Analyzing Bifurcation, Stability and Chaos for a Passive Walking Biped Model with a Sole Foot

Abstract

Human walking is an action with low energy consumption. Passive walking models (PWMs) can present this intrinsic characteristic. Simplicity in the biped helps to decrease the energy loss of the system. On the other hand, sufficient parts should be considered to increase the similarity of the model’s behavior to the original action. In this paper, the dynamic model for passive walking biped with unidirectional fixed flat soles of the feet is presented, which consists of two inverted pendulums with L-shaped bodies. This model can capture the effects of sole foot in walking. By adding the sole foot, the number of phases of a gait increases to two. The nonlinear dynamic models for each phase and the transition rules are determined, and the stable and unstable periodic motions are calculated. The stability situations are obtained for different conditions of walking. Finally, the bifurcation diagrams are presented for studying the effects of the sole foot. Poincaré section, Lyapunov exponents, and bifurcation diagrams are used to analyze stability and chaotic behavior. Simulation results indicate that the sole foot has such a significant impression on the dynamic behavior of the system that it should be considered in the simple PWMs.

Further Analysis of the Passive Dynamics of the Compass Biped Walker and Control of Chaos via Two Trajectory Tracking Approaches

“…Note that in (29), u(q, _ q, q d , _ q d , € q d ) is the FF + PD controller u(t) defined by expression (23). Moreover, we stress that in this first control approach based on the passive dynamic walking of the compass robot, and relying on the desired model defined by expression (28) to track, the feedforward controller u FF in (24) is zero (u FF � 0) and then has no action on the tracking problem of the period-1 passive gait.…”

“…e first research group that reported such behavior in the passive dynamic walking of the compass-gait walker was that of Prof. Goswami and his coworkers [27,28]. After that, several works have shown the presence of such scenarios in other related biped models, see the review paper [26] and also the recent papers [29][30][31][32][33][34][35][36][37]. Moreover, it was shown in [38] that the passive dynamic walking of the compass robot exhibits the cyclic-fold bifurcation.…”

Frontal-Sagittal Dynamic Coupling in the Optimal Design of a Passive Bipedal Walker

“…With regards to assembling the PBW joint, an element B n which represents a cylindrical portion of material that is removed from the Leg-L is considered; this material portion is geometrically described by the radius B r,o and length L l . Furthermore, to simulate the PBW behavior in sagittal plane, the mass of one leg m sag is calculated in (12), where the vector of the PBW structural element masses is represented as m = [m F , m A , m L , m B , −m Bn , m H , m S ] and the vector 1 ∈ R 7 represents a vector with elements set as one. For the simulation of the frontal plane, the total mass m fro is determined in (13).…”

Modelling and analysis of Parkinsonian gait

“…McGeer et al's early work in passive gait models [48] shows the stability and effect of external inputs on passive gait. Several others have also studied the passive gait models and their bifurcations [10,16,17,23,35,44,46,66,81]. The fully actuated bipedal robots are in the other end of the spectrum where typically feedback linearisation is used to nullify the 'natural' dynamics of the robot [26].…”