足式机器人 运动控制

Legged robot is designed for more flexibility when navigating in complex unstructured environment. When the end-effectors of the robot contacting non-rigid ground, the robot sinks due to different stiffness of the ground. This presents a challenge for accurate and robust control of the upper platform. In this paper, a real-time muti-sensor fusion method Dual Parallelizable Particle Filter (DPPF) is proposed to estimate ground stiffness. DPPF utilized RGB-D camera, IMU and 3-DoF force sensors. Meanwhile, we established a ground material database and trained a real-time ground segmentation network to assist the stiffness estimation of the ground. Then the information of ground material is utilized as a prior distribution for DPPF to achieve faster stiffness estimation. The experiments on synthetic data and on six-legged robot show that DPPF has faster computing speed, fewer convergent steps than previous state estimation methods. The estimated stiffness can be utilized for legged robot impedance control, posture control and trajectory planning.

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Model Predictive Control (MPC) and Reinforcement Learning (RL) are two prominent strategies for controlling legged robots. RL learns control policies through system interaction, adapting to various scenarios, whereas MPC relies on a predefined mathematical model to solve optimization problems in real-time. Despite their widespread use, there is a lack of direct comparative analysis under standardized conditions. This work addresses this gap by benchmarking MPC and RL controllers on a Unitree Go1 quadruped robot within the MuJoCo simulation environment, focusing on a standardized task, straight walking at a constant velocity. Performance is evaluated based on disturbance rejection, energy efficiency, and terrain adaptability. The results show that RL excels in handling disturbances and maintaining energy efficiency but struggles with generalization to new terrains due to its dependence on learned policies tailored to specific environments. In contrast, MPC shows enhanced recovery capabilities from larger perturbations by leveraging its optimization-based approach, allowing for a balanced distribution of control efforts across the robot’s joints. The results present the advantages and limitations of both RL and MPC, offering insights into selecting an appropriate control strategy for legged robotic applications.

This paper proposes a height adaptive Linear Quadratic Regulator (LQR) control framework for a two-wheel legged robot, using closed-loop frequency domain identification to enhance stability and adaptability across varying robot heights. Conventional LQR control strategies often employ a linearized model obtained from a single operating condition, such as a fixed height. While these strategies can maintain stability when height changes are small, they struggle to maintain optimal performance under significant height variations due to either the robot's own adjustments or additional loads, which are common for two-wheel legged robots. To address this challenge, we propose a novel approach that integrates closed-loop frequency identification with an adaptive LQR controller. The frequency identification module identifies the system's closed-loop natural frequency, enabling the identification of the robot system's height. This information allows the LQR controller to adjust its parameters for optimized performance. Experimental results demonstrate that the proposed method significantly improves the robot's stability and robustness during height transitions, outperforming conventional fixed-parameter LQR controllers.

In this paper, we propose a novel control strategy called the self-adaptive velocity allocation strategy (SVAS), which integrates kinematic models into reinforcement learning (RL). This strategy has been implemented on a wheeled-legged robot named “Pegasus” with four ankle joints to enable its ability to self-adapt to complex terrains. SVAS separates Pegasus' motion system into two subsystems: the legged and wheeled subsystem. Legged motion is controlled directly by the output of the RL agent, while wheeled motion is achieved by the vehicle model we derived. By utilizing historical data as input for the RL agent, which has an asymmetric actor-critic structure, we enable Pegasus to have terrain inference capabilities without relying on visual perception. With the terrain inference capabilities, the RL agent can produce a coefficient that allocates velocities for two subsystems, respectively. Simulation results revealed that, with SVAS, the robot achieves higher rewards in fewer training iterations. It is also shown that Pegasus is able to self-optimize its motion and reduce energy consumption with SVAS. In addition, experimental results demonstrated Pegasus' ability to adaptively traverse complex terrains.

A wheeled–legged robot has the advantage of stable and agile movement on flat ground and an excellent ability to overcome obstacles. However, when faced with a narrow footprint, there is a limit to its ability to move. We developed the control moment gyroscope (CMG) unicycle–legged robot to solve this problem. A scissored pair of CMGs was applied to control the roll balance, and the pitch balance was modeled as a double-inverted pendulum. We performed Linear Quadratic Regulator (LQR) control and model predictive control (MPC) in a system in which the control systems in the roll and pitch directions were separated. We also devised a method for controlling the rotation of the robot in the yaw direction using torque generated by the CMG, and the performance of these controllers was verified in the Gazebo simulator. In addition, forward driving control was performed to verify mobility, which is the main advantage of the wheeled–legged robot; it was confirmed that this control enabled the robot to pass through a narrow space of 0.15 m. Before implementing the verified controllers in the real world, we built a CMG test platform and confirmed that balancing control was maintained within ±1∘.

This paper proposes a technical framework that can simultaneously control the hip's horizontal motion and vertical load force with the ability of bilateral force perception for the heavy-legged robot driven by electric cylinders in grounded phases. We observed unknown load through the inverse modeling approach and the Nonlinear disturbances observer (NDOB). Based on the preset height constraints, the robot leg dynamics' order is reduced. With this in mind, the force/motion hybrid controller is designed to realize the hip joint's horizontal motion and vertical force control under unknown load force. Meantime we obtain the cylinder acceleration signal through a high-order acceleration observer (HOAO) with external force compensation to build the vertical bilateral force balance relation. The combination of centroidal dynamics (CD) and estimated cylinder acceleration signal enables the observation of single-dimension ground reaction force (GRF) without force sensors. Verification results show that the proposed framework has a promising force/position hybrid control and bilateral force perception ability.

This paper presents an optimized wheel-legged biped robot for inspection tasks in semi-structured urban environments. By integrating the stability of wheeled locomotion with the terrain adaptability of legged systems, the robot achieves efficient traversal across diverse surfaces. Key innovations include a unified control framework combining balance, state estimation, and terrain adaptation via multi-sensor fusion. The mechanical design supports impact resilience from jumps and falls, while the electronic and software architecture delivers real-time and robust operation. Experimental results validate the robot’s capabilities in velocity tracking, disturbance rejection, and traversability over complex terrains. The proposed approach enhances robustness and adaptability, demonstrating strong potential for practical deployment in public service scenarios. This work advances autonomous mobile robotics and lays a foundation for future research in urban inspection applications.

Legged robots are promising candidates for exploring challenging areas on low-gravity bodies such as the Moon, Mars, or asteroids, thanks to their advanced mobility on unstructured terrain. However, as planetary robots'power and thermal budgets are highly restricted, these robots need energy-efficient control approaches that easily transfer to multiple gravity environments. In this work, we introduce a reinforcement learning-based control approach for legged robots with gravity-scaled power-optimized reward functions. We use our approach to develop and validate a locomotion controller and a base pose controller in gravity environments from lunar gravity (1.62 m/s2) to a hypothetical super-Earth (19.62 m/s2). Our approach successfully scales across these gravity levels for locomotion and base pose control with the gravity-scaled reward functions. The power-optimized locomotion controller reached a power consumption for locomotion of 23.4 W in Earth gravity on a 15.65 kg robot at 0.4 m/s, a 23 % improvement over the baseline policy. Additionally, we designed a constant-force spring offload system that allowed us to conduct real-world experiments on legged locomotion in lunar gravity. In lunar gravity, the power-optimized control policy reached 12.2 W, 36 % less than a baseline controller which is not optimized for power efficiency. Our method provides a scalable approach to developing power-efficient locomotion controllers for legged robots across multiple gravity levels.

Bipedal Wheel-Legged robots often encounter challenges such as strong nonlinearity, model uncertainty, and external disturbances when conducting complex task. This paper proposes a control method based on a self-tuning single-input interval type-1 fuzzy PID (ST-SIT1-FPID) and an improved particle swarm optimization (Improved PSO) algorithm. The proposed method adaptively adjusts key fuzzy mapping (FM) parameters online in response to state variations, thereby improving the controller“s adaptability to varying operating conditions. In addition, the improved PSO is used to optimize the fuzzy control gains, further improving overall control performance. Finally, simulation and experimental results are conducted to verify performance of the proposed method.

Soft robots have gained significant attention across various fields for their adaptability to complex environments, enabled by their inherent flexibility. In this study, we present improvements to the previous robot, TAOYAKA-S IV, enhancing stability through structure redesign by repositioning the arms and augmenting trunks. Additionally, we improved maneuverability through the integration of wireless control for left and right bending locomotion. These modifications introduce new capabilities, including the ability to climb pipes of various shapes and navigating in the desired direction when climbing Y-shaped pipes. The robot successfully replicated this behavior in trees, demonstrating its better applicability in real-world environments.

Purpose Determining how to enhance foot walking ability and maintain pose control in unstructured terrain is a significant technological challenge for multilegged robots. To address this problem, the authors have devised a stable walking control strategy for the whole body to improve the stable pose control of multilegged robots. This paper studies a whole-body stable walking control strategy to improve walking ability and maintain a stable posture in order to enhance the stable posture control of quadruped robots. Design/methodology/approach In this work, a stable walking control strategy for the whole body is investigated to improve the stable pose control of multilegged robots. First, the perception system is used to obtain the terrain height in the environment and send it to the trajectory planner to generate the foot-end trajectory for different terrains. Next, a stability control strategy is developed to constrain the body posture and motion space, including the main support triangle-based posture controller, foot-end force distribution-based barycentric horizontal position controller and barycentric height controller. Finally, simulations and real-world demonstrations using the robot are conducted on unstructured terrain (bulge and concave terrain). Findings Results indicate that the control method designed in this work can effectively maintain posture stability and maximum movement space of the legs when the robot walks on unstructured terrain. Originality/value This research can provide technical guidance and reference for wheel-legged robot foot walking on unstructured terrain. In the future, it can be used in various fields such as industrial rescue, material transportation and earthquake relief, greatly improving transportation efficiency.

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Animals possess a remarkable ability to navigate challenging terrains, achieved through the interplay of various pathways between the brain, central pattern generators (CPGs) in the spinal cord, and musculoskeletal system. Traditional bioinspired control frameworks often rely on a singular control policy that models both higher (supraspinal) and spinal cord functions. In this work, we build upon our previous research by introducing two distinct neural networks: one tasked with modulating the frequency and amplitude of CPGs to generate the basic locomotor rhythm (referred to as the spinal policy), and the other responsible for receiving environmental perception data and directly modulating the rhythmic output from the spinal policy to execute precise movements on challenging terrains (referred to as the descending modulation policy). This division of labor more closely mimics the hierarchical locomotor control systems observed in legged animals, thereby enhancing the robot’s ability to navigate various uneven surfaces, including steps, high obstacles, and terrains with gaps. Additionally, we investigate the impact of sensorimotor delays within our framework, validating several biological assumptions about animal locomotion systems. Specifically, we demonstrate that spinal circuits play a crucial role in generating the basic locomotor rhythm, while descending pathways are essential for enabling appropriate gait modifications to accommodate uneven terrain. Notably, our findings also reveal that the multi-layered control inherent in animals exhibits remarkable robustness against sensorimotor delays. These findings advance our understanding of the fundamental principles governing the interplay between spinal and supraspinal mechanisms in biological locomotion. Moreover, they inform the design of bioinspired locomotion controllers that emulate these biological structures, facilitating natural movement in complex and realistic environments.

This paper introduces a new wheel-legged robot and develops motion controllers based on central pattern generators (CPGs) for the robot to navigate over a range of terrains. A transformable leg-wheel design is considered and characterized in terms of key locomotion characteristics as a function of the design. Kinematic analysis is conducted based on a generalized four-bar mechanism driven by a coaxial hub arrangement. The analysis is used to inform the design of a central pattern generator to control the robot by mapping oscillator states to wheel-leg trajectories and implementing differential steering within the oscillator network. Three oscillator models are used as the basis of the CPGs, and their performance is compared over a range of inputs. The CPG-based controller is used to drive the developed robot prototype on level ground and over obstacles. Additional simulated tests are performed for uneven terrain negotiation and obstacle climbing. Results demonstrate the effectiveness of CPG control in transformable wheel-legged robots.

Compared with traditional wheeled or legged robots, wheeled legged robots can maintain better stability in unstable terrain, but nonlinear system control of wheeled legged robots in strongly disturbed terrain still faces challenges. This article proposes an integrated adaptive control strategy that integrates two control models based on the kinematic characteristics of a five link legged robot. Firstly, in order to streamline the complex overall motion model of the robot, the motion of the robot is decoupled into two parts: the body and the legs. Secondly, an MPC-LADRC (Model Predictive Control Linear Active Disturbance Rejection Control) integrated adaptive control method is proposed, which uses LADRC to correct the MPC optimization problem and improve the stability of the wheeled legged robot in complex terrain. The simulation comparison verification results show that compared with the Anti-interference controller based on LQR (linear quadratic regulator), the MPC-LADRC controller has better control accuracy, speed tracking response, and anti-interference ability.

This paper introduces a wheel-legged metamorphic robot, that combines elements from both wheeled vehicles and legged mechanisms. It can switch between driving on wheels and walking on legs. One major challenge is ensuring stability during steering and reconfiguration, as the robot can become unstable during this process. To address this, we designed the steering and reconfiguration motion based on the robot’s design and established a spatial kinematic model using the homogeneous coordinate transformation method. A virtual prototype was created for verification results. Additionally, an electromechanical dynamic model was developed for the robot’s center-of-mass position-adjusting mechanism. To enhance system stability during steering and reconfiguration, a fuzzy-PID algorithm based on zero-moment-point theory was designed to control the slider movement within this mechanism. The effectiveness of this stability control has been verified through simulation and experimental validation.

Abstract The two-wheeled legged robot combines the advantages of legged robot and wheeled robot and has high terrain adaptability. Spherical robots are highly resistant to interference during detection. In this paper, a new sphere-wheel-legged robot is designed by combining these three motion modes. This paper begins by introducing the mechanical design, hardware, and software. Then, kinematics and dynamics of wheel-legged motion and spherical motion are analyzed in detail. Subsequently, the controllers for wheel-legged balancing motion, wheel-legged jumping motion, and sphere rolling motion are developed, respectively. Finally, experiments are carried out for different modes. The results demonstrate that the designed robot has excellent locomotor capabilities over different terrains.

—Dynamic fast adaptation is one of the basic capabilities that enables the animals to timely and properly adjust its locomotion reacting to the unpredictable changes. Such capability is also essential for the quadruped robot, when working in the unforseen environment. While reinforcement learning (RL) has achieved a significant progress in locomotion control, rapid adaptation to the model uncertainties remains a challenge. In this paper, we seek to ascertain the control mechanism behind the locomotion RL policy, from which we propose a new RL-based Rapid onLine Adaptive Control (RL2AC) algorithm to complementarily combine the RL policy and the adaptive control together. RL2AC is run at a frequency of 1000Hz without the need for simultaneous training with RL. It presents a strong capability against the external disturbances or the sim-to-real gap, resulting in a robust locomotion, which is achieved through proper torque compensation derived from a novel adaptive controller. Various simulation and experiments have demonstrated the effectiveness of the proposed RL2AC against the heavy load, disturbances acted on one leg, lateral torque, sim-to-real gap and various terrains.

In order to solve the problem that the existing lunar rover has too large volume, too rudimentary algorithm and too weak structure to be flexible and sturdy, this paper has carried out the research on the electromechanical system design and motion control of it, and designed a new prototype of the biped wheeled legged robot. This paper adopts the calculation of degrees of freedom, the solution of inverse kinematics, the dynamic analysis and the design of extended circuit to obtain the hardware structure of the robot; A control algorithm for the robot was designed through Kalman filter, fuzzy control, PID control and LQR control. After landing experiments and testing, the feasibility of the design under different gravity conditions was verified.

Wheeled-legged humanoid robots combine the mobility of wheels with the versatility of legs, offering significant advantages for locomotion. This paper proposes a hierarchical control framework for such robots, using a two-wheeled inverted pendulum with a roll joint (TWIP-R) as a template model. The framework integrates a motion planner and a whole-body controller. The motion planner utilizes a linear quadratic regulator (LQR) to dynamically adjust the zero moment point (ZMP), counteracting centrifugal forces and enabling stable, dynamic movements. Meanwhile, the whole-body controller (WBC), based on centroidal momentum, solves an optimization problem via quadratic programming (QP) while incorporating constraints from the rolling contact condition of the wheels and the dynamics of the humanoid robot. This framework generates optimized torque commands, enabling feasible and stable motion even in dynamic scenarios. Simulations featuring challenging maneuvers, such as slalom, demonstrate its ability to enhance stability and dynamic performance compared to traditional two-wheeled inverted pendulum (TWIP)-based methods by leveraging a dynamics model with roll motion. This framework demonstrates the potential of wheeled-legged humanoid robots to achieve dynamic, stable, and efficient locomotion in a variety of scenarios.

To address the tracking problem of the hybrid‐driven underwater legged robot, a control strategy is proposed that decomposes the whole tracking control system into two subsystems: body‐level and actuator‐level. The body‐level subsystem uses a central pattern generator (CPG)‐based controller to plan suitable gaits to meet the required heading and the forward velocity, crucial for accurate tracking in underwater environments. The actuators‐level subsystem employs a cooperative approach between the C‐shaped legs and thrusters of the robot. To execute the intended gait while adhering to actuation constraints and the no‐slip requirement, the torques of the legs are calculated by a model predictive control and feedback compensation (MPCF)‐based controller. Simultaneously, the calculation of the thrusters concerns four aspects to keep the legs attached to the ground and maintain the stable locomotion of the robot. Simulations on the ROS‐Gazebo platform verify the mobility of the robot and demonstrate the effectiveness of the proposed CPG‐MPCF strategy.

Designing Hybrid energy storage system (HESS) for a legged robot is significant to improve the motion performance and energy efficiency of the robot. However, switching between the driving mode and regenerative braking mode in the HESS may generate a torque bump, which has brought significant challenges to the stability of the robot locomotion. To address this issue, an AI‐enabled control strategy for bumpless transfer switching is proposed, which is composed of a Proximal Policy Optimisation (PPO)‐based non‐linear active disturbance rejection controller and Deep neural network (DNN)‐based bumpless transfer strategy. We indicate that the proposed intelligent strategy solves bumpless transfer by reducing the torque bump during the system switching process. Meanwhile, the authors analyse the energy driving subsystem and the energy regenerating subsystem based on operational modes and the energy flow. Via the parameters tuning criterion, the authors adopt a PPO algorithm to adaptively tune the parameters of the non‐linear active disturbance rejection controller, which can improve the performance of the primary torque control. A DNN‐based intelligent bumpless transfer strategy is proposed to set the initial value of two switching controllers at a switching moment. Simulations and experimental results validate the effectiveness of the proposed control strategy.

The present work in bio-inspired robotics explores the design and implementation of a novel-legged robotic system featuring a modified Peaucellier-Lipkin mechanism with three control points for a single degree of freedom. The emphasis is placed on the robot’s adaptability to various walking gaits in different environments. The paper delves into the robot’s design, construction, and control system, which includes the application of PID control for enhanced stability and efficiency in mimicking biological locomotion. The primary aim is to demonstrate a robot capable of adjusting its form and function for diverse operational challenges, enhancing robotic mobility. The design also addresses repeatability issues, ensuring consistent performance across various tasks and conditions, contributing to the robot’s reliability and practical applicability in real-world scenarios.

Legged robots equipped with manipulators exhibit significant benefits in unstructured manipulation tasks. How-ever, ensuring that these robots successfully perform locomotion and manipulation tasks remains challenging, especially when both the center-of-mass (CoM) state and object parameters are unknown and time-varying. Based on the whole-body model predictive control (MPC), a novel online payload identification method is proposed to compensate for modeling errors and tackle loco-manipulation tasks. The whole-body MPC integrates the robot state and user inputs in the robot centroidal dynamics. Moreover, the online payload identification method facilitates evaluating the time-varying CoM state and the parameters of unknown objects. Additionally, the MPC efficacy of the centroidal dynamics model is evaluated by a single rigid body dynamics model, considering both scenarios: with and without the object dynamics. The robot can manipulate an unknown and time-varying load of up to 20kg, equivalent to 50% of the manipulator's mass. After online identification, the linear and rotational root mean square errors of the manipulator are reduced by approximately 70%. The results indicate that the proposed strategy can effectively coordinate the interaction force/moment between the robot and the manipulated object while maintaining stability.

Affected by the Moon's low gravity and complex terrain, traditional wheeled and legged robots struggle to balance terrain adaptability and locomotion efficiency, thereby limiting their applicability in lunar missions. To enhance multitask operation capabilities on the lunar surface, this paper proposes a novel biomimetic wheeled-legged mobile robotic system, adopting a four-bar linkage wheeled-leg structure driven by a dual-motor coaxial nested configuration. This design ensures high terrain adaptability and locomotion efficiency. Furthermore, the kinematic and dynamic models of the robot are established, and the safe operational workspace of the wheeled-leg system is analyzed. To address attitude disturbances caused by complex lunar terrains, a self-balancing control strategy based on position feedback and force feedback loops is developed. Simulation results show that, under biaxial asymmetric sinusoidal disturbances, the robot's attitude fluctuations are effectively controlled within $\pm 1^{\circ}$ (pitch) and $\pm 0.6^{\circ}$ (roll), demonstrating good dynamic leveling performance of the system. Overall, this work provides a novel mobile system design and control strategy for achieving high mobility and stability operation in lunar terrain environments, offering theoretical guidance for future research on autonomous and adaptive locomotion systems in extraterrestrial exploration missions.

As a novel robotic architecture, the wheeled–biped robot (WBR) offers the potential for rapid locomotion in complex environments. However, existing control schemes still exhibit limited mobility on rugged terrain with vertical drops and obstacles. This work proposes an Integrated Motion Control Framework (IMCF) based on a wheeled–legged floating base dynamic model, incorporating a three-phase balance control system and a composite motion state estimator. The IMCF was implemented on the WBR platform Lustin, and experimental results demonstrate outstanding robustness and superior trajectory-tracking performance. The robot achieves high-speed, stable traversal of rugged terrain with drops and obstacles while maintaining stability and exhibiting strong adaptability and robustness to external disturbances.

Wheeled-legged humanoid robots combine the rough terrain compliance of humanoid robots with the high efficiency of wheeled robots, enabling the robot to achieve flexible and stable locomotion over multiple terrains. However, the stability control of the wheel-legged humanoid robot in dealing with rough terrains and unexpected external disturbances remains unsolved. In the current investigation, a compliant balance control framework (CBCF) is proposed, which can absorb ground shocks, withstand unexpected external disturbances, and remain stable posture during motion. The CBCF connects the control of legs movement and the wheels balance control through the movement of the robot's center of mass. The wheel balance control employs the inverted pendulum model and controls the two wheels through model prediction. The leg posture control utilizes a whole-body dynamic compensator to realize the compliant motion and remain a stable posture. Cooperating with the high-level motion planner, the CBCF can allow the BHR-WI to move quickly and perform excellent adaptation in unmodeled rough terrains, and it is able to appropriately handle unexpected external disturbances as well. It is also worth mentioning that the BHR-WI is capable of remaining balance and quickly recovering stability in the event of a disturbance, even if one of the legs leaves the ground. Finally, tests confirm that the BHR-WI could withstand sustained unexpected disturbances, could smoothly cross the grass and steps and even realize high maneuverability in jumping.

Reinforcement learning has developed as a promising approach for robot locomotion control, which can save engineering effort compared to conventional approaches. This article presents the implementation of Reinforcement learning on a low-cost, 12 degree of freedom robot known as a quadruped (spider) to optimize locomotion control, enabling the robot to move on different surfaces such as flat surfaces, ramps, speed bumps, and rough terrain. A MATLAB Simulink model is developed as a digital twin of the spider robot. The dynamics of the model were studied and validated using an open-loop algorithm. Then, the model is utilized in a training simulation environment to apply the reinforcement learning algorithm, showing its ability to move along a predefined path as a replacement for conventional motion control systems. Moreover, the work compares the performance and effectiveness of machine learning-based locomotion control with traditional motion control systems regarding navigation accuracy, speed, and adaptability in challenging environments. The minimum hardware requirements are also studied to move the experiment from simulation to reality.

The robot used for disaster rescue or field exploration requires the ability of fast moving on flat road and adaptability on complex terrain. The hybrid wheel-legged robot (WLR-3P, prototype of the third-generation hydraulic wheel-legged robot) has the characteristics of fast and efficient mobility on flat surfaces and high environmental adaptability on rough terrains. In this paper, 3 design requirements are proposed to improve the mobility and environmental adaptability of the robot. To meet these 3 requirements, 2 design principles for each requirement are put forward. First, for light weight and low inertia with high stiffness, 3-dimensional printing technology and lightweight material are adopted. Second, the integrated hydraulically driven unit is used for high power density and fast response actuation. Third, the micro-hydraulic power unit achieves power autonomy, adopting the hoseless design to strengthen the reliability of the hydraulic system. What is more, the control system including hierarchical distributed electrical system and control strategy is presented. The mobility and adaptability of WLR-3P are demonstrated with a series of experiments. Finally, the robot can achieve a speed of 13.6 km/h and a jumping height of 0.2 m.

The development of robots with the ability to adapt and perform effectively across a wide range of different environments has been a significant challenge in the field of robotics. The conventional approach often relies on complex environmental state estimation and distinct control strategies tailored for each environment. As an initial step towards creating robotic systems for multiple environments, we aim to address this challenge by focusing on two fundamentally different dynamic settings: land and water. Our research introduces ‘implicit control’ based robot design philosophy, which views the interaction between the robot's body and the environment as components that can be utilized to achieve the control purpose, rather than treating them as disturbances to be overcome. By harnessing this interaction for our control purposes, we envision a mobile robot capable of autonomously adapting to the dynamics of vastly different environments. In this study, we present our ‘implicit control’ based designed amphibious multi-legged mobile robot called ‘i-AMMR,’ which shows remarkable adaptability with simple and single movement for both environments without external sensory feedback. Our study underscores the potential of ‘implicit control’ as an innovative approach to pursuing adaptable and versatile robotic systems for multiple environments. GRAPHICAL ABSTRACT

The wheel-legged robot combines the functions of wheeled vehicles and legged robots: high speed and high passability. However, the limited performance of existing joint actuators has always been the bottleneck in the actual applications of large wheel-legged robots. This paper proposed a highly integrated electro-hydrostatic actuator (EHA) to enable high-dynamic performance in giant wheel-legged robots (>200 kg). A prototype with a high force-to-weight ratio was developed by integrating a micropump, a miniature spring accumulator, and a micro-symmetrical cylinder. The prototype achieves a large output force of more than 9400 N and a high force-to-weight ratio of more than 2518 N/kg. Compared with existing EHA-based robots, it has a higher force-to-weight ratio and can bear larger loads. A detailed EHA model was presented, and controllers were designed based on sliding mode control and PID methods to control the output position and force of the piston. The model’s accuracy is improved by identifying uncertain parameters such as friction and leakage coefficient. Finally, both simulations and experiments were carried out. The results verified the fast response of force control (step response within 50 ms, the force tracking control frequency about 6.7 Hz) and the developed EHA’s good potential for future large wheel-legged robots.

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This article mainly introduces a dynamically stabilized and balanced point-footed bipedal robot, which we named MPLBR (Multi-locomotion Parallel-Legged Bipedal Robot). The robot structure is designed with a symmetric architecture, including six motors and a five-link double crank mechanism for extending the legs. Based on the inverted pendulum model, a robot control process framework is proposed using the virtual model control (VMC) algorithm and the decoupling balance control algorithm, combined with robot gait planning and state detection, to achieve stable walking and hopping control of the robot. Through simulation experiments and physical prototype experiments to verify the algorithm, the robot can achieve stable walking control and bounce motion modes.

Reduced-order models (ROMs) provide a powerful means of synthesizing dynamic walking gaits on legged robots. Yet this approach lacks the formal guarantees enjoyed by methods that utilize the full-order model (FOM) for gait synthesis, e.g., hybrid zero dynamics. This paper aims to unify these approaches through a layered control perspective. In particular, we establish conditions on when a ROM of locomotion yields stable walking on the full-order hybrid dynamics. To achieve this result, given an ROM we synthesize a zero dynamics manifold encoding the behavior of the ROM—controllers can be synthesized that drive the FOM to this surface, yielding hybrid zero dynamics. We prove that a stable periodic orbit in the ROM implies an input-to-state stable periodic orbit of the FOM’s hybrid zero dynamics, and hence the FOM dynamics. This result is demonstrated in simulation on a linear inverted pendulum ROM and a 5-link planar walking FOM.

This paper presents an extension to the periodic autoencoder architecture for learning compact representations of locomotion gaits in legged robots. Unlike conventional time-series models that require storing full joint trajectories, our approach exploits the inherent periodicity of locomotion to encode motion patterns using a minimal set of Fourier parameters: frequency, amplitude, offset, and phase difference. The encoder-decoder framework employs 1D convolutional layers to transform high-dimensional joint angle sequences into low-dimensional latent embeddings, which are then decomposed via a differentiable Fast Fourier Transform (FFT) module. A carefully designed multi-term loss function enforces accurate trajectory reconstruction, sinusoidal consistency in the latent space, and consistency of Fourier parameters within each gait type. Experimental evaluation on the Loco-MuJoCo Unitree H1 dataset demonstrates that our method accurately reconstructs both walking and running gaits while significantly reducing storage requirements. The learned periodic representations effectively capture the coordination dynamics of legged locomotion and provide a compact foundation for efficient motion generation and control.

Hybrid dynamical systems, which include continuous flow and discrete mode switching, can model robotics tasks like legged robot locomotion. Model-based methods usually depend on predefined gaits, while model-free approaches lack explicit mode-switching knowledge. Current methods identify discrete modes via segmentation before regressing continuous flow, but learning high-dimensional complex rigid body dynamics without trajectory labels or segmentation is a challenging open problem. This paper introduces Discrete-time Hybrid Automata Learning (DHAL), a framework to identify and execute mode-switching without trajectory segmentation or event function learning. Besides, we embedded it in reinforcement learning pipeline and incorporates a beta policy distribution and a multi-critic architecture to model contact-guided motions, exemplified by a challenging quadrupedal robot skateboard task. We validate our method through sufficient real-world tests, demonstrating robust performance and mode identification consistent with human intuition in hybrid dynamical systems.

The performance of a model-based controller can severely suffer when its model inaccurately represents the real world dynamics. We propose to learn a time-varying, locally linear residual model along the robot’s current trajectory, to compensate for the prediction errors of the controller’s model. Supervised learning is performed online, as the robot is running in the unknown environment, using data collected from its immediate past. We theoretically investigate our method in its general formulation, then apply it to a bipedal controller derived from the full-order dynamics of virtual constraints, and a quadrupedal controller derived from a simplified model of contact forces. For a biped in simulation, our method consistently outperforms the baseline and a recent learning-based method. We also experiment with a 12 kg quadruped in simulation and real world, where the baseline fails to walk with 10 kg of payload but our method succeeds.

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

Generalizing locomotion policies across diverse legged robots with varying morphologies is a key challenge due to differences in observation/action dimensions and system dynamics. In this work, we propose Multi-Loco, a novel unified framework combining a morphology-agnostic generative diffusion model with a lightweight residual policy optimized via reinforcement learning (RL). The diffusion model captures morphology-invariant locomotion patterns from diverse cross-embodiment datasets, improving generalization and robustness. The residual policy is shared across all embodiments and refines the actions generated by the diffusion model, enhancing task-aware performance and robustness for real-world deployment. We evaluated our method with a rich library of four legged robots in both simulation and real-world experiments. Compared to a standard RL framework with PPO, our approach -- replacing the Gaussian policy with a diffusion model and residual term -- achieves a 10.35% average return improvement, with gains up to 13.57% in wheeled-biped locomotion tasks. These results highlight the benefits of cross-embodiment data and composite generative architectures in learning robust, generalized locomotion skills.

Humanoid soccer dribbling is a highly challenging task that demands dexterous ball manipulation while maintaining dynamic balance. Traditional rule-based methods often struggle to achieve accurate ball control due to their reliance on fixed walking patterns and limited adaptability to real-time ball dynamics. To address these challenges, we propose a two-stage curriculum learning framework that enables a humanoid robot to acquire dribbling skills without explicit dynamics or predefined trajectories. In the first stage, the robot learns basic locomotion skills; in the second stage, we fine-tune the policy for agile dribbling maneuvers. We further introduce a virtual camera model in simulation that simulates the field of view and perception constraints of the real robot, enabling realistic ball perception during training. We also design heuristic rewards to encourage active sensing, promoting a broader visual range for continuous ball perception. The policy is trained in simulation and successfully transferred to a physical humanoid robot. Experiment results demonstrate that our method enables effective ball manipulation, achieving flexible and visually appealing dribbling behaviors across multiple environments. This work highlights the potential of reinforcement learning in developing agile humanoid soccer robots. Additional details and videos are available at https://zhuoheng0910.github.io/dribble-master/.

We provide an algorithm for adaptive legged locomotion via online learning and model predictive control. The algorithm is composed of two interacting modules: model predictive control (MPC) and online learning of residual dynamics. The residual dynamics can represent modeling errors and external disturbances. We are motivated by the future of autonomy where quadrupeds will autonomously perform complex tasks despite real-world unknown uncertainty, such as unknown payload and uneven terrains. The algorithm uses random Fourier features to approximate the residual dynamics in reproducing kernel Hilbert spaces. Then, it employs MPC based on the current learned model of the residual dynamics. The model is updated online in a self-supervised manner using least squares based on the data collected while controlling the quadruped. The algorithm enjoys sublinear <italic>dynamic regret</italic>, defined as the suboptimality against an optimal clairvoyant controller that knows how the residual dynamics. We validate our algorithm in Gazebo and MuJoCo simulations, where the quadruped aims to track reference trajectories. The Gazebo simulations include constant unknown external forces up to <inline-formula><tex-math notation="LaTeX">$12\boldsymbol{g}$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\boldsymbol{g}$</tex-math></inline-formula> is the gravity vector, in flat terrain, slope terrain with <inline-formula><tex-math notation="LaTeX">$20^{\circ }$</tex-math></inline-formula> inclination, and rough terrain with <inline-formula><tex-math notation="LaTeX">$0.25\,\text{m}$</tex-math></inline-formula> height variation. The MuJoCo simulations include time-varying unknown disturbances with payload up to <inline-formula><tex-math notation="LaTeX">$\text{8}\,\text{kg}$</tex-math></inline-formula> and time-varying ground friction coefficients in flat terrain.

Wheel-legged robots integrate the adaptability of legged locomotion with the efficiency of wheeled movement, enabling agile traversal across diverse terrains. However, abrupt terrain transitions introduce substantial state variations, including velocity fluctuations, posture shifts, and slippage, which pose significant challenges to locomotion stability. To address these issues, we propose a state error compensation framework that integrates a residual network with a trust-region mechanism. The residual network implicitly captures nonlinear contact dynamics, enabling real-time correction of slippage-induced state deviations, while the trust-region mechanism regulates compensation amplitude to maintain stable locomotion. Furthermore, we introduce a dual-source contrastive learning strategy, which explicitly differentiates terrain-induced transitions from external perturbations, facilitating context-aware error recovery. The proposed framework is integrated into a model-free reinforcement learning pipeline, ensuring adaptability to previously unseen environments. To further enhance robustness, an uncertainty-aware calibration module is introduced. This module dynamically adjusts the trust region boundary in real time, leveraging sensory feedback to adaptively constrain residual corrections and prevent over-adjustment, thereby maintaining stability during diverse terrain transitions. Experimental results demonstrate that the proposed framework achieves a 96.7% terrain traversal success rate and 92% velocity tracking accuracy under dynamic disturbances. On unstructured and mixed terrains, it maintains a mean velocity tracking error of 0.15 m/s and stable posture, with pitch and roll angles constrained to ±0.04 rad and ±0.02 rad, respectively. Note to Practitioners—This work addresses the challenge of maintaining stable locomotion for wheel-legged robots when encountering abrupt terrain transitions. Conventional control methods often struggle with state variations such as slippage and posture shifts, leading to instability. To tackle this issue, we propose a state error compensation framework that integrates a residual network with a trust-region mechanism, ensuring real-time correction while maintaining control robustness. The framework is designed for seamless deployment in model-free reinforcement learning settings, requiring no prior terrain modeling. Our method is expected to facilitate real-world applications where reliable locomotion across diverse terrains is essential, such as autonomous exploration and disaster response.

Most terrestrial animals move with a specific number of propulsive legs, which differs between clades. The reasons for these differences are often unknown and rarely queried, despite the underlying mechanisms being indispensable for understanding the evolution of multilegged locomotor systems in the animal kingdom and the development of swiftly moving robots. Moreover, when speeding up, a range of species change their number of propulsive legs. The reasons for this behaviour have proven equally elusive. In animals and robots, the number of propulsive legs also has a decisive impact on the movement dynamics of the centre of mass. Here, I use the leg force interference model to elucidate these issues by introducing gradually declining ground reaction forces in locomotor apparatuses with varying numbers of leg pairs in a first numeric approach dealing with these measures’ impact on locomotion dynamics. The effects caused by the examined changes in ground reaction forces and timing thereof follow a continuum. However, the transition from quadrupedal to a bipedal locomotor system deviates from those between multilegged systems with different numbers of leg pairs. Only in quadrupeds do reduced ground reaction forces beneath one leg pair result in increased reliability of vertical body oscillations and therefore increased energy efficiency and dynamic stability of locomotion.

Accurate proprioceptive odometry is fundamental for legged robot navigation in GPS-denied and visually degraded environments where conventional visual odometry systems fail. Current approaches face critical limitations: analytical filtering methods suffer from modeling uncertainties and cumulative drift, hybrid learning-filtering approaches remain constrained by their analytical components, while pure learning-based methods struggle with simulation-to-reality transfer and demand extensive real-world data collection. This paper introduces AutoOdom, a novel autoregressive proprioceptive odometry system that overcomes these challenges through an innovative two-stage training paradigm. Stage 1 employs large-scale simulation data to learn complex nonlinear dynamics and rapidly changing contact states inherent in legged locomotion, while Stage 2 introduces an autoregressive enhancement mechanism using limited real-world data to effectively bridge the sim-to-real gap. The key innovation lies in our autoregressive training approach, where the model learns from its own predictions to develop resilience against sensor noise and improve robustness in highly dynamic environments. Comprehensive experimental validation on the Booster T1 humanoid robot demonstrates that AutoOdom significantly outperforms state-of-the-art methods across all evaluation metrics, achieving 57.2% improvement in absolute trajectory error, 59.2% improvement in Umeyama-aligned error, and 36.2% improvement in relative pose error compared to the Legolas baseline. Extensive ablation studies provide critical insights into sensor modality selection and temporal modeling, revealing counterintuitive findings about IMU acceleration data and validating our systematic design choices for robust proprioceptive odometry in challenging locomotion scenarios.

In the field of robotics research involving mechatronics and advanced control technology, four-link tandem wheel-legged robots frequently face control instability issues. This study proposes a control strategy that combines LQR (Linear Quadratic Regulator) real-time feedback with VMC (Virtual Model Control) system mapping. The system dynamics model is established using the Newton-Euler method, then simplified and transformed into an inverted-pendulum model. The LQR algorithm is employed to control the simplified system, while VMC is utilized to map the leg virtual torque to the hip-joint motor output torque. MATLAB/Simulink simulations demonstrate the high adaptability of the model in complex environments and its practical application potential.

Model predictive control is one of the most common methods for stabilizing the dynamics of a legged robot. Yet, it remains unclear which level of complexity should be considered for modeling the system dynamics. On the one hand, most embedded pipelines for legged locomotion rely on reduced models with low computational load in order to ensure real-time capabilities at the price of not exploiting the full potential of the whole-body dynamics. On the other hand, recent numerical solvers can now generate whole-body trajectories on the fly while still respecting tight time constraints. This paper compares the performances of common dynamic models of increasing complexity (centroidal, kino-dynamics, and whole-body models) in simulation over locomotion problems involving challenging gaits, stairs climbing and balance recovery. We also present a 3-D kino-dynamics model that reformulates centroidal dynamics in the coordinates of the base frame by efficiently leveraging the centroidal momentum equation at the acceleration level. This comparative study uses the humanoid robot Talos and the augmented Lagrangian-based solver Aligator to enforce hard constraints on the optimization problem.

Legged aerial-terrestrial robots have garnered significant research attention in recent years due to their enhanced environmental adaptability through combined aerial and terrestrial locomotion. However, existing passive spring-legged aerial robots exhibit limited motion versatility, demonstrating single stance gait during ground impacts, which constrains their task adaptability and creates substantial challenges in hybrid trajectory optimization and switching control. To address these difficulties, this work presents a systematic solution to achieve diverse hybrid locomotion. We innovatively establish the differential flatness property for spring-legged quadrotors in both aerial and terrestrial domains, and propose a unified hybrid trajectory optimization framework that generates smooth, agile, and dynamically feasible multi-modal trajectories incorporating diverse stance gait patterns. Furthermore, a hybrid nonlinear model predictive controller with a trajectory extension strategy is developed to enhance hybrid tracking precision and mode transition execution. Compared to existing methods, we achieve a 27% reduction in tracking error during hybrid locomotion while maintaining high-precision foot placement. The source code will be released to benefit the community1

Loco-manipulation demands coordinated whole-body motion to manipulate objects effectively while maintaining locomotion stability, presenting significant challenges for both planning and control. In this work, we propose a whole-body model predictive control (MPC) framework that directly optimizes joint torques through full-order inverse dynamics, enabling unified motion and force planning and execution within a single predictive layer. This approach allows emergent, physically consistent whole-body behaviors that account for the system’s dynamics and physical constraints. We implement our MPC formulation using open software frameworks (Pinocchio and CasADi), along with the state-of-the-art interior-point solver Fatrop. In real-world experiments on a Unitree B2 quadruped equipped with a Unitree Z1 manipulator arm, our MPC formulation achieves real-time performance at 80 Hz. We demonstrate loco-manipulation tasks that demand fine control over the end-effector’s position and force to perform real-world interactions like pulling heavy loads, pushing boxes, and wiping whiteboards.

In this paper, we introduce a novel approach to centroidal state estimation, which plays a crucial role in predictive model-based control strategies for dynamic legged locomotion. Our approach uses the Koopman operator theory to transform the robot’s complex nonlinear dynamics into a linear system, by employing dynamic mode decomposition and deep learning for model construction. We evaluate both models on their linearization accuracy and capability to capture both fast and slow dynamic system responses. We then select the most suitable model for estimation purposes, and integrate it within a moving horizon estimator. This estimator is formulated as a convex quadratic program to facilitate robust, real-time centroidal state estimation. Through extensive simulation experiments on a quadruped robot executing various dynamic gaits, our data-driven framework outperforms conventional Extended Kalman Filtering technique based on nonlinear dynamics. Our estimator addresses challenges posed by force/torque measurement noise in highly dynamic motions and accurately recovers the centroidal states, demonstrating the adaptability and effectiveness of the Koopman-based linear representation for complex locomotive behaviors. Importantly, our model based on dynamic mode decomposition, trained with two locomotion patterns (trot and jump), successfully estimates the centroidal states for a different motion (bound) without retraining.

This manuscript primarily aims to enhance the performance of whole-body controllers(WBC) for underactuated legged locomotion. We introduce a systematic parameter design mechanism for the floating-base feedback control within the WBC. The proposed approach involves utilizing the linearized model of unactuated dynamics to formulate a Linear Quadratic Regulator(LQR) and solving a Riccati gain while accounting for potential physical constraints through a second-order approximation of the log-barrier function. And then the user-tuned feedback gain for the floating base task is replaced by a new one constructed from the solved Riccati gain. Extensive simulations conducted in MuJoCo with a point bipedal robot, as well as real-world experiments performed on a quadruped robot, demonstrate the effectiveness of the proposed method. In the different bipedal locomotion tasks, compared with the user-tuned method, the proposed approach is at least 12% better and up to 50% better at linear velocity tracking, and at least 7% better and up to 47% better at angular velocity tracking. In the quadruped experiment, linear velocity tracking is improved by at least 3% and angular velocity tracking is improved by at least 23% using the proposed method.

For decades, the field of biologically inspired robotics has leveraged insights from animal locomotion to improve the walking ability of legged robots. Recently, ‘biomimetic’ robots have been developed to model how specific animals walk. By prioritizing biological accuracy to the target organism rather than the application of general principles from biology, these robots can be used to develop detailed biological hypotheses for animal experiments, ultimately improving our understanding of the biological control of legs while improving technical solutions. In this work, we report the development and validation of the robot Drosophibot II, a meso-scale robotic model of an adult fruit fly, Drosophila melanogaster. This robot is novel for its close attention to the kinematics and dynamics of Drosophila, an increasingly important model of legged locomotion. Each leg’s proportions and degrees of freedom have been modeled after Drosophila 3D pose estimation data. We developed a program to automatically solve the inverse kinematics necessary for walking and solve the inverse dynamics necessary for mechatronic design. By applying this solver to a fly-scale body structure, we demonstrate that the robot’s dynamics fit those modeled for the fly. We validate the robot’s ability to walk forward and backward via open-loop straight line walking with biologically inspired foot trajectories. This robot will be used to test biologically inspired walking controllers informed by the morphology and dynamics of the insect nervous system, which will increase our understanding of how the nervous system controls legged locomotion.

In this letter, we present a fast and decentralized state estimation framework for the control of legged locomotion. The nonlinear estimation of the floating base states is decentralized to an orientation estimation via Extended Kalman Filter (EKF) and a linear velocity estimation via Moving Horizon Estimation (MHE). The EKF fuses the inertia sensor with vision to estimate the floating base orientation. The MHE uses the estimated orientation with all the sensors within a time window in the past to estimate the linear velocities based on a time-varying linear dynamics formulation of the interested states with state constraints. More importantly, a marginalization method based on the optimization structure of the full information filter (FIF) is proposed to convert the equality-constrained FIF to an equivalent MHE. This decoupling of state estimation promotes the desired balance of computation efficiency, accuracy of estimation, and the inclusion of state constraints. The proposed method is shown to be capable of providing accurate state estimation to several legged robots, including the highly dynamic hopping robot PogoX, the bipedal robot Cassie, and the quadrupedal robot Unitree Go1, with a frequency at 200 Hz and a window interval of 0.1 s.

This letter presents a control framework that combines model-based optimal control and reinforcement learning (RL) to achieve versatile and robust legged locomotion. Our approach enhances the RL training process by incorporating on-demand reference motions generated through finite-horizon optimal control, covering a broad range of velocities and gaits. These reference motions serve as targets for the RL policy to imitate, leading to the development of robust control policies that can be learned with reliability. Furthermore, by utilizing realistic simulation data that captures whole-body dynamics, RL effectively overcomes the inherent limitations in reference motions imposed by modeling simplifications. We validate the robustness and controllability of the RL training process within our framework through a series of experiments. In these experiments, our method showcases its capability to generalize reference motions and effectively handle more complex locomotion tasks that may pose challenges for the simplified model, thanks to RL's flexibility. Additionally, our framework effortlessly supports the training of control policies for robots with diverse dimensions, eliminating the necessity for robot-specific adjustments in the reward function and hyperparameters.

Mobile robots can replace humans to fulfill tasks in dangerous environments, which has been a research focus in recent years. This paper proposes a wheel-legged mobile robot with multi-locomotion and a low-energy consumption planning method. Different from the existing wheel-legged mobile robots, it can realize low-energy movement in different terrains with simple structures, and it can realize three modes: synchronous, tumbling, and curl–stretch. Then, based on the kinematics and dynamics model, a low-energy planning method is proposed, and low-energy motion planning is carried out for the three modes to obtain the low-energy driving law in each mode. A robot prototype is developed, and the experimental results show that the robot can move through the three modes with lower energy consumption in all three terrains. The planning method provides an effective reference for applying wheel-legged mobile robots.

Legged robot control has improved in recent years with the rise of deep reinforcement learning, however, much of the underlying neural mechanisms remain difficult to interpret. Our aim is to leverage bio-inspired methods from computational neuroscience to better understand the neural activity of robust robot locomotion controllers. Similar to past work, we observe that terrain-based curriculum learning improves agent stability. We study the biomechanical responses and neural activity within our neural network controller by simultaneously pairing physical disturbances with targeted neural ablations. We identify an agile hip reflex that enables the robot to regain its balance and recover from lateral perturbations. Model gradients are employed to quantify the relative degree that various sensory feedback channels drive this reflexive behavior. We also find recurrent dynamics are implicated in robust behavior, and utilize sampling-based ablation methods to identify these key neurons. Our framework combines model-based and sampling-based methods for drawing causal relationships between neural network activity and robust embodied robot behavior.

In this paper, we propose an improved and effective control framework for wheel-legged vehicles. Our framework allows the vehicle to navigate uneven terrains while maintaining balance and tracking the trajectory of the center of mass (CoM). In detail, single rigid body dynamics (SRBD) and whole body dynamics (WBD) are introduced to describe the body motion at first. Then, based on the SRBD, by introducing the output of the longitudinal speed controller as a constraint, a body balance controller is designed, which can simultaneously optimize the wheel driving force and the support force required by the body. The joint controller is designed through a WBD model and the overall dynamics compensation is performed. Ultimately, by adjusting gain weights to coordinate different control computation results, whole vehicle control is achieved. Validation in simulation demonstrates that our wheel-legged vehicle can navigate uneven terrain at a maximum of 3.0m/s, showing the feasibility of this framework.

This paper presents a nonlinear model predictive control (NMPC) toward versatile motion generation for the telescopic-wheeled-legged robot Tachyon 3, the unique hardware structure of which poses challenges in control and motion planning. We apply the full-centroidal NMPC formulation with dedicated constraints that can capture the accurate kinematics and dynamics of Tachyon 3. We have developed a control pipeline that includes an internal state integrator to apply NMPC to Tachyon 3, the actuators of which employ high-gain position-controllers. We conducted simulation and hardware experiments on the perceptive locomotion of Tachyon 3 over structured terrains and demonstrated that the proposed method can achieve smooth and dynamic motion generation under harsh physical and environmental constraints.

Designing control policies for legged locomotion11In this work, we specifically consider quadruped locomotion. is complex due to the under-actuated and non-continuous robot dynamics. Model-free reinforcement learning provides promising tools to tackle this challenge. However, a major bottleneck of applying model-free reinforcement learning in real world is safety. In this paper, we propose a safe reinforcement learning framework that switches between a safe recovery policy that prevents the robot from entering unsafe states, and a learner policy that is optimized to complete the task. The safe recovery policy takes over the control when the learner policy violates safety constraints, and hands over the control back when there are no future safety violations. We design the safe recovery policy so that it ensures safety of quadruped locomotion while minimally intervening in the learning process. Furthermore, we theoretically analyze the proposed framework and provide an upper bound on the task performance. We verify the proposed framework in four tasks on a simulated and real quadrupedal robot: efficient gait, catwalk, two-leg balance, and pacing. On average, our method achieves 48.6% fewer falls and comparable or better rewards than the baseline methods in simulation. When deployed it on real-world quadruped robot, our training pipeline enables 34% improvement in energy efficiency for the efficient gait, 40.9% narrower of the feet placement in the catwalk, and two times more jumping duration in the two-leg balance. Our method achieves less than five falls over the duration of 115 minutes of hardware time.22Video is included in the submission and the project website: https://sites.google.com/view/saferlleggedlocomotion/

Deep reinforcement learning produces robust locomotion policies for legged robots over challenging terrains. To date, few studies have leveraged model-based methods to combine these locomotion skills with the precise control of manipulators. Here, we incorporate external dynamics plans into learning-based locomotion policies for mobile manipulation. We train the base policy by applying a random wrench sequence on the robot base in simulation and add the noisified wrench sequence prediction to the policy observations. The policy then learns to counteract the partially-known future disturbance. The random wrench sequences are replaced with the wrench prediction generated with the dynamics plans from model predictive control to enable deployment. We show zero-shot adaptation for manipulators unseen during training. On the hardware, we demonstrate stable locomotion of legged robots with the prediction of the external wrench.

Understanding and predicting how mechanical systems respond to environmental variability is essential for advancing next-generation robotic systems with physical intelligence. In this study, we investigated the use of echo state networks (ESNs), a representative class of reservoir computing (RC) models, to predict the bifurcation structures of real-world mechanical systems from limited observations. We examined two representative cases: a simulated passive dynamic walking (PDW) robot with hybrid continuous-discrete dynamics and a real-world soft pneumatic artificial muscle (PAM) actuator whose electrical resistance undergoes complex changes under varying loads. To address the challenges posed by the PDW's hybrid dynamics, we proposed a hybrid ESN (HESN) model that integrates a knowledge-based touchdown detection mechanism with an ESN module. The HESN successfully reproduced the route-to-chaos bifurcation structure of the PDW, captured its multi-attractor dynamics, and demonstrated robustness against imperfect domain knowledge. For the PAM, where no reliable physical model is available, a purely data-driven ESN accurately predicted resistance bifurcations across changing environmental conditions. These results highlight the potential of RC models as flexible digital twins for mechanical systems, enabling parameter-aware modeling of bifurcations with limited training data and supporting the design of robust, adaptive robots capable of operating in complex environments.

Recurrent neural network-based reinforcement learning systems are capable of complex motor control tasks such as locomotion and manipulation, however, much of their underlying mechanisms still remain difficult to interpret. Our aim is to leverage computational neuroscience methodologies to understanding the population-level activity of robust robot locomotion controllers. Our investigation begins by analyzing topological structure, discovering that fragile controllers have a higher number of fixed points with unstable directions, resulting in poorer balance when instructed to stand in place. Next, we analyze the forced response of the system by applying targeted neural perturbations along directions of dominant population-level activity. We find evidence that recurrent state dynamics are structured and low-dimensional during walking, which aligns with primate studies. Additionally, when recurrent states are perturbed to zero, fragile agents continue to walk, which is indicative of a stronger reliance on sensory input and weaker recurrence.

Optimal control is a successful approach to generate motions for complex robots, in particular for legged locomotion. However, these techniques are often too slow to run in real time for model predictive control or one needs to drastically simplify the dynamics model. In this work, we present a method to learn to predict the gradient and hessian of the problem value function, enabling fast resolution of the predictive control problem with a one-step quadratic program. In addition, our method is able to satisfy constraints like friction cones and unilateral constraints, which are important for high dynamics locomotion tasks. We demonstrate the capability of our method in simulation and on a real quadruped robot performing trotting and bounding motions.

Legged robot locomotion on a dynamic rigid surface (i.e., a rigid surface moving in the inertial frame) involves complex full-order dynamics that is high-dimensional, nonlinear, and time-varying. Towards deriving an analytically tractable dynamic model, this study theoretically extends the reduced-order linear inverted pendulum (LIP) model from legged locomotion on a stationary surface to locomotion on a dynamic rigid surface (DRS). The resulting model is herein termed as DRS-LIP. Furthermore, this study introduces an approximate analytical solution of the proposed DRS-LIP that is computationally efficient with high accuracy. To illustrate the practical uses of the analytical results, they are used to develop a hierarchical planning framework that efficiently generates physically feasible trajectories for DRS locomotion. The effectiveness of the proposed theoretical results and motion planner is demonstrated both through simulations and experimentally on a Laikago quadrupedal robot that walks on a rocking treadmill.

Locomotion through resistive media is an organic occurrence during traversal of the natural world. Due to the complexities required to analyze the effect of these media on the dynamics of locomotion, controllers of legged robots generally neglect or treat them as disturbances. In this paper, we address the challenge of producing optimal locomotion control in resistive media. We do so by applying trajectory optimization techniques within a direct collocation framework onto a reduced-order resistive model of legged locomotion: the Fluid Field SLIP model. The results of this optimization led to five different optimal gaits being found for hopping in air, fluidized media, and at the interface between these fluids. When applying the optimal control policies to a single leg robotic hopper in mixed fluid it was found that the new controller was able to improve its efficiency by 54% over the previous controller. It achieved this by employing a novel "kickback" and retraction maneuver found by the optimizer. This maneuver was found to improve efficiency even in un-optimized controllers when hopping in deep fluid.

The legged locomotion manipulation system (LMS) combines the powerful movement of the legged platform and the operation ability of the robotic manipulation arm, which can replace humans to perform dangerous tasks. This paper focuses on the legged locomotion manipulation system, composed of a quadruped locomotion platform with a six-degree-of-freedom (Dof) robotic arm. After the LMS completes movement tasks and reaches the designated position, the supporting foot point are not moved. The overall kinematics model is established with the parallel kinematics model of the legged LMS and the serial kinematics model of the robotic arm. In whole-body collaborative planning, the goal is to minimize the sum of the squares of all joint torques. With the constraints of whole-body kinematics and statics, dynamics, and stability, the trajectory optimization problem is converted into a nonlinear planning problem. By solving the nonlinear problem, we can get the optimal joint trajectory. Compared with independently planning trajectories of the quadruped locomotion platform and robotic arm, this algorithm can coordinate the motion of the whole joints, optimize the distribution, and reduce the system's power. Simulation experiments verify all algorithms.

The complex dynamics of agile robotic legged locomotion requires motion planning to intelligently adjust footstep locations. Often, bipedal footstep and motion planning use mathematically simple models such as the linear inverted pendulum, instead of dynamically-rich models that do not have closed-form solutions. We propose a real-time optimization method to plan for dynamical models that do not have closed form solutions and experience irrecoverable failure. Our method uses a data-driven approximation of the step-to-step dynamics and of a failure margin function. This failure margin function is an oriented distance function in state-action space where it describes the signed distance to success or failure. The motion planning problem is formed as a nonlinear program with constraints that enforce the approximated forward dynamics and the validity of state-action pairs. For illustration, this method is applied to create a planner for an actuated spring-loaded inverted pendulum model. In an ablation study, the failure margin constraints decreased the number of invalid solutions by between 24 and 47 percentage points across different objectives and horizon lengths. While we demonstrate the method on a canonical model of locomotion, we also discuss how this can be applied to data-driven models and full-order robot models.

Fast constraint satisfaction, frontal dynamics stabilization, and avoiding fallovers in dynamic, bipedal walkers can be pretty challenging. The challenges include underactuation, vulnerability to external perturbations, and high computational complexity that arise when accounting for the system full-dynamics and environmental interactions. In this work, we study the potential roles of thrusters in addressing some of these locomotion challenges in bipedal robotics. We will introduce a thruster-assisted bipedal robot called Harpy. We will capitalize on Harpy’s unique design to propose an optimization-free approach to satisfy gait feasibility conditions. In this thruster-assisted legged locomotion, the reference trajectories can be manipulated to fulfill constraints brought on by ground contact and those prescribed for states and inputs. Unintended changes to the trajectories, especially those optimized to produce periodic orbits, can adversely affect gait stability and hybrid invariance. We will show our approach can still guarantee stability and hybrid invariance of the gaits by employing the thrusters in Harpy. We will also show that the thrusters can be leveraged to robustify the gaits by dodging fallovers or jumping over large obstacles.

In this manuscript, we present a novel method for estimating the stochastic stability characteristics of metastable legged systems using the unscented transformation. Prior methods for stability analysis in such systems often required high-dimensional state space discretization and a broad set of initial conditions, resulting in significant computational complexity. Our approach aims to alleviate this issue by reducing the dimensionality of the system and utilizing the unscented transformation to estimate the output distribution. This technique allows us to account for multiple sources of uncertainty and high-dimensional system dynamics, while leveraging prior knowledge of noise statistics to inform the selection of initial conditions for experiments. As a result, our method enables the efficient assessment of controller performance and analysis of parametric dependencies with fewer experiments. To demonstrate the efficacy of our proposed method, we apply it to the analysis of a one-dimensional hopper and an underactuated bipedal walking simulation with a hybrid zero dynamics controller.

Learning controllers that reproduce legged locomotion in nature has been a longtime goal in robotics and computer graphics. While yielding promising results, recent approaches are not yet flexible enough to be applicable to legged systems of different morphologies. This is partly because they often rely on precise motion capture references or elaborate learning environments that ensure the naturality of the emergent locomotion gaits but prevent generalization. This work proposes a generic approach for ensuring realism in locomotion by guiding the learning process with the spring-loaded inverted pendulum model as a reference. Leveraging on the exploration capacities of Reinforcement Learning (RL), we learn a control policy that fills in the information gap between the template model and full-body dynamics required to maintain stable and periodic locomotion. The proposed approach can be applied to robots of different sizes and morphologies and adapted to any RL technique and control architecture. We present experimental results showing that even in a model-free setup and with a simple reactive control architecture, the learned policies can generate realistic and energy-efficient locomotion gaits for a bipedal and a quadrupedal robot. And most importantly, this is achieved without using motion capture, strong constraints in the dynamics or kinematics of the robot, nor prescribing limb coordination. We provide supplemental videos for qualitative analysis of the naturality of the learned gaits4.

Dynamic locomotion for legged robots is difficult because the system dynamics are highly nonlinear and complex, nominally underactuated and unstable, multi-input and multi-output, as well as time-variant and hybrid. One usually faces the choice between the intricate full-body dynamics which remains computationally expensive and sometimes even intractable, and the empirically simplified model which inevitably limits the locomotion capability. In this paper, we explore the legged robot dynamics from a different perspective. By decomposing the robot into the body and the legs, with interaction forces and moments connecting them, we enjoy a novel method called Dynamic Model Decomposition that involves lower-dimensional dynamics for each subsystem while their composition maintaining the equivalence to the original full-order robot model. Based on that, we further propose a corresponding model predictive control framework via quadratic programming, which con-siders linearly approximated body dynamics with constrained leg reaction forces as inputs. The overall methodology was successfully applied to a planar five-link biped robot. The simulation results show that the robot is capable of body reference tracking, push recovery, velocity tracking, and even blind locomotion on fairly rough terrain. This suggests a promising dynamic motion control scheme in the future.

No abstract available

Walking robot has a strong ability to adapt to the environment: its low requirements on the road, not only can overcome obstacles, but also can go up and down the stairs. A novel quadruped/biped reconfigurable parallel legged walking chair robot is presented. The 6-UPS parallel mechanism is selected as the basic leg, and its kinematics is analyzed by establishing the robot's coordinate system. The gait planning of the walking chair is carried out, and the foot path of the swinging leg is calculated. The length of each branch rod is obtained by simulation, which lays the foundation for motion control of the walking chair.

Walking with load is a common task in daily life and disaster rescue. Long-term load carriage may cause irreversible damage to the human body. Although remarkable progress has been made in the field of wearable robots, it is still far from avoiding interference to human legs, which will lead to energy consumption. In this paper, a novel wearable robot, Centaur, for assisting load carriage has been proposed. The Centaur system consists of two rigid robotic legs of two degrees-of-freedom (DOFs) to transfer load weight to the ground. Different from exoskeletons, the robotic legs of the Centaur are placed behind the human rather than attached to human limbs, which can provide a larger support polygon and avoid additional interference to the wearer. Additionally, the Centaur can attain the locomotion stability of the quadruped while maintaining the motion agility of the biped itself. This paper also presents an interactive motion control strategy based on the human-robot interaction force. This control strategy incorporates legged robotics walking controller and real-time walking trajectory planning to realize the cooperative walking with human beings. Finally, experiments of human walking with load carriage have been conducted on flat terrain to verify the concept of the Centaur system. The result demonstrates that the Centaur system can effectively reduce 70.03% of load weight during the single stance phase, which indicates that the Centaur system provides a new solution for assisting human walking with load-carriage.

Quadrupedal robots often display a rhythmic and cyclical pattern of movement. Central pattern generators (CPG), which can generate rhythmic signals, are extensively used to control biomimetic robots because of their resemblance to the locomotion mechanisms of actual vertebrates. Nevertheless, achieving stable walking in the desired direction can be challenging when solely relying on CPG signals to control joints for quadruped robots. This study suggests combining CPG and deep reinforcement learning (DRL) as a means to achieve stable locomotion in quadrupedal robots. By setting various parameters of the CPG, we generate a library containing various joint trajectories. Through DRL, we optimize the locomotion capabilities of quadruped robot for diverse motion styles generated by different set of CPG parameters. The experimental results in the simulation indicate that our method can maintain a steady forward motion on flat ground while executing a wide variety of motion styles, and it also possesses good exploratory capabilities in unstructured environments.

In this article, we propose an adaptive control method for quadruped robots carrying unknown static loads and realizing dynamic human–robot interaction motion. To enhance the load capacity of quadruped robots, the control method based on pulse impulse and recursion is proposed to identify the actual Center of Mass (CoM) position of the trunk. The actual CoM position of the quadruped robot is estimated according to the moment generated by the pulse, so as to realize the stability control of the quadruped robot after carrying unknown load. On this basis, to enhance the human–robot interaction ability of quadruped robot, the human arm is modeled, combined with the complexity of the manipulator, through the adaptive control method to reduce the negative external disturbances of the human arm, and make good use of the positive contribution. Finally, we carry out load and human–robot interaction experiments in multiple scenarios, and prove that the adaptive control method achieves fast and stable results in the process of quadruped robot carrying loads and human–robot interaction motion.

Vision-based locomotion in outdoor environments presents significant challenges for quadruped robots. Accurate environmental prediction and effective handling of depth sensor noise during real-world deployment remain difficult, severely restricting the outdoor applications of such algorithms. To address these deployment challenges in vision-based motion control, this letter proposes the Redundant Estimator Network (RENet) framework. The framework employs a dual-estimator architecture that ensures robust motion performance while maintaining deployment stability during onboard vision failures. Through an online estimator adaptation, our method enables seamless transitions between estimation modules when handling visual perception uncertainties. Experimental validation on a real-world robot demonstrates the framework's effectiveness in complex outdoor environments, showing particular advantages in scenarios with degraded visual perception. This framework demonstrates its potential as a practical solution for reliable robotic deployment in challenging field conditions.

A humanoid robot has a similar form to humans, allowing it to be deployed directly into many existing infrastructures built for people, making it highly applicable to future industries or services. However, generating motion is challenging, and it is highly affected by disturbances due to its complex dynamic characteristics. This paper proposes a method to enable a biped robot to achieve stable motion in various environments. The capture point (CP) control is used to modify the zero moment point (ZMP) to stabilize the naturally divergent dynamics of the simplified linear inverted pendulum model (LIPM) and a model predictive control (MPC) framework is implemented to generate a center of mass (COM) trajectory that tracks the adjusted ZMP. To minimize angular momentum caused by disturbances and discrepancies between the actual robot and the dynamics model, arm motion is generated through a momentum controller. Various walking simulations were conducted to verify stable whole-body motion.

The quadruped mobile manipulator (QMM) combines the environmental adaptability of a quadruped robot and the operation capabilities of a robotic arm. The design of the hydraulic QMM with an emphasis on large operational workspace considerations is reported, covering both mechanical design and motion planning aspects. A joint optimization method for determining the structural parameters under two operating conditions is introduced to balance the motion range and force output efficiency, reducing the cylinder drive force by 44.5 % 44.5% compared with before optimization. Furthermore, this paper proposes a redundant motion planning method for the QMM, based on null‐space projection and task feasibility assessment to expand the operational workspace. Through coordinated leg–arm motion, the robot's reachable workspace increases by 30.8 % 30.8% . Physical experiments are conducted to demonstrate the effectiveness of the proposed method and the field capabilities of the QMM.

Reinforcement learning (RL)-based motion imitation methods trained on demonstration data can effectively learn natural and expressive motions with minimal reward engineering but often struggle to generalize to novel environments. We address this by proposing a hierarchical RL framework in which a low-level policy is first pre-trained to imitate animal motions on flat ground, thereby establishing motion priors. A subsequent high-level, goal-conditioned policy then builds on these priors, learning residual corrections that enable perceptive locomotion, local obstacle avoidance, and goal-directed navigation across diverse and rugged terrains. Simulation experiments illustrate the effectiveness of learned residuals in adapting to progressively challenging uneven terrains while still preserving the locomotion characteristics provided by the motion priors. Furthermore, our results demonstrate improvements in motion regularization over baseline models trained without motion priors under similar reward setups. Real-world experiments with an ANYmal-D quadruped robot confirm our policy's capability to generalize animal-like locomotion skills to complex terrains, demonstrating smooth and efficient locomotion and local navigation performance amidst challenging terrains with obstacles.

Biped truss climbing robots (BTCRs) are employed to perform high-rise tasks within truss environments, benefiting from their superior transition capabilities and flexible mobility. However, the intricate geometry of these structures poses challenges for robot navigation and operation. To tackle this issue, this paper proposes a novel BTCRs climbing path planning framework based on a progressive multi-layer architecture. The robot’s transition regions between adjacent members are determined efficiently by unfolding three-dimensional truss members onto two-dimensional planes and discretizing them. Initially, leveraging transition analysis, a global truss member route using graph search methods is generated. Subsequently, a mathematical optimization model is introduced to determine transition grips along the global route, minimizing the total number of grips. Finally, the single-step path planner employs an improved rapidly-exploring random tree (RRT)-connect algorithm, guaranteeing collision-free motion between adjacent grips. By integrating these three layers, the framework demonstrates the feasibility and effectiveness of the proposed analysis and algorithms for climbing path planning in simulation tests, using a self-developed BTCR, Climbot.

The configuration of limb joints has a significant impact on the sprawling quadruped locomotion. Specifically, the arrangement of joints in the limbs can affect the effectiveness and efficiency of movement. This paper presents an analysis of the motion performance of sprawling quadruped robots with different joint configurations. A primitive robot model is provided based on morphological design principles, and an inverse kinematics-based planner is developed. Subsequently, simulations are conducted using an open-source robotic simulator, Webots, to test the robot models with various joint configurations. The effectiveness and efficiency of the robots under different conditions are compared to validate the differences in motion performance. Finally, results demonstrate the importance of joint configuration in achieving desirable motion performance for sprawling quadruped robots.

The utilization of single degree-of-freedom (DOF) closed-chain mechanisms as leg mechanisms of quadruped robots has been studied because of its advantages, such as a lightweight structure and simplified control resulting from the decreased number of actuators needed for operation. Furthermore, to improve the scalability of these approaches, researchers have investigated the use of reconfigurable structures. Nevertheless, the traditional method that relies on adjusting the length of links presents a substantial challenge in achieving precise production of various trajectories while maintaining the centroid positions of those trajectories. We propose a novel reconfigurable structure that utilizes the phase difference between the crank gears to generate desired trajectories accurately. The proposed mechanism guarantees that the centroid positions of multiple trajectories remain confined inside the specified border. Comparison with conventional reconfigurable mechanisms demonstrates the ability of the proposed mechanism to achieve high trajectory accuracy over multiple trajectories, even in environments with limited centroid position changes. Additionally, a gait stability analysis of a quadruped robot with the proposed leg mechanism in a simulation environment was conducted to verify its suitability.

This work presents RobotSpot, a novel low-cost quadruped robot designed to address the growing need for accessible and robust platforms in robotics education and research Background. The main aims are to develop a system that combines intelligent control with economic feasibility, emphasizing ease of reproduction and modification. The methods involve integrating hybrid control algorithms and conducting experimental evaluations to assess stability and energy efficiency. The results demonstrate reliable operation with an average stability of 88.1% and energy efficiency of 77.5%, highlighting RobotSpot’s potential as a practical and affordable tool for hands-on learning and innovation, especially in resource-constrained academic environments.

Robust locomotion of quadruped robots in unstructured environments necessitates precise body balance and adaptive leg compliance to mitigate impacts and ensure safe human-robot interactions. We propose a hierarchical control framework that integrates model predictive control with Cartesian impedance control to address these challenges. The MPC module solves an online quadratic programming optimization problem to compute optimal ground reaction forces and joint torques, while adhering to constraints such as system dynamics, friction cones, actuator limits, and safety requirements. Concurrently, a Cartesian impedance controller models the end-effector dynamics as a virtual mass-spring-damper system, enabling compliant responses to external perturbations. By incorporating the impedance-generated torque reference as a soft constraint within the QP formulation, our approach effectively balances compliance objectives with global constraint enforcement. The quadruped robot prototype leverages biomimetic leg kinematics to enhance maneuverability and stability across uneven and irregular terrains. The efficacy of this framework is demonstrated through comprehensive simulations, with gait experiments conducted in the Gazebo environment validating its ability to maintain operational stability and compliance under external disturbances.

Legged robots have great potential in complex environments, but achieving robust and natural locomotion remains difficult due to challenges in generating smooth gaits and resisting disturbances. This article presents a novel reinforcement learning framework that integrates a skeleton-aware graph neural network (GNN), a single-stage teacher–student architecture, a system-response model, and a Wasserstein Adversarial Motion Priors (wAMP) module. The skeleton-aware GNN enriches observations by encoding key node information and link properties, providing structured body information and better spatial awareness on irregular terrains. Unlike conventional two-stage approaches, this method jointly trains teacher and student policies to accelerate learning and improve sim-to-real transfer using hybrid advantage estimation (HAE). The system-response model further enhances robustness by predicting future observations from historical states via contrastive learning, enabling the policy to anticipate terrain variations and external disturbances. Finally, wAMP provides a more stable adversarial imitation method for fitting expert datasets of both flat ground and stair locomotion. Experiments on quadruped robots demonstrate that the proposed approach achieves more natural gaits and stronger robustness than existing baselines.

In robotics, the ability of quadruped robots to perform tasks in industrial, mining, and disaster environments has already been demonstrated. To ensure the safe execution of tasks by the robot, meticulous planning of its foot placements and precise leg control are crucial. Traditional motion planning and control methods for quadruped robots often rely on complex models of both the robot itself and its surrounding environment. Establishing these models can be challenging due to their nonlinear nature, often entailing significant computational resources. However, a more simplified approach exists that focuses on the kinematic model of the robot’s floating base for motion planning. This streamlined method is easier to implement but also adaptable to simpler hardware configurations. Moreover, integrating impedance control into the leg movements proves advantageous, particularly when traversing uneven terrain. This article presents a novel approach in which a quadruped robot employs impedance control for each leg. It utilizes sixth-degree Bézier curves to generate reference trajectories derived from leg velocities within a planar kinematic model for body control. This scheme effectively guides the robot along predefined paths. The proposed control strategy is implemented using the Robot Operating System (ROS) and is validated through simulations and physical experiments on the Go1 robot. The results of these tests demonstrate the effectiveness of the control strategy, enabling the robot to track reference trajectories while showing stable walking and trotting gaits.

Controlling a biped robot to walk through rough terrains is crucial to the robot’s field application. For a human in the workplace, the ability to flexibly transfer motion while walking in some urgent circumstances is necessary. Explicitly, the according scenario can be dodging an approaching object or instantly modifying the target place to step on. The function is also important for humanoid robot workers. Therefore, we proposed a walking control framework that achieves three-dimensional (3-D) walking and transfers the whole body motion when the target stepping location is urgently changed. The proposed framework contains a motion planner which outputs the desired center of mass (CoM) and center of pressure (CoP) trajectories in 3-D space and a hierarchical whole body controller (WBC) that outputs corresponding whole body joints’ trajectories. In the motion planner, the CoM jerk for each loop is calculated by the Linear-Quadratic-Tracker (LQT), a variation of the Linear-Quadratic-Regulator (LQR). The LQT coefficients adapt to the adjusted step length, making the desired CoM and CoP trajectories respond flexibly to the change of target step-stone. In WBC, three levels of tasks are defined, which meet dynamic, kinematic, and viable contact constraints, respectively. The optimal joints’ angular accelerations are obtained by exploiting the nullspace of the first two levels tasks and by quadratic programming (QP) for the third-level task. In the simulations, our method is demonstrated to be effective for the robot to transfer the motion under urgent change of the target step-stone.

In real-world environments, quadruped robots often need to traverse unstructured terrains and may encounter various unexpected impacts. To enhance the motion balance performance of quadruped robots when facing unknown disturbances in such environments, we propose a fuzzy adaptive weight controller based on Model Predictive Control (MPC). By conducting an in-depth analysis of the impact of weight coefficients in the MPC's objective function on control effectiveness, we have designed an adaptive fuzzy algorithm. This algorithm dynamically adjusts the weight coefficients according to the current errors in the robot's roll and pitch angles, as well as their rates of change. Subsequently, the MPC controller calculates the optimal control torques using these adjusted weight coefficients. To validate the effectiveness of this strategy, we conducted simulations of lateral impacts and walking on unstructured terrains in Gazebo. The simulation results demonstrate that, under various test conditions, the proposed adaptive weight MPC controller significantly improves the robot's motion stability, showcasing its strong capability to handle unknown disturbances.

Large language models (LLMs) have shown dominant performance in various language tasks, including code-writing, machine translation, and semantic comprehension. With prompt engineering, LLMs can also comprehend complex tasks and translate them into executable code. These powers offer great potential for controlling the motion of robots. In this paper, we focus on leveraging the ability of LLMs, prompt engineering, and predefined robot action APIs to facilitate high-level motion planning for quadruped robots. With LLMs, we enable the robot to autonomously plan and execute sophisticated actions based on the comprehension of effective prompts. Through various experiments and evaluations, we demonstrate the effectiveness and adaptability of our approach in handling intricate motion tasks. Our research contributes to the advancement of intelligent robotics and paves the way for more versatile quadruped robots in real-world scenarios.

Installing a multi-degree-of-freedom manipulator on a quadruped robot is an important way to expand the application capability of quadruped robots. However, the introduction of the manipulator will greatly increase the difficulty of motion control of the quadruped robot, such as the poorer stability of the system and the difficulty of motion coordination between the quadruped and the manipulator. Aiming at the above problems, a biomimetic intelligent motion control method based on reinforcement learning for a quadruped robot with a manipulator is proposed, which draws on the idea that quadrupeds swing their tails to increase stability. The policy network and the state estimation network are trained separately. The robustness of the algorithm is improved by adding state-space noise and randomizing the robot dynamics parameters. Curriculum learning is introduced to obtain a more robust training direction. Simulation results demonstrate that the control strategy obtained through training effectively improves the robot’s motion stability and enhances the robot’s ability to resist external disturbances by actively moving the manipulator. During rapid motion, the strategy of dynamically adjusting the manipulator improves robot survivability by about 20% compared to the static strategy of the manipulator.

Bionic amphibious quadruped robots are a huge leap in robotics, with the ability to navigate various terrains with agility and efficiency. In this study, the researchers utilized numerical simulations of motion tests to assess the locomotion skills of these extraordinary robotic systems. They learned a lot about their functioning and problems by rigorously analyzing performance parameters across various terrains and conditions, such as terrain adaptability, traversal speed, energy efficiency, stability, and manoeuvrability. The results they obtained highlighted the importance of adaptive control algorithms and mechanical designs in maximizing traversal speed and energy economy, especially while negotiating difficult terrains. In addition, the study demonstrated the interdisciplinary character of research in bionic amphibious quadruped robots, drawing on insights from biomechanics, control theory, robotics, and computational modelling. By combining previous research and utilizing computational simulation approaches, they opened the road for the creation of versatile and durable robotic systems capable of navigating difficult terrains and surroundings with precision and efficiency. Future research efforts could focus on improving design and control, investigating novel propulsion mechanisms, adaptive locomotion tactics, and biomimetic control algorithms to improve performance and adaptability. Addressing the challenges identified in the research and seizing opportunities for innovation will drive the development of robotic systems that push the limits of locomotion capabilities and open up new possibilities for applications in exploration, search and rescue, environmental monitoring, and beyond. 

This paper presents an algorithm that finds a centroidal motion and footstep plan for a Spring-Loaded Inverted Pendulum (SLIP)-like bipedal robot model substantially faster than real-time. This is achieved with a novel representation of the dynamic footstep planning problem, where each point in the environment is considered a potential foothold that can apply a force to the center of mass to keep it on a desired trajectory. For a biped, up to two such footholds per time step must be selected, and we approximate this cardinality constraint with an iteratively reweighted l1-norm minimization. Along with a linearizing approximation of an angular momentum constraint, this results in a quadratic program can be solved for a contact schedule and center of mass trajectory with automatic gait discovery. A 2 s planning horizon with 13 time steps and 20 surfaces available at each time is solved in 142 ms, roughly ten times faster than comparable existing methods in the literature. We demonstrate the versatility of this program in a variety of simulated environments.

This paper introduces a motion planning and cooperative formation control approach for quadruped robots and multi-agent systems. First, in order to improve the efficiency and safety of quadruped robots navigating in complex environments, this paper proposes a new planning method that combines the dynamic model of quadruped robots and a gradient-optimized obstacle avoidance strategy without Euclidean Signed Distance Field. The framework is suitable for both static and slow dynamic obstacle environments, aiming to achieve multiple goals of obstacle avoidance, minimizing energy consumption, reducing impact, satisfying dynamic constraints, and ensuring trajectory smoothness. This approach differs in that it reduces energy consumption throughout the movement from a new perspective. Meanwhile, this method effectively reduces the impact of the ground on the robot, thus mitigating the damage to its structure. Second, we combine the dynamic control barrier function and the virtual leader-follower model to achieve efficient and safe formation control through model predictive control. Finally, the proposed algorithm is validated through both simulations and real-world scenarios testing.

The Wheeled Quadruped Robot (WQR) combines the agility of wheels with the stability of legs for precise landings from elevated platforms. However., achieving a fast and stable descent remains a key challenge. This paper presents a novel method for landing motion planning of a moving WQR from a platform using Virtual Model Control (VMC). The proposed Motion Planning VMC (MPVMC) approach utilizes kinematic principles to ensure a desired descent. During landing., the front legs are controlled by VMC., while the hind legs switch control strategies based on feedback from contact sensors in the wheels. When the front legs detach and the hind legs remain in contact with the platform., the ideal distance between the hind wheels and the torso is adjusted based on the pitch angle from the inertial measurement unit mounted in the torso. Once all wheels are off the platform., the hind legs use PD control in joint space to maximize torque for stabilizing the WQR. As the torso's pitch angle decreases to a desired value., the hind legs switch to VMC in Cartesian space for a soft landing. Simulations confirm the effectiveness of this method., advancing motion planning and control strategies for WQRs in various challenging environments.

To improve the precision of the biped robot’s walking control system, a fuzzy PID controller with a variable theory domain was added to the design of the biped robot’s lower limb control system to optimize the tracking process of joint angular displacement and improve the motion effect. Firstly, the motion capture experiment is carried out to obtain the angular displacement data of the joint. The theoretical angle was calculated according to the biped robot’s lower limb diagram model and kinematics equation, and the expected trajectory was obtained by simple gait planning and using MATLAB to verify the validity of the angle parameters. Then the fuzzy PID controller of the lower limb joint was established, and the variable theory domain fuzzy controller was established by introducing the expansion factor. Finally, the Simulink test bench is built to simulate the two controllers, and the expected data and simulation data are compared. The results show that the error range of the variable theory domain fuzzy controller is only ±2°, and the variable theory domain adaptive fuzzy controller can accurately track the desired trajectory, reduce the joint angular displacement error of the biped robot, ensure the precision and stability of the biped robot motion, and basically meet the requirements of practical production.

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No abstract available

Crawling robots are the focus of intelligent inspection research, and the main feature of this type of robot is the flexibility of in-plane attitude adjustment. The crawling robot HIT_Spibot is a new type of steam generator heat transfer tube inspection robot with a unique mobility capability different from traditional quadrupedal robots. This paper introduces a hierarchical motion planning approach for HIT_Spibot, aiming to achieve efficient and agile maneuverability. The proposed method integrates three distinct planners to handle complex motion tasks: a nonlinear optimization-based base motion planner, a TOPSIS-based base orientation planner, and a Mask-D3QN (MD3QN) algorithm-based gait motion planner. Initially, the robot’s base and foot workspace were delineated through envelope analysis, followed by trajectory computation using Larangian methods. Subsequently, the TOPSIS algorithm was employed to establish an evaluation framework conducive to foundational turning planning. Finally, the MD3QN algorithm trained foot-points to facilitate robot movement along predefined paths. Experimental results demonstrated the method’s adaptability across diverse tube structures, showcasing robust performance even in environments with random obstacles. Compared to the D3QN algorithm, MD3QN achieved a 100% success rate, enhanced average overall scores by 6.27%, reduced average stride lengths by 39.04%, and attained a stability rate of 58.02%. These results not only validate the effectiveness and practicality of the method but also showcase the significant potential of HIT_Spibot in the field of industrial inspection.

To address the limitations and instability against disturbances of the walking gait of existing bipedal robots based on the inverted pendulum model, this paper proposes a new control method based on the Central Pattern Generator (CPG) model. By incorporating Hopf oscillators and Kuramoto terms into the original inverted pendulum model, and by adjusting parameters, this method enables the controller to produce stable and gait-periodic oscillations that match the robot's stride, thereby making the robot's center of gravity more stable and enhancing its disturbance resistance during walking. This study also connects the improved controller with the robot's joint motors and adjusts the robot's leg-lifting height and foot trajectory using Bezier curve interpolation, based on the Zero Moment Point (ZMP) gait generation method, to plan a complete robot gait. Finally, through simulations and experiments of the bipedal humanoid robot walking continuously and under sinusoidal disturbances, it was found that the new model can maintain a smaller fluctuation range of the center of gravity during walking, stabilize more quickly against external disturbances, and produce less noise when disturbed. This verifies the feasibility and effectiveness of the proposed control strategy.

Dynamic jumping with legged robots poses a challenging problem in planning and control. Formulating the jump optimization to allow fast online execution is difficult; efficiently using this capability to generate long-horizon motion plans further complicates the problem. In this work, we present a hierarchical planning framework to address this problem. We first formulate a real-time tractable trajectory optimization for performing omnidirectional jumping. We then embed the results of this optimization into a low dimensional jump feasibility classifier. This classifier is leveraged to produce geometric motion plans that select dynamically feasible jumps while mitigating the effects of the process noise. We deploy our framework on the Mini Cheetah Vision quadruped, demonstrating the robot's ability to generate and execute reliable, goal-oriented plans that involve forward, lateral, and rotational jumps onto surfaces as tall as the robot's nominal hip height. The ability to plan through omnidirectional jumping greatly expands the robot's mobility relative to planners that restrict jumping to the sagittal or frontal planes.

Quadruped robots working in jungles, mountains or factories should be able to move through challenging scenarios. In this paper, we present a control framework for quadruped robots walking over rough terrain. The planner plans the trajectory of the robot's center of gravity by using the normalized energy stability criterion, which ensures that the robot is in the most stable state. A contact detection algorithm based on the probabilistic contact model is presented, which implements event‐based state switching of the quadruped robot legs. And an on‐line detection of contact force based on generalized momentum is also showed, which improves the accuracy of proprioceptive force estimation. A controller combining whole body control and virtual model control is proposed to achieve precise trajectory tracking and active compliance with environment interaction. Without any knowledge of the environment, the experiments of the quadruped robot SDUQuad‐144 climbs over significant obstacles such as 38 cm high steps and 22.5 cm high stairs are designed to verify the feasibility of the proposed method.

Generating natural and physically feasible motions for legged robots has been a challenging problem due to its complex dynamics. In this work, we introduce a novel learning-based framework of autoregressive motion planner (ARMP) for quadruped locomotion and navigation. Our method can generate motion plans with an arbitrary length in an autore-gressive fashion, unlike most offline trajectory optimization algorithms for a fixed trajectory length. To this end, we first construct the motion library by solving a dense set of trajectory optimization problems for diverse scenarios and parameter settings. Then we learn the motion manifold from the dataset in a supervised learning fashion. We show that the proposed ARMP can generate physically plausible motions for various tasks and situations. We also showcase that our method can be successfully integrated with the recent robot navigation frameworks as a low-level controller and unleash the full capability of legged robots for complex indoor navigation.

This paper presents a motion planning algorithm for quadruped locomotion based on density functions. We decompose the locomotion problem into a high-level density planner and a model predictive controller (MPC). Due to density functions having a physical interpretation through the notion of occupancy, it is intuitive to represent the environment with safety constraints. Hence, there is an ease of use to constructing the planning problem with density. The proposed method uses a simplified the model of the robot into an integrator system, where the high-level plan is in a feedback form formulated through an analytically constructed density function. We then use the MPC to optimize the reference trajectory, in which a low-level PID controller is used to obtain the torque level control. The overall framework is implemented in simulation, demonstrating our feedback density planner for legged locomotion. The implementation of work is available at https://github.com/AndrewZheng-1011/legged_planner.

With the rapid advancement of mobile robot technology, its application in power inspection has become increasingly prevalent. Among these applications, legged robots have garnered significant attention due to their exceptional environmental adaptability. This paper focuses on studying the motion controller for quadruped robots. Firstly, a Central Pattern Generator (CPG) based on Van der Pol nonlinear oscillator is proposed. Secondly, a motion controller based on the above CPG is designed for the walk gait and the trot gait of the quadruped robot. The controller can achieve stable gait control by modifying the coupling topology coefficients and the coupling phase differences. Finally, Simulink-Adams joint simulation is employed to validate the effectiveness of the above motion controllers.

This paper deals with a preliminary study of swing trajectory optimization of ground walking quadruped robot, ST-Quad. We aim to generate an optimal foot trajectory with minimal torque operating on the joints during the swing motion. Based on the dynamics of the robot leg with 4-DOF during swinging between arbitrary initial and final positions, we calculate a trajectory that minimizes the summation of joint torques. Through this, it can be known that the torque tendency at an arbitrary position and the optimal trajectory at that position. Finally, we validate the trajectory optimization algorithm through gazebo simulations and experiments using ST-Quad.

The quadruped robots have a good prospect in substation inspection applications because of their stronger terrain adaptability. This paper focuses on the gait planning and motion control of small intelligent quadruped robots. Firstly, the mechanical structure design of a quadruped robot is completed based on SolidWorks software. Secondly, the kinematics of the designed quadruped robot is analyzed, and gait planning of the walk gait and the steering gait is designed. Finally, the experiments of the walk gait and the steering gait are completed on the physical platform of the quadruped robot designed in this paper. When the robot performs the walk gait, the average forward speed is 0.04 m/s. When the robot performs steering gait, the average steering speed is 9°/s. The experimental results show that the gait control method designed in this paper can achieve stable control of quadruped robot motion.

Multi-legged robots exhibit exceptional adaptability and stability in complex environments. However, maintaining locomotion capability when legs are damaged or missing poses significant challenges. This study focuses on a hexapod biomimetic spider robot and proposes a gait planning and stability analysis method applicable to varying numbers of legs. For hexapod robots, three gaits are considered; for pentapod robots (one leg missing), three gaits are analyzed; and for quadruped robots (two legs missing), two gaits are examined. The Center of Gravity (CoG) projection range for stable postures is determined by integrating the CoG projection method with the SP method. Corresponding gait optimization and adjustment strategies are proposed. To validate the proposed method, both simulation platforms and physical robots were developed and tested. Experimental results demonstrate that the proposed approach enables the hexapod robot to maintain good locomotion stability even when legs are missing.

This article explores the integration of fully autonomous legged robots in obstacle filled environments, simultaneously addressing the challenges of navigation and control. Despite the potential of legged robots for dynamic tasks, their deployment in complex environments has been hindered by the difficulty of developing effective autonomous control systems. In particular, the motion planning problem is addressed in this article, by formulating it as a Partially Observable Markov Decision Process (POMDP) and applying Proximal Policy Optimization (PPO), a model-free Deep Reinforcement Learning (DRL) algorithm. To improve sample efficiency and real-world applicability, the proposed method incorporates a Central Pattern Generator (CPG) for motion planning and a Variational Autoencoder (VAE) for terrain representation, reducing the complexity of action and observation spaces. Referred to as the VAE-CPG architecture, its performance is demonstrated using the Unitree Laikago robot within the PyBullet simulation environment, aiming to show its effectiveness in simulated construction sites. Our findings indicate that by reducing the legged action space to periodic gait patterns and optimizing the gait based on sensory feedback, we achieve enhanced adaptability and efficiency. This work presents a viable means towards the deployment of autonomous legged robots and their improved efficiency in real applications.

The wheel-legged quadruped robot has high mobility efficiency and strong obstacle-crossing ability, which combines the advantages of both wheeled and legged robots. Most robots do not have satisfactory collaborative planning strategies for wheeled and legged mobile mechanisms. This article proposes a wheel step composite gait planning algorithm for quadruped robots, which utilizes the advantage of wheel mechanisms providing support during movement. When a single leg is in the swing phase, the positions of the remaining three wheels are adjusted through the wheeled mobile mechanisms. Analyze the change of the center of mass (COM) of the robot when a single leg is moving, and project the COM on the plane formed by the other three footholds. By optimizing the best foothold of the supporting legs, the stability margin and the robot’s anti-overturning ability are improved. Compared with the original method, the climbing slope of the robot during the single-leg swing is increased by 14.6 degrees, and the stability margin is increased from -52.8mm to 130mm.

To realize continuous legged locomotion between different walls, especially the large-angled walls, we present a gait planning method for a underwater legged robot based on central pattern generator (CPG) and back propagation (BP) neural network in this paper. We use CPG as the signal generator for hip joint of each leg, and collect the data set in Gazebo. After that, we set up a BP neural network to fit the mapping relationship between the joint rotation angle and the output of CPG. Then, we use the trained network to generate adaptive gait autonomously for our robot. The test results in Gazebo verify the effectiveness of our method.

In this letter, we address path planning for the quasi-static locomotion of a multi-legged walking robot on terrains with limited available footholds, such as passing a water stream over rocks. The task is to find a feasible sequence of steps to navigate the robot in environments where precise foot placement and order of the leg movements are necessary for successful traversal. A finite set of the considered footholds forms a state-space search domain, where states are defined by pairing the robot legs with footholds. The actions represent the connectivity of submanifolds of the robot configuration space approximating the robot's kinematic constraints indicating possible steps in a given stance. We propose a novel heuristic that significantly reduces the number of expanded states in the A* planner by avoiding local minima exhibited by commonly used heuristics. The computational requirements are nearly an order of magnitude lower than for the existing contact-driven solutions reported in the literature for similarly formulated planning problems. The viability of the proposed approach is further supported by an experimental deployment.

In order to meet the requirements of keeping the body posture level with the ground during the movement of the quadruped robots, this paper designs a quadruped robot with three joints and three degrees of freedom in each leg, which is a 3-RRP kinematic chain. The legs and body of the quadruped robot are virtually modeled by SolidWorks. Combining the forward and inverse kinematics analysis, the reasonable and ideal foot trajectory and body motion trajectory are planned in Matlab. The gait stability is based on the ZMP stability criterion and is quantified by the stability margin. According to the structural characteristics of the quadruped robot in this paper, the body posture adjustment strategy is proposed. Finally, the simulation is carried out under the two gait plannings of straight driving and turning using ADAMS. The simulation results show that the quadruped robot has a reasonable walking gait on the flat road surface with little impact on the body. The minimum stability margin of the strategy with body adjustment is improved compared with the strategy without body adjustment.

Robots, as a form of technology with various designs, have infinite potential to serve in multiple areas. Hexapod robots are a design that mimics the six-legged structure of many insects. The six-legged structure offers significant advantages in stability, adaptability to diverse environments, and fault tolerance compared to other widely used robots. A well-designed algorithm for gait planning is required to control this complex system. Fixed pattern gait planning is limited in multiple ways, so advanced gait planning methods are developed. The paper compares the advantages of two commonly used gait planning algorithms: Reinforcement Learning (RL) and Central Pattern Generators (CPGs). RL enables hexapod robots to discover the best gaits by employing a process of trial and error. This enables them to adapt dynamically to complex and changing environments. However, flexibility comes at the cost of requiring significant computational resources and extensive training time. CPGs leverage biological principles to produce rhythmic and stable movement patterns through simpler, oscillatory control mechanisms, offering robust and energy-efficient gait generation. While CPGs provide quick and reliable solutions with minimal computation, they lack the adaptability to sudden disturbance without enough data for the environment. Future work will focus on developing hybrid approaches that effectively combine the strengths of both methods.

Traditional rovers cannot maneuver easily through challenging environments, which limits their movement and exploration in planetary missions. Legged robots are potential solutions to replace them, but a detailed investigation is required to evaluate the mobility and the system efficiency. Control and gait-generation issues will have to be tackled to enhance the locomotion of a legged robot in such adverse environments. The paper will discuss the design and development of a Quadruped legged robot with emphasis on solving the most critical problems with mobility, control, gait generation, and power systems. The paper points to the possibilities of multifunction multi-modal mobile quadruped robots to be used in the exploration of planetary terrains, where their velocity and energy consumption are more advantageous than conventional rovers. Raising the state of the art in morphologies of leggings on different terrains and the use of sustainable and efficient on-board power generation systems are the key areas of development. The findings of the simulation suggest that the quadruped robot model was highly versatile as it was capable of climbing a 0.5-meter obstacle and traversing a 30-degree slope without losing its balance. The model showed that the gaits with full flight phases, like running trot/gait, become more efficient with reduced gravity. The findings confirm that the proposed system can afford high-speed and energy-efficient mobility to explore planets.

Based on the ankylosaurus, a four-legged robot structure with 14 degrees of freedom was designed in this study. The kinematic model was established using the Denavit-Hartenberg (D-H) method. The inverse kinematics of the robot was analyzed, and the angle equations of each leg joint were obtained. In order to reduce the kinetic energy generated when leaving the ground and landing, the compound cycloid was modified. The modified foot curve effectively reduces energy and meets the kinematic requirements. On the basis of the foot trajectory, the diagonal leg phase difference was set to 0.5, and the diagonal gait was adopted as the gait of the bionic quadrupled robot. Simulink software was used to construct the simulation environment, and the maximum error of the simulation results and the theoretical step height was 2.05%, and the step length maximum error was 2.39%. During walking, the thigh joint angle was −92.82° to −80.21°, and the hip joint angle was 22.23° to 43.83°. Compared with the simulation trajectory and the theoretical curve, because the experiment did not consider the balance of the fuselage, there was a certain error when the leg was raised to the highest point. Overall, the gait planning strategy designed in this study basically achieves the expected effect, lays a certain foundation for the practical application of the bionic quadruped dinosaur robot, and also provides a reference for the subsequent research.

No abstract available

This study investigates the impact of gait control on the mobility performance of curved-leg hexapod robots across rugged terrains. By integrating bio-inspired design with dynamic gait planning, three locomotion modes—tripod, quadruped, and hexapod gaits—are systematically analyzed. The tripod gait maximizes speed on flat terrain through synchronized leg alternation, reducing friction and optimizing stride frequency. The quadruped gait enhances stability on stairs via a larger support polygon and adaptive phase coordination, minimizing tipping risks. The hexapod gait excels on slopes by leveraging full-leg synchronization for high torque output and anti-overturning capability. Unity-based simulations validate terrain-specific adaptability: tripod gait achieves 20%-35% higher speed on flat ground, quadruped gait maintains directional stability on stairs, and hexapod gait ensures robust ascent on inclines. Results demonstrate that context-aware gait switching significantly improves terrain negotiation, highlighting the potential for adaptive algorithms to advance multilegged robots in extreme environments.

A kinematics analysis of a new hybrid mechanical leg suitable for bipedal robots was carried out and the gait of the robot walking on flat ground was planned. Firstly, the kinematics of the hybrid mechanical leg were analyzed and the applicable relevant models were established. Secondly, based on the preliminary motion requirements, the inverted pendulum model was used to divide the robot walking into three stages for gait planning: mid-step, start and stop. In the three stages of robot walking, the forward and lateral robot centroid motion trajectories and the swinging leg joint trajectories were calculated. Finally, dynamic simulation software was used to simulate the virtual prototype of the robot, achieving its stable walking on flat ground in the virtual environment, and verifying the feasibility of the mechanism design and gait planning. This study provides a reference for the gait planning of hybrid mechanical legged bipedal robots and lays the foundation for further research on the robots involved in this thesis.

In recent years, significant progress has been made in employing reinforcement learning for controlling legged robots. However, a major challenge arises with quadruped robots due to their continuous states and vast action space, making optimal control using simple reinforcement learning controllers particularly challenging. This paper introduces a hierarchical reinforcement learning framework based on the Deep Deterministic Policy Gradient (DDPG) algorithm to achieve optimal motion control for quadruped robots. The framework consists of a high-level planner responsible for generating ideal motion parameters, a low-level controller using model predictive control (MPC), and a trajectory generator. The agents within the high-level planner are trained to provide the ideal motion parameters for the low-level controller. The low-level controller uses MPC and PD controllers to generate the foot-end force and calculates the joint motor torque through inverse kinematics. The simulation results show that the motion performance of the trained hierarchical framework is superior to that obtained using only the DDPG method.

Legged robotic systems can play an important role in real-world applications due to their superior load-bearing capabilities, enhanced autonomy, and effective navigation on uneven terrain. They offer an optimal trade-off between mobility and payload capacity, excelling in diverse environments while maintaining efficiency in transporting heavy loads. However, planning and optimizing gaits and gait sequences for these robots presents significant challenges due to the complexity of their dynamic motion and the numerous optimization variables involved. Traditional trajectory optimization methods address these challenges by formulating the problem as an optimization task, aiming to minimize cost functions, and to automatically discover contact sequences. Despite their structured approach, optimization-based methods face substantial difficulties, particularly because such formulations result in highly nonlinear and difficult to solve problems. To address these limitations, we propose CrEGOpt, a bi-level optimization method that combines traditional trajectory optimization with a black-box optimization scheme. CrEGOpt at the higher level employs the Mixed Distribution Cross-Entropy Method to optimize both the gait sequence and the phase durations, thus simplifying the lower level trajectory optimization problem. This approach allows for fast solutions of complex gait optimization problems. Extensive evaluation in simulated environments demonstrates that CrEGOpt can find solutions for biped, quadruped, and hexapod robots in under 10 seconds. This novel bi-level optimization scheme offers a promising direction for future research in automatic contact scheduling.

Abstract Real-time gait trajectory planning is challenging for legged robots walking on unknown terrain. In this paper, to realize a more efficient and faster motion control of a quadrupedal robot, we propose an optimized gait planning generator (GPG) based on the decision tree (DT) and random forest (RF) model of the robot leg workspace. First, the framework of this embedded GPG and some of the modules associated with it are illustrated. Aiming at the leg workspace model described by DT and RF used in GPG, this paper introduces in detail how to collect the original data needed for training the model and puts forward an Interpolation Labeling with Dilation and Erosion (ILDE) data processing algorithm. After the DT and RF models are trained, we preliminarily evaluate their performance. We then present how these models can be used to predict the location relation between a spatial point and the leg workspace based on its distributional features. The DT model takes only 0.00011 s to process a sample, while the RF model can give the prediction probability. As a complement, the PID inverse kinematic model used in GPG is also mentioned. Finally, the optimized GPG is tested during a real-time single-leg trajectory planning experiment and an unknown terrain recognition simulation of a virtual quadrupedal robot. According to the test results, the GPG shows a remarkable rapidity for processing large-scale data in the gait trajectory planning tasks, and the results can prove it has an application value for quadruped robot control.

Legged soccer robots present a significant challenge in robotics owing to the need for seamless integration of perception, manipulation, and dynamic movement. While existing models often depend on external perception or static techniques, our study aims to develop a robot with dynamic and untethered capabilities. We have introduced a motion planner that allows the robot to excel in dynamic shooting and dribbling. Initially, it identifies and predicts the position of the ball using a rolling model. The robot then pursues the ball, using a novel optimization-based cycle planner, continuously adjusting its gait cycle. This enables the robot to kick without stopping its forward motion near the ball. Each leg is assigned a specific role (stance, swing, pre-kick, or kick), as determined by a gait scheduler. Different leg controllers were used for tailored tiptoe trajectory planning and control. We validated our approach using real-world penalty shot experiments (5 out of 12 successful), cycle adjustment tests (11 out of 12 successful), and dynamic dribbling assessments. The results demonstrate that legged robots can overcome onboard capability limitations and achieve dynamic mobility and manipulation.

The legged robot has a strong ability to adapt to complex terrain. There are few researches on the adaptive fault tolerance of motion joint actuators of quadruped robots. Therefore, based on the basic principles of foot trajectory planning and gait planning, this paper plans the quadruped robot's foot trajectory and walking gait. Secondly, an adaptive fault-tolerant control law is designed for a quadruped robot's leg joint actuator fault. Finally, mathematical proof and simulation results show that the system can still safely complete the control task within the fault-tolerant range of the expected performance index after some actuator faults. This study provides direction for further research on the compliant motion of quadruped robots with partial actuator faults.

Legged robots made of rigid links have disadvantages like poor energy efficiency and large impact forces during foot/terrain contact while walking on 3D uneven terrain. This paper proposes a unique design of an 18 DOF quadruped robot with compliant shanks. Compliance has been added to the quadruped by introducing a “c-section“ in the shank of each leg. The robot dynamics has been modeled using the projected Newton-Euler method, while the “Craig-Bampton Method” has been utilized to model the compliant shanks. Optimal body trajectory and foot placements while maintaining dynamic stability is obtained using an NLP based optimization strategy. A torque-based inverse dynamics control technique ensures that the quadruped follows the desired trajectory. The robot was simulated using Simscape Multibody, which utilizes the robot model’s “.stl,”“.step,” and “.xml” files to follow the optimized joint trajectories. The total energy consumed and joint torques are compared between the complaint and rigid link quadruped robots for walk on different types of terrain with different gaits and C-sections of 3, 4, 6, and 8 mm thickness. The simulation results show a significant reduction in joint torque spikes (more than 70% for all cases) due to the addition of compliance in each leg. This also leads to a reduction in total energy consumption during the walk especially over an uneven terrain. Hence, the results verify that the proposed design of quadruped is functionally more efficient than its rigid counterpart for walking on uneven terrain.

No abstract available

Environment sensing for legged robots can improve their mobility performance. A foot-ground contact model can be used to evaluate the properties of the robot's contact with the environment. A contact parameter estimator based on artificial neural networks was developed to guide the gait planning of a hexapod robot so that it can achieve higher mobility efficiency. This contact parameter estimator can use acceleration, velocity, and contact force data as inputs and output contact parameters. The estimator avoided using deformations as input which are difficult to measure. Meanwhile, the robot's cost of transport with different contact parameters is tested and recorded. Accordingly, the hexapod robot was guided to choose a better gait shape between cycloid and rectangular. The simulation proved that changes in the gait shape according to the contact parameters can reduce the hexapod's cost of transport.

How does a wheeled robot move and turn? The answer is straightforward for a conventional wheeled robot, but it is not so easy for a robot with a discrete wheel design. Regular wheeled robots always have four contact points, resulting in static stability during locomotion. However, QuadRunner's novel leg mechanism provides only a semi-circular wheel shape, and proper gait planning is needed to go straight or turn. Therefore, this paper presents a dual frequency gait planning method which controls the robot's gait cycle's duty factor and generates unique turning gait patterns for wheel locomotion. Describing requirements and limitations, we found sets of solutions that can achieve turning. Results show that the smallest turning radius QuadRunner achieved is 1.05m, and the biggest is 1.86m. In addition, detailed experiments were made to observe the performance and stability of straight and turning wheel behaviors. Finally, a gait verification is made using high-speed cameras.

Abstract This paper presents a kinematics modeling and hybrid motion planning framework for wheeled-legged rovers. It is a unified solution for wheeled-legged rovers to traverse multiple challenging terrains using hybrid locomotion. A kinematic model is first established to describe the rover’s motions. Then, a hybrid motion planning framework is proposed to determine the rover’s gait patterns and parameterize the legs’ and the body’s trajectories. Furthermore, an optimization algorithm based on B-spline is utilized to minimize the motors’ energy dissipation and generate smooth trajectories. The wheeled and legged hybridization allows the rover for faster locomotion while maintaining high stability. Besides, it also improves the rover’s ability to overcome obstacles. Prototype experiments are carried out in more complex environments to verify the rover’s flexibility and maneuverability to traverse irregular terrains. The proposed algorithm reduces the swing amplitude by 83.3% compared to purely legged locomotion.

This paper presents a novel formulation of motion planning for multi-legged walking robots. In the proposed method, a single-step motion is formulated as a nonlinear equation problem (NLE): including kinematic, stability, and collision constraints. For the given start and goal configurations, the robot's path is parametrized as Bézier curve in the configuration space. The resulting NLE is solved using Levenberg-Marquardt optimization implemented using a sparse matrix solver. We propose handling the trigonometric kinematic constraints with the polynomial path parametrization. A relaxation of the constraint is used while guaranteeing a desired tolerance along the planned path. Although the proposed method does not explicitly optimize any criterion, it produces high-quality paths. The method is deployed in transforming a sequence of discrete configurations produced by a step sequence planner into a valid path for a multi-legged walking robot in challenging planning scenarios where a regular locomotion gait cannot be used because of sparse footholds.

Contact planning is crucial in locomoting systems.Specifically, appropriate contact planning can enable versatile behaviors (e.g., sidewinding in limbless locomotors) and facilitate speed-dependent gait transitions (e.g., walk-trot-gallop in quadrupedal locomotors). The challenges of contact planning include determining not only the sequence by which contact is made and broken between the locomotor and the environments, but also the sequence of internal shape changes (e.g., body bending and limb shoulder joint oscillation). Most state-of-art contact planning algorithms focused on conventional robots (e.g.biped and quadruped) and conventional tasks (e.g. forward locomotion), and there is a lack of study on general contact planning in multi-legged robots. In this paper, we show that using geometric mechanics framework, we can obtain the global optimal contact sequence given the internal shape changes sequence. Therefore, we simplify the contact planning problem to a graph optimization problem to identify the internal shape changes. Taking advantages of the spatio-temporal symmetry in locomotion, we map the graph optimization problem to special cases of spin models, which allows us to obtain the global optima in polynomial time. We apply our approach to develop new forward and sidewinding behaviors in a hexapod and a 12-legged centipede. We verify our predictions using numerical and robophysical models, and obtain novel and effective locomotion behaviors.

This article addresses the problem of legged locomotion in unstructured environments, and a novel hierarchical multicontact motion planning method for hexapod robots is proposed by combining free gait motion planning and deep reinforcement learning. We structurally decompose the complex free gait multicontact motion planning task into path planning in discrete state space and gait planning in a continuous state space. First, the soft deep Q-network is used to obtain the global prior path information in the path planner (PP). Second, a free gait planner (FGP) is proposed to obtain the gait sequence. Finally, based on the PP and the FGP, the center-of-mass sequence is generated by the trained optimal policy using the designed deep reinforcement learning algorithm. Experimental results in different environments demonstrate the feasibility, effectiveness, and advancement of the proposed method.

In order to enable the quadruped robot to walk according to the preset scheme, avoiding the occurrence of rollover, discontinuity of walking, etc., this paper conducts gait planning for a quadruped robot prototype with a four-bar shock absorbing mechanism in the laboratory. Using a zero-impact static foot trajectory, the forward kinematics, inverse kinematics and other methods are used to analyze the position of the four-legged robot foot in the body coordinate system and the driving joint angle and the foot end. The relationship between the trajectories further obtains the operation of the quadruped robot in the above gait. The calculated results were brought into the Adams software for simulation, which verified the feasibility of the static gait during the prototype operation. Finally, through the prototype running test, it was confirmed that the four-legged robot prototype with four-bar shock absorbing mechanism can operate stably under the planned gait.

No abstract available

Traditional mobile robots have limitations in obstacle-crossing ability, motion stability and load-bearing capacity in complex environments, which make them difficult to be applied on a large scale. Based on the Rubik’s Cube mechanism (RCM) with strong coupling and variable topology, a polymorphous wheel-legged mobile robot (WLMR) is proposed. Aiming at the problems of the classical three-order RCM, such as small internal space, difficult processing and demanding precision, a new type of chute third-order RCM is designed, and its mechanical characteristics analysis and feasibility analysis are carried out. What’s more, a driving configuration analysis method is established according to different driving configuration relationships, and the configuration of WLMR is determined by the configuration stability analysis. Then, a WLMR with polymorphism is designed, and gait planning and gait stability analysis are conducted. Eventually, the co-simulation and prototype experiments are performed to verify the efficiency of the WLMR’s straight motion, in-situ rotation, obstacle-crossing and morphology transformation in complex environments. This research not only provides a reference for the design of polymorphous mobile robots, but also opens up ideas for the application of the RCM in daily production and life.

In order to make the quadruped robot better adapt to the complex environment and maintain stability during the movement, the movement of the robot should be planned. In the currently existing research, the gaits and footholds of the quadruped robot are pre-defined, and it is based on the accurate prior knowledge of the terrain. In this paper, the concept of the feasible force set is introduced, and the ability of the foot end during the supporting period is quantified , considering the different configurations at different joint angles and physical constraints of the legs. Secondly, the momentum theorem is used to determine the required foot forces during the robot movement, thereby determining the leg configuration that satisfies the requirements, and realizing the online selection of the footholds when velocity changes. The simulation results verify the rationality of the method. The results show that in the trot gait, the quadruped robot can find the feasible forces to meet the acceleration motion so that the robot can choose the footholds online, improving the robot’s adaptability to the environment

Abstract Wheel-leg composite robots exhibit robust mobility and exceptional obstacle-crossing capabilities in complex environments. This paper proposes a novel transformable wheel-leg composite structure and presents the design of a wheel-leg composite obstacle-crossing robot, fundamentally configured as a two-wheeled quadruped. The research encompasses a comprehensive analysis of the robot’s overall mechanical structure, a detailed kinematic investigation of its body and obstacle-crossing gait planning, virtual prototype dynamics simulation, and field experimentation. Utilizing advanced modeling software, a 3D model of the robot was established. The kinematic characteristics of the robot in both wheeled and legged modes were thoroughly examined. Specifically, for the legged mode, the Denavit-Hartenberg coordinate system was established, and a detailed kinematic model was analyzed. The obstacle-crossing gait was planned based on the robot’s leg action mechanism. Furthermore, the Lagrangian method was employed to develop a mathematical model for the dynamics of the robot in both wheel-foot modes, allowing for a comprehensive force analysis. To validate the feasibility and rationality of the robot’s obstacle-crossing capabilities under various conditions, extensive simulations and prototype tests were conducted across diverse terrains. The results provide valuable insights and practical guidance for the structural design of wheel-leg composite obstacle-crossing robots, contributing to advancements in this promising field.

This study focuses on the external mechanical structure design, gait planning, and servo control system development of the cross-legged robot. By equipping pitch degrees of freedom at the hip, knee, and ankle joints, the system effectively addresses error accumulation caused by the absence of sensor feedback, significantly enhancing the robot’s stability and locomotion capabilities. The servo control system achieves precise execution of critical gait movements, including center of gravity transfer and leg swing, through accurate angle regulation and motion sequence coordination, thereby improving operational stability during complex maneuvers. The sensor-free architecture endows the robot with enhanced adaptability to extreme environments, indicating significant engineering potential in practical applications.

The wheel-legged composite robot combines the advantages of efficient wheeled movement and legged obstacle crossing, but its state estimation accuracy is limited by sensors dependent on contact detection and the challenge of multi-source data fusion. Aiming at the four-legged wheel-legged robot, we proposed a contact estimator based on multi-probability model fusion and a position and velocity estimator based on odometry to achieve high-precision state estimation with low hardware cost. The contact estimator constructed a three-layer probability model: the gait planning model dynamically estimated the contact probability based on phase information, the knee flexion and extension joint torque model used the mutation characteristics of joint torque to establish a Gaussian distribution mapping, the wheel bottom height model constructed a probability relationship through the geometric constraint of ground clearance height, and then fused the three-layer output through Kalman filter to solve the problems of high cost and difficult wiring of traditional sensors. The position and velocity estimator fused motor encoder and IMU data, constructs Kalman filter state equation and observation equation, calculated the pose state of the wheel bottom and the centroid in the world coordinate system, and improved the estimation efficiency under nonholonomic constraints. Simulation results showed that the contact state discrimination results of the contact estimator in pure rolling and mixed motion are highly consistent with the real contact state, and the estimation results of the position and velocity estimator in the X, Y and Z directions are in good agreement with the real values, and the estimation error is always kept within a small range. This method provides reliable state feedback for the dynamic control of the wheel-legged robot.

To enhance the terrain adaptability and intelligence of the closed-chain legged robot, this study introduces a design of a closed-chain legged robot with a reconfigurable trunk and presents its autonomous obstacle-crossing strategy. The design integrates four coupled Watt-I type linkages to form a quadruped unit. Four quadruped units are linked to the robot's trunk through motors, enabling the overall reconfigurability of the unit. This paper introduces the analysis of the kinematic model, optimizes the dimensions of the leg components. Finally, the analysis of the foot trajectory is completed. To enable autonomous obstacle-crossing, this paper introduces, for the first time, a continuous and collision-free autonomous obstacle-crossing strategy. Based on the analysis of foot-endpoint trajectories, obstacles are categorized by scale. For small-scale obstacles, gait is planned through pre-planned posture mapping relationships. For large-scale obstacles, a stride length planning algorithm has been designed to plan the obstacle-crossing gait. Collision-free is achieved by controlling the heights of the maximum and minimum effective crossing points. Feasibility of this strategy has been confirmed through simulations and experimental studies.

Introduction. Walking robots are widely used in industry due to their unique capabilities for moving on uneven and complex surfaces. To provide high precision in controlling their movement, it is required to develop mathematical models and algorithms for planning the robot movement along various trajectories. A key aspect of the motion control system of walking robots is the planning of their leg movements. Despite significant advances in the field of modeling the kinematics of quadruped robots, existing scientific publications do not provide a complete kinematic model for robots similar to the Mini Cheetah. This research was aimed at the development of a kinematic model of a quadruped robot based on Mini Cheetah, as well as the formulation of recommendations for optimizing its gait to provide rotation around various axes. The creation of such a model will improve the smoothness and accuracy of the robot movements, which, in turn, will increase its efficiency under real production conditions.Materials and Methods. The process of constructing a kinematic model of the robot was based on the use of formulas for the geometry of spatial motion of solids. To test the efficiency of the proposed algorithms for moving the robot legs when performing rotational movements of its body, numerical modeling of the robot kinematics was used. Numerical calculations were performed using the Wolfram Mathematica package.Results. The laws of changing the endpoints of the robot legs during its rotation around the vertical axis were proposed. The conducted numerical modeling of the robot kinematics covered the rotation of the body at the course, roll and pitch angles. Based on the simulation results, it was established that the dependences of the rotation angles of the leg links were periodic functions. The considered rotational movements of the robot platform could take place without the occurrence of singular configurations.Discussion and Conclusion. The results of numerical modeling of the robot platform rotation movements confirmed the operability of the proposed leg transfer plan, which allowed for smooth movement of the robot body and avoidance of singular configurations. The resulting kinematic model can be used to control the robot motion at the kinematic level when moving along curvilinear trajectories. As a prospect for further research, it is worth highlighting the development of a mathematical model of the dynamics of a four-legged robot, as well as the creation of laws for controlling its movement at a dynamic level. This will significantly expand the functionality of the robot and increase its efficiency under various operating conditions.

Animals are capable of precise and agile locomotion using vision. Replicating this ability has been a long-standing goal in robotics. The traditional approach has been to decompose this problem into elevation mapping and foothold planning phases. The elevation mapping, however, is susceptible to failure and large noise artifacts, requires specialized hardware, and is biologically implausible. In this paper, we present the first end-to-end locomotion system capable of traversing stairs, curbs, stepping stones, and gaps. We show this result on a medium-sized quadruped robot using a single front-facing depth camera. The small size of the robot necessitates discovering specialized gait patterns not seen elsewhere. The egocentric camera requires the policy to remember past information to estimate the terrain under its hind feet. We train our policy in simulation. Training has two phases - first, we train a policy using reinforcement learning with a cheap-to-compute variant of depth image and then in phase 2 distill it into the final policy that uses depth using supervised learning. The resulting policy transfers to the real world and is able to run in real-time on the limited compute of the robot. It can traverse a large variety of terrain while being robust to perturbations like pushes, slippery surfaces, and rocky terrain. Videos are at https://vision-locomotion.github.io

A wheel-legged robot is equipped with Stewart parallel mechanism, constituting a reconfigurable robot which can change its wheelbase, robot body height, and achieve omnidirectional steering. The legged character effectively improves the terrain adaptability, which concerns our planning concentration. We introduced an optimization-based whole-body trajectory planning algorithm to navigate robot in rugged terrain. The planner combines terrain data and stability, allowing lower-level motion generator and controller to operate more efficiently. The Model Predictive Control(MPC)-based method updates the footholds and CoG trajectories, which builds upon the support polygon constraints on optimization. The simulations of methodology working in several structure-obstacle scene demonstrated and compared the availability of approach.

No abstract available

Nonlinear model predictive locomotion controllers based on the reduced centroidal dynamics are nowadays ubiquitous in legged robots. These schemes, even if they assume an inherent simplification of the robot's dynamics, were shown to endow robots with a step-adjustment capability in reaction to small pushes, and in the case of uncertain parameters - as unknown payloads - they were shown to provide some “practical”, albeit limited, robustness. In this work, we provide rigorous certificates of their closed-loop stability via reformulating the online centroidal MPC controller. This is achieved thanks to a systematic procedure inspired by the machinery of adaptive control, together with ideas coming from Control Lyapunov Functions. Our reformulation, in addition, provides robustness for a class of unmeasured constant disturbances. To demonstrate the generality of our approach, we validated our formulation on a new generation of humanoid robots - the <inline-formula><tex-math notation="LaTeX">$\text{56.7 kg}$</tex-math></inline-formula> ergoCub, as well as on the commercially available <inline-formula><tex-math notation="LaTeX">$\text{21 kg}$</tex-math></inline-formula> quadruped robot Aliengo.

This article presents a hierarchical model predictive control (MPC) framework for the end-effector tracking control problem of a legged mobile manipulator. In the high-level part, a kinematic MPC over a long-time horizon computes both base and joint trajectories, then a quadratic program (QP) based optimization solves ground-reaction-forces (GRFs) satisfying the robot’s centroidal dynamics and the friction cone constraints. In the low-level part, a kinodynamic MPC over a short-time horizon tracks command end-effector trajectories and outputs from the high-level part while satisfying all nonlinear dynamics constraints. Due to the complexity of MPC formulations and high real-time requirements, traditional MPC for legged mobile manipulators can only generate short-time horizon solutions for tracking tasks over longer time horizons, which may lead to the optimization falling into bad local minima. In our method, the long-term trajectories from the high-level part can guide the optimization of the short-term kinodynamic MPC to generate a better solution. We validate the effectiveness of our method through several simulation and hardware experiments. In comparison to traditional MPC, the proposed method improves the trajectory tracking accuracy of the robot’s end-effector while reducing the violations of the system’s physical limit constraints and environment-collision avoidance constraints.Note to Practitioners—Legged mobile manipulators have received increasing research attention in recent years due to their potential applications. In this paper, we particularly focus on the high-efficiency end-effector tracking control of legged mobile manipulators. We propose a hierarchical MPC framework incorporating a high-level part generating long-term whole-body motions and a low-level kinodynamic MPC replanning at a high frequency. Simulation and hardware experiments show that the proposed method improves the trajectory tracking accuracy while reducing the violations of the system’s physical limit constraints and environment-collision avoidance constraints in comparison to traditional MPC. In future research, we will address the end-effector tracking control problem when the end-effector contacts the environment.

Ensuring safe and effective collaboration between humans and autonomous legged robots is a fundamental challenge in shared autonomy, particularly for teleoperated systems navigating cluttered environments. Conventional shared-control approaches often rely on fixed blending strategies that fail to capture the dynamics of legged locomotion and may compromise safety. This paper presents a teleoperator-aware, safety-critical, adaptive nonlinear model predictive control (ANMPC) framework for shared autonomy of quadrupedal robots in obstacle-avoidance tasks. The framework employs a fixed arbitration weight between human and robot actions but enhances this scheme by modeling the human input with a noisily rational Boltzmann model, whose parameters are adapted online using a projected gradient descent (PGD) law from observed joystick commands. Safety is enforced through control barrier function (CBF) constraints integrated into a computationally efficient NMPC, ensuring forward invariance of safe sets despite uncertainty in human behavior. The control architecture is hierarchical: a high-level CBF-based ANMPC (10 Hz) generates blended human-robot velocity references, a mid-level dynamics-aware NMPC (60 Hz) enforces reduced-order single rigid body (SRB) dynamics to track these references, and a low-level nonlinear whole-body controller (500 Hz) imposes the full-order dynamics via quadratic programming to track the mid-level trajectories. Extensive numerical and hardware experiments, together with a user study, on a Unitree Go2 quadrupedal robot validate the framework, demonstrating real-time obstacle avoidance, online learning of human intent parameters, and safe teleoperator collaboration.

This paper presents the design and analysis of Pegasus, a quadrupedal wheeled robot grounded in biomimicry principles. Pegasus offers two distinct motion modes, including a wheeled motion and a hybrid wheeled-legged motion, enabling adaptability across various tasks and environmental conditions. The robot draws inspiration from the joint structures of quadruped animals and incorporates biomimetic features. At the robot’s ankle joint, we imitate the articulation of a radiusulna joint to enhance the wheeled motion’s agility. Additionally, we establish comprehensive mathematical models for adaptive dynamics model, providing a robust theoretical foundation for subsequent motion planning and high-precision control. A novel telescopic vehicle mode is also proposed for complex wheel-leg hybrid motion, offering optimized solutions for intricate robot locomotion. Furthermore, we employ parallel underactuated MPC controllers for each leg at the control level, contributing to heightened motion precision and stability. Extensive validation through physical platform experiments highlights the effectiveness and feasibility of the proposed controllers, offering substantial support for real-world applications in robotics.

Serving the Stewart mechanism as a wheel-legged structure, the most outstanding superiority of the proposed wheel-legged hybrid robot (WLHR) is the active vibration isolation function during rolling on rugged terrain. However, it is difficult to obtain its precise dynamic model, because of the nonlinearity and uncertainty of the heavy robot. This paper presents a dynamic control framework with a decentralized structure for single wheel-leg, position tracking based on model predictive control (MPC) and adaptive impedance module from inside to outside. Through the Newton-Euler dynamic model of the Stewart mechanism, the controller first creates a predictive model by combining Newton-Raphson iteration of forward kinematic and inverse kinematic calculation of Stewart. The actuating force naturally enables each strut to stretch and retract, thereby realizing six degrees-of-freedom (6-DOFs) position-tracking for Stewart wheel-leg. The adaptive impedance control in the outermost loop adjusts environmental impedance parameters by current position and force feedback of wheel-leg along Z -axis. This adjustment allows the robot to adequately control the desired support force tracking, isolating the robot body from vibration that is generated from unknown terrain. The availability of the proposed control methodology on a physical prototype is demonstrated by tracking a Bezier curve and active vibration isolation while the robot is rolling on decelerate strips. By comparing the proportional and integral (PI) and constant impedance controllers, better performance of the proposed algorithm was operated and evaluated through displacement and force sensors internally-installed in each cylinder, as well as an inertial measurement unit (IMU) mounted on the robot body. The proposed algorithm structure significantly enhances the control accuracy and vibration isolation capacity of parallel wheel-legged robot.

This paper reports on a new error-state Model Predictive Control (MPC) approach to connected matrix Lie groups for robot control. The linearized tracking error dynamics and the linearized equations of motion are derived in the Lie algebra. Moreover, given an initial condition, the linearized tracking error dynamics and equations of motion are globally valid and evolve independently of the system trajectory. By exploiting the symmetry of the problem, the proposed approach shows faster convergence of rotation and position simultaneously than the state-of-the-art geometric variational MPC based on variational-based linearization. Numerical simulation on tracking control of a fully-actuated 3D rigid body dynamics confirms the benefits of the proposed approach compared to the baselines. Furthermore, the proposed MPC is also verified in pose control and locomotion experiments on a quadrupedal robot MIT Mini Cheetah.

Recent progress in legged locomotion has rendered quadruped manipulators a promising solution for performing tasks that require both mobility and manipulation (loco-manipulation). In the real world, task specifications and/or environment constraints may require the quadruped manipulator to be equipped with high redundancy as well as whole-body motion coordination capabilities. This work presents an experimental evaluation of a whole-body Model Predictive Control (MPC) framework achieving real-time performance on a dual-arm quadruped platform consisting of 37 actuated joints. To the best of our knowledge this is the legged manipulator with the highest number of joints to be controlled with real-time whole-body MPC so far. The computational efficiency of the MPC while considering the full robot kinematics and the centroidal dynamics model builds upon an open-source DDP-variant solver and a state-of-the-art optimal control problem formulation. Differently from previous works on quadruped manipulators, the MPC is directly interfaced with the low-level joint impedance controllers without the need of designing an instantaneous whole-body controller. The feasibility on the real hardware is showcased using the CENTAURO platform for the challenging task of picking a heavy object from the ground. Dynamic stepping (trotting) is also showcased for first time with this robot. The results highlight the potential of replanning with whole-body information in a predictive control loop.

Wheel-legged robots (WLRs) with knee-wheel placement can switch motion mode into wheeled motion and legged motion depending on the road condition. However, in wheeled motion mode, robots are energy-efficiency and high maneuverability but struggle to handle rough terrain. To address this challenge, we propose a quarter-car model predictive controller (MPC) that utilizes both thigh and calf leg actuation to track sprung mass displacement and regulate tire force during unknown road disturbances. To overcome the high computational demands of whole-body MPC-based posture control for the entire robot, we propose a distributed MPC-based framework with inverse kinematics. Simulations, including quarter-car wheel-legged system simulation, posture tracking, braking, and steering with load transfer, were conducted to evaluate the effectiveness of our proposed control framework and the promising performance of multi-actuation involving the calf leg. The results of this paper demonstrate that our proposed approach is effective in improving the posture control and stability of knee-wheeled WLRs, particularly in rough terrain conditions.

In the field of wheel-legged robots (WLRs), two types of configurations exist: toe-wheeled and knee-wheeled. In the knee-wheeled configuration, the robot is capable of switching between leg walking and wheel driving modes. However, the participation of the calf leg in driving control is often limited during the switch to the wheel driving mode, which makes it similar to a rocker-arm suspension vehicle with an additional unsprung mass. The presence of redundant calf legs in knee-wheeled WLRs can increase the unsprung mass, potentially impacting the overall performance of the robot. To address this issue, this paper presents a collaborative control method based on road preview for knee-wheeled WLRs. The proposed approach simplifies the wheeled system to a quarter car model and designs a Model Predictive Controller (MPC). By effectively integrating the redundant calf legs into the control strategy, the proposed approach aims to mitigate the negative effects of increased unsprung mass and enhance the overall performance of the robot in terms of vertical comfort and road holding ability. Simulation results demonstrate that the vertical performance of the wheel leg system can be effectively improved through the control of the calf leg when traversing undulating terrains. The proposed anticipatory control method and collaborative controller were found to effectively enhance the overall performance of the robot in terms of comfort and road holding ability. The findings of this study provide insights into the development of control strategies for knee-wheeled legged robots and contribute to the advancement of wheeled legged locomotion research.

The field of legged robots has seen tremendous progress in the last few years. Locomotion trajectories are commonly generated by optimization algorithms in a Model Predictive Control (MPC) loop. To achieve online trajectory optimization, the locomotion community generally makes use of heuristic-based contact planners due to their low computation times and high replanning frequencies. In this work, we propose ContactNet, a fast acyclic contact planner based on a multi-output regression neural network. ContactNet ranks discretized stepping locations, allowing to quickly choose the best feasible solution, even in complex environments. The low computation time, in the order of 1 ms, enables the execution of the contact planner concurrently with a trajectory optimizer in a MPC fashion. In addition, the computational time does not scale up with the configuration of the terrain. We demonstrate the effectiveness of the approach in simulation in different scenarios with the quadruped robot Solo12. To the best knowledge of the authors, this is the first time a contact planner is presented that does not exhibit an increasing computational time on irregular terrains with an increasing number of gaps.

Deep learning and model predictive control (MPC) can play complementary roles in legged robotics. However, integrating learned models with online planning remains challenging. When dynamics are learned with neural networks, three key difficulties arise: (1) stiff transitions from contact events may be inherited from the data; (2) additional non-physical local nonsmoothness can occur; and (3) training datasets can induce non-Gaussian model errors due to rapid state changes. We address (1) and (2) by introducing the smooth neural surrogate, a neural network with tunable smoothness designed to provide informative predictions and derivatives for trajectory optimization through contact. To address (3), we train these models using a heavy-tailed likelihood that better matches the empirical error distributions observed in legged-robot dynamics. Together, these design choices substantially improve the reliability, scalability, and generalizability of learned legged MPC. Across zero-shot locomotion tasks of increasing difficulty, smooth neural surrogates with robust learning yield consistent reductions in cumulative cost on simple, well-conditioned behaviors (typically 10-50%), while providing substantially larger gains in regimes where standard neural dynamics often fail outright. In these regimes, smoothing enables reliable execution (from 0/5 to 5/5 success) and produces about 2-50x lower cumulative cost, reflecting orders-of-magnitude absolute improvements in robustness rather than incremental performance gains.

We present a model predictive controller (MPC) that automatically discovers collision-free locomotion while simultaneously taking into account the system dynamics, friction constraints, and kinematic limitations. A relaxed barrier function is added to the optimization’s cost function, leading to collision avoidance behavior without increasing the problem’s computational complexity. Our holistic approach does not require any heuristics and enables legged robots to find whole-body motions in the presence of static and dynamic obstacles. We use a dynamically generated euclidean signed distance field for static collision checking. Collision checking for dynamic obstacles is modeled with moving cylinders, increasing the responsiveness to fast-moving agents. Furthermore, we include a Kalman filter motion prediction for moving obstacles into our receding horizon planning, enabling the robot to anticipate possible future collisions. Our experiments1 demonstrate collision-free motions on a quadrupedal robot in challenging indoor environments. The robot handles complex scenes like overhanging obstacles and dynamic agents by exploring motions at the robot’s dynamic and kinematic limits.

No abstract available

No abstract available

Autonomous navigation through unstructured and dynamically-changing environments is a complex task that continues to present many challenges for modern roboticists. In particular, legged robots typically possess manipulable asymmetric geometries which must be considered during safetycritical trajectory planning. This work proposes a predictive safety filter: a nonlinear model predictive control (MPC) algorithm for online trajectory generation with geometry-aware safety constraints based on control barrier functions (CBFs). Critically, our method leverages Poisson safety functions to numerically synthesize CBF constraints directly from perception data. We extend the theoretical framework for Poisson safety functions to incorporate temporal changes in the domain by reformulating the static Dirichlet problem for Poisson's equation as a parameterized moving boundary value problem. Furthermore, we employ Minkowski set operations to lift the domain into a configuration space that accounts for robot geometry. Finally, we implement our real-time predictive safety filter on humanoid and quadruped robots in various safetycritical scenarios. The results highlight the versatility of Poisson safety functions, as well as the benefit of CBF constrained model predictive safety-critical controllers.

Electrically driven legged robots have become popular in recent years. However, the development of reliable energy supply systems and effective energy management strategies for legged robots with dramatically varying power requirements still needs to be explored. This article proposes a learning-based model predictive control (MPC) energy management strategy for legged robots with battery–supercapacitor hybrid energy storage systems containing a power prediction unit and an MPC with learning-based adaptive weights. Firstly, the mathematical model of the legged robot is established, and a dual-layer long short-term memory network is constructed to predict the load power demand, providing the model and measurable disturbance for the MPC. Secondly, a multi-objective optimization objective function is established for the MPC-based energy management strategy. Three normalized terms, battery capacity loss, battery power fluctuation, and supercapacitor state-of-charge regulation, are balanced in the objective function. Finally, a deep learning algorithm is proposed to adaptively adjust the three weighting factors to meet the diverse operation conditions. Hardware-in-the-loop experimental implementations demonstrate that the proposed method can improve the kinematic performance of the legged robot by maintaining the supercapacitor state of charge at a relatively high level and reducing the battery capacity loss by 12.7% compared with the conventional MPC method.

This letter introduces a novel Model Predictive Control (MPC) implementation for legged robot locomotion that leverages GPU parallelization. Our approach enables both temporal and state-space parallelization by incorporating a parallel associative scan to solve the primal-dual Karush–Kuhn–Tucker (KKT) system. In this way, the optimal control problem is solved in <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(\log ^{2}(n)\log {N} + \log ^{2}(m))$</tex-math></inline-formula> complexity, instead of <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(N(n + m)^{3})$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula>, and <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> are the dimension of the system state, control vector, and the length of the prediction horizon. We demonstrate the advantages of this implementation over two state-of-the-art solvers (acados and crocoddyl), achieving up to a 60% improvement in runtime for Whole Body Dynamics (WB)-MPC and a 700% improvement for Single Rigid Body Dynamics (SRBD)-MPC when varying the prediction horizon length. The presented formulation scales efficiently with the problem state dimensions as well, enabling the definition of a centralized controller for up to 16 legged robots that can be computed in less than 25 ms. Furthermore, thanks to the JAX implementation, the solver supports large-scale parallelization across multiple environments, allowing the possibility of performing learning with the MPC in the loop directly in GPU.

This paper presents a system for enabling real-time synthesis of whole-body locomotion and manipulation policies for real-world legged robots. Motivated by recent advancements in robot simulation, we leverage the efficient parallelization capabilities of the MuJoCo simulator on a multi-core CPU to achieve fast sampling over the robot state and action trajectories. Our results show surprisingly effective real-world locomotion and manipulation capabilities with a very simple control strategy. We demonstrate our approach on several hardware and simulation experiments: robust locomotion over flat and uneven terrains, climbing over a box whose height is comparable to the robot, and pushing a box to a goal position. To our knowledge, this is the first successful deployment of whole-body sampling-based MPC on real-world legged robot hardware. Experiment videos and code can be found at: whole-body-mppi.github.io.

Legged robots employ multiple joint actuators that are susceptible to abrupt partial failures during prolonged operation. Joint failures take multiple forms. They are not limited to complete lockout failures, which are the main focus of existing literature. They also include partial torque tracking failures, where actuators only partially respond to control commands. This letter investigates fault joints detection and fault rates estimation for legged robots with partial joint failures under model predictive control (MPC) and whole-body control frameworks. To handle these problems, a control framework named GRUFD-FTC is proposed. Firstly, we have created a dataset for Unitree A1 robots under different numbers of fault joints and different torque retention rates, and have open-sourced it. In addition, a gait recurrent unit based fault detector is proposed to simultaneously detect multiple joints partial failure during robot movement and then output initial joint torque retention rates. Subsequently, a fault-tolerant controller based on joint states is proposed to accurately estimate the joint torque retention rates. Finally, simulations and real-world experiments have verified the effectiveness of the proposed algorithms.

In this paper, we introduce a kinodynamic model predictive control (MPC) framework that exploits unidirectional parallel springs (UPS) to improve the energy efficiency of dynamic legged robots. The proposed method employs a hierarchical control structure, where the solution of MPC with simplified dynamic models is used to warm-start the kinody-namic MPC, which accounts for nonlinear centroidal dynamics and kinematic constraints. The proposed approach enables energy efficient dynamic hopping on legged robots by using UPS to reduce peak motor torques and energy consumption during stance phases. Simulation results demonstrated a 38.8% reduction in the cost of transport (CoT) for a monoped robot equipped with UPS during high-speed hopping. Additionally, preliminary hardware experiments show a 14.8% reduction in energy consumption.

This paper presents a novel approach to enhance Model Predictive Control (MPC) for legged robots through Distributed Optimization. Our method focuses on decomposing the robot dynamics into smaller, parallelizable subsystems, and utilizing the Alternating Direction Method of Multipliers (ADMM) to ensure consensus among them. Each subsystem is managed by its own Optimal Control Problem, with ADMM facilitating consistency between their optimizations. This approach not only decreases the computational time but also allows for effective scaling with more complex robot configurations, facilitating the integration of additional subsystems such as articulated arms on a quadruped robot. We demonstrate, through numerical evaluations, the convergence of our approach on two systems with increasing complexity. In addition, we showcase that our approach converges towards the same solution when compared to a state-of-the-art centralized whole-body MPC implementation. Moreover, we quantitatively compare the computational efficiency of our method to the centralized approach, revealing up to a 75% reduction in computational time. Overall, our approach offers a promising avenue for accelerating MPC solutions for legged robots, paving the way for more effective utilization of the computational performance of modern hardware. Accompanying video at https://youtu.be/Yar4W-Vlh2A. The related code can be found at https://github.com/iit-DLSLab/DWMPC

The paper presents a method to stabilize dynamic gait for a legged robot with embodied compliance. Our approach introduces a unified description for rigid and compliant bodies to approximate their deformation and a formulation for deformable multibody systems. We develop the centroidal composite predictive deformed inertia (CCPDI) tensor of a deformable multibody system and show how to integrate it with the standard-of-practice model predictive controller (MPC). Simulation shows that the resultant control framework can stabilize trot stepping on a quadrupedal robot with both rigid and compliant spines under the same MPC configurations. Compared to standard MPC, the developed CCPDI-enabled MPC distributes the ground reactive forces closer to the heuristics for body balance, and it is thus more likely to stabilize the gaits of the compliant robot. A parametric study shows that our method preserves some level of robustness within a suitable envelope of key parameter values.

The study of slippage in legged robots addresses slip detection, avoidance strategies, and the development of mechanisms to manage slippage. However, limited research focuses on analyzing the phenomenon itself despite its potential to advance robotic locomotion significantly. This study introduces a method for estimating slip direction in legged locomotion, which can indicate terrain type and serve as a quantitative factor for scaling slip reaction. To this end, simulations of the quadruped robot Go1 were conducted across various terrains, including flat surfaces, icy patches, and slopes, using both trot and static walk gaits. A Model Predictive Control (MPC) system was used for trajectory planning, and a Whole-Body Controller (WBC) was employed for trajectory tracking. From the simulation data, foot velocity was extracted and compared to a virtual direction aligned with the robot’s forward movement. This analysis enabled the detection of slippage and the determination of its direction. A low-pass filter was applied to minimize noise in the data, ensuring negligible impact on the algorithm’s performance. The results revealed distinct slip direction patterns associated with different terrains and gaits, providing a valuable quantitative metric for refining and scaling slip-response strategies in legged systems.

Achieving both target accuracy and robustness in dynamic maneuvers with long flight phases, such as high or long jumps, has been a significant challenge for legged robots. To address this challenge, we propose a novel learning-based control approach consisting of model learning and model predictive control (MPC) utilizing a variable-frequency scheme. Compared to existing MPC techniques, we learn a model directly from experiments, accounting not only for leg dynamics but also for modeling errors and unknown dynamics mismatch in hardware and during contact. Additionally, learning the model with variable-frequency allows us to cover the entire flight phase and final jumping target, enhancing the prediction accuracy of the jumping trajectory. Using the learned model, we also design variable-frequency to effectively leverage different jumping phases and track the target accurately. In a total of 92 jumps on Unitree A1 robot hardware, we verify that our approach outperforms other MPCs using fixed-frequency or nominal model, reducing the jumping distance error $2-8$ times. We also achieve jumping distance errors of less than 3% during continuous jumping on uneven terrain with randomly-placed perturbations of random heights (up to 4 cm or 27% the robot's standing height). Our approach obtains distance errors of $1-2$ cm on 34 single and continuous jumps with different jumping targets and model uncertainties.

Legged robot proved their capability to cross complex terrain in recent research, yet the autonomy of robots on discrete terrain still needs to be enhanced since it requires a full stack framework. This paper introduces a real-time motion and foothold planning framework tailored for legged robots navigating uneven terrains, such as stepping stones. Our approach addresses the critical challenges of determining feasible global paths and local footholds to enhance autonomous mobility across complex landscapes. By using a sampling-based global path planner integrated with terrain segmentation and the robot’s kinematic model, our framework swiftly generates viable navigation paths. Concurrently, it utilizes a Mixed Integer Programming (MIP) methodology for real-time foothold optimization, ensuring the robot’s stability and safety through dynamic terrain interaction. Finally, an execution layer including Model Predictive Control (MPC) and Whole-Body Control (WBC) generates the robots’ motion. Simulation and real-world experiments demonstrate that our framework improves legged robots’ adaptability on discrete terrains.

We present a model-predictive control (MPC) framework for legged robots that avoids the singularities associated with common three-parameter attitude representations like Euler angles during large-angle rotations. Our method parameterizes the robot's attitude with singularity-free unit quaternions and makes modifications to the iterative linear-quadratic regulator (iLQR) algorithm to deal with the resulting geometry. The derivation of our algorithm requires only elementary calculus and linear algebra, deliberately avoiding the abstraction and notation of Lie groups. We demonstrate the performance and computational efficiency of quaternion MPC in several experiments on quadruped and humanoid robots.

For legged robots, achieving dynamic motion in challenging terrains requires precise foot positioning and careful planning of underactuated system dynamics. Compared to robots with point feet, those with surface feet face a more complex task in selecting landing areas, and the surface contact with the ground further complicates the dynamics due to its varying interaction forms. We propose an online perception-control framework that optimizes the robot's full-degree-of-freedom motion in real time based on perceived terrain information. To manage the complexity of surface contact, we employ a contact point model that simplifies the interaction between the foot and the ground into multiple discrete point contacts, which are treated as landing points. An elevation map is created, and plane segmentation is precomputed to select the most suitable landing plane. Inequality constraints are extracted as approximations of the selected plane region and integrated into a nonlinear model predictive controller. Simulation experiments on bipedal robots were conducted in various complex terrains, including slopes, stairs, and uneven surfaces resembling plum blossom piles. The results demonstrate that our framework effectively enhances the robot's adaptability to diverse terrains.

We present a modeling and hybrid locomotion control method for a novel wheeled quadruped robot, Pegasus, featuring a unique lightweight linear-actuator-driven ankle structure that enables four-wheel independent steering. A quadruped model, which only activates 12 joints(hip, thigh, and knee), and a vehicle model are implemented on Pegasus simultaneously, with a hybrid velocity allocation strategy aiming to combine both legged and wheeled movements, distributing desired linear and angular velocity to both locomotion modes. Moreover, a model predictive control (MPC) controller is designed to generate optimal torque for the quadruped model in real-time, and a "telescopic vehicle model" is derived to calculate the commands for ankles and wheels. Our experimental results demonstrate that the proposed modeling method and velocity allocation strategy with our controller can enable Pegasus to track the desired commands and adapt to various terrains, such as ramps, curves, and gravel roads. Furthermore, the utilization of ankle joints has led to a considerable reduction in the cost of transport (CoT) by surpassing 20% compared to scenarios lacking ankle joints.

As legged robots are deployed in industrial and autonomous construction tasks requiring collaborative manipulation, they must handle object manipulation while maintaining stable locomotion. The challenge intensifies in real-world environments, where they should traverse discrete terrain, avoid obstacles, and coordinate with other robots for safe loco-manipulation. This work addresses safe motion planning for collaborative manipulation of an unknown payload on discrete terrain while avoiding obstacles. Our approach uses two sets of model predictive controllers (MPCs) as motion planners: a global MPC generates a safe trajectory for the team with obstacle avoidance, while decentralized MPCs for each robot ensure safe footholds on discrete terrain as they follow the global trajectory. A model reference adaptive whole-body controller (MRA-WBC) then tracks the desired path, compensating for model uncertainties from the unknown payload. We validated our method in simulation and hardware on a team of Unitree robots. The results demonstrate that our approach successfully guides the team through obstacle courses, requiring planar positioning and height adjustments, and all happening on discrete terrain such as stepping stones.

Wheeled bipedal robots (WBRs) have attracted increasing research attention due to their capability to integrate the agility of legged locomotion with the efficiency of wheeled mobility. However, their underactuated and nonlinear dynamics pose significant control challenges. This paper presents a balance-motion control framework for a novel eight-degree-of-freedom WBR with hip joints actuated in roll and pitch directions. The framework consists of balance and motion control: a single rigid body model and model predictive control (MPC) are used to compute ground reaction forces for maintaining body stability, while a wheeled linear inverted pendulum (WLIP) model and an offline linear quadratic regulator (LQR) are applied to calculate wheel torques for executing motion commands. The proposed method is validated through both simulation and real-world experiments, demonstrating fine performance in dynamic balance and locomotion tasks.

Humanoid robots, with their anthropomorphic structure, dexterous manipulation capabilities, and high environmental compatibility with human habitats, have emerged as a research focus in legged robotics. However, contact impacts during dynamic walking and associated stability challenges continue to limit their practical applications. Traditional control methods relying on fixed reference ground reaction forces struggle to effectively address these issues. This paper proposes a hierarchical control framework integrating zero-moment point (ZMP) constraints with whole-body control (WBC). The proposed method involves: 1) establishing both a simplified model and a full-body dynamic model for humanoid robots; 2) real-time generation of ZMP-stabilized reference contact forces; and 3) computation of desired joint torques through WBC optimization. Simulation results demonstrate that compared to conventional WBC approaches, the proposed method significantly reduces peak foot impact forces while enhancing walking stability and terrain adaptability.

Legged robots have high mobility for application with uneven terrain. Bipedal robots require a lower number of actuators with less complication of control system. The five-bar mechanisms are widely applied in robotic legs to enable the leg movement to be actuated by the motors located on the robot body (instead of at the leg joints) to minimize the inertia of the moving legs. For attaining a desired foot position with respect to the two actuated joints of each leg, there are two non-trivial solutions obtained from the inverse kinematics. For both solutions, the parameters of the five-bar linkage are optimized using the derivative-free method by considering the energy consumption of the stance leg. Based on the optimized leg parameters, the zero-moment point (ZMP) along the foot support is derived by using the table-cart model. The ZMP generator for shifting the center of mass (COM) forward according to the stride is simulated with the dynamic model of the leg for predicting the motor torques varying in the stance phase and comparing between both solutions. Based on the optimal five-bar parameters, the small-scaled leg prototype was built, and the experiment was conducted to evaluate the trajectory and the actuation torque of each joint varying against the stride.

An optimization framework for upward jumping motion based on quadratic programming (QP) is proposed in this paper, which can simultaneously consider constraints such as the zero moment point (ZMP), limitation of angular accelerations, and anti-slippage. Our approach comprises two parts: the trajectory generation and real-time control. In the trajectory generation for the launch phase, we discretize the continuous trajectories and assume that the accelerations between the two sampling intervals are constant and transcribe the problem into a nonlinear optimization problem. In the real-time control of the stance phase, the over-constrained control objectives such as the tracking of the center of moment (CoM), angle, and angular momentum, and constraints such as the anti-slippage, ZMP, and limitation of joint acceleration are unified within a framework based on QP optimization. Input angles of the actuated joints are thus obtained through a simple iteration. The simulation result reveals that a successful upward jump to a height of 16.4 cm was achieved, which confirms that the controller fully satisfies all constraints and achieves the control objectives.

Compared with wheeled and tracked robots, legged robot has significant advantages in unstructured terrain, but it is difficult for them to achieve the stability of wheeled and tracked robots. Therefore, it is of great significance to study the stability of legged robot. The traditional ZMP method is mostly used in the balance control of biped robot, but ZMP is also applicable to quadruped robot. This paper presents a strategy based on ZMP stability theory to plan the foothold of quadruped robot in trotting gait, to improve the stability of quadruped robot motion. This paper first introduces the calculation method of quadruped robot ZMP. Then, the optimal foothold to maintain the theoretical stability is derived from simple to complex motion from one direction to any direction. Finally, the stability criterion of ZMP is used to compare the situation before and after using this strategy. In this paper, the stone path experiment is carried out on the Unitree A1 robot platform. Finally, it is found that compared with before using this strategy, the distance from ZMP to diagonal leg support line is reduced to a certain extent, and the stability is improved.

No abstract available

Legged robots have high mobility for application with uneven terrain. Bipedal robots require the lower number of actuators with less complication of control system, but their walking stability is the main challenge. This paper presents a design of the bipedal robot having two actuated joints for the hip and the knee and one passive joint for the ankle of each leg. To minimize the inertia of the moving legs, the four brushless motors with integrated controller and planetary gearbox are concentrically located at the hip axis while the knee joints are driven through the parallel linkages. The zero-moment point (ZMP) along the foot support contacting the floor was derived based on the table-cart model. During the stance phase of stable walking, the desired ZMP must be located within the foot boundary. The ZMP generator for shifting the center of mass (COM) forward according to the stride was simulated along with kinematics of the legs. The hip and the knee joint trajectories were implemented on the bipedal robot prototype. The desired leg kinematics was validated by the experiment. As the load supported by the feet, the effect of gravity on the leg joint positions and torques was also studied. Additional sensing and the control of joint stiffnesses will be applied for achieving the dynamic walking of the bipedal robot.

This paper aims to provide robust dynamic motion control and eliminate parameter uncertainty in single-legged robots with a centroidal momentum-based control algorithm. In order to ensure the continuity of the movement despite external disturbances, the disturbance observer (DOB), which is frequently used in the motion control literature, is used. A one-legged robot model was simulated using MSC ADAMS, and then a centroidal momentum-based control algorithm and Zero Moment Point (ZMP)-based control algorithm were developed. The performances of the controllers were tested in the simulation environment in line with three different scenarios, unknown load, external forces and external momentum disturbances. The controllers were evaluated by comparing the selected reference orbital positions with the center of mass (CoM) positions on the x and z axes and the ZMP positions on the x-axis. The simulation results satisfactorily validated the proposed centroidal momentum observer as it performed well in all tested conditions.

This paper presents a reactive legged locomotion generation scheme that enables our quadruped robot CEN-TAURO to adapt to varying payloads while walking. The center-of-mass (CoM) trajectories are generated in real time in a model predictive control (MPC) fashion, trading off large stability margins against evenly stretched legs. Vertex-based zero-moment-point (ZMP) constraints are imposed to ensure quasi-static walking stability. A Kalman filter is then implemented to estimate the CoM states and the impact of external payloads which can vary online and affect/disturb the locomotion differently. The CoM estimation is used to update the MPC motion planner at every replanning instant so that the robot can react to unknown and time-varying payloads on the fly.We validate the proposed scheme through experimental trials where the robot walks on flat ground or steps on different surface levels while carrying heavy payloads. It is shown that the proposed reactive locomotion strategy enables the robot to carry 20 kg payloads, which is close to the maximum capacity of the robot arms.

The realization of running motion can greatly improve the athletic performance of the legged mobile robot. The biologically inspired deadbeat (BID) controller realizes robust bipedal running, allows versatile running patterns and is realtime capable. However, the lack of precise position adjustment strategy of swing leg leads to large center of mass (CoM) tracking error and slow convergence to steady state when perturbed. This paper proposed a simple event-based controller of swing leg position on the basis of the BID controller, using feedback of the system state at the lift off moment. And a linear quadratic regulator (LQR) is used to adjust the feedback gains. The controllers are then embeded into a whole-body control framework, which unifies the task space objectives and the physical constraints of dynamics, contact, ZMP and friction. Meanwhile, the physical limitations of the real robot such as joint torque amplitude and foot size are guaranteed. The proposed control framework was implemented on a torque-controlled bipedal robot and realize steady running in a simulated environment. And the swing leg control is proved under push recovery which decreases the CoM tracking error from 0.076m to 0.045m and recovery time from 2.34s to 1.22s.

Using proprioception to reproduce multiple gaits is nontrivial for legged robots, especially when encountering different terrains and velocity commands. Recently, reinforcement learning (RL) has been utilized to design powerful blind locomotion controllers. However, many RL-based blind locomotion methods are studied based on single gait. In this work, we propose an end-to-end training framework capable of learning multiple gaits for a quadruped robot. A latent space is constructed concurrently by a gait encoder and a gait generator, which helps the robot to reuse multiple gait skills to achieve adaptive gait (AG) behaviors. The trained controller enables smooth transitions between gaits and generates an AG. This means that the robot can effectively apply different gaits based on the current state and velocity command. To learn natural behaviors for multiple gaits, the reward explicitly constructed from gait parameters and the reward implicitly constructed from conditional adversarial motion priors together form the gait-dependent reward, which is subsequently added to the total reward. In the experiment, we demonstrate good performance of the multiple gaits control on a quadruped robot with only proprioceptive sensors.

This paper proposes to solve the problem of Vision-and-Language Navigation with legged robots, which not only provides a flexible way for humans to command but also allows the robot to navigate through more challenging and cluttered scenes. However, it is non-trivial to translate human language instructions all the way to low-level leg joint actions. We propose NaVILA, a 2-level framework that unifies a Vision-Language-Action model (VLA) with locomotion skills. Instead of directly predicting low-level actions from VLA, NaVILA first generates mid-level actions with spatial information in the form of language, (e.g.,"moving forward 75cm"), which serves as an input for a visual locomotion RL policy for execution. NaVILA substantially improves previous approaches on existing benchmarks. The same advantages are demonstrated in our newly developed benchmarks with IsaacLab, featuring more realistic scenes, low-level controls, and real-world robot experiments. We show more results at https://navila-bot.github.io/

Legged robots, designed to emulate human functions, have greatly influenced numerous sectors. However, the focus on continuously improving the joint motors and control systems of existing legged robots not only increases costs and complicates maintenance but also results in failure to accurately mimic the functionality of the human skeletal‒muscular system. This study introduces a bionic legged robot structure that leverages the tensegrity principle, drawing inspiration from the human leg's structural morphology and kinematic mechanisms. By designing a system that distinguishes between rolling and sliding movements, the human knee's variable instantaneous center of rotation (ICR), is successfully replicated showcasing its capabilities in achieving gait resemblance and vibration absorption. The tensegrity unit's features, including remarkable deformability, self‐recovery, and the four‐bar mechanism's singular position characteristic, alongside a rope unlocking mechanism reminiscent of human muscles, facilitate in situ compliance–rigid–compliance transitions of the knee joint without the need for knee joint motors, relying solely on ground contact through the foot. This innovation overcomes the conventional dependency of legged robots on joint motors, as the system requires only a single DC motor positioned at the hip joint and a straightforward control program to seamlessly execute a complete cycle of a single leg's movement.

Legged locomotion animals produce various gaits, e.g., skipping, walking, running, and crawling, depending upon the environmental situation. Such an autonomous selection of distinct gait patterns is highly advantageous for energy efficiency, stress minimization, and postural stability. It should be of interest to introduce such a remarkably flexible mechanism to robotics research. This study addresses the mechanism of generating various gait patterns in a passive legged locomotion system by introducing a tensegrity structure. Building upon the classical model of the rimless wheel, we propose a novel model, called rimless wheel-like tensegrity walker (RTW). Numerical simulations show that the RTW system can generate skipping, walking, and crawling gaits depending upon the strength of the body-leg coupling, which controls independence level of the leg movements. Smooth gait transition can also be realized by a change in the body-leg coupling or the environmental parameter. An experimental study using physical models of the RTW confirmed the validity of the numerical results. The RTW may provide a minimal locomotion model to generate various gaits and to induce their transitions.

Sim-to-real discrepancies hinder learning-based policies from achieving high-precision tasks in the real world. While Domain Randomization (DR) is commonly used to bridge this gap, it often relies on heuristics and can lead to overly conservative policies with degrading performance when not properly tuned. System Identification (Sys-ID) offers a targeted approach, but standard techniques rely on differentiable dynamics and/or direct torque measurement, assumptions that rarely hold for contact-rich legged systems. To this end, we present SPI-Active (Sampling-based Parameter Identification with Active Exploration), a two-stage framework that estimates physical parameters of legged robots to minimize the sim-to-real gap. SPI-Active robustly identifies key physical parameters through massive parallel sampling, minimizing state prediction errors between simulated and real-world trajectories. To further improve the informativeness of collected data, we introduce an active exploration strategy that maximizes the Fisher Information of the collected real-world trajectories via optimizing the input commands of an exploration policy. This targeted exploration leads to accurate identification and better generalization across diverse tasks. Experiments demonstrate that SPI-Active enables precise sim-to-real transfer of learned policies to the real world, outperforming baselines by 42-63% in various locomotion tasks.

Rigid robots were extensively researched, whereas soft robotics remains an underexplored field. Utilizing soft-legged robots in performing tasks as a replacement for human beings is an important stride to take, especially under harsh and hazardous conditions over rough terrain environments. For the demand to teach any robot how to behave in different scenarios, a real-time physical and visual simulation is essential. When it comes to soft robots specifically, a simulation framework is still an arduous problem that needs to be disclosed. Using the simulation open framework architecture (SOFA) is an advantageous step. However, neither SOFA's manual nor prior public SOFA projects show its maximum capabilities the users can reach. So, we resolved this by establishing customized settings and handling the framework components appropriately. Settling on perfect, fine-tuned SOFA parameters has stimulated our motivation towards implementing the state-of-the-art (SOTA) reinforcement learning (RL) method of proximal policy optimization (PPO). The final representation is a well-defined, ready-to-deploy walking, tripedal, soft-legged robot based on PPO-RL in a SOFA environment. Robot navigation performance is a key metric to be considered for measuring the success resolution. Although in the simulated soft robots case, an 82\% success rate in reaching a single goal is a groundbreaking output, we pushed the boundaries to further steps by evaluating the progress under assigning a sequence of goals. While trailing the platform steps, outperforming discovery has been observed with an accumulative squared error deviation of 19 mm. The full code is publicly available at \href{https://github.com/tarekshohdy/PPO_SOFA_Soft_Legged_Robot.git}{github.com/tarekshohdy/PPO$\textunderscore$SOFA$\textunderscore$Soft$\textunderscore$Legged$\textunderscore$ Robot.git}

This article proposes an invariant smoother for legged robot state estimation with the measurement of an inertial measurement unit and leg kinematics while assuming static foot contact. Because the proposed smoother is formulated with the residual functions with group-affine property, their Jacobians become independent from current state estimates. These state-independent Jacobians lead to better convergence properties in optimizing the cost in the smoother, especially under dynamic contact events. The proposed slip rejection method increases the uncertainty of static contact assumption when the robot has dynamic contact events. The estimated foot velocity, which is utilized to detect the dynamic contact events, is re-evaluated within the preserving time window. We also propose the contact loop method, a new measurement model asserting that foot position remains constant over multiple timesteps during stable contact. The proposed estimator is tested through online experiments, including indoor and 160 m-long outdoor experiments, and compared against state-of-the-art algorithms.

Legged robots are physically capable of navigating a diverse variety of environments and overcoming a wide range of obstructions. For example, in a search and rescue mission, a legged robot could climb over debris, crawl through gaps, and navigate out of dead ends. However, the robot's controller needs to respond intelligently to such varied obstacles, and this requires handling unexpected and unusual scenarios successfully. This presents an open challenge to current learning methods, which often struggle with generalization to the long tail of unexpected situations without heavy human supervision. To address this issue, we investigate how to leverage the broad knowledge about the structure of the world and commonsense reasoning capabilities of vision-language models (VLMs) to aid legged robots in handling difficult, ambiguous situations. We propose a system, VLM-Predictive Control (VLM-PC), combining two key components that we find to be crucial for eliciting on-the-fly, adaptive behavior selection with VLMs: (1) in-context adaptation over previous robot interactions and (2) planning multiple skills into the future and replanning. We evaluate VLMPC on several challenging real-world obstacle courses, involving dead ends and climbing and crawling, on a Go1 quadruped robot. Our experiments show that by reasoning over the history of interactions and future plans, VLMs enable the robot to autonomously perceive, navigate, and act in a wide range of complex scenarios that would otherwise require environmentspecific engineering or human guidance.

This paper presents an algorithm to improve state estimation for legged robots. Among existing model-based state estimation methods for legged robots, the contact-aided invariant extended Kalman filter defines the state on a Lie group to preserve invariance, thereby significantly accelerating convergence. It achieves more accurate state estimation by leveraging contact information as measurements for the update step. However, when the model exhibits strong nonlinearity, the estimation accuracy decreases. Such nonlinearities can cause initial errors to accumulate and lead to large drifts over time. To address this issue, we propose compensating for errors by augmenting the Kalman filter with an artificial neural network serving as a nonlinear function approximator. Furthermore, we design this neural network to respect the Lie group structure to ensure invariance, resulting in our proposed Invariant Neural-Augmented Kalman Filter (InNKF). The proposed algorithm offers improved state estimation performance by combining the strengths of model-based and learning-based approaches. Project webpage: https://seokju-lee.github.io/innkf_webpage

In this work, we propose the integration of a mechanism to enable smooth transitions between different locomotion patterns in a hexapod robot. Specifically, we utilize a spiking neural network (SNN) functioning as a Central Pattern Generator (CPG) to generate three distinct locomotion patterns, or gaits: walk, jog, and run. This network produces coordinated spike trains, mimicking those generated in the brain, which are translated into synchronized robot movements via PWM signals. Subsequently, these spike trains are compared using a similarity metric known as SPIKE-synchronization to identify the optimal point for transitioning from one gait to another. This approach aims to achieve three main objectives: first, to maintain the robot’s balance during transitions; second, to ensure that gait transitions are almost imperceptible; and third, to improve energy efficiency by reducing abrupt changes in the robot’s actuators (servomotors). To validate our proposal, we incorporated FSR sensors on the robot’s legs to detect the rigidity of the terrain it navigates. Based on the terrain’s rigidity, the robot dynamically transitions between gaits. The system was tested in real time on a physical hexapod robot across four different types of terrain. Although the method was validated exclusively on a hexapod robot, it can be extended to any legged robot.

In legged robotics locomotion, extracting comprehensive local information from terrain is essential for generating specific leg motions and navigating through unstructured areas. This involves identifying the environment and obstacles, thoroughly characterizing these elements, and defining the best areas to place the legs. Most state-of-the-art methods focus on navigating unstructured terrain only using height analysis, which, although reliable, does not consider the steadiness of the elements of the ground. This paper aims to enhance legged robot motion in unstructured terrain by precisely defining stability zones and leg support points. The primary method for obstacle identification and optimal foothold selection relies on a semantic-based criterion that considers the stability probabilities of each terrain element. A CNN has been trained to address probabilistic characterization. For applicability in a quadrupedal robot, methodology includes discretizing image regions, grouping pixels according to detections, associating discretized regions with the actual depth of the environment, and transforming coordinate systems from RGB-D camera to world-robot. Algorithms of the proposed method are found in the authors' GitHub repository.

This work introduces a model-free reinforcement learning framework that enables various modes of motion (quadruped, tripod, or biped) and diverse tasks for legged robot locomotion. We employ a motion-style reward based on a relaxed logarithmic barrier function as a soft constraint, to bias the learning process toward the desired motion style, such as gait, foot clearance, joint position, or body height. The predefined gait cycle is encoded in a flexible manner, facilitating gait adjustments throughout the learning process. Extensive experiments demonstrate that KAIST HOUND, a 45 kg robotic system, can achieve biped, tripod, and quadruped locomotion using the proposed framework; quadrupedal capabilities include traversing uneven terrain, galloping at 4.67 m/s, and overcoming obstacles up to 58 cm (67 cm for HOUND2); bipedal capabilities include running at 3.6 m/s, carrying a 7.5 kg object, and ascending stairs-all performed without exteroceptive input.

We present a Morphology-Informed Heterogeneous Graph Neural Network (MI-HGNN) for learning-based contact perception. The architecture and connectivity of the MI-HGNN are constructed from the robot morphology, in which nodes and edges are robot joints and links, respectively. By incorporating the morphology-informed constraints into a neural network, we improve a learning-based approach using model-based knowledge. We apply the proposed MI-HGNN to two contact perception problems, and conduct extensive experiments using both real-world and simulated data collected using two quadruped robots. Our experiments demonstrate the superiority of our method in terms of effectiveness, generalization ability, model efficiency, and sample efficiency. Our MI-HGNN improved the performance of a state-of-the-art model that leverages robot morphological symmetry by 8.4 % with only 0.21 % of its parameters. Although MI-HGNN is applied to contact perception problems for legged robots in this work, it can be seamlessly applied to other types of multi-body dynamical systems and has the potential to improve other robot learning frameworks. Our code is made publicly available at https://github.com/lunarlab-gatech/Morphology-Informed-HGNN.

We present SpaceHopper, a three-legged, small-scale robot designed for future mobile exploration of asteroids and moons. The robot weighs 5.2 kg and has a body size of 245 mm while using space-qualifiable components. Furthermore, SpaceHopper’s design and controls make it well-adapted for investigating dynamic locomotion modes with extended flight-phases. Instead of gyroscopes or fly-wheels, the system uses its three legs to reorient the body during flight in preparation for landing. We control the leg motion for reorientation using Deep Reinforcement Learning policies. In a simulation of Ceres’ gravity (0.029 g), the robot can reliably jump to commanded positions up to 6 m away. Our real-world experiments show that SpaceHopper can successfully reorient to a safe landing orientation within 9.7 deg inside a rotational gimbal and jump in a counterweight setup in Earth’s gravity. Overall, we consider SpaceHopper an important step towards controlled jumping locomotion in low-gravity environments.

We present VAPOR, a novel method for autonomous legged robot navigation in unstructured, densely vegetated outdoor environments using offline Reinforcement Learning (RL). Our method trains a novel RL policy using an actor-critic network and arbitrary data collected in real outdoor vegetation. Our policy uses height and intensity-based cost maps derived from 3D LiDAR point clouds, a goal cost map, and processed proprioception data as state inputs, and learns the physical and geometric properties of the surrounding obstacles such as height, density, and solidity/stiffness. The fully-trained policy’s critic network is then used to evaluate the quality of dynamically feasible velocities generated from a novel contextaware planner. Our planner adapts the robot’s velocity space based on the presence of entrapment including vegetation, and narrow passages in dense environments. We demonstrate our method’s capabilities on a Spot robot in complex real-world outdoor scenes, including dense vegetation. We observe that VAPOR’s actions improve success rates by up to 40%, decrease the average current consumption by up to 2.9%, and decrease the normalized trajectory length by up to 11.2% compared to existing end-to-end offline RL and other outdoor navigation methods.

Navigating rugged landscapes poses significant challenges for legged locomotion. Multi-legged robots (those with 6 and greater) offer a promising solution for such terrains, largely due to their inherent high static stability, resulting from a low center of mass and wide base of support. Such systems require minimal effort to maintain balance. Recent studies have shown that a linear controller, which modulates the vertical body undulation of a multi-legged robot in response to shifts in terrain roughness, can ensure reliable mobility on challenging terrains. However, the potential of a learning-based control framework that adjusts multiple parameters to address terrain heterogeneity remains underexplored. We posit that the development of an experimentally validated physics-based simulator for this robot can rapidly advance capabilities by allowing wide parameter space exploration. Here we develop a MuJoCo-based simulator tailored to this robotic platform and use the simulation to develop a reinforcement learning-based control framework that dynamically adjusts horizontal and vertical body undulation, and limb stepping in real-time. Our approach improves robot performance in simulation, laboratory experiments, and outdoor tests. Notably, our real-world experiments reveal that the learning-based controller achieves a 30\% to 50\% increase in speed compared to a linear controller, which only modulates vertical body waves. We hypothesize that the superior performance of the learning-based controller arises from its ability to adjust multiple parameters simultaneously, including limb stepping, horizontal body wave, and vertical body wave.

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Legged robots are supposed to traverse complicated environments, which makes it challenging to design a model-based controller due to their functional complexity. Currently, using deep reinforcement learning to improve the adaptability of robots in complex scenarios has been a major research trend. In this letter, we propose Adaptive Latent Aggregation for Reliable Mimicry (ALARM), a reinforcement learning framework that enables safe and robust locomotion in legged robots using only proprioception. This work features a one-step teacher-student training paradigm by constructing an adaptive aggregation strategy, which integrates the merits of imitation learning and reinforcement learning effectively. The framework integrates normalized penalized proximal policy optimization, which penalizes constraint-violating behaviors while optimizing locomotion policy. Our method facilitates efficient sim-to-real transfer, offering a promising approach for real-world legged robot applications.

In the field of legged robot motion control, reinforcement learning (RL) holds great promise but faces two major challenges: high computational cost for training individual robots and poor generalization of trained models. To address these problems, this paper proposes a novel framework called Prior Transfer Reinforcement Learning (PTRL), which improves both training efficiency and model transferability across different robots. Drawing inspiration from model transfer techniques in deep learning, PTRL introduces a fine-tuning mechanism that selectively freezes layers of the policy network during transfer, making it the first to apply such a method in RL. The framework consists of three stages: pre-training on a source robot using the Proximal Policy Optimization (PPO) algorithm, transferring the learned policy to a target robot, and fine-tuning with partial network freezing. Extensive experiments on various robot platforms confirm that this approach significantly reduces training time while maintaining or even improving performance. Moreover, the study quantitatively analyzes how the ratio of frozen layers affects transfer results, providing valuable insights into optimizing the process. The experimental outcomes show that PTRL achieves better walking control performance and demonstrates strong generalization and adaptability, offering a promising solution for efficient and scalable RL-based control of legged robots.

Thanks to recent explosive developments of data-driven learning methodologies, reinforcement learning (RL) emerges as a promising solution to address the legged locomotion problem in robotics. In this letter, we propose CTS, a novel Concurrent Teacher-Student reinforcement learning architecture for legged locomotion over uneven terrains. Different from conventional teacher-student architecture that trains the teacher policy via RL first and then transfers the knowledge to the student policy through supervised learning, our proposed architecture trains teacher and student policy networks concurrently under the reinforcement learning paradigm. To this end, we develop a new training scheme based on a modified proximal policy gradient (PPO) method that exploits data samples collected from the interactions between both the teacher and the student policies with the environment. The effectiveness of the proposed architecture and the new training scheme is demonstrated through substantial quantitative simulation comparisons with the state-of-the-art approaches and extensive indoor and outdoor experiments with quadrupedal and point-foot bipedal robot platforms, showcasing robust and agile locomotion capability. Quantitative simulation comparisons show that our approach reduces the average velocity tracking error by up to 20% compared to the two-stage teacher-student, demonstrating significant superiority in addressing blind locomotion tasks.

In recent years, legged and wheeled-legged robots have gained prominence for tasks in environments predominantly created for humans across various domains. One significant challenge faced by many of these robots is their limited capability to navigate stairs, which hampers their functionality in multi-story environments. This study proposes a method aimed at addressing this limitation, employing reinforcement learning to develop a versatile controller applicable to a wide range of robots. In contrast to the conventional velocity-based controllers, our approach builds upon a position-based formulation of the RL task, which we show to be vital for stair climbing. Furthermore, the methodology leverages an asymmetric actor-critic structure, enabling the utilization of privileged information from simulated environments during training while eliminating the reliance on exteroceptive sensors during real-world deployment. Another key feature of the proposed approach is the incorporation of a boolean observation within the controller, enabling the activation or deactivation of a stair-climbing mode. We present our results on different quadrupeds and bipedal robots in simulation and showcase how our method allows the balancing robot Ascento to climb 15cm stairs in the real world, a task that was previously impossible for this robot.

Deep Reinforcement Learning (RL) has demonstrated impressive results in solving complex robotic tasks such as quadruped locomotion. Yet, current solvers fail to produce efficient policies respecting hard constraints. In this work, we advocate for integrating constraints into robot learning and present Constraints as Terminations (CaT), a novel constrained RL algorithm. Departing from classical constrained RL formulations, we reformulate constraints through stochastic terminations during policy learning: any violation of a constraint triggers a probability of terminating potential future rewards the RL agent could attain. We propose an algorithmic approach to this formulation, by minimally modifying widely used off-the-shelf RL algorithms in robot learning (such as Proximal Policy Optimization). Our approach leads to excellent constraint adherence without introducing undue complexity and computational overhead, thus mitigating barriers to broader adoption. Through empirical evaluation on the real quadruped robot Solo crossing challenging obstacles, we demonstrate that CaT provides a compelling solution for incorporating constraints into RL frameworks. Videos and code are available at constraints-as-terminations.github.io.

The parallel dual-slider telescopic leg bipedal robot (L04) is characterized by its simple structure and low leg rotational inertia, which contribute to its walking efficiency. However, end-to-end methods often overlook the robot’s physical structure, leading to difficulties in maintaining the parallel alignment of the dual sliders, which in turn compromises walking stability. One potential solution to this issue involves utilizing imitation learning to replicate human motion data. However, the dual telescopic leg structure of the L04 robot makes it difficult to perform motion retargeting of human motion data. To enable L04 walking, we design a method that integrates prior feedforward with reinforcement learning (PFRL), specifically tailored for the parallel dual-slider structure. We utilize prior knowledge as a feedforward action to compensate for system nonlinearities; meanwhile, the feedback action generated by the policy network adaptively regulates dynamic balance and, combined with the feedforward action, jointly controls the robot’s walking. PFRL enforces constraints within the motion space to mitigate the chaotic behavior of the parallel dual sliders. Experimental results show that our method successfully achieves sim2real transfer on a real bipedal robot without the need for domain randomization techniques or intricate reward functions. L04 achieves omnidirectional walking with minimal energy consumption and exhibits robustness against external disturbances.

Advancing the dynamic loco-manipulation capabilities of quadruped robots in complex terrains is crucial for performing diverse tasks. Specifically, dynamic ball manipulation in rugged environments presents two key challenges. The first is coordinating distinct motion modalities to integrate terrain traversal and ball control seamlessly. The second is overcoming sparse rewards in end-to-end deep reinforcement learning, which impedes efficient policy convergence. To address these challenges, we propose a hierarchical reinforcement learning framework. A high-level policy, informed by proprioceptive data and ball position, adaptively switches between pre-trained low-level skills such as ball dribbling and rough terrain navigation. We further propose Dynamic Skill-Focused Policy Optimization to suppress gradients from inactive skills and enhance critical skill learning. Both simulation and real-world experiments validate that our methods outperform baseline approaches in dynamic ball manipulation across rugged terrains, highlighting its effectiveness in challenging environments. Videos are on our website: dribble-hrl.github.io.

In this paper, the One-Legged robot is designed to stabilize itself and stand upright at the desired location after being thrown from a different heights. The 5-DOF planar underactuated main body is driven by Reaction wheels, and adaptive Cartesian impedance control has been implemented to effectively manage hard impacts. Evolutionary Reinforcement Learning based AI Agent have been used to adapt to different launch conditions, such as varying speed and altitude. The learning process was performed in real-time using the Matlab simulation program, which models the system dynamics of the robot. The graphical results of the simulation confirm that, with the assistance of the AI agent, the dynamic robot has successfully maintained its stability without tipping over after the launch and has been able to make the desired correction.

In this paper, we propose a reinforcement learning-based control strategy for multi-terrain walking in a wheel-legged robot. The robot employs a five-bar linkage mechanism and interacts with its environment without the use of vision. The reward function considers both the smoothness of movement and the target velocity, and the proximal policy optimization algorithm is used for reinforcement learning. Simulations on NVIDIA's Isaac Lab platform demonstrate that the robot can navigate slopes, stairs, and uneven terrain using convex and concave pentagonal wheel-leg configurations.

Hydraulic legged robots have potential for high-dynamic motion due to their large power-to-weight ratios. However, it is challenging to ensure both stability and continuity in the motion of such robots. In this study, we propose a jumping motion control framework based on deep reinforcement learning that enables hydraulic limb leg units to perform stable and continuous jumping motions. First, to accurately represent the performance of a physical prototype, a quasi-realistic model incorporating physical feasibility constraints is constructed. This model is informed by analysis of the relevant fluid dynamics, and incorporates a trajectory generator and a motion tracking controller. To achieve stable and continuous jumping performance, a deep reinforcement learning algorithm is developed, which jointly optimizes the trajectory generator and the motion tracking controller. Through validation on the physical prototype, we demonstrate that the proposed method reduces the maximum deviation and the average deviation by over 47% and 60%, respectively, and improves landing compliance by up to 7.7% compared to a baseline optimization algorithm, the non-dominated sorting genetic algorithm (NSGA-II). The proposed control framework may serve as a reference for high-dynamic motion control of legged robots and multi-objective optimization across several decision variables.

Precise close-contact inspections are critical in underwater environments, where complex dynamics and biofouling present significant challenges for conventional vehicles. To address these issues, this study proposes a Reinforcement Learning (RL)-based framework to optimize the gait of an underwater legged robot for accurate and stable locomotion. A simulation environment was developed by modeling buoyancy, added mass, seabed interactions, and data-driven leg hydrodynamic forces. The control framework was designed for close-contact inspections by using a structured action space and stability-promoting rewards. The trained policy was validated through both simulation and real-world experiments, demonstrating effective mitigation of hydrodynamic disturbances. Results showed reduced pitch and altitude oscillations during high-speed forward walking and improved heading accuracy in curved trajectories.

Robotics plays a pivotal role in planetary science and exploration, where autonomous and reliable systems are crucial due to the risks and challenges inherent to space environments. The establishment of permanent lunar bases demands robotic platforms capable of navigating and manipulating in the harsh lunar terrain. While wheeled rovers have been the mainstay for planetary exploration, their limitations in unstructured and steep terrains motivate the adoption of legged robots, which offer superior mobility and adaptability. This paper introduces a constrained reinforcement learning framework designed for autonomous quadrupedal mobile manipulators operating in lunar environments. The proposed framework integrates whole-body locomotion and manipulation capabilities while explicitly addressing critical safety constraints, including collision avoidance, dynamic stability, and power efficiency, in order to ensure robust performance under lunar-specific conditions, such as reduced gravity and irregular terrain. Experimental results demonstrate the framework's effectiveness in achieving precise 6D task-space end-effector pose tracking, achieving an average positional accuracy of 4 cm and orientation accuracy of 8.1 degrees. The system consistently respects both soft and hard constraints, exhibiting adaptive behaviors optimized for lunar gravity conditions. This work effectively bridges adaptive learning with essential mission-critical safety requirements, paving the way for advanced autonomous robotic explorers for future lunar missions.

Legged locomotion in constrained spaces (called crawl spaces) is challenging. In crawl spaces, current proprioceptive locomotion learning methods are difficult to achieve traverse because only ground features are inferred. In this study, a point cloud supervised RL framework for proprioceptive locomotion in crawl spaces is proposed. A state estimation network is designed to estimate the robot’s collision states as well as ground and spatial features for locomotion. A point cloud feature extraction method is proposed to supervise the state estimation network. The method uses representation of the point cloud in polar coordinate frame and MLPs for efficient feature extraction. Experiments demonstrate that, compared with existing methods, our method exhibits faster iteration time in the training and more agile locomotion in crawl spaces. This study enhances the ability of legged robots to traverse constrained spaces without requiring exteroceptive sensors.

Recent advances of locomotion controllers utilizing deep reinforcement learning (RL) have yielded impressive results in terms of achieving rapid and robust locomotion across challenging terrain, such as rugged rocks, non-rigid ground, and slippery surfaces. However, while these controllers primarily address challenges underneath the robot, relatively little research has investigated legged mobility through confined 3D spaces, such as narrow tunnels or irregular voids, which impose all-around constraints. The cyclic gait patterns resulted from existing RL-based methods to learn parameterized locomotion skills characterized by motion parameters, such as velocity and body height, may not be adequate to navigate robots through challenging confined 3D spaces, requiring both agile 3D obstacle avoidance and robust legged locomotion. Instead, we propose to learn locomotion skills end-to-end from goal-oriented navigation in confined 3D spaces. To address the inefficiency of tracking distant navigation goals, we introduce a hierarchical locomotion controller that combines a classical planner tasked with planning waypoints to reach a faraway global goal location, and an RL-based policy trained to follow these waypoints by generating low-level motion commands. This approach allows the policy to explore its own locomotion skills within the entire solution space and facilitates smooth transitions between local goals, enabling long-term navigation towards distant goals. In simulation, our hierarchical approach succeeds at navigating through demanding confined 3D environments, outperforming both pure end-to-end learning approaches and parameterized locomotion skills. We further demonstrate the successful real-world deployment of our simulation-trained controller on a real robot.

Falling is inevitable for legged robots when deployed in unstructured and unpredictable real-world scenarios, such as uneven terrain in the wild. Therefore, to recover dynamically from a fall without unintended termination of locomotion, the robot must possess the complex motor skills required for recovery maneuvers. However, this is exceptionally challenging for existing methods, since it involves multiple unspecified internal and external contacts. To go beyond the limitation of existing methods, we introduced a novel deep reinforcement learning framework to train a learning-based state estimator and a proprioceptive history policy for dynamic fall recovery under external disturbances. The proposed learning-based framework applies to different fall cases indoors and outdoors. Furthermore, we show that the learned fall recovery policies are hardware-feasible and can be implemented on real robots. The performance of the proposed approach is evaluated with extensive trials using a quadruped robot, which shows good effectiveness in recovering the robot after a fall on flat surfaces and grassland.

Small-scale legged robots have found widespread utilization in various industrial and biomedical applications due to their compact size and superior locomotion capabilities. Reducing the number of actuators is often desirable to decrease the robot's size and weight, which comes at the expense of the robot's workspace. Our study proposes a method to enhance the mobility of small-scale legged robots with limited degrees of actuators (DoAs) by co-optimizing both morphology parameters and control policy. The co-optimization is formulated as a bi-level optimization problem, where the control policy is designed using deep reinforcement learning algorithms and central pattern generators (CPGs) at the lower level. The inclusion of CPGs significantly speeds up training and enables the application of simulation results in real-world scenarios. At the upper level, morphology optimization is achieved through Bayesian optimization based on dual-networks. This approach eliminates the need to train a policy for each morphology candidate from scratch, leveraging previous experience to enhance efficiency. Through simulation and physical experiments, the effectiveness of our proposed approach is demonstrated, showcasing its ability to discover optimal morphology and gait for small-scale legged robots with limited DoAs. These findings have potential long-term impacts on small-scale legged robot design and locomotion control.

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This paper presents a novel approach to developing control strategies for mobile robots, specifically the Pegasus, a bionic wheel-legged quadruped robot with unique chassis mechanics that enable four-wheel independent steering and diverse gaits. A multi-agent (MA) reinforcement learning (RL) controller is proposed, treating each leg as an independent agent with the goal of autonomous learning. The framework involves a multi-agent setup to model torso and leg dynamics, incorporating motion guidance optimization signal in the policy training and reward function. By doing so, we address leg schedule patterns for the complex configuration of the Pegasus, the requirement for various gaits, and the design of reward functions for MA-RL agents. Agents were trained using two variations of policy networks based on the framework, and real-world tests show promising results with easy policy transfer from simulation to the actual hardware. The proposed framework models acquired higher rewards and converged faster in training than other variants. Various experiments on the robot deployed framework showed fast response (0.8 s) under disturbance and low linear, angular velocity, and heading error, which was 2.5 cm/s, 0.06 rad/s, and 4°, respectively. Overall, the study demonstrates the feasibility of the proposed MA-RL control framework.

No abstract available

Controlling Wheeled-legged robots is challenging especially on slippery surfaces due to their dependence on continuous ground contact. Unlike quadrupeds or bipeds, which can leverage multiple fixed contact points for recovery, wheeled-legged robots are highly susceptible to slip, where even momentary loss of traction can result in irrecoverable instability. Anticipating ground physical properties such as friction before contact would allow proactive control adjustments, reducing slip risk. In this paper, we propose a frictionaware safety locomotion framework that integrates VisionLanguage Models (VLMs) with a Reinforcement Learning (RL) policy. Our method employs a Retrieval-Augmented Generation (RAG) approach to estimate the Coefficient of Friction (CoF), which is then explicitly incorporated into the RL policy. This enables the robot to adapt its speed based on predicted friction conditions before contact. The framework is validated through experiments in both simulation and on a physical customized Wheeled Inverted Pendulum (WIP). Experimental results show that our approach successfully completes trajectory tracking tasks on slippery surfaces, whereas baseline methods relying solely on proprioceptive feedback fail. These findings highlight the importance and effectiveness of explicitly predicting and utilizing ground friction information for safe locomotion. They also point to a promising research direction in exploring the use of VLMs for estimating ground conditions, which remains a significant challenge for purely vision-based methods.

The advances in wheeled-legged robots in recent years have led to their proliferation in human environments. However, these robots still face significant challenges regarding non-flat terrain. Stair climbing is particularly difficult for these robots and restricts their functionality in human facilities. We implemented a goal conditioned deep reinforcement learning algorithm to develop a position-based controller that allows Wheely, a wheeled-legged quadrupedal robot, to blindly climb stairs up to a record breaking 10 cm in height or 33% of Wheely's maximum body height. By training the reinforcement learning algorithm on a single environment each time, we create specialized policies that excel at their own specific terrain type. Through the independent training, we create a controller with a mixture of experts architecture that does not need to compromise between different environments, leading to a more optimal policy for each scenario. To maintain robustness, a selector neural network selects the policy based on past observations, allowing the robot to function independently in dynamic environments.

Crawling robots are becoming increasingly prevalent in both industrial and private applications. Despite their many advantages over other robot types, they have complex movement mechanics. Artificial intelligence can simplify this by reinforcement learning. This process requires configuring the training environment and defining input parameters, including a robot model for movement training. To translate the virtual results into real-world scenarios, a 3D model with appropriate mechanical parameters must be developed.These parameters can vary significantly between multiple mechanical configurations, which will further impact the reinforcement learning process of such a robot. For this reason, it was decided to test which limb configurations would work best in this process. Initially, various kinematic types of walking robots were analysed, drawing on the anatomy of mammals, reptiles, and insects for the biological model. The reptilian model was chosen for its balance of stability, dynamics, and energy efficiency. The article reviews the preparation of robot models and the configuration of the Unity3D development environment using the ML-Agents toolkit. The experiment examined how different limb lengths affect training, resulting in movement algorithms for various quadruped robot configurations using artificial neural networks. Based on the numerical results, the best configuration was the default, with the same length of the tibia as the thigh, achieving a reward function value of 883.9 and an episode length of 245.5. Taking into account the same criteria, the least efficient configuration was definitely the one characterised by the shortest thigh and the longest tibia among those considered. In its case, the reward function reached a value of only 526.2 with an episode lasting 999.0, which means that it never achieved the intended goal.

In this scholarly investigation, the study focuses on scrutinizing the locomotion and control mechanisms governing a single-legged robot. The analysis encompasses the robot’s movement dynamics pertaining to two primary objectives: executing jumps and sustaining equilibrium throughout successive jump sequences. Diverse concepts of this robot model have been scrutinized, leading to the introduction of a distinctive semi-active model devised for maintaining the robot’s balance. The research involves an initial design for the robot model followed by the introduction of a multi-phase composite control system. As per the proposed model, the jumping action is facilitated through a four-link mechanism augmented by a spring, while balance preservation is achieved through the independent operation of two arms connected to the upper body. To address the successive jumps within the four-link mechanism, a multi-phase feedback controller is engineered. Additionally, a hybrid control strategy, incorporating the Deep Deterministic Policy Gradient algorithm (DDPG) along with a feedback controller, is proposed to sustain balance throughout the robot’s contact and flight phases. The research outcomes, acquired through a series of comprehensive tests conducted within the Simulink simulator environment, demonstrate the robot’s capacity to maintain balance over 80 consecutive jumps. The evaluations encompassed various simulated external disturbances, including 1- horizontal impacts on the upper body, 2- disparities in ground height, and 3- alterations in ground angle between consecutive steps. Notably, the findings showcase the robot’s adeptness in maintaining balance despite an impact with an amplitude of 25 N for a duration of 0.1 seconds, as well as its resilience in managing ground height disparities up to 3 cm and ground angle variations of up to 3°.

Shifting from traditional control strategies to Deep Reinforcement Learning (RL) for legged robots poses inherent challenges, especially when addressing real-world physical constraints during training. While high-fidelity simulations provide significant benefits, they often bypass these essential physical limitations. In this paper, we experiment with the Constrained Markov Decision Process (CMDP) framework instead of the conventional unconstrained RL for robotic applications. We perform a comparative study of different constrained policy optimization algorithms to identify suitable methods for practical implementation. Our robot experiments demonstrate the critical role of incorporating physical constraints, yielding successful sim-to-real transfers, and reducing operational errors on physical systems. The CMDP formulation streamlines the training process by separately handling constraints from rewards. Our findings underscore the potential of constrained RL for the effective development and deployment of learned controllers in robotics.

We propose ReinFlow, a simple yet effective online reinforcement learning (RL) framework that fine-tunes a family of flow matching policies for continuous robotic control. Derived from rigorous RL theory, ReinFlow injects learnable noise into a flow policy's deterministic path, converting the flow into a discrete-time Markov Process for exact and straightforward likelihood computation. This conversion facilitates exploration and ensures training stability, enabling ReinFlow to fine-tune diverse flow model variants, including Rectified Flow [35] and Shortcut Models [19], particularly at very few or even one denoising step. We benchmark ReinFlow in representative locomotion and manipulation tasks, including long-horizon planning with visual input and sparse reward. The episode reward of Rectified Flow policies obtained an average net growth of 135.36% after fine-tuning in challenging legged locomotion tasks while saving denoising steps and 82.63% of wall time compared to state-of-the-art diffusion RL fine-tuning method DPPO [43]. The success rate of the Shortcut Model policies in state and visual manipulation tasks achieved an average net increase of 40.34% after fine-tuning with ReinFlow at four or even one denoising step, whose performance is comparable to fine-tuned DDIM policies while saving computation time for an average of 23.20%. Project webpage: https://reinflow.github.io/

Agile and adaptive maneuvers such as fall recovery, high-speed turning, and sprinting in the wild are challenging for legged systems. We propose a Curricular Hindsight Reinforcement Learning (CHRL) that learns an end-to-end tracking controller that achieves powerful agility and adaptation for the legged robot. The two key components are (i) a novel automatic curriculum strategy on task difficulty and (ii) a Hindsight Experience Replay strategy adapted to legged locomotion tasks. We demonstrated successful agile and adaptive locomotion on a real quadruped robot that performed fall recovery autonomously, coherent trotting, sustained outdoor running speeds up to 3.45 m/s, and a maximum yaw rate of 3.2 rad/s. This system produces adaptive behaviors responding to changing situations and unexpected disturbances on natural terrains like grass and dirt.

This study examines the problem of hopping robot navigation planning to achieve simultaneous goal-directed and environment exploration tasks. We consider a scenario in which the robot has mandatory goal-directed tasks defined using Linear Temporal Logic (LTL) specifications as well as optional exploration tasks represented using a reward function. Additionally, there exists uncertainty in the robot dynamics which results in motion perturbation. We first propose an abstraction of 3D hopping robot dynamics which enables high-level planning and a neural-network-based optimization for low-level control. We then introduce a Multi-task Product IMDP (MT-PIMDP) model of the system and tasks. We propose a unified control policy synthesis algorithm which enables both task-directed goal-reaching behaviors as well as task-agnostic exploration to learn perturbations and reward. We provide a formal proof of the trade-off induced by prioritizing either LTL or RL actions. We demonstrate our methods with simulation case studies in a 2D world navigation environment.

Human motion driven control (HMDC) is an effective approach for generating natural and compelling robot motions while preserving high-level semantics. However, establishing the correspondence between humans and robots with different body structures is not straightforward due to the mismatches in kinematics and dynamics properties, which causes intrinsic ambiguity to the problem. Many previous algorithms approach this motion retargeting problem with unsupervised learning, which requires the prerequisite skill sets. However, it will be extremely costly to learn all the skills without understanding the given human motions, particularly for high-dimensional robots. In this work, we introduce CrossLoco, a guided unsupervised reinforcement learning framework that simultaneously learns robot skills and their correspondence to human motions. Our key innovation is to introduce a cycle-consistency-based reward term designed to maximize the mutual information between human motions and robot states. We demonstrate that the proposed framework can generate compelling robot motions by translating diverse human motions, such as running, hopping, and dancing. We quantitatively compare our CrossLoco against the manually engineered and unsupervised baseline algorithms along with the ablated versions of our framework and demonstrate that our method translates human motions with better accuracy, diversity, and user preference. We also showcase its utility in other applications, such as synthesizing robot movements from language input and enabling interactive robot control.

Model-based approaches for planning and control for bipedal locomotion have a long history of success. It can provide stability and safety guarantees while being effective in accomplishing many locomotion tasks. Model-free reinforcement learning, on the other hand, has gained much popularity in recent years due to computational advancements. It can achieve high performance in specific tasks, but it lacks physical interpretability and flexibility in re-purposing the policy for a different set of tasks. For instance, we can initially train a neural network (NN) policy using velocity commands as inputs. However, to handle new task commands like desired hand or footstep locations at a desired walking velocity, we must retrain a new NN policy. In this work, we attempt to bridge the gap between these two bodies of work on a bipedal platform. We formulate a model-based reinforcement learning problem to learn a reduced-order model (ROM) within a model predictive control (MPC). Results show a 49% improvement in viable task region size and a 21% reduction in motor torque cost. All videos and code are available at https://sites.google.com/view/ymchen/research/rl-for-roms.

Designing a single mobile platform that can traverse diverse terrain both effectively and efficiently remains a significant challenge in robotics. In this work, we propose an optimized wheel-like walking (WLW) locomotion method that integrates terrain-based foothold planning to adapt to rugged environments. Our gait optimization framework incorporates multiple criteria, including tip-over stability, collision avoidance, leg–wheel mechanism kinematics, motion continuity, and energy consumption. Both the stance and swing phases of the leg trajectories are separately planned and optimized. We validate the proposed trajectory planning strategy on a leg–wheel transformable robot across various types of uneven terrain. Experimental results demonstrate that our approach effectively maintains body stability, minimizes foot–hip height variation, and stabilizes the body pitch near zero. Under equivalent conditions, the optimized WLW gait achieves better terrain adaptation, reduces energy consumption by up to 10%, and maintains minimal pitch variation, allowing the robot to remain nearly horizontal while moving forward on uneven terrain.

This letter presents an approach for auto-tuning feedback controllers and online trajectory planners to achieve robust locomotion of a legged robot. The auto-tuning approach uses an Unscented Kalman Filter (UKF) formulation, which adapts/calibrates control parameters online using a recursive implementation. In particular, this letter shows how to use the auto-tuning approach to calibrate cost function weights of a Model Predictive Control (MPC) stance controller and feedback gains of a swing controller for a quadruped robot. Furthermore, this letter extends the auto-tuning approach to calibrating parameters of an online trajectory planner, where the height of a swing leg and the robot’s walking speed are optimized, while minimizing its energy consumption and foot slippage. This allows us to generate stable reference trajectories online and in real time. Results using a high-fidelity Unitree A1 robot simulator in Gazebo provided by the robot manufacturer show the advantages of using auto-tuning for calibrating feedback controllers and for computing reference trajectories online for reduced development time and improved tracking performance.

Traversing across adjacent planes is an important ability for legged climbing robots. While many robots can achieve autonomous ground-to-wall transitions, most are limited to scenarios where the angle between the planes has a certain value. In some cases, however, the robot needs to traverse planes with a wide variety of angles. To enhance the adaptability of the robot in such diverse scenarios, we analyze the plane transition process and propose a universal methodology for hexapod climbing robots with a two-stage workflow. In the first stage, we plan a trajectory of body without considering configuration of legs, within a reachable map. This low-dimensional map can be efficiently sampled and explored to identify feasible transitions. In the second stage, we use a motion prediction to generate landing points, as well as swing and stance trajectories for each leg. By tracking these trajectories, the robot can autonomously transition from one plane to another. Guided by this methodology, we design a hexapod climbing robot capable of autonomously traversing planes with angles ranging from 30° to 270°. For further validation, we build the physical prototype of the robot and conduct a series of plane transition experiments. The results demonstrate the feasibility of both our methodology and the robot.

Abstract In this paper, a novel Central Pattern Generator (CPG) network topology based locomotion control strategy for a smooth gait transition of a biomimetic hexapod robot is proposed. Some preliminaries and correlations have been discussed to provide more suitable CPG network topology for both gait patterns that adapt to different environments, both in terms of transient state time and amplitude overshoot. The design network structure is developed with bidirectional diffusive coupling topologies to obtain robustness and efficient gait transitions. The stability of the proposed network is proved using coupling analyses. In contrast to conventional methods in the CPG network, the proposed method provides remarkable results that could generate four typical hexapod gaits transitions under rapid transient-state and steady-state conditions depending on the frequency, amplitude, and phase relationships among neurons. In order to govern the swing and stance phases according to the proposed network, the leg trajectory generator is designed and an inverse kinematics module is added to compute the link angles of the legs. By applying the proposed locomotion control strategy, the hexapod robot is capable of performing stable and rapid walking gaits. The simulation and experimental results show the effectiveness of the proposed method. High motion ability with the proposed network topology is provided considering walking frequency, forward speed, gait transition time, transient-state time, and steady-state comparisons with the literature.

Crabs are adept at traversing natural terrains that are challenging for mobile robots. Curved dactyls are a characteristic feature that engage terrain in order to resist wave forces in surf zones. Inward gripping motions at the onset of the stance could increase stability. Here, we add inward gripping motions to the foot trajectories of walking gaits to determine the energetic costs and speed for our 12 degree of freedom (DOF) crab-like robot, Sebastian. Specifically, we compared two gaits in which the step size (stance length) was the same, but the swing trajectories were either triangular (to minimize trajectory length) or quadrilateral (in which the leg deliberately oversteps in order to perform a distributed inward grip). The resulting gripping quadrilateral gait significantly outperformed the nongripping triangular gait on diverse terrains (hard linoleum, soft mats, and underwater sand), providing between 15% and 34% energy savings. Using this gait eliminates the advantage of spherical end effectors for slip reduction on hard linoleum, which may lead to a better understanding of how to use crab-like morphology for more efficient locomotion. Finally, we subjected the walking robot to lab-generated waves with a wave height approximately 166% of the dactyl length. Both gaits enabled the robot to walk undisturbed by the waves. Taken together, these results suggest that impact trajectory will be key for future amphibious robots. Future work can provide a deeper understanding of the relationships between dactyls, gaits, and substrates in biology and robots.

In this paper, a 6-DOF biped’s gait is generated based on the motion of the center of mass and the contact point of a nonlinear inverted pendulum (NIP). For walking on deformable terrain, the foot-terrain interaction is modeled using terrain stiffness and damping. The impact dynamics of the NIP model is used to obtain the curve of capture to stop the motion and the curve of equal local energy for walking with equal minimum velocity. The step length obtained from the intersection of the energy curve of the NIP and the terrain is used for the swing leg ankle trajectory. The center of mass and contact point trajectories of the NIP are used in the trajectory generation of the hip-link and the stance leg’s ankle, and the joint angles of the biped are obtained from the inverse kinematics. The floating-base dynamics is used to compare the actual and desired center of mass trajectories of the biped robot. The energy analysis of the biped validates the footstep planning on deformable terrain. The joint torques of the actuated joints are obtained from the inverse dynamics, and the actuation energy is shown to be negligible for energy-efficient walking. The dynamic stability of the biped’s gait is also analyzed based on the rotation of the foot on the deformable terrain.

Robots operating in human environments need various skills, like slow and fast walking, turning, side-stepping, and many more. However, building robot controllers that can exhibit such a large range of behaviors is a challenging problem that requires tedious investigation for every task. We present a unified model-based control algorithm for imitating different animal gaits without expensive simulation training or real-world fine-tuning. Our method consists of stance and swing leg controllers using a centroidal dynamics model augmented with online adaptation techniques. We also develop a whole-body trajectory optimization procedure to fix the kinematic infeasibility of the reference animal motions. We demonstrate that our universal data-driven model-based controller can seamlessly imitate various motor skills, including trotting, pacing, turning, and side-stepping. It also shows better tracking capabilities in simulation and the real world against several baselines, including another model-based imitation controller and a learning-based motion imitation technique.