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1. From Bipedal Walking to Quadrupedal Locomotion: Full-Body Dynamics Decomposition for Rapid Gait Generation

   https://doi.org/10.1109/ICRA40945.2020.9196841  

   Ma, Wen-Loong; Ames, Aaron D. (January 2020, IEEE International Conference on Robotics and Automation (ICRA))

   null (Ed.)

   This paper systematically decomposes a quadrupedal robot into bipeds to rapidly generate walking gaits and then recomposes these gaits to obtain quadrupedal locomotion. We begin by decomposing the full-order, nonlinear and hybrid dynamics of a three-dimensional quadrupedal robot, including its continuous and discrete dynamics, into two bipedal systems that are subject to external forces. Using the hybrid zero dynamics (HZD) framework, gaits for these bipedal robots can be rapidly generated (on the order of seconds) along with corresponding controllers. The decomposition is achieved in such a way that the bipedal walking gaits and controllers can be composed to yield dynamic walking gaits for the original quadrupedal robot - the result is the rapid generation of dynamic quadruped gaits utilizing the full-order dynamics. This methodology is demonstrated through the rapid generation (3.96 seconds on average) of four stepping-in-place gaits and one diagonally symmetric ambling gait at 0.35 m/s on a quadrupedal robot - the Vision 60, with 36 state variables and 12 control inputs - both in simulation and through outdoor experiments. This suggested a new approach for fast quadrupedal trajectory planning using full-body dynamics, without the need for empirical model simplification, wherein methods from dynamic bipedal walking can be directly applied to quadrupeds. 

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2. Real‐Time Remodeling of Granular Terrain for Robot Locomotion

   https://doi.org/10.1002/aisy.202200119  

   Karsai, Andras; Kerimoglu, Deniz; Soto, Daniel; Ha, Sehoon; Zhang, Tingnan; Goldman, Daniel I. (December 2022, Advanced Intelligent Systems)

   Terrain irregularities in natural environments present mobility challenges for autonomous robots and vehicles. Loosely consolidated sandy slopes flow unpredictably when perturbed, often leading to locomotion failure. Systematic experiments with various robot morphologies on flowable terrains feature open‐loop quasistatic gait strategies that remodel the terrain to aid locomotor kinematics. On a sloped terrain of granular media near the critical angle, a laboratory‐scale rover robot induces a flow via a localized fluidization gait to remodel local terrain and succeed in locomotion. A Bayesian optimization machine learning approach that modulates this gait strategy then finds a pattern of selectively fluidizing and solidifying terrain to climb slopes rapidly. In a biped walker robot, a cleated foot design dynamically manipulates the stress fields of flowable slopes. The deeply submerged cleats remodel the shear response of the material by creating jammed regions behind them which then improve forward progression by reducing slip when compared to a flat foot. The “robophysics” approach of systematic experiments exploring terrain reconfiguration combined with future machine learning models of flowable terrain evolution can augment gait discovery for future robots. 

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3. Adaptive Step Duration for Accurate Foot Placement: Achieving Robust Bipedal Locomotion on Terrains with Restricted Footholds

   https://doi.org/10.1109/IROS60139.2025.11246171  

   Xiang, Zhaoyang; Paredes, Victor; Castillo, Guillermo A; Hereid, Ayonga (October 2025, IEEE)

   Traditional one-step preview planning algorithms for bipedal locomotion struggle to generate viable gaits when walking across terrains with restricted footholds, such as stepping stones. To overcome such limitations, this paper introduces a novel multi-step preview foot placement planning algorithm based on the step-to-step discrete evolution of the Divergent Component of Motion (DCM) of walking robots. Our proposed approach adaptively changes the step duration and the swing foot trajectory for optimal foot placement under constraints, thereby enhancing the long-term stability of the robot and significantly improving its ability to navigate environments with tight constraints on viable footholds. We demonstrate its effectiveness through various simulation scenarios with complex stepping-stone configurations and external perturbations. These tests underscore its improved performance for navigating foothold-restricted terrains, even with external disturbances. 

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4. Dynamic Walking: Toward Agile and Efficient Bipedal Robots

   https://doi.org/10.1146/annurev-control-071020-045021  

   Reher, Jenna; Ames, Aaron D. (May 2021, Annual Review of Control, Robotics, and Autonomous Systems)

   null (Ed.)

   Dynamic walking on bipedal robots has evolved from an idea in science fiction to a practical reality. This is due to continued progress in three key areas: a mathematical understanding of locomotion, the computational ability to encode this mathematics through optimization, and the hardware capable of realizing this understanding in practice. In this context, this review outlines the end-to-end process of methods that have proven effective in the literature for achieving dynamic walking on bipedal robots. We begin by introducing mathematical models of locomotion, from reduced-order models that capture essential walking behaviors to hybrid dynamical systems that encode the full-order continuous dynamics along with discrete foot-strike dynamics. These models form the basis for gait generation via (nonlinear) optimization problems. Finally, models and their generated gaits merge in the context of real-time control, wherein walking behaviors are translated to hardware. The concepts presented are illustrated throughout in simulation, and experimental instantiations on multiple walking platforms are highlighted to demonstrate the ability to realize dynamic walking on bipedal robots that is both agile and efficient. 

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5. Template Model Inspired Task Space Learning for Robust Bipedal Locomotion

   https://doi.org/10.1109/IROS55552.2023.10341263  

   Castillo, Guillermo A.; Weng, Bowen; Yang, Shunpeng; Zhang, Wei; Hereid, Ayonga (October 2023, IEEE)

   This work presents a hierarchical framework for bipedal locomotion that combines a Reinforcement Learning (RL)-based high-level (HL) planner policy for the online generation of task space commands with a model-based low-level (LL) controller to track the desired task space trajectories. Different from traditional end-to-end learning approaches, our HL policy takes insights from the angular momentum-based linear inverted pendulum (ALIP) to carefully design the observation and action spaces of the Markov Decision Process (MDP). This simple yet effective design creates an insightful mapping between a low-dimensional state that effectively captures the complex dynamics of bipedal locomotion and a set of task space outputs that shape the walking gait of the robot. The HL policy is agnostic to the task space LL controller, which increases the flexibility of the design and generalization of the framework to other bipedal robots. This hierarchical design results in a learning-based framework with improved performance, data efficiency, and robustness compared with the ALIP model-based approach and state-of-the-art learning-based frameworks for bipedal locomotion. The proposed hierarchical controller is tested in three different robots, Rabbit, a five-link underactuated planar biped; Walker2D, a seven-link fully-actuated planar biped; and Digit, a 3D humanoid robot with 20 actuated joints. The trained policy naturally learns human-like locomotion behaviors and is able to effectively track a wide range of walking speeds while preserving the robustness and stability of the walking gait even under adversarial conditions. 

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