SCIENTIA SINICA Informationis, Volume 49, Issue 5: 570-584(2019) https://doi.org/10.1360/N112018-00341

## Multi-source passive localization via multiple unmanned aerial vehicles

• AcceptedMar 10, 2019
• PublishedMay 8, 2019
Share
Rating

### Abstract

Passive source localization via multiple unmanned aerial vehicles (UAVs) is a key technology for the practical application of military reconnaissance. Previous studies primarily focused on the localization of a single source. This study explores the multi-source passive localization problem using time difference of arrival (TDOA) measurements. We formulate the localization problem as a constrained weighted least squares problem. The formulation is an indefinite quadratically constrained quadratic programming problem, which is non-convex and NP-hard. To obtain approximate programming with linear constraints, an iterative constrained weighted least squares (CWLS) algorithm is proposed to perform a linearization procedure on the quadratic equality constraints. Theoretical analysis reveals that the proposed algorithm, if converges, can lead to a global optimal solution of the formulated problem. The results of the Monte Carlo experiment indicate that the proposed algorithm quickly converges in most situations and offers better localization accuracy compared with the previous two-step weighted least squares method.

### Supplement

Appendix

Ben-Israel A, Greville T N E. Generalized Inverses: Theory and Applications. 2nd ed. Hoboken: Wiley, 2002.

Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press, 2004.

### References

[1] Zeng Y, Zhang R, Lim T J. Wireless communications with unmanned aerial vehicles: opportunities and challenges. IEEE Commun Mag, 2016, 54: 36-42 CrossRef Google Scholar

[2] Zhu S Q, Wang D W, Low C B. Cooperative control of multiple UAVs for moving source seeking. In: Proceedings of IEEE International Conference on Unmanned Aircraft Systems, New York, 2013. 193--202. Google Scholar

[3] Kwon H, Pack D J. A Robust Mobile Target Localization Method for Cooperative Unmanned Aerial Vehicles Using Sensor Fusion Quality. J Intell Robot Syst, 2012, 65: 479-493 CrossRef Google Scholar

[4] Khelifi F, Bradai A, Singh K. Localization and Energy-Efficient Data Routing for Unmanned Aerial Vehicles: Fuzzy-Logic-Based Approach. IEEE Commun Mag, 2018, 56: 129-133 CrossRef Google Scholar

[5] Zhang J, Yuan H. Analysis of unmanned aerial vehicle navigation and height control system based on GPS. J Syst Eng Electron, 2010, 21: 643-649 CrossRef Google Scholar

[6] Seifeldin M, Saeed A, Kosba A E. Nuzzer: A Large-Scale Device-Free Passive Localization System for Wireless Environments. IEEE Trans Mobile Comput, 2013, 12: 1321-1334 CrossRef Google Scholar

[7] Kehu Yang , Gang Wang , Zhi-Quan Luo . Efficient Convex Relaxation Methods for Robust Target Localization by a Sensor Network Using Time Differences of Arrivals. IEEE Trans Signal Process, 2009, 57: 2775-2784 CrossRef ADS Google Scholar

[8] Lu L, Wu H C, Chang S Y. New direction-of-arrival-based source localization algorithm for wideband signals. IEEE Trans Wirel Commun, 2012, 11: 3850--3859. Google Scholar

[9] Yeredor A, Angel E. Joint TDOA and FDOA Estimation: A Conditional Bound and Its Use for Optimally Weighted Localization. IEEE Trans Signal Process, 2011, 59: 1612-1623 CrossRef ADS Google Scholar

[10] Smith J, Abel J. Closed-form least-squares source location estimation from range-difference measurements. IEEE Trans Acoust Speech Signal Process, 1987, 35: 1661-1669 CrossRef Google Scholar

[11] Chan Y T, Ho K C. A simple and efficient estimator for hyperbolic location. IEEE Trans Signal Process, 1994, 42: 1905-1915 CrossRef ADS Google Scholar

[12] Stoica P, Li J. Lecture Notes - Source Localization from Range-Difference Measurements. IEEE Signal Process Mag, 2006, 23: 63-66 CrossRef ADS Google Scholar

[13] Ho K C, Lu X, Kovavisaruch L. Source Localization Using TDOA and FDOA Measurements in the Presence of Receiver Location Errors: Analysis and Solution. IEEE Trans Signal Process, 2007, 55: 684-696 CrossRef ADS Google Scholar

[14] Beck A, Stoica P, Jian Li P. Exact and Approximate Solutions of Source Localization Problems. IEEE Trans Signal Process, 2008, 56: 1770-1778 CrossRef ADS Google Scholar

[15] Lui K, Chan F, So H C. Semidefinite Programming Approach for Range-Difference Based Source Localization. IEEE Trans Signal Process, 2009, 57: 1630-1633 CrossRef ADS Google Scholar

[16] Picard J S, Weiss A J. Time difference localization in the presence of outliers. Signal Processing, 2012, 92: 2432-2443 CrossRef Google Scholar

[17] Yu H, Huang G, Gao J. An Efficient Constrained Weighted Least Squares Algorithm for Moving Source Location Using TDOA and FDOA Measurements. IEEE Trans Wireless Commun, 2012, 11: 44-47 CrossRef Google Scholar

[18] Lin L, So H C, Chan F K W. A new constrained weighted least squares algorithm for TDOA-based localization. Signal Processing, 2013, 93: 2872-2878 CrossRef Google Scholar

[19] Qu X M, Xie L H. Source localization by TDOA with random sensor position errors -- part I: static sensors. In: Proceedings of the 15th IEEE International Conference on Information Fusion, Singapore, 2012, 48--53. Google Scholar

[20] Qu X, Xie L. An efficient convex constrained weighted least squares source localization algorithm based on TDOA measurements. Signal Processing, 2016, 119: 142-152 CrossRef Google Scholar

[21] Liu M, Quan T F, Yao T B, et al. Multi-sensor multi-target passive locating and tracking. Acta Electron Sin, 2006, 34: 991--997. Google Scholar

[22] Le Yang , Ho K C. An Approximately Efficient TDOA Localization Algorithm in Closed-Form for Locating Multiple Disjoint Sources With Erroneous Sensor Positions. IEEE Trans Signal Process, 2009, 57: 4598-4615 CrossRef ADS Google Scholar

[23] Ming Sun , Ho K C. An Asymptotically Efficient Estimator for TDOA and FDOA Positioning of Multiple Disjoint Sources in the Presence of Sensor Location Uncertainties. IEEE Trans Signal Process, 2011, 59: 3434-3440 CrossRef ADS Google Scholar

[24] Xiu J J, Wang W S, Sun P. Multiple target passive location of TDOA based on bidirectional elect and nearest neighbor method. J Astronautics, 2015, 36: 483--488. Google Scholar

[25] Sundar H, Sreenivas T V, Seelamantula C S. TDOA-Based Multiple Acoustic Source Localization Without Association Ambiguity. IEEE/ACM Trans Audio Speech Lang Process, 2018, 26: 1976-1990 CrossRef Google Scholar

[26] Luo Z, Ma W, So A. Semidefinite Relaxation of Quadratic Optimization Problems. IEEE Signal Process Mag, 2010, 27: 20-34 CrossRef ADS Google Scholar

• Figure 1

(Color online) Comparison of the localization accuracy in scenario 1 when fixing $\sigma_s$ and varying $\sigma_j$

• Figure 2

(Color online) Comparison of the localization accuracy in scenario 1 when fixing $\sigma_j$ and varying $\sigma_s$

• Figure 3

(Color online) Comparison of the localization accuracy of source 1 in scenario 2

• Figure 4

(Color online) Comparison of the localization accuracy of source 2 in scenario 2

• Table 1   True positions of UAVs in scenario 1 (m)
 UAV number $i$ $x_i^0$ $y_i^0$ $z_i^0$ 1 300 100 150 2 400 150 100 3 300 500 200 4 350 200 100 5 $-$100 $-$100 $-$100 6 200 $-$300 $-$200
•

Algorithm 1 Recursive CWLS multi-source localization algorithm

Input${\boldsymbol~s}$, $r_{j,i1}~(j=1,\ldots,m,~i=1,\ldots,n)$, ${\boldsymbol~Q}_s$, ${\boldsymbol~Q}_t$;

Algorithm process:

Initialize: $k=0$, $\hat{{ u}}^k=({ G}'{G})^{-1}{G}'({ h}-{G}\bar{{ s}}_1)$;

Update: update $\bar{{\boldsymbol~W}}^k=\bar{{\boldsymbol~W}}(\hat{{\boldsymbol~u}}^k)$ and $\bar{{\boldsymbol~h}}^k=\bar{{\boldsymbol~h}}(\hat{{\boldsymbol~u}}^k)$ following (20) and (21), and then update ${\boldsymbol~P}^k={\boldsymbol~P}(\hat{{\boldsymbol~u}}^k)$ following (24) and (25);

Calculate: $\bar{{\boldsymbol~u}}^k=({\boldsymbol~P}^k\bar{{\boldsymbol~W}}^k{\boldsymbol~P}^k)^\dagger\bar{{\boldsymbol~h}}^k$;

$k=k+1$;

Update: $\hat{{\boldsymbol~u}}^k=(\hat{{\boldsymbol~u}}^{k-1}+\bar{{\boldsymbol~u}}^{k-1})/2$;

while $\frac{\|\hat{{\boldsymbol~u}}^{k}-\hat{{\boldsymbol~u}}^{k-1}\|}{\|\hat{{\boldsymbol~u}}^{k}\|}~\leq~\delta$ do

Update: $\bar{{\boldsymbol~W}}^k=\bar{{\boldsymbol~W}}(\hat{{\boldsymbol~u}}^k)$, $\bar{{\boldsymbol~h}}^k=\bar{{\boldsymbol~h}}(\hat{{\boldsymbol~u}}^k)$, ${\boldsymbol~P}^k={\boldsymbol~P}(\hat{{\boldsymbol~u}}^k)$;

Calculate: $\bar{{\boldsymbol~u}}^k=({\boldsymbol~P}^k\bar{{\boldsymbol~W}}^k{\boldsymbol~P}^k)^\dagger\bar{{\boldsymbol~h}}^k$;

$k=k+1$;

Update: $\hat{{\boldsymbol~u}}^k=(\hat{{\boldsymbol~u}}^{k-1}+\bar{{\boldsymbol~u}}^{k-1})/2$;

end while

Output:$\hat{{ u}}^k$, $k.$
• Table 2   The true positions of UAVs in scenario 2 (m)
 UAV number $i$ $x_i^0$ $y_i^0$ $z_i^0$ 1 510 $-$480 30 2 510 520 30 3 $-$490 $-$480 30 4 $-$490 520 30 5 10 20 $30+500\sqrt{2}$

Citations

• #### 0

Altmetric

Copyright 2019 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有