In this work we present a trajectory Optimization framework for whole-body motion planning through contacts. We demonstrate how the proposed approach can be applied to automatically discover different gaits and dynamic motions on a quadruped robot. In contrast to most previous methods, we do not pre-specify contact switches, timings, points or gait patterns, but they are a direct outcome of the optimization. Furthermore, we optimize over the entire dynamics of the robot, which enables the optimizer to fully leverage the capabilities of the robot. To illustrate the spectrum of achievable motions, here we show eight different tasks, which would require very different control structures when solved with state-of-the-art methods. Using our trajectory Optimization approach, we are solving each task with a simple, high level cost function and without any changes in the control structure. Furthermore, we fully integrated our approach with the robot's control and estimation framework such that optimization can be run online. By demonstrating a rough manipulation task with multiple dynamic contact switches, we exemplarily show how optimized trajectories and control inputs can be directly applied to hardware.
It is often overlooked by roboticists when designing locomotion controllers for their legged machines, that energy consumption plays an important role in selecting the best gaits for locomotion at high speeds or over long distances. The purpose of this study is to examine four similar asymmetrical quadrupedal gaits that are frequently observed in legged animals in nature. To understand how a specific footfall pattern will change the energetics of a legged system, we first developed a full body model of a quadrupedal robot called A1. And for each gait we created a hybrid system with desired foo…
This paper systematically decomposes a quadrupedal robot into bipeds to rapidly generate walking gaits and then recomposes these gaits to obtain quadrupedal locomotion. We begin by decomposing the full-order, nonlinear and hybrid dynamics of a three-dimensional quadrupedal robot, including its continuous and discrete dynamics, into two bipedal systems that are subject to external forces. Using the hybrid zero dynamics (HZD) framework, gaits for these bipedal robots can be rapidly generated (on the order of seconds) along with corresponding controllers. The decomposition is achieved in such a w…
Zhejiang University（浙江大学）；ZJU-Hangzhou Global Scientific and Technological Innovation Center（浙江大学-杭州市全球科技创新中心）；State Key Laboratory of Fluid Power and Mechatronic System, Zhejiang University（浙江大学流体动力与机电系统国家重点实验室）；Institute of Applied Mechanics, Zhejiang University（浙江大学应用力学研究所）
Gaits and transitions are key components in legged locomotion. For legged robots, describing and reproducing gaits as well as transitions remain longstanding challenges. Reinforcement learning has become a powerful tool to formulate controllers for legged robots. Learning multiple gaits and transitions, nevertheless, is related to the multi-task learning problems. In this work, we present a novel framework for training a simple control policy for a quadruped robot to locomote in various gaits. Four independent phases are used as the interface between the gait generator and the control policy, …
Robotic Systems Lab, ETH Zürich（苏黎世联邦理工学院机器人系统实验室）
This paper addresses the problem of legged locomotion in non-flat terrain. As legged robots such as quadrupeds are to be deployed in terrains with geometries which are difficult to model and predict, the need arises to equip them with the capability to generalize well to unforeseen situations. In this work, we propose a novel technique for training neural-network policies for terrain-aware locomotion, which combines state-of-the-art methods for model-based motion planning and reinforcement learning. Our approach is centered on formulating Markov decision processes using the evaluation of dynam…
Birla Institute of Technology and Science, Pilani（皮拉尼比特拉科技学院）
Legged robots can traverse challenging terrain, use perception to plan their safe foothold positions, and navigate the environment. Such unique mobility capabilities make these platforms a perfect candidate for scenarios such as search and rescue, inspection, and exploration tasks. While traversing through such terrains, the robot's instability is a significant concern. Many times the robot needs to switch gaits depending on its environment. Due to the complex dynamics of quadruped robots, classical PID control fails to provide high stability. Thus, there is a need for advanced control methods…
ETH Zurich（苏黎世联邦理工学院）；Max Planck Institute for Solar System Research（马克斯·普朗克太阳系研究所）
Celestial bodies such as the Moon and Mars are mainly covered by loose, granular soil, a notoriously challenging terrain to traverse with (wheeled) robotic systems. Here, we present experimental work on traversing steep, granular slopes with the dynamically walking quadrupedal robot SpaceBok. To adapt to the challenging environment, we developed passive-adaptive planar feet and optimized grouser pads to reduce sinkage and increase traction on planar and inclined granular soil. Single-foot experiments revealed that a large surface area of 110cm2 per foot reduces sinkage to an acceptable level e…
University of Washington（华盛顿大学）；Robotics at Google（Google机器人部门）
We focus on the problem of developing energy efficient controllers for quadrupedal robots. Animals can actively switch gaits at different speeds to lower their energy consumption. In this paper, we devise a hierarchical learning framework, in which distinctive locomotion gaits and natural gait transitions emerge automatically with a simple reward of energy minimization. We use evolutionary strategies (ES) to train a high-level gait policy that specifies gait patterns of each foot, while the low-level convex MPC controller optimizes the motor commands so that the robot can walk at a desired vel…
Planning locomotion trajectories for legged microrobots is challenging because of their complex morphology, high frequency passive dynamics, and discontinuous contact interactions with their environment. Consequently, such research is often driven by time-consuming experimental methods. As an alternative, we present a framework for systematically modeling, planning, and controlling legged microrobots. We develop a three-dimensional dynamic model of a 1.5 gram quadrupedal microrobot with complexity (e.g., number of degrees of freedom) similar to larger-scale legged robots. We then adapt a recen…
End-to-End Multi-Task Policy Learning from NMPC for Quadruped Locomotion
Institute for Artificial Intelligence and Autonomous Systems (A2S)（人工智能与自主系统研究所）；Bonn-Rhein-Sieg University of Applied Sciences（波恩-莱茵-锡格应用科学大学）；Humanoid Robots Lab（人形机器人实验室）；University of Bonn（波恩大学）；Lamarr Institute for Machine Learning and Artificial Intelligence（Lamarr机器学习与人工智能研究所）；Center for Robotics（机器人中心）；Fraunhofer Institute for Intelligent Analysis and Information Systems（弗劳恩霍夫智能分析与信息系统研究所）
Italian Institute of Technology（意大利技术研究院）；University of Genoa（热那亚大学）
This paper presents a simplified model-based trajectory optimization (TO) formulation for motion planning on quadruped mobile manipulators that carry heavy payload of known mass. The proposed payload-aware formulation simultaneously plans locomotion, payload manipulation and considers both robot and payload model dynamics while remaining computationally efficient. At the presence of heavy payload, the approach exhibits reduced leg outstretching (thus increased manipulability) in kinematically demanding motions due to the contribution of payload manipulation in the optimization. The framework's…
Shenyang Institute of Automation, Chinese Academy of Sciences（中国科学院沈阳自动化研究所）；Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences（中国科学院机器人与智能制造创新研究院）；Westlake University（西湖大学）；University of Chinese Academy of Sciences（中国科学院大学）
The essence of quadrupeds' movements is the movement of the center of gravity, which has a pattern in the action of quadrupeds. However, the gait motion planning of the quadruped robot is time-consuming. Animals in nature can provide a large amount of gait information for robots to learn and imitate. Common methods learn animal posture with a motion capture system or numerous motion data points. In this paper, we propose a video imitation adaptation network (VIAN) that can imitate the action of animals and adapt it to the robot from a few seconds of video. The deep learning model extracts key …
Korea Advanced Institute of Science and Technology（韩国科学技术院）；Istituto Italiano di Tecnologia（意大利理工学院）
In this work, a non-gaited framework for legged system locomotion is presented. The approach decouples the gait sequence optimization by considering the problem as a decision-making process. The redefined contact sequence problem is solved by utilizing a Monte Carlo Tree Search (MCTS) algorithm that exploits optimization-based simulations to evaluate the best search direction. The proposed scheme has proven to have a good trade-off between exploration and exploitation of the search space compared to the state-of-the-art Mixed-Integer Quadratic Programming (MIQP). The model predictive control (…
California Institute of Technology（加州理工学院）；Virginia Polytechnic Institute and State University（弗吉尼亚理工学院暨州立大学）
The hybrid zero dynamics (HZD) approach has become a powerful tool for the gait planning and control of bipedal robots. This paper aims to extend the HZD methods to address walking, ambling and trotting behaviors on a quadrupedal robot. We present a framework that systematically generates a wide range of optimal trajectories and then provably stabilizes them for the full-order, nonlinear and hybrid dynamical models of quadrupedal locomotion. The gait planning is addressed through a scalable nonlinear programming using direct collocation and HZD. The controller synthesis for the exponential sta…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Virginia Polytechnic Institute and State University（弗吉尼亚理工学院暨州立大学）；Korea Advanced Institute of Science and Technology（韩国科学技术院）
This paper presents a novel Representation-Free Model Predictive Control (RF-MPC) framework for controlling various dynamic motions of a quadrupedal robot in three dimensional (3D) space. Our formulation directly represents the rotational dynamics using the rotation matrix, which liberates us from the issues associated with the use of Euler angles and quaternion as the orientation representations. With a variation-based linearization scheme and a carefully constructed cost function, the MPC control law is transcribed to the standard Quadratic Program (QP) form. The MPC controller can operate a…
Southern University of Science and Technology（南方科技大学）；Guangdong University of Technology（广东工业大学）
This paper studies real-time motion planning and control for ball bumping motion with quadruped robots. To enable the quadruped to bump the flying ball with different initializations, we develop a nonlinear trajectory optimization-based planning scheme that jointly identifies the take-off time and state to achieve accurate ball hitting during the flight phase. Such a planning scheme employs a two-dimensional single rigid body model that achieves a satisfactory balance between accuracy and efficiency for the highly time-sensitive task. To precisely execute the planned motion, the tracking contr…
Institute for Robotics and Intelligent Machines, Georgia Institute of Technology（佐治亚理工学院机器人与智能机器研究所）；MIT Biomimetic Robotics Lab（麻省理工学院仿生机器人实验室）；NAVER LABS（NAVER实验室）
In this letter, we present a versatile hierarchical offline planning algorithm, along with an online control pipeline for agile quadrupedal locomotion. Our offline planner alternates between optimizing centroidal dynamics for a reduced-order model and whole-body trajectory optimization, with the aim of achieving dynamics consensus. Our novel momentum-inertia-aware centroidal optimization, which uses an equimomental ellipsoid parameterization, is able to generate highly acrobatic motions via ``inertia shaping". Our whole-body optimization approach significantly improves upon the quality of stan…
Istituto Italiano di Tecnologia (IIT)（意大利理工学院）；LAAS-CNRS（法国国家科学研究中心自动化系统实验室）
We present experimental results using a passive whole-body control approach for quadruped robots that achieves dynamic locomotion while compliantly balancing the robot's trunk. We formulate the motion tracking as a Quadratic Program (QP) that takes into account the full robot rigid body dynamics, the actuation limits, the joint limits and the contact interaction. We analyze the controller's robustness against inaccurate friction coefficient estimates and unstable footholds, as well as its capability to redistribute the load as a consequence of enforcing actuation limits. Additionally, we prese…
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
In this paper, WALK-VIO, a novel visual-inertial odometry (VIO) with walking-motion-adaptive leg kinematic constraints that change with body motion for localization of quadruped robots, is proposed. Quadruped robots primarily use VIO because they require fast localization for control and path planning. However, since quadruped robots are mainly used outdoors, extraneous features extracted from the sky or ground cause tracking failures. In addition, the quadruped robots' walking motion cause wobbling, which lowers the localization accuracy due to the camera and inertial measurement unit (IMU). …
Hybrid Dynamics Modeling and Trajectory Planning for a Cable-Trailer System with a Quadruped Robot
School of Artificial Intelligence and Automation, Huazhong University of Science and Technology（人工智能与自动化学院，华中科技大学）；Department of Data Science, City University of Hong Kong（数据科学系，城市大学）；Embodied AI Center of Shanghai AI Lab（上海人工智能实验室具身智能中心）
University of Edinburgh（爱丁堡大学）；McGill University（麦吉尔大学）
Legged robot locomotion requires the planning of stable reference trajectories, especially while traversing uneven terrain. The proposed trajectory optimization framework is capable of generating dynamically stable base and footstep trajectories for multiple steps. The locomotion task can be defined with contact locations, base motion or both, making the algorithm suitable for multiple scenarios (e.g., presence of moving obstacles). The planner uses a simplified momentum-based task space model for the robot dynamics, allowing computation times that are fast enough for online replanning.This fa…
Southern University of Science and Technology（南方科技大学）；University of Notre Dame（圣母大学）
This paper studies capturability and push recovery for quadrupedal locomotion. Despite the rich literature on capturability analysis and push recovery control for legged robots, existing tools are developed mainly for bipeds or humanoids. Distinct quadrupedal features such as point contacts and multiple swinging legs prevent direct application of these methods. To address this gap, we propose a switched systems model for quadruped dynamics, and instantiate the abstract viability concept for quadrupedal locomotion with a time-based gait. Capturability is characterized through a novel specificat…
This work presents a two part framework for online planning and execution of dynamic aerial motions on a quadruped robot. Motions are planned via a centroidal momentum-based nonlinear optimization that is general enough to produce rich sets of novel dynamic motions based solely on the user-specified contact schedule and desired launch velocity of the robot. Since this nonlinear optimization is not tractable for real-time receding horizon control, motions are planned once via nonlinear optimization in preparation of an aerial motion and then tracked continuously using a variational-based optima…
Shandong University（山东大学）；Shandong Youbaote Intelligent Robot Co., Ltd（山东优宝特智能机器人有限公司）；China North Vehicle Research Institute（中国北方车辆研究所）
An intuitive control method for the flying trot, which combines offline trajectory planning with real-time balance control, is presented. The motion features of running animals in the vertical direction were analysed using the spring-load-inverted-pendulum (SLIP) model, and the foot trajectory of the robot was planned, so the robot could run similar to an animal capable of vertical flight, according to the given height and speed of the trunk. To improve the robustness of running, a posture control method based on a foot acceleration adjustment is proposed. A novel kinematic based CoM observati…
Dynamic legged locomotion is a challenging topic because of the lack of established control schemes which can handle aerial phases, short stance times, and high-speed leg swings. In this paper, we propose a controller combining whole-body control (WBC) and model predictive control (MPC). In our framework, MPC finds an optimal reaction force profile over a longer time horizon with a simple model, and WBC computes joint torque, position, and velocity commands based on the reaction forces computed from MPC. Unlike existing WBCs, which attempt to track commanded body trajectories, our controller i…
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 autoregressive 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 settin…
Robotic Systems Lab, ETH Zurich（苏黎世联邦理工学院机器人系统实验室）；Robotics and Artificial Intelligence Lab, KAIST（韩国科学技术院机器人与人工智能实验室）；Intelligent Systems Lab, Intel（英特尔智能系统实验室）
Legged robots that can operate autonomously in remote and hazardous environments will greatly increase opportunities for exploration into under-explored 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, utilizing 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 …
Tandon School of Engineering, New York University（纽约大学坦登工程学院）；Max-Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Oxford Robotics Institute, University of Oxford（牛津大学牛津机器人研究所）
Online planning of whole-body motions for legged robots is challenging due to the inherent nonlinearity in the robot dynamics. In this work, we propose a nonlinear MPC framework, the BiConMP which can generate whole body trajectories online by efficiently exploiting the structure of the robot dynamics. BiConMP is used to generate various cyclic gaits on a real quadruped robot and its performance is evaluated on different terrain, countering unforeseen pushes and transitioning online between different gaits. Further, the ability of BiConMP to generate non-trivial acyclic whole-body dynamic moti…
The Indirect Method for Generating Libraries of Optimal Periodic Trajectories and Its Application to Economical Bipedal Walking
Institute for Adaptive Mechanical Systems, University of Stuttgart（斯图加特大学自适应机械系统研究所）；Systems Modeling and Simulation, Saarland University（萨尔大学系统建模与仿真研究所）
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…
Non-Gaited Legged Locomotion with Monte-Carlo Tree Search and Supervised Learning
Dynamic Legged Systems Laboratory, Istituto Italiano di Tecnologia (IIT), Genova, Italy（动态腿系统实验室，意大利技术研究院（IIT），意大利热那亚）；D AI Laboratory, Technical University of Munich (TUM), Germany（3D人工智能实验室，慕尼黑技术大学（TUM），德国）；ATARI Laboratory, MIRMI, Technical University of Munich (TUM), Germany（ATARI实验室，MIRMI，慕尼黑技术大学（TUM），德国）
Autonomous navigation has played an increasingly significant role in quadruped robot system. However, most existing works on quadruped robots navigation using traditional search-based or sample-based methods do not consider the kinodynamic characteristics of quadruped robots, generating kinodynamically infeasible parts, that are difficult to track. In this paper, we introduce a complete navigation system considering the omnidirectional abilities of quadruped robots. First, we use kinodynamic path finding method to obtain smooth, dynamically feasible, time-optimal initial paths and add collisio…
Robust Nonprehensile Object Transport with Quadruped Robots
Technical University of Munich (TUM)（慕尼黑工业大学）；Tampere University（坦佩雷大学）；University of Naples Federico II（那不勒斯费德里科二世大学）；Mohamed Bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
