# Consensus-Based Track Association with Multistatic Sensors under a Nested Probabilistic-Numerical Linguistic Environment

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Assumptions and Abbreviations

#### 2.1. Assumptioins

- The earth is a sphere with a radius of 6371 km.
- The gravitational acceleration is $g=10\text{}\mathrm{m}/{\mathrm{s}}^{2}$.
- The interference factors obey Gaussian white noise.

- Ignore the time of transmitting and receiving light waves of the sensor.
- The speed of light is infinite.
- The maneuvering target and the sensor are particles.

#### 2.2. Abbreviations

Variables | Descriptions |

${s}_{q}$ | $q$-th sensor, $q=1,2,\cdots ,k$ |

${t}_{i}$ | $i$-th track point, $i=1,2,\cdots ,p$ |

${c}_{j}$ | $j$-th attribute, $j=1,2,\cdots ,m$ |

${x}_{l}$ | $l$-th maneuvering target, $l=1,2,\cdots ,n$ |

$D$ | Distance between target and the sensor |

$\alpha $ | Azimuth angle |

$\beta $ | Pitch angle |

$\xi $ | The consensus threshold |

$\tau $ | The adjustment coefficient |

${w}_{q}$ | $q$-th weight with respect to the sensor ${s}_{q}$, $q=1,2,\cdots ,k$ |

${\omega}_{j}$ | $j$-th weight with respect to the attribute ${c}_{j}$, $j=1,2,\cdots ,m$ |

Acronyms | Full name |

MAGDM | Multi-attribute group decision making |

NPNLTSs | Nested probabilistic-numerical linguistic term sets |

GMM | Gaussian mixture model |

MHT | Multiple hypothesis tracking |

JPDA | Joint probabilistic data association |

HFLTSs | Hesitant fuzzy linguistic term sets |

PLTSs | Probabilistic linguistic term sets |

DHHFLTSs | Double hierarchy hesitant fuzzy linguistic term sets |

## 3. Methodology

#### 3.1. MAGDM with NPNLTSs

#### 3.1.1. NPNLTSs

#### 3.1.2. MAGDM Problem

#### 3.2. Consensus Model in NPNLTSs

#### 3.2.1. Consensus Checking Process

- (1)
- $CD\left({s}_{1},{s}_{2},\cdots ,{s}_{k}\right)\in \left[0,0.2\right]$ indicates extremely strong consensus;
- (2)
- $CD\left({s}_{1},{s}_{2},\cdots ,{s}_{k}\right)\in \left(0.2,0.4\right]$ indicates strong consensus;
- (3)
- $CD\left({s}_{1},{s}_{2},\cdots ,{s}_{k}\right)\in \left(0.4,0.6\right]$ indicates moderate degree consensus;
- (4)
- $CD\left({s}_{1},{s}_{2},\cdots ,{s}_{k}\right)\in \left(0.6,0.8\right]$ indicates weak consensus;
- (5)
- $CD\left({s}_{1},{s}_{2},\cdots ,{s}_{k}\right)\in (0.8,1]$ indicates extremely weak consensus or no consensus.

#### 3.2.2. Consensus Modifying Process

#### 3.3. Track Association Algoritm

Algorithm 1. The track association algorithm based on consensus model with NPNLTSs. | |

Step 1. | (1) Input the parameters: $k,p,m,\xi ,\tau $. (2) Determine the attributes, OLTS, ILTS and the weight vectors $w={\left({w}_{1},{w}_{2},\cdots ,{w}_{k}\right)}^{T}$ and $\omega ={\left({\omega}_{1},{\omega}_{2},\cdots ,{\omega}_{m}\right)}^{T}$. (3) Collect the corresponding data at each track point $T=\left\{{t}_{1},{t}_{2},\cdots ,{t}_{p}\right\}\left(p\ge 2\right)$ measured by a set of sensors $S=\left\{{s}_{1},{s}_{2},\cdots ,{s}_{k}\right\}\left(k\ge 2\right)$. Go to the next step. |

Step 2. | Based on the OLTS and the ILTS, establish the individual evaluation matrix $NP{N}^{q}={\left(NP{N}_{ij}^{q}\right)}_{p\times m}$ with the sensor ${s}_{q}\in S\left(q=1,2,\cdots ,k\right)$ for the track point ${t}_{i}\in T$ with respect to the attribute ${c}_{j}\in C$. Go to the next step. |

Step 3. | (1) Calculate the collective evaluation matrix $NP{N}^{c}=\left(NP{N}_{ij}^{c}\right)$ using Equation (5). $\mathrm{Go}\text{}\mathrm{to}\text{}\mathrm{the}\text{}\mathrm{next}\text{}\mathrm{step}.\text{}(2)\text{}\mathrm{Aggregate}\text{}\mathrm{the}\text{}\mathrm{individual}\text{}\mathrm{overall}\text{}\mathrm{evaluation}\text{}\mathrm{vectors}\text{}{z}^{q}={\left({z}_{1}^{q},{z}_{2}^{q},\cdots ,{z}_{p}^{q}\right)}^{T}$ using Equation (6), with the associated weight vector $w={\left({w}_{1},{w}_{2},\cdots ,{w}_{k}\right)}^{T}$ over sensors $S$. Go to the next step. |

Step 4. | Aggregate the collective overall evaluation vector ${z}^{c}=\left({z}_{1}^{c},{z}_{2}^{c},\cdots ,{z}_{p}^{c}\right)$ using Equation (6), with the associated weight vector $\omega ={\left({\omega}_{1},{\omega}_{2},\cdots ,{\omega}_{m}\right)}^{T}$ over attributes $C$. Go to the next step. |

Step 5. | Determine the consensus threshold $\xi $, which is in the range of [0.4, 0.8] generally, and the adjustment coefficient $\tau $, in this paper, we let $\tau =1/2$. Go to the next step. |

Step 6. | Calculate the consensus degree $CD\left({s}_{1},{s}_{2},\cdots ,{s}_{k}\right)$ by Equation (7). If $CD\left({s}_{1},{s}_{2},\cdots ,{s}_{k}\right)>\xi $, then go to the next step; Otherwise, go to Step 8. |

Step 7. | Adjust the individual evaluation matrix $NP{N}^{q}={\left(NP{N}_{ij}^{q}\right)}_{p\times m}\left(q=1,2,\cdots ,k\right)$ by Equation (8) until $CD\left({s}_{1},{s}_{2},\cdots ,{s}_{k}\right)\le \xi $. Go to the next step. |

Step 8. | Aggregate all the individual evaluation matrices into a final group evaluation matrix $NPN={\left(NP{N}_{ij}\right)}_{p\times m}$ by Equation (5). Go to the next step. |

Step 9. | Obtain the most likely maneuvering target at each track point ${t}_{i}\in T$ based on Equation (6). Go to the next step. |

Step 10. | End. |

Pseudo-code. The pseudo-code of the track association algorithm. |

Input parameters: k—the number of sensors; p—the number of track points; m—the number of attributes; $\xi $—the consensus threshold; $\tau $—the adjustment coefficient; $w$—the weight of the sensors; $\omega $—the weight of the attributes. 1. // Calculate the collective evaluation matrix 2. for i: = 1 to p 3. for j: = 1 to k 4. collective. element (i, j): = sum (sensor (i, j) * $w$(j)) 5. // Calculate the consensus degree 6. for i: = 1 to p 7. for j: = 1 to m 8. individual. element (i, j): = sum (attribute (i, j) * $\omega $(j)) 9. overall. element (i): = sum (individual. element (i, j) * $\omega $(j)) 10. consensus = sum (abs (individual. element (i)—overall. element (i)))/p 11. // Calculate the final result with a consensus 12. while (consensus < $\xi $) 13. ad_individual. element (i, j): = (individual. element (i, j) + collective. element (i, j))/$\tau $ 14. group. element (i, j): = ad_individual. element (i, j) * $w$(j)) 15. final. element (i): = group. element (i, j) *$\omega $(j) |

16. return max_final. element (i) |

## 4. A Case Study

#### 4.1. Problem Description

#### 4.2. Establishing the Proposed Model

#### 4.3. Solving the Problem

## 5. Comparison and Discussion

#### 5.1. Comparative Analysis

- (1)
- The average root-mean-square error (RMSE) of the key parameters.
- (2)
- The impact of the number of the track points on the average RMSE.
- (3)
- The average operation time (AOT).

#### 5.2. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Attari, M.; Habibi, S.; Gadsden, S.A. Target tracking formulation of the SVSF with data association techniques. IEEE Trans. Aerosp. Electron. Syst.
**2017**, 53, 12–25. [Google Scholar] [CrossRef] - Zhu, S.H.; Shi, Z.; Sun, C.J. Tracklet association based multi-target tracking. Multimedia Tools Appl.
**2016**, 75, 9489–9506. [Google Scholar] [CrossRef] - Hu, X.Q.; Bao, M.; Zhang, X.P.; Wen, S.; Li, X.D.; Hu, Y.H. Quantized kalman filter tracking in directional sensor networks. IEEE Trans. Mob. Comput.
**2018**, 17, 871–883. [Google Scholar] [CrossRef] - Fortino, G.; Galzarano, S.; Gravina, R.; Li, W.F. A framework for collaborative computing and multisensor data fusion in body sensor networks. Inf. Fusion
**2015**, 22, 50–70. [Google Scholar] [CrossRef] - Gravina, R.; Alinia, P.; Ghassemzadeh, H.; Fortino, G. Multisensor fusion in body sensor networks: State-of-the-art and research challenges. Inf. Fusion
**2017**, 35, 68–80. [Google Scholar] [CrossRef] - Kazimierski, W. Proposal of neural approach to maritime radar and automatic identification system tracks association. IET Radar Sonar Navig.
**2017**, 11, 729–735. [Google Scholar] [CrossRef] - Yoon, J.H.; Lee, C.R.; Yang, M.H.; Yoon, K.J. Structural constraint data association for online multi-object tracking. Int. J. Comput. Vis.
**2019**, 127, 1–21. [Google Scholar] [CrossRef] - Yang, M.; Wu, Y.W.; Jia, Y.D. A hybrid data association framework for robust online multi-object tracking. IEEE Trans. Image Process.
**2017**, 26, 5667–5679. [Google Scholar] [CrossRef] - Ma, J.Y.; Jiang, J.J.; Liu, C.Y.; Li, Y.S. Feature guided Gaussian mixture model with semi-supervised EM and local geometric constraint for retinal image registration. Inf. Sci.
**2017**, 417, 128–142. [Google Scholar] [CrossRef] - Coraluppi, S.P.; Carthel, C.A. Multiple-hypothesis tracking for targets producing multiple measurements. IEEE Trans. Aerosp. Electron. Syst.
**2018**, 54, 1485–1498. [Google Scholar] [CrossRef] - Chen, X.; Li, Y.A.; Li, Y.X.; Yu, J.; Li, X.H. A novel probabilistic data association for target tracking in a cluttered environment. Sensors
**2016**, 16, 2180. [Google Scholar] [CrossRef] [PubMed] - Vivone, G.; Braca, P. Joint probabilistic data association tracker for extended target tracking applied to X-band marine radar data. IEEE J. Ocean. Eng.
**2016**, 41, 1007–1019. [Google Scholar] [CrossRef] - Fan, L.X.; Fan, E.; Yuan, C.H.; Hu, K.L. Weighted fuzzy track association method based on Dempster-Shafer theory in distributed sensor networks. Int. J. Distrib. Sens. Netw.
**2016**, 12. [Google Scholar] [CrossRef] - Li, J.; Xie, W.X.; Li, L.Q. Online visual multiple target tracking by intuitionistic fuzzy data association. Int. J. Fuzzy Syst.
**2017**, 19, 355–366. [Google Scholar] - Yang, D.; Ji, H.B.; Gao, Y.C. A robust D-S fusion algorithm for multi-target multisensor with higher reliability. Inf. Fusion
**2019**, 47, 32–44. [Google Scholar] - Yoon, K.; Kim, D.Y.; Yoon, Y.C.; Jeon, M. Data association for multi-object tracking via deep neural networks. Sensors
**2019**, 19, 559. [Google Scholar] [CrossRef] [PubMed] - Scott, S.L.; Blocker, A.W.; Bonassi, F.V.; Chipman, H.A.; George, E.I.; McCulloch, R.E. Bayes and big data: The consensus Monte Carlo algorithm. Int. J. Eng. Sci.
**2016**, 11, 78–88. [Google Scholar] [CrossRef] - Luengo, D.; Martino, L.; Elvira, V.; Bugallo, M.F. Efficient linear fusion of partial estimators. Digit. Signal Process.
**2018**, 78, 265–283. [Google Scholar] [CrossRef] - Olfati-Saber, R.; Fax, J.A.; Murray, R.M. Consensus and cooperation in networked multi-agent systems. Proc. IEEE
**2007**, 95, 215–233. [Google Scholar] [CrossRef] - Dimakis, A.G.; Kar, S.; Moura, J.F.; Rabbat, M.G.; Scaglione, A. Gossip algorithms for distributed signal processing. Proc. IEEE
**2010**, 98, 1847–1864. [Google Scholar] [CrossRef] - Yu, Y.H. Distributed target tracking in wireless sensor networks with data association uncertainty. IEEE Commun. Lett.
**2017**, 21, 1281–1284. [Google Scholar] [CrossRef] - Wang, X.X.; Xu, Z.S.; Gou, X.J. Nested probabilistic-numerical linguistic term sets in two-stage multi-attribute group decision making. Appl. Intell.
**2019**, 49, 1–21. [Google Scholar] [CrossRef] - Wang, X.X.; Xu, Z.S.; Gou, X.J.; Trajkovic, L. Tracking a maneuvering target by multiple sensors using extended kalman filter with nested probabilistic-numerical linguistic information. IEEE Trans. Fuzzy Syst.
**2019**. [Google Scholar] [CrossRef] - Liao, H.C.; Xu, Z.S.; Zeng, X.J.; Merigo, J.M. Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets. Knowl.-Based Syst.
**2015**, 76, 127–138. [Google Scholar] [CrossRef] - Wei, C.P.; Zhao, N.; Tang, X.J. Operators and comparisons of hesitant fuzzy linguistic term sets. IEEE Trans. Fuzzy Syst.
**2014**, 22, 575–585. [Google Scholar] [CrossRef] - Zhang, Y.X.; Xu, Z.S.; Liao, H.C. A consensus process for group decision making with probabilistic linguistic preference relations. Inf. Sci.
**2017**, 414, 260–275. [Google Scholar] [CrossRef] - Rodriguez, R.M.; Martinez, L.; Herrera, F. Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst.
**2012**, 20, 109–119. [Google Scholar] [CrossRef] - Pang, Q.; Wang, H.; Xu, Z.S. Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci.
**2016**, 369, 128–143. [Google Scholar] [CrossRef] - Gou, X.J.; Liao, H.C.; Xu, Z.S.; Herrera, F. Double hierarchy hesitant fuzzy linguistic term set and MULTIMOORA method: A case of study to evaluate the implementation status of haze controlling measures. Inf. Fusion
**2017**, 38, 22–34. [Google Scholar] [CrossRef] - Wang, X.X.; Xu, Z.S.; Gou, X.J. Distance and similarity measures for nested probabilistic-numerical linguistic term sets applied to evaluation of medical treatment. Int. J. Fuzzy Syst.
**2019**. [Google Scholar] [CrossRef] - Karplus, P.A.; Diederichs, K. Linking crystallographic model and data quality. Science
**2012**, 336, 1030–1033. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chiclana, F.; Tapia Garcia, J.M.; del Moral, M.J.; Herrera-Viedma, E. A statistical comparative study of different similarity measures of consensus in group decision making. Inf. Sci.
**2013**, 221, 110–123. [Google Scholar] [CrossRef] [Green Version] - Herrera-Viedma, E.; Herrera, F.; Chiclana, F. A consensus model for multiperson decision making with different preference structures. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.
**2002**, 32, 394–402. [Google Scholar] [CrossRef] [Green Version] - Herrera-Viedma, E.; Martinez, L.; Mata, F.; Chiclana, F. A consensus support system model for group decision-making problems with multigranular linguistic preference relations. IEEE Trans. Fuzzy Syst.
**2005**, 13, 644–658. [Google Scholar] [CrossRef] - Dong, Q.X.; Cooper, O. A peer-to-peer dynamic adaptive consensus reaching model for the group AHP decision making. Eur. J. Oper. Res.
**2016**, 250, 521–530. [Google Scholar] [CrossRef] - Ma, L.C. A new group ranking approach for ordinal preferences based on group maximum consensus sequences. Eur. J. Oper. Res.
**2016**, 251, 171–181. [Google Scholar] [CrossRef]

**Figure 1.**The flowchart of the track association algorithm based on the consensus model with nested probabilistic-numerical linguistic term sets (NPNLTSs).

**Figure 3.**The corresponding tracks of two maneuvering targets. (

**a**) The corresponding tracks in two dimensions; (

**b**) The corresponding tracks in three dimensions.

**Figure 4.**The tracking of two maneuvering targets by Extended Kalman Filter (EKF). (

**a**) The tracking of target 1; (

**b**) The tracking of target 2.

Distance (m) | Azimuth Angle (degree) | Pitch Angle (degree) | Time (s) | Sensor Label |
---|---|---|---|---|

84,626.83 | 89.99 | 1.74 | 0.10 | 1 |

85,016.58 | 89.50 | 3.07 | 0.70 | 1 |

… | … | … | … | 1 |

53,481.28 | 87.24 | 3.93 | 272.50 | 1 |

48,556.38 | 233.38 | 3.02 | 272.60 | 2 |

48,538.89 | 227.05 | 3.50 | 273.10 | 2 |

… | … | … | … | 2 |

20,360.67 | 248.65 | 7.49 | 542.50 | 2 |

25,166.65 | 331.88 | 8.21 | 543.00 | 3 |

25,229.30 | −8.92 | 9.50 | 543.10 | 3 |

… | … | … | … | 3 |

32,031.73 | −104.41 | 29.11 | 808.90 | 3 |

Sensor Label | $\mathbf{Longitude}\text{}\mathit{\omega}$ (degree) | $\mathbf{Latitude}\text{}\mathit{\varphi}$ (degree) | $\mathbf{Height}\text{}\mathit{h}$ (m) | $\mathbf{Error}\text{}\mathbf{of}\text{}\mathit{D}$ (m) | $\mathbf{Error}\text{}\mathbf{of}\text{}\mathit{\alpha}$ (degree) | $\mathbf{Error}\text{}\mathbf{of}\text{}\mathit{\beta}$ (degree) |
---|---|---|---|---|---|---|

1 | 102.1 | 30.5 | 0 | 50 | 0.4 | 0.4 |

2 | 102.4 | 31.5 | 0 | 60 | 0.5 | 0.5 |

3 | 102.7 | 31.9 | 0 | 60 | 0.5 | 0.5 |

Sensor 1 | Location | Height | Speed |
---|---|---|---|

${t}_{1}$ | $\left\{\begin{array}{l}{s}_{0}(0.5)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.5)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.5)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.5)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.5)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.5)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ |

${t}_{2}$ | $\left\{\begin{array}{l}{s}_{0}(0.8)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.2)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.7)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.3)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ | |

… | … | … | … |

${t}_{2649}$ | $\left\{\begin{array}{l}{s}_{0}(0.2)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.8)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.4)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.6)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.3)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.7)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ |

Sensor 2 | Location | Height | Speed |
---|---|---|---|

${t}_{1}$ | $\left\{\begin{array}{l}{s}_{0}(0.3)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.7)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.3)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.7)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.4)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.6)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ |

${t}_{2}$ | $\left\{\begin{array}{l}{s}_{0}(0.6)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.4)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.7)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.3)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.7)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.3)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ |

… | … | … | … |

${t}_{2649}$ | $\left\{\begin{array}{l}{s}_{0}(0.8)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.2)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.6)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.4)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.8)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.2)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ |

Sensor 3 | Location | Height | Speed |
---|---|---|---|

${t}_{1}$ | $\left\{\begin{array}{l}{s}_{0}(0.5)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.5)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.4)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.6)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.3)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.7)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ |

${t}_{2}$ | $\left\{\begin{array}{l}{s}_{0}(0.6)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.4)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.5)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.5)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.7)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.3)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ |

… | … | … | … |

${t}_{2649}$ | $\left\{\begin{array}{l}{s}_{0}(0.5)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.5)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ |

Location | Height | Speed | |
---|---|---|---|

${t}_{1}$ | $\left\{\begin{array}{l}{s}_{0}(0.43)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.57)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.40)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.60)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.40)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.60)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ |

${t}_{2}$ | $\left\{\begin{array}{l}{s}_{0}(0.57)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.43)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.67)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.33)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.70)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.30)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ |

… | … | … | … |

${t}_{2649}$ | $\left\{\begin{array}{l}{s}_{0}(0.53)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.47)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.57)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.43)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.53)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.47)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ |

Sensor 1 | Sensor 2 | Sensor 3 | |
---|---|---|---|

${t}_{1}$ | $\left\{\begin{array}{l}{s}_{0}(0.50)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.50)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.34)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.66)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.38)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.62)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ |

${t}_{2}$ | $\left\{\begin{array}{l}{s}_{0}(0.70)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.30)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.68)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.32)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.60)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.40)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ |

… | … | … | … |

${t}_{2649}$ | $\left\{\begin{array}{l}{s}_{0}(0.32)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\},\\ {s}_{1}(0.68)\left\{{n}_{0}\left(50\right),{n}_{1}\left(0.4\right),{n}_{2}\left(0.4\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.72)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.28)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.60)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\},\\ {s}_{1}(0.40)\left\{{n}_{0}\left(60\right),{n}_{1}\left(0.5\right),{n}_{2}\left(0.5\right)\right\}\end{array}\right\}$ |

${t}_{1}$ | $\left\{{s}_{0}(0.406)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},{s}_{1}(0.594)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\right\}$ |

${t}_{2}$ | $\left\{{s}_{0}(0.662)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},{s}_{1}(0.338)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\right\}$ |

… | … |

${t}_{2649}$ | $\left\{{s}_{0}(0.546)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},{s}_{1}(0.454)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\right\}$ |

Location | Height | Speed | |
---|---|---|---|

${t}_{1}$ | $\left\{\begin{array}{l}{s}_{0}(0.35)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.65)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.30)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.70)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.28)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.72)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ |

${t}_{2}$ | $\left\{\begin{array}{l}{s}_{0}(0.60)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.40)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.71)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.29)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.80)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.20)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ |

… | … | … | … |

${t}_{2649}$ | $\left\{\begin{array}{l}{s}_{0}(0.58)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.42)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.67)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.33)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ | $\left\{\begin{array}{l}{s}_{0}(0.68)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},\\ {s}_{1}(0.32)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\end{array}\right\}$ |

${t}_{1}$ | $\left\{{s}_{0}(0.302)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},{s}_{1}(0.698)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\right\}$ |

${t}_{2}$ | $\left\{{s}_{0}(0.724)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},{s}_{1}(0.276)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\right\}$ |

… | … |

${t}_{2649}$ | $\left\{{s}_{0}(0.656)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\},{s}_{1}(0.344)\left\{{n}_{0}\left(57\right),{n}_{1}\left(0.47\right),{n}_{2}\left(0.47\right)\right\}\right\}$ |

Average RMSE | Method 1 [12] | Method 2 [27] | Method 3 [28] | Proposed Method |
---|---|---|---|---|

Distance (m) | 66.93 | 63.21 | 57.32 | 52.12 |

Azimuth angle (degree) | 0.60 | 0.55 | 0.52 | 0.42 |

Pitch angle (degree) | 0.58 | 0.52 | 0.50 | 0.41 |

Average RMSE | Number | Method 1 [12] | Method 2 [27] | Method 3 [28] | Proposed Method |
---|---|---|---|---|---|

Distance (m) | 1000 | 75.73 | 74.25 | 64.72 | 59.13 |

2000 | 66.94 | 65.98 | 58.25 | 53.72 | |

3000 | 64.78 | 62.56 | 56.29 | 50.44 | |

Azimuth angle (degree) | 1000 | 0.63 | 0.59 | 0.57 | 0.46 |

2000 | 0.60 | 0.55 | 0.52 | 0.42 | |

3000 | 0.58 | 0.54 | 0.50 | 0.40 | |

Pitch angle (degree) | 1000 | 0.63 | 0.55 | 0.53 | 0.45 |

2000 | 0.59 | 0.52 | 0.50 | 0.41 | |

3000 | 0.58 | 0.50 | 0.48 | 0.40 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, X.; Xu, Z.; Gou, X.
Consensus-Based Track Association with Multistatic Sensors under a Nested Probabilistic-Numerical Linguistic Environment. *Sensors* **2019**, *19*, 1381.
https://doi.org/10.3390/s19061381

**AMA Style**

Wang X, Xu Z, Gou X.
Consensus-Based Track Association with Multistatic Sensors under a Nested Probabilistic-Numerical Linguistic Environment. *Sensors*. 2019; 19(6):1381.
https://doi.org/10.3390/s19061381

**Chicago/Turabian Style**

Wang, Xinxin, Zeshui Xu, and Xunjie Gou.
2019. "Consensus-Based Track Association with Multistatic Sensors under a Nested Probabilistic-Numerical Linguistic Environment" *Sensors* 19, no. 6: 1381.
https://doi.org/10.3390/s19061381