Enhanced Kalman filter algorithm using fuzzy inference for improving position estimation in indoor navigation

Abstract

In recent few years, the widespread applications of indoor navigation have compelled the research community to propose novel solutions for detecting objects position in the Indoor environment. Various approaches have been proposed and implemented concerning the indoor positioning systems. This study propose an fuzzy inference based Kalman filter to improve the position estimation in indoor navigation. The presented system is based on FIS based Kalman filter aiming at predicting the actual sensor readings from the available noisy sensor measurements. The proposed approach has two main components, i.e., multi sensor fusion algorithm for positioning estimation and FIS based Kalman filter algorithm. The position estimation module is used to determine the object location in an indoor environment in an accurate way. Similarly, the FIS based Kalman filter is used to control and tune the Kalman filter by considering the previous output as a feedback. The Kalman filter predicts the actual sensor readings from the available noisy readings. To evaluate the proposed approach, the next-generation inertial measurement unit is used to acquire a three-axis gyroscope and accelerometer sensory data. Lastly, the proposed approach’s performance has been investigated considering the MAD, RMSE, and MSE metrics. The obtained results illustrate that the FIS based Kalman filter improve the prediction accuracy against the traditional Kalman filter approach.

Keywords

ANN FIS based Kalman Filter navigation system inertial measurement unit indoor navigation sensors fusion

1 Introduction

With the advancement of new technologies, the use of navigation has reshaped the traditional way of navigation. The innovation of smartphones introduced various location-based services, which allows the user to find the user’s interest location. Over the last several years, the enhancement in GPS (Global Positioning System) increase the use of location-based services to find the object position [1, 2]. Presently, GPS is considered the only technology used to calculate the user’s location through satellite. Many location-based services use GPS in several domains, such as navigation, aviation, agriculture, military, etc. [3]. As GPS is considered the recognized technology for estimation of user’s location in an outdoor environment, it also has several drawbacks when it came to finding the position of the object in an indoor environment. Moreover, finding the location of the user in the indoor environment using GPS is not practicable as it requires a constant connection with the satellite [4]. The unreliability of GPS in an indoor environment is also due to signal attenuation, which sometimes breaks the signal from satellite [5]. Thus, GPS is not consistent to accurately find the object’s position in the indoor environment. Many researchers conclude that GPS technology is not suitable for IPS (Indoor Positioning System) [6, 7]. IPS technology is utilized for computing the location of the object in the indoor environment. Many miniature sensors are used to acquire data, such as sensor data, radio waves, magnetic field, and WLAN nodes [8]. Presently, several studies have been published in the field of indoor localization. However, many shortcomings are the current system as they are not consistent and either not follow the standard protocol for indoor localization [1]. Moreover, theses system does not directly focus on calculating the person location, which makes them unreliable. Therefore, many miniaturized devices are introduced which are responsible for calculating the objects’ position in the indoor environment [9, 10] using the accelerometer and gyroscope readings acquired from the Inertial Measurement Unit (IMU).

The IMU is a microelectronic device used to find the body’s orientation, angular rate, and other factors using the fusion of sensor data obtained from the gyroscope, magnetometer, and accelerometer. The double integration method is commonly used to find the objects’ position considering the accelerometer sensor data. Likewise, the Euler angle is utilized for calculating the orientation of the object concerning roll, yaw, and pitch, with the sensor data obtained from the gyroscope. Nevertheless, these sensor data contains some bias and noise that also affect the output. Many prediction algorithms, such as alpha-beta and Kalman filter, are used to remove the noise to smooth the sensor readings [11, 12].

Several research has been done to improve the accuracy of position using machine learning and control [13 –22]. Many researchers used machine learning models to position the object based on historical data accurately. Moreover, the Fuzzy Inference System is also used to enhance the position accuracy through defined rules. A fuzzy inference system (FIS) is similar to artificial neural networks containing a set of linguistic rules obtained from expert users. In this study, we present the hybrid model for computing the object’s position in the indoor environment. We have used a Next-Generation IMU to acquire gyroscope and accelerometer data. For experiment analysis, the data is collected from several locations while the object moves on 3^rd floor of the Engineering building of Jeju National University, Jeju, South Korea. The time for acquiring one sample of data is 60 second, where the first ten seconds remains stationary so that the algorithm gets in a stable state. The proposed hybrid model consists of two-part, i.e., fuzzy controller-based Kalman filter and position calculation using sensor fusion in an indoor environment. The FIS continuously controls the Kalman filter performance by tuning its parameters and considering the output as feedback. The tuning of statistical metrics using FIS increases the Kalman filter’s performance and accuracy. Likewise, the Kalman filter is utilized for noise removal from the sensory data, such as bias and drifting error, and predict the actual sensor reading. Similarly, for position estimation, the three-axis accelerometer and gyroscope are used in the indoor environment to calculate the object’s position and orientation.

The main contribution of the proposed model is followed as:

The main objective of this work is an accurate calculation of the objects’ position while minimizing the error using FIS based Kalman filter.

The FIS based Kalman filter is used to control and tune the Kalman filter by considering the output as feedback (the Kalman filter is responsible for predicting the actual sensor reading from the available noisy readings).

In the case of position estimation, the three-axis sensor reading is used to calculate the object position (i.e., noise and drift-free) in the indoor environment.

Lastly, the performance of the proposed approach has been investigated considering the MAD, RMSE, and MSE metrics. The obtained results illustrate that the FIS-based Kalman filter improves prediction accuracy compared to the traditional Kalman filter approach.

The rest of the paper is organized as follows: The state-of-art related studies are discussed and described in Section 3. The proposed methodology is discussed in Section 4. Similarly, the experimental setup, comparative analysis with detailed result visualization is delineated in Section 4. Lastly, the conclusions and future work is presented in Section 5.

2 Related work

Tracking and indoor navigation is a challenging task due to the unavailability of resources, such as GPS signal, satellite availability, etc. During the past several years, IoT-based [23 –25],and many location-based algorithms have been developed which aims to calculate the object location and position in an outdoor and indoor environment [26 –29]. These location-based approaches are segregated into the following categories: motion and inertial sensor, dead reckoning, fingerprinting, connectivity and triangulation [30]. Nonetheless, our primary focus is to investigate the state-of-arts inertial-based navigation approaches in this study.

2.1 Inertial and motion sensor

Inertial measurement unit or inertial and motion sensors are the microelectronic devices that are used to calculate the position of the objection in an indoor environment. These devices have the following sensor information: accelerometer, gyroscope, magnetometer, etc. The accelerometer is used in the double integration method to find its position. In contrast, the gyroscope is used to find the object’s orientation, such as roll, pitch, and yaw.

In [31], the MEME gyroscope is used to find the body’s orientation. The extended Kalman filter is added to remove the error in the measurements. The presented system performance is measured using VICON Optical in terms of root mean square error.

In [32], a MUSE approach is presented to improve the object’s orientation by considering the output of the inertial measurement unit. The experiments were performed on a human elbow and arm rotation to investigate the orientation and angle.

In [33], a machine learning-based approach is presented to predict the precise pedestrian positioning using the inertial measurement unit. The system is based on two states, i.e., stationary state when the object is at a fixed position with respect to orientation and the object is in a moving state. Several experiments are carried out in different states, i.e., running, jogging, jumping, etc., to accessed system accuracy.

In [34], the author presented an integrated approach to adjust the Kalman filter parameters using fuzzy logic to minimize the error in the position estimation.

In [35], the author proposed a model to measure the distance of the object by double integration method and step count along with the angle between legs. Moreover, the gyroscope is used to calculate the angle between the legs. The results show that the presented model performs well in terms of cost and precise angle calculation.

2.2 Navigation using fuzzy inference approaches

In [36], the author presented an indoor positioning model based on IEEE 802.11 pattern recognition. The model comprised of two models, i.e., operational and calibration mode. The main aim of this model is to find the Euclidean distance. In [37], the author presented a fuzzy logic-based real-time robot navigation model for a mission-critical scenario. The developed robot is embedded with motion and location sensors to avoid colliding with hurdles. A fuzzy inference engine is used to define a robot’s behavior in multiple scenarios. In [38], a wireless indoor localization system is developed for disabled persons’ navigation. The presented model uses Bluetooth and fingerprint technology to find the Euclidean distance using the fuzzy inference system. In [39], a low-cost controller is developed based on fuzzy rules for the indoor navigation system. The implemented model achieved all the hurdles in the time span of 70 seconds autonomously. Moreover, an interface is developed in order to establish interaction between the robot and the human.

Table 1 outline the above-mentioned indoor navigation algorithm based on fuzzy inference system and machine learning. The comparative studies are divided into the following groups, i.e., inertial measurement unit data [40, 41], radio signal strength [42 –46], channel state information [47], and angle of arrival [48].

Table 1
Analytical comparison of machine learning based approaches for indoor localization

Methods Link Data ML algorithm Hidden layer Outcomes

IMU [40] INS input AI 2–4 Length of steps

(acclerometer, gyroscope, magnetometer) Recurrent Neural Network 4 Static Detection

[42] FFNN 1–3 Location

[43] GANN 3 Distance

[44] ANN 1 Location

[45] RFNN 1 Location

Radio Signal Strength [46] WiFi input data (Ap, nodes) Adaptive Neural Fuzzy Inference System 3 Distance

CSI [47] WiFi Information (AP, nodes) Generalized Cross-correlation 1–-2 Location

AoA [48] Radio, Optical or Acoustic CNN 8 Location

FIS based Proposed IMU input data Fuzzy Inference – Position

Prediction Algorithm Approach (acc, gyro, mag) System Estimation

Methods	Link	Data	ML algorithm	Hidden layer	Outcomes
IMU	[40]	INS input	AI	2–4	Length of steps
		(acclerometer, gyroscope, magnetometer)	Recurrent Neural Network	4	Static Detection
	[42]		FFNN	1–3	Location
	[43]		GANN	3	Distance
	[44]		ANN	1	Location
	[45]		RFNN	1	Location
Radio Signal Strength	[46]	WiFi input data (Ap, nodes)	Adaptive Neural Fuzzy Inference System	3	Distance
CSI	[47]	WiFi Information (AP, nodes)	Generalized Cross-correlation	1–-2	Location
AoA	[48]	Radio, Optical or Acoustic	CNN	8	Location
FIS based	Proposed	IMU input data	Fuzzy Inference	–	Position
Prediction Algorithm	Approach	(acc, gyro, mag)	System		Estimation

The table mentioned above 1 shows the comparative analysis of related terminologies, such as indoor navigation, fuzzy control based navigation system, inertial, and motion sensor. These presented approaches have many shortcomings in terms of performance and accuracy. Some of the presented approaches directly use sensor data to estimate the object’s position. These sensing data has drifting and bias error, which degrade the system’s performance and accuracy. However, in the proposed system, we minimize the error by turning the traditional Kalman filter using the FIS based Kalman filter. The FIS enabled Kalman filter which is used to control and tune the prediction algorithm by considering the output as feedback. According to the authors’ best information, no functional hybrid position estimation system based on FIS enabled the Kalman filter for an indoor navigation system.

3 Proposed methodology

In our proposed system, the accelerometer and gyroscope are obtained from the next-generation IMU for computing the linear acceleration (L_a) and angular velocity (L_w). The FIS-based Kalman filter and sensor fusion algorithm are part of the designed system where the FIS is used to fine-tune the Kalman filter algorithm. The sensor fusion algorithm is used for position estimation in the indoor navigation system. In the next section, the step-by-step working of these two modules is delineated.

3.1 Scenario of position estimation in indoor navigation

For accurate estimation of the position, the Position estimation module is sub-divided into four simpler modules that are IMU acceleration. This sensor fusion exploits the Kalman filter algorithm, Integrator, and actual position estimation. Figure 1 shows how these components are joined together to perform the position estimation.

Fig. 1

Proposed architecture of position estimation model in indoor navigation.

The IMU process the data and results in generating three outputs (i.e., angular velocity, magnetic field vector, and linear acceleration), which are then passed to the sensor fusion module. The sensor fusion module used the Kalman filter to fuse together the IMU output to obtain the object in the indoor environment based on noise-free sensor readings. The Kalman filter is used to predict the output and perform in two steps that are first predicting and then correcting [49, 50]. To execute the prediction, the integral is calculated for the gyroscope measurement, and then in the next step, the accelerometer and the magnetic readings are used for correcting the obtained prediction. An important point that is to be noted here is that fusion in the case of the Kalman filter is based on probability calculation that first puts together the inputs, and then the inputs are corrected based on the maximum likelihood. After this, the attained orientation matrix is moved to the accelerator component for removing the centripetal force and gravity.

The next module is the integrator module that intercepts the processed data and then calculates the velocity taking integral of acceleration. After extracting the velocity, the object’s position can be obtained by taking the second integral of the velocity. Another important function of the integrator is to utilize a mathematical function for integration related to the time and the position and velocity calculation.

A non-linear matrix is utilized for determining the relative position and orientation of the object in the context of the indoor environment. In previous studies, the gyroscope outputs, magnetometer, and accelerometer correspond to the orientation estimation that is very handy in finding the orientation of the object. In Equation (1), it can be seen that the gyroscope, the complete adjustment may not be obtained while the gyroscopes reading is taken into consideration. One solution is to get the object’s orientation considering the integration of the angular velocity. Though this seems a reasonable solution; however, it may also introduce a drifting error. $L_{ω} = L_{ω_{true}} + b_{g} + v_{g} .$ (1)

In Equation (1), the angular velocity vector is represented by L_ω of each local sensor frame, the b_g corresponds to the sensor bias, which is constant with respect to a small time frame, the v_g is noise in the frequency, and L_{ω
_true} parameter that is the actual angular velocity obtained from the sensors. Another important point is that for acceleration, it can be assumed that the body of motion does not exist, and thus only gravity is used here, divided by the three components. Then, some trigonometry calculation is applied and obtain the respective roll and pitch, considering the vertical axis. However, it will lead to a high-frequency noise in the accelerometer, as shown in Equation (2). $L_{a} = L_{a_{body}} + L_{g} + b_{a} + v_{a},$ (2)

In Equation (2), the L_a corresponds to the acceleration vector (i.e, in the local frame), the linear acceleration also called true acceleration is represented by L_{m
_true}. The gravitational acceleration is represented by L_g, b_a is the low frequency noise bias and finally v_a represents the actual frequency noise. In this way, we can combine the magnetometer and accelerator for calculating the orientation estimation. Again the problem of noise is introduced, estimated by Equation (3). $L_{m} = L_{m_{true}} + L_{m_{int}} + b_{m} + v_{m},$ (3)

In equation (3), the b_m represents the bias and v_m corresponds to the noise. The actual magnetic field called true magnetic field is represented by L_{m
_true} that is the earth’s magnetic field. The L_{m
_int} corresponds to the magnetic disturbance. As a solution to the two main issues, i.e., noise and bias issues, the best estimation out of the two is chosen. Thus the solution which has minimum noise and drift is picked up.

The proposed system exploits the Kalman filter approach for applying the fusion. The first step would be to perform prediction and extract data from the IMU sensor and apply a mathematical model for calculating the estimation. The obtained estimation is then corrected in the next step. The algorithm is performed in two steps: first, predicting and then correcting. In order to execute the prediction, the integral is calculated for the gyroscope measurement, and then in the next step, the accelerometer and the magnetic readings are used for correcting the obtained prediction. An important point that is to be noted here is that fusion in the case of the Kalman filter is based on probability calculation that first puts together the inputs, and then the inputs are corrected based on the maximum likelihood. The symbol represents the uncertainty associated with the gyroscope measurement, σ_ω corresponds to the angular velocity. Moreover, the acceleration is symbolized as a, and the uncertainty associated with the acceleration measurement is represented by σ_a. Finally, m and σ_m corresponds to the magnetic field and the uncertainty associated with the acceleration measurement, respectively.

3.1.1 Sensor fusion using Kalman filter

In this research study, the non-linear version of the Kalman filter algorithm is employed. The mean is linearized by the Kalman filter approach using the state-space, and then the covariance is computed, keeping in view the current state estimation. The Kalman filter works in two phases; In the first phase, the prediction is performed; the second phase employs the measurement updating. The Kalman filter is part of the sensor fusion module. The linear acceleration, magnetic field, and angular velocity are provided as an input to the Kalman filter through the sensor and then produce a corrected predicted parameter. Figure 2 delineates the Kalman filter algorithm’s working. Equation (4) calculates the error covariance estimation whereas equation (5) determines the initial state. Apart from inputs, other parameters such as the values of measurments and the noise covariance are also provided as an input. The again achieved by the Kalman filter is estimated using Equation (5). Equation (7) and equation (8) is used to find the estimation error covariance and the estimation of state to correct these values.

Fig. 2

Indoor navigation system configuration using Kalman filter.

Following is the equation used for updating the time in the context of Kalman filter: ${\hat{x}}_{\bar{k}} = {\hat{x}}_{k - 1} + B_{k}$ (4) $P_{\bar{k}} = A P_{k - 1} A^{T} + Q .$ (5)

Equation given below performs the state updating in the context of the Kalman filter: $K_{i} = P_{\bar{i}} H^{T} (H P_{\bar{i}} H^{T} + R)^{- 1}$ (6) ${\hat{x}}_{i} = {\hat{x}}_{\bar{i}} + k_{i} (z_{i} - h_{i})$ (7) $P_{i} = (I - K_{i} H) P_{\bar{i}},$ (8)

Here the ${\hat{x}}_{\bar{i}}$ parameter is used for state vector prediction that is comprised of 3-axis coordinates and heading primarily considering the sensor fusion approach employed by the proposed system. Bk corresponds to the state matrix. Moreover, the error covariance matrix is represented by $P_{\bar{i}}$ that consists of two portions; firstly, the error noise predicted represented by Q; and the State space matrix symbolized as (A). During each iteration the value of P_i is modified considering the three parameters (i.e., Kalman Gain (K_i), predicted error covariance matrix ( $P_{\bar{i}}$ ), and measurements (H) that are updated.

3.1.2 IMU acceleration

The IMU accelerometer output that is divided into three parts (i.e., linear acceleration of IMU, gravity correction, and finally the centripetal acceleration force correction.

To find the position in an indoor environment, the proposed system employs linear acceleration. Therefore, it is essential to eliminate the gravitational and centripetal forces for the input sensor data. The gravity acceleration of the IMU frame can be calculated using (9). $[\begin{matrix} {acc}_{x_{gravity}} \\ {acc}_{y_{gravity}} \\ {acc}_{z_{gravity}} \end{matrix}] (t) = {Ori}^{- 1} (t) [\begin{matrix} 0 \\ 0 \\ ga \end{matrix}],$ (9) where ga represents the gravitational acceleration matrix represents the acceleration considering the earth fixed frame. $[\begin{matrix} 0 \\ 0 \\ ga \end{matrix}]$ . Moreover, matrix corresponds to the gravity acceleration considering the IMU fixed frame. $[\begin{matrix} {acc}_{x_{gravity}} \\ {acc}_{y_{gravity}} \\ {acc}_{z_{gravity}} \end{matrix}]$ .

The object rotation can occur in two different ways that are; the rotation of the object around some point in space and the rotation of object around itself. Thus, the object rotation can be obtained using Equation (10): $[\begin{matrix} {acc}_{x_{centripetal}} \\ {acc}_{y_{centripetal}} \\ {acc}_{z_{centripetal}} \end{matrix}] = [\begin{matrix} 0 & - ω_{z} & ω_{y} \\ ω_{z} & 0 & - ω_{x} \\ - ω_{y} & ω_{x} & 0 \end{matrix}] . [\begin{matrix} V_{x} \\ V_{y} \\ V_{z} \end{matrix}] .$ (10)

The centripetal force is defined as the linear velocity and angular velocity cross product.

3.1.3 Integrator module

The integrator is the third module of the proposed system. This module employs the mathematical formulation to find the integration considering the time and linear acceleration to calculate the respective position and velocity [51]. Following are the steps used to perform the calculation of the integration module. Firstly, the object’s velocity (in the indoor environment) is finding out by applying the obtained measured acceleration, as shown in Equation (12). The parameter a as in equation (11) represents the measured acceleration extracted from the IMU sensor. $a_{cc} = Const$ (11) $vel = \int a_{cc} dt = v_{o} + a_{cc} t .$ (12)

The next step is to calculate the object position (i.e., in the indoor environment) by taking the integral of the previous step’s velocity. Next, the integration is applied second time as can be seen in Equation (12),(13) and (14). In this case, the position is symbolized as y, vel₀ represents the computed velocity, acceleration is represented by a_cc, and finally, the t corresponds to the time. $y = \int veldt$ (13) $y = \int ({vel}_{o} + a_{cc} t) dt$ (14) $y = y_{o} + {vel}_{o} t + \frac{1}{2} a_{cc} t^{2} .$ (15)

3.2 The adaptive Kalman filter algorithm

Owning optimality for the Kalman filter algorithm assumes that the systems’ noise and equations are constant and already known. However, the factors involving the environment, motion state, and running time make the said constraints impractical. The distance for moving tends to increase according to NGIMU, due to which the process noise covariance matrix Q change in a dynamic manner.

Q’s value causes state deviation of system estimation, and filters to be applied to the one-step process gain more weight. The errors caused by this factor may lead to sub-optimality of the Kalman filter, which further exhibits declining behavior. The state-of-the-art in this area has focused on IAE to enhance the robustness of EKF [25]. The adaptive accurateness of Q is ensured by forming a scaling factor.

Some of the studies have focused on the fuzzy inference system (FIS) to diminish the noise covariance matrix’s adverse effect and presented approaches to deal with amendments in measurement noise. In this study, we present FIS based Kalman filter (FISKF), which is the coupling of innovation adaptive estimation (IAE) and fuzzy logic methods for the adjustment of Q in the Kalman filter for the indoor navigation system (INS). The INS based FISKF is shown in Fig. 3.

Fig. 3

Proposed block diagram of FIS based Kalman filter.

Kalman filter’s robust behavior is improved through FISKF without decreasing the accuracy. This is possible because FIS holds the potential to find incorrect and dynamic Q variations. The linguistic and human thinking is captured and mimicked via a rule-based FIS [27]. The FIS comprises three main modules: (1) fuzzification, (2) fuzzy logic inference, and (3) defuzzification as shown in Fig. 4.

Fig. 4

Fuzzy inference system (FIS).

3.2.1 Fuzzification

A process wherein mapping between numerical input variables and fuzzy variables is known as Fuzzification. A linguistic variable represents a subset from a series of fuzzy subsets and functions from MFs. A membership that lies between 0 and 1, denoting the grade of membership for an element with an association with fuzzy set. The function has a potential role in the fuzzy set. The terms like "mine", "yours" or "his", which are simply words from the English language, are values of fuzzy subsets. The ratio among theoretical co-variance at each chunk and innovations’ sample co-variance is given as an input to FIS. It is assumed that covariance matrix of noise (R) is constant and already known. The states including observation error, estimation error at time k are denoted with r_k. ${\hat{C}}_{K}$ represents statistical sample covariance, which is computed by considering the restricted quantity of innovation samples. C_k represents theoretical covariance, which is associated with Q and R, is given as: $r_{k} = Z_{k} - H_{k} δ X_{\bar{k}}$ (16) ${\hat{C}}_{k} = \frac{1}{N} \sum_{i = k - N + 1}^{K}$ (17) $C_{k} = H_{k} (φ_{k} P_{k - 1} φ_{k}^{T} + G_{k} Q_{k} G_{k}^{T}) H_{K}^{T} + R_{k}$ (18)

The moving window has a size, which is denoted as N. In this experiment, FIS for Kalman filter is adopted to match ${\hat{C}}_{k}$ with C_k.

3.2.2 Fuzzy inference

Considering the set of fuzzy rules like the premise and fuzzy interferential method’s exertion for drawing the fuzzy conclusion is known as fuzzy inference. The FIS holds some characteristics which follow fuzzy rules, which affects the performance of FIS based Kalman filter. There are three fuzzy rules followed by the FIS. If the value of ${\hat{C}}_{K}$ lies within Ck, then it indicates that FIS output ΔQk is equal to 1, and the values of two covariances are matched. If the value of ${\hat{C}}_{K}$ is greater than C_k, then it indicates that FIS output ΔQk should be more than 1. Conversely, If the value of ${\hat{C}}_{K}$ is less than C_k, then it indicates that FIS output ΔQk should be less than 1. The fuzzy formed rules or fuzzy implications are represented in the form of if-then, as follows:

3.2.3 Defuzzification

The output produced in the preceding section forms a fuzzy set. However, the deterministic value Q should be scaled down in the INS system. Defuzzification is the last step in FIS wherein Δ $Q_{k}^{*}$ is transformed into ΔQk. There exist different procedures for fuzzification that produce different results. In this experiment, we opted for the centroid method for computing the deterministic value ΔQk., representing the center of the area via the MF curve. ΔQk is used to update all the elements of Q_k, that holds an adaptive behavior for system noise variance, after completion of FIS at different time intervals k. $Q_{k} (m, m) = Δ Q_{k} \times Q_{k} (m, m)$ (19) Figure 5 shows the working flow of FIS based Kalman filter.

Fig. 5

Proposed flowchart of FIS based Kalman filter.

4 Experimental results and discussion

4.1 Development environment

The environment employed to develop the proposed work is classified into two categories: (1) FIS based Kalman filter and (2) estimate position via sensor fusion algorithm. We have employed a next-generation inertial measurement unit (NGIMU) for the stochastic model. The measurement unit for NGIMU is formed to gather data with a data processing algorithm and onboard sensors. The three axes, magnetometer sensor, and accelerometer are part of on-board sensor that is harnessed to compute indoor environment position.

To get data for the indoor environment’s inertial navigation, we employed next-generation inertial measurement unit sensor. The data values were captured during the object’s movement to room D242 from the corridor of a building in Jeju National University (JNU), Republic of Korea. The time duration considered for sample data acquisition comprised of 1 minute whereof first 10 seconds were inactive for algorithm stability. Table 2 contains details about the tools and technologies used to develop the stochastic model. We have harnessed a fuzzy inference system as a predicting model wherein the prediction algorithm is tuned to improve the model’s accuracy. Table 3 recapitulates the summary of the development environment for the FIS based Kalman filter model.

Table 2
Tool and technologies for the development environment of stochastic model

Tool and technologies Description

Integrated development toolkit MATLAB R2018a

OS Window 10

Central processing unit Intel(R) Core(TM) i5-8500 CPU @ 3.00GHz

Memory 8GB

Prediction and Noise removal algorithm Kalman Filter

Application programming Interface Next generation inertial measurement unit

Tool and technologies	Description
Integrated development toolkit	MATLAB R2018a
OS	Window 10
Central processing unit	Intel(R) Core(TM) i5-8500 CPU @ 3.00GHz
Memory	8GB
Prediction and Noise removal algorithm	Kalman Filter
Application programming Interface	Next generation inertial measurement unit

Table 3

FIS based Kalman filter module development environment

Component	Description
Integrated development toolkit	MATLAB R2018a
Operating System	Win 10
Central processing unit	Intel(R) Core(TM) i5-8500 CPU @ 3.00GHz
Memory	8Gigabyte
Artificial Neural Network	Fuzzy Inference System
Input	3
Prediction and Noise removal algorithm	Kalman Filter

The experiments have been performed on a 64-bit windows operating system with 8 GB memory, 3.00GHz processor, and Intel(R) Core i5-8500. The NGIMU application programming interface (API) and MATLAB are used to develop the model.

4.2 Implementation

We have developed the proposed system to assess the Kalman filter’s performance. The experiments were carried out in the real data set that is collected from Jeju national university (JNU), Republic of Korea. Initially, NGIMUP API was used to load data into the application. The data set comprises of ten input parameters, enlisting a 3-axis magnetometer, 3-axis gyroscope, 3-axis accelerometer, and time interval in which data records were captured. Thereafter, sensor fusion using the Kalman filter was used to compute orientation. The centripetal and gravitational forces were removed further to apply double integration to calculate the object’s position in an indoor environment. The RMSE for IMPU sensor reading is calculated by performing comparisons between gyroscope and accelerometer sensor values. The reported value for IMU sensor reading was too high, i.e., 5.25. The actual sensor reading was forecasted using the Kalman filter algorithm. The Kalman filter’s internal parameter was manually trained using the estimated error in measurement (R) via developed interface. Different values of R are considered for the evaluation. The reported values of RMSE was reported as 2.30 for gyroscope and accelerometer at R=20. This predicted value resulted in 55% less error than RMSE for sensor readings.

The evaluation measures, including root mean squared error (RMSE), mean square error (MSE), and mean absolute deviation (MAD), were used to evaluate the accuracy of the model. The equations of the measures are given as Equations (20)–(22). $MAD = \sum_{n}^{i + 1} | T_{i} - \hat{P_{i}} |$ (20) $MAE = \sum_{n}^{i + 1} (T_{i} - \hat{P_{i}})^{2}$ (21) $RMSE = \sqrt{\frac{\sum_{i + 1}^{n} (T_{i} - \hat{P_{i}})^{2}}{n}},$ (22)

Where n determines total observations, T determines the target value and estimated value is denoted as $\hat{P}$ .

4.3 Results and discussion

To compute the object’s position in an indoor environment, we used open-source NGIMU for data acquisition. For evaluation, the Kalman filter algorithm and predicting model’s prediction results were compared. The raw accelerometer data were collected from the next-generation inertial measurement unit. The time interval whereof data values were captured is also collected, as shown in Fig. 6. The x, y, and z determine the accelerometer data’s 3-axis representation. The Butterworth filter based on the defined cut-off measurements is used to filter the data, denoted as a dotted line. The stationary data is denoted via solid black lines. If the magnitude value is less than 0.05, then stationary data is deemed to be the object’s donated state; otherwise, the object is deemed to be walking.

Fig. 6

Acceleration.

The walking object’s angular velocity is calculated by harnessing the value of the gyroscope, as shown in Fig. 7. X, y and z are used to represent 3-axis gyroscope. The Euler angle is formed via integrating the angular velocity w.r.t time. The angle defines object’s orientation. The Roll, Yaw and Pitch is normally computed using the Euler angle. The rate of change reported with the movement of the object is reported as angular velocity. The Equation (23) illustrates the formula to compute angular velocity.

Fig. 7

Gyroscope.

ω = \frac{θ_{f} - θ_{i}}{t},

(23)

The initial and final angles are denoted as θ_f and θ_i, respectively. Δθ, denotes the change of angle, and time is denoted as t.

The object’s acceleration m/s² in an indoor environment is shown in Fig. 8. The acceleration is computed as the rate of velocity measured during the captured time interval. The formula for acceleration computation is shown in Equation (24).

Fig. 8

Acceleration m/s².

a_{cc} = \frac{Δ vel}{time},

(24)

The change in velocity is denoted via Δvel, the acceleration in m/s² is denoted as a_cc.

The object’s velocity determines the speed at which an object moves in an indoor environment. The object’s 3-axis velocity is shown in Fig. 9. The formula for the velocity computation is shown in Equation (25).

Fig. 9

Velocity.

vel = \frac{Δ x_{p}}{time}

(25)

Δx_p denotes the change of an object’s position in an indoor environment, and velocity is denoted as vel. The time directs to that time wherein the object’s position was changed.

The two-dimensional graph is used to represent object’s 3-axis position, as shown in Fig. 10.

Fig. 10

Position.

The predicted values of accelerometer sensor, computed using traditional Kalman filter without the FIS based Kalman filter model, is shown in Fig. 11. The comparisons are performed using varying values of R and original sensing data. The optimal value for R depends on the data set. As it is impractical to manually obtain optimal values for R, therefore, varying values of R are considered. It can be noted from the graph that a change in R leads to a change in the prediction accuracy of the Kalman filter.

Fig. 11

Position.

Likewise, the predicted values for angular velocity, computed without applying FIS based Kalman filter model to the conventional Kalman filter, are shown in Fig. 12. The graph denotes the varying predicted results obtained for different values of R in configuration of Kalman filter. Table 4 recapitulates the prediction values for sensor reading using three different configurations.

Fig. 12

Position.

Table 4

Results of Kalman filter prediction with and without FIS

Metric	Kalman Filter without FIS Module	FIS based Kalman Filter
	R = 20	R = 15	R = 10	F = 0.1	F = 0.02	F = 0.01
RMSE	2.125	2.225	2.445	2.441	2.265	2.321
MAD	0.124	0.145	0.154	0.126	0.135	0.136
MSE	6.136	6.125	6.255	5.770	5.358	5.656

The graph depicts the data of position in 60 seconds interval wherein the first 10 seconds are deemed stationary. The position plot’s observation yields that the proposed model has minimized the error value and drift from the sensing data. The trajectory of a person moving to corridor from room D242 is shown in Fig. 13.

Fig. 13

Tracking of person in an indoor environment.

The person commences movement from the starting point in both the scenarios wherein the initial 10 seconds are considered stationary for algorithm convergence, and ends movement at the main corridor. The person’s trajectory is computed via a stochastic model and shown using the black line. The model estimates position by harnessing sensor fusion in which predicted trajectory using a predicting model, is shown via green line. The drifting is significantly reduced on post tunning of the prediction algorithm using fuzzy inference system. The starting point (13, 2) and the ending point (0, 24) is computed via conventional Kalman filter according to defined reference point (0, 0). Likewise, the indoor system’s accuracy is also enhanced wherein the starting point is (14, 5) and ending point is (-2, 24) in the case of FIS based Kalman filter model. The defined reference point (0, 0) is considered for mapping the calculated coordinates.

5 Conclusions and future work

We have presented a study that focuses on improving the prediction algorithm’s accuracy in an indoor environment. The position estimation and FIS based Kalman filter models have been combined by harnessing a sensor fusion algorithm. The Kalman filter’s sensor fusion technique is employed to fuse the measurements for three sensors. This has been done to attain the noise and non-drifted orientation estimation to compute the correct indoor navigation position. Similarly, fuzzy inference system has been employed to improve the accuracy of the prediction algorithm in the FIS-based Kalman filter module. The system has been designed considering a scenario wherein gyroscope and accelerometer impact some explicit conditions in which the conventional Kalman filter cannot find noise-free readings out of real readings. The outcomes suggested that the proposed system resulted in improved accuracy of a prediction algorithm for tuned values of R. The standard evaluation measures like MSE, MAD and RMSE have been harnessed for comparative analysis of the proposed model. The comparisons have yielded that FIS based Kalman filter model for the indoor navigation system attained adequate results (i.e., RMSE=2.26) for 0.02 error factor. The overall outcome indicates that the proposed model has been proven useful in improving prediction accuracy. This model may further be extended for improving more prediction algorithms in indoor navigation systems.

Conflicts of interest

The authors declare no conflict of interest.

Footnotes

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2018R1D1A1A09082919), and this work was supported by the project “Standardization of Networking of Everything Architecture and Protocols” funded by KEIT (No. 20002532).

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