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.
Legged robot locomotion on a dynamic rigid surface (i.e., a
rigid surface moving in the inertial frame) involves
complex full-order dynamics that is high-dimensional,
nonlinear, and time-varying. Towards deriving an
analytically tractable dynamic model, this study
theoretically extends the reduced-order linear inverted
pendulum (LIP) model from legged locomotion on a stationary
surface to locomotion on a dynamic rigid surface (DRS). The
resulting model is herein termed as DRS-LIP. Furthermore,
this study introduces an approximate analytical solution of
the proposed DRS-LIP that is computationally efficient with
high accuracy. To illustrate the practical uses of the
analytical results, they are used to develop a hierarchical
planning framework that efficiently generates physically
feasible trajectories for DRS locomotion.
The effectiveness of the proposed theoretical results and
motion planner is demonstrated both through simulations and
experimentally on a Laikago quadrupedal robot that walks on
a rocking treadmill.
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