BISWAL P , MOHANTY P K . Development of quadruped walking robots: a review [J ] . Ain Shams Engineering Journal , 2021 , 12 ( 2 ): 2017 - 2031 . DOI: 10.1016/j.asej.2020.11.005 http://doi.org/10.1016/j.asej.2020.11.005 https://linkinghub.elsevier.com/retrieve/pii/S2090447920302501 https://linkinghub.elsevier.com/retrieve/pii/S2090447920302501

WINKLER A W , FARSHIDIAN F , PARDO D , et al. Fast trajectory optimization for legged robots using vertex-based ZMP constraints [J ] . IEEE Robotics and Automation Letters , 2017 , 2 ( 4 ): 2201 - 2208 . DOI: 10.1109/LRA.2017.2723931 http://doi.org/10.1109/LRA.2017.2723931 http://ieeexplore.ieee.org/document/7970144/ http://ieeexplore.ieee.org/document/7970144/

CHEN H , HONG Z J , YANG S P , et al. Quadruped capturability and push recovery via a switched-systems characterization of dynamic balance [J ] . IEEE Transactions on Robotics , 2023 , 39 ( 3 ): 2111 - 2130 . DOI: 10.1109/TRO.2023.3240622 http://doi.org/10.1109/TRO.2023.3240622 https://ieeexplore.ieee.org/document/10077785/ https://ieeexplore.ieee.org/document/10077785/

FAWCETT R T , PANDALA A , KIM J , et al. Real-time planning and nonlinear control for quadrupedal locomotion with articulated tails [J ] . Journal of Dynamic Systems, Measurement, and Control , 2021 , 143 ( 7 ): 071004 . DOI: 10.1115/1.4049555 http://doi.org/10.1115/1.4049555 https://asmedigitalcollection.asme.org/dynamicsystems/article/143/7/071004/1094100/Real-Time-Planning-and-Nonlinear-Control-for https://asmedigitalcollection.asme.org/dynamicsystems/article/143/7/071004/1094100/Real-Time-Planning-and-Nonlinear-Control-for The primary goal of this paper is to develop a formal foundation to design nonlinear feedback control algorithms that intrinsically couple legged robots with bio-inspired tails for robust locomotion in the presence of external disturbances. We present a hierarchical control scheme in which a high-level and real-time path planner, based on an event-based model predictive control (MPC), computes the optimal motion of the center of mass (COM) and tail trajectories. The MPC framework is developed for an innovative reduced-order linear inverted pendulum (LIP) model that is augmented with the tail dynamics. At the lower level of the control scheme, a nonlinear controller is implemented through the use of quadratic programming (QP) and virtual constraints to force the full-order dynamical model to track the prescribed optimal trajectories of the COM and tail while maintaining feasible ground reaction forces at the leg ends. The potential of the analytical results is numerically verified on a full-order simulation model of a quadrupedal robot augmented with a tail with a total of 20 degrees-of-freedom. The numerical studies demonstrate that the proposed control scheme coupled with the tail dynamics can significantly reduce the effect of external disturbances during quadrupedal locomotion.

LAKATOS D , PLOEGER K , LOEFFL F , et al. Dynamic locomotion gaits of a compliantly actuated quadruped with slip-like articulated legs embodied in the mechanical design [J ] . IEEE Robotics and Automation Letters , 2018 , 3 ( 4 ): 3908 - 3915 . DOI: 10.1109/LRA.2018.2857511 http://doi.org/10.1109/LRA.2018.2857511 https://ieeexplore.ieee.org/document/8413123/ https://ieeexplore.ieee.org/document/8413123/

KOCO E , MUTKA A , KOVACIC Z . New variable passive-compliant element design for quadruped adaptation to stiffness-varying terrain [J ] . International Journal of Advanced Robotic Systems , 2016 , 13 ( 3 ): 90 . DOI: 10.5772/63893 http://doi.org/10.5772/63893 http://journals.sagepub.com/doi/10.5772/63893 http://journals.sagepub.com/doi/10.5772/63893 This paper presents the design of a novel variable passive-compliant ( VPC) element utilized as a lower-leg implant of a fully electrically driven quadruped robot. It is designed as a slider-piston mechanism which ensures that the force produced during a foot-ground contact is directly perpendicular to the contact surface of an actuated revolute spring. In this way, by altering the stiffness of quadruped legs in a closed-loop manner, the VPC element enables the quadruped robot to adapt to varying terrain characteristics, ensuring a constant hopping frequency over a wide range of terrain-stiffness variations. The designed VPC element and its beneficial characteristics are described in detail. Mathematical relations are formulated that help to describe the influence of the VPC element during vertical hopping of a quadruped robot. The properties of the quadruped research platform with integrated VPC element were verified in simulation and through experiments.

HAN B , YI H Y , XU Z Y , et al. 3D-slip model based dynamic stability strategy for legged robots with impact disturbance rejection [J ] . Scientific Reports , 2022 , 12 ( 1 ): 5892 . DOI: 10.1038/s41598-022-09937-9 http://doi.org/10.1038/s41598-022-09937-9 Inspired by biomechanical studies, the spring-loaded inverted pendulum model is an effective behavior model to describe the running movement of animals and legged robots in the sagittal plane. However, when confronted with external lateral disturbances, the model has to move out of the 2-D plane and be extended to 3-D locomotion. With the degree of freedom increasing, the computational complexity is higher and the real-time control is more and more difficult, especially when considering the complex legged model. Here, we construct a control strategy based on the classical Raibert controller for legged locomotion under lateral impact disturbances. This strategy, named 3D-HFC, is composed of three core modules: touchdown angle control, body attitude angle control and energy compensation. The energy loss in each step is taken into consideration, and the real-time measured energy loss of the current step is adopted to predict that of the next step. We demonstrate the efficiency of the proposed control strategy on a simulated 3D-SLIP lower order model and a simulated running quadruped, which are perturbed by different impact forces. Furthermore, a quadruped bionic prototype named MBBOT was set up, on which lateral impact experiments were designed and implemented. Both simulation and experimental results show that the proposed approach can realize the impact disturbance rejection.© 2022. The Author(s).

GONG D W , WANG P , ZHAO S Y , et al. Bionic quadruped robot dynamic gait control strategy based on twenty degrees of freedom [J ] . IEEE/CAA Journal of Automatica Sinica , 2017 , 5 ( 1 ): 382 - 388 . DOI: 10.1109/JAS.2017.7510790 http://doi.org/10.1109/JAS.2017.7510790 https://ieeexplore.ieee.org/document/8232606/ https://ieeexplore.ieee.org/document/8232606/

WANG L , MENG L B , KANG R , et al. Design and dynamic locomotion control of quadruped robot with perception-less terrain adaptation [J ] . Cyborg and Bionic Systems , 2022 ( 2 ): 9816495 .

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KIM D , DI CARLO J , KATZ B , et al. Highly dynamic quadruped locomotion via whole-body impulse control and model predictive control:arXiv:1909.06586 [R ] . Ithaca,NY , US : Cornell University , 2019 : 1909 .06586.

YUAN S D , ZHOU Y J , LUO C . Crawling gait planning based on foot trajectory optimization for quadruped robot [C ] // Proceedings of the 16th IEEE International Conference on Mechatronics and Automation. Tianjing , China : IEEE , 2019 : 1490 - 1495 .

LI X S , ZHANG X H , NIU J K , et al. A stable walking strategy of quadruped robot based on ZMP in trotting gait [C ] // Proceedings of the 19th IEEE International Conference on Mechatronics and Automation (ICMA). Guilin , China : IEEE , 2022 : 858 - 863 .

王艳琴 , 许威 , 周川 , 等 . 基于质心调整的四足机器人转弯步态规划 [J ] . 控制工程 , 2021 , 28 ( 11 ): 2280 - 2285 .

WANG Y Q , XU W , ZHOU C , et al. Turning gait strategy for quadruped robot based on adjustment of center of mass [J ] . Control Engineering of China , 2021 , 28 ( 11 ): 2280 - 2285 . (in Chinese)

CHEN G R , GUO S , HOU B W , et al. Virtual model control for quadruped robots [J ] . IEEE Access , 2020 , 8 : 140736 - 140751 . DOI: 10.1109/Access.6287639 http://doi.org/10.1109/Access.6287639 https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6287639 https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=6287639

谭永营 , 晁智强 , 金毅 , 等 . 基于虚拟元件的负载型四足步行平台静步态行走控制 [J ] . 兵工学报 , 2019 , 40 ( 12 ): 2570 - 2579 . DOI: 10.3969/j.issn.1000-1093.2019.12.023 http://doi.org/10.3969/j.issn.1000-1093.2019.12.023 传统的基于逆运动学或逆动力学静步态行走控制方法易导致步行平台足端与地面产生较大冲击力、机体轨迹产生较大跟踪误差，使步行平台出现稳定性问题。为实现负载型四足步行平台静步态柔顺稳定行走，提出一种基于虚拟元件的静步态行走控制方法。将步行平台静步态行走控制分为机体运动控制及摆动腿运动控制两部分，分别在机体各自由度及摆动腿各自由度添加虚拟弹簧阻尼元件，将机体与摆动腿控制转换为虚拟力控制。运用序列二次规划方法将机体虚拟力分配到支撑腿足端。结合各腿的雅克比矩阵，得到支撑腿与摆动腿的驱动关节力矩，并设计了步行平台静步态行走状态机。运用MATLAB与ADAMS软件建立四足步行平台仿真模型，对平台静步态行走进行联合仿真。仿真结果表明，虚拟元件控制实现了步行平台复杂地形静步态平稳行走，平台能够适应复杂地形变化，足端与地面冲击力较小，证实了所提虚拟元件控制方法的有效性。

TAN Y Y , CHAO Z Q , JIN Y , et al. Control of static walking gait of load carrying quadruped walking vehicle based on virtual components [J ] . Acta Armamentarii , 2019 , 40 ( 12 ): 2570 - 2579 . (in Chinese) DOI: 10.3969/j.issn.1000-1093.2019.12.023 http://doi.org/10.3969/j.issn.1000-1093.2019.12.023 The traditional static gait control method based on inverse kinematics or inverse dynamics is liable to produce a large impact between the feet and the ground, which causes a large tracking error of vehicle body trajectory. A virtual-components-based static gait control method is proposed to achieve the static gait compliance of load-carrying quadruped walking vehicle. The proposed method is divided into two parts: vehicle body motion control and swinging leg control. The virtual spring damper elements are added to the vehicle body's and swing leg's degrees of freedom, respectively, so as to convert the control of vehicle body and the swing leg to the control of virtual forces. And the sequential quadratic programming method is used to assign the virtual forces of vehicle body to the supporting legs. Based on the Jacobian matrix of each leg, the joint torques of supporting legs and swing leg are obtained. A state machine of vehicle’s static gait is designed. MATLAB and ADAMS are used to establish the simulation models for the quadruped walking vehicle, and the co-simulations are carried out for the quadruped walking vehicle. The simulated results show that the virtual-components-based control method enables the vehicle to walk on rough terrain in a stable gait. The vehicle is able to adapt to the change of terrain, and a impact between the feet and the ground is reduced by using the proposed control method.Key

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CUI J W , LI Z , QIU J , et al. Fault-tolerant motion planning and generation of quadruped robots synthesised by posture optimization and whole body control [J ] . Complex & Intelligent Systems , 2022 , 8 ( 4 ): 2991 - 3003 .

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CHOI J , NA B , OH S , et al. A disturbance observer for robust position tracking control and ground contact detection of aCheetaroid-I leg [C ] // Proceedings of the 13rd IEEE/ASME International Conference on Advanced Intelligent Mechatronics. Besacon , France : IEEE , 2014 : 72 - 75 .

朱晓璐 , 万锦晓 , 许威 , 等 . 侧向推搡下的四足机器人复合抗扰控制策略 [J ] . 信息与控制 , 2021 , 50 ( 1 ): 119 - 128 . DOI: 10.13976/j.cnki.xk.2020.0158 http://doi.org/10.13976/j.cnki.xk.2020.0158 针对四足机器人侧向推搡下的平衡恢复问题，提出了一种复合抗扰反应式鲁棒控制策略.该策略由摆动相的自适应侧摆规划策略和支撑相的关节抗扰控制构成.摆动相自适应侧摆规划策略通过四足机器人足端落地点的力平衡条件进行主动式步态规划以保证机器人在侧向推搡下的姿态稳定，并基于关节输出力矩给出了侧摆的启动条件.支撑相关节抗扰控制通过带扰动项的四足机器人完整动力学模型设计了基于干扰观测器的鲁棒滑模控制器，实现对侧向推搡扰动的补偿.最后，通过Matlab与ADAMS联合仿真验证了提出的控制策略的有效性.

ZHU X L , WAN J X , XU W , et al. Composite disturbance rejection control strategy for quadruped robot under lateral disturbance [J ] . Information and Control , 2021 , 50 ( 1 ): 119 - 128 . (in Chinese) DOI: 10.13976/j.cnki.xk.2020.0158 http://doi.org/10.13976/j.cnki.xk.2020.0158 We propose a composite robust reactive control strategy with disturbance rejection to solve the balance recovery problem of the quadruped robot under lateral disturbance. The proposed strategy has two parts:An adaptive lateral motion planning strategy for the swinging phase and joint robust control for the supporting phase. The swing-phase adaptive lateral motion planning strategy gives the starting conditions of the lateral motion based on the joint output torque, and performs active gait planning based on the force balance conditions of the foot of the quadruped robot to ensure the stability of the robot's posture under lateral disturbance. The supporting-phase joint control compensates for the lateral disturbance of the supporting leg joint via a robust sliding mode controller, which has a disturbance observer based on a complete dynamic model of the quadruped robot that includes a disturbance term to ensure the stability of the quadruped robot under lateral disturbance. The effectiveness of the proposed control strategy is verified by a co-simulation of Matlab and ADAMS.

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