds.

And repeat this. There are three tunnels, which cannot

receive GPS signals. The Monte-Carlo simulation

was performed on this trajectory, and the number of

simulations was 10 times. Fig. 2 shows the RMSE and

standard deviation (SD) obtained through the

covariance matrix of the filter. In the section where

the GPS signal can be received, it is seen that the

RMSE and SD are similar, and it is confirmed that the

Monte-Carlo simulation is performed normally. It can

be seen that accelerometer bias and gyro bias can be

estimated according to time and train rotation.

However, when the GPS signal cannot be received,

the RMSE values of the position and velocity of the

horizontal axis increase greatly. Therefore, it can be

Figure 2: RMSE and standard deviation in INS/GPS.

Reliability Analysis of the Kalman Filter for INS/GPS Integrated Navigation System Applied to Train

239

confirmed that there is a limitation of INS/GPS

integrated navigation in the section where the tunnel

exists.

3 NON-HOLONOMIC

CONSTRAINT FILTER AND

ANALYSIS

3.1 Filter Design

Trains have six degree of freedom movement.

However, it only moves on railways with non-

holonomic constraints. In other words, there is

velocity only in the longitudinal direction on the body

frame, and the velocities are zero in the lateral and

vertical directions. Using this information as an

additional measurement, a new measurement matrix

for the filter can be constructed. To do this, the

velocity on the body frame is calculated.

bbn

VCV

(7)

The linear perturbation method is applied to this

equation.



()( )

bbb n nn

VVCI VV



  

(8)

Therefore, the velocity error model on the y and z

axis of the body frame can be approximated as

follows:

(:) (:)( )

bnbnn

CyzV CyzV

















(9)

where

010

(:)

001

Cyz C















(10)

Based on this equation, the following two

measurement matrices can be derived.

33 33 33 36

23 26

000

0(:)(:)()0

bbn

Cyz CyzV

  



























(11)

23 26

0(:)(:)()0

bbn

knn

HCyzCyzV













(12)

We define two filters as follows.

- INS/GPS with NHC-1: The measurement matrix (16)

is used where GPS signals can be received and the

measurement matrix (12) is used where GPS signals

cannot be received.

- INS/GPS with NHC-2: (11) is used in an open space

where GPS signals can be received, and (12) is used

in places where GPS signals cannot be received like

a tunnel.

3.2 Simulation Analysis

Three types of INS/GPS integrated navigation were

performed: INS/GPS, INS/GPS with NHC-1, and

INS/GPS with NHC-2. Fig. 3 shows the RMSE for

each type as a result of Monte-Carlo simulations.

Figure 3: RMSEs according to the filters.

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

240

Figure 4: RMSE and standard deviation in INS with NHC.

First, in the case of NHC-2, the x-axis

accelerometer bias is estimated faster than other

methods. As a result, the E-axis attitude error is also

estimated quickly. And the z-axis gyro bias

estimation speed is increased. Based on this effect,

the horizontal-axis velocity and position error

estimation performance is improved. In particular, the

increase in the error is also reduced in tunnels because

the filter can be driven through the NHC.

In case of NHC-1, the sensor bias estimation

performance is not significantly different from

INS/GPS. However, overall performance is better

than INS/GPS because it has the advantage of driving

the filter even in tunnels.

What should be considered here is that until the

train meets the tunnel after departure, the errors are

reduced by driving the filter sufficiently through the

GPS signal. Therefore, the errors in tunnels relatively

small.

To further analyse this problem, the NHC filter

was driven without a GPS signal for the first 5

minutes after departure and the results are shown in

Fig. 4. Comparing RMSE and SD, we can see that the

accuracy of the actual estimate of the gyro bias is

within the covariance of the filter, which is the

performance index of the filter. In the case of the

horizontal-axis accelerometer bias, however, the two

values seem to be somewhat different. The horizontal

–axis attitude errors also has the same tendency. A

problem lies in the D-axis attitude error. In the NHC

filter, the covariance of the D-axis attitude error falls

to a value close to zero. In other words, it is ensured

that the observability is obtained. However, when

starting with a relatively large initial attitude

estimation error, the estimates do not converge to true

values. This is a crucial factor, which results in

velocity and position estimation errors. This is a hoe

in the NHC filter.

3.3 Reliability Analysis

The performance of the filter can be expressed in

terms of observability. In a time-varying system such

as INS/GPS, the observability of the filter can be

confirmed by covariance analysis rather than rank-

based analysis. It is determined that the state variable

having the covariance converging to a value close to

0 according to the filter update is estimable. However,

the important point is that estimability does not

always mean convergence to true values. As shown in

Fig. 4, the state variables converge but may converge

to the wrong values.

Therefore, it is difficult to judge the performance

of the filter only by the observability. In this case, it

is necessary to set various environments and judge the

performance of the filter based on the RMSE analysis

through Monte-Carlo simulations. It is important to

carry out simulations using various trajectories for

trains. In addition, reliability analysis of GPS signal

and INS/GPS performance analysis according to IMU

specification must be done.

Reliability Analysis of the Kalman Filter for INS/GPS Integrated Navigation System Applied to Train

241

4 CONCLUSION

In this paper, we designed the navigation filters and

analysed its performance when applying the INS/GPS

integrated navigation system to the train. NHC filters

for NHC-based trains were designed and Monte-

Carlo simulations were performed to performance

analysis. Tunnels were set on a trajectory and

simulations were performed. Here, the case of

performing pure INS in the tunnels and the case of

driving the NHC filter were considered, respectively.

By analysing covariance and RMSE together, it was

verified that the use of NHC filter regardless of GPS

signal availability is good for filter performance. And

the loophole in covariance analysis was point out.

Based on this, it can be concluded that it is necessary

to analyse covariance and RMSE together based on

the Monte-Carlo simulations performed under

various environment settings to analyse the reliability

of the filter, therefore.

ACKNOWLEDGEMENTS

This research was supported by a grant from R&D

Program (PK1904A1) of the Korea Railroad

Research Institute, Republic of Korea.

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