Indoor positioning is a key technology in today’s intelligent environments, and it plays a crucial role in many application areas. This paper proposed an unscented Kalman filter (UKF) based on the maximum correntropy criterion (MCC) instead of the minimum mean square error criterion (MMSE). This innovative approach is applied to the loose coupling of the Inertial Navigation System (INS) and UltraWideband (UWB). By introducing the maximum correntropy criterion, the MCCUKF algorithm dynamically adjusts the covariance matrices of the system noise and the measurement noise, thus enhancing its adaptability to diverse environmental localization requirements. Particularly in the presence of nonGaussian noise, especially heavytailed noise, the MCCUKF exhibits superior accuracy and robustness compared to the traditional UKF. The method initially generates an estimate of the predicted state and covariance matrix through the unscented transform (UT) and then recharacterizes the measurement information using a nonlinear regression method at the cost of the MCC. Subsequently, the state and covariance matrices of the filter are updated by employing the unscented transformation on the measurement equations. Moreover, to mitigate the influence of nonlineofsight (NLOS) errors positioning accuracy, this paper proposes a kmedoid clustering algorithm based on bisection kmeans (Bikmeans). This algorithm preprocesses the UWB distance measurements to yield a more precise position estimation. Simulation results demonstrate that MCCUKF is robust to the uncertainty of UWB and realizes stable integration of INS and UWB systems.
Indoor localization technology has important applications in modern society. As the need for indoor positioning accuracy and reliability increases dramatically, researchers are committed to developing more efficient and accurate indoor positioning methods. The Inertial Navigation System (INS) and UltraWideband (UWB) techniques are two common methods for indoor localization, each with its own set of advantages and limitations.
The Inertial Navigation System (INS) is a selfsufficient navigation technology that operates independently without reliance on external information or the emission of external energy. This autonomy endows it with unique advantages, such as being inconspicuous and immune to external electromagnetic interference. Inertial Navigation utilizes an inertial measurement unit (IMU), a technology that estimates position, velocity, and attitude by fusing acceleration and angular velocity. The Inertial Navigation System (INS) relies on sensors like accelerometers and gyroscopes to gauge an object’s acceleration and angular velocity. This data allows for the continuous and realtime tracking of position and attitude. Nonetheless, as time progresses, cumulative errors begin to accumulate, becoming increasingly significant with extended durations. UltraWideband technology can realize centimeterlevel highprecision distance measurement. However, in indoor environments, the path of signal propagation is nonlineofsight stemming from various obstacles such as furniture, decorations, pillars, doors, and partitions. At this point, UWB is unable to achieve accurate distance measurement and position estimation. The fusion localization scheme based on UWB/INS, on the one hand, compensates for the loss of accuracy of UWB localization in the nonlineofsight (NLOS) environment and smoothes the localization trajectory of UWB; on the other hand, it also eliminates the aggregated error of INS and augments the localization precision of INS. The integrated system can also output multidimensional data such as position, velocity and attitude, which enriches the positioning information. In this research paper, the application of the Kalman filter is explored. Nonetheless, these filters, which are developed using the minimum mean square error (MMSE) approach, may not adequately accommodate the unpredictable noise encountered in UWB scenarios, leading to potentially significant errors. To triumph over this dilemma, this paper presents a novel approach utilizing the unscented Kalman filter (UKF) algorithm incorporating the maximum correntropy criterion (MCC).
Linear dynamic systems usually use the Kalman filter (KF), but when dealing with nonlinear problems, the extended Kalman filter (EKF) [
The paper presents the following contributions:
We proposed a new clustering algorithm: kmedoid clustering based on bisection k means to preprocess UWB data and remove outliers caused by ranging errors or noise interference, making the positioning results more accurate.
Applying the unscented Kalman filter based on the maximum correntropy criterion to indoor positioning to complete the fusion of ultrawideband and inertial Navigation.
A dual filtering algorithm based on maximum correntropy unscented Kalman filter and extended Kalman filter is proposed to loosely couple the localization results of INS and UWB while achieving more accurate and stable indoor localization.
The subsequent sections of the paper are structured as follows. In the next section, we present the work on the localization algorithm.
Source  Objective  Advantages 

Cho et al. [ 
Uses the maximum correntropy criterion (MCC) to improve the performance of UKF in the presence of nonGaussian, heavytailed impulsive noises.  Excelling in performance and robustness in nonGaussian, nonzero mean noise scenarios. 
Bu et al. [ 
Proposed an INS/UWB integration using robust unbiased finite impulse response (UFIR) filters for reconciling INS and UWB positional measurements.  Offers enhanced accuracy, robustness against measurement errors, realtime positional updates and improved navigation in complex areas. 
Zeng et al. [ 
Apply EKF to integrate IMU and UWB and the channel impulses (CIRS) of UWB are utilized for NLOS detection in practical field tests.  EKF allows for realtime data fusion and processing, critical for dynamic applications like navigation and tracking. 
Zhao et al. [ 
A robust stabilized iterative maximum correntropy unscented Kalman filter is brought up by directly utilizing nonlinear measurement functions and numerical stability methods.  When the noise is nonGaussian, the algorithm demonstrates better localization accuracy and robustness. 
Hafez et al. [ 
Proposed an adaptive variational bayesian maximum correntropy kalman filter (VBMCCKF) algorithm to achieve highly accurate SOC estimation.  Proposes a method for estimating battery stateofcharge that compensates for noisy data and parameter uncertainties to enhance accuracy. 
Liu et al. [ 
Introduced the variable center maximum correntropy criterion (MCCVC) into the kernel space and developed the kernel recursive variable center maximum correntropy (KRMCVC) algorithm.  The KRMCVC merges MCCVC and KRMC features, excelling in performance and robustness in nonGaussian, nonzero mean noise scenarios. 
Wang et al. [ 
Proposed a novel indoor fingerprint recognition system utilizing channel state information (CSI), employing deep learning techniques, as feature values for localization.  Leverages CSI data and deep learning for robust, accurate indoor localization through advanced feature extraction and pattern recognition. 
Target tracking and localization are necessary in the fields of satellite network orbiting and autonomous Navigation, network communication, mobile position estimation, and wireless sensor network motion target tracking [
Compared with RSSI, channel status information (CSI) can effectively avoid the adverse effect of the multipath effect on the localization results. Therefore, the value of CSI is used as the feature value of localization, the location fingerprint database of Radio Map is established, and the weighted proximity algorithm matches the recent fingerprint database data of KNN to estimate the location of the localization point. Reference [
In recent research, numerous scholars have incorporated the concept of correntropy into the field of indoor localization. The maximum correntropy principle argues that in the absence of a priori knowledge, we should choose the probability distribution with maximum uncertainty. In indoor localization, the maximum correntropy criterion can be employed to estimate the uncertainty of the location and incorporate it into the design of the filter. Reference [
The intention of this paper is to accommodate a loosely coupled integrated highaccuracy localization system for INS and UWB in the indoor environment. Generally, the inertial navigation system mounts the inertial element on the mobile node (MN). As the node moves, its angular velocity changes constantly. The inertial element’s gyroscope is capable of measuring the MN’s angular momentum. By applying knowledge of dynamics, this data can be used to determine the heading angle and attitude of the MN during movement. Additionally, the accelerometer can measure the acceleration of MN through primary and secondary integration over time. One can obtain the velocity and position information of the MN during its motion. For each UWB subsystem, the distance connecting MN and BS is measured using the timeofarrival (TOA) technique. This distance data is then preprocessed using the kmedoid clustering method based on bisection k, which means discarding the outliers with large errors and attenuating the ramification of NLOS on the positioning validity. Afterward, the UWB position estimation is obtained by utilizing the trilateral measurement method. Finally, the dual filtering algorithms, namely the extended Kalman filter (EKF) and the maximum correntropybased unscented Kalman filter (MCCUKF), are employed to fuse INS and UWB data. The overall framework is portrayed in
INS is an environmentally independent navigation technique. In INS, the coordinate system for attitude detection is commonly used as a carrier coordinate system and navigation coordinate system. In the theory of rigid body fixed point rotation, the attitude modes describing the kinematic coordinate system are Euler angles, quaternions, and directional cosines. The inertial measurement unit (IMU) collects acceleration and angular velocity data in three axes through the accelerometer and gyroscope built into the INS, respectively. It should be emphasized that the acceleration data is the information in connection with the carrier’s coordinate system. Therefore, to accurately determine the position and velocity of the Mobile Node (MN), it is essential to transform the acceleration data into information applicable to the navigational coordinate system. Angular velocity information is typically used to update the attitude of the Mobile Node (MN) and thus does not require conversion between coordinate systems. Euler angles are used to provide information about the MN’s orientation.
where,
where,
where,
The available UWB ranging models are Received Signal Strength (RSS), Time of Arrival/Time Difference of Arrival/Round Trip Time of Flight (TOA/TDOA/RTOF), and Phase Difference of Arrival (PDOA). In this paper, the Time of Arrival (TOA) is used for UWB distance measurement. Assuming that the number of base stations involved in localization is M and the coordinates are j = 1, 2,…, M, the base station receives signals from the mobile node, the TOA is modeled as follows:
where,
where
The distance measurements of UWB contain NLOS errors and measurement errors, which can cause serious errors in the localization results if this result is directly used for subsequent localization. It is imperative to mitigate the impact of NLOS on the localization results following the distance measurement. This paper introduces a novel algorithm for NLOS suppression to preprocess the obtained distance values, replacing some of the measurement values containing NLOS errors with those not interfered with by NLOS. In the kmeans algorithm, we repeatedly select the cluster’s mean as the new center and continue iterating until the cluster’s objects’ distribution remains unchanged. In contrast, the kmedoid algorithm selects the points in the sample as the clustering center each time. This paper’s objective is to enhance the precision and stability of clustering outcomes by presenting a kmedoid clustering approach that utilizes bisection kmeans. The method first performs bisection kmean clustering on the ranging values, takes the clustering result of the bisection kmean as a known point, then calculates the distance from this known point to each point in the sample, and selects the sample point closest to the known point as the initial clustering center. By adopting this approach, we can enhance the accuracy of selecting the initial cluster center and thus improve the quality of the clustering results.
To implement bisection kmedoid clustering on the data, we set the number of clusters k to 4. Select the two clusters with the smallest sum of squared errors (SSE) and compute the centers of the clusters. It is important to note that in kmedoid clustering, the center of mass for each cluster must be an actual data point, and these initial centers are chosen randomly. However, this random selection can lead to instability and the problem of local optima in the kmedoid algorithm. To address this, our paper introduces an algorithm for selecting the kmedoid cluster centers, with the process outlined as follows:
Step 1: At the
Step 2: The L measurements are grouped into a single cluster, which is then divided into two subclusters, after which the cluster
where,
Step 3: From the final four clusters, the two clusters
Step 4: Calculate the distance from
The flowchart of our clustering algorithm is portrayed in
Trilateral measurements and least squares are introduced to deduce the coordinates of MN. The location coordinates of the base station and MN are
We represent the above equation using a linear matrix
where,
Finally, the position of MN at moment n is estimated as
Correntropy, a novel concept, is employed to evaluate the overall similarity between two random variables. Considering two specified random variables
where,
where,
Correntropy can be gauged by utilizing the sample mean estimator at N specified points.
where,
The concept of correntropy becomes apparent. Correntropy is defined as the summation of weighted evenorder moments of the error variable. Additionally, the kernel bandwidth plays a significant role as a parameter that determines the importance of statistical instances of order two and higher. It is important to note that when the kernel bandwidth greatly surpasses the data’s dynamic range. Considering a time series of residuals
A comparison of the formulas for the minimum mean square error criterion and the maximum correntropy criterion is as follows:
It is evident that the minimum mean square error criterion is a quadratic function in random space in the joint probability density along the
Navigation information from INS and UWB is fused by quadratic filtering based on EKF and MCCUKF. Specifically, the state transfer equation and observation equation of MCCUKF are first constructed based on the output of EKF. Then, the nonlinear function is mapped onto a Gaussian distribution using the traceless transform, and the state estimation and update are performed through the prediction and update steps. Finally, more accurate position estimation results are obtained by MCCUKF fusion. The error state prediction and prediction covariance are given below:
where,
where,
where,
The Kalman gain is giving by
The error state is updated as below:
Covariance update:
The error equations of state and observation equations for discretetime nonlinear systems are
where,
When performing MCCUKF prediction, the initial state is the output state of the first stage of INS/UWB fusion using EKF, i.e.,
where,
Using the
where,
A collection of sigma points is derived from the priori error state prediction mean
Afterwards, the forecasted measurement mean is determined as shown below:
Furthermore, the state measurement crosscovariance matrix is as follows:
Subsequently, we utilize MCC to execute the observation update. Initially, we denote the priori estimation error of the state as
Combining
where,
with
where,
where,
We proceed by formulating a cost function utilizing MCC.
where,
where,
The optimal solution of
This can alternatively be stated below:
The
where,
where,
Define
Next, write
The actual state
The revised measurement covariance is
The state equation and measurement equation can be further written as
Status update:
Covariance update:
where,
The autocorrelation matrix
As a result, the nominal state of the MT is calibrated as follows and the final state estimate is produced:
In this chapter, the effectiveness of the suggested solution is evaluated by simulation results.
In the conducted experiment, five beacon nodes are randomly arranged in the scene, using NaveGo as a simulation tool. The target node follows a predetermined trajectory in the simulation scene. The range of the simulation scene is
where,
Subsystems  Parameter values  

IMU  Gyroscope  Random walk (ARW): 
Bias: 

Correlation time: 

Frequency: 

Accelerometer  Random Walk (ARW): 

Bias: 

Correlation time: 

Frequency: 

UWB  Anchor nodes  
Frequency 
We make the assumption that the measurement noise’s distribution is Gaussian, and when the nonlineofsight error obeys a gamma distribution, the components for the simulation are set as presented in
Component  Notation  Default values 

Probability of NLOS noise  
LOS noise  
Probability density function of NLOS noise  
The NLOS mean  4  
The NLOS standard deviation  0.5 
The following simulation results are analyzed:
By referring to
As depicted in
When the measurement noise and the NLOS error both follow a Gaussian distribution, the corresponding parameters are set, as portrayed in
Component  Notation  Default values 

Probability of NLOS occurrence  
LOS noise  
The NLOS error 
In
The CDF of the location error was depicted in
Assuming that the LOS error follows a Gaussian distribution
Component  Notation  Default values 

Probability of NLOS error  5  
LOS noise  
The NLOS error 
In
At higher NLOS probability, the localization error is also increasing. However, the solution suggested can lower the impact of NLOS error on the localization effectiveness, so that the effectiveness remains better under higher NLOS probability, which indicates that the proposed solution demonstrates greater robustness and adaptability.
In
In
To verify the efficacy of the suggested solution in localization, we established an authentic experimental setting within the library and the classroom. The distance separating the mobile node from the base station is measured by the TOAbased ranging technique through UWB wireless communication technology utilizing narrowband pulse transmission data characteristics. Considering the requirements of low cost and high effectiveness. The INS sensor comprises a triaxial gyroscope and a triaxial accelerometer, which can gauge the acceleration and angular velocity of the corresponding axes.
As depicted in
In this research, a novel approach for enhancing the precision and reliability of indoor localization is introduced. The proposed method combines maximum correntropy with unscented Kalman filtering for inertial Navigation and an integrated ultrawideband localization system. We first proposed a kmedoid clustering algorithm based on bisection kmeans to preprocess UWB data and remove outliers caused by ranging errors or noise interference. Then, the concept of maximum correntropy criterion is implemented in the unscented Kalman filter, and the solutions of the inertial navigation system and the ultrawideband system are fused using a hybrid filter based on EKF and MCCUKF to derive the ultimate state estimation of the target. The algorithm described in this paper exhibits enhanced precision and robustness in localization. Although the proposed algorithm has good performance in singletarget localization, in multitarget localization, the performance of the algorithm is degraded due to the interactions between targets. Future research will concentrate on assessing the algorithm’s effectiveness in complex environments, such as multitarget tracking.
The authors wish to express their appreciation to the editors and reviewers for their helpful review and recommendations which greatly improved the presentation of this paper.
This work was supported by the National Natural Science Foundation of China under Grant Nos. 62273083 and 61803077; Natural Science Foundation of Hebei Province under Grant No. F2020501012.
The authors confirm contribution to the paper as follows: Guidance and oversight throughout the project: Yan Wang; provide critical feedback on the manuscript: Yan Wang; study conception and design: You Lu; data collection: You Lu; analysis and interpretation of results: You Lu; experimental design and perform data analysis: You Lu, Yuqing Zhou, Zhijian Zhao; draft manuscript preparation: You Lu. All authors reviewed the results and approved the final version of the manuscript.
All the reviewed research literature and used data in this research paper consists of publicly available scholarly articles, conference proceedings and reports. The references and citations are contained in the reference list of this manuscript. Any further inquiries about the data can be directed to the corresponding author.
The authors declare that they have no conflicts of interest to report regarding the present study.