Coal mine personnel positioning algorithm based on improved adaptive unscented Kalman filter with wireless channel fading and unknown noise statistics

Abstract

This paper is concerned with the problem of personnel localization in the complex coal mine environment with wireless channel fading and unknown noise statistics. Considering the random channel fading caused by signal fluctuation and transmission fault, an improved adaptive unscented Kalman filter (IAUKF) algorithm is proposed. The mean and error covariances of noise are estimated adaptively by adopting the improved Sage–Husa noise estimation method. In order to save energy and improve energy utilization, the multi-sensor clustering is performed to divide the spatial distribution of sensors into multiple clusters. The sensors in the same cluster can communicate with each other to maintain the consistency of estimation. The simulation results show that the IAUKF algorithm is better than extended Kalman filter (EKF), unscented Kalman filter (UKF), and improved unscented Kalman filter (IUKF) algorithms.

Keywords

Personnel positioning wireless sensor networks channel fading unknown noise statistics unscented Kalman filter

Introduction

Coal mine is one of the most dangerous industrial environments (Bai et al., 2020; Ivaz et al., 2020; Zhao et al., 2020), which is prone to severe safety problem for both workers and the surrounding environments (Min et al., 2020). Although the number of major accidents and casualties in coal mining enterprises is on the decline, the frequent occurrence of accidents is still a persistent problem in the industry (Zhao et al., 2020). For example, the results obtained from the former SAWS (State Administration of Work Safety) annual accident survey and reports have revealed that there were about 495 major accidents and 10,546 fatalities occurring in the coal mine industry from 2001 to 2018 (Zhang et al., 2020). Allowing for this severe situation, it is urgent to develop an effective personnel monitoring system in the underground working environment, which is conducive to obtaining the position information of underground personnel timely and accurately, so as to facilitate the efficient rescue under abnormal conditions (Li et al., 2017a; Min et al., 2020; Yuan et al., 2015; Zhang et al., 2020). In the past, the personnel positioning in coal mine mainly used the wired communication, which has the defects of complicated wiring, high cost, strong line dependence, and so on. When there is a dangerous accident in the mine, sensors and cables will be fatally damaged, which cannot provide the reliable information for search and rescue work (Zhang et al., 2018).

With the development of wireless communication technology, the potential applications of wireless sensor networks (WSNs) in the personnel positioning have attracted the significant attention (Chen et al., 2013; Vaseghi et al., 2018; Xie et al., 2019). In general, WSNs are comprised of a large number of sensor nodes deployed in the monitoring area, which are highly compatible and easy to deploy (Kang et al., 2020; Zhang and Xing Zheng, 2018). In a specific monitoring area, sensor nodes can share the local information with each other through wireless information transmission, which help the system to execute complex tasks cooperatively (Ge et al., 2017). In the actual engineering application, the personnel positioning with WSNs can be deployed in the underground working sites of coal mine to track the personnel status in real time, thus realizing the reliable identification of personnel safety, and facilitating the safe management of coal mine production efficiently (Li et al., 2020; Zheng et al., 2019a).

It is quite common that in the complex industrial environment, the wireless signal transmission may be affected by the limits on bandwidth, multipath interference, and shadowing. Besides, the channel fading phenomenon is likely to occur, which would lead to transmission delay and packet loss (Cai and Lau, 2021; Geng et al., 2017). For measurement degradation and network induced phenomena, there are many concerns to reduce the impact on estimation performance (Bai et al., 2018; Leong and Quevedo, 2015; Li et al., 2017b, 2017c; Li and Xia, 2015). For instance, in Bai et al. (2018), a recursive filtering algorithm has been proposed, which is characterized by random parameter matrix. In addition, the quantization effect has been modeled by sector bounded uncertainty. In Li et al. (2018), an unbiased insensitive filter with unknown input has been designed, which relies on robust state estimation to deal with multiplicative noise and unknown inputs over the fading channel. In Liu (2016), with consideration given to the fading measurement and time-dependent channel noise, the fading phenomenon has been described by a random variable, and the recursive filtering strategy for discrete linear systems with fading measurement and time-dependent channel noise has been proposed. In Shen et al. (2018), the measurement transmission between the sensor and the filter is conducted through a fading channel characterized by the Rice fading model. An event-based transmission mechanism is adopted to determine whether the sensor measurement should be transmitted to the filter. Besides, a study has been carried out on the problem of recursive filtering for a class of time-varying nonlinear systems with random parameter matrix. However, in the above literature, the coexistence of transmission failure and signal fluctuation in nonlinear systems have not been fully investigated yet.

With respect to the personnel positioning with WSNs, the state estimation can be regarded as a problem of nonlinear filtering. For example, the extended Kalman filter (EKF), unscented Kalman filter (UKF), and particle filter (PF) (Demir and Barut, 2018; Han et al., 2020; Song et al., 2019; Wu et al., 2013; Zhang et al., 2017) are commonly used for the personnel positioning. As for the EKF algorithm, the linearization process is aimed to expand the nonlinear state equation and the measured equation with Taylor series, which retains the linear part and omits other order terms. However, this would reduce the filtering accuracy significantly and even lead to filtering divergence. In Zhang et al. (2017), an adaptive EKF algorithm has been proposed, which can update the noise covariance at the time of estimation. Although the adaptive EKF has been used to estimate the covariance matrix, the algorithm is not applicable in case of strong nonlinearity. Also known as sequential Monte Carlo estimation (Wu et al., 2013), PF can deal with the nonlinear problem by estimating the posterior density distribution of a large number of particles, but this requires a high computational cost. In addition, there is a problem of sample dilution. It is well known that UKF algorithm does not require the nonlinear state and the measured model to be linearized. Instead, it selects a group of sigma points to complete the nonlinear transformation and applies Kalman filter framework for recursive filtering (Ge et al., 2019; Peng et al., 2017). For any Gaussian and nonlinear systems, the posterior mean and covariance accuracy can reach the third order. Generally speaking, for nonlinear problems, the filtering accuracy of UKF is higher than that of EKF, while its computational complexity is lower than that of PF. Although UKF has clear advantages in solving the filtering problem of nonlinear systems, the estimation accuracy of UKF algorithm is not high when the noise statistics are unknown. For the state estimation of high accuracy, the prior knowledge of measurement noise is required. In Li et al. (2020), a solution to the joint state estimation problem of jump Markov nonlinear systems without knowing the measurement noise covariance has been developed. In Ge et al. (2019), the novel adaptive UKF algorithms have been proposed to address the time-varying noise covariance problem. In Peng et al. (2017), an adaptive UKF with a noise statistics estimator has been proposed. Based on the modified Sage–Husa maximum posterior, the noise statistics estimator has been designed to flexibly estimate the mean and error covariances of system process noises online, especially when the prior noise covariance is unknown or inaccurate. However, the accurate estimation results cannot be obtained when the channel fading and the noise covariance are not accurate at the same time.

Inspired by the discussion as mentioned above, we conduct investigation in this paper into the improved adaptive unscented Kalman filter (IAUKF) based on wireless channel fading and unknown noise statistics. The main contributions of this paper are detailed as follows. First of all, in order to save energy and improve energy efficiency, the multi-sensor clustering method is used to divide the spatial distribution of sensors into multiple clusters. The sensors in the same cluster can communicate with each other to maintain the consistency of estimation. Second, an IAUKF algorithm is proposed for the reliable state estimation of underground personnel location with WSNs. Third, the statistical characteristics of noise may be unknown or inaccurate due to the dynamic changes of the system. An improved Sage–Husa noise estimation method is proposed. Finally, the proposed filtering algorithm is applied to the personnel localization in dangerous coal mine environment.

The remainder of the paper is organized as follows. Section “Problem formulation” describes WSNs clustering and wireless channel fading in coal mine roadway. Section “Personnel positioning based on improved algorithm” illustrates the proposed IAUKF algorithm. In section “Simulation results and discussion,” the simulation and evaluation analyses are presented. In section “Conclusion,” the conclusion is drawn.

Problem formulation

In order to achieve the accurate personnel position in the context of coal mine production, the WSNs are deployed in the monitoring area, as shown in Figure 1. The sensor nodes are distributed in the mine tunnel and divided into multiple clusters. The sensors in the same cluster can communicate with each other. Moreover, the node clusters are denoted as Z₀ = {1,…,M}, where $s_{r, i}$ represents the ith sensor in the rth cluster. Let M_r = {1,…,m} represents the rth cluster in the sensor network, where r∈Z₀ and m indicate the number of sensors in M_r.

In addition, the underground personnel motion model is expressed as follows

x_{k} = A_{k - 1} x_{k - 1} + ω_{k - 1}

(1)

A_{k - 1} = [\begin{matrix} 1 & T & \frac{T^{2}}{2} & 0 & 0 & 0 \\ 0 & 0 & 1 & T & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & T & \frac{T^{2}}{2} & 0 \\ 0 & 0 & 0 & 0 & 1 & T \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]

where $A_{k - 1}$ is a known matrix with appropriate dimensions, and $x_{k} \in R^{n}$ is the state vector at discrete time k, which can be described as follows

x_{k} = [{x_{p}}_{k}, {x_{v}}_{k}, {x_{a}}_{k}, {y_{p}}_{k}, {y_{v}}_{k}, {y_{a}}_{k}]^{T}

(2)

where ${x_{p}}_{k}$ and ${y_{p}}_{k}$ represent the position coordinates of the personnel on the x-axis and y-axis, respectively. ${x_{v}}_{k}$ and ${y_{v}}_{k}$ represent the velocity coordinates of the personnel on the x-axis and y-axis, respectively. ${x_{a}}_{k}$ and ${y_{a}}_{k}$ represent the acceleration coordinates of the personnel on the x-axis and y-axis, respectively. In addition, $ω_{k - 1}$ is zero mean Gaussian white noise satisfying the following equations

E [ω_{k - 1}] = 0

(3)

E [ω_{k - 1} ω_{k - 1}^{T}] = Q_{k - 1}

(4)

And, the initial state $x_{0}$ is a random variable expressed as follows

E [x_{0}] = {\bar{x}}_{0}

(5)

E {[x_{0} - {\bar{x}}_{0}] {[x_{0} - {\bar{x}}_{0}]}^{T}} = {\bar{P}}_{0}

(6)

where $E (\cdot)$ is expectation mean value.

The measurement value of rth cluster is denoted as $y_{k, r}$ , and the measurement output is given by

y_{k, r} = h (x_{k, r}) + v_{k, r}, r \in Z_{0}

(7)

where $y_{k, r}$ is the measured value, $y_{k, r} = [y_{k, r, 1}, y_{k, r, 2}, \dots, y_{k, r, i}, \dots y_{k, r, m}]^{T}$ , and $h (\cdot)$ is the expression of the nonlinear measurement equation, which is expressed as follows

h (x_{k, r}) = {[h (x_{k, r, 1}), h (x_{k, r, 2}), . . ., h (x_{k, r, 3}), . . ., h (x_{k, r, m})]}^{T}

(8)

h (x_{k, r, i}) = \sqrt{{(x_{pk} - P_{ri, x})}^{2} + {(y_{pk} - P_{ri, y})}^{2}}

(9)

where $P_{ri, x}$ and $P_{ri, y}$ represent the position of the sensor on the x-axis and y-axis, respectively. $ω_{k}$ and $v_{k, r}$ represent the process noise and the measurement noise respectively, and $v_{k, r} = [v_{k, r, 1}, v_{k, r, 2}, . . ., v_{k, r, i}, . . . v_{k, r, m}]^{T}$ . $ω_{k}$ and $v_{k, r}$ are uncorrelated with each other. It is shown as follows

E [v_{k, r}] = 0

(10)

E [v_{k, r} v_{k, r}^{T}] = R_{k, r}

(11)

E [ω_{k} v_{k, r}^{T}] = 0

(12)

In this paper, our consideration is given to the case that the process noise covariance matrix $Q_{k - 1}$ and the measurement noise covariance matrix $R_{k, r}$ are unknown or imprecise. In addition, as for the wireless communication in the complex environment, the channel fading will lead to serious instability of signal transmission, which degrades the monitoring performance of system (Li and Xia, 2015). Figure 2 shows the channel fading process. When channel fading occurs, the link noise and quantization noise may be produced in the communication channel of sensor and estimator, which can be represented by $ζ_{k, r}$ .

And, the received signal $z_{k, r}$ at the estimator can be expressed as

z_{k, r} = Ξ_{k, r} h (x_{k, r}) + {\bar{v}}_{k, r}

(13)

where $Ξ_{k, r} = μ_{k, r} κ_{k, r}$ , $μ_{k, r}$ and $κ_{k, r}$ denote signal fluctuation and transmission failure, respectively. $Ξ_{k, r}$ is fading channel gain, which can be expressed as $diag {Ξ_{k, r, 1}, Ξ_{k, r, 2}, . . ., Ξ_{k, r, m}}$ . Also, the channel fading gain $Ξ_{k, r}$ is fully known at the estimator

{\begin{matrix} μ_{k, r} = diag {μ_{k, r, 1}, μ_{k, r, 2}, . . ., μ_{k, r, m}} \\ κ_{k, r} = diag {κ_{k, r, 1}, κ_{k, r, 2}, . . ., κ_{k, r, m}} \\ ζ_{k, r} = {[ζ_{k, r, 1}, ζ_{k, r, 2}, . . ., ζ_{k, r, m}]}^{T} \end{matrix}

(14)

where $μ_{k, r}$ is nonzero, white and identically distributed with a given continuous distribution function such as Nakagami. $ζ_{k, r}$ is assumed to be white with zero mean and covariance matrix $\sum^{ζ}$ . $ζ_{k, r}$ and $ζ_{k, s}$ are mutually independent for $r \neq s$ . $\sum^{ζ}$ is expressed as

\sum ζ = diag {ℜ_{k, r, 1}, ℜ_{k, r, 2}, \dots, ℜ_{k, r, m}}

(15)

Moreover

κ_{k, r, i} = {\begin{matrix} 1, the Packet transfer successful \\ 0, otherwise \end{matrix}

(16)

where $κ_{k, r, i}$ is white and identically distributed with probability distribution as follows

{\begin{matrix} P (κ_{k, r, i} = 0) = 1 - π_{r, i} \\ P (κ_{k, r, i} = 1) = π_{r, i}, 0 \leq π_{r, i} \leq 1 \end{matrix}

(17)

Without loss of generality, $x_{0}$ , $μ_{k, r}$ , $κ_{k, r}$ , $ζ_{k, r}$ , $ω_{k}$ , and $v_{k, r}$ are uncorrelated with each other. The nonlinear model together with the wireless fading channel can be rewritten as

{\begin{matrix} x_{k} = A_{k - 1} x_{k - 1} + ω_{k - 1} \\ z_{k, r} = Ξ_{k, r} h (x_{k, r}) + {\bar{v}}_{k, r} \\ = μ_{k, r} κ_{k, r} h (x_{k, r}) + {\bar{v}}_{k, r} \end{matrix}

(18)

where ${\bar{v}}_{k, r} = Ξ_{k, r} v_{k, r} + ζ_{k, r}$ . It can be seen clearly that ${\bar{v}}_{k, r}$ is Gaussian distributed with zero mean and covariance matrix ${\bar{R}}_{k, r} = Ξ_{k, r} R_{k, r} Ξ_{k, r}^{T} + \sum^{ζ}$ .

Personnel positioning based on improved algorithm

Standard UKF algorithm

Unscented transformation (UT) is a method used to approximate the probability distribution of nonlinear transformation. Under the assumption that the independent variable is Gaussian distribution, the mean and covariance of the dependent variable can be approximated by 2n + 1 sigma point. The structure of the standard UKF algorithm is presented as follows:

1. Initialization ${\bar{x}}_{0}$ and ${\bar{P}}_{0}$ .

2. Select sigma points

χ_{k - 1, r}^{l} = {\hat{x}}_{\frac{k - 1}{k - 1}, r} + {\bar{x}}_{k - 1, r}^{l}, l = 0, 1, . . ., 2 n

(19)

{\begin{matrix} {\bar{x}}_{k - 1, r}^{l} = 0, & l = 0 \\ {\bar{x}}_{k - 1, r}^{l} = {(\sqrt{(n + μ) P_{\frac{k - 1}{k - 1}, r}})}_{l}, & l = 1, 2, . . ., n \\ {\bar{x}}_{k - 1, r}^{l} = - {(\sqrt{(n + μ) P_{\frac{k - 1}{k - 1}, r}})}_{l}, & l = n + 1, . . ., 2 n \end{matrix}

(20)

where n is the state of the system, $(\sqrt{(n + μ) P_{\frac{k - 1}{k - 1}, r}})$ is the square root of $(n + μ) P_{\frac{k - 1}{k - 1}, r}$ , and ${(\sqrt{(n + μ) P_{\frac{k - 1}{k - 1}, r}})}_{l}$ is the l column of $(\sqrt{(n + μ) P_{\frac{k - 1}{k - 1}, r}})$ . The weighting coefficients are expressed as

{\begin{matrix} W_{m_{l}} = \frac{μ}{n + μ} \\ W_{c_{l}} = \frac{μ}{n + μ} (1 - ϕ^{2} + φ) \\ W_{m_{l}} = W_{c_{l}} = \frac{1}{2 (n + μ)}, l = 1, 2, . . ., 2 n \end{matrix}

(21)

where $μ = ϕ^{2} (n + ξ) - n$ , $ϕ$ and $ξ$ are two adjustable parameters, and $μ$ is a scaling parameter. $φ$ is a parameter which restrains error caused by the higher order terms. In this case, it is assumed that $φ = 2$ .

3. Prediction

{\hat{x}}_{\frac{k}{k - 1}, r}^{l} = A_{k - 1} χ_{k - 1, r}^{l}, l = 0, . . ., 2 n

(22)

{\hat{x}}_{\frac{k}{k - 1}, r} = \sum_{l = 0}^{2 n} W_{m_{l}} {\hat{x}}_{\frac{k}{k - 1}, r}^{l} + q_{k - 1}

(23)

where $q_{k - 1}$ is the mean of system process noise and set to 0 in standard UKF

P_{\frac{k}{k - 1}, r} = \sum_{l = 0}^{2 n} W_{c_{l}} ({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r}) {({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r})}^{T} + Q_{k - 1}

(24)

where ${\hat{x}}_{\frac{k}{k - 1}, r}^{l}$ is the prediction value of sigma point. ${\hat{x}}_{\frac{k}{k - 1}, r}$ is the priori state estimates, and $P_{\frac{k}{k - 1}, r}$ is the covariance of the priori estimation error.

4. Update

{\hat{z}}^{l}_{\frac{k}{k - 1}, r} = Ξ_{k, r} h (χ_{k - 1, r}^{l})

(25)

{\hat{z}}_{\frac{k}{k - 1}, r} = \sum_{l = 0}^{2 n} W_{m_{l}} {\hat{z}}^{l}_{\frac{k}{k - 1}, r} + r_{k}

(26)

where ${\hat{z}}^{l}_{\frac{k}{k - 1}, r}$ is the measurement prediction value of sigma point, and ${\hat{z}}_{\frac{k}{k - 1}, r}$ is the weighted measurement prediction value. $r_{k}$ is the mean of measurement noise and set to 0 in standard UKF

P_{z z} = \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T} + R_{k, r}^{-}

(27)

where $P_{zz}$ is the measurement prediction covariance matrix. The cross-covariance matrix of sensors in the r cluster at k time is defined as $P_{xz}$ . The estimated value is ${\hat{x}}_{\frac{k}{k}, r}$ . The estimation error covariance is $P_{\frac{k}{k}, r}$ , and the gain is $K_{k}$

P_{xz} = \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T}

(28)

{\hat{x}}_{\frac{k}{k}, r} = {\hat{x}}_{\frac{k}{k - 1}, r} + K_{k} (z_{k, r} - {\hat{z}}_{\frac{k}{k - 1}, r})

(29)

P_{\frac{k}{k}, r} = {\hat{P}}_{\frac{k}{k - 1}, r} - K_{k} P_{zz} K_{k}^{T}

(30)

K_{k} = P_{xz} P_{zz}^{- 1}

(31)

The overall procedure of the standard UKF algorithm is summarized in Algorithm 1.

Algorithm 1: Standard unscented Kalman filter (UKF) algorithm
Initialization of ${\bar{x}}_{0}$ and ${\bar{P}}_{0}$ .
Sigma points values $χ_{k - 1, r}^{l}$ are generated by equations (19) and (20).
The state prediction value ${\hat{x}}_{\frac{k}{k - 1}, r}^{l}$ of sigma point, the weighted state prediction value ${\hat{x}}_{\frac{k}{k - 1}, r}$ , and the state prediction error covariance matrix $P_{\frac{k}{k - 1}, r}$ are generated by equations (22), (23), and (24).
The measurement prediction value ${\hat{z}}^{l}_{\frac{k}{k - 1}, r}$ of sigma point, weighted measurement prediction value ${\hat{z}}_{\frac{k}{k - 1}, r}$ , and measurement prediction error covariance matrix $P_{zz}$ are generated by equations (25), (26), and (27).
The cross-covariance matrix $P_{xz}$ is calculated by equation (28).
The estimation value ${\hat{x}}_{\frac{k}{k}, r}$ , estimation error covariance matrix $P_{\frac{k}{k}, r}$ , and gain value $K_{k}$ at k time are calculated by equations (29), (30), and (31).
Return to step 2.

As for the standard UKF algorithm, it is difficult to deal with channel fading and unknown noise covariance matrix. The estimation result of UKF algorithm may lead to the decline or even divergence of filtering accuracy when the channel fading or noise covariance is unknown. To solve this problem, an improved algorithm is required to solve the problem of poor estimation performance caused by channel fading and noise covariance matrix.

IAUKF algorithm

As for the standard UKF, it can be seen from equations (24) and (27) that the calculation of prediction covariance $P_{\frac{k}{k - 1}, r}$ and $P_{zz}$ is affected by the varying process noise covariance matrix $Q_{k - 1}$ and the measurement noise covariance matrix ${\bar{R}}_{k, r}$ . In light of this, a new IAUKF scheme is proposed to ensure that the process noise covariance matrix $Q_{k - 1}$ and measurement noise covariance matrix $R_{k, r}^{-}$ can be adaptively estimated accurately (Ge et al., 2019; Li et al., 2017a; Sun et al., 2016). For better online estimation, the $q_{k}$ , $Q_{k}$ , $r_{k}$ , and ${\bar{R}}_{k, r}$ of the UKF are replaced with the ${\hat{q}}_{k}$ , ${\hat{Q}}_{k}$ , ${\hat{r}}_{k}$ , and ${\hat{\bar{R}}}_{k, r}$ of the IAUKF, respectively, which are adaptively estimated using the noise statistics estimator.

Considering the phenomenon of channel fading, the state predicted value ${\hat{x}}_{\frac{k}{k - 1}, r}^{l}$ of sigma point is calculated as sigma point $χ_{k - 1, r}^{l}$ when calculating the measurement prediction value ${\hat{z}}^{l}_{\frac{k}{k - 1}, r}$ of sigma point. The IAUKF algorithm can not only deal with unknown of noise covariance matrix but also solve the problem of channel fading. The calculation of the noise characteristics is detailed as follows:

1. Calculate the estimated mean value of process noise

{\hat{q}}_{k} = (1 - l_{k}) {\hat{q}}_{k - 1} + l_{k} [{\hat{x}}_{\frac{k}{k}, r} - \sum_{l = 0}^{2 n} W_{m_{l}} {\hat{x}}_{\frac{k}{k - 1}, r}^{l}]

(32)

where $l_{k}$ is the correction factor

l_{k} = \frac{1 - b}{1 - b^{k + 1}}

(33)

where b is a forgetting factor (usually between 0.95 and 0.99).

2. Calculate estimated covariance value of process noise

\begin{matrix} {\hat{Q}}_{k} = (1 - l_{k}) {\hat{Q}}_{k - 1} + l_{k} \\ [\begin{matrix} K_{k} e_{k} e_{k}^{T} K_{k}^{T} + P_{\frac{k}{k}, r} - \sum_{l = 0}^{2 n} W_{c_{l}} ({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r}) \\ \times {({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r})}^{T} \end{matrix}] \end{matrix}

(34)

where $e_{k}$ is error term, which is expressed as

e_{k} = z_{k, r} - {\hat{z}}_{\frac{k}{k - 1}, r}

(35)

3. Calculate estimated mean value of measurement noise

{\hat{r}}_{k} = (1 - l_{k}) {\hat{r}}_{k - 1} + l_{k} [z_{k, r} - \sum_{l = 0}^{2 n} W_{m_{l}} {\hat{z}}^{l}_{\frac{k}{k - 1}, r}]

(36)

4. Calculate the estimation results ${\hat{\bar{R}}}_{k, r}$ including the measurement noise and communication noise caused by channel fading, as follows

\begin{matrix} {\hat{\bar{R}}}_{k, r} = (1 - l_{k}) {\hat{\bar{R}}}_{k - 1, r} + l_{k} \\ [e_{k} e_{k}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T}] \end{matrix}

(37)

In general, the standard UKF algorithm has higher accuracy when the characteristics of process noise and measurement noise are known. However, as for the mine personnel positioning system on WSNs, the mean and covariance of noise are often unknown or imprecise, which would reduce the estimation performance of UKF algorithm. In this paper, given the shortcomings of UKF algorithm and the advantages of adaptive unscented Kalman filter algorithm in Li and Xia (2015), a new IAUKF method is developed for unknown noise statistical characteristics of personnel positioning. In contrast to the methods based on covariance matching mechanisms, the ${\hat{q}}_{k - 1}$ , ${\hat{Q}}_{k - 1}$ , ${\hat{r}}_{k - 1}$ , and ${\hat{\bar{R}}}_{k, r}$ can be obtained online by the noise estimator based on the improved Sage–Husa filter. With process noise covariance ${\hat{\bar{R}}}_{k, r}$ , a brief derivation is shown as follows

\begin{matrix} {\hat{\bar{R}}}_{k, r} = (1 - l_{k}) {\hat{\bar{R}}}_{k - 1, r} + l_{k} \\ [e_{k} e_{k}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T}] \\ = (1 - l_{k}) (1 - l_{k - 1}) {\hat{\bar{R}}}_{k - 2, r} + (1 - l_{k}) l_{k - 1} \\ [\begin{matrix} e_{k - 1} e_{k - 1}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{k - 1}{k - 2}, r} - {\hat{z}}_{\frac{k - 1}{k - 2}, r}) \\ \times {({\hat{z}}^{l}_{\frac{k - 1}{k - 2}, r} - {\hat{z}}_{\frac{k - 1}{k - 2}, r})}^{T} \end{matrix}] \\ + l_{k} [e_{k} e_{k}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T}] \\ = (1 - l_{k}) (1 - l_{k - 1}) \dots (1 - l_{1}) {\hat{\bar{R}}}_{0, r} \\ + (1 - l_{k}) (1 - l_{k - 1}) \dots (1 - l_{2}) l_{1} \\ [e_{1} e_{1}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{1}{0}, r} - {\hat{z}}_{\frac{1}{0}, r}) {({\hat{z}}^{l}_{\frac{1}{0}, r} - {\hat{z}}_{\frac{1}{0}, r})}^{T}] \\ + (1 - l_{k}) (1 - l_{k - 1}) \dots (1 - l_{3}) l_{2} \\ [e_{2} e_{2}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{2}{1}, r} - {\hat{z}}_{\frac{2}{1}, r}) {({\hat{z}}^{l}_{\frac{2}{1}, r} - {\hat{z}}_{\frac{2}{1}, r})}^{T}] \\ + \dots + l_{k} [e_{k} e_{k}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T}] \end{matrix}

(38)

Substituting $l_{k} = (1 - b) / (1 - b^{k + 1})$ and ${\hat{\bar{R}}}_{0, r} = {\hat{\bar{R}}}_{0, r}$ into equation (38), we can obtain

\begin{matrix} {\hat{\bar{R}}}_{k, r} = \frac{b^{k} (1 - b)}{1 - b^{k + 1}} {\bar{R}}_{0, r} + \frac{b^{k - 1} (1 - b)}{1 - b^{k + 1}} \\ [e_{1} e_{1}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{1}{0}, r} - {\hat{z}}_{\frac{1}{0}, r}) {({\hat{z}}^{l}_{\frac{1}{0}, r} - {\hat{z}}_{\frac{1}{0}, r})}^{T}] \\ + \frac{b^{k - 2} (1 - b)}{1 - b^{k + 1}} [e_{2} e_{2}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{2}{1}, r} - {\hat{z}}_{\frac{2}{1}, r}) {({\hat{z}}^{l}_{\frac{2}{1}, r} - {\hat{z}}_{\frac{2}{1}, r})}^{T}] \\ + \dots + \frac{1 - b}{1 - b^{k + 1}} \\ [e_{k} e_{k}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T}] \end{matrix}

(39)

From the above formula, it can be seen that the weight gradually tends to be a constant value of 1−b over time. Similarly, with the increase of k, the distribution weight of initial value distribution gradually decreases and approaches zero, suggesting that the adaptive degree of the estimator decreases with the filtering process.

Through the above analysis, we can obtain a modified noise statistics estimator, which is expressed as

{\begin{matrix} {\hat{q}}_{k} = (1 - l_{k}) {\hat{q}}_{k - 1} + l_{k} [{\hat{x}}_{\frac{k}{k}, r} - \sum_{l = 0}^{2 n} W_{m_{l}} {\hat{x}}_{\frac{k}{k - 1}, r}^{l}] \\ {\hat{Q}}_{k} = (1 - l_{k}) {\hat{Q}}_{k - 1} + l_{k} [\begin{matrix} K_{k} e_{k} e_{k}^{T} K_{k}^{T} + P_{\frac{k}{k}, r} - \sum_{l = 0}^{2 n} W_{c_{l}} ({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r}) \\ \times {({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r})}^{T} \end{matrix}] \\ {\hat{r}}_{k} = (1 - l_{k}) {\hat{r}}_{k - 1} + l_{k} [z_{k, r} - \sum_{l = 0}^{2 n} W_{m_{l}} {\hat{z}}^{l}_{\frac{k}{k - 1}, r}] \\ {\hat{\bar{R}}}_{k, r} = (1 - l_{k}) {\hat{\bar{R}}}_{k - 1, r} + l_{k} [e_{k} e_{k}^{T} - \sum_{j = 0}^{2 n} W_{c_{l}} ({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T}] \end{matrix}

(40)

The filter convergence criterion is applied to judge whether the measurement noise has a significant change. The criterion formula is presented as follows

e_{k}^{T} e_{k} \leq ι^{2} Tr {E [e_{k} e_{k}^{T}]}

(41)

where $Tr {E [e_{k} e_{k}^{T}]}$ denotes the trace of the matrix $E [e_{k} e_{k}^{T}]$ ; $ι$ is used to control the strictness degree of criterion, and its range is $ι \geq 1$ .

Through the above analysis, the mean and covariance matrix of process and measurement noise can be obtained, while the IUKF is used to estimate the system state variables in the iterative process. The main steps of IAUKF are detailed as follows

{\begin{matrix} {\hat{x}}_{\frac{k}{k - 1}, r}^{l} = A_{k - 1} χ_{k - 1, r}^{l}, l = 1, 2, . . ., 2 n \\ {\hat{x}}_{\frac{k}{k - 1}, r} = \sum_{l = 0}^{2 n} W_{m_{l}} {\hat{x}}_{\frac{k}{k - 1}, r}^{l} + {\hat{q}}_{k - 1} \end{matrix}

(42)

P_{\frac{k}{k - 1}, r} = \sum_{l = 0}^{2 n} W_{c_{l}} ({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r}) {({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r})}^{T} + {\hat{Q}}_{k - 1}

(43)

where ${\hat{q}}_{k - 1}$ and ${\hat{Q}}_{k - 1}$ are the mean and covariance matrices of process noise estimated by equation (40).

In order to further improve the estimation accuracy, IUKF algorithm is introduced to improve the UKF algorithm considering the unknown noise covariance matrix. In this paper, the sigma value $χ_{k - 1, r}^{l}$ is replaced with the sigma state prediction value ${\hat{x}}_{\frac{k}{k - 1}, r}^{l}$ as follows

{\begin{matrix} {\hat{z}}^{l}_{\frac{k}{k - 1}, r} = Ξ_{k, r} h ({\hat{x}}_{\frac{k}{k - 1}, r}^{l}) \\ {\hat{z}}_{\frac{k}{k - 1}, r} = \sum_{l = 0}^{2 n} W^{l} {\hat{z}}^{l}_{\frac{k}{k - 1}, r} + {\hat{r}}_{k} \end{matrix}

(44)

P_{zz} = \sum_{j = 0}^{2 n} W^{l} ({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T} + {\hat{\bar{R}}}_{k, r}

(45)

P_{xz} = \sum_{j = 0}^{2 n} W^{l} ({\hat{x}}_{\frac{k}{k - 1}, r}^{l} - {\hat{x}}_{\frac{k}{k - 1}, r}) {({\hat{z}}^{l}_{\frac{k}{k - 1}, r} - {\hat{z}}_{\frac{k}{k - 1}, r})}^{T}

(46)

Similarly, ${\hat{r}}_{k}$ and ${\hat{\bar{R}}}_{k, r}$ are the mean and covariance matrices of process noise estimated by equation (40)

{\begin{matrix} K_{k} = P_{xz} P_{zz}^{- 1} \\ {\hat{x}}_{\frac{k}{k}, r} = {\hat{x}}_{\frac{k}{k - 1}, r} + K_{k} (z_{k, r} - {\hat{z}}_{\frac{k}{k - 1}, r}) \\ P_{\frac{k}{k}, r} = {\hat{P}}_{\frac{k}{k - 1}, r} - K_{k} P_{zz} K_{k}^{T} \end{matrix}

(47)

From equations (43) and (45), it can be found out that the improved algorithm combines the advantages of improved Sage–Husa filtering algorithm and IUKF algorithm, which can not only improve the accuracy of estimation but also reduce the computational burden. Besides, to do so is more friendly to sensors with limited computing power and storage capacity. The overall procedure of the proposed IAUKF algorithm is summarized in Algorithm 2.

Algorithm 2. Improved adaptive unscented Kalman filter algorithm (IAUKF).
Initialization of ${\bar{x}}_{0}$ and ${\bar{P}}_{0}$ .
Sigma points values $χ_{k - 1, r}^{l}$ are generated by equations (19) and (20).
The mean values of state and measurement noise ${\hat{q}}_{k - 1}$ and ${\hat{r}}_{k}$ , covariance matrices ${\hat{Q}}_{k - 1}$ and ${\hat{\bar{R}}}_{k, r}$ are estimated by equation (40).
The state prediction value ${\hat{x}}_{\frac{k}{k - 1}, r}^{l}$ of sigma point, the weighted state prediction value ${\hat{x}}_{\frac{k}{k - 1}, r}$ , and the state prediction error covariance matrix $P_{\frac{k}{k - 1}, r}$ are generated by equations (42) and (43). The measurement prediction value ${\hat{z}}^{l}_{\frac{k}{k - 1}, r}$ of sigma point, weighted measurement prediction value ${\hat{z}}_{\frac{k}{k - 1}, r}$ , and measurement prediction error covariance matrix $P_{zz}$ are generated by equations (44) and (45).
The cross-covariance matrix $P_{xz}$ is calculated by equation (46).
The estimation value ${\hat{x}}_{\frac{k}{k}, r}$ , estimation error covariance matrix $P_{\frac{k}{k}, r}$ , and gain value $K_{k}$ at k time are calculated by equation (47).
Return to step 2.

Simulation results and discussion

In this section, the advantages of the proposed IAUKF algorithm are illustrated through an application example of wireless positioning in the mine industrial environment. To improve the positioning accuracy of personnel, these sensor nodes are divided into several clusters, with each cluster including four sensors and a cluster head node. All sensors in the same cluster can communicate with each other. Figure 3 shows the network structure of the wireless network. When the tag carried by personnel is in the sensing range of sensor node, the sensor can detect the signal from the tag and then send the data to the cluster head node. The cluster head node further transmits the data to the filter. Then, the position estimation value of the tag is obtained by the proposed filtering algorithm and transmitted to the monitoring center for further collaborative control.

In order to validate the proposed algorithm, the nonlinear stochastic system of coal mine personnel positioning is considered as follows

[\begin{matrix} {x_{p}}_{k} \\ {x_{v}}_{k} \\ {x_{a}}_{k} \\ {y_{p}}_{k} \\ {y_{v}}_{k} \\ y_{ak} \end{matrix}] = [\begin{matrix} 1 & T & \frac{T^{2}}{2} & 0 & 0 & 0 \\ 0 & 0 & 1 & T & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & T & \frac{T^{2}}{2} & 0 \\ 0 & 0 & 0 & 0 & 1 & T \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] \times [\begin{matrix} {x_{p}}_{k - 1} \\ {x_{v}}_{k - 1} \\ {x_{a}}_{k - 1} \\ {y_{p}}_{k - 1} \\ {y_{v}}_{k - 1} \\ y_{ak - 1} \end{matrix}] + ω_{k - 1}

(48)

z_{k, r} = Ξ_{k, r} (\sqrt{{(x_{pk} - P_{ri, x})}^{2} + {(y_{pk} - P_{ri, y})}^{2}} + v_{k, r}) + ζ_{k, r}

(49)

where $ω_{k - 1}$ and $v_{k, r}$ are Gaussian white noise, and their statistical characteristics are expressed as follows

{\begin{matrix} Q_{0} = 10^{- 4} * diag {5, 1, 1, 1, 1, 1} \\ R_{0, r} = diag {0.1, 0.2, 0.1, 0.1} \\ \sum^{ζ} = 10^{- 3} * diag {2, 2, 2, 2} \end{matrix}

(50)

where $Q_{0}$ and $R_{0, r}$ represent the initial state and measurement noise covariance matrix, respectively. It is assumed that sampling period $T = 0.1$ , and initial states

x_{0} = {[\begin{matrix} 0 & 2 & 1.5 & 3 & 0 & 0.5 \end{matrix}]}^{T}

(51)

In the following part, we will compare the EKF, UKF, IUKF, and IAUKF algorithm with channel fading. It is supposed that $μ_{k, r}$ conforms to Rayleigh distribution with the channel power gain of 0.75. The transmission failure rate is 0.2, which is expressed as follows

{\begin{matrix} P (κ_{k, r, i} = 0) = 0.2 \\ P (κ_{k, r, i} = 1) = 0.8 \end{matrix}

(52)

The root mean square error (RMSE) is used to evaluate the performances of the filters

RMS E_{k} = \sqrt{\frac{1}{Λ} \sum_{j = 1}^{Λ} {(x_{j, k} - {\hat{x}}_{j, k})}^{2}}

(53)

where $Λ$ is the number of Monte Carlo simulation, which is set to 30 times.

It is supposed that the personnel make constant acceleration in 0–20 seconds, variable acceleration in 21–30 seconds, and constant acceleration in 31–50 seconds. The actual trajectory of personnel movement is shown in Figure 4, and the actual acceleration curve is shown in Figure 5.

In order to demonstrate the superiority of the proposed method, the comparisons in estimation of several typical algorithms are presented with a given trajectory. Figure 4 shows the real trajectory of the coal mine personnel in the two-dimensional plane. As shown in Figure 5, the coal mine personnel have made a dynamic maneuver between 21 and 30 seconds. Notably, maneuvering acceleration will lead to the mismatch of system model on which the tracking filter depends. Therefore, the potential change of process noise covariance is caused.

Figure 6 shows the trajectory comparison of the real value with EKF, UKF, IUKF, and IAUKF algorithm. It can be seen from the figure that the motion trajectory of IAUKF algorithm is closer to the true trajectory. Figure 7 shows the RMSE comparison of EKF, UKF, IUKF, and IAUKF algorithm. IAUKF has higher accuracy and produces more obvious effect.

Figures 8 –10 show the performance RMSE comparison of EKF, UKF, IUKF, and IAUKF in the case of channel fading. It can be seen from these figures that the accuracy of EKF is the worst, even beyond the range of coordinate axes, and that the RMSE of IAUKF is the smallest and the accuracy is the highest. Obviously, IAUKF can overcome the shortcomings of traditional UKF when the noise statistics are unknown or inaccurate, such as the decline of filtering accuracy and even divergence. Besides, the feasibility of adaptive IAUKF is verified.

Conclusion

In this paper, a personnel safety localization method has been developed under the complex coal mine environment. First, the multi-sensor clustering method is used to divide the spatial distribution of sensors into multiple clusters to save energy and improve energy efficiency. Moreover, considering the random channel fading, the system model with wireless channel fading has been established. Besides, to solve the problem of unknown and time-varying noise statistical parameters, an improved Sage–Husa noise estimator has been proposed. Finally, a new IAUKF algorithm has been designed to guarantee the stability and accuracy of personnel location. The simulation results have shown the satisfactory estimation performance of the proposed method.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 62073198, No. 51807134) and the Natural Science Foundation of Shandong Province of China (No. ZR2020MF071).

ORCID iDs

Xingzhen Bai

Xuehui Gao

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