[Paper Review] 3D Underactuated Bipedal Walking via H-LIP based Gait Synthesis and Stepping Stabilization

Xiaobin Xiong, Aaron D. Ames|arXiv (Cornell University)|Jan 23, 2021

Robotic Locomotion and Control4 citations

TL;DR

This paper presents a Hybrid-Linear Inverted Pendulum (H-LIP) framework for synthesizing and stabilizing 3D underactuated bipedal walking on the Cassie robot. By modeling step-to-step dynamics as a linear approximation and using state-feedback stepping control, the method achieves robust, dynamic walking through precise step size regulation, validated both in simulation and on hardware with high versatility and stability.

ABSTRACT

In this paper, we holistically present a Hybrid-Linear Inverted Pendulum (H-LIP) based approach for synthesizing and stabilizing 3D foot-underactuated bipedal walking, with an emphasis on thorough hardware realization. The H-LIP is proposed to capture the essential components of the underactuated and actuated part of the robotic walking. The robot walking gait is then directly synthesized based on the H-LIP. We comprehensively characterize the periodic orbits of the H-LIP and provably derive the stepping stabilization via its step-to-step (S2S) dynamics, which is then utilized to approximate the S2S dynamics of the horizontal state of the center of mass (COM) of the robotic walking. The approximation facilities a H-LIP based stepping controller to provide desired step sizes to stabilize the robotic walking. By realizing the desired step sizes, the robot achieves dynamic and stable walking. The approach is fully evaluated in both simulation and experiment on the 3D underactuated bipedal robot Cassie, which demonstrates dynamic walking behaviors with both high versatility and robustness.

Motivation & Objective

• To address the challenge of stabilizing dynamic, 3D underactuated bipedal walking on robots with passive feet and limited actuation.
• To develop a low-dimensional, analytically tractable model (H-LIP) that captures essential dynamics of underactuated gait while enabling systematic gait synthesis.
• To design a stepping controller based on H-LIP's step-to-step dynamics that stabilizes the robot's center of mass by regulating step sizes.
• To validate the approach experimentally on the 3D underactuated robot Cassie, demonstrating robust and versatile dynamic walking.

Proposed method

• Propose the Hybrid-Linear Inverted Pendulum (H-LIP) model as a hybrid, passive-actuated simplification of underactuated bipedal dynamics, with step size as the control input.
• Derive the linear step-to-step (S2S) dynamics of the H-LIP to approximate the S2S dynamics of the actual robot’s center of mass (COM).
• Design a state-feedback H-LIP stepping controller that uses COM position and velocity errors to compute desired step sizes for stabilization.
• Treat model-robot discrepancies as bounded disturbances, ensuring convergence to invariant sets via the S2S approximation.
• Implement the H-LIP-based gait on the Cassie robot using joint-level controllers, with trajectories generated via Bézier polynomials for smooth foot and COM motion.
• Validate the approach through extensive simulation and hardware experiments, including walking on mildly uneven terrain.

Experimental results

Research questions

• RQ1 Can a simplified hybrid model (H-LIP) accurately approximate the step-to-step dynamics of a 3D underactuated bipedal robot?
• RQ2 Can a linear S2S dynamics model of H-LIP enable provably stabilizing stepping control for real robotic systems?
• RQ3 How effectively can H-LIP-based step size regulation stabilize dynamic walking on a real 3D underactuated robot like Cassie?
• RQ4 To what extent can the H-LIP framework be extended to rough or non-flat terrains?
• RQ5 How does the H-LIP approach compare in performance and robustness to existing HZD or SLIP-based methods for underactuated walking?

Key findings

• The H-LIP model successfully captures the essential dynamics of 3D underactuated bipedal walking through a hybrid, passive-actuated formulation with step size as control input.
• The linear S2S dynamics of the H-LIP provide a provably stabilizing stepping controller that regulates step sizes to stabilize the robot’s COM trajectory.
• The H-LIP-based approach enables robust, dynamic walking on the Cassie robot in both simulation and hardware, demonstrating high versatility and stability.
• The method achieves stabilization despite model-robot discrepancies by treating them as bounded disturbances, ensuring convergence to invariant sets.
• The approach is validated on mildly uneven terrain, showing potential for extension to stairs, slopes, and general rough terrains with further vertical COM control.
• The use of Bézier polynomials enables smooth, high-fidelity trajectory tracking for foot and COM motion, supporting stable gait execution.