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The International Journal of Robotics Research
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Running in Three Dimensions: Analysis of a Point-mass Sprung-leg Model

Justin E. Seipel

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

Philip Holmes

Department of Mechanical and Aerospace Engineering, and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA

We analyze a simple model for running: a three-dimensional spring-loaded inverted pendulum carrying a point mass (3D-SLIP). Our formulation reduces to the sagittal plane SLIP and horizontal plane lateral leg spring (LLS) models in the appropriate limits. Using the intrinsic geometry and symmetries and appealing to the case of stiff springs, in which gravity may be neglected during stance, we derive an explicit approximate mapping describing stride-to-stride behavior. We thereby show that all left-right symmetric periodic gaits are unstable, deriving a particularly simple mapping for sagittal plane dynamics. Continuation to fixed points for the "exact" mapping confirms instability of these gaits, and we describe a simple feedback stabilization scheme for leg placement at touchdown.

Key Words: legged locomotion • spring loaded inverted pendulum • three-dimensional motions • Poincaré mapping • stability • periodic gaits

The International Journal of Robotics Research, Vol. 24, No. 8, 657-674 (2005)
DOI: 10.1177/0278364905056194
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