Bipedal Locomotion

Combining wheeled and legged locomotion within a single robotic platform offers the potential to harness the advantages of both systems. The development of such hybrid systems remains an active area of research. While legged robots exhibit remarkable adaptability for traversing uneven and rugged terrains, their relatively low speed limits their practicality in indoor environments. Conversely, wheeled robots demonstrate higher efficiency on flat surfaces but often encounter difficulties when navigating obstacles such as steps. This paper introduces OmniQuad, a novel hybrid robot that integrates mecanum wheels with legged locomotion to exploit the benefits of both modalities. Mecanum wheels provide omnidirectional mobility, enhancing the robot's manoeuvrability by enabling agile movement in confined spaces and the ability to maintain a consistent orientation during task execution. In this work, we first present the design of a custom wheeled-foot featuring mecanum wheels. Subsequently, we describe the complete robot assembly, comprising these wheeled feet and EM-Act actuators. Finally, we detail the control scheme developed to govern the OmniQuad and evaluate its performance through experimental trials conducted in both indoor and outdoor environments.

We present RAIWAY, an in-house wheeled bipedal robot with 20-inch wheels, together with a contact-informed control approach for rolling locomotion across irregular and discontinuous terrain. Such locomotion is fundamentally contact-dominated, governed by local wheel--terrain interaction that is impractical to instrument directly and depends on more than terrain geometry alone. The proposed framework exploits the contact information available in simulation, distilling multi-contact events into a latent representation. At deployment, the controller infers this representation online from proprioception, requiring no contact sensing. In real-world experiments, the proposed system traverses diverse environments including steep slopes, gravel beds, studded surfaces, and rocky terrain, and demonstrates continuous ascent on the tested outdoor staircases with riser heights up to 15~cm. On our platform, the proposed controller further outperforms baselines without the contact-informed mechanism across challenging indoor and outdoor settings. The video can be found at https://youtu.be/Ch0divLg64w.

Learning energy-efficient humanoid locomotion requires discovering mechanically economical gait coordination, not merely reducing actuator effort. Reinforcement learning promotes efficiency through effort-related reward penalties, which guide the step-to-step mechanics of walking only indirectly. This article proposes a framework inspired by passive dynamic walking (PDW) that temporarily creates slope-equivalent conditions favorable to economical gait discovery and removes all PDW-specific guidance before nominal-dynamics optimization. During early training, a tilted-gravity field assists sagittal progression on flat collision geometry, complemented by curriculum-coupled reward terms. The core framework requires no reference trajectories, gait phases, or contact schedules. In a five-seed forward-locomotion study on a 29-DoF Unitree G1, the framework reduces mechanical cost of transport by 6.8-15.2% over commanded speeds of 0.5-2.0m/s without degrading velocity tracking. Mechanical-work decomposition attributes the reduction to positive actuator work, and reward-matched comparisons separate the guided regime's faster gait acquisition from the tilt's additional benefit to converged economy. The framework extends to unassisted omnidirectional locomotion, where its benefit persists once a walking-specific motion prior supplies kinematic coordination, the combination reducing speed-matched cost of transport by 18.7%. On hardware, forward cost of transport falls by 16.3% with the motion prior and by 4.5% without it, the latter within the trial-to-trial spread.

General purpose humanoids require locomotion controllers that are multi-skill, perceptive, dynamic, and robust enough to go anywhere humans can. In this work, we present a two layer locomotion architecture: (1) a perceptive flow matching motion generator plans whole body trajectories from raw depth images while a (2) perceptive tracking policy trained with control-guided RL follows these motions. Both policies are trained on a library of terrain consistent motion clips created with dynamically optimized human data which yields both accurate velocity tracking and terrain consistent references. Our central contribution is a simple yet effective off-policy RL fine tuning loop that improves the motion generator. A structured search method is used with the generator to gather data for advantage weighted regression. This off-policy loop is much more sample efficient than on-policy residual fine tuning and improves terrain consistency on unseen geometries and skill compositions. We find that successful terrain traversals increased by up to 25 percentage points and skill selection improved by up to 80 percentage points. By using raw depth images to perceive the environment no odometry or height maps are needed, and outdoor deployment is easy. With two cameras, the policy can see terrain coming from further away and adjust its velocity regardless of the commanded speed so it can traverse the terrain. A single policy pair enables a Unitree G1 humanoid to walk, run, stand, jump on and off of boxes, and traverse stairs in outdoor environments. Project page: https://zolkin1.github.io/generate-track-improve/

Humanoid locomotion over complex terrain requires anticipating footholds that may no longer be visible at touchdown. Limited camera coverage and self-occlusion make it necessary to retrieve relevant terrain information from earlier observations. We present FootQuery, a perceptive locomotion framework that queries depth history using each foot's predicted next touchdown. The policy predicts touchdown locations and uncertainty from proprioception and uses these distributions, together with per-foot features, to query sparsely sampled historical depth frames. During training, realized contacts are projected into historical images to supervise retrieval at the regions where those contacts were visible. The retrieved per-foot features are fused with global visual memory to generate control actions. A progressive force-assistance curriculum supports early exploration, while event-consistent tread-midline shaping encourages coordinated stair contacts. Deployment requires only proprioception and onboard depth images. In simulation, the complete framework outperforms its component ablations on the most challenging tested stairs, gaps, and platforms. Real-world experiments on a Unitree G1 demonstrate continuous traversal with a single policy across outdoor stairs and indoor routes combining stair ascent and descent, platforms, and gaps. These results support organizing visual history around anticipated contacts for perceptive humanoid locomotion.

Low-friction surfaces challenge bipedal locomotion by limiting the contact forces available during stepping. Inspired by penguin waddling, we investigate how lateral torso motion and center of mass (COM) placement affect locomotion as surface friction changes. Using a five-actuator biped, we compare an upright-gait strategy with a penguin-inspired torso-over-stance-leg strategy across multiple COM placements in simulation and hardware. In the 3-D simulator MuJoCo, we sweep through sinusoidal leg and hip actuation parameters across four friction coefficients mu = 0.1, 0.3, 0.5, 0.7. In simulation, torso-over-stance-leg motion produces more successful controllers and higher forward speeds at low friction, with the highest speed occurring for the high-COM configuration. Hardware experiments show the same low-friction speed trend: at mu=0.12, torso-over-stance-leg motion increases forward speed and reduces cost of transport at both tested COM ratios, and the higher COM also improves both measures. The high-COM penguin configuration is the fastest and most energy efficient while maintaining low sideways foot motion. At mu=0.45, the COM trend reverses: the lower-COM configurations are faster and more energy efficient, while gait strategy has little effect on forward speed but still changes sideways foot motion. These results show that the effects of lateral torso motion and COM placement depend on the available friction, and that forward speed, energy use, and slip-related foot motion can be modulated with a penguin-inspired torso motion on hardware.

This paper presents the kinematic and dynamic modeling, trajectory generation, and stability analysis of an 8-degree-of-freedom (DOF) biped robot walking on flat and inclined terrain. Denavit-Hartenberg (DH) parameters and homogeneous transformations are used to derive the forward kinematics, while closed-form inverse kinematics maps the desired hip and swing-foot Cartesian trajectories, generated with cubic splines, to joint angles. Joint torques are computed using the Newton-Euler iterative algorithm, and dynamic stability is evaluated using the zero moment point (ZMP) criterion. A genetic algorithm (GA) optimizes the hip height, maximum swing-foot lift, and frontal-plane tilt angle by minimizing the work done by the joints subject to a ZMP feasibility penalty. Simulation results in MATLAB show that the nominal 8-DOF model remains ZMP-stable for step completion times down to 0.5 s and for slope inclinations up to 22.5 degrees with the given foot geometry. Beyond these limits, the ZMP leaves the support polygon, and either the foot dimensions or the trajectory parameters must be modified. The results also show that ZMP stability is governed by the mass distribution among the links rather than the total mass of the robot.

Enabling robots to walk on yielding terrain is vital for applications ranging from disaster response to planetary exploration. While bipedal robots hold immense potential, their locomotion on deformable surfaces remains limited as current simulators fail to capture the spatiotemporal heterogeneity of such yielding substrates. We present MILD, featuring a physics-grounded discrete-element contact solver that accurately simulates spatially varying foot-terrain interactions. Complementing this model, we train a terrain-aware locomotion controller via deep reinforcement learning with latent modulation and proprioceptive estimation. Quantitative comparisons against state-of-the-art methods show our approach generates more diverse and realistic contact scenarios during training, resulting in controllers that exhibit natural adaptation on real deformable surfaces. Through hardware experiments, we demonstrate the system's capability for online terrain identification and adaptation across a wide range of surface stiffness.

Bipedal robots are challenging to control because they operate close to instability, where small variations in foot-terrain contact can rapidly destabilize locomotion. On rigid terrain, bipedal robots mitigate this fragility by using well-established contact mechanics and control strategies. On flowable surfaces such as granular slopes, foot contact can induce large surface deformations and solid-fluid-like transitions, coupling terrain effects with robot dynamics, leading to underperformance or failure. This is partly due to the lack of reliable methods to represent the dynamics of flowable terrain, making it difficult to account for terrain effects in locomotion design. Here, we investigate how controlling terrain response can improve bipedal locomotion on granular slopes by studying the terradynamics of cleated feet, thin plates emanating from the foot soles. Systematic studies of a small-scale (1.4 kg) robophysical biped reveal that cleats with sparse and dense spacing lead to excessive terrain yielding and resistance, respectively, degrading performance and leading to failure. An intermediate cleat spacing distributes interaction forces to maintain substrate stresses near (or below) the yield threshold, enabling walking on granular slopes up to 30 degrees. Guided by these principles, we design a foot that actively adjusts cleat depth and accommodates both rigid and granular terrain. We also demonstrate that the principles of effective foot-terrain interaction translate to a larger (15 kg) autonomous biped. Our study presents an alternative to conventional body-centric robot control approaches, which regulate terrain-induced effects through body motion, by instead regulating terrain interactions through limb-centric approach.

Wheeled-legged robots possess both agile mobility for traversing complex terrains and high efficiency, making them suitable for long-distance transportation applications. Conventional actuated wheeled robots require specialized hardware and electrical design due to the incorporation of wheel components. We propose a novel and low-cost passive wheeled legged robot equipped with standard casters on each leg to obtain omnidirectional mobility. The control method employs an asymmetric actor-critic structure, enabling the utilization of the privileged information of the passive caster's angles and velocities. We develop a caster base posture adjustment strategy based on velocity commands, utilizing actuated joints to modify the caster base joint axis posture and thereby adjust the propulsion direction of the casters. Moreover, we implemented multiple propulsion modes to achieve varying degrees of caster twisting oscillation, converting these into propulsive force. We conducted a slalom test and mode switch experience, which shows the passive wheeled quadruped could achieve omnidirectional movement versatility, and reduce the cost of transport (COT) by up to 89.1% with respect to legged motion.

We propose a unified reinforcement learning framework that enables a single policy to perform walking, running, and fall recovery on the Unitree G1 humanoid robot, validated on physical hardware without any explicit mode-switching command at deployment. The framework extends Adversarial Motion Priors (AMP) by replacing the conventional global reference distribution with a state-dependent gate that routes each training transition to one of two discriminators: a dedicated recovery discriminator and a velocity-conditioned locomotion discriminator that jointly covers walking and running. The gate is defined by a single fixed threshold on projected gravity: the recovery discriminator is activated when body tilt exceeds approximately $37^\circ$ from vertical ($|g_z+1|>0.6$); otherwise the locomotion discriminator is used, with the normalized commanded velocity serving as a condition that selects the appropriate reference trajectory between walk and run clips. Only three LAFAN1 reference clips are required to regularize the complete behavior set. At deployment, a single frozen ONNX policy executes at 50,Hz with no runtime mode logic; hardware experiments demonstrate successful recovery from both prone and supine falls and smooth walk-to-run transitions under the same controller.

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.

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.

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.

A class of planar bipedal robots with unique mechanical properties has been proposed, where all links are balanced around the hip joint, preventing natural swinging motion due to gravity. A common property of their equations of motion is that the inertia matrix is a constant matrix, there are no nonlinear velocity terms, and the gravity term contains simple nonlinear terms. By performing a Taylor expansion of the gravity term and making a linear approximation, it is easy to derive a linearized model, and calculations for future states or walkability determination can be performed instantaneously without the need for numerical integration. This paper extends the method to a planar biped robot model with knees. First, we derive the equations of motion, constraint conditions, and inelastic collisions for a planar 6-DOF biped robot, design its control system, and numerically generate a stable bipedal gait on a horizontal plane. Next, we reduce the equations of motion to a 3-DOF model, and derive a linearized model by approximating the gravity term as linear around the expansion point for the thigh frame angle. Through numerical simulations, we demonstrate that calculations for future states and walkability determination can be completed in negligible time. By applying control inputs to the obtained model, performing state-space realization, and then discretizing it, instantaneous walkability determination through iterative calculation becomes possible. Through detailed gait analysis, we discuss how the knee joint flexion angle and the expansion point affect the accuracy of the linear approximation, and the issues that arise when descending a small step.

Bipeds have demonstrated high agility and mobility in unstructured environments such as sand. The yielding of such granular media brings significant sinkage and slip of the bipedal feet, leading to uncertainty and instability of walking locomotion. We present a new dynamics-modeling approach to capture and predict bipedal-walking locomotion on granular media. A dynamic foot-terrain interaction model is integrated to compute the ground reaction force (GRF). The proposed granular dynamic model has three additional degree-of-freedom (DoF) to estimate foot sinkage and slip that are critical to capturing robot-walking kinematics and kinetics such as cost of transport (CoT). Using the new model, we analyze bipedal kinetics, CoT, and foot-terrain rolling and intrusion affects. Experiments are conducted using a biped robotic walker on sand to validate the proposed dynamic model with robot-gait profiles, media-intrusion prediction, and GRF estimations. This new dynamics model can further serve as an enabling tool for locomotion control and optimization of bipedal robots to efficiently walk on granular terrains.

Foot slip is a major source of instability in bipedal locomotion on low-friction or uncertain terrain. Standard control approaches typically assume no-slip contact and therefore degrade when slip occurs. We propose a control framework that explicitly incorporates slip into the locomotion model through virtual nonholonomic constraints, which regulate the tangential stance-foot velocity while remaining compatible with the virtual holonomic constraints used to generate the walking gait. The resulting closed-loop system is formulated as a hybrid dynamical system with continuous swing dynamics and discrete impact events. A nonlinear feedback law enforces both classes of constraints and yields a slip-compatible hybrid zero dynamics manifold for the reduced-order locomotion dynamics. Stability of periodic walking gaits is characterized through the associated Poincaré map, and numerical results illustrate stabilization under slip conditions.

Legged robots operate across a wide range of physical scales, but how their designs should be adapted as size changes remains unclear. Here, we tackle this question in two ways. First, we survey existing legged robots to provide a broad context for the key scaling variables, robot mass m and leg length L. We find the surprising result that bipedal robot mass generally scales with the length squared, L^2, rather than the isometric prediction L^3. Then, to reduce the variance in design choices, we focus on a pair of previously developed bipeds that share the same quasi-passive morphology but differ by a factor of six in leg length, use different feet and controllers, and achieve different relative speeds. We reconstruct both robots in a common 3-D simulation environment and scale each design over leg lengths from 0.02 to 1.2 meters under both mass models (mass is proportional to L^2 and is proportional to L^). The controlled comparison shows that velocity follows dynamic similarity, velocity is proportional to L^{1/2}, across designs and mass models, while the torque needed to sustain walking follows that tau is proportional to mL. Consequently, torque scales approximately with L^3 when m is proportional to L^2 and L^4 when mass is proportional to L^3. A 3-D foot-shape sweep further shows that foot dimensions that permit walking scale approximately linearly with leg length, but the speed-maximizing shape and the mechanism by which each robot moves do not transfer by scaling alone. Overall, the results provide practical insights for rescaling legged systems that leverage natural body dynamics.

Planetary bodies characterized by low gravitational acceleration, such as the Moon and near-Earth asteroids, impose unique locomotion constraints due to diminished contact forces and extended airborne intervals. Among traversal strategies, hopping locomotion offers high energy efficiency but is prone to mid-flight attitude instability caused by asymmetric thrust generation and uneven terrain interactions. This paper presents an underactuated bipedal hopping robot that employs an internal reaction wheel to regulate body posture during the ballistic flight phase. The system is modeled as a gyrostat, enabling analysis of the dynamic coupling between torso rotation and reaction wheel momentum. The locomotion cycle comprises three phases: a leg-driven propulsive jump, mid-air attitude stabilization via an active momentum exchange controller, and a shock-absorbing landing. A reduced-order model is developed to capture the critical coupling between torso rotation and reaction wheel dynamics. The proposed framework is evaluated in MuJoCo-based simulations under lunar gravity conditions (g = 1.625 m/s^2). Results demonstrate that activation of the reaction wheel controller reduces peak mid-air angular deviation by more than 65% and constrains landing attitude error to within 3.5 degrees at touchdown. Additionally, actuator saturation per hop cycle is reduced, ensuring sufficient control authority. Overall, the approach significantly mitigates in-flight attitude excursions and enables consistent upright landings, providing a practical and control-efficient solution for locomotion on irregular extraterrestrial terrains.

Legged locomotion benefits from embodied sensing, where perception emerges from the physical interaction between body and environment. We present a soft-surfaced, vision-based tactile foot sensor that endows a bipedal robot with a skin-like deformable layer that captures contact deformations optically, turning foot-ground interactions into rich haptic signals. From a contact image stream, our method estimates contact pose (position and orientation), visualizes shear, computes center of pressure (CoP), classifies terrain, and detects geometric features of the contact patch. We validate these capabilities on a tilting platform and in visually obscured conditions, showing that foot-borne tactile feedback improves balance control and terrain awareness beyond proprioception alone. These findings suggest that integrating tactile perception into legged robot feet improves stability, adaptability, and environmental awareness, offering a promising direction toward more compliant and intelligent locomotion systems. For the supplementary video, please visit: https://youtu.be/ceJiy9q_2Aw

Trajectory optimization is a fundamental tool for generating efficient, dynamically consistent periodic motions, commonly referred to as gaits, in legged locomotion. This paper explores the indirect method of periodic trajectory optimization, emphasizing its application in creating optimal gaits for legged systems and contrasting it with the more common direct method. While the direct method offers implementation flexibility, it is limited by its need for an input space parameterization. In contrast, the indirect method achieves higher accuracy by computing the control input from states and costates along the optimal trajectory. In this work, we tackle the convergence challenges associated with indirect shooting methods by utilizing numerical continuation methods. This is particularly useful for the systematic development of gait libraries. Our contributions include: (1) the formalization of a general periodic trajectory optimization problem that extends existing first-order necessary conditions to a broader range of cost functions and operating conditions; (2) a methodology for efficiently generating optimal gait libraries using single shooting and numerical continuation; (3) a novel approach for reconstructing Lagrange multipliers and costates from passive gaits; (4) a comparative analysis of the indirect and direct shooting methods using a compass-gait walker as a case study, demonstrating the higher accuracy of the indirect method; and (5) demonstrating applicability to the more complex legged robot RABBIT, with ten dynamic states and four inputs. These results establish a systematic indirect optimal control framework for synthesizing families of optimal periodic trajectories in legged locomotion.

Despite decades of research, the dynamic mechanisms governing human locomotion remain incompletely understood. Current motor control models only partially capture these complex dynamics, a theoretical gap that likely limits the clinical efficacy of modern lower-limb rehabilitation therapies. Recently, it has been proposed that human locomotion is planned in task-space by explicitly exploiting the gravitational pull acting on the Center of Mass (CoM). Building upon this concept, this paper presents a novel deterministic task-space planner for straight, constant-speed locomotion. Our model formulates the CoM trajectory in the transverse plane as a harmonic oscillator moving at a constant forward velocity, while demonstrating that the vertical CoM trajectory is actively regulated through localized ankle strategies. We derive closed-form equations that successfully generate human-like trajectories for both the CoM and the swing foot. The model's predictive accuracy is empirically validated against the kinematic behavior of 12 healthy subjects walking at three self-selected speeds. Furthermore, the proposed planner is mathematically compatible with an extended formulation of the Passive Motion Paradigm ($λ_0$-PMP), establishing a comprehensive theoretical foundation for decentralized, hierarchical architectures of semi-autonomous controllers in both human motor control and assistive robotics.