Robust Dynamic Walking for a 3D Dual-SLIP Model under One-Step Unilateral Stiffness Perturbations: Towards Bipedal Locomotion over Compliant Terrain

TL;DR

Addresses robust bipedal locomotion on compliant terrains and proposes a 3D Dual-SLIP extension using a Hertzian-contact ground model. Compares a standard LQR controller to a biomechanics-inspired stiffness-control strategy that adjusts leg stiffness in response to one-step unilateral perturbations, supported by nonlinear gait optimization and LQR stabilization. Findings show the LQR works only at moderate stiffness (≥200 kN/m) whereas the proposed controller maintains stable walking down to 30 kN/m with significant leg penetration but rapid recovery. The work has practical implications for humanoid robots and prosthetics with tunable stiffness, enabling robust walking over a wide range of compliant surfaces.

Abstract

Bipedal walking is one of the most important hallmarks of human that robots have been trying to mimic for many decades. Although previous control methodologies have achieved robot walking on some terrains, there is a need for a framework allowing stable and robust locomotion over a wide range of compliant surfaces. This work proposes a novel biomechanics-inspired controller that adjusts the stiffness of the legs in support for robust and dynamic bipedal locomotion over compliant terrains. First, the 3D Dual-SLIP model is extended to support for the first time locomotion over compliant surfaces with variable stiffness and damping parameters. Then, the proposed controller is compared to a Linear-Quadratic Regulator (LQR) controller, in terms of robustness on stepping on soft terrain. The LQR controller is shown to be robust only up to a moderate ground stiffness level of 174 kN/m, while it fails in lower stiffness levels. On the contrary, the proposed controller can produce stable gait in stiffness levels as low as 30 kN/m, which results in a vertical ground penetration of the leg that is deeper than 10% of its rest length. The proposed framework could advance the field of bipedal walking, by generating stable walking trajectories for a wide range of compliant terrains useful for the control of bipeds and humanoids, as well as by improving controllers for prosthetic devices with tunable stiffness.

Figures (7)

• Figure 1: The extended 3D Dual-SLIP model in compliant terrain during DS at Touchdown.
• Figure 2: Sagittal plane view of nominal "human-like” CoM trajectory in a full walking step of the 3D Dual-SLIP model.
• Figure 3: Timing of the one-step unilateral low stiffness perturbation. Top and bottom figures illustrate the ground stiffness values underneath the legs $B$ and $A$, respectively. Rigid ground stiffness corresponds to $50 \; MN/m$, while low ground stiffness can take any value lower than that. The color of the label for each gait event indicates the related leg (blue for leg $B$, red for leg $A$).
• Figure 4: Overview of the biomechanics-inspired proposed controller. The model is initiated with the optimal state-control pair ($\bm{x_{0}^{*}}, \bm{u_{0}^{*}}$) to achieve periodic gait with a desired forward velocity at MS. The {MS, TD} and {LH, LO} gait events are identified using SS and DS dynamics, respectively. At every MS event, the LQR controller is implemented and the $\bm{u_{n}}$ feedback law is derived. During the perturbation step ($n=n_{p}$), the stiffness of both legs is initially determined by the LQR controller at MS (Q1) and then it is amplified at TD by the control gains $k_{1}$ and $k_{2}$ (Q2). At the next step ($n=n_{p}+1$), the leg experiencing the perturbation ($A$) maintains an increased stiffness at MS, while the stiffness of the leg about to land on rigid terrain ($B$) is determined based on the LQR controller with no adjustment (Q3). Then, at TD, the stiffness of both legs is not altered (Q4). Finally, the stiffness for both legs is determined based on the LQR for all other steps.
• Figure 5: State error response for stiffness perturbations of 50 $MN/m$, 1 $MN/m$, 500 $kN/m$ and 200 $kN/m$ using the standard LQR controller.