Intelligent Service Robotics

, Volume 12, Issue 1, pp 69–86 | Cite as

Range-based relative localization using a fixed number of measurements

  • Lin ZhangEmail author
  • Li Yu
Original Research Paper


Storms block the global positioning system and reduce visibility. The key to success in rescuing a wrecked robot is to find it first; however, in hostile territory, broadcasting distress signals is not an option. In this paper, we design an iterative algorithm for a rescue robot that identifies the relative pose of a wrecked robot using five range measurements and communications at different time points. Just as architects build real frameworks to hold up a building, we use the above information as building blocks to construct a hypothetical geometric framework, which consists of vertices bounded by straight, stiff edges. Then, the relative pose can be calculated from the coordinates of vertices in the framework. Theoretical analysis shows that measuring and communicating four or more times can form only one framework; thus, this framework provides one relative pose. Therefore, we use only one more measurement than the theoretical lower bound. To restrain noise, we suppress the fact that there are many possible frameworks. The algorithm finds a framework whose edge lengths are close to the expectations of the edge lengths of these frameworks. The algorithm is guaranteed to give an error-bounded estimate with an adjustable possibility rather than gradually stabilizing estimates or numerous possible estimates. The entire positioning scheme contains no matrix operations and does not require a polynomial toolbox. Additionally, this approach requires no prior information of the relative pose, no coordinated motion of the two robots, and no parameter must be adjusted through experience. In summary, a robot in distress can be located if five range measurements and communications can be performed between the two robots.


Relative pose estimation Range-only measurement Randomized gradient descent Rigidity Framework Wireless sensor network 



  1. 1.
    Gilmour JH, Godleski JS, Carter RL et al (1992) Aircraft rendezvous using low data rate two-way TACAN bearing information. U.S. Patent 5,128,873Google Scholar
  2. 2.
    Uttam BJ, Amos DH, Covino JM et al (2007) Terrestrial radio-navigation systems. Kayton M, Fried WR Avionics navigation systems, 2nd edn. Wiley: Hoboken, pp 99–177Google Scholar
  3. 3.
    Freitag L, Grund M, Singh S et al (2005) The WHOI micro-modem: An acoustic communications and navigation system for multiple platforms. In: Proceedings of OCEANS MTS/IEEE, Vol. 2, pp 1086–1092Google Scholar
  4. 4.
    Xu B, Xiao YP, Gao W et al (2014) Dual-model reverse CKF algorithm in cooperative navigation for USV. Math Probl Eng
  5. 5.
    Webster SE, Walls JM, Whitcomb LL (2013) Decentralized extended information filter for single-beacon cooperative acoustic navigation: theory and experiments. IEEE Trans Robot 29(4):957–974. CrossRefGoogle Scholar
  6. 6.
    Martinelli A, Siegwart R (2005) Observability analysis for mobile robot localization. In: Proceedings of IEEE/RSJ international conference on intelligent robots and systems, pp 1471–1476Google Scholar
  7. 7.
    Nilsson JO, Handel P (2013) Recursive bayesian initialization of localization based on ranging and dead reckoning. In: Proceedings of IEEE/RSJ international conference on intelligent robots and systems, pp 1399–1404Google Scholar
  8. 8.
    Strader J, Gu Y, Gross JN et al (2016) Cooperative relative localization for moving UAVs with single link range measurements. In: Proceedings of IEEE/ION position, location and navigation symposium, pp 336–343Google Scholar
  9. 9.
    Trawny N, Roumeliotis SI (2010) On the global optimum of planar, range-based robot-to-robot relative pose estimation. In: IEEE international conference on robotics and automation, pp 3200–3206Google Scholar
  10. 10.
    Cornejo A, Nagpal R (2015) Distributed range-based relative localization of robot swarms. Springer Tracts Adv Robot 107:91–107. MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zhou XS, Roumeliotis SI (2008) Robot-to-robot relative pose estimation from range measurements. IEEE Trans Robot 24(6):1379–1393. CrossRefGoogle Scholar
  12. 12.
    Evensen G (2009) Data assimilation: the ensemble Kalman filter, 2nd edn. Springer, BergenCrossRefzbMATHGoogle Scholar
  13. 13.
    Aliev FA, Ozbek L (1999) Evaluation of convergence rate in the central limit theorem for the Kalman filter. IEEE Trans Autom Control 44(10):1905–1909. MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Delyon B (2001) A note on uniform observability. IEEE Trans Autom Control 46(8):1326–1327. MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dulmage J, Cioffi R, Fitz MP et al (2010) Characterization of distance error with received signal strength ranging. In: Proceedings of IEEE wireless communications and networking conference, pp 1–6Google Scholar
  16. 16.
    Jackson B, Jordán T, Szabadka Z (2006) Globally linked pairs of vertices in equivalent realizations of graphs. Discrete Comput Geom 35(3):493–512. MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mesbahi M, Egerstedt M (2010) Graph theoretic methods in multiagent networks. Princeton University Press, Princeton. zbMATHGoogle Scholar
  18. 18.
    Jackson B, Jordán T, Szabadka Z (2014) Globally linked pairs of vertices in rigid frameworks. Robert C. Asia IW and Walter W Rigidity and symmetry. Springer, New York, pp 177–203Google Scholar
  19. 19.
    Gluck H (1975) Almost all simply connected closed surfaces are rigid. In: Proceedings of geometric topology conference, pp 225–239Google Scholar
  20. 20.
    Alfakih AY (2014) Local, dimensional and universal rigidities: a unified gram matrix approach. Robert C. Asia IW and Walter W Rigidity and symmetry. Springer, New York, pp 41–60Google Scholar
  21. 21.
    Steven JG, Alexander DH, Dylan PT (2010) Characterizing generic global rigidity. Am J Math 132(4):897–939. MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jackson B, Jordán T (2005) Connected rigidity matroids and unique realizations of graphs. J Comb Theory Ser B 94(1):1–29. MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jacobs DJ, Hendrickson B (1997) An algorithm for two-dimensional rigidity percolation: the pebble game. J Comput Phys 137(2):346–365. MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
  25. 25.
    Diao Y, Hu G, Marelli D et al (2014) Cooperative localization of a cascading quadrilateral network. In: Proceedings of 11th IEEE international conference on control & automation (ICCA), pp 13–18Google Scholar
  26. 26.
    Eren T, Goldenberg DK, Whiteley W et al (2004) Rigidity, computation, and randomization in network localization. In: Proceedings of twenty-third annual joint conference of the IEEE computer and communications societies, pp 2673–2684Google Scholar
  27. 27.
    Naraghi-Pour M, Rojas GC (2014) A novel algorithm for distributed localization in wireless sensor networks. ACM Trans Sens Netw 11(1):1–25. CrossRefGoogle Scholar
  28. 28.
    Strydom R, Thurrowgood S, Srinivasan MV (2014) Visual odometry: autonomous UAV navigation using optic flow and stereo. In: Proceedings of Australasian conference on robotics & automationGoogle Scholar
  29. 29.
    Mouats T, Aouf N, Chermak L et al (2015) Thermal stereo odometry for UAVs. IEEE Sens J 15(11):6335–6347. CrossRefGoogle Scholar
  30. 30.
    Scaramuzza BD, Fraundorfer F, Fraundorfer BF (2011) IEEE Robot Autom Mag. Visual odometry 18:80–92Google Scholar
  31. 31.
    Papoulis A, Pillai SU (2002) One function of two random variables. In: Probability, random variables, and stochastic processes, 4th edn. The McGraw-Hill Companies, pp 190–192Google Scholar
  32. 32.
    Arfken GB, Weber HJ, Harris FE (2013) Probability and statistics. In: Mathematical methods for physicists, 7th edn. Academic Press, pp 1125–1179.Google Scholar
  33. 33.
    Ploskas N, Samaras N (2016) Parallel computing toolbox. In: GPU program MATLAB, Todd Green, pp 37–70.Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Information EngineeringZhejiang University of TechnologyHangzhouPeople’s Republic of China
  2. 2.Zhejiang University of TechnologyHangzhouPeople’s Republic of China

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