A Covariance Matching-Based Adaptive Measurement Differencing

Kalman Filter for INS’s Error Compensation

CHINGIZ HAJIYEV1, ULVIYE HACIZADE2

1Faculty of Aeronautics and Astronautics,

Istanbul Technical University,

Maslak, 34469, Istanbul,

TURKEY

2Department of Computer Engineering,

Halic University,

Güzeltepe Neighborhood, Temmuz Şehitler Street, No.15

34060 Eyüp-Istanbul,

TURKEY

Abstract: - In this study, a covariance matching-based adaptive measurement differencing Kalman filter

(AMDKF) for the case of time-correlated measurement errors is proposed. The solution to the state estimation

problem involves deriving a filter that accounts for measurement differences. Specifically, the measurement

noise in the generated measurements is assumed to be correlated with the process noise. To address this issue in

the context of correlated process and measurement noise, we propose an adaptive measurement differencing

Kalman filter that is robust to measurement faults. We also evaluate the robustness of the suggested AMDKF

through an analysis. When noise increment type sensor faults are present in the time-correlated inertial

navigation systems (INS) measurements, the states of a multi-input/output aircraft model were estimated using

both the previously developed measurement differencing Kalman filter (MDKF) and the suggested AMDKF

and the results were compared.

Key-Words: - differencing Kalman filter, time-correlated error, process noise, adaptive Kalman filter, Inertial

Navigation System, aircraft model.

Received: February 16, 2023. Revised: November 18, 2023. Accepted: December 3, 2023. Published: December 31, 2023.

1 Introduction

The primary sources of error in Inertial Navigation

Systems (INS) are connected with insufficient initial

knowledge and the gradual propagation of

inaccuracies. The accelerometers' signals are

integrated twice to determine location and velocity,

in that order, subsequent to adjustments made for

sensor error and gravity. The chief factors of

velocity inaccuracies include the inexactness of

accelerometer readings (generally due to bias and

scale factors), slip-ups in local gravity calculation,

and significant attitude inaccuracies resulting from

gyroscope precession, [1].

Linear differential equations can characterize

minor INS errors. Therefore, for the linear error

analysis to remain reliable, the INS errors must

remain small.

The accuracy of an INS's position and velocity

estimates decays over time, [2]. The INS's

drawbacks include its unbounded error growth. The

use of more precise inertial sensors like

accelerometers and gyroscopes can boost the INS's

accuracy. However, there are reasonable boundaries

to its performance. The cost of an inertial navigation

system may be excessive.

Achieving high precision and low cost are

fundamental considerations for numerous types of

vehicles. This has been achieved in several works,

[3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13],

[14], [15], through the utilization of integrated

navigation systems that combine inertial navigation

systems with additional navigational aids. This

allows for the comparison of independent

measurements from external sensors that produce

comparable values with the output of the INS. The

discrepancies measured between multiple navigation

systems are used to correct the inertial navigation

system.

Inexpensive sensors, known for their low

precision, have been the focus of research in this

field for the past two decades. Consequently, many

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algorithms have been developed to mitigate errors in

inertial sensors. These algorithms are founded on

either a mathematical model designed for the self-

damping of inertial sensor errors or the integration

of inertial sensors with external information sources

including GPS receivers, magnetometers,

barometers, and more. The study, [5], proposes a

specific approach for compensating gyroscope drift

via a PI controller based on magnetometer data in

addition to a method for compensating inaccuracies

in the horizontal channel of the navigation system.

An approach to refine the performance of the

vertical accelerometer through a specific aircraft

manoeuvre was analyzed in, [10], to implement

supplementary in-flight calibration measures. Based

on simulation results, conducting the proposed

calibration manoeuvre before capturing the glide

slope using a standalone INS feedback signal

ensures satisfactory precision and safety standards

(in the event of a barometric altimeter (BA) failure).

Additionally, the suggested scheme for generating

the measurement signal used in the Kalman filter

allows for precise calibration of the vertical

accelerometer without the requirement to estimate

the BA bias. This confirms that the calibration

accuracy is unaffected by the influence of BA bias.

The study, [11], proposes self-calibrated visual-

inertial odometry (VIO), which applies a stereo

camera and eliminates the need for calibration

boards to estimate the intrinsic parameters of an

inertial measurement unit (IMU), including scale

factor and misalignment. Most visual-inertial

navigation algorithms presume that the inertial and

visual sensors are accurately calibrated.

Nonetheless, modeling errors can impair navigation

performance. To enhance the accuracy of the ego-

motion modelling, we use an extended Kalman filter

(EKF)-based pose estimator, with the addition of the

IMU intrinsic parameter to the filter state. These

variables are crucial to the effectiveness of ego-

motion tracking since the intrinsic parameter is

responsible for converting raw IMU readings.

There has been considerable debate surrounding

the accuracy of estimating with a combined GPS

and inertial navigation system using different types

of Kalman filters. Because classical Kalman filters

were incapable of withstanding noise and

environmental disturbances, sensor fusion

approaches employ adaptive and resilient structures.

The study, [12], proposes numerous adaptable

structures have been proposed in this context. The

adaptation of the measurement covariance matrix, a

factor for refining the process covariance matrix,

and the Chi-square approach to identify and limit

disturbances, are all beneficial to the fuzzy inference

system.

External disturbances and imprecise noise

statistics can cause non-Gaussian and unknown

measurement noise to affect SINS/GPS integrated

systems. To address this issue, a robust SINS/GPS

integrated system based on the variational Bayesian

method has been developed in, [13]. The unknown

measurement noise covariance is initially estimated

using the variational Bayesian-based Kalman filter.

To mitigate non-Gaussian noise interference, the

nonlinear robust filter is further enhanced by

integrating the maximum correntropy criterion.

Subsequently, the robust variational Bayesian

approach leveraging the interacting multiple model

is developed. This approach eliminates non-

Gaussian noise interference to the measurement

noise covariance estimation outcome.

In the study, [14], an integrated MEMS system

is built that serves as a supporting system for the

inertial navigation system and the Kalman filter

when it comes to locating flying objects. The paper

addresses the phenomena of unbalance in the INS

system, as determined by an accelerometer, and

summarizes the fundamentals of MEMS technology

utilized in accelerometric measurements.

The study, [15], investigates the use of low-cost

sensors, combining GNSS and IMU, as well as the

impact of GNSS signal errors and different fusion

algorithm designs. In order to enhance the accuracy

and reliability of GNSS and IMU sensors for

localization purposes, this paper introduces a

segmented Rau-Tung-Striebel (RTS) smoothing

algorithm and an error state extended Kalman filter

algorithm. These technologies enable the INS to

overcome the accumulated error over time, in case

the GNSS signal is disrupted. The proposed method

surpasses the traditional EKF algorithm in

localization accuracy and linearity as revealed by

the simulation results. Moreover, it displays

remarkable robustness to achieve improved

precision even in unfavorable GPS signal

conditions.

To address the attitude of robots with high

precision while employing a low-cost inertial

measurement unit, calibration methods, and an

attitude fusion filter were developed in, [16], to

make better use of measurement data from several

sensors. This calibration procedure accurately

calibrates the accelerometer, magnetometer, and

gyroscope.

The paper, [17], describes a method for self-

calibrating IMUs using distributed sensor

topologies. The IMU is made up of modules that are

arranged along the measurement axes. A single-axis

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gyroscope and three-axis accelerometer sensors are

included in each module. Each module has its

signal-conditioning circuits and processors. This

enables the module to be calibrated on a single axis

using a servo system based on a piezoelectric

actuator. To calibrate the gyroscope sensor and

accelerometer, the servo system produces high

angular velocity and tangential acceleration.

The disadvantages of the IMU calibration methods

presented in, [16] and [17], are that these methods

require auxiliary tools and equipment.

The study, [18], presents an inertial navigation

system error compensation method for the condition

of time-correlated measurement errors. The

proposed measurement differencing Kalman filter

(MDKF) uses a measurement differencing

approach-based Kalman filter to handle correlated

systems with measurement noise. Due to the

consideration of measurement differences,

measurement biases are compensated for in this

filter. This allows INS to be utilized for extended

periods during flight, enabling autonomous

navigation for real-time applications without

requiring external navigation sources.

In this study, an adaptive version of the MDKF

proposed in, [18], is developed. A covariance

matching-based adaptive measurement differencing

Kalman filter (AMDKF) for the case of time-

correlated measurement errors is proposed. The

robustness properties of the proposed AMDKF are

investigated. Measurement differencing Kalman

filter and proposed AMDKF were applied to

estimate the states of a multi-input /output aircraft

model in the presence of noise increment type

sensor faults in the time-correlated INS

measurements, and the obtained results were

compared.

2 Problem Statement

The mathematical models for system and

measurement can be expressed as follows:

( ) ( 1) ( 1) ( 1)k k k k     x Ax Bu Gw

(1)

( ) ( ) ( ) ( )k k k k  z Hx v λ

(2)

where

()kx

is the system's state vector,

is the

system transition matrix,

is the control

distribution matrix,

()ku

is the control input vector,

()kw

is the random system noise,

is the system

noise transition matrix,

()kz

is the measurement

vector,

is the system measurement matrix,

()kv

is the measurement noise vector,

()kλ

is the time-

correlated INS errors process.

It is assumed that the random vectors

()kw

and

()kv

are zero-mean Gaussian white noise. Their

covariance matrices are expressed as follows:

( ) ( ) ( ) ( )

( ) ( ) 0

E k j k kj

E k j





















w w Q

v v Q

(3)

where

()kj



is the Kronecker delta symbol.

The INS errors

()kλ

are correlated and considered

to be, [19]:

( ) ( 1) ( 1)

k k k   λ A λ B U

(4)

where

and

are proper dimension matrices and

( 1)jU

is the Gaussian white noise vector with

zero mean and covariance

 

( ) 0; ( ) ( ) ( ) ( )

E k E k j k kj









U U U Q

(5)

The equations (2) and (4) are unsuitable for

state estimation of the system (1) via optimal

discrete Kalman filter when time-correlated

measurement noise is present. To address this

problem, the optimal discrete Kalman filter should

be adjusted.

3 The Measurement Differencing

Kalman Filter

The measurement differencing approach is proposed

as a means of developing a filter for estimating

system states in the presence of time-correlated

measurements (2). This technique utilizes a linear

combination of measurements

( 1)kz

and

()kz

a system measurement, and excludes the correlated

measurement errors

()kλ

. The appropriate linear

combination is presented below, [1]:

 

( ) ( 1) ( ) ( )

( ) ( ) ( ) ( 1) ( )

k k k k

k k k k k

    

     

μ z A z HA A H x

HBu HGw B U v A v

(6)

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Instead of the correlated sequence

()jλ

, the

measurement

()jμ

only includes the purely random

sequence

( ) ( ) ( 1) ( )

j j j j   HGw B U v A v

In this case, it is practical to formulate the state

estimation problem in a structure that facilitates the

application of the optimal discrete Kalman filter:

( ) ( 1) ( 1) ( 1)

( 1) ( 1) ( 1) ( 1)

j j j j

     

      

x Ax Bu Gw

μ H x HBu ξ

(7)

where

( 1) ( ) ( 1)

k k k



   

H HA A H

μ z A z

( 1) ( 1)

( 1) ( ) ( 1)

k k k

   

   

ξ HGw

B U v A v

(8)

It is shown in, [18], that the measurement noise

()kξ

in the system (7) is the purely random process

(white noise) with an expected value:

 

( 1)

( 1) ( 1) ( ) ( 1) 0

E k k k k



      

HGw B U v A v

(9)

and covariance

 

{ ( 1) ( 1) ( ) ( 1)

( 1) ( 1) ( ) ( 1) }

T T T T

w n U n v n v n

E k k k k

k k k k

      

      

   

R HGw B U v A v

HGw B U v A v

HGQ G H B Q B Q A Q A

(10)

As seen from the expressions (7) and (8), the

process noise

( 1)kGw

and the measurement noise

( 1)kξ

are correlated:

 

 

( 1) ( 1)

( 1) ( 1) ( 1) ( ) ( 1)

( 1) ( 1)

T T T T T

E k k

E k k k k k

E k k



   



      



   



C Gw ξ

Gw HGw B U v A v

G w w G H GQ G H

(11)

Formula (7) shows that a known deterministic

input

( 1)jHBu

is added to the output.

The measurement differencing Kalman filter

(MDKF) formulas for estimation of the system (7)

states are obtained in [18] in the following form.

The estimation equation of the MDKF

ˆˆ

( / ) ( / 1) ( )

{ ( ) ( ) ( / 1)}

k k k k k

k k k k

   

  

x x K

μ HBu H x

(12)

where

ˆ( / 1)kkx

is the extrapolation value.

ˆˆ

( / 1) ( 1/ 1) ( 1)k k k k k     x Ax Bu

(13)

The optimal gain of the MDKF

 

( ) ( / 1) ( )

( / 1) ( ) ( )

T T T

d d d d

k k k k

k k k k 

   



   



K P H C

H P H R H C C H

(14)

The extrapolation error covariance matrix

( / 1) ( 1/ 1) TT

k k k k    P AP A GQ G

. (15)

The expression for the covariance matrix of

estimation error

()kP

( ) ( / 1) ( )

( / 1) ( ) ( ) ( )

T T T T

d d d d

k k k k

k k k k k

   



   



P P K

H P H R H C C H K

(16)

As seen from expressions (14) and (16), the gain

matrix of MDKF

()kK

and covariance matrix of

estimation error

()kP

involve the cross-correlation

term

()kC

The measurement differencing Kalman filter for

time-correlated measurement errors is represented

by the formulas (10)-(16).

4 Adaptive Measurement Differencing

Kalman Filter

A measurement noise covariance matching-based

adaptive measurement differencing Kalman filter

for the case of time-correlated measurement errors

is presented below.

Statement 1: The measurement differencing Kalman

filter (10)-(16) innovation sequence:

( ) ( ) ( ) ( / 1)

j j j j j   Δ μ HBu H x

(17)

is a zero mean Gaussian random process with

covariance:

( ) ( ) ( / 1)

T T T T

d d d d

E j k j j



    



Δ Δ H P H R H C C H

(18)

The proof of Statement 1 is given in, [18].

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An innovative sample covariance matrix:

ˆ( ) ( ) ( )

j k M

k j j



  



SΔΔ

(19)

will be used in the MDKF's R-adaptation technique.

Here

is the number of implementations used (the

width of the "sliding window").

The real and theoretical values of the MDKF

innovation covariance can be compared to

determine the multiple measurement noise scale

factors (MMNSFs) for the R-adaptation. The

multiple measurement noise scale matrix

()kS

can

be calculated from the equality of the real and

theoretical innovation covariances as follows:

1( ) ( ) ( / 1)

( ) ( )

kTT

j k M

j j k k

  







  

Δ Δ H P H

S R H C C H

(20)

()

1( ) ( ) ( / 1)

()

kT T T T

d d d d

j k M

j j k k

  





   





Δ Δ H P H H C C H

(21)

However, due to the limited number of

measurements

and the possibility of errors such

as approximation errors and rounding errors in

computer calculations, the matrix found by using

(21) may not be diagonal and may contain diagonal

elements that are "negative" or less than "one"

(these are physically impossible). Thus, the

following rule should be used when creating the

scale matrix in order to avoid such a circumstance:

 

, ,..., n

diag s s s

   

S

, (22)

 

max 1,

i ii



1,in

. (23)

where

is the ith diagonal element of the matrix

In this way, in the case of measurement alfunction,

*()kS

will alter and have an impact on the Kalman

gain:

 

( ) ( / 1) ( )

( / 1) ( ) ( ) ( )

T T T

d d d d

k k k k

k k k k k 

   



   



K P H C

H P H S R H C C H

(24)

As a consequence of any form of malfunction,

the relevant element of the scale matrix (which

corresponds to the faulty measurement vector

component) will increase. This subsequently

reduces the Kalman gain, thereby minimizing the

effect of innovation on the state update procedure

and leading to greater accuracy of estimation results.

5 INS’s Error Compensation using

AMDKF

To compensate for INS time-correlated errors, the

proposed adaptive measurement differencing

Kalman filter is employed in a multi-input multi-

output model for aircraft. The lateral and

longitudinal motion of an aircraft is modeled using

the state-space model based on, [20], which supplies

information for the BRAVO - a twin-engine, jet

fighter aircraft.

5.1 Simplified INS Error Model

This section outlines the semi-analytical error model

for INS. Note that the INS error model as a whole

poses a significant challenge. To simplify the

model, certain assumptions can be made. For

instance, in some scenarios, inter-channel

connections can be disregarded. Therefore, to

represent the INS error model, we used a simplified

system of difference equations, as specified in, [3].

In the literature, [3], accelerometer errors and

errors related to gravitational indetermination were

classified as simple white noise inputs, random walk

processes, first-order Gauss-Markov processes, and

similar methods. This work utilizes the first-order

Gauss-Markov process to model the INS errors,

which are presented as follows:

( ) ( 1) 1 ( 1)

a j a j t tU j







        



(25)

( ) ( 1) 1 ( 1)

g j g j t tU j







        



(26)

Here

x y z

a a a  

are the measurement errors of

accelerometers,

x y z

g g g  

are the terms of error

due to gravitational indetermination,

t

is the

discretization interval,

U

and

U

are the white

Gauss noises with zero mean;



and



are the

terms for the correlation period.

5.2 Results of the AMDKF Simulation for the

Error Compensation of the INS

The longitudinal and lateral dynamics of aircraft

BRAVO have been examined through simulations.

To estimate the aircraft's state vector, the presented

in, [18], MDKF and proposed in this study AMDKF

were employed.

Measurement noise increment type sensor faults

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are simulated by multiplying the standard deviation

of the pitch rate and pitch angle gyro measurement

noises with a constant term in interval

80 150k

( ) ( ) ( ) 8

q q q

z k z k v k  

, (

80 150k

). (27)

( ) ( ) ( ) 5z k z k v k

  

  

, (

80 150k

). (28)

Figure 1, Figure 2, Figure 3, Figure 4, Figure 5,

Figure 6, Figure 7 and Figure 8 show a subset of the

simulation findings. Figure 1, Figure 3, Figure 5,

Figure 7 demonstrate the estimation results for

forward velocity

()u

, vertical velocity

()w

, pitch

rate

 

, and pitch angle

 



when the presented

AMDKF was employed. Figure 2, Figure 4, Figure

6, Figure 8 show the MDKF estimation results for

the same aircraft states. The first section of Figure 1,

Figure 2, Figure 3, Figure 4, Figure 5, Figure 6,

Figure 7 and Figure 8 compares the estimation

findings with actual values of the aircraft states. The

error of the estimates is shown in the second half of

the Figures. The state estimate errors are relatively

minimal, as demonstrated by the graphs. The

proposed adaptive measurement differencing

Kalman filter allows the aircraft state vector to be

estimated at each step while compensating for INS

errors and noise increment type sensor faults.

Fig. 1: AMDKF results for forward velocity

estimation

Fig. 2: MDKF results for forward velocity

estimation

Fig. 3: AMDKF results for vertical velocity

estimation

Fig. 4: MDKF results for vertical velocity

estimation

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Fig. 5: AMDKF results for pitch rate estimation

Fig. 6: MDKF results for pitch rate estimation

Fig. 7: AMDKF results for pitch angle estimation

Fig. 8: MDKF results for pitch angle estimation

The Root Mean Square Errors (RMSE) of the

estimation values, calculated between 80-150

iterations (after filter convergence), are likewise

consistent with these findings. The following

equation was used to calculate the RMSE values for

the MDKF and AMDK estimates:

 

150

ˆ( / ) ( )

x k k x k

RMSE







(29)

Table 1. RMSE of the state estimations in case of

time-correlated bias and noise increment at INS

measurements

RMSE

MDKF

AMDKF

(m/s)

1.8245

0.9845

(m/s)

3.8745

3.7517

(deg/ )qs

0.0379

0.0308

(deg)



0.1891

0.1229

(deg)



0.0206

0.0234

(deg/ )ps

0.0360

0.0608

(deg/ )rs

0.0287

0.0303

(deg)



0.0743

0.0863

We can see that the suggested strategy improves

estimation accuracy in the presence of noise

increment type sensor faults.

In the presence of time-correlated bias and noise

increment at INS measurements, the proposed

AMDKF estimates for the longitudinal motion

parameters outperform the simulation results in

Table 1. For the lateral motion parameters, MDKF

estimates are more accurate than AMDKF.

The proposed adaptive measurement

differencing Kalman filter provides more accurate

estimates for the faulty measurement channels in the

presence of time-correlated INS’s errors.

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Furthermore, AMDKF estimation accuracy of the

rest of longitudinal motion parameters, such as

forward velocity (

) and vertical velocity (

), is

better than MDKF, because these parameters are

highly affected by faulty sensor data. This is

because the suggested AMDKF compensates for

INS errors as well as noise increment type sensor

faults.

This research demonstrates that AMDKF is

robust against noise increment type sensor faults.

This is explained by the fact that the AMDKF

measurement noise covariance rises as the scale

matrix expression (21) is applied. As a result, the

gain of AMDKF decreases and the weight of the

measurements in the Kalman estimates is reduced,

and the effect of the measurement result on the filter

is less. The filter adapts to the noise increment type

sensor fault. Table 1 displays the RMSE of the

estimation values generated for the AMDKF and

MDKF. As can be seen from Table 1, AMDKF is

more accurate than MDKF for the faulty

measurement channels and channels that are highly

affected by faulty sensor data. Figure 1, Figure 2,

Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and

Figure 8 show that the proposed AMDKF provides

accurate estimates of the aircraft's longitudinal

states while being unaffected by INS measurement

biases and noise increment type sensor faults. As a

result, the INS can be used for extended periods in

flight.

6 Conclusion

In this study, we present a covariance matching-

based adaptive measurement differencing Kalman

filter for time-correlated measurement errors and

noise increment type sensor faults. Measurement

differences are calculated in the filter to solve the

state estimation problem. In this scenario, the

measurement noise for the derived measurements is

correlated with the process noise. AMDKF, robust

to noise increment-type measurement faults is

designed for correlated process and measurement

noise situations. The developed AMDKF's

robustness properties are studied. The proposed

AMDKF and the previously developed MDKF were

used to estimate the states of a multi-input/output

aircraft model in the presence of noise increment

type sensor faults in the time-correlated INS

measurements and the results were compared.

Simulation results show that, in the presence of

noise increment type sensor faults in the time-

correlated INS measurements, AMDKF provides

more accurate estimates for the faulty measurement

channels and channels that are highly affected by

faulty sensor data. The proposed AMDKF is robust

to the time-correlated measurement errors and noise

increment type sensor faults simultaneously.

Using only sensor error models, the proposed

AMDKF can correct INS errors without a need for

hardware redundancy. Thanks to this method,

autonomous navigation is possible without the need

for external navigation resources.

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Chingiz Hajiyev, Ulviye Hacizade

E-ISSN: 2224-2856

485

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Chingiz Hajiyev carried out the conceptualization,

investigation, methodology, validation, writing -

review & editing.

- Ulviye Hacizade has organized and executed the

simulation, data curation, formal analysis.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.51

Chingiz Hajiyev, Ulviye Hacizade

E-ISSN: 2224-2856

486

Volume 18, 2023