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1. Time-Varying Foot Placement Control for Humanoid Walking on Swaying Rigid Surface

   Gao, Yuan; Paredes, Victor; Gong, Yukai Gong; He, Zijian; Hereid, Ayonga; Gu, Yan (August 2025, IEEE transactions on robotics)

   Locomotion on dynamic rigid surface (i.e., rigid surface accelerating in an inertial frame) presents complex challenges for controller design, which are essential to address for deploying humanoid robots in dynamic real-world environments such as moving trains, ships, and airplanes. This paper introduces a real-time, provably stabilizing control approach for humanoid walking on periodically swaying rigid surface. The first key contribution is an analytical extension of the classical angular momentum-based linear inverted pendulum model from static to swaying grounds whose motion period may be different than the robot’s gait period. This extension results in a time-varying, nonhomogeneous robot model, which is fundamentally different from the existing pendulum models. We synthesize a discrete footstep control law for the model and derive a new set of sufficient stability conditions that verify the controller’s stabilizing effect. Finally, experiments conducted on a Digit humanoid robot, both in simulations and on hardware, demonstrate the framework’s effectiveness in addressing bipedal locomotion on swaying ground, even under uncertain surface motions and unknown external pushes. 

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2. Time-Varying ALIP Model and Robust Foot-Placement Control for Underactuated Bipedal Robotic Walking on a Swaying Rigid Surface

   https://doi.org/10.23919/ACC55779.2023.10156254  

   Gao, Yuan; Gong, Yukai; Paredes, Victor; Hereid, Ayonga; Gu, Yan (May 2023, Proceedings of the American Control Conference)

   Controller design for bipedal walking on dynamic rigid surfaces (DRSes), which are rigid surfaces moving in the inertial frame (e.g., ships and airplanes), remains largely underexplored. This paper introduces a hierarchical control approach that achieves stable underactuated bipedal walking on a horizontally oscillating DRS. The highest layer of our approach is a real-time motion planner that generates desired global behaviors (i.e., center of mass trajectories and footstep locations) by stabilizing a reduced-order robot model. One key novelty of this layer is the derivation of the reduced-order model by analytically extending the angular momentum based linear inverted pendulum (ALIP) model from stationary to horizontally moving surfaces. The other novelty is the development of a discrete-time foot-placement controller that exponentially stabilizes the hybrid, linear, time-varying ALIP. The middle layer translates the desired global behaviors into the robot’s full-body reference trajectories for all directly actuated degrees of freedom, while the lowest layer exponentially tracks those reference trajectories based on the full-order, hybrid, nonlinear robot model. Simulations confirm that the proposed framework ensures stable walking of a planar underactuated biped under different swaying DRS motions and gait types. 

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3. Analytical solution to a time-varying LIP model for quadrupedal walking on a vertically oscillating surface

   https://doi.org/10.1016/j.mechatronics.2023.103073  

   Iqbal, Amir; Veer, Sushant; Gu, Yan (December 2023, Mechatronics)

   Sun, Weichao; Yao, Bin (Ed.)

   This paper introduces an analytically tractable and computationally efficient model for legged robot dynamics during locomotion on a dynamic rigid surface (DRS), along with an approximate analytical solution and a real-time walking pattern generator synthesized based on the model and solution. By relaxing the static-surface assumption, we extend the classical, time-invariant linear inverted pendulum (LIP) model for legged locomotion on a static surface to dynamic-surface locomotion, resulting in a time-varying LIP model termed as “DRS-LIP”. Sufficient and necessary stability conditions of the time-varying DRS-LIP model are obtained based on the Floquet theory. This model is also transformed into Mathieu’s equation to derive an approximate analytical solution that provides reasonable accuracy with a relatively low computational cost. Using the extended model and its solution, a walking pattern generator is developed to efficiently plan physically feasible trajectories for quadrupedal walking on a vertically oscillating surface. Finally, simulations and hardware experiments from a Laikago quadrupedal robot walking on a pitching treadmill (with a maximum vertical acceleration of 1 m/s ) confirm the accuracy and efficiency of the proposed analytical solution, as well as the efficiency, feasibility, and robustness of the pattern generator, under various surface motions and gait parameters. 

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4. Real-Time Safe Bipedal Robot Navigation using Linear Discrete Control Barrier Functions

   https://doi.org/10.1109/ICRA55743.2025.11128366  

   Peng, Chengyang; Paredes, Victor; Castillo, Guillermo A; Hereid, Ayonga (May 2025, IEEE)

   Safe navigation in real-time is an essential task for humanoid robots in real-world deployment. Since humanoid robots are inherently underactuated thanks to unilateral ground contacts, a path is considered safe if it is obstacle-free and respects the robot's physical limitations and underlying dynamics. Existing approaches often decouple path planning from gait control due to the significant computational challenge caused by the full-order robot dynamics. In this work, we develop a unified, safe path and gait planning framework that can be evaluated online in real-time, allowing the robot to navigate clustered environments while sustaining stable locomotion. Our approach uses the popular Linear Inverted Pendulum (LIP) model as a template model to represent walking dynamics. It incorporates heading angles in the model to evaluate kinematic constraints essential for physically feasible gaits properly. In addition, we leverage discrete control barrier functions (DCBF) for obstacle avoidance, ensuring that the subsequent foot placement provides a safe navigation path within clustered environments. To guarantee real-time computation, we use a novel approximation of the DCBF to produce linear DCBF (LDCBF) constraints. We validate the proposed approach in simulation using a Digit robot in randomly generated environments. The results demonstrate that our approach can generate safe gaits for a nontrivial humanoid robot to navigate environments with randomly generated obstacles in real-time. 

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5. Data-driven Identification of a Non-homogeneous Inverted Pendulum Model for Enhanced Humanoid Control

   https://doi.org/10.1109/Humanoids57100.2023.10375164  

   Hinojosa, Ernesto Hernandez; Bhounsule, Pranav A (December 2023, IEEE)

   This work aims to enhance the linear inverted pendulum model (LIPM) for bipedal robot control. While the LIPM simplifies the dynamics by assuming homogeneity, it fails to capture critical nonlinear dynamics observed in real-world scenarios. To address this limitation, we propose the non-homogeneous LIPM (NH-LIPM), which incorporates a non-homogeneous term in the traditional LIPM dynamics. The NH-LIPM is augmented with controllable inputs, allowing for greater parameter control compared to the LIPM. Through regression analysis and the use of the Recursive Least Squares algorithm with forgetting, we extract and adaptively tune the NH-LIPM parameters. Evaluation through high-fidelity simulation and experimentation on a 30-degree-of-freedom humanoid demonstrates that the NH-LIPM offers improved velocity tracking control, particularly when ankle torque with damping control is added. This model provides a flexible framework for simultaneously controlling the center of mass velocity and position, enabling precise reference tracking and enhanced bipedal locomotion. A video is in this shortened link: http://tiny.cc/NHLIPM 

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