Optimal Beacon Placement for Self-Localization Using Three Beacon Bearings
<p>Beacon geometry for self-localization.</p> "> Figure 2
<p>Two equivalent optimal beacon geometries.</p> "> Figure 3
<p>Histograms showing the distribution of the errors between analytically maximizing <math display="inline"><semantics> <mrow> <mo>|</mo> <mo mathvariant="bold-sans-serif">Φ</mo> <mo>|</mo> </mrow> </semantics></math> and numerically maximizing <math display="inline"><semantics> <mrow> <mo>|</mo> <mo mathvariant="bold-sans-serif">Φ</mo> <mo>|</mo> </mrow> </semantics></math> using (<b>a</b>) the MATLAB function <tt>ga</tt> or (<b>b</b>) the MATLAB function <tt>fminsearch</tt>.</p> "> Figure 4
<p><math display="inline"><semantics> <mrow> <mo>|</mo> <mo mathvariant="bold-sans-serif">Φ</mo> <mo>|</mo> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>β</mi> <mn>2</mn> </msub> </semantics></math> where <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>31.9025</mn> <mo>,</mo> <mn>25.7053</mn> <mo>,</mo> <mn>62.9409</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. The maxima <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>β</mi> <mn>1</mn> <mo>*</mo> </msubsup> <mo>,</mo> <msubsup> <mi>β</mi> <mn>2</mn> <mo>*</mo> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mo>±</mo> <mrow> <mo>(</mo> <mn>1.8463</mn> <mo>,</mo> <mn>2.1373</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> radians are indicated with ‘×’.</p> "> Figure 5
<p>The determinant of the inverse estimation covariance <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msubsup> <mo>Σ</mo> <mi>m</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> plotted as a surface function for a grid of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>β</mi> <mn>2</mn> </msub> </semantics></math> with resolution <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>15</mn> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>31.9025</mn> <mo>,</mo> <mn>25.7053</mn> <mo>,</mo> <mn>62.9409</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <msup> <mn>1</mn> <mo>°</mo> </msup> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
- A simplified expression for the determinant of the FIM for vehicle self-localization using AoA measurements for an arbitrary number of beacons.
- An analytical method for calculating angular separations between beacons that satisfy the D-optimality criterion when three beacons are used.
- A mathematical proof that our solution satisfies the sufficient and necessary conditions for optimality.
- Simulations that confirm the optimality of the proposed approach.
2. Related Work
3. Problem Definition
4. Analysis of Mean Square Error and Determinant of Fisher Information Matrix
5. Three Beacons and One Vehicle
6. Simulation Results
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Appendix A. Proof for Theorem 1
References
- Bulusu, N.; Heidemann, J.; Estrin, D. GPS-less low-cost outdoor localization for very small devices. IEEE Pers. Commun. 2000, 7, 28–34. [Google Scholar] [CrossRef] [Green Version]
- Loevsky, I.; Shimshoni, I. Reliable and efficient landmark-based localization for mobile robots. Robot. Auton. Syst. 2010, 58, 520–528. [Google Scholar] [CrossRef]
- Niculescu, D.; Nath, B. Ad hoc positioning system (APS) using AOA. In Proceedings of the INFOCOM 2003, Twenty-Second Annual Joint Conference of the IEEE Computer and Communications, IEEE Societies, San Francisco, CA, USA, 30 March–3 April 2003; Volume 3, pp. 1734–1743. [Google Scholar]
- Bishop, A.N.; Fidan, B.; Anderson, B.D.; Doğançay, K.; Pathirana, P.N. Optimality analysis of sensor-target localization geometries. Automatica 2010, 46, 479–492. [Google Scholar] [CrossRef]
- Doğançay, K.; Hmam, H. Optimal angular sensor separation for AOA localization. Signal Process. 2008, 88, 1248–1260. [Google Scholar] [CrossRef]
- Shimshoni, I. On mobile robot localization from landmark bearings. IEEE Trans. Robot. Autom. 2002, 18, 971–976. [Google Scholar] [CrossRef]
- Doğançay, K. Self-localization from landmark bearings using pseudolinear estimation techniques. IEEE Trans. Aerosp. Electron. Syst. 2014, 50, 2361–2368. [Google Scholar] [CrossRef]
- Esteves, J.S.; Carvalho, A.; Couto, C. Generalized geometric triangulation algorithm for mobile robot absolute self-localization. In Proceedings of the ISIE’03, 2003 IEEE International Symposium on Industrial Electronics, Rio de Janeiro, Brazil, 9–11 June 2003; Volume 1, pp. 346–351. [Google Scholar]
- Melo, J.; Matos, A. Survey on advances on terrain based navigation for autonomous underwater vehicles. Ocean Eng. 2017, 139, 250–264. [Google Scholar] [CrossRef] [Green Version]
- LaPointe, C.E. Virtual Long Baseline (VLBL) Autonomous Underwater Vehicle Navigation Using a Single Transponder. Master’s Thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2006. [Google Scholar]
- Jourdan, D.B.; Dardari, D.; Win, M.Z. Position error bound for UWB localization in dense cluttered environments. IEEE Trans. Aerosp. Electron. Syst. 2008, 44, 613–628. [Google Scholar] [CrossRef]
- Miles, J.; Kamath, G.; Muknahallipatna, S.; Stefanovic, M.; Kubichek, R.F. Optimal trajectory determination of a single moving beacon for efficient localization in a mobile ad-hoc network. Ad Hoc Netw. 2013, 11, 238–256. [Google Scholar] [CrossRef]
- Ucinski, D. Optimal sensor location for parameter estimation of distributed processes. Int. J. Control 2000, 73, 1235–1248. [Google Scholar] [CrossRef]
- Bishop, A.N.; Fidan, B.; Anderson, B.D.; Pathirana, P.N.; Doğançay, K. Optimality analysis of sensor-target geometries in passive localization: Part 2-Time-of-arrival based localization. In Proceedings of the 2007 3rd IEEE International Conference on Intelligent Sensors, Sensor Networks and Information, Melbourne, Australia, 3–6 December 2007; pp. 13–18. [Google Scholar]
- Hammel, S.; Liu, P.; Hilliard, E.; Gong, K. Optimal observer motion for localization with bearing measurements. Comput. Math. Appl. 1989, 18, 171–180. [Google Scholar] [CrossRef] [Green Version]
- Oshman, Y.; Davidson, P. Optimization of observer trajectories for bearings-only target localization. IEEE Trans. Aerosp. Electron. Syst. 1999, 35, 892–902. [Google Scholar] [CrossRef]
- Passerieux, J.M.; Van Cappel, D. Optimal observer maneuver for bearings-only tracking. IEEE Trans. Aerosp. Electron. Syst. 1998, 34, 777–788. [Google Scholar] [CrossRef]
- Zhang, H.; Dufour, F.; Anselmi, J.; Laneuville, D.; Nègre, A. Piecewise optimal trajectories of observer for bearings-only tracking of maneuvering target. In Proceedings of the 2018 IEEE Aerospace Conference, Big Sky, MT, USA, 4–11 March 2018; pp. 1–7. [Google Scholar]
- Sabet, M.; Fathi, A.; Daniali, H.M. Optimal design of the own ship maneuver in the bearing-only target motion analysis problem using a heuristically supervised extended Kalman filter. Ocean Eng. 2016, 123, 146–153. [Google Scholar] [CrossRef]
- Xu, S.; Doğançay, K.; Hmam, H. Distributed path optimization of multiple UAVs for AOA target localization. In Proceedings of the 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Shanghai, China, 20–25 March 2016; pp. 3141–3145. [Google Scholar]
- Wang, W.; Bai, P.; Zhou, Y.; Liang, X.; Wang, Y. Optimal configuration analysis of AOA localization and optimal heading angles generation method for UAV swarms. IEEE Access 2019, 7, 70117–70129. [Google Scholar] [CrossRef]
- Hernandez, M.L. Optimal sensor trajectories in bearings-only tracking. In Proceedings of the Seventh International Conference on Information Fusion, Stockholm, Sweden, 28 June–1 July 2004; Volume 2, pp. 893–900. [Google Scholar]
- Doğançay, K. Single-and multi-platform constrained sensor path optimization for angle-of-arrival target tracking. In Proceedings of the 2010 18th European IEEE Signal Processing Conference, Aalborg, Denmark, 23–27 August 2010; pp. 835–839. [Google Scholar]
- Roh, H.; Cho, M.H.; Tahk, M.J. Trajectory optimization using Cramér-Rao lower bound for bearings-only target tracking. In Proceedings of the 2018 AIAA Guidance, Navigation, and Control Conference, Grapevine, TX, USA, 9–13 January 2018; p. 1591. [Google Scholar]
- Moreno-Salinas, D.; Pascoal, A.; Aranda, J. Sensor networks for optimal target localization with bearings-only measurements in constrained three-dimensional scenarios. Sensors 2013, 13, 10386–10417. [Google Scholar] [CrossRef] [Green Version]
- Xu, S.; Doğançay, K. Optimal sensor placement for 3D angle-of-arrival target localization. IEEE Trans. Aerosp. Electron. Syst. 2017, 53, 1196–1211. [Google Scholar] [CrossRef]
- Zhao, S.; Chen, B.M.; Lee, T.H. Optimal sensor placement for target localisation and tracking in 2D and 3D. Int. J. Control 2013, 86, 1687–1704. [Google Scholar] [CrossRef]
- Ucinski, D. Optimal Measurement Methods for Distributed Parameter System Identification; CRC Press: Boca Raton, FL, USA, 2004. [Google Scholar]
- Dette, H. Designing experiments with respect to ‘standardized’optimality criteria. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 1997, 59, 97–110. [Google Scholar] [CrossRef]
- Betke, M.; Gurvits, L. Mobile robot localization using landmarks. IEEE Trans. Robot. Autom. 1997, 13, 251–263. [Google Scholar] [CrossRef] [Green Version]
- Bernstein, D.S. Matrix Mathematics: Theory, Facts, and Formulas; Princeton University Press: Princeton, NJ, USA, 2009. [Google Scholar]
- Wang, X. A simple proof of Descartes’s rule of signs. Am. Math. Mon. 2004, 111, 525. [Google Scholar] [CrossRef]
- Doğançay, K. Bias compensation for the bearings-only pseudolinear target track estimator. IEEE Trans. Signal Process. 2006, 54, 59–68. [Google Scholar] [CrossRef]
- Yang, P.; Freeman, R.A.; Lynch, K.M. Distributed cooperative active sensing using consensus filters. In Proceedings of the 2007 IEEE International Conference on Robotics and Automation, Roma, Italy, 10–14 April 2007; pp. 405–410. [Google Scholar]
- Chung, T.H.; Gupta, V.; Burdick, J.W.; Murray, R.M. On a decentralized active sensing strategy using mobile sensor platforms in a network. In Proceedings of the 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No. 04CH37601), Nassau, Bahamas, 14–17 December 2004; Volume 2, pp. 1914–1919. [Google Scholar]
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McGuire, J.; Law, Y.W.; Chahl, J.; Doğançay, K. Optimal Beacon Placement for Self-Localization Using Three Beacon Bearings. Symmetry 2021, 13, 56. https://doi.org/10.3390/sym13010056
McGuire J, Law YW, Chahl J, Doğançay K. Optimal Beacon Placement for Self-Localization Using Three Beacon Bearings. Symmetry. 2021; 13(1):56. https://doi.org/10.3390/sym13010056
Chicago/Turabian StyleMcGuire, John, Yee Wei Law, Javaan Chahl, and Kutluyıl Doğançay. 2021. "Optimal Beacon Placement for Self-Localization Using Three Beacon Bearings" Symmetry 13, no. 1: 56. https://doi.org/10.3390/sym13010056