# Optimal Beacon Placement for Self-Localization Using Three Beacon Bearings

^{1}

^{2}

^{*}

## 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

**Theorem**

**1.**

## 6. Simulation Results

`ga`, which implements a genetic algorithm; and (ii) the MATLAB function

`fminsearch`, which implements a derivative-free method for finding the minimum of an unconstrained multivariable function. The histograms in Figure 3a,b help us visualize the distributions of differences ${|\mathsf{\Phi}|}_{\mathrm{a}}^{*}-{|\mathsf{\Phi}|}_{\mathrm{ga}}^{*}$ and ${|\mathsf{\Phi}|}_{\mathrm{a}}^{*}-{|\mathsf{\Phi}|}_{\mathrm{fmin}}^{*}$, where ${|\mathsf{\Phi}|}_{\mathrm{a}}^{*}$, ${|\mathsf{\Phi}|}_{\mathrm{ga}}^{*}$, and ${|\mathsf{\Phi}|}_{\mathrm{fmin}}^{*}$ are the maximum values of $|\mathsf{\Phi}|$ calculated using the proposed analytical method, the MATLAB function

`ga`, and the MATLAB function

`fminsearch`, respectively. Our observation that ${|\mathsf{\Phi}|}_{\mathrm{a}}^{*}-{|\mathsf{\Phi}|}_{\mathrm{ga}}^{*}$ and ${|\mathsf{\Phi}|}_{\mathrm{a}}^{*}-{|\mathsf{\Phi}|}_{\mathrm{fmin}}^{*}$ are always positive confirms that the maximum value calculated analytically is always larger than the maximum value calculated using either of the other methods.

`fminsearch`implements. For this simulation task, the beacon distances associated with Figure 4 were used, and the angular separations of the beacons were varied. As ${\beta}_{1}$ and ${\beta}_{2}$ were varied from $-\pi $ to $\pi $ at a step size of $\Delta =\frac{\pi}{15}$ radians, $31\times 31=961$ pairs of $({\beta}_{1},{\beta}_{2})$ values were generated. For each $({\beta}_{1},{\beta}_{2})$ value pair, 1000 MLEs were performed to estimate the vehicle states $({\textstyle p},\varphi )$ using AoA measurements corrupted by a zero-mean Gaussian noise with standard deviation $\sigma ={1}^{\xb0}$. The maximum likelihood estimator was initialized at the true vehicle state values to ensure convergence. For each $({\beta}_{1},{\beta}_{2})$ pair, the determinant of the inverse estimation error covariance matrix $|{\Sigma}_{m}^{-1}|$ was calculated and used to produce Figure 5.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof for Theorem 1

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

## 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]

**Figure 3.**Histograms showing the distribution of the errors between analytically maximizing $|\mathsf{\Phi}|$ and numerically maximizing $|\mathsf{\Phi}|$ using (

**a**) the MATLAB function

`ga`or (

**b**) the MATLAB function

`fminsearch`.

**Figure 4.**$|\mathsf{\Phi}|$ as a function of ${\beta}_{1}$ and ${\beta}_{2}$ where $({d}_{1},{d}_{2},{d}_{3})=(31.9025,25.7053,62.9409)$. The maxima $({\beta}_{1}^{*},{\beta}_{2}^{*})=\pm (1.8463,2.1373)$ radians are indicated with ‘×’.

**Figure 5.**The determinant of the inverse estimation covariance $|{\Sigma}_{m}^{-1}|$ plotted as a surface function for a grid of ${\beta}_{1}$ and ${\beta}_{2}$ with resolution $\Delta =\pi /15$. $({d}_{1},{d}_{2},{d}_{3})=(31.9025,25.7053,62.9409)$, $\sigma ={1}^{\xb0}$.

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

McGuire, 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