Boston Dynamics . Introducing Spot (previouslySpotMini [EB/OL ] . ( 2016-06-23 )[ 2023-12-03 ] . https://www.youtube.com/watch?v=tf7IEVTDjng https://www.youtube.com/watch?v=tf7IEVTDjng https://www.youtube.com/watch?v=tf7IEVTDjng.

Boston Dynamics . Cheetah robot runs 28.3 mph; a bit faster than Usain Bolt [EB/OL ] . ( 2012-09-06 ) [ 2023-12-03 ] . https://www.youtube.com/watch?v=chPanW0QWhA&t=7s https://www.youtube.com/watch?v=chPanW0QWhA&t=7s https://www.youtube.com/watch?v=chPanW0QWhA & t=7s.

Boston Dynamics . LS3-Legged Squad Support System [EB/OL ] . ( 2012-09-11 )[ 2023-12-03 ] . https://www.youtube.com/watch?v=R7ezXBEBE6U https://www.youtube.com/watch?v=R7ezXBEBE6U https://www.youtube.com/watch?v=R7ezXBEBE6U.

PARK H W , WENSING P M , KIM S . Online planning for autonomous running jumps over obstacles in high-speed quadrupeds [C ] // Proceedings of the 11st Robotics Science and Systems . Rome, Italy : MIT , 2015 : 1 - 9 .

HYUN D J , SEOK S , LEE J , et al. High speed trot-running: Implementation of a hierarchical controller using proprioceptive impedance control on the MIT Cheetah [J ] . The International Journal of Robotics Research , 2015 , 33 ( 11 ): 1417 - 1445 . DOI: 10.1177/0278364914532150 http://doi.org/10.1177/0278364914532150 http://journals.sagepub.com/doi/10.1177/0278364914532150 http://journals.sagepub.com/doi/10.1177/0278364914532150 This paper presents implementation of a highly dynamic running gait with a hierarchical controller on the MIT Cheetah. The developed controller enables high-speed running of up to 6 m/s (Froude number of Fr ≈ 7.34) incorporating proprioceptive feedback and programmable virtual leg compliance of the MIT Cheetah. To achieve a stable and fast trot gait, we applied three control strategies: (a) programmable virtual leg compliance that provides instantaneous reflexes to external disturbance and facilitates the self-stabilizing shown in the passive dynamics of locomotion; (b) tunable stance-trajectory design, intended to adjust impulse at each foot-end in the stance phase in a high speed trot-running according to the equilibrium-point hypothesis; and (c) a gait-pattern modulation that imposes a desired cyclic gait-pattern taking cues from proprioceptive TD feedback. Based on three strategies, the controller is hierarchically structured. The control parameters for forward speeds, a specific gait-pattern, and desired leg trajectories are managed by a high-level controller. It consists of both a gait-pattern modulator with proprioceptive leg TD detection and a leg-trajectory generator using a Bèzier curve and a tunable amplitude sinusoidal wave. Instead of employing physical spring/dampers in the robot’s leg, the programmable virtual leg compliance is realized using proprioceptive impedance control in individual low-level leg controllers.

PARK H W , CHUAH M Y , KIM S , et al. Quadruped bounding control with variable duty cycle via vertical impulse scaling [C ] // Proceedings of the 26th Intelligent Robots and Systems. Chicago, IL , US : IEEE , 2014 : 3245 - 3252 .

DI C J , WENSING P M , KATZ B , et al. Dynamic locomotion in the mit cheetah 3 through convex model-predictive control [C ] // Proceedings of the 30th Intelligent Robots and Systems. Madrid, Spain:IEEE , 2018 : 1 - 9 .

KIM D , DI C J , KATZ B , et al. Highly dynamic quadruped locomotion via whole-body impulse control and model predictive control [DB/OL ] . ( 2019-09-14 ) [ 2023-12-03 ] . https://arxiv.org/abs/2110.02799 https://arxiv.org/abs/2110.02799 https://arxiv.org/abs/2110.02799.

GEHRING C , COROS S , HUTTER M , et al. Control of dynamic gaits for a quadrupedal robot [C ] // Proceedings of the 29th International Conference on Robotics and Automation. Karlsruhe , Germany : IEEE , 2013 : 3287 - 3292 .

GEHRING C , BELLICOSO C D , COROS S , et al. Dynamic trotting on slopes for quadrupedal robots [C ] // Proceedings of the 27th Intelligent Robots and Systems. Hamburg , Germany : IEEE , 2015 : 5129 - 5135 .

BJELONIC M , GRANDIA R , GEILINGER M , et al. Offline motion libraries and online MPC for advanced mobility skills [J ] . The International Journal of Robotics Research , 2022 , 41 ( 9/10 ): 903 - 924 . DOI: 10.1177/02783649221102473 http://doi.org/10.1177/02783649221102473 http://journals.sagepub.com/doi/10.1177/02783649221102473 http://journals.sagepub.com/doi/10.1177/02783649221102473 We describe an optimization-based framework to perform complex locomotion skills for robots with legs and wheels. The generation of complex motions over a long-time horizon often requires offline computation due to current computing constraints and is mostly accomplished through trajectory optimization (TO). In contrast, model predictive control (MPC) focuses on the online computation of trajectories, robust even in the presence of uncertainty, albeit mostly over shorter time horizons and is prone to generating nonoptimal solutions over the horizon of the task’s goals. Our article’s contributions overcome this trade-off by combining offline motion libraries and online MPC, uniting a complex, long-time horizon plan with reactive, short-time horizon solutions. We start from offline trajectories that can be, for example, generated by TO or sampling-based methods. Also, multiple offline trajectories can be composed out of a motion library into a single maneuver. We then use these offline trajectories as the cost for the online MPC, allowing us to smoothly blend between multiple composed motions even in the presence of discontinuous transitions. The MPC optimizes from the measured state, resulting in feedback control, which robustifies the task’s execution by reacting to disturbances and looking ahead at the offline trajectory. With our contribution, motion designers can choose their favorite method to iterate over behavior designs offline without tuning robot experiments, enabling them to author new behaviors rapidly. Our experiments demonstrate complex and dynamic motions on our traditional quadrupedal robot ANYmal and its roller-walking version. Moreover, the article’s findings contribute to evaluating five planning algorithms.

MIKI T , LEE J , HWANGBO J , et al. Learning robust perceptive locomotion for quadrupedal robots in the wild [J ] . Science Robotics , 2022 , 7 ( 62 ): eabk2822 . DOI: 10.1126/scirobotics.abk2822 http://doi.org/10.1126/scirobotics.abk2822 https://www.science.org/doi/10.1126/scirobotics.abk2822 https://www.science.org/doi/10.1126/scirobotics.abk2822 Legged robots that can operate autonomously in remote and hazardous environments will greatly increase opportunities for exploration into underexplored areas. Exteroceptive perception is crucial for fast and energy-efficient locomotion: Perceiving the terrain before making contact with it enables planning and adaptation of the gait ahead of time to maintain speed and stability. However, using exteroceptive perception robustly for locomotion has remained a grand challenge in robotics. Snow, vegetation, and water visually appear as obstacles on which the robot cannot step or are missing altogether due to high reflectance. In addition, depth perception can degrade due to difficult lighting, dust, fog, reflective or transparent surfaces, sensor occlusion, and more. For this reason, the most robust and general solutions to legged locomotion to date rely solely on proprioception. This severely limits locomotion speed because the robot has to physically feel out the terrain before adapting its gait accordingly. Here, we present a robust and general solution to integrating exteroceptive and proprioceptive perception for legged locomotion. We leverage an attention-based recurrent encoder that integrates proprioceptive and exteroceptive input. The encoder is trained end to end and learns to seamlessly combine the different perception modalities without resorting to heuristics. The result is a legged locomotion controller with high robustness and speed. The controller was tested in a variety of challenging natural and urban environments over multiple seasons and completed an hour-long hike in the Alps in the time recommended for human hikers.

HWANGBO J , LEE J , DOSOVITSKIY A , et al. Learning agile and dynamic motor skills for legged robots [J ] . Science Robotics , 2019 , 4 ( 26 ): eaau5872 . DOI: 10.1126/scirobotics.aau5872 http://doi.org/10.1126/scirobotics.aau5872 https://www.science.org/doi/10.1126/scirobotics.aau5872 https://www.science.org/doi/10.1126/scirobotics.aau5872 A method for learning agile control policies uses simulated data to enable precise, efficient movements in a complex physical robot.

LEE J , HWANGBO J , WELLHAUSEN L , et al. Learning quadrupedal locomotion over challenging terrain [J ] . Science Robotics , 2020 , 5 ( 47 ): eabc5986 . DOI: 10.1126/scirobotics.abc5986 http://doi.org/10.1126/scirobotics.abc5986 https://www.science.org/doi/10.1126/scirobotics.abc5986 https://www.science.org/doi/10.1126/scirobotics.abc5986 A learning-based locomotion controller enables a quadrupedal ANYmal robot to traverse challenging natural environments.

梁启星 , 李彬 , 李志 , 等 . 基于模型预测控制的四足机器人斜坡自适应调整算法与实现 [J ] . 山东大学学报 (工学版) , 2021 , 51 ( 3 ): 37 - 44 .

LIANG Q X , LI B , LI Z , et al. Algorithm of adaptive slope adjustment of quadruped robot based on model predictive control and its application [J ] . Journal of Shandong University(Engineering Science) , 2021 , 51 ( 3 ): 37 - 44 . (in Chinese)

HAN K C , KIM J Y . Posture stabilizing control of quadruped robot based on cart-inverted pendulum model [J ] . Intelligent Service Robotics , 2023 , 16 : 521 - 536 . DOI: 10.1007/s11370-023-00480-8 http://doi.org/10.1007/s11370-023-00480-8

SOMBOLESTAN M , NGUYEN Q . Hierarchical adaptive loco-manipulation control for quadruped robots [C ] // Proceedings of the 39th International Conference on Robotics and Automation.London , UK : IEEE , 2023 : 12156 - 12162 .

张国腾 , 荣学文 , 李贻斌 , 等 . 基于虚拟模型的四足机器人对角小跑步态控制方法 [J ] . 机器人 , 2016 , 38 ( 1 ): 64 - 74 . DOI: 10.13973/j.cnki.robot.2016.0064 http://doi.org/10.13973/j.cnki.robot.2016.0064 为提高四足机器人对角小跑运动的稳定性，实现机器人躯干 6 维运动方向控制的解耦，提出了一种基于虚拟模型的对角小跑步态控制方法．控制器主要包括支撑相虚拟模型控制和摆动相虚拟模型控制．在支撑相，建立了作用于躯干质心的虚拟力与对角支撑腿关节扭矩之间的数学关系，通过调整躯干虚拟力的大小控制躯干的高度与姿态，控制机器人前进速度和自转角速度．在摆动相，将机器人侧向速度控制引入到足端轨迹规划中，并通过虚拟的“弹簧－阻尼”元件驱动摆动足沿给定轨迹运动．此外，在控制器设计过程中，引入了状态机，用于监控机器人各腿的状态，并输出对角小跑步态相位切换指令．仿真实验结果表明，机器人能够以对角小跑步态在平地上进行全方位移动，跨越不平坦地形，并能够抵抗外部冲击，证明了文中控制方法的有效性和鲁棒性．

ZHANG G T , RONG X W , LI Y B , et al. Control of the quadrupedal trotting based on virtual model [J ] . Robot , 2016 , 38 ( 1 ): 64 - 74 .(in Chuese) DOI: 10.13973/j.cnki.robot.2016.0064 http://doi.org/10.13973/j.cnki.robot.2016.0064 In order to improve the stability of the trotting quadruped robot and to decouple the control of the robot torso motion along six directions, an approach based on virtual model is presented for trot gait control. The controller mainly consists of two main modules: the virtual model control at support phase and the virtual model control at flight phase. During the support phase, the mathematical relationship are mapped between the joint torques of diagonal support legs and the virtual forces acted on the center-of-mass of the torso. And the values of virtual torso forces are regulated to control the torso attitude and height, as well as the forward velocity and the yaw angular velocity of the robot. During the flight phase, lateral velocity is introduced to plan the toe trajectory. And virtual spring-damper sections are implemented to drive the flight toes to track the planned trajectories. In addition, while designing the controller, a state machine is introduced to monitor the legs' states and output phase switching commands for trot gait regulation. The simulations show that the robot is able to trot omni-directionally on flat ground as well as uneven terrains, even suffering from external impacts. And thus the effectiveness and robustness of the controller are proved.

ZHAO J X , YAO Q C , XING B Y , et al. One-legged hop of compliance control based on minimum-jerk [J ] . Journal of Physics Conference Series , 2020 , 1507 ( 5 ): 052012 . DOI: 10.1088/1742-6596/1507/5/052012 http://doi.org/10.1088/1742-6596/1507/5/052012 Aiming at the trajectory planning and compliance of the one-legged hop of a quadruped robot, a method for designing the optimal trajectory of the bouncing by minimizing the jerk index is proposed, and active compliance control is used to reduce the impact of the end of the foot and improve the compliance of the one-legged. First, design a one-legged bounce strategy. The bounce is divided into three phases: take-off phase, vacant phase, and buffer phase, and trajectory optimization is performed for each phase. Secondly, a compliance control method is designed. During the bouncing process, the impact of the foot end is large, which will affect the stability of the robot and even destroy the mechanical structure of the leg. The joint level adopts force-position mixed control, and the one-legged level adopts impedance control to eliminate the impact of ground impact on one-legged. Finally, a one-legged model of a quadruped robot was established in the Vortex multi-body dynamics system. Simulation experiments have proved the feasibility of trajectory optimization and compliance control.

孟健 , 李贻斌 , 李彬 . 四足机器人跳跃步态控制方法 [J ] . 山东大学学报(工学版) , 2015 , 45 ( 3 ): 28 - 34 .

MENG J , LI Y B , LI B , et al . Bound gait controlling method of quadruped robot [J ] . Journal of Shandong University (Engineering Science) , 2015 , 45 ( 3 ): 28 - 34 . (in Chinese)

PARK H W , PARK S , KIM S . Variable-speed quadrupedal bounding using impulse planning:Untethered high-speed 3D Running of MIT Cheetah 2 [C ] // Proceedings of the 31st International Conference on Robotics and Automation. Seattle, WA , US : IEEE , 2015 : 5163 - 5170 .

BJELONIC M . Planning and control for hybrid locomotion of wheeled-legged robots [D ] . ETH Zurich , 2021 .

HOELLER D , RUDIN N , SAKO D , et al . Learning agile navigation for quadrupedal robots [DB/OL ] . ( 2023-06-26 ) [ 2023-12-03 ] . https://arxiv.org/abs/2306.14874 https://arxiv.org/abs/2306.14874 https://arxiv.org/abs/2306.14874.

HAN L , ZHU Q , SHENG J , et al . Lifelike agility and play on quadrupedal robots using reinforcement learning and generative pre-trained models [DB/OL ] . ( 2023-08-29 ) [ 2023-12-03 ] . https://arxiv.org/abs/2308.15143 https://arxiv.org/abs/2308.15143 https://arxiv.org/abs/2308.15143.

NGUYEN Q , POWELL M J , KATZ B , et al. Optimized jumping on the mit cheetah 3 robot [C ] // Proceedings of the 35th International Conference on Robotics and Automation. Montreal , Canada : IEEE , 2019 : 7448 - 7454 .

CHIGNOLI M M T . Trajectory optimization for dynamic aerial motions of legged robots [D ] . Cambridge, MA, US:MIT , 2021 .