  This study introduces an analytically tractable and computationally efficient
model of the legged robot dynamics associated with locomotion on a dynamic
rigid surface (DRS), and develops a real-time motion planner based on the
proposed model and its analytical solution. This study first theoretically
extends the classical linear inverted pendulum (LIP) model from legged
locomotion on a static surface to DRS locomotion, by relaxing the LIP's
underlying assumption that the surface is static. The resulting model, which we
call "DRS-LIP", is explicitly time-varying. After converting the DRS-LIP into
Mathieu's equation, an approximate analytical solution of the DRS-LIP is
obtained, which is reasonably accurate with a low computational cost.
Furthermore, to illustrate the practical uses of the analytical results, they
are exploited to develop a hierarchical motion planner that efficiently
generates physically feasible trajectories for DRS locomotion. Finally, the
effectiveness of the proposed theoretical results and motion planner is
demonstrated both through PyBullet simulations and experimentally on a Laikago
quadrupedal robot that walks on a rocking treadmill. The videos of simulations
and hardware experiments are available at https://youtu.be/u2Q_u2pR99c.
