  Contact planning is crucial in locomoting systems.Specifically, appropriate
contact planning can enable versatile behaviors (e.g., sidewinding in limbless
locomotors) and facilitate speed-dependent gait transitions (e.g.,
walk-trot-gallop in quadrupedal locomotors). The challenges of contact planning
include determining not only the sequence by which contact is made and broken
between the locomotor and the environments, but also the sequence of internal
shape changes (e.g., body bending and limb shoulder joint oscillation). Most
state-of-art contact planning algorithms focused on conventional robots
(e.g.biped and quadruped) and conventional tasks (e.g. forward locomotion), and
there is a lack of study on general contact planning in multi-legged robots. In
this paper, we show that using geometric mechanics framework, we can obtain the
global optimal contact sequence given the internal shape changes sequence.
Therefore, we simplify the contact planning problem to a graph optimization
problem to identify the internal shape changes. Taking advantages of the
spatio-temporal symmetry in locomotion, we map the graph optimization problem
to special cases of spin models, which allows us to obtain the global optima in
polynomial time. We apply our approach to develop new forward and sidewinding
behaviors in a hexapod and a 12-legged centipede. We verify our predictions
using numerical and robophysical models, and obtain novel and effective
locomotion behaviors.
