State-of-the-art perceptive Reinforcement Learning controllers for legged robots either (i) impose oscillator or IK-based gait priors that constrain the action space, add bias to the policy optimization and reduce adaptability across robot morphologies, or (ii) operate "blind", which struggle to anticipate hind-leg terrain, and are brittle to noise. In this paper, we propose Phase-Guided Terrain Traversal (PGTT), a perception-aware deep-RL approach that overcomes these limitations by enforcing gait structure purely through reward shaping, thereby reducing inductive bias in policy learning compared to oscillator/IK-conditioned action priors. PGTT encodes per-leg phase as a cubic Hermite spline that adapts swing height to local heightmap statistics and adds a swing-phase contact penalty, while the policy acts directly in joint space supporting morphology-agnostic deployment. Trained in MuJoCo (MJX) on procedurally generated stair-like terrains with curriculum and domain randomization, PGTT achieves the highest success under push disturbances (median +7.5% vs. the next best method) and on discrete obstacles (+9%), with comparable velocity tracking, and converging to an effective policy roughly 2x faster than strong end-to-end baselines. We validate PGTT on a Unitree Go2 using a real-time LiDAR elevation-to-heightmap pipeline, and we report preliminary results on ANYmal-C obtained with the same hyperparameters. These findings indicate that terrain-adaptive, phase-guided reward shaping is a simple and general mechanism for robust perceptive locomotion across platforms.

Abstract-- Vision-based locomotion has shown great promise in enabling legged robots to perceive and adapt to complex environments. However, visual information is inherently fragile, being vulnerable to occlusions, reflections, and lighting changes, which often cause instability in locomotion. Inspired by animal sensorimotor integration, we propose KiVi, a Ki nesthetic-Vi suospatial integration framework, where kinesthetics encodes proprioceptive sensing of body motion and visuospatial reasoning captures visual perception of surrounding terrain. This modality-balanced, yet integrative design, combined with memory-enhanced attention, allows the robot to robustly interpret visual cues while maintaining fallback stability through proprioception. Extensive experiments show that our method enables quadruped robots to stably traverse diverse terrains and operate reliably in unstructured outdoor environments, remaining robust to out-of-distribution (OOD) visual noise and occlusion unseen during training, thereby highlighting its effectiveness and applicability to real-world legged locomotion. Kinesthetic sense and visuospatial perception constitute two fundamental modalities that allow legged animals to achieve effective locomotion.

This paper presents a scalable and adaptive control framework for legged robots that integrates Iterative Learning Control (ILC) with a biologically inspired torque library (TL), analogous to muscle memory. The proposed method addresses key challenges in robotic locomotion, including accurate trajectory tracking under unmodeled dynamics and external disturbances. By leveraging the repetitive nature of periodic gaits and extending ILC to nonperiodic tasks, the framework enhances accuracy and generalization across diverse locomotion scenarios. The control architecture is data-enabled, combining a physics-based model derived from hybrid-system trajectory optimization with real-time learning to compensate for model uncertainties and external disturbances. A central contribution is the development of a generalized TL that stores learned control profiles and enables rapid adaptation to changes in speed, terrain, and gravitational conditions-eliminating the need for repeated learning and significantly reducing online computation. The approach is validated on the bipedal robot Cassie and the quadrupedal robot A1 through extensive simulations and hardware experiments. Results demonstrate that the proposed framework reduces joint tracking errors by up to 85% within a few seconds and enables reliable execution of both periodic and nonperiodic gaits, including slope traversal and terrain adaptation. Compared to state-of-the-art whole-body controllers, the learned skills eliminate the need for online computation during execution and achieve control update rates exceeding 30x those of existing methods. These findings highlight the effectiveness of integrating ILC with torque memory as a highly data-efficient and practical solution for legged locomotion in unstructured and dynamic environments.

Dynamic locomotion of legged robots is a critical yet challenging topic in expanding the operational range of mobile robots. It requires precise planning when possible footholds are sparse, robustness against uncertainties and disturbances, and generalizability across diverse terrains. While traditional model-based controllers excel at planning on complex terrains, they struggle with real-world uncertainties. Learning-based controllers offer robustness to such uncertainties but often lack precision on terrains with sparse steppable areas. Hybrid methods achieve enhanced robustness on sparse terrains by combining both methods but are computationally demanding and constrained by the inherent limitations of model-based planners. To achieve generalized legged locomotion on diverse terrains while preserving the robustness of learning-based controllers, this paper proposes to learn an attention-based map encoding conditioned on robot proprioception, which is trained as part of the end-to-end controller using reinforcement learning. We show that the network learns to focus on steppable areas for future footholds when the robot dynamically navigates diverse and challenging terrains. We synthesize behaviors that exhibit robustness against uncertainties while enabling precise and agile traversal of sparse terrains. Additionally, our method offers a way to interpret the topographical perception of a neural network. We have trained two controllers for a 12-DoF quadrupedal robot and a 23-DoF humanoid robot respectively and tested the resulting controllers in the real world under various challenging indoor and outdoor scenarios, including ones unseen during training.

Reinforcement learning (RL) has demonstrated impressive performance in legged locomotion over various challenging environments. However, due to the sim-to-real gap and lack of explainability, unconstrained RL policies deployed in the real world still suffer from inevitable safety issues, such as joint collisions, excessive torque, or foot slippage in low-friction environments. These problems limit its usage in missions with strict safety requirements, such as planetary exploration, nuclear facility inspection, and deep-sea operations. In this paper, we design a hierarchical optimization-based whole-body follower, which integrates both hard and soft constraints into RL framework to make the robot move with better safety guarantees. Leveraging the advantages of model-based control, our approach allows for the definition of various types of hard and soft constraints during training or deployment, which allows for policy fine-tuning and mitigates the challenges of sim-to-real transfer. Meanwhile, it preserves the robustness of RL when dealing with locomotion in complex unstructured environments. The trained policy with introduced constraints was deployed in a hexapod robot and tested in various outdoor environments, including snow-covered slopes and stairs, demonstrating the great traversability and safety of our approach.

Legged locomotion over various terrains is challenging and requires precise perception of the robot and its surroundings from both proprioception and vision. However, learning directly from high-dimensional visual input is often data-inefficient and intricate. To address this issue, traditional methods attempt to learn a teacher policy with access to privileged information first and then learn a student policy to imitate the teacher's behavior with visual input. Despite some progress, this imitation framework prevents the student policy from achieving optimal performance due to the information gap between inputs. Furthermore, the learning process is unnatural since animals intuitively learn to traverse different terrains based on their understanding of the world without privileged knowledge. Inspired by this natural ability, we propose a simple yet effective method, World Model-based Perception (WMP), which builds a world model of the environment and learns a policy based on the world model. We illustrate that though completely trained in simulation, the world model can make accurate predictions of real-world trajectories, thus providing informative signals for the policy controller. Extensive simulated and real-world experiments demonstrate that WMP outperforms state-of-the-art baselines in traversability and robustness. Videos and Code are available at: https://wmp-loco.github.io/.

Thanks to the explosive developments of data-driven learning methodologies recently, reinforcement learning (RL) emerges as a promising solution to address the legged locomotion problem in robotics. In this manuscript, we propose a novel concurrent teacher-student reinforcement learning architecture for legged locomotion over challenging terrains, based only on proprioceptive measurements in real-world deployment. Different from convectional teacher-student architecture that trains the teacher policy via RL and transfers the knowledge to the student policy through supervised learning, our proposed architecture trains teacher and student policy networks concurrently under the reinforcement learning paradigm. To achieve this, we develop a new training scheme based on conventional proximal policy gradient (PPO) method to accommodate the interaction between teacher policy network and student policy network. The effectiveness of the proposed architecture as well as the new training scheme is demonstrated through extensive indoor and outdoor experiments on quadrupedal robots and point-foot bipedal robot, showcasing robust locomotion over challenging terrains and improved performance compared to two-stage training methods.

While most recent advancements in legged robot control have been driven by model-free reinforcement learning, we explore the potential of differentiable simulation. Differentiable simulation promises faster convergence and more stable training by computing low-variant first-order gradients using the robot model, but so far, its use for legged robot control has remained limited to simulation. The main challenge with differentiable simulation lies in the complex optimization landscape of robotic tasks due to discontinuities in contact-rich environments, e.g., quadruped locomotion. This work proposes a new, differentiable simulation framework to overcome these challenges. The key idea involves decoupling the complex whole-body simulation, which may exhibit discontinuities due to contact, into two separate continuous domains. Subsequently, we align the robot state resulting from the simplified model with a more precise, non-differentiable simulator to maintain sufficient simulation accuracy. Our framework enables learning quadruped walking in minutes using a single simulated robot without any parallelization. When augmented with GPU parallelization, our approach allows the quadruped robot to master diverse locomotion skills, including trot, pace, bound, and gallop, on challenging terrains in minutes. Additionally, our policy achieves robust locomotion performance in the real world zero-shot. To the best of our knowledge, this work represents the first demonstration of using differentiable simulation for controlling a real quadruped robot. This work provides several important insights into using differentiable simulations for legged locomotion in the real world.

Multi-legged robots with six or more legs are not in common use, despite designs with superior stability, maneuverability, and a low number of actuators being available for over 20 years. This may be in part due to the difficulty in modeling multi-legged motion with slipping and producing reliable predictions of body velocity. Here we present a detailed measurement of the foot contact forces in a hexapedal robot with multiple sliding contacts, and provide an algorithm for predicting these contact forces and the body velocity. The algorithm relies on the recently published observation that even while slipping, multi-legged robots are principally kinematic, and employ a friction law ansatz that allows us to compute the shape-change to body-velocity connection and the foot contact forces. This results in the ability to simulate motion plans for a large number of contacts, each potentially with slipping. Furthermore, in homogeneous environments, this kind of simulation can run in (parallel) logarithmic time of the planning horizon.

Discrete and periodic contact switching is a key characteristic of steady-state legged locomotion. This paper introduces a framework for modeling and analyzing this contact-switching behavior through the framework of geometric mechanics on a toy robot model that can make continuous limb swings and discrete contact switches. The kinematics of this model form a hybrid shape-space and by extending the generalized Stokes' theorem to compute discrete curvature functions called \textit{stratified panels}, we determine average locomotion generated by gaits spanning multiple contact modes. Using this tool, we also demonstrate the ability to optimize gaits based on the system's locomotion constraints and perform gait reduction on a complex gait spanning multiple contact modes to highlight the method's scalability to multilegged systems.