The complex dynamic behaviors of legged locomotion on stationary terrain have been extensively analyzed using a simplified dynamic model called the Spring-Loaded Inverted Pendulum (SLIP) model. However, legged locomotion on dynamic platforms has not been thoroughly investigated even by using a simplified dynamic model such as SLIP. In this paper, we present the modeling, analysis, and control of a SLIP model running on dynamic platforms. Three types of dynamic platforms are considered: a) a sinusoidally excited rigid-body platform; b) a spring-supported rigid-body platform; and c) an Euler-Bernoulli beam. These platforms capture some important domains of real-world locomotion terrain (e.g., harmonically excited platforms, suspended floors, and bridges). The interaction force model and the equations of motion of the SLIP-platform systems are derived. Numerical simulations of SLIP running on the three types of dynamic platforms reveal that the platform movement can destabilize the SLIP even when the initial conditions of the SLIP motion are within the domain of attraction of its motion on flat, stationary platforms. A simple control strategy that can sustain the forward motion of a SLIP on dynamic platforms is then synthesized. The effectiveness of the proposed control strategy in sustaining SLIP motion on dynamic platforms is validated through simulations.

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