This paper presents a digital twin system that facilitates robot operation and combined prediction of state and dynamics of a biped robot with imperfections such as parasitic compliance. The key contribution lies in utilising proprioceptive sensor feedback together with a contact-aware forward-dynamics (shadow) simulation, augmented by a contact-implicit inverse-dynamics controller—assuming MuJoCo's rigid-body contact formulation—to improve the replication of the robot's posture and dynamics. The speed and accuracy of the developed method is demonstrated through physical experiments on a small biped robot standing on flat, steps or inclined ground. The predicted contact forces are validated using the feedback of a force-sensing robot foot via the concept of zero-tilting moment point. For the first time, a systematic design of a robot foot using force-sensing resistors is presented to achieve repeatability, linearity, and sensitivity to small loads.

Nowadays, the adaptability of bipedal robots to different terrains becomes a critical topic because the application environments of robots often are complex, variable, and often unstructured. Examining this adaptability not only reveals the current progress of gait control and perception methods but also highlights key challenges that must be addressed to enable reliable, real-world deployment of bipedal robotic systems. This essay will first introduce the mechanism, types, and controlling methods of bipedal robots then provide numerous popular robots and finally discuss some highly controversial discussions. Overall, after widely searching for different types of newly invented bipedal robots, a significant progress was seen in bipedal robot's adaptability when meeting distinct uneven terrains, which most of the bipedal robots showed strong adaptability to uneven terrains. This conclusion is crucial because it demonstrates the readiness of bipedal robots for real-world environments, where surfaces are often rarely flat or predictable. After improving the advancement of robot's adaptability, a safer and more effective applications in areas such as post-disaster rescue and exploration work.

The robust rejection of unforeseen external perturbations by bipedal robots during walking poses a significant challenge. This article presents a sequential centroidal model predictive control (SC-MPC) framework based on foothold and contact wrench decomposition, enabling bipedal robots to robustly reject external perturbations during dynamic walking. Unlike methods that simultaneously optimize footholds and contact wrenches, which lead to nonlinearity and computational complexity, we decouple this complex problem into two sequential lightweight MPC problems to ensure an efficient online solution. The SC-MPC framework first employs a low-fidelity linear inverted pendulum model to predict the reactive foothold sequence, incorporating velocity tracking, kinematic reachability, and slack constraints for heuristic foothold predictions. Subsequently, using the optimized footholds, a high-fidelity variable-inertia centroidal dynamics model is used to predict the continuously varying contact wrenches, with closed-form contact stability constraints. The proposed SC-MPC method enhances the prediction frequency of multistep planning from typically below 100 Hz to at least 200 Hz, enabling the robot to respond more quickly to unknown disturbances. With the torque-controlled bipedal robot BHR8TC, the proposed method is validated through extensive simulations and experiments involving external force and terrain perturbations. Comparative results with different control methods demonstrate the superior performance of SC-MPC in perturbation rejection.

Cable-driven robots face significant challenges in achieving precise motion control due to the inherent nonlinearity, strong coupling, and time-varying transmission dynamics. Traditional model-based methods require precise parameter identification and offline calibration, while existing data-driven approaches often fail to adapt to the evolving physical couplings in real time. To bridge this gap, this paper proposes a novel Dynamic Transmission Learning (DTL) framework. Its core innovation is an Evolving Graph Structure (EGS) that explicitly models the robot's motor-joint transmission as a dynamic graph. The EGS continuously learns and adapts the time-varying coupling coefficients (represented by the mapping matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {A}_{t}$</tex-math></inline-formula>) directly from sensor data, enabling a calibration-free representation of the transmission dynamics. The framework thereby captures complex deviations such as cable elasticity variations, backlash, and nonlinear damping effects. Through extensive experiments on bipedal robots, we demonstrate that our framework significantly outperforms a range of sequential and graph-based baseline models in prediction accuracy. Furthermore, integrated into a real-time Proximal Policy Optimization (PPO) control loop, the DTL framework effectively compensates for transmission errors, reduces joint tracking error by over 50%, and demonstrates robust performance across diverse locomotion scenarios.

Obstacle crossing in revolute and prismatic knee underactuated biped robots

Reinforcement Learning Framework for Improving Real-World Performance of the Bipedal Robot SUBO-2 with Low-Backdrivability

Wheeled bipedal robots are promising for industrial mobility because they combine tight turning, agile balancing, and efficient rolling. Their inherently unstable and underactuated dynamics make reliable reference tracking challenging, particularly in the presence of sustained external disturbances and modeling errors. This paper presents a systematic modeling and control study using a three-degrees-of-freedom sagittal plane representation derived from the original six-degrees-of-freedom dynamics. Two linear tracking controllers are designed and compared: a full state feedback tracking controller and a linear quadratic servo controller with integral action. Practical performance is validated through real-time hardware in the loop simulation, where the controller runs on embedded hardware and the plant is executed on a real-time target including discrete time-sampling effects and analog input output communication noise associated with signal transmission. The results show that both controllers achieve stabilization, while the comparative HILS results reveal a trade-off rather than a uniformly superior controller. The full state feedback controller often yields lower finite-horizon position tracking errors, whereas the linear quadratic servo controller provides tighter body-pitch regulation and the more reliable removal of steady-state offset under sustained constant disturbances. These results demonstrate the feasibility of optimal servo control on cost-effective embedded platforms and indicate that controller selection should depend on the desired balance, considering tracking accuracy, disturbance rejection, convergence behavior, and actuator usage.

Posture-Aware Standing Stability Control of a Bipedal Wheel-Legged Robot Using Hybrid Adaptive Optimization

Human bipedal locomotion arises from continuous, closed-loop interactions between neural control and biomechanical structure-collectively referred to as neuromechanics. The relationship between human locomotion and robotic locomotion is deeply interconnected through shared principles of neuromechanics, thereby providing a comprehensive framework for understanding human movement and informing robotic system design. In this review, we synthesize insights from neuroscience, biomechanics, computational modelling and robotics to establish a cohesive perspective on human-inspired bipedal locomotion. We begin by outlining essential anatomical and physiological principles, such as spinal circuits, supraspinal coordination and musculoskeletal structure. Next, we analyse mathematical models-ranging from simplified neural oscillators to complex musculoskeletal simulations-that formalize these mechanisms. Finally, we discuss the embodiment of these models in bipedal robots, which promotes reciprocal advancements in both biological understanding and engineering innovation. Rather than offering a comprehensive literature survey, we focus on pivotal developments, emerging trends and unresolved questions that shape this interdisciplinary domain. By integrating diverse fields, this review aims to enhance the design of agile, energy-efficient robots and deepen our understanding of human locomotion.

The development of robust locomotion strategies for bipedal robots remains a significant challenge in robotics, particularly when navigating unstructured and unknown environments. The Cassie robot, a dynamic bipedal platform with high degrees of freedom and underactuated passive dynamics, presents specific control difficulties that classical model-based approaches often fail to address adequately due to modeling mismatches and computational latency. This paper proposes a novel framework utilizing Soft Actor-Critic (SAC), an off-policy deep reinforcement learning algorithm, to generate stable gait patterns and ensure robust balance control. Unlike standard on-policy methods, SAC optimizes a maximum entropy objective, which encourages substantial exploration and provides greater robustness to external disturbances. We introduce a comprehensive reward function design and a domain randomization strategy that enables the policy to generalize across varying terrain irregularities without requiring exteroceptive mapping during the training phase. Extensive simulation results demonstrate that the proposed SAC-based controller outperforms baseline algorithms in terms of convergence speed, energy efficiency, and stability on uneven terrain. The learned policy exhibits emergent behaviors capable of recovering from significant perturbations, suggesting a promising pathway for deploying autonomous bipedal systems in real-world scenarios.

Robotic jumping research advances engineering and biomimicry frontiers, prioritizing range, precision, and predictability to navigate unstructured environments. Earth’s gravity necessitates powerful actuators and lightweight bodies in robotic designs for maximal jump height. While many robots excel in statical environments, precise, predictable jumps in dynamic settings remain challenging. We realized this with a bipedal robot leveraging thrust-induced hypogravity, alongside dual regulation of aerial attitude and parabolic trajectory via thrust vectoring. Hypogravity multiplies leap range (max: 6.9 m) despite leg force saturation, enabling the robot to clear multi-level stairs, a 2.35-m-high wall, and 3-m-wide stream. Parabolic trajectory regulation allows leap distance precision/consistency surpassing existing thrust-assisted hybrids and leg-only jumpers. It enables pre-jump prediction of aerial/landing positions and timing, facilitating leaps in dynamic scenarios: through fast-moving windows (3.8 m/s), onto shifting, confined targets, and against wind disturbance. This research establishes extended range, precise, and predictable jumping through self-generated hypogravity and parabolic trajectory regulation.

Like humans, all birds adopt a strictly bipedal posture. However, unlike humans, birds have such good balance that they can sleep while standing up, which must require minimal energy. This makes them an interesting model for studying bipedalism in robotics. In this study, we examine balance and postural stability via a tensegrity system (assembly in parallel of rigid bodies and cables). To test this hypothesis, we created mathematical models based on anatomical observations of the legs of various birds (zebra finch, little egret, mallard and military macaw) to investigate different configurations. Building on a previous model, we demonstrate that tensegrity systems can achieve passive stability under simplified loading. Here, we aim to establish whether this model can be generalized, to determine stability, and to identify the impact of certain kinematic, dynamic and material parameters. Our results enabled us to identify the parameters that allow the model to be generalized. We determined that adding two cables corresponding to tendinous and muscular sets generalizes the model to a varied range of configurations and exploits the rear part of the foot when present. These findings offer new insights into avian bipedalism and could inspire the design of bipedal robots with passive stability for greater autonomy.

The development of highly dynamic bipedal robots demands sensing capable of capturing key contact-related signals in real time, particularly the Center of Pressure (CoP). CoP is fundamental for locomotion control and state estimation and is also of interest in biomedical applications such as gait analysis and lower-limb assistive devices. To enable reliable CoP acquisition under dynamic walking, this paper presents a foot-mounted measurement system and an online self-calibration method that adapts sensor scale and bias parameters during locomotion using both external foot sensors and the robot’s proprioceptive measurements. We demonstrate an online self-calibration pipeline that updates foot-sensor scale and bias parameters during a walking experiment on a NAO-V5 platform using a sliding window optimization. The reported results indicate improved within-trial consistency relative to an offline-calibrated reference baseline under the tested walking conditions. In addition, the framework reconstructs a digitized estimate of the vertical ground reaction force (vGRF) from load-cell readings; due to ADC quantization and the discrete offline calibration dataset, the vGRF signal may exhibit stepwise behavior and should be interpreted as a reconstructed (digitized) quantity rather than laboratory-grade continuous force metrology. Overall, the proposed sensing-and-calibration pipeline offers a practical solution for dynamic CoP acquisition with low-cost hardware.

This paper proposes a nonlinear optimal control approach for the three-link biped robot. In this humanoid robot there is actuation only in the two legs while the third link which is the robot’s torso is unactuated. Due to nonlinearities and the unactuated torso in the robot’s dynamics, the treatment of the stabilization and trajectories tracking problem is a non-trivial task. To solve the associated nonlinear optimal control problem, the state-space model of the three-link biped robot undergoes approximate linearization based on Taylor series expansion and the associated Jacobian matrices. For the linearized state-space model of the biped robot a stabilizing optimal (H-infinity) feedback controller is designed. To compute the controller’s feedback gains, an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis.

With the rapid development of mobile robotics, wheeled bipedal robots, which combine the terrain adaptability of legged robots with the high mobility of wheeled systems, have attracted increasing research attention. To address the balance control problem during both standing and locomotion while reducing the influence of noise on control performance, this paper proposes a balance control framework based on a Linear Quadratic Regulator integrated with an Extended Kalman Filter (KLQR). Specifically, a baseline LQR controller is designed using the robot’s dynamic model, where the control input is generated in the form of wheel-hub motor torques. To mitigate measurement noise and suppress oscillatory behavior, an Extended Kalman Filter is applied to smooth the LQR torque output, which is then used as the final control command. Filtering experiments demonstrate that, compared with median filtering and other baseline methods, the proposed EKF-based approach significantly reduces high-frequency torque fluctuations. In particular, the peak-to-peak torque variation is reduced by more than 60%, and large-amplitude torque spikes observed in the baseline LQR controller are effectively eliminated, resulting in continuous and smooth torque output. Static balance experiments show that the proposed KLQR algorithm reduces the pitch-angle oscillation amplitude from approximately ±0.03 rad to ±0.01 rad, corresponding to an oscillation reduction of about threefold. The estimated RMS value of the pitch angle is reduced from approximately 0.010 rad to 0.003 rad, indicating improved convergence and steady-state stability. Furthermore, experiments involving constant-speed straight-line locomotion and turning indicate that the KLQR algorithm maintains stable motion with velocity fluctuations limited to within ±0.05 m/s. The lateral displacement deviation during locomotion remains below 0.02 m, and no abrupt acceleration or deceleration is observed throughout the experiments. Overall, the results demonstrate that applying Extended Kalman filtering to smooth the control torque effectively improves the smoothness and stability of LQR-based balance control for wheeled bipedal robots.

The presence of sensor noise, missing states and inadequate future prediction capabilities imposes significant limitations on the locomotion performance of bipedal robots operating in unstructured terrain. Conventional methods generally depend on long-term history observations to reconstruct single-frame privileged information. However, these methods fail to acknowledge the pivotal function of short-term history in rapid state responses and the significance of future state prediction in anticipating potential risks. The proposed framework is a Long-Short World Model (LSWM), which integrates state reconstruction and future state prediction to enhance the locomotion capabilities of bipedal robots in complex environments. The LSWM framework comprises two modules: a state reconstruction module (SRM) and a future state prediction module (SPM). The state reconstruction module employs long-term history observations to reconstruct privileged information in the current short-term history, thereby effectively improving the system's robustness to sensor noise and enhancing state observability. The future state prediction module enhances the robot's adaptability to complex environments and unpredictable scenarios by predicting the robot's future short-term privileged information. We conducted extensive comparative experiments in simulation as well as in a variety of real-world indoor and outdoor environments. In the indoor stair-climbing task, LSWM achieved a 94% success rate, outperforming the current state-of-the-art baseline methods by at least 34%, thereby demonstrating its substantial performance advantages in complex and dynamic environments.

This article proposes a Nash game-optimized adaptive robust control framework for bipedal parallel wheel-legged robots. Specifically, the framework targets a balance between tracking accuracy and control effort under underactuation, constraint coupling, and substantial uncertainties. To address these challenges, a constraint following adaptive robust controller is adopted, while gain selection is posed as a two-player Nash game between performance and cost objectives. Consequently, the resulting equilibrium yields controller gains without heuristic tuning. Furthermore, Lyapunov analysis establishes uniform ultimate boundedness of the closed-loop trajectories under bounded disturbances and model errors. Finally, numerical simulations verify that the proposed approach achieves a superior balance, concurrently enhancing tracking accuracy while significantly reducing control effort compared to conventional methods.

The bipedal wheel-legged robot combines the high energy efficiency of wheeled movement with the terrain adaptability of legged locomotion. However, achieving a smooth transition between these two heterogeneous motion modes within a unified control framework remains challenging. This study proposes a reinforcement learning control framework that integrates the Mixture of Experts (MoE) architecture. This approach employs a "divide and conquer" strategy by introducing a dynamic gating network and a Top-K sparse activation mechanism, which automatically allocates different motion modes to specific expert subnetworks, effectively decoupling conflicting gradients. Simulation results demonstrate that, compared to the single-network PPO method, the MoE-enhanced algorithm exhibits significant improvements in training stability and rewards. The learned policy successfully achieved smooth rolling on flat surfaces and transitioned to dynamic leg-lifting gaits when confronted with obstacles. In various test terrains, it showed a markedly higher success rate compared to the single-network PPO method.

A miniature bipedal piezoelectric robot with high payload-to-weight ratio

This paper addresses the problem of orbital stabilization of a periodic walking gait for a model or a digital twin of a three-link planar biped robot with a single actuator. A Lyapunov equation-based approach is proposed for the synthesis of a stabilizing controller for the corresponding impulsive mechanical system. The method ensures exponential vanishing of transverse coordinates, defining deviations from the nominal periodic trajectory, by solving Lyapunov matrix inequalities, which provide sufficient conditions for orbital stability of the closed-loop dynamics in the nominal case of no disturbances. The proposed approach allows systematic