Biological Cybernetics

, Volume 99, Issue 3, pp 197–217 | Cite as

Animal navigation: general properties of directed walks

  • Allen CheungEmail author
  • Shaowu Zhang
  • Christian Stricker
  • Mandyam V. Srinivasan
Original Paper


The ability to locomote is a defining characteristic of all animals. Yet, all but the most trivial forms of navigation are poorly understood. Here we report and discuss the analytical results of an in-depth study of a simple navigation problem. In principle, there are two strategies for navigating a straight course. One is to use an external directional reference and to continually reorient with reference to it. The other is to monitor body rotations from internal sensory information only. We showed previously that, at least for simple representations of locomotion, the first strategy will enable an animal or mobile agent to move arbitrarily far away from its starting point, but the second strategy will not do so, even after an infinite number of steps. This paper extends and generalizes the earlier results by demonstrating that these findings are true even when a very general model of locomotion is used. In this general model, error components within individual steps are not independent, and directional errors may be biased. In the absence of a compass, the expected path of a directed walk in general approximates a logarithmic spiral. Some examples are given to illustrate potential applications of the quantitative results derived here. Motivated by the analytical results developed in this work, a nomenclature for directed walks is proposed and discussed. Issues related to path integration in mammals and robots, and measuring the curvature of a noisy path are also addressed using directed walk theory.


Navigation Compass Cumulative errors Sensory noise Motor noise Directed walk Elementary step General Simple Bias Path integration Straightness Curvature Correlated random walk Persistent random walk Logarithmic spiral Dead reckoning Vectorial navigation Central limit theorem Positional moments Mean position Expected position Positional variance Axis of intended locomotion 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Allen Cheung
    • 1
    • 2
    • 3
    Email author
  • Shaowu Zhang
    • 3
  • Christian Stricker
    • 4
  • Mandyam V. Srinivasan
    • 1
    • 2
    • 3
  1. 1.Thinking Systems, Queensland Brain InstituteUniversity of QueenslandBrisbaneAustralia
  2. 2.School of Information Technology and Electrical EngineeringUniversity of QueenslandBrisbaneAustralia
  3. 3.Centre of Excellence in Vision Science, Research School of Biological SciencesAustralian National UniversityCanberraAustralia
  4. 4.The John Curtin School of Medical ResearchAustralian National UniversityCanberraAustralia

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