Nonlinear Dynamics

, Volume 62, Issue 4, pp 729–742 | Cite as

Approximate analytic solutions to non-symmetric stance trajectories of the passive Spring-Loaded Inverted Pendulum with damping

  • Uluc̣ Saranlı
  • Ömür Arslan
  • M. Mert Ankaralı
  • Ömer Morgül
Original Paper


This paper introduces an accurate yet analytically simple approximation to the stance dynamics of the Spring-Loaded Inverted Pendulum (SLIP) model in the presence of non-negligible damping and non-symmetric stance trajectories. Since the SLIP model has long been established as an accurate descriptive model for running behaviors, its careful analysis is instrumental in the design of successful locomotion controllers. Unfortunately, none of the existing analytic methods in the literature explicitly take damping into account, resulting in degraded predictive accuracy when they are used for dissipative runners. We show that the methods we propose not only yield average predictive errors below 2% in the presence of significant damping, but also outperform existing alternatives to approximate the trajectories of a lossless model. Finally, we exploit both the predictive performance and analytic simplicity of our approximations in the design of a gait-level running controller, demonstrating their practical utility and performance benefits.


Legged locomotion Hybrid dynamical systems Spring-Loaded Inverted Pendulum Analytic approximations Damping Gait control 


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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Uluc̣ Saranlı
    • 1
  • Ömür Arslan
    • 2
  • M. Mert Ankaralı
    • 3
  • Ömer Morgül
    • 2
  1. 1.Dept. of Computer Eng.Bilkent Univ.Bilkent, AnkaraTurkey
  2. 2.Dept. of Electrical and Electronics Eng.Bilkent Univ.Bilkent, AnkaraTurkey
  3. 3.Dept. of Electrical and Electronics Eng.Middle East Technical Univ.AnkaraTurkey

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