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The Effect of Limb Kinematics on the Speed of a Legged Robot on Granular Media

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An Erratum to this article was published on 30 April 2011

An Erratum to this article was published on 30 April 2011

Abstract

Achieving effective locomotion on diverse terrestrial substrates can require subtle changes of limb kinematics. Biologically inspired legged robots (physical models of organisms) have shown impressive mobility on hard ground but suffer performance loss on unconsolidated granular materials like sand. Because comprehensive limb–ground interaction models are lacking, optimal gaits on complex yielding terrain have been determined empirically. To develop predictive models for legged devices and to provide hypotheses for biological locomotors, we systematically study the performance of SandBot, a small legged robot, on granular media as a function of gait parameters. High performance occurs only in a small region of parameter space. A previously introduced kinematic model of the robot combined with a new anisotropic granular penetration force law predicts the speed. Performance on granular media is maximized when gait parameters utilize solidification features of the granular medium and minimize limb interference.

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Notes

  1. When tripods move in phase (e.g. d c  = 1) this approximation is exact as the body rests on the surface during the swing phase.

  2. In our previous study [11] we measured leg depth to the bottom of the c-leg; here we measure it to the point on the c-leg furthest from the motor axle to simplify the expression for penetration force vs. angle. As mg is comparable to kz for this study, the smallest penetration depth where rotary walking begins is close to the maximum possible value of 2R − h so that the simplified expression for the leg depth is nearly identical to the exact value.

  3. In rotary walking when s < R, a limb encounters material disturbed by its previous step. If the initial volume fraction exceeds the critical value φ ≈ 0.605, disturbed granular material dilates to a lower volume fraction after each step; if initially φ < 0.605, disturbed granular material compacts to a higher volume fraction after each step [19]. The dilation of disturbed granular material above φ ≈ 0.605 results in premature transition from rotary walking to swimming if s < R, as the disturbed granular material is weaker which increases penetration depth and reduces step length. Here we choose φ = 0.605 to ensure that the φ encountered by the leg is unchanged even for s < R.

  4. The previous study did not err in considering \(\overline{v}_x\) as a function of ω alone since for fixed gait parameters ω s scales with ω. The only difference would be Δt →Δt ω s /ω.

  5. For d c  = 1, tripods are in phase so that each c-leg needs to provide just 1/6 the required total force compared to 1/3 when the tripods act independently which could affect the step size. However, we found that for {θ s , θ 0, d c } = {1.5, − 0.5, 1}, decreasing m from 1/3 to 1/6 the body mass left the expression for s unchanged.

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Acknowledgements

We thank Daniel Koditschek, Ryan Maladen, Yang Ding, Nick Gravish, and Predrag Cvitanović for helpful discussion. This work was supported by the Burroughs Wellcome Fund (D.I.G., C.L., and P.B.U.), the Army Research Laboratory (ARL) Micro Autonomous Systems and Technology (MAST) Collaborative Technology Alliance (CTA) under cooperative agreement number W911NF-08-2-0004 (D.I.G. and P.B.U.), and the National Science Foundation (H.K.).

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Authors and Affiliations

  1. School of Physics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    C. Li & D. I. Goldman

  2. Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208, USA

    P. B. Umbanhowar

  3. Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA

    H. Komsuoglu

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  1. C. Li
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  2. P. B. Umbanhowar
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  3. H. Komsuoglu
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  4. D. I. Goldman
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Color images can be found in the online version of this article.

An erratum to this article can be found at http://dx.doi.org/10.1007/s11340-011-9497-9

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Li, C., Umbanhowar, P.B., Komsuoglu, H. et al. The Effect of Limb Kinematics on the Speed of a Legged Robot on Granular Media. Exp Mech 50, 1383–1393 (2010). https://doi.org/10.1007/s11340-010-9347-1

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