A Study on Stability of Limit Cycle Walking Model with Feet: Parameter Study

Abstract

In this paper, two kinds of feet, namely, curved and flat feet, are added to limit cycle walking model to investigate its stability properties. Although both models are already proposed and are investigated, most previous works are focused on efficiency and gait behaviors. Only the stability properties of the simplest walking model conceived Garcia et al. are well defined. Therefore, there is still a need for a precise research on the effect of feet, especially in the view of local stability, bifurcation route to chaos, global stability, falling boundary and energy balance line. Therefore, this article revisits the stability analysis of limit cycle walking model with various foot shape. To analyze the effects of feet, we re-derive the equation of motion of modified models by adding one more parameter of foot shape than the simplest walking model. Also, the falling boundary and energy balance line of modified models are derived to get proper initial conditions for stable walking and to explain global stability. Simulation results show us that the curved feet can enlarge both stable walking range and area of basin of attraction while the case of flat feet depends on foot shape parameter.

An Improved Method for Estimating the Domain of Attraction of Passive Biped Walker

“…Some researchers have tried different ways to solve these problems. Jeon et al selected the DOI of the simplest biped walker by calculating its theoretical falling boundary [14], but this method was difficult to apply for those complicated biped walkers which had complex falling boundaries. Zhang et al proposed a bisection method to quickly determine the approximate edge of the simplest passive biped walker's DOA, but this method must know the fixed point at the beginning and the obtained edge was too rough [15].…”

On the stability of passive dynamic walker with flat foot and series ankle spring

“…Through both simulations and experiments, Mizuno et al 19 demonstrated that energy storage device such as a spring or rubber placed between two legs can improve the walking stability of a compass-like PDW walker. Recently, Jeon et al 20 demonstrated that the size of BoA of the simplest walking model with rolling foot or flat foot is small and the maximum slope angle and walking speed that can form stable walking are also very limited.…”

Determination of the initial conditions by solving boundary value problem method for period-one walking of a passive biped walking robots

“…Other researchers proposed a simple search algorithm to find a set of the initial conditions that result in stable limit cycles for various parameter values. 18 Recently, Jeon et al 19 applied falling boundary and energy balance line to estimate the proper initial conditions. These set of initial conditions are within the basin of attraction of the stable gait cycle.…”