Estimation of Minimum Stride Frequency for the Frontal Plane Stability of Bipedal Systems

TL;DR

The paper addresses frontal-plane stability in bipedal locomotion under feedforward leg retraction/extension by developing three SLIP-based models (Fixed-Hip, Fixed-Ankle, Free-Ankle-and-Hip) and deriving a minimum stabilizing stride frequency. It introduces a dimensionless framework showing $\hat{\omega}_s \approx \sqrt{\hat{k}}+1$, leading to the practical estimate $\omega_{s,min} \approx \omega_n + \omega_p$ with $\omega_p=\sqrt{g/l}$, and validates this across 50 random models. The study also demonstrates a threshold hip width below which stability vanishes, analyzes the influence of mass, leg stiffness, and leg length, and compares simplified models with an extended 7-DoF model. Finally, it shows that combining feedforward stability with simple PD stance-phase control can substantially reduce the required minimum stride frequency, suggesting a robust, low-energy approach to designing lateral-stable bipeds.

Abstract

Stability of bipedal systems in frontal plane is affected by the hip offset, to the extent that adjusting stride time using feedforward retraction and extension of the legs can lead to stable oscillations without feedback control. This feedforward stabilization can be leveraged to reduce the control effort and energy expenditure and increase the locomotion robustness. However, there is limited understanding of how key parameters, such as mass, stiffness, leg length, and hip width, affect stability and the minimum stride frequency needed to maintain it. This study aims to address these gaps through analyzing how individual model parameters and the system's natural frequency influence the minimum stride frequency required to maintain a stable cycle. We propose a method to predict the minimum stride frequency, and compare the predicted stride frequencies with actual values for randomly generated models. The findings of this work provide a better understanding of the frontal plane stability mechanisms and how feedforward stabilization can be leveraged to reduce the control effort.

Figures (11)

• Figure 1: Schematics of the models used; (a) fixed hip, (b) fixed ankle, and (c) both ankle and hip are free to move. The model parameters and variables: $m_T, m_L,k,b,w,d,l_i,l_m,\theta_{a_i},\theta_{h_i}$ denote torso mass, leg mass, leg stiffness, leg damping, hip width, offset of torso mass from hip center, offset of leg mass from the foot, angle of the ankle joint, and angle of the hip joint, respectively. Index $i$ in the variables refers to the leg index (left or right).
• Figure 2: Profile of the neutral spring length for a stride time of 0.3 seconds and the maximum (stance phase) resting length of $l_0=0.9$ m. The maximum leg retraction is 0.18 m (1/5 of the maximum resting leg length).
• Figure 3: Effect of mass on the minimum stride frequency for stability; (a) fixed hip, and (b) fixed ankle configurations. The mass of each model varies from 50 kg to 100 kg.
• Figure 4: Variation of the minimum stride frequency without and with torso inertia for the fixed-hip and fixed-ankle models.
• Figure 5: Effect of leg stiffness on the minimum stabilizing stride frequency; (a) for hip-fixed and (b) for ankle-fixed configurations.