nlin.CDOct 1, 2026
SaveThe Effect of Gait Stability Based on Two Types of Impact Strategies for Two-Link Walking and Brachiating Robots
Organizations: Illinois Institute of Technology, Chicago IL 60616, USA
Abstract
In this paper, we explore the impulsive dynamics common to single-joint, two-link models of walking and brachiating gaits with respect to slope and switching time. In particular, we investigate how the stability of a gait and bifurcations encountered within a family of gaits change under time-based and state-based switching of the impulsive dynamics.
Figures & tables
Figure 1 : Example period-one walking and brachiating gaits of a two-link robot.
Figure 2 : (left plot) Curves representing two symmetric gait families projected onto the - (the configuration at ) plane under a TBS strategy. Red represents unstable gaits and blue stable gaits. The black arrows depict unstable gaits on one family converging to stable gaits on the mirrored family. Each curve has NS and FD bifurcation points. Finally, the top-left quadrant shows the model, and the bottom-right quadrant shows an example brachiating gait on a curve. Its mirrored gait (not shown) brachiates in the opposite direction. (right plot) The evolution of the example brachiating gait, which is unstable. Under SBS, it converges to a stable period-one fixed point. With TBS, it converges to a stable period-seven fixed point.
Figure 3 : (top) Example gait families with their stability properties (red unstable, blue stable) and bifurcation points under SBS. (below) A similar plot, but with gaits subject to TBS. To avoid overlapping curves, we plot joint angles beyond a range of on the left axes of each plot. The corresponding values for under modulo arithmetic are shown on the right axes.
Figure 4 : (top left) A brachiating branch of gaits (part of the top-most curve in Figure 3 ) undergoing a PD bifurcation under an SBS strategy. (top right) The same branch undergoing an NS bifurcation under a TBS strategy. (bottom left) The corresponding eigenvalues are displayed for each bifurcation. (bottom right) A state-space projection of the evolution of an unstable period-one gait (red) converging to a closed invariant curve (blue) during the NS bifurcation, with 1,000 impacts plotted (purple).
Figure 5 : Convergence of unstable points after an FD bifurcation for the brachiating branch under a TBS method.
Figure 6 : The evolution of unstable period-one walking gaits (UWGs) along a segment in red after 1,000 impacts (points in cyan) under SBS (gaits evolve along lines of constant slope in the plot). Stable period-one walking and brachiating gaits are in blue. The points in cyan are also stable periodic orbits. The insets show zoomed in regions of interest. For example, the inset of the bottom-left corner shows UWGs converging to stable walking gaits of period- ( ), and the inset in the bottom-right corner shows UWGs converging to stable period- brachiating gaits. Most UWGs converge to stable period-one brachiating gaits.