  Inertia-dominated mechanical systems can achieve net displacement by 1)
periodically changing their shape (known as kinematic gait) and 2) adjusting
their inertia distribution to utilize the existing nonzero net momentum (known
as momentum gait). Therefore, finding the gait that most effectively utilizes
the two types of locomotion in terms of the magnitude of the net momentum is a
significant topic in the study of locomotion. For kinematic locomotion with
zero net momentum, the geometry of optimal gaits is expressed as the equilibria
of system constraint curvature flux through the surface bounded by the gait,
and the cost associated with executing the gait in the metric space. In this
paper, we identify the geometry of optimal gaits with nonzero net momentum
effects by lifting the gait description to a time-parameterized curve in
shape-time space. We also propose the variational gait optimization algorithm
corresponding to the lifted geometric structure, and identify two distinct
patterns in the optimal motion, determined by whether or not the kinematic and
momentum gaits are concentric. The examples of systems with and without
fluid-added mass demonstrate that the proposed algorithm can efficiently solve
forward and turning locomotion gaits in the presence of nonzero net momentum.
At any given momentum and effort limit, the proposed optimal gait that takes
into account both momentum and kinematic effects outperforms the reference
gaits that each only considers one of these effects.
