Geometrically Modulable Gait Design for Quadrupeds

TL;DR

This work develops a geometric-mechanics-based framework for designing modulable two-beat gaits in planar quadrupeds under no-slip constraints. It models a trot as two decoupled subgaits, each governed by a four-bar stance mechanism, and uses a local connection and stratified panels to compute net body displacement from infinitesimal shape changes. By exploiting translational invariants and rotational skew-symmetries, the authors construct gaits with controllable average speed, course, and steering using simple flow-control inputs: a scaling input for path-length modulation and a sliding input for steering adjustments. The approach yields offline, open-loop gait primitives and gain-schedules that can be cached for online planners, offering a scalable solution for resource-constrained legged robots and enabling future integration with optimal maneuverability analyses and MPC-based control on miniature platforms.

Abstract

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.

Figures (6)

• Figure 1: Overview: Two-beat quadrupedal gaits (top-left) like trot decouple into subgaits based on shared stance phase (top-right) where each subgait stance phase is a four-bar mechanism. Using geometric mechanics, we solve for the net displacement generated by integrating the stratified panel, a measure of the effective body velocity of the system. Then, by exploiting the design symmetries, we generate steerable two-beat gaits by modulating the subgaits (bottom-right), resulting in real-time gait control (bottom-left) of rigid quadrupeds.
• Figure 2: Geometric mechanics of a four-bar mechanism: (a) System with attached $\mathit{SE}(2)$ frames. (b) The local connection vector fields ($\vec{\mathbf{A}}$) model four-bar mechanism locomotion during the stance phase of feet $g_1$ and $g_2$. (c) The virtual link length (a conserved quantity under the no-slip constraint) is squared ($F$) and plotted as a function of the shape space. Its extrema (inset with configuration) are singular points in the shape space. (d) The derived unit shape velocity basis ($\left( \Delta \alpha \right)_F$, red quivers) corresponds to the nonslip directions tangential to the level sets of $F$. A non-slip gait example (e) with its stance trajectory embedded in (b), (c) (with flow parameters), and (f) produces the system trajectory shown in (g). (f) The stratified panels ($dz$) shown as heatmaps encode effective body velocity from an infinitesimal stance-swing gait cycle in the shape space. The embedded stance trajectory provides a visual representation of each displacement component accrued.
• Figure 3: (a) A generic planar quadrupedal system is defined using parameters for each leg module. For our choice, we obtain a (b) fiducial rigid quadrupedal system used in §\ref{['sec:geom_quad']}.
• Figure 4: Geometric trotting: Forward displacing gait constituted by subgaits ($\phi_{13}$ and $\phi_{24}$) with highlighted stance phases with negative orientation. (a) The contours of $F_{ij}$ in each shape subspace represent the domain for nonslip shape paths. (b) The corresponding robot trajectory. The stratified panels, $dz$ of the two contact states (c) and (d), (refer to (a) for axes information) correspond to anticlockwise trajectories around the inset singularity. The full stratified panel between the two stance paths in reduced shape spaces (e) and as a function of gait phase, $\tau$ in (f).
• Figure 5: Flow control of $\hat{\phi}_{13}$ while keeping $\hat{\phi}_{24}$ fixed: All the gait cycles are derived by modulating the flow control inputs of the fiducial forward gait cycle. The scaling input modifies the path length whereas the sliding input modifies the path location relative to the reference point ($\alpha_{*}$).