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.

翻译：本文提出了一种用于生成无滑动四足两节拍步态的几何框架。我们将四杆机构视为接触状态的替代模型，并发展了形状变化基等几何工具以辅助步态生成，引入局部连接作为运动方程矩阵，采用分层面板对净运动进行建模，这与先前研究\cite{prasad2023contactswitch}一致。四足系统（如溜蹄步）中的标准两节拍步态将形状空间划分为两个相等且解耦的子空间。每个子空间中生成的子步态独立设计，当与适当相位结合时，可产生一个两节拍步态，其中位移因系统的几何特性而叠加。通过为定义为形状变化基上流动的子步态添加“缩放”和“滑动”控制旋钮，我们能够连续操控任意平面四足系统。当使用缩放输入进行调制时，该系统表现出平移各向异性。为表征由滑动输入引起的转向效应，我们解析定义了平均路径曲率函数，并证明了转向步态可通过几何无滑动接触建模框架生成。