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Research Article | | Peer-Reviewed

Interacting Multi-Model Strong Robust Adaptive Unscented Kalman Filter to Bearing only Tracking of Underwater Vehicle Approaching the Observer

Kang Song Ju,  Myong Hyok Sin*
Published in Engineering Mathematics (Volume 9, Issue 2)
Received: 4 November 2025     Accepted: 22 November 2025     Published: 29 December 2025
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Abstract

We have designed an interacting multi-model strong robust adaptive unscented Kalman filter for bearing only tracking of an underwater vehicle approaching the observer. To solve the problem of tracking an approaching underwater vehicle to the observer based on only its bearing, an interactive multi-model robust adaptive unscented Kalman filter is proposed in this paper. First, a new model of the bearing sense motion towards the observer is proposed to construct a set of realistic target motion modes consisting of linear and curved motion modes. In addition, to account for the influence of outliers in the target bearing measurements, the distribution of measurement noise is assumed to have a Student’s t-distribution, and the probability distribution of the degree of function and the scaling matrix of this distribution is assumed to have a gamma distribution and an inverse Wishart distribution. Thus, the model interaction step is to factorize the mixed probability density function using variational Bayesian method and, based on this, a predictive update method is proposed. In the measurement update phase, the posterior probability density function is obtained in factorization form using variational Bayesian method, and based on this, a posteriori mode probability calculation method is proposed. Simulation results show that our proposed method greatly improves the convergence rate of target tracking error.

Published in Engineering Mathematics (Volume 9, Issue 2)
DOI 10.11648/j.engmath.20250902.13
Page(s) 46-67
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2025. Published by Science Publishing Group
Previous article
Keywords

Bearing only Tracking, Unscented Kalman Filter, Inverse Wishart Distribution, Interactive Multi-Model

1. Introduction
For its wide application to civilian and military fields, target tracking has been deeply studied in the last decades.
[1-5]
. In particular, the problem of increasing the convergence rate of target tracking is considered very difficult due to the uncertainty of the motion model of target and the non-Gaussianity of measurement noise
[6-8]
. So far, many methods have been proposed for target tracking. According to the structure of the motion model, the methods for target tracking can be classified into multi-segment analysis
[9]
, reinforcement learning
[10]
, data-driven method
[11]
, filter-based estimation method
[12-16]
, Singer model-based method
[17]
, multi-model-based method, interactive multi-model-based method
[18, 19]
, model-switching method
[19]
, and modern statistical model-based method
[20]
, etc. Especially, the most remarkable method is to use various updates of the Unscented Kalman Filter (UKF) and Cubature Kalman Filter (CKF) which are designed based on nonlinear motion models and observation models
[21-27, 31, 32]
. However, many methods are based on the simultaneous measurement of range as well as the bearing of the target, or on the use of more than two sensors. In general, it is an important issue to develop a method that guarantees the desired tracking accuracy within a minimum time in the problem of tracking the state of a target (range, magnitude and direction of velocity) with only bearing measurements by a single sensor. An important factor affecting the tracking accuracy is the uncertainty of the motion model and the non-Gaussianity of the measurement noise. In water, the non-Gaussianity of the measurement noise is much stronger than in the air. In underwater target tracking, outliers have a great impact. The uncertainty of the motion model has the greatest influence on tracking accuracy.
Various motion models have been proposed in
[1]
, and two-dimensional underwater target tracking methods based on these models
[2, 4, 5, 13, 14]
with only bearing have been proposed. However, the simulation results in these methods do not have enough convergence rate of tracking error.
In real situations, underwater targets approach observers with various tactical maneuvers. One way to ensure the convergence rate of state tracking in such environments is to construct a set of motion modes as possible motion models and to perform state tracking based on the Interacting Multiple Models (IMM). In many literatures, IMM-based state tracking methods have been introduced
[12, 23, 28, 29]
. Considering the simulation results of
[12]
which introduced IMM-based two-dimensional (2D) underwater target tracking method, the error convergence time is too long and the simulation conditions in
[29]
did not fully reflect the realistic condition because the measurement noise was set as zero-mean Gaussian white noise. In
[3, 29]
, Krameo-Rao bounds are analyzed according to the characteristics of measurement noise in target tracking with only bearing. In real environments, we often know all possible modes of motion of a 2D underwater target. If this condition is assumed to hold, a state tracking method with sufficient error convergence rate should be proposed, taking into account the more realistic noise of the target bearing measurements.
In real environments, the measurement data of underwater targets include outliers. The outliers have the characteristic of thickening the tail of the probability density function (PDF) of the measurement noise. In
[30]
, a method to improve the robustness of IMM filter is proposed. To do this, we assume the measurement noise as a Student’s t-distributed random variable and apply the IMM method and Variational Bayesian (VB) approximation to estimate the state of the target.
However, this filter is not suitable for tracking 2D underwater target with nonlinear measurement equations, since it is designed for linear systems. Therefore, we aimed to design a robust IMM filter with fast error convergence considering the measurement noise with outliers and the nonlinearity of the system. For this purpose, we assume that the measurement noise is a Student’s t-distribution, and the unknown degrees of freedom and scale matrix of this distribution are assumed to follow the gamma distribution and inverse Wishart distribution as in
[30]
. To address the nonlinearity of the system, an UKF using the Unscented transform (UT) is used, and to enable state estimation, we obtain the factorization form of the mixed PDF and the posterior PDF using VB approximation. In addition, we have proposed a new method to approximate computational expression of Gaussian integral using the UT to simplify the expression, and using constants, such as Euler constant, during the expression derivation to improve the approximation accuracy. Simulation results confirm that the convergence rate of the model tracking error increases remarkably, provided that the entire motion mode of the underwater target is known.
The paper is organized as follows;
Section 2 gives the basic concepts of the method of the UT and VB approximation, and Section 3 presents the design of an Interacting Multi-Model Strong Robust Adaptive Unscented Kalman filter (IMM-SRAUKF) over a set of modes consisting of two motion modes, newly constructing the motion equations for the approaching target, always with the direction of the observer. Section 4 shows the simulation analysis and Section 5 concludes.
2. Basic Concepts
2.1. UT and Gaussian Integral Approximation Functions
Consider the following nonlinear dynamic system.
(1)
where represents the moment (simply called moment), is the state vector and the measurement of the system. is the sampling period, is the independent process noise vector and measurement noise.
Given a possible non-Gaussian filter density function with mean and co-variance , UKF approximates by samples of the Gaussian density function respectively to perform a one-step prediction of the state quantity and co-variance , respectively, as follows
[33].
(2)
(3)
where
is the scale parameter, controls the deviation of sigma points and is usually set to a small parameter. is usually set to zero as a second-order scale parameter.
is a dirac-delta function. For a Gaussian distribution, is optimal.
is obtained as follows.
(4)
where represents the -th column vector of the square root matrix of the matrix . Equation (4) is called the UT for computing the sigma points of from . is called a sigma point.
Then, the posterior density function for the state quantity and co-variance prediction is respectively described as follows.
Approximating the state transition density function as , the predicted state becomes
(5)
Likewise, the predicted co-variance becomes
(6)
where
For convenience, we predefine the following notation.
(7)
where is the sigma points distributed with co-variance centered around and is the weight vector. This function is called the Gaussian integral approximation function using the UT and denoted by GU.
Using Eq. (7), Eqs. (5) and (6) can be written as follows:
(8)
where is a vector with as a component.
2.2. Kullback-Leibler (KL) Deviation
The KL deviation can be used as a similarity measure of two PDFs
[26]
.
Definition 1. Assuming to be a PDF on , the KL deviation of from is defined as follows.
(9)
Some authors call the KL range, but since , in fact, KL deviation does not satisfy the range axiom.
Also, some authors associate Eq. (9) with to call KL divergence.
Lemma 1 (Theorem 2.1 of
[37]
)
Assuming in the probability space is a PDF on with mean and co-variance , is a Gaussian PDF, we have
(10)
(The proof is omitted).
The lemma above proves that the Gaussian PDF with the smallest KL deviation from the PDF with some mean and co-variance is a Gaussian PDF with the same mean and co-variance.
Definition 2
[33]
For PDFs defined in the probability space , we define the weighted KL deviation as follows:
(11)
where
Lemma 2
[33]
The weighted KL deviation (11) can be written as follows:
(12)
3. Interacting Multi-Model Strong Robust Adaptive Unscented Kalman Filter Design
3.1. Construction and Assumptions of Motion Sets
To construct the set of motion of underwater vehicles, we make the following assumptions.
Assumption 1. The magnitude of the underwater vehicle velocity is constant and the direction of motion of the underwater vehicle at any time during tactical maneuvers is directed towards the observer.
Assumption 1 captures the characteristics of the underwater vehicle controlled by sensing the direction of the observer's acoustic signal generation.
3.1.1. Non-Maneuvering Motion Model
In the uncontrolled region of underwater vehicles, the non-maneuvering motion model is assumed to follow a white noise acceleration model moving at constant velocity in a 2D plane
[1]
.
(13)
where is the state transition matrix and is the noise gain matrix.
The components of the state vector are the position and the velocity vector components of target in the 2D Cartesian coordinate system ; .
, (14)
The components of the noise vector are mutually independent zero-mean white noises whose variances are all .
3.1.2. Maneuvering Motion Model
In this paper, according to the motion characteristics of vehicles such as underwater vehicles approaching the observer at a constant speed, the motion model is newly modeled based on Figure 1 as follows:
In Figure 1, is the bearing, and , are the constant velocity of the underwater vehicle and the observer, respectively.
When the vehicle and observer are placed in the same position as Figure 1, .
Figure 1. Motion Characteristics of Underwater Vehicle Approaching the Observer.
According to Assumption 1, the velocity component of the underwater vehicle without noise at any time can be written as follows:
In the absence of noise, if the observer is stationary, , and if it is moving, .
When , the vehicle does not generally have a straight line motion, so that all components of acceleration are nonzero. Thus
(15)
The corresponding discrete model of Eq. (15) is
Assumption 1, which is constant in magnitude at any time, gives the following expression;
(16)
Thus, we can see that at any time the velocity vector of the vehicle moving toward the observer at a constant velocity rotates by during .
Then, by the acceleration vector obtained from Eq. (15) at time , the equation of motion of the underwater vehicle can be expressed as follows:
(17)
Here, the components of the noise vector are mutually independent zero-mean white noises whose variances are all , and the state transition matrix is obtained by .
The results are as follows;
, (18)
The measurement process can be modeled as:
(19)
Remark 1. Since the bearing in Eq. (16) varies with the position of the observer, as in Eq. (19), by means of the position vector of the observer, we can treat the position of the observer as the optimal state estimation problem of the underwater vehicle with the control signal.
However, the tactical maneuvering models, such as the rotational motion models and the curve motion models, presented in
[1]
, do not present such a control signal. The model uncertainty in the state estimation problem of the plant under study is only represented by the model selection uncertainty in a given set of motion models, unlike the uncontrolled underwater vehicle state estimation problem. In this paper, Interacting multi-model strong robust adaptive unscented Kalman filter is designed to solve the optimal state estimation problem of underwater vehicles with model selection uncertainty.
3.1.3. Assumptions on Measurement Noise
We can summarize Eqs.(13) and (17) and (19) as follows:
(20)
where , , , , and is the process noise co-variance.
Remark 2. To reflect the non-Gaussian character of the thickening of the tail of the PDF due to outliers in the measurement sequence, we assume that the measurement noise follows the Student’ t-distribution.
That is, we assume that the following is true for .
Here is the mean, scale matrix and the degree of function (DOF), respectively.
Since the Student’ t-distribution is expressed by a Gaussian PDF, the likelihood PDF under the condition that it is in the motion mode can be written as follows
[34]
:
(21)
Where is the gamma PDF. Where is an auxiliary parameter.
According to
[36]
, for the parameter , we assume as follows:
Assumption 3. The posterior distribution of the parameter is gamma distributed.
(22)
Assumption 4. To overcome the thickening of the tail of the Student’ t-PDF when the DOF has a large value, the posterior distribution of DOF is assumed to be gamma distributed.
(23)
We also assume the posterior distribution of the scaling matrix as follows.
Assumption 5. The posterior distribution of the scaling matrix is the inverse Wishart distribution.
where is the inverse Wishart PDF, is the DOF, and is the inverse scaling matrix.
The inverse Wishart PDF is
(24)
where is the Di-gamma function and is the output number.
Assumption 6. Given the mode of motion at time , the state quantities , the scaling matrix , and the DOF are independent of each other.
This assumption is set by the condition that the measurement noise at time in Eq. (1) is independent of the state quantity at time , enabling the design of the new IMM filter we are going to propose.
3.2. Model Interaction
The posterior mode probability , the predicted mode probability and the mixed probability have the following relations:
(25)
(26)
Here is the probability that the -th motion is converted to the -th motion
[32]
. By Assumption 6, the mixed PDF of and becomes as follows:
(27)
The mixed PDF (27) can be approximated in a more convenient form based on the VB approximation.
The following theorem is given for this.
Theorem 1. The Gaussian PDF minimizing the KL deviation from the marginal density function for in the mixed PDF (27) is
(28)
Proof.
The marginal density function for is
(29)
Then the average is
(30)
Also, using , the co-variance becomes
(31)
And by Lemma 1, it is proved that the Gaussian PDF (28) with the smallest KL deviation from the marginal PDF (29) for with the mean (30) and co-variance (31).
Theorem 2
The weighted KL deviations from the marginal density functions for and in the mixed PDF (27) are obtained, respectively, as follows:
(32)
(33)
Then the mean and co-variance of are respectively
(34)
(35)
Proof.
First we prove (32). The marginal density function for is
(36)
Then, since , the weighted KL deviation for two inverse Wishart PDFs with weight is given by Definition 2 and Lemma 2 as follows.
(37)
Thus, Eq. (32) is proved.
Next we prove (33).
The marginal density function for is
(38)
Since
(39)
as in Eq. (37), Eq. (33) is proved.
Equations (34) and (35) can be proved as follows considering that the mean and co-variance of the gamma distribution PDF are respectively.
By Eqs. (38) and (39), we have
(40)
(41)
The weighted KL deviation from the peripheral density function for by Eq. (39) has a gamma distribution, and hence from Eqs. (40) and (41) we obtain
(42)
Proof. End.
By Theorems 1 and 2, the mixed PDF (27) of and is approximated as follows.
(43)
3.3. Prediction Step
In the prediction step, the state and co-variance are predicted on the condition given the mode of motion. The predictions in the filters are obtained by the predictive PDF at time , and this predictive PDF is obtained by the champhman-Kolmogorov equation
[35]
, the propagation equation of the posterior state PDF at moment.
(44)
Here, is a state transition PDF, and the distribution of the dynamic model, scale matrix and DOF that give the state transition can be considered independent of each other. Also, since the posterior PDF in (44) is a mixed PDF at the time shown in (43), we can write (44) as follows.
(45)
This indicates that, given the mode of motion at the time , the predicted state, the scaling matrix and the probability distribution of DOF are independent of each other and maintain the previous distribution patterns.
Thus, Eq. (45) can be written as
(46)
Using Eqs. (45) and (46) and Gaussian integral approximation function (7) using the unscented transformation, we obtain
(47)
(48)
In the champhman-Kolmogorov model (46), the prediction parameters and of the scaling matrix and the DOF are difficult to find directly by Eq. (45). This is because the dynamic model of the scale matrix and the DOF is usually unknown in practice, and the dynamics of the scale matrix and the DOF in Eq. (45) are not known exactly. Therefore, we propose heuristic dynamics for scaling matrices and DOF . This is the dynamics that simply spreads the previous approximation posterior of the scaling matrix and the DOF
[34]
. Based on Eq. (43) which is obtained based on the VB approximation, the prediction parameters , that must be determined in the prediction step are determined by the following discovery dynamics such that in the measurement update step the posterior parameters , can be determined accurately by the fixed point iteration method. (see the IMM-SRAUKF algorithm)
(49)
where . We usually set . The IMM-SRAUKF cannot estimate the process noise co-variance shown in Eq. (20) as the total mean of . One method is to dynamically adjust by fusing the past and the current of the co-variance matrix through the forgetting factor
[36]
.
(50)
3.4. Measurement Update Step
The aim of the measurement update step is to obtain the posterior PDF on the condition that the new momentary motion mode and the measurement is given, and based on this, to find the state quantity and co-variance matrix of the new moments. IMM filters must first obtain the posterior PDF and the posterior mode probability based on it at this stage. First, we obtain the posterior PDF .
Since the motion mode of the time and the measurement are difficult to obtain analytically the posterior PDF, the measurement update step is to approximate the posterior PDF by using the VB approximation in the following factorized form, given the motion mode of the time and the new measurement .
(51)
For convenience, we will have the following notation.
(52)
The posterior PDFs are obtained by the VB approximation as the PDFs that minimize the KL deviation as follows:
The VB marginal density functions satisfying the above equation have the following forms:
(53)
(54)
(55)
(56)
To derive a method for calculating Eqs. (53)-(56), we can first write the following expression using Eqs. (21) and (51):
(57)
where .
Using Eq. (24), Eq. (57) becomes as follows:
(58)
First, let us look at the corrective form of Eq. (53).
Substituting Eq. (58) into Eq. (53), we obtain
(59)
where
(60)
is the modified co-variance matrix. By Eq. (59), we can write the following relation:
Thus, can be written in the form of Gaussian PDF as
(61)
where
(62)
(63)
The in Eqs. (61) and (62) are obtained as follows:
(64)
(65)
(66)
Next, we consider the corrective form of Eq. (54). Substituting Eq. (58) into Eq. (54), we obtain
(67)
where
(68)
By Assumption 5, is an inverse Wishart PDF because has an inverse Wishart distribution. Thus, we have the following form of PDF:
(69)
Comparing Eqs. (68) and (66) by (24), the following relation is obtained:
(70)
(71)
Substituting these results into Eq. (66), we obtain
(72)
Next, we consider the corrective form of Eq. (55). By Assumption 3, has a gamma distribution, so is a gamma PDF as follows:
(73)
Substituting Eq. (58) into Eq. (55) and comparing with Eq. (72), the following relationship is obtained:
(74)
(75)
Next, we consider the corrective form of Eq. (56). Substituting Eq. (58) into Eq. (56), we use the following approximation
[21]
.
(76)
Here is the Euler gamma constant and is the least squares estimate. For example, for , , . Then Eq. (56) is obtained as follows:
(77)
By Assumption 4, has the gamma PDF, so let us write this density function as
(78)
Comparing Eqs. (78) and (77), we have the following relation:
(79)
By Eqs. (61), (69), (73), and (79), the unknown quantities , , , , present in Eqs. (60), (68), (71), (74), and (75) are determined as follows:
(80)
(81)
(82)
(83)
(84)
(85)
In Eq. (86), is a di-gamma function.
When using the results obtained above, Eqs. (62)-(66), (70), (71), (74), (75), (79)-(85), except for the methods of finding and in Eqs. (70) and (80), the problem of finding can be solved using the fixed point iteration method.
Figure 2 shows the fixed point iterative scheme.
Figure 2. A Fixed Point Iterative Scheme for Measurement Update Parameter Determination. A Fixed Point Iterative Scheme for Measurement Update Parameter Determination.
The posterior mode probability is calculated as
(86)
Therefore, to calculate the posterior mode probability, we must determine the predictive likelihood function .
Here, the predicted mode probability is determined by Eq. (25).
The predicted likelihood function can be found as follows:
However, due to (57), we cannot find the analytical solution of the above integral.
To overcome this problem, we can approximate the predicted likelihood function using the VB approximation as shown in
[30]
as follows:
Since the second term on the right-hand side of the above equation is minimized by the VB approximation, we can write the above equation as
(87)
Substituting Eqs. (58) and (61), (69), (73), and (78) in Eq. (87), we obtain the following expression:
(88)
where
By Eq. (88), we can determine the posterior mode probability shown in Eq. (88). Thus, the final state estimator and co-variance matrix are determined as follows:
(89)
(90)
Summarizing the above results, Figure 3 shows the block diagram of the IMM-RAUKF algorithm.
Figure 3. Schematic Diagram of IMM-SRAUKF Algorithm.
The IMM-SRAUKF algorithm is as follows.
IMM-SRAUKF algorithm
Step 1: We set the initial estimates for each mode of motion and the forgetting factor of Eqs. (49) and (50). Let and go to step 2.
Step 2: Using Eqs. (25) and (26), the prediction mode probability and the mixed mode probability are calculated. Then, the mixing quantities are calculated using Eqs. (30)-(32), (42).
Step 3: Using Eqs. (47)-(50), we calculate the following predictions for each mode.
Step 4: We update the posterior quantities in each mode using the fixed point iteration method shown in Figure 2.
Step 5: We update the posterior probability using Eqs. (86) and (88).
Step 6: Calculate using Eqs. (89) and (90). Go to step 2.
4. Simulation and Analysis
To evaluate the performance of the proposed filter in tracking an underwater vehicle approaching the observer in two dimensions, the root mean square errors (RMSE) is used as follow:
(91)
(92)
The underwater vehicle is assumed to approach the observer as two modes of motion shown in Section 2. It is assumed that the underwater vehicle approach observer by the linear motion modes with constant velocity for the non-maneuvering motion model and the bearing sense motion modes for maneuvering motion model. In the time , the linear motion mode and the Bearing sense motion mode . The sampling period . The simulation time interval is 0~200s. Figure 4 shows the motion trajectory of the observer and the target. The sight range of the day (SRD) was given as 6000 m.
Figure 4. Target Motion to the Observer's Motion.
The tactical maneuver of the observer is assumed to be circular with center and radius SRD in the coordinate system shown in Figure 4, and the target is assumed to have a linear motion for 100 s and then a bearing sense motion towards the observer. To evaluate the performance of the proposed filter in tracking an underwater vehicle approaching the observer in two dimensions, we compare the tracking performance in three measurement noises.
The modal transition probability is given by and the initial posterior mode probability by respectively. The initial state and covariance matrix are given as follows:
,
In the Eqs. (13) and (17), acceleration noise for process noise is the Gaussian noise with zero and co-variance . In the simulation, The Co-variance is assumed unknown. The forgetting factor of Eqs. (49) and (50) is given by , respectively, the initial parameter of the inverse Wishart distribution is given by , respectively, and all other initial parameters are given by 0.5.
The observer velocity is assumed to be constant and the target velocity is constant .
To analyze the effectiveness of the proposed method, it is compared with Interacting Multi-Model Unscented Kalman filter (IMM-UKF)
[25]
, Interacting Multi-Model Cubature Kalman Filter (IMM-CKF)
[19]
, and Variational Bayesian -based Interacting Multi-Model Filter (IMM-VBF)
[30]
.
First, to see if the real covariance is different from the standard covariance, we simulated Gaussian noise with the following time-varying co-variance:
Here . In Figure 5, the modal probabilities for the linear and bearing sense motions obtained by the proposed method are shown.
Figure 5. Mode Probabilities for Constant Velocity Models.
As shown in Figure 5, it is shown that the linear motion with constant velocity between 0 and 100 s and the bearing sense motion toward the observer between 100 and 200 s are almost tracked. In Figure 6, the performance of the proposed filter is shown by comparison with IMM-CKF, IMM-UKF and IMM-VBF. As shown in the figure, the proposed method can track the position of the real target accurately compared to other methods.
Figure 7 shows the reduction of the range measurement error with time. It can also be seen in Figure 7 that the proposed method has the highest reduction rate of range tracking error compared to other methods. The detailed numerical data are shown in Table 1.
Figure 6. Comparison with Various Methods for Target Tracking.
Figure 7. Comparison of Range Measurement Error of Various Methods for Target Tracking.
Table 1 shows the mean values of the RMSE with respect to the position of the measurement noise for the cases.
(a)
(a)
(b)
(b)
(c)
(c)
respectively, averaged through 500 Monte Carlo simulations.
Table 1. Root Mean Square Error of the Position.

Case

IMM-CKF

IMM-UKF

IMM-VBF

Proposed method

a

16.3853

12.5860

15.7735

6.7743

b

15.1209

10.9820

15.0893

5.0908

c

19.4562

13.8467

17.3328

7.0835
As shown in Table 1, the proposed method can see a significant reduction in RMSE compared to other methods.
Compared with the case with time-varying co-variance (a) and the case with Gaussian hybrid noise (b), the RMSE is large in the case of (c), and the case of (b) with Gaussian hybrid noise is the smallest.
Table 2 shows the mean values of the RMSE with respect to the magnitude of velocity.
Table 2. Root Mean Square Error of the Magnitude of Velocity.

Case

IMM-CKF

IMM-UKF

IMM-VBF

Proposed method

a

0.1092

0.1078

0.1001

0.0733

b

0.1352

0.9677

0.0992

0.0912

c

0.2921

0.8110

0.1191

0.1164
As shown in Table 2, also the proposed method can see a significant reduction in RMSE compared to other methods.
Simulation results show that our method is robust against measurement noise with outliers and the target tracking performance is very effective compared to other methods.
5. Conclusions
We design and validate the effectiveness of a IMM-SRAUKF for tracking of the bearing of an underwater vehicle approaching the observer only with a tactical maneuver. In particular, given the definite set of motion model of the underwater vehicle approaching the observer, the tracking performance is significantly higher when using our proposed model than when using a conventional model such as the past constant turn models or the constant speed curvature model, especially when the IMM-SRAUKF designed in this paper is very robust to the outliers which are included in the measurements. The reason for the high performance of the proposed filter, even for very severe measurement noise, is that it has the effective application of various techniques that can significantly improve the approximation accuracy in the calculation of the posterior probability density by VB approximation in the measurement updating step of the filter design, and the new motion equation that accurately reflects the motion characteristics of the underwater vehicle approaching the observer. In general, the problem of accurately and quickly tracking the behavior of underwater vehicles with bearing only measurements is very difficult, and it is expected that in the future, it will be important to improve the target tracking performance in real environments, taking into account the effect of process noise in the equations of motion caused by more severe measurement noise, measurement noise of devices such as GPS on the observer, and measurement delay due to the speed of sound waves in the fluid.
Abbreviations

UKF

Unscented Kalman Filter

CKF

Cubature Kalman Filter

IMM

Interacting Multiple Models

2D

Two-Dimensional

PDF

Probability Density Function

VB

Variational Bayesian

UT

Unscented Transform

IMM-SRAUKF

Interacting Multi-Model Strong Robust Adaptive Unscented Kalman Filter

KL

Kullback-Leibler

DOF

Degree of Function

RMSE

Root Mean Square Errors

SRD

Sight Range of the Day

IMM-UKF

Interacting Multi-Model Unscented Kalman Filter

IMM-CKF

Interacting Multi-Model Cubature Kalman Filter

IMM-VBF

Variational Bayesian-based IMM Filter
Conflicts of Interest
The author declares no conflicts of interest.
References
[1] X. Rong Li, Vesselin P. Jilkov, “Survey of Maneuvering Target Tracking. Part I: Dynamic Models,” IEEE Transactions on Aerospace and Electronic Systems, vol. 39, no. 4, pp. 1333- 1364, October 2003. IEEE Log No. T-AES/39/4/822061.
[2] Dongliang Peng, Yunfei Guo, “Fuzzy-logic adaptive variable structure multiple-model algorithm for tracking a high maneuvering target,” Journal of the Franklin Institute, vol. 351, pp. 3837-3846, 2014.
[3] A. B. Miller, B. M. Miller, “Underwater Target Tracking Using Bearing-Only Measurements”, Journal of Communications Technology and Electronics, Vol. 63, No. 6, pp. 643–649, 2018.

https://doi.org/10.1134/S106422 6918060207

[4] Benlian Xu, Zhengyi Wu, Zhiquan Wang, “Theoretic Performance Bound for Bearings-Only Tracking”, Proceedings of the Sixth International Conference on Machine Learning and Cybernetics, Hong Kong, PP. 19-22, August 2007.
[5] Kutluyıl Dogancay, “Bias Compensation for the Bearings-Only Pseudo linear Target Track Estimator”, IEEE Transactions on Signal Processing, vol. 54, no. 1, January 2006.
[6] Zhao Li, Yidi Wang, Wei Zheng, “Adaptive Consensus-Based Unscented Information Filter for Tracking Target with Maneuver and Colored Noise”, Sensors, vol. 19, 2019.

https://doi.org/10.3390/s19143069

[7] Zhuoran Zhang, Changqiang Huang, Dali Ding, Shangqin Tang, Bo Han, Hanqiao Huang, “Hummingbirds optimization algorithm-based particle filter for maneuvering target tracking”, Nonlinear Dynamics, vol. 13, June 2019.

https://doi.org/10.1007/s11071-019-05043-0

[8] Jincheng Wang, Lei Xia, Lei Peng, Huiyun Li, Yunduan Cui, “Efficient Uncertainty Propagation in Model-Based Reinforcement Learning Unmanned Surface Vehicle Using Unscented Kalman Filte”, Drones, vol. 7, 2023.

https://doi.org/10.3390/drones7040228

[9] J. Tague, “Detection of Target Maneuver from Bearings-Only Measurements”, IEEE Transactions on Aerospace and Electronic Systems”, vol. 49, no. 3, July 2013. IEEE Log No. T-AES/49/3/944629.
[10] Dong Hui, Yang Wang Fang, Xiang Gao, “Target Tracking Method with Bearing-only measurements based on Reinforcement Learning”, IEICE Communication Express, vol. 5, no. 1, pp. 19-26, 2016.

https://doi.org/10.1587/comex.2015XBL01564

[11] Waqas Aftab, Lyudmila Mihaylova, “A Learning Gaussian Process Method for Maneuvering Target Tracking and Smoothing”, IEEE Transactions on Aerospace and Electronic Systems, vol. 57, no. 1, February 2021.
[12] M. T. Sabet, A. R. Fathi, H. R. Mohammadi Daniali, “Optimal design of the Own Ship maneuver in the bearing-only target motion analysis problem using a heuristically super vised Extended Kalman Filter”, Ocean Engineering, vol. 123, pp. 146–153, 2016.
[13] Wu Hao, Chen Shu Xin, Yang BFeng, Luo Xi, “Robust Range Parameterized Cubature Kalman Filter for Bearings-only Tracking”, J. Cent. South Univ., vol. 23, pp. 1399-1405, 2016.

https://doi.org/10.1007/s11771-016-3192-z

[14] Renke He, Shuxin Chen, Hao Wu, Zhuowei Liu, Jianhua Chen, “Robust maneuver strategy of observer for bearings‐only tracking”, Asian J. Control, pp. 1-13, 2013.

https://doi.org/10.1002/asjc. 2101

[15] Thomas Northardt and Steven C. Nardone, “Track-Before-Detect Bearings-Only Localization Performance in Complex Passive Sonar Scenarios: A Case Study”, IEEE Journal of Oceanic Engineering,

https://doi.org/10.1109/JOE.2018.2811419

[16] S. Koteswara Rao, “Bearings-only Passive Target Tracking: Range Uncertainty Ellipse Zone”, IETE Journal of Research,

https://doi.org/10.1080/03 772063.2020.1739571

[17] Singer, R. A, “Estimating optimal tracking filter performance for manned maneuvering target.” IEEE Transactions on Aerospace and Electronic Systems, Vol. 6, pp. 473-483, 1970.
[18] H. A. P. Blom, Y. Bar-Shalom, “The interacting multiple model algorithm for systems with markovian switching coefficients.” IEEE Trans. on Automatic Control, Vol. 33, No. 8, pp. 780-783, 1988.
[19] Ahmed A. Bahnasawi, “A switching models gain rotation algorithm for tracking a maneuvering target.” Simulation Practice and Theory, Vol. 7, pp. 71-89, 1999.
[20] Zhou Hong-ren, Jing Zhong-liang, Wang Pei-de, Maneuvering Target Tracking. Beijing: National Defense Industry Press, 1991. (in Chinese).
[21] Qie Liu, Yingming Tian, Yi Chai, Min Liu, Li Sun, “Design of unscented Kalman filter based on the adjustments of the number and placements of the sampling points”, ISA Transactions,

https://doi.org/10.1016/j.isatra.2020.08.013

[22] Bing Zhu, Lubin Chang, Jiangning Xu, Feng Zha, Jingshu Li, “Huber-Based Adaptive Unscented Kalman Filter with Non-Gaussian Measurement Noise”, Circuits Syst Signal Process,

https://doi.org/10.1007/s00034-017-0736-x

[23] Ye Tian, Zhe Chen, Fuliang Yin, “Distributed IMM-Unscented Kalman Filter for Speaker Tracking in Microphone Array Networks”, IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 23, no. 10, October 2015.

https://doi.org/10.1109/TASLP.2015.2442418

[24] Jian Wang, Tao Zhang, Xiang Xu, Yao Li, “A Variational Bayesian Based Strong Tracking Interpolatory Cubature Kalman Filter for Maneuvering Target Tracking”, IEEE Acess, vol. 6, 2008.

https://doi.org/10.1109/ACCESS.2018.2869020

[25] Soheil Sadat Hosseini, Mohsin M. Jamali, Simo Särkkä, “Variational Bayesian adaptation of noise co-variances in multiple target tracking problems”, Measurement, vol. 122, pp. 14-19, 2018.

https://doi.org/10.1016/j.measurement.2018.02.055

[26] Chen Hu, Haoshen Lin, Zhenhua Li, Bing He, Gang Liu, “Kullback–Leibler Deviation Based Distributed Cubature Kalman Filter and Its Application in Cooperative Space Object Tracking”, Entropy, vol. 20, 2018,

https://doi.org/10.3390/e20020116

[27] Jian Wan, Peiwen Ren, Qiang Guo, “Application of Interactive Multiple Model Adaptive Five-Degree Cubature Kalman Algorithm Based on Fuzzy Logic in Target Tracking”, symmetry, vol. 11, 2019,

https://doi.org/10.3390/sym11060767

[28] Ying Xu, Wenjie Zhang, Wentao Tang, Chengxiang Liu, Rong Yang, Li He, Yun Wang, “Estimation of Vehicle State Based on IMM-AUKF”, symmetry, vol. 12, 2022,

https://www.mdpi.com/journal/symmetry

[29] Xu Zhigang, Sheng Andong, Li Yinya, “Cramer-Rao Lower Bounds for Bearings-Only Maneuvering Target Tracking with Incomplete Measurements”, 2009 Chinese Control and Decision Conference.
[30] Shen, C.; Xu, D.; Huang, W.; Shen, F. “An interacting multiple model method for state estimation with non-Gaussian noise using a variational Bayesian method”, Asian J. Control, vol. 17, pp. 1424–1434, 2015.
[31] A. sitz, U. schwarz, J. kurths, “The Unscented Kalman Filter, a Powerful Tool for Data Analysis”, International Journal of Bifurcation and Chaos, Vol. 14, No. 6, pp. 2093-2105, 2004.
[32] Wenling Li, Yingmin Jia, “Adaptive filtering for jump Markov systems with unknown noise co-variance”, IET Control Theory and Applications, vol. 7, Iss. 13, pp. 1765-1772, 2013.

https://doi.org/10.1049/iet-cta.2013.0162

[33] Wenling Li, Yingmin Jia, “Kullback-Leibler divergence for interacting multiple model estimation with random matrices”, IET Signal Processing, Vol. 10, Iss. 1, pp. 12-18, 2016.

https://doi.org/10.1049/iet-spr.2015.0149www.ietdl.org

[34] Hao Zhu, Henry Leung, Zhongshi He, A variational Bayesian method to robust sensor fusion based on Student-t distribution”, Information Sciences, vol. 221, pp. 201-214, 2013.
[35] Särkkä, S., Bayesian Filtering and Smoothing, Cambridge Univ. Press, Cambridge, U. K., pp. 17-26, 2013.
[36] Simo Särkkä, Aapo Nummenmaa, “Recursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations”, IEEE Transactions on Automatic Control, vol. 54, no. 3, march 2009.
[37] Ying Xiong, Yindi Jing, Tongwen Chen, “Abnormality detection based on the Kullback-Leibler divergence for generalized Gaussian data” Control Engineering Practice, vol. 85, pp. 257-270, 2019.

https://doi.org/10.1016/j.conengprac.2019.01.007

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    Ju, K. S., Sin, M. H. (2025). Interacting Multi-Model Strong Robust Adaptive Unscented Kalman Filter to Bearing only Tracking of Underwater Vehicle Approaching the Observer. Engineering Mathematics, 9(2), 46-67. https://doi.org/10.11648/j.engmath.20250902.13

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    Ju, K. S.; Sin, M. H. Interacting Multi-Model Strong Robust Adaptive Unscented Kalman Filter to Bearing only Tracking of Underwater Vehicle Approaching the Observer. Eng. Math. 2025, 9(2), 46-67. doi: 10.11648/j.engmath.20250902.13

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    Ju KS, Sin MH. Interacting Multi-Model Strong Robust Adaptive Unscented Kalman Filter to Bearing only Tracking of Underwater Vehicle Approaching the Observer. Eng Math. 2025;9(2):46-67. doi: 10.11648/j.engmath.20250902.13

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  • @article{10.11648/j.engmath.20250902.13,
  author = {Kang Song Ju and Myong Hyok Sin},
  title = {Interacting Multi-Model Strong Robust Adaptive Unscented Kalman Filter to Bearing only Tracking of Underwater Vehicle Approaching the Observer},
  journal = {Engineering Mathematics},
  volume = {9},
  number = {2},
  pages = {46-67},
  doi = {10.11648/j.engmath.20250902.13},
  url = {https://doi.org/10.11648/j.engmath.20250902.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20250902.13},
  abstract = {We have designed an interacting multi-model strong robust adaptive unscented Kalman filter for bearing only tracking of an underwater vehicle approaching the observer. To solve the problem of tracking an approaching underwater vehicle to the observer based on only its bearing, an interactive multi-model robust adaptive unscented Kalman filter is proposed in this paper. First, a new model of the bearing sense motion towards the observer is proposed to construct a set of realistic target motion modes consisting of linear and curved motion modes. In addition, to account for the influence of outliers in the target bearing measurements, the distribution of measurement noise is assumed to have a Student’s t-distribution, and the probability distribution of the degree of function and the scaling matrix of this distribution is assumed to have a gamma distribution and an inverse Wishart distribution. Thus, the model interaction step is to factorize the mixed probability density function using variational Bayesian method and, based on this, a predictive update method is proposed. In the measurement update phase, the posterior probability density function is obtained in factorization form using variational Bayesian method, and based on this, a posteriori mode probability calculation method is proposed. Simulation results show that our proposed method greatly improves the convergence rate of target tracking error.},
 year = {2025}
}

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  • TY  - JOUR
T1  - Interacting Multi-Model Strong Robust Adaptive Unscented Kalman Filter to Bearing only Tracking of Underwater Vehicle Approaching the Observer
AU  - Kang Song Ju
AU  - Myong Hyok Sin
Y1  - 2025/12/29
PY  - 2025
N1  - https://doi.org/10.11648/j.engmath.20250902.13
DO  - 10.11648/j.engmath.20250902.13
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 46
EP  - 67
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20250902.13
AB  - We have designed an interacting multi-model strong robust adaptive unscented Kalman filter for bearing only tracking of an underwater vehicle approaching the observer. To solve the problem of tracking an approaching underwater vehicle to the observer based on only its bearing, an interactive multi-model robust adaptive unscented Kalman filter is proposed in this paper. First, a new model of the bearing sense motion towards the observer is proposed to construct a set of realistic target motion modes consisting of linear and curved motion modes. In addition, to account for the influence of outliers in the target bearing measurements, the distribution of measurement noise is assumed to have a Student’s t-distribution, and the probability distribution of the degree of function and the scaling matrix of this distribution is assumed to have a gamma distribution and an inverse Wishart distribution. Thus, the model interaction step is to factorize the mixed probability density function using variational Bayesian method and, based on this, a predictive update method is proposed. In the measurement update phase, the posterior probability density function is obtained in factorization form using variational Bayesian method, and based on this, a posteriori mode probability calculation method is proposed. Simulation results show that our proposed method greatly improves the convergence rate of target tracking error.
VL  - 9
IS  - 2
ER  -

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