Geometric Mechanics of Simultaneous Nonslip Contact in a Planar Quadruped

In this paper, we develop a geometric framework for generating non-slip quadrupedal two-beat gaits. We consider a four-bar mechanism as a surrogate model for a contact state and develop the geometric tools such as shape-change basis to aid in gait generation, local connection as the matrix-equation of motion, and stratified panels to model net locomotion in line with previous work\cite{prasad2023contactswitch}. Standard two-beat gaits in quadrupedal systems like trot divide the shape space into two equal, decoupled subspaces. The subgaits generated in each subspace space are designed independently and when combined with appropriate phasing generate a two-beat gait where the displacements add up due to the geometric nature of the system. By adding ``scaling" and ``sliding" control knobs to subgaits defined as flows over the shape-change basis, we continuously steer an arbitrary, planar quadrupedal system. This exhibits translational anisotropy when modulated using the scaling inputs. To characterize the steering induced by sliding inputs, we define an average path curvature function analytically and show that the steering gaits can be generated using a geometric nonslip contact modeling framework.