Legged Robots

We study the problem of scaling dynamic loco-manipulation from a single free-rolling sphere to multiple spheres transported simultaneously on the back of a wheel-legged quadruped, without fences, grippers, or mechanical stops. Multiple identical free-rolling spheres form an unordered set with no persistent identity: their ordering may change independently at each history frame, creating a \emph{per-frame permutation symmetry} that standard history-concatenation set encoders do not explicitly enforce -- these encoders impose only a shared, diagonal permutation symmetry over the full history. We show that this symmetry mismatch leads to a concrete failure mode in curriculum-based reinforcement learning. Within the same PPO training budget, flat MLPs and branch-wise encoders plateau at or below the two-sphere stage, while a history-concatenation Deep Sets baseline (\HCDS) fails to progress past the two-sphere stage in our runs unless ball-to-slot assignments are randomised during training, suggesting that it exploits slot indices as a curriculum shortcut rather than learning identity-free multi-sphere dynamics. We propose \textbf{Per-Frame Deep Sets (\PFDS)}, which performs permutation-invariant pooling within each history frame before temporal readout; we prove that \PFDS is $\Gframe$-invariant and universally approximates continuous $\Gframe$-invariant policies. A $2{\times}2$ ablation over encoder architecture and slot randomisation separates the architectural and data-augmentation pathways, and \PFDS reaches the five-sphere stage with 100% no-drop transport in simulation across all five random seeds. We further distill the \PFDS teacher into \TactSet via DAgger, replacing privileged sphere-state observations with a $16{\times}16$ Boolean union contact map, yielding a compact and naturally $\Gframe$-invariant tactile representation.

To enable reliable climbing locomotion of quadruped robots on ferromagnetic surfaces, this paper presents a high-load-density electro-permanent magnetic foot with controllable adhesion, featuring force-feedback circular Halbach-net electro-permanent magnet (CHN-EPM) adhesion units and a magnetization control system. Due to its three-dimensional magnetic circuit structure and flux-concentration effect, the CHN-EPM enables a distributed parallel magnetic flux path with enhanced flux utilization, resulting in reduced sensitivity to air-gap variations and allowing effective adhesion to be maintained even under partial contact conditions. The proposed CHN-EPM generates a maximum adhesion force exceeding 1000 N with a load-to-weight ratio over 200:1. A magnetization driver and a two-stage pulse current control strategy are developed to regulate the excitation current amplitude and duration, enabling accurate and reliable magnetization. By incorporating a flexible pressure sensor for contact force feedback, the system can effectively monitor attachment and detachment states, ensuring robust adhesion switching under uncertain contact conditions. The proposed system is integrated into a commercial quadruped robot (Unitree GO2), demonstrating high-load adhesion on ceiling and vertical-wall surfaces and stable locomotion on painted, perforated, and curved ferromagnetic surfaces.

The biological spine of quadrupeds enables sagittal flexion/extension, lateral bending, and axial rotation, playing a crucial role in highly agile and dexterous locomotion. While numerous studies have integrated active spinal joints into quadrupedal robots to enhance agility, most designs simplify control complexity by reducing spinal degrees of freedom (DOF), failing to achieve the spatial tri-axial rotation characteristic of biological spines. Consequently, replicating a multi-DOF biomimetic spine and effectively leveraging it to empower the agile locomotion of quadrupedal robots remains a significant research challenge. In this study, we present S-Cheetah, a quadrupedal robot featuring a 3-DOF bio-inspired serial active spine capable of biomimetic spatial tri-axial rotation. To empower the robot to fully utilize this active spine, we developed a specialized reinforcement learning framework to actively promote the engagement of the introduced spine and maximize the robot's locomotive capabilities by integrating an acceleration curriculum learning strategy with tailored reward functions, such as a gallop gait reward, a spine undulation reward, and a spine steering reward. Experimental results demonstrate that S-Cheetah can achieve a peak speed of 6.9 m/s using the rotary G2 gallop gait and an in-place turning rate of 7.2 rad/s. Besides, the system exhibits an emergent, feline-inspired aerial self-righting capability, allowing it to land stably on four feet from arbitrary orientations during free fall. Finally, through extensive evaluations across diverse locomotion tasks, we prove that the introduction of the proposed 3-DOF spine comprehensively enhances the locomotive agility of quadrupedal robots. Project website: himmy-robotics.github.io/scheetah

Proximal Policy Optimization (PPO) has become the de facto standard for training legged robots, thanks to its robustness and scalability in massively parallel simulation environments like IsaacLab. However, its on-policy nature makes it inherently sample-inefficient, preventing its use for continuous adaptation and fine-tuning on real hardware. Soft Actor-Critic (SAC), by contrast, is an off-policy algorithm that can reuse past experience, making it a natural candidate for sim-to-real transfer workflows where the same algorithm can be used both in simulation and for online learning on the real robot. Despite these advantages, SAC has consistently failed to match PPO's empirical performance in massively parallel training settings. This work identifies the root causes of this gap and introduces targeted modifications, covering policy initialization, timeout-aware critic targets, and multi-step return estimation, that enable SAC to train stably at scale. Evaluated across multiple legged robot platforms and diverse locomotion tasks, our approach closes the performance gap with PPO entirely.

Diffusion models excel at generating diverse and multimodal trajectories for robotic planning, yet their iterative denoising process introduces latency that is incompatible with high-frequency closed-loop control. To address this problem, we propose Dynamic Neural Koopman Distillation, a framework that distills multistep diffusion inference into a single forward pass while retaining the multimodal expressivity of the teacher model. Specifically, we introduce a Factorized Dynamic Koopman layer that models the denoising process through a factorized latent transition with state-dependent modal gains. We evaluate the proposed method on standard D4RL MuJoCo locomotion benchmarks and a physical Kinova manipulator, comparing against one-step baselines. The results show that our method significantly outperforms existing one-step distillation approaches on the reported locomotion tasks, and reduces the inference latency to the millisecond regime compared with the teacher policy. Hardware experiments further demonstrate that our method enables smooth and fast closed-loop execution while maintaining task success and comparable accuracy. A project page is available at https://fdkoopman.github.io/.

We present MuJoCoUni, a downstream MuJoCo distribution for online robot learning and batched physics evaluation. Alongside the open-loop batched trajectory generation already provided by upstream mujoco.rollout, MuJoCoUni supplies runtime primitives for stateful environment execution. The target workloads need high-throughput parallel execution while retaining upstream CPU MuJoCo semantics for models, sensors, contact, and constraints. Its core object, BatchEnvPool, is a C++/pybind11 executor that owns per-environment mjModel copies, per-thread mjData workers, and an internal thread pool. It provides final-state-only short stepping, sparse reset, reset-lifecycle domain randomization, batched sensor forward evaluation without advancing dynamics, and batched Jacobian and height-field queries. The implementation is confined to the Python binding layer; MuJoCo's solver, contact model, integrator, and core source tree retain upstream semantics. This report describes the BatchEnvPool API, implementation boundary, relationship to rollout, and the validation and benchmark scripts shipped with the open-source mujoco-uni package, which is installed with \texttt{pip install mujoco-uni}.

Streaming reinforcement learning has emerged as an online learning paradigm that conforms to the restrictions of natural learning agents that process data incrementally, i.e. with a batch size of 1 and no replay buffer. While streaming RL has recently been shown to scale with deep function approximation with full observability, partially observable settings have remained out of reach. Truncated backpropagation through time collapses to a one-step gradient horizon under the streaming setting, and exact real-time recurrent learning is prohibitively expensive. We close this gap using recurrent trace units, a diagonal recurrent architecture that enables exact RTRL with linear time and memory complexity in the parameter count, and show that they integrate cleanly into existing streaming algorithms across both discrete and continuous control. On a MemoryChain diagnostic with chain lengths from 2 to 128, our method sustains performance where streaming TBPTT(1) baselines using feedforward, GRU, and RTU networks collapse. On five POPGym tasks and on partially observable MuJoCo continuous control, the streaming approach is competitive with batched PPO on POPGym and recovers a substantial fraction of batched performance on masked MuJoCo, despite using no replay buffer or batched updates.

Legged robots carry an IMU, but the inertial solution drifts because consumer-grade IMUs are noisy. However, the feet create intermittent contacts with the environment that can be used to mitigate that drift. This report develops a sequence of increasingly expressive legged robot state estimators that leverage this. In all cases, the floating-base state comprises attitude, position, velocity, and IMU biases. To model foot contacts, we start from the contact-aided invariant EKF of Hartley et al., albeit at a reduced contact update rate. This is then augmented by replacing the measurement update by a small factor graph. Finally, we turn the same factors into a fixed-lag smoother with contact-episode footholds, with and without an evolving IMU bias. To facilitate reproducibility and further research in proprioceptive legged odometry, all four variants are available in GTSAM (Dellaert et. al), and we additionally provide a ROS2-compatible implementation.

One of the significant challenges in legged robotics is achieving accurate odometry using only onboard proprioceptive sensors. In this study, we present a complete leg odometry pipeline based on an Error-State EKF (ESEKF) that relies exclusively on proprioceptive data: a body fixed IMU, joint encoders, and force sensors, where filter's state is corrected by feet determined to be in a stationary stance. The core of our contribution is fused contact detection and an uncertainty quantification module designed to explicitly identify and reject slippage. This module runs two detectors in parallel for each foot, 1) a debounced, force-based Gaussian Mixture Model (GMM) guided Finite State Machine (FSM) to confirm physical contact, and 2) a kinematic-based Generalized Likelihood Ratio Test (GLRT) on the estimated velocity of the foot. The continuous quality scores from both estimators are fused to detect if the foot is both physically loaded and kinematically stationary and served as an uncertainty signal for each contact. To validate our approach, we collected a multi-modal dataset of 29 sequences spanning diverse indoor and outdoor terrains (e.g., concrete, grass, pebble, and rock) total of 2.4 km long. We benchmarked our approach against both proprioceptive and exteroceptive methods. The results demonstrate our method's efficacy in providing accurate odometry estimates, robustly handling slippage-prone environments. We also share our code and real-time ROS2 package as open-source.

Wheeled-legged robots, which have wheels at their feet and achieve high mobility by coordinating wheel drive and leg drive, have been developed. These robots have been developed purely as platforms specialized for locomotion. Therefore, they do not have a means to repurpose their legs for roles other than locomotion, such as object manipulation or tool utilization. In this paper, we address the problem of how to draw out the potential task-execution capability of the legs by freeing them from the roles of locomotion through external body support. To this end, we propose and develop a new robot, WiXus, which fuses a wheeled-legged mechanism with a wire-driven mechanism that utilizes the external environment. The developed WiXus demonstrates not only planar locomotion with wheeled-legged drive, but also three-dimensional mobility such as cliff climbing by coordinating wire-driven and wheeled-legged actuation. Furthermore, by suspending the body with wire-driven actuation, WiXus successfully repurpose its legs as arms to perform object manipulation, (e.g., rescuing a dog (stuffed animal)), and tool utilization (e.g., harvesting an apple (mockup) with loppers). This study demonstrates that the approach of utilizing the environment with wire-driven actuation is a new design principle that extends the operational domain of wheeled-legged robots.

Robust state estimation for highly dynamic motion of legged robots remains challenging, especially in dynamic, contact-rich scenarios. Traditional approaches often rely on binary contact states that fail to capture the nuances of partial contact or directional slippage. This paper presents CoCo-InEKF, a differentiable invariant extended Kalman filter that utilizes continuous contact velocity covariances instead of binary contact states. These learned covariances allow the method to dynamically modulate contact confidence, accounting for more nuanced conditions ranging from firm contact to directional slippage or no contact. To predict these covariances for a set of predefined contact candidate points, we employ a lightweight neural network trained end-to-end using a state-error loss. This approach eliminates the need for heuristic ground-truth contact labels. In addition, we propose an automated contact candidate selection procedure and demonstrate that our method is insensitive to their exact placement. Experiments on a bipedal robot demonstrate a superior accuracy-efficiency tradeoff for linear velocity estimation, as well as improved filter consistency compared to baseline methods. This enables the robust execution of challenging motions, including dancing and complex ground interactions -- both in simulation and in the real world.

Quadruped robots are often designed with rigid feet to simplify control and maintain stable contact during locomotion. While this approach is straightforward, it limits the ability of the legs to absorb impact forces and reuse stored elastic energy, leading to higher energy expenditure during locomotion. To explore whether compliant feet can provide an advantage, we integrate foot compliance into a reinforcement learning (RL) locomotion controller and study its effect on walking efficiency. In simulation, we train eight policies corresponding to eight different spring stiffness values and then cross-evaluate their performance by measuring mechanical energy consumed per meter traveled. In experiments done on a developed quadruped, the energy consumption for the intermediate stiffness spring is lower by ~ 17% when compared to a very stiff or a very flexible spring incorporated in the feet, with similar trends appearing in the simulation results. These results indicate that selecting an appropriate foot compliance can improve locomotion efficiency without destabilizing the robot during motion.

Quadruped robots demonstrate exceptional potential for navigating complex terrain in critical applications such as search and rescue missions and infrastructure inspection However autonomous traversal of confined 3D environments including tunnels caves and collapsed structures remains a significant challenge Existing methods often struggle with rigid gait patterns limited adaptability to diverse geometries and reliance on oversimplified environmental assumptions This paper introduces a Reinforcement Learning RL framework that combines procedural environment generation with policy distillation to enable robust locomotion across various tunnel configurations Our approach leverages a teacher student training paradigm where specialized expert policies trained on procedurally generated tunnel geometries transfer their knowledge to a unified student policy This strategy eliminates the need for complex reward shaping in end-to-end RL training simplifying the process by breaking down complicated tasks into smaller more manageable components that are easier for the robot to learn By synthesizing diverse tunnel structures during training and distilling navigation strategies into a generalizable policy our method achieves consistent traversal across complex spatial constraints where conventional approaches fail We demonstrate through both simulation and real world experiments that our method enables quadruped robots to successfully traverse challenging confined tunnel environments

Wheeled-legged robots hold promise for traversing complex terrains and offer superior mobility compared to legged robots. However, wheeled-legged robots must effectively balance both wheeled driving and legged control. Furthermore, due to noisy proprioceptive sensing and real-world motor constraints, realizing robust and adaptive locomotion at peak performance of motors remains challenging. We propose the Multi-skill Unified Joint Integration of Control Architecture (MUJICA), a unified, fully proprioceptive control framework for wheeled-legged robots that integrates diverse low-level skills-including omnidirectional moving, high platform climbing, and fall recovery-within a single policy. All skills, distinguished by unique indicator variables, are trained jointly with accurate DC-motor constraint modeling. Additionally, a high-level skill selector is learned to dynamically choose the optimal skill based solely on proprioceptions, enabling adaptive responses to the surrounding environment. Therefore, MUJICA enhances sim-to-real robustness and enables seamless transitions across diverse locomotion modes, facilitating autonomous adjustment to the environment. We validate our framework in both simulation and real-world experiments on the Unitree Go2-W robot, demonstrating significant improvements in adaptability and task success in unstructured environments.

Stable, low-latency whole-body teleoperation of humanoid robots is an open research challenge, complicated by kinematic mismatches between human and robot morphologies, accumulated inertial sensor noise, non-trivial control latency, and persistent sim-to-real transfer gaps. This paper presents a complete real-time whole-body teleoperation system that maps human motion, recorded with a Virdyn IMU-based full-body motion capture suit, directly onto a Unitree G1 humanoid robot. We introduce a custom motion-processing, kinematic retargeting, and control pipeline engineered for continuous, low-latency operation without any offline buffering or learning-based components. The system is first validated in simulation using the MuJoCo physics model of the Unitree G1 (sim2sim), and then deployed without modification on the physical platform (sim2real). Experimental results demonstrate stable, synchronized reproduction of a broad motion repertoire, including walking, standing, sitting, turning, bowing, and coordinated expressive full-body gestures. This work establishes a practical, scalable framework for whole-body humanoid teleoperation using commodity wearable motion capture hardware.

We compare three state-of-the-art proprioceptive state estimators for quadruped robots: MUSE [1], the Invariant Extended Kalman Filter (IEKF) [2], and the Invariant Smoother (IS) [3], on the CYN-1 sequence of the GrandTour Dataset [4]. Our goal is to give practitioners clear guidance on accuracy and computation time: we report long-term accuracy (Absolute Trajectory Error, ATE), short-term accuracy (translational and rotational Relative Pose Error, RPE), and per-update computation time on a fixed hardware/software stack. On this dataset, RPEs are broadly similar across methods, while IEKF and IS achieve a lower ATE than MUSE. Runtime results highlight the accuracy-latency trade-offs across the three approaches. In the discussion, we outline the evaluation choices used to ensure a fair comparison and analyze factors that influence short-horizon metrics. Overall, this study provides a concise snapshot of accuracy and cost, helping readers choose an estimator that fits their application constraints, with all evaluation code and documentation released open-source at https://github.com/iit-DLSLab/state_estimation_benchmark for full reproducibility.

Learning dynamic whole-body motions for legged robots through reinforcement learning (RL) remains challenging due to the high risk of failure, which makes efficient exploration difficult and often leads to unstable learning. In this paper, we propose External Force Guided Curriculum Learning (EFGCL), a guided RL approach based on the principle of physical guidance, in which external assistive forces are introduced during training. Inspired by spotting in artistic gymnastics, EFGCL enables agents to physically experience successful motion executions without relying on task-specific reward shaping or reference trajectories. Experiments on a quadrupedal robot performing Jump, Backflip, and Lateral-Flip tasks demonstrate that EFGCL accelerates learning of the Jump task by approximately a factor of two and enables the acquisition of complex whole body motions that conventional RL methods fail to learn. We further show that the learned policies can be deployed on real robot, reproducing motions consistent with those observed in simulation. These results indicate that physically guided exploration, which allows agents to experience success early in training, is an effective and general strategy for improving learning efficiency in dynamic whole-body motion tasks.

We present an active embodiment identification method for legged robots that jointly learns information-seeking behavior and explicit embodiment prediction. Using a history-augmented URMA architecture, the method infers joint-level and global embodiment parameters through interaction with the environment in simulation across different morphologies.

Stochastic trajectory optimization methods like STOMP enable planning with non-differentiable costs, offering substantial flexibility over gradient-based approaches. We show that STOMP implicitly minimizes the KL divergence from a Boltzmann trajectory distribution, revealing an elegant Variational Inference (VI) structure underlying its updates. Building on this insight, we propose the \textit{Proximal Inference for Stochastic Trajectory Optimization} (PISTO) algorithm that stabilizes the updates by augmenting the objective with a KL regularization between successive Gaussian proposals. This proximal formulation admits a trust-region interpretation and yields closed-form mean updates computable as expectations under a surrogate distribution. We estimate these expectations via importance-weighted Monte Carlo sampling, producing a simple, derivative-free algorithm that inherits STOMP's ability to handle non-differentiable and discontinuous costs without modification. On robot arm motion planning benchmarks, PISTO achieves an 89% success rate -- outperforming CHOMP (63%) and STOMP (68%) -- while producing shorter, smoother paths at twice the speed of competing stochastic methods. We further validate PISTO on contact-rich MuJoCo locomotion and manipulation tasks, where it consistently outperforms both CEM and MPPI baselines in reward.

This paper presents a multi-phase whole-body model predictive control (MPC) approach for bipedal walking, combining a detailed whole-body model in the near horizon with a simplified single-rigid-body model in the later prediction steps. This reduces computational complexity while retaining prediction capabilities. The resulting nonlinear optimal control problem is solved entirely within the general-purpose, off-the-shelf nonlinear MPC framework acados, using sequential quadratic programming (SQP). Given a contact schedule and a target walking speed, the controller optimizes joint torques without depending on preselected footstep locations. The controller is validated in MuJoCo simulation on the 18-DoF bipedal robot HyPer-2.

This paper mainly studies the accurate height jumping control of wheeled-bipedal robots based on torque planning and energy consumption optimization. Due to the characteristics of underactuated, nonlinear estimation, and instantaneous impact in the jumping process, accurate control of the wheeled-bipedal robot's jumping height is complicated. In reality, robots often jump at excessive height to ensure safety, causing additional motor loss, greater ground reaction force and more energy consumption. To solve this problem, a novel wheeled-bipedal jumping dynamical model(W-JBD) is proposed to achieve accurate height control. It performs well but not suitable for the real robot because the torque has a striking step. Therefore, the Bayesian optimization for torque planning method(BOTP) is proposed, which can obtain the optimal torque planning without accurate dynamic model and within few iterations. BOTP method can reduce 82.3% height error, 26.9% energy cost with continuous torque curve. This result is validated in the Webots simulation platform. Based on the torque curve obtained in the W-JBD model to narrow the searching space, BOTP can quickly converge (40 times on average). Cooperating W-JBD model and BOTP method, it is possible to achieve the height control of real robots with reasonable times of experiments.

Legged robots maintain dynamic feasibility through multicontact interactions with terrain. Learned foothold prediction can provide feasibility-aware costs for motion planning and path selection, but accurately predicting future contacts from perceptual inputs such as height scans remains challenging on highly unstructured terrain, even with a repetitive gait cycle. In this work, we show that modeling epistemic uncertainty in predicted footholds, conditioned on terrain observations and commanded motion, distinguishes in-distribution from out-of-distribution operating regimes in simulation and real-world settings. This allows a single learned model, trained on limited data distributions, to express uncertainty caused by missing training coverage. We use this learned uncertainty to detect OOD regions and incorporate them into a unified costmap-generation framework for uncertainty-aware path planning. Using these uncertainty-aware costmaps, we evaluate feasibility error across in-distribution and OOD terrains in simulation and real-world settings. The results show improved OOD detection, up to a 37% reduction in simulation feasibility error, and more reliable planning behavior than geometry-only baselines.

Navigating quadruped robots in unstructured 3D environments poses significant challenges, requiring goal-directed motion, effective exploration to escape from local minima, and posture adaptation to traverse narrow, height-constrained spaces. Conventional approaches employ a sequential mapping-planning pipeline but suffer from accumulated perception errors and high computational overhead, restricting their applicability on resource-constrained platforms. To address these challenges, we propose Hierarchical Posture-Adaptive Navigation (HiPAN), a framework that operates directly on onboard depth images at deployment. HiPAN adopts a hierarchical design: a high-level policy generates strategic navigation commands (planar velocity and body posture), which are executed by a low-level, posture-adaptive locomotion controller. To mitigate myopic behaviors and facilitate long-horizon navigation, we introduce Path-Guided Curriculum Learning, which progressively extends the navigation horizon from reactive obstacle avoidance to strategic navigation. In simulation, HiPAN achieves higher navigation success rates and greater path efficiency than classical reactive planners and end-to-end baselines, while real-world experiments further validate its applicability across diverse, unstructured 3D environments.

Wheeled-legged robots combine the efficiency of rolling with the adaptability of legged locomotion, offering unique agility for dynamic environments. However, enabling rapid reflexive evasion remains challenging due to the coexistence of heterogeneous wheel-leg dynamics, hybrid locomotion modes, and non-holonomic constraints. In this work, we investigate how wheeled-legged robots can exploit their hybrid morphology for high-dynamic obstacle avoidance. We propose AWARE, a hierarchical reinforcement learning framework that decomposes avoidance into navigation-avoidance and reflexive-evasion regimes coordinated by a threat-conditioned high-level policy. By learning specialized low-level experts, AWARE autonomously discovers distinct rolling-, stepping-, and hybrid-dominated evasive behaviors, including forward lunges and lateral dodges. Simulation experiments across different reaction times and approach directions, together with real-robot evaluations on the M20 platform, demonstrate improved evasion capability and effective online transition between locomotion regimes. These results highlight the potential of exploiting hybrid wheel-leg actuation for agile and reactive mobility in dynamic environments. Paper homepage: https://aware-ral-2026.github.io/.

Legged robots have demonstrated remarkable agility on rigid, stationary ground, but their locomotion reliability remains limited in non-inertial environments, where the supporting ground moves, tilts, or accelerates. Such conditions arise in ground transportation, maritime platforms, and aerospace settings, and they introduce persistent time-varying disturbances that break the stationary-ground assumptions underlying conventional legged locomotion. This survey reviews the state of the art in modeling, state estimation, and control for legged robots in non-inertial environments. We summarize representative application domains and motion characteristics, analyze the root causes of locomotion performance degradation, and review existing methods together with their key assumptions and limitations. We further identify open problems in robot-environment coupling, observability, robustness, and experimental validation, and discuss future directions in autonomy, system-level design, bio-inspired strategies, safety, and testing. The survey aims to clarify the technical foundations of this emerging area and support the development of reliable legged robots for real-world dynamic environments.

Developing bipedal football robots in dynamiccombat environments presents challenges related to motionstability and deep coupling of multiple tasks, as well ascontrol switching issues between different states such as up-right walking and fall recovery. To address these problems,this paper proposes a modular reinforcement learning (RL)framework for achieving adaptive multi-task control. Firstly,this framework combines an open-loop feedforward oscilla-tor with a reinforcement learning-based feedback residualstrategy, effectively separating the generation of basic gaitsfrom complex football actions. Secondly, a posture-driven statemachine is introduced, clearly switching between the ballseeking and kicking network (BSKN) and the fall recoverynetwork (FRN), fundamentally preventing state interference.The FRN is efficiently trained through a progressive forceattenuation curriculum learning strategy. The architecture wasverified in Unity simulations of bipedal robots, demonstratingexcellent spatial adaptability-reliably finding and kicking theball even in restricted corner scenarios-and rapid autonomousfall recovery (with an average recovery time of 0.715 seconds).This ensures seamless and stable operation in complex multi-task environments.

The physicality of exercise makes the role of athletic trainers unique. Their physical presence allows them to guide a student through a motion, demonstrate an exercise, and give intuitive feedback. Robot quadrupeds are also embodied agents with robust agility and athleticism. In our work, we investigate whether a robot quadruped can serve as an effective and enjoyable personal trainer device. We focus on a case study of interval training for runners: a repetitive, long-horizon task where precision and consistency are important. To meet this challenge, we propose SNOOPIE, an autonomous robot quadruped pacer capable of running interval training exercises tailored to challenge a user's personal abilities. We conduct a set of user experiments that compare the robot trainer to a wearable trainer device--the Apple Watch--to investigate the benefits of a physical embodiment in exercise-based interactions. We demonstrate 60.6% better adherence to a pace schedule and were 45.9% more consistent across their running speeds with the quadruped trainer. Subjective results also showed that participants strongly preferred training with the robot over wearable devices across many qualitative axes, including its ease of use (+56.7%), enjoyability of the interaction (+60.6%), and helpfulness (+39.1%). Additional videos and visualizations can be found on our website: https://sites.google.com/view/snoopie

Kalman filter-based algorithms are fundamental for mobile robots, as they provide a computationally efficient solution to the challenging problem of state estimation. However, they rely on two main assumptions that are difficult to satisfy in practice: (a) the system dynamics must be linear with Gaussian process noise, and (b) the measurement model must also be linear with Gaussian measurement noise. Previous works have extended assumption (a) to nonlinear spaces through the Invariant Extended Kalman Filter (IEKF), showing that it retains properties similar to those of the classical Kalman filter when the system dynamics are group-affine on a Lie group. More recently, the counterpart of assumption (b) for the same nonlinear setting was addressed in [1]. By means of the proposed Iterated Invariant Extended Kalman Filter (IterIEKF), the authors of that work demonstrated that the update step exhibits several compatibility properties of the classical linear Kalman filter. In this work, we introduce a novel open-source state estimation algorithm for legged robots based on the IterIEKF. The update step of the proposed filter relies solely on proprioceptive measurements, exploiting kinematic constraints on foot velocity during contact and base-frame velocity, making it inherently robust to environmental conditions. Through extensive numerical simulations and evaluation on real-world datasets, we demonstrate that the IterIEKF outperforms the vanilla IEKF, the SO(3)-based Kalman Filter, and its iterated variant in terms of both accuracy and consistency.

Embodiment is a significant keyword in recent machine learning fields. This study focused on the passive nature of the body of a biped robot to generate walking and running locomotion using model-based deep reinforcement learning. We constructed two models in a simulator, one with passive elements (e.g., springs) and the other, which is similar to general humanoids, without passive elements. The training of the model with passive elements was highly affected by the attractor of the system. This lead that although the trajectories quickly converged to limit cycles, it took a long time to obtain large rewards. However, thanks to the attractor-driven learning, the acquired locomotion was robust and energy-efficient. The results revealed that robots with passive elements could efficiently acquire high-performance locomotion by utilizing stable limit cycles generated through dynamic interaction between the body and ground. This study demonstrates the importance of implementing passive properties in the body for future embodied AI.

This paper presents a sim-to-real approach that enables legged robots to dynamically manipulate large and heavy objects with whole-body dexterity. Our key insight is that by performing test-time steering of a pre-trained whole-body control policy with a sample-based planner, we can enable these robots to solve a variety of dynamic loco-manipulation tasks. Interestingly, we find our method generalizes to a diverse set of objects and tasks with no additional tuning or training, and can be further enhanced by flexibly adjusting the cost function at test time. We demonstrate the capabilities of our approach through a variety of challenging loco-manipulation tasks on a Spot quadruped robot in the real world, including uprighting a tire heavier than the robot's nominal lifting capacity and dragging a crowd-control barrier larger and taller than the robot itself. Additionally, we show that the same approach can be generalized to humanoid loco-manipulation tasks, such as opening a door and pushing a table, in simulation. Project code and videos are available at https://sumo.rai-inst.com/.

Quadrupedal robots show great potential for valuable real-world applications such as fire rescue and industrial inspection. Such applications often require urgency and the ability to navigate agilely, which in turn demands the capability to change directions smoothly while running in high speed. Existing approaches for agile navigation typically learn a single-goal reaching policy by encouraging the robot to stay at the target position after reaching there. As a result, when the policy is used to reach sequential goals that require changing directions, it cannot anticipate upcoming maneuvers or maintain momentum across the switch of goals, thereby preventing the robot from fully exploiting its agility potential. In this work, we formulate the task as sequential local navigation, extending the single-goal-conditioned local navigation formulation in prior work. We then introduce SmoothTurn, a learning-based control framework that learns to turn smoothly while running rapidly for agile sequential local navigation. The framework adopts a novel sequential goal-reaching reward, an expanded observation space with a lookahead window for future goals, and an automatic goal curriculum that progressively expands the difficulty of sampled goal sequences based on the goal-reaching performance. The trained policy can be directly deployed on real quadrupedal robots with onboard sensors and computation. Both simulation and real-world empirical results show that SmoothTurn learns an agile locomotion policy that performs smooth turning across goals, with emergent behaviors such as controlling momentum when switching goals, facing towards the future goal in advance, and planning efficient paths.

Litter pollution represents a growing environmental problem affecting natural and urban ecosystems worldwide. Waste discarded in public spaces often accumulates in areas that are difficult to access, such as uneven terrains, coastal environments, parks, and roadside vegetation. Over time, these materials degrade and release harmful substances, including toxic chemicals and microplastics, which can contaminate soil and water and pose serious threats to wildlife and human health. Despite increasing awareness of the problem, litter collection is still largely performed manually by human operators, making large-scale cleanup operations labor-intensive, time-consuming, and costly. Robotic solutions have the potential to support and partially automate environmental cleanup tasks. In this work, we present a quadruped robotic system designed for autonomous litter collection in challenging outdoor scenarios. The robot combines the mobility advantages of legged locomotion with a manipulation system consisting of a robotic arm and an onboard litter container. This configuration enables the robot to detect, grasp, and store litter items while navigating through uneven terrains. The proposed system aims to demonstrate the feasibility of integrating perception, locomotion, and manipulation on a legged robotic platform for environmental cleanup tasks. Experimental evaluations conducted in outdoor scenarios highlight the effectiveness of the approach and its potential for assisting large-scale litter removal operations in environments that are difficult to reach with traditional robotic platforms. The code associated with this work can be found at: https://github.com/iit-DLSLab/trash-collection-isaaclab.

Traditional safety-critical control methods, such as control barrier functions, suffer from semantic blindness, exhibiting the same behavior around obstacles regardless of contextual significance. This limitation leads to the uniform treatment of all obstacles, despite their differing semantic meanings. We present Safe-SAGE (Social-Semantic Adaptive Guidance for Safe Engagement), a unified framework that bridges the gap between high-level semantic understanding and low-level safety-critical control through a Poisson safety function (PSF) modulated using a Laplace guidance field. Our approach perceives the environment by fusing multi-sensor point clouds with vision-based instance segmentation and persistent object tracking to maintain up-to-date semantics beyond the camera's field of view. A multi-layer safety filter is then used to modulate system inputs to achieve safe navigation using this semantic understanding of the environment. This safety filter consists of both a model predictive control layer and a control barrier function layer. Both layers utilize the PSF and flux modulation of the guidance field to introduce varying levels of conservatism and multi-agent passing norms for different obstacles in the environment. Our framework enables legged robots to safely navigate semantically rich, dynamic environments with context-dependent safety margins.

Accurate state estimation and multi-modal perception are prerequisites for autonomous legged robots in complex, large-scale environments. To date, no large-scale public legged-robot dataset captures the real-world conditions needed to develop and benchmark algorithms for legged-robot state estimation, perception, and navigation. To address this, we introduce the GrandTour dataset, a multi-modal legged-robotics dataset collected across challenging outdoor and indoor environments, featuring an ANYbotics ANYmal-D quadruped equipped with the Boxi multi-modal sensor payload. GrandTour spans a broad range of environments and operational scenarios across distinct test sites, ranging from alpine scenery and forests to demolished buildings and urban areas, and covers a wide variation in scale, complexity, illumination, and weather conditions. The dataset provides time-synchronized sensor data from spinning LiDARs, multiple RGB cameras with complementary characteristics, proprioceptive sensors, and stereo depth cameras. Moreover, it includes high-precision ground-truth trajectories from satellite-based RTK-GNSS and a Leica Geosystems total station. This dataset supports research in SLAM, high-precision state estimation, and multi-modal learning, enabling rigorous evaluation and development of new approaches to sensor fusion in legged robotic systems. With its extensive scope, GrandTour represents the largest open-access legged-robotics dataset to date. The dataset is available at https://grand-tour.leggedrobotics.com on HuggingFace (ROS-independent), and in ROS formats, along with tools and demo resources.

Reliable odometry for legged robots without cameras or LiDAR remains challenging due to IMU drift and noisy joint velocity sensing. This paper presents a purely proprioceptive state estimator that uses only IMU and motor measurements to estimate body pose and velocity, with a unified formulation applicable to quadruped and wheel-legged robots and extensible to other legged morphologies. The key idea is to treat each reliable contact as a kinematic anchor: joint-torque--based foot wrench estimation selects stance contacts, and the corresponding footfall records provide intermittent world-frame constraints that suppress long-term drift. To prevent elevation drift during extended traversal, we introduce a lightweight height clustering and time-decay correction that snaps newly recorded footfall heights to previously observed support planes. For wheel-legged platforms, the recorded contact is further propagated by effective wheel rolling displacement with shank-motion compensation and a slope-aware rolling direction. To improve foot velocity observations under encoder quantization, we retain an inverse-kinematics cubature Kalman filter as an optional velocity-enhancement module that filters foot-end velocities from joint angles and velocities. The implementation further mitigates yaw drift through multi-contact geometric consistency, which is injected as a soft heading prior rather than as a hard reset of the attitude state. The method is evaluated on four quadruped platforms.

Deployment of legged robots for navigating challenging terrains (e.g., stairs, slopes, and unstructured environments) has gained increasing preference over wheel-based platforms. In such scenarios, accurate odometry estimation is a preliminary requirement for stable locomotion, localization, and mapping. Traditional proprioceptive approaches, which rely on leg kinematics sensor modalities and inertial sensing, suffer from irrepressible vertical drift caused by frequent contact impacts, foot slippage, and vibrations, particularly affected by inaccurate roll and pitch estimation. Existing methods incorporate exteroceptive sensors such as LiDAR or cameras. Further enhancement has been introduced by leveraging gravity vector estimation to add additional observations on roll and pitch, thereby increasing the accuracy of vertical pose estimation. However, these approaches tend to degrade in feature-sparse or repetitive scenes and are prone to errors from double-integrated IMU acceleration. To address these challenges, we propose GaRLILEO, a novel gravity-aligned continuous-time radar-leg-inertial odometry framework. GaRLILEO decouples velocity from the IMU by building a continuous-time ego-velocity spline from SoC radar Doppler and leg kinematics information, enabling seamless sensor fusion which mitigates odometry distortion. In addition, GaRLILEO can reliably capture accurate gravity vectors leveraging a novel soft S2-constrained gravity factor, improving vertical pose accuracy without relying on LiDAR or cameras. Evaluated on a self-collected real-world dataset with diverse indoor-outdoor trajectories, GaRLILEO demonstrates state-of-the-art accuracy, particularly in vertical odometry estimation on stairs and slopes. We open-source both our dataset and algorithm to foster further research in legged robot odometry and SLAM. https://garlileo.github.io/GaRLILEO

This work focuses on sampling strategies of configuration variations for generating robust universal locomotion policies for quadrupedal robots. We investigate the effects of sampling physical robot parameters and joint proportional-derivative gains to enable training a single reinforcement learning policy that generalizes to multiple parameter configurations. Three fundamental joint gain sampling strategies are compared: parameter sampling with (1) linear and polynomial function mappings of mass-to-gains, (2) performance-based adaptive filtering, and (3) uniform random sampling. We improve the robustness of the policy by biasing the configurations using nominal priors and reference models. All training was conducted using the RaiSim simulation environment, tested in simulation on a range of diverse quadrupeds, and zero-shot deployed onto hardware using the ANYmal quadruped robot. Compared to multiple baseline implementations, our results demonstrate the need for significant joint controller gains randomization for robust closing of the sim-to-real gap.

Quadruped mammals coordinate sagittal spinal bending with axial extension and compression during dynamic locomotion. Yet most robotic quadrupeds use rigid trunks, passively compliant spines with fixed properties, or actively controlled spines that track prescribed trajectories. Whether actively regulated spinal compliance can support faster dynamic locomotion remains unclear. We present SPARC, a compact 1.26-kg, 3-DoF sagittal-plane spine that combines revolute and prismatic motion with independently tunable task-space stiffness and damping. A floating-base impedance controller renders the desired task-space compliance, and benchtop tests show that the fitted axial stiffness matches commanded values within 1.5%. We integrate SPARC into an 8-DoF quadruped and evaluate it across 97 bounding trials under three spine configurations: impedance-controlled SPARC, the same SPARC module held near a fixed pose using position control, and a lightweight rigid spine. Impedance-controlled SPARC reaches 1.029 m/s, compared with 0.769 m/s for position-controlled SPARC and 0.673 m/s for the rigid spine. Impedance-controlled SPARC reaches higher speeds with larger axial motion and greater mechanical power exchange, while at matched speed it has a higher electrical cost of transport than the rigid spine, revealing an energetic trade-off. Code and hardware are available at: https://github.com/YueWang996/sparc

In standard reinforcement learning (RL) settings, the interaction between the agent and the environment is typically modeled as a Markov decision process (MDP), which assumes that the agent observes the system state instantaneously, selects an action without delay, and executes it immediately. In real-world dynamic environments, such as cyber-physical systems, this assumption often breaks down due to delays in the interaction between the agent and the system. These delays can vary stochastically over time and are typically unobservable when deciding on an action. Existing methods deal with this uncertainty conservatively by assuming a known fixed upper bound on the delay, even if the delay is often much lower. In this work, we introduce the interaction layer, a general framework that enables agents to adaptively handle unobservable and time-varying delays. Specifically, the agent generates a matrix of possible future actions, anticipating a horizon of potential delays, to handle both unpredictable delays and lost action packets sent over networks. Building on this framework, we develop a model-based algorithm, Actor-Critic with Delay Adaptation (ACDA), which dynamically adjusts to delay patterns. Our method significantly outperforms state-of-the-art approaches across a wide range of locomotion benchmark environments, including real-world measured delays.

Robust and accurate proprioceptive state estimation of the main body is crucial for legged robots to execute tasks in extreme environments where exteroceptive sensors, such as LiDARs and cameras, may become unreliable. In this paper, we propose DogLegs, a state estimation system for legged robots that fuses the measurements from a body-mounted inertial measurement unit (Body-IMU), joint encoders, and multiple leg-mounted IMUs (Leg-IMU) using an extended Kalman filter (EKF). The filter system contains the error states of all IMU frames. The Leg-IMUs are used to detect foot contact, thereby providing zero-velocity measurements to update the state of the Leg-IMU frames. Additionally, we compute the relative position constraints between the Body-IMU and Leg-IMUs by the leg kinematics and use them to update the main body state and reduce the error drift of the individual IMU frames. Field experimental results have shown that our proposed DogLegs system achieves better state estimation accuracy compared to the traditional leg odometry method (using only Body-IMU and joint encoders) across various terrains. We make our code and datasets publicly available to benefit the research community (https://github.com/YibinWu/DogLegs).