  Miniature-legged robots are constrained by their onboard computation and
control, thus motivating the need for simple, first-principles-based geometric
models that connect \emph{periodic actuation or gaits} (a universal robot
control paradigm) to the induced average locomotion. In this paper, we develop
a \emph{modulable two-beat gait design framework} for sprawled planar
quadrupedal systems under the no-slip using tools from geometric mechanics. We
reduce standard two-beat gaits into unique subgaits in mutually exclusive shape
subspaces. Subgaits are characterized by a locomotive stance phase when limbs
are in ground contact and a non-locomotive, instantaneous swing phase where the
limbs are reset without contact. During the stance phase, the contacting limbs
form a four-bar mechanism. To analyze the ensuing locomotion, we develop the
following tools: (a) a vector field to generate nonslip actuation, (b) the
kinematics of a four-bar mechanism as a local connection, and (c) stratified
panels that combine the kinematics and constrained actuation to encode the net
change in the system's position generated by a stance-swing subgait cycle.
Decoupled subgaits are then designed independently using flows on the
shape-change basis and are combined with appropriate phasing to produce a
two-beat gait. Further, we introduce ``scaling" and ``sliding" control inputs
to continuously modulate the global trajectories of the quadrupedal system in
gait time through which we demonstrate cycle-average speed, direction, and
steering control using the control inputs. Thus, this framework has the
potential to create uncomplicated open-loop gait plans or gain schedules for
robots with limited resources, bringing them closer to achieving autonomous
control.
