This paper extends the forced-oscillation-based reduced-order model of walking to a model with ankles and feet. A human-inspired paradigm was designed for the ankle dynamics, which results in improved gait characteristics compared to the point-foot model. In addition, it was shown that while the proposed model can stabilize against large errors in initial conditions through combination of foot placement and ankle strategies, the model is able to stabilize against small perturbations without relying on the foot placement control and solely through the designed proprioceptive ankle scheme. This novel property, which is also observed in humans, can help in better understanding of anthropomorphic walking and its stabilization mechanisms.

Abstract-- Traversing narrow paths is challenging for humanoid robots due to the sparse and safety-critical footholds required. Purely template-based or end-to-end reinforcement learning-based methods suffer from such harsh terrains. This paper proposes a two-stage training framework for such narrow path traversing tasks, coupling a template-based foothold planner with a low-level foothold tracker from Stage-I training and a lightweight perception aided foothold modifier from Stage-II training. With the curriculum setup from flat ground to narrow paths across stages, the resulted controller in turn learns to robustly track and safely modify foothold targets to ensure precise foot placement over narrow paths. This framework preserves the interpretability from the physics-based template and takes advantage of the generalization capability from reinforcement learning, resulting in easy sim-to-real transfer . The learned policies outperform purely template-based or reinforcement learning-based baselines in terms of success rate, centerline adherence and safety margins.

Typically, learned robot controllers are trained via relatively unsystematic regimens and evaluated with coarse-grained outcome measures such as average cumulative reward. The typical approach is useful to compare learning algorithms but provides limited insight into the effects of different training regimens and little understanding about the richness and complexity of learned behaviors. Likewise, human infants and other animals are "trained" via unsystematic regimens, but in contrast, developmental psychologists evaluate their performance in highly-controlled experiments with fine-grained measures such as success, speed of walking, and prospective adjustments. However, the study of learned behavior in human infants is limited by the practical constraints of training and testing babies. Here, we present a case study that applies methods from developmental psychology to study the learned behavior of the simulated bipedal robot Cassie. Following research on infant walking, we systematically designed reinforcement learning training regimens and tested the resulting controllers in simulated environments analogous to those used for babies--but without the practical constraints. Results reveal new insights into the behavioral impact of different training regimens and the development of Cassie's learned behaviors relative to infants who are learning to walk. This interdisciplinary baby-robot approach provides inspiration for future research designed to systematically test effects of training on the development of complex learned robot behaviors.

Traversing risky terrains with sparse footholds poses a significant challenge for humanoid robots, requiring precise foot placements and stable locomotion. Existing approaches designed for quadrupedal robots often fail to generalize to humanoid robots due to differences in foot geometry and unstable morphology, while learning-based approaches for humanoid locomotion still face great challenges on complex terrains due to sparse foothold reward signals and inefficient learning processes. To address these challenges, we introduce BeamDojo, a reinforcement learning (RL) framework designed for enabling agile humanoid locomotion on sparse footholds. BeamDojo begins by introducing a sampling-based foothold reward tailored for polygonal feet, along with a double critic to balancing the learning process between dense locomotion rewards and sparse foothold rewards. To encourage sufficient trail-and-error exploration, BeamDojo incorporates a two-stage RL approach: the first stage relaxes the terrain dynamics by training the humanoid on flat terrain while providing it with task terrain perceptive observations, and the second stage fine-tunes the policy on the actual task terrain. Moreover, we implement a onboard LiDAR-based elevation map to enable real-world deployment. Extensive simulation and real-world experiments demonstrate that BeamDojo achieves efficient learning in simulation and enables agile locomotion with precise foot placement on sparse footholds in the real world, maintaining a high success rate even under significant external disturbances.

This paper presents a Discrete-Time Model Predictive Controller (MPC) for humanoid walking with online footstep adjustment. The proposed controller utilizes a hierarchical control approach. The high-level controller uses a low-dimensional Linear Inverted Pendulum Model (LIPM) to determine desired foot placement and Center of Mass (CoM) motion, to prevent falls while maintaining the desired velocity. A Task Space Controller (TSC) then tracks the desired motion obtained from the high-level controller, exploiting the whole-body dynamics of the humanoid. Our approach differs from existing MPC methods for walking pattern generation by not relying on a predefined foot-plan or a reference center of pressure (CoP) trajectory. The overall approach is tested in simulation on a torque-controlled Humanoid Robot. Results show that proposed control approach generates stable walking and prevents fall against push disturbances.

In this work, we introduce a control framework that combines model-based footstep planning with Reinforcement Learning (RL), leveraging desired footstep patterns derived from the Linear Inverted Pendulum (LIP) dynamics. Utilizing the LIP model, our method forward predicts robot states and determines the desired foot placement given the velocity commands. We then train an RL policy to track the foot placements without following the full reference motions derived from the LIP model. This partial guidance from the physics model allows the RL policy to integrate the predictive capabilities of the physics-informed dynamics and the adaptability characteristics of the RL controller without overfitting the policy to the template model. Our approach is validated on the MIT Humanoid, demonstrating that our policy can achieve stable yet dynamic locomotion for walking and turning. We further validate the adaptability and generalizability of our policy by extending the locomotion task to unseen, uneven terrain. During the hardware deployment, we have achieved forward walking speeds of up to 1.5 m/s on a treadmill and have successfully performed dynamic locomotion maneuvers such as 90-degree and 180-degree turns.

In the context of legged robots, adaptive behavior involves adaptive balancing and adaptive swing foot reflection. While adaptive balancing counteracts perturbations to the robot, adaptive swing foot reflection helps the robot to navigate intricate terrains without foot entrapment. In this paper, we manage to bring both aspects of adaptive behavior to quadruped locomotion by combining RL and MPC while improving the robustness and agility of blind legged locomotion. This integration leverages MPC's strength in predictive capabilities and RL's adeptness in drawing from past experiences. Unlike traditional locomotion controls that separate stance foot control and swing foot trajectory, our innovative approach unifies them, addressing their lack of synchronization. At the heart of our contribution is the synthesis of stance foot control with swing foot reflection, improving agility and robustness in locomotion with adaptive behavior. A hallmark of our approach is robust blind stair climbing through swing foot reflection. Moreover, we intentionally designed the learning module as a general plugin for different robot platforms. We trained the policy and implemented our approach on the Unitree A1 robot, achieving impressive results: a peak turn rate of 8.5 rad/s, a peak running speed of 3 m/s, and steering at a speed of 2.5 m/s. Remarkably, this framework also allows the robot to maintain stable locomotion while bearing an unexpected load of 10 kg, or 83\% of its body mass. We further demonstrate the generalizability and robustness of the same policy where it realizes zero-shot transfer to different robot platforms like Go1 and AlienGo robots for load carrying. Code is made available for the use of the research community at https://github.com/DRCL-USC/RL_augmented_MPC.git

This paper presents an online framework for synthesizing agile locomotion for bipedal robots that adapts to unknown environments, modeling errors, and external disturbances. To this end, we leverage step-to-step (S2S) dynamics which has proven effective in realizing dynamic walking on underactuated robots -- assuming known dynamics and environments. This paper considers the case of uncertain models and environments and presents a data-driven representation of the S2S dynamics that can be learned via an adaptive control approach that is both data-efficient and easy to implement. The learned S2S controller generates desired discrete foot placement, which is then realized on the full-order dynamics of the bipedal robot by tracking desired outputs synthesized from the given foot placement. The benefits of the proposed approach are twofold. First, it improves the ability of the robot to walk at a given desired velocity when compared to the non-adaptive baseline controller. Second, the data-driven approach enables stable and agile locomotion under the effect of various unknown disturbances: additional unmodeled payload, large robot model errors, external disturbance forces, biased velocity estimation, and sloped terrains. This is demonstrated through in-depth evaluation with a high-fidelity simulation of the bipedal robot Cassie subject to the aforementioned disturbances.

A new control paradigm using angular momentum and foot placement as state variables in the linear inverted pendulum model has expanded the realm of possibilities for the control of bipedal robots. This new paradigm, known as the ALIP model, has shown effectiveness in cases where a robot's center of mass height can be assumed to be constant or near constant as well as in cases where there are no non-kinematic restrictions on foot placement. Walking up and down stairs violates both of these assumptions, where center of mass height varies significantly within a step and the geometry of the stairs restrict the effectiveness of foot placement. In this paper, we explore a variation of the ALIP model that allows the length of the virtual pendulum formed by the robot's stance foot and center of mass to follow smooth trajectories during a step. We couple this model with a control strategy constructed from a novel combination of virtual constraint-based control and a model predictive control algorithm to stabilize a stair climbing gait that does not soley rely on foot placement. Simulations on a 20-degree of freedom model of the Cassie biped in the SimMechanics simulation environment show that the controller is able to achieve periodic gait.

Humans can balance very well during walking, even when perturbed. But it seems difficult to achieve robust walking for bipedal robots. Here we describe the simplest balance controller that leads to robust walking for a linear inverted pendulum (LIP) model. The main idea is to use a linear function of the body velocity to determine the next foot placement, which we call linear foot placement control (LFPC). By using the Poincar\'e map, a balance criterion is derived, which shows that LFPC is stable when the velocity-feedback coefficient is located in a certain range. And that range is much bigger when stepping faster, which indicates "faster stepping, easier to balance". We show that various gaits can be generated by adjusting the controller parameters in LFPC. Particularly, a dead-beat controller is discovered that can lead to steady-state walking in just one step. The effectiveness of LFPC is verified through Matlab simulation as well as V-REP simulation for both 2D and 3D walking. The main feature of LFPC is its simplicity and inherent robustness, which may help us understand the essence of how to maintain balance in dynamic walking.