This work explores an innovative algorithm designed to enhance the mobility of underactuated bipedal robots across challenging terrains, especially when navigating through spaces with constrained opportunities for foot support, like steps or stairs. By combining ankle torque with a refined angular momentum-based linear inverted pendulum model (ALIP), our method allows variability in the robot's center of mass height. We employ a dual-strategy controller that merges virtual constraints for precise motion regulation across essential degrees of freedom with an ALIP-centric model predictive control (MPC) framework, aimed at enforcing gait stability. The effectiveness of our feedback design is demonstrated through its application on the Cassie bipedal robot, which features 20 degrees of freedom. Key to our implementation is the development of tailored nominal trajectories and an optimized MPC that reduces the execution time to under 500 microseconds--and, hence, is compatible with Cassie's controller update frequency. This paper not only showcases the successful hardware deployment but also demonstrates a new capability, a bipedal robot using a moving walkway.
Global position control for underactuated bipedal walking is a challenging problem due to the lack of actuation on the feet of the robots. In this paper, we apply the Hybrid-Linear Inverted Pendulum (H-LIP) based stepping on 3D underactuated bipedal robots for global position control. The step-to-step (S2S) dynamics of the H-LIP walking approximates the actual S2S dynamics of the walking of the robot, where the step size is considered as the input. Thus the feedback controller based on the H-LIP approximately controls the robot to behave like the H-LIP, the differences between which stay in an…
We present a framework to generate periodic trajectory references for a 3D under-actuated bipedal robot, using a linear inverted pendulum (LIP) based controller with adaptive neural regulation. We use the LIP template model to estimate the robot's center of mass (CoM) position and velocity at the end of the current step, and formulate a discrete controller that determines the next footstep location to achieve a desired walking profile. This controller is equipped on the frontal plane with a Neural-Network-based adaptive term that reduces the model mismatch between the template and physical rob…
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…
This paper presents a novel Adaptive-frequency MPC framework for bipedal locomotion over terrain with uneven stepping stones. In detail, we intend to achieve adaptive foot placement and gait period for bipedal periodic walking gait with this MPC, in order to traverse terrain with discontinuities without slowing down. We pair this adaptive-frequency MPC with a kino-dynamics trajectory optimization for optimal gait periods, center of mass (CoM) trajectory, and foot placements. We use whole-body control (WBC) along with adaptive-frequency MPC to track the optimal trajectories from the offline opt…
Extended Hybrid Zero Dynamics for Bipedal Walking of the Knee-less Robot SLIDER
Robot Intelligence Lab, Dyson School of Design Engineering, Imperial College London, UK（机器人智能实验室，设计工程学院，伦敦帝国理工学院，英国）；Department of Aeronautics, Imperial College London, UK（航空系，伦敦帝国理工学院，英国）
Southern University of Science and Technology（南方科技大学）；National University of Singapore（新加坡国立大学）；University of Notre Dame（圣母大学）
This paper studies jumping for wheeled-bipedal robots, a motion that takes full advantage of the benefits from the hybrid wheeled and legged design features. A comprehensive hierarchical scheme for motion planning and control of jumping with wheeled-bipedal robots is developed. Underactuation of the wheeled-bipedal dynamics is the main difficulty to be addressed, especially in the planning problem. To tackle this issue, a novel wheeled-spring-loaded inverted pendulum (W-SLIP) model is proposed to characterize the essential dynamics of wheeled-bipedal robots during jumping. Relying on a differe…
University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）；University of Michigan（密歇根大学）；Ohio State University（俄亥俄州立大学）；Purdue University（普渡大学）
Controller design for bipedal walking on dynamic rigid surfaces (DRSes), which are rigid surfaces moving in the inertial frame (e.g., ships and airplanes), remains largely uninvestigated. This paper introduces a hierarchical control approach that achieves stable underactuated bipedal robot walking on a horizontally oscillating DRS. The highest layer of our approach is a real-time motion planner that generates desired global behaviors (i.e., the center of mass trajectories and footstep locations) by stabilizing a reduced-order robot model. One key novelty of this layer is the derivation of the …
Time-Varying Foot-Placement Control for Underactuated Humanoid Walking on Swaying Rigid Surfaces
College of Engineering, University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校工程学院）；The Ohio State University（俄亥俄州立大学）；University of Michigan（密歇根大学）；School of Mechanical Engineering, Purdue University（普渡大学机械工程学院）
The ability to track a general walking path with specific timing is crucial to the operational safety and reliability of bipedal robots for avoiding dynamic obstacles, such as pedestrians, in complex environments. This paper introduces an online, full-body motion planner that generates the desired impact-aware motion for fully-actuated bipedal robotic walking. The main novelty of the proposed planner lies in its capability of producing desired motions in real-time that respect the discrete impact dynamics and the desired impact timing. To derive the proposed planner, a full-order hybrid dynami…
Bipedal Robust Walking on Uneven Footholds: Piecewise Slope LIPM with Discrete Model Predictive Control
Faculty of Computing, Harbin Institute of Technology（哈尔滨工业大学计算学部）；State Key Laboratory of Robotics and System, Harbin Institute of Technology（哈尔滨工业大学机器人技术与系统国家重点实验室）
Delft University of Technology（代尔夫特理工大学）；California Institute of Technology（加州理工学院）
Humans are able to negotiate downstep behaviors -- both planned and unplanned -- with remarkable agility and ease. The goal of this paper is to systematically study the translation of this human behavior to bipedal walking robots, even if the morphology is inherently different. Concretely, we begin with human data wherein planned and unplanned downsteps are taken. We analyze this data from the perspective of reduced-order modeling of the human, encoding the center of mass (CoM) kinematics and contact forces, which allows for the translation of these behaviors into the corresponding reduced-ord…
We present a theoretical analysis of a recent whole body motion planning method, the Randomized Possibility Graph, which uses a high-level decomposition of the feasibility constraint manifold in order to rapidly find routes that may lead to a solution. These routes are then examined by lower-level planners to determine feasibility. In this paper, we show that this approach is probabilistically complete for bipedal robots performing quasi-static walking in "semi-unstructured" environments. Furthermore, we show that the decomposition into higher and lower level planners allows for a considerably…
In this paper, we propose a novel framework on force-and-moment-based Model Predictive Control (MPC) for dynamic legged robots. Specifically, we present a formulation of MPC designed for 10 degree-of-freedom (DoF) bipedal robots using simplified rigid body dynamics with input forces and moments. This MPC controller will calculate the optimal inputs applied to the robot, including 3-D forces and 2-D moments at each foot. These desired inputs will then be generated by mapping these forces and moments to motor torques of 5 actuators on each leg. We evaluate our proposed control design on physical…
In this paper, previous works on the Model Predictive Control (MPC) and the Divergent Component of Motion (DCM) for bipedal walking control are extended. To this end, we employ a single MPC which uses a combination of Center of Pressure (CoP) manipulation, step adjustment, and Centroidal Moment Pivot (CMP) modulation to design a robust walking controller. Furthermore, we exploit the concept of time-varying DCM to generalize our walking controller for walking in uneven surfaces. Using our scheme, a general and robust walking controller is designed which can be implemented on robots with differe…
We present a new walking controller based on 3LP, a 3D model of bipedal walking that is composed of three pendulums to simulate falling, swing and torso dynamics. Taking advantage of linear equations and closed-form solutions of 3LP, the proposed controller projects intermediate states of the biped back to the beginning of the phase for which a discrete LQR controller is designed. After the projection, a proper control policy is generated by this LQR controller and used at the intermediate time. The projection controller reacts to disturbances immediately and compared to the discrete LQR contr…
A safety-critical measure of legged locomotion performance is a robot's ability to track its desired time-varying position trajectory in an environment, which is herein termed as "global-position tracking". This paper introduces a nonlinear control approach that achieves asymptotic global-position tracking for three-dimensional (3-D) bipedal robot walking. Designing a global-position tracking controller presents a challenging problem due to the complex hybrid robot model and the time-varying desired global-position trajectory. Towards tackling this problem, the first main contribution is the c…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Trento（特伦托大学）；New York University（纽约大学）
Linear Model Predictive Control (MPC) has been successfully used for generating feasible walking motions for humanoid robots. However, the effect of uncertainties on constraints satisfaction has only been studied using Robust MPC (RMPC) approaches, which account for the worst-case realization of bounded disturbances at each time instant. In this letter, we propose for the first time to use linear stochastic MPC (SMPC) to account for uncertainties in bipedal walking. We show that SMPC offers more flexibility to the user (or a high level decision maker) by tolerating small (user-defined) probabi…
We present a highly reactive controller which enables bipedal robots to blindly walk over various kinds of uneven terrains while resisting pushes. The high level motion planner does fast online optimization for footstep locations and Center of Mass (CoM) height using the decoupled actuated Spring Loaded Inverted Pendulum (aSLIP) model. The decoupled aSLIP model simplifies the original aSLIP with Linear Inverted Pendulum (LIP) dynamics in horizontal states and spring dynamics in the vertical state. The motion planning can be formulated as a discrete-time Model Predictive Control (MPC) and solve…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Amirkabir University of Technology（阿米尔卡比尔理工大学）；Tandon School of Engineering, New York University（纽约大学坦登工程学院）
In this paper, we present a novel two-level variable Horizon Model Predictive Control (VH-MPC) framework for bipedal locomotion. In this framework, the higher level computes the landing location and timing (horizon length) of the swing foot to stabilize the unstable part of the center of mass (CoM) dynamics, using feedback from the CoM states. The lower level takes into account the swing foot dynamics and generates dynamically consistent trajectories for landing at the desired time as close as possible to the desired location. To do that, we use a simplified model of the robot dynamics project…
California Institute of Technology（加州理工学院）；Technion - Israel Institute of Technology（以色列理工学院）
Dynamic bipedal robot locomotion has achieved remarkable success due in part to recent advances in trajectory generation and nonlinear control for stabilization. A key assumption utilized in both theory and experiments is that the robot's stance foot always makes no-slip contact with the ground, including at impacts. This assumption breaks down on slippery low-friction surfaces, as commonly encountered in outdoor terrains, leading to failure and loss of stability. In this work, we extend the theoretical analysis and trajectory optimization to account for stick-slip transitions at point foot co…
Korea Institute of Science and Technology（韩国科学技术研究院）；UNIVERSITY OF SCIENCE AND TECHNOLOGY（科学技术大学）
In this work, the hierarchical control strategy of template-based control for a bipedal robot is described. The axial force of a compliant leg is redirected to a point, called the virtual pivot point (VPP), of a 2D biped robot, which is located above the CoM of the model, to generate a restoring moment for the trunk motion. The resulting behavior of the model would resemble a virtual pendulum rotating around this VPP, thus aiming for an upright trunk during walking. Inspired by this analysis, we propose a new force redirecting method as a controller for robot walking. Then, these key features …
This paper presents a framework that leverages both control theory and machine learning to obtain stable and robust bipedal locomotion without the need for manual parameter tuning. Traditionally, gaits are generated through trajectory optimization methods and then realized experimentally -- a process that often requires extensive tuning due to differences between the models and hardware. In this work, the process of gait realization via hybrid zero dynamics (HZD) based optimization is formally combined with preference-based learning to systematically realize dynamically stable walking. Importa…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Technical University of Munich（慕尼黑工业大学）
Legged locomotion in humans is governed by natural dynamics of the human body and neural control. One mechanism that is assumed to contribute to the high efficiency of human walking is the impulsive ankle push-off, which potentially powers the swing leg catapult. However, the mechanics of the human lower leg with its complex muscle-tendon units spanning over single and multiple joints is not yet understood. Legged robots allow testing the interaction between complex leg mechanics, control, and environment in real-world walking gait. We developed a 0.49m tall, 2.2kg anthropomorphic bipedal robo…
Ohio State University（俄亥俄州立大学）；National Research Council of Canada（加拿大国家研究委员会）；University of Waterloo（滑铁卢大学）；Southern University of Science and Technology（南方科技大学）
This paper presents a neural-network based adaptive feedback control structure to regulate the velocity of 3D bipedal robots under dynamics uncertainties. Existing Hybrid Zero Dynamics (HZD)-based controllers regulate velocity through the implementation of heuristic regulators that do not consider model and environmental uncertainties, which may significantly affect the tracking performance of the controllers. In this paper, we address the uncertainties in the robot dynamics from the perspective of the reduced dimensional representation of virtual constraints and propose the integration of an …
We present a trajectory planning and control architecture for bipedal locomotion at a variety of speeds on a highly underactuated and compliant bipedal robot. A library of compliant walking trajectories are planned offline, and stored as compact arrays of polynomial coefficients for tracking online. The control implementation uses a floating-base inverse dynamics controller which generates dynamically consistent feedforward torques to realize walking using information obtained from the trajectory optimization. The effectiveness of the controller is demonstrated in simulation and on hardware fo…
The Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Google X（谷歌X）；The University of Southern California（南加利福尼亚大学）；New York University（纽约大学）
Cost functions have the potential to provide compact and understandable generalizations of motion. The goal of Inverse Optimal Control (IOC) is to analyze an observed behavior which is assumed to be optimal with respect to an unknown cost function, and infer this cost function. Here we develop a method for characterizing cost functions of legged locomotion, with the goal of representing complex humanoid behavior with simple models. To test this methodology we simulate walking gaits of a simple 5 link planar walking model which optimize known cost functions, and assess the ability of our IOC me…
Central pattern generators (CPGs), with a basis is neurophysiological studies, are a type of neural network for the generation of rhythmic motion. While CPGs are being increasingly used in robot control, most applications are hand-tuned for a specific task and it is acknowledged in the field that generic methods and design principles for creating individual networks for a given task are lacking. This study presents an approach where the connectivity and oscillatory parameters of a CPG network are determined by an evolutionary algorithm with fitness evaluations in a realistic simulation with ac…
A Hybrid passive Linear Inverted Pendulum (HLIP) model is proposed for characterizing, stabilizing and composing periodic orbits for 3D underactuated bipedal walking. Specifically, Period-1 (P1) and Period-2 (P2) orbits are geometrically characterized in the state space of the H-LIP. Stepping controllers are designed for global stabilization of the orbits. Valid ranges of the gains and their optimality are derived. The optimal stepping controller is used to create and stabilize the walking of bipedal robots. An actuated Spring-loaded Inverted Pendulum (aSLIP) model and the underactuated robot …
ETH Zürich（苏黎世联邦理工学院）；California Institute of Technology（加州理工学院）
The ability to generate dynamic walking in real-time for bipedal robots with input constraints and underactuation has the potential to enable locomotion in dynamic, complex and unstructured environments. Yet, the high-dimensional nature of bipedal robots has limited the use of full-order rigid body dynamics to gaits which are synthesized offline and then tracked online. In this work we develop an online nonlinear model predictive control approach that leverages the full-order dynamics to realize diverse walking behaviors. Additionally, this approach can be coupled with gaits synthesized offlin…
University of California Berkeley（加州大学伯克利分校）；Carnegie Mellon University（卡内基梅隆大学）
Recent advances in legged locomotion have enabled quadrupeds to walk on challenging terrains. However, bipedal robots are inherently more unstable and hence it's harder to design walking controllers for them. In this work, we leverage recent advances in rapid adaptation for locomotion control, and extend them to work on bipedal robots. Similar to existing works, we start with a base policy which produces actions while taking as input an estimated extrinsics vector from an adaptation module. This extrinsics vector contains information about the environment and enables the walking controller to …
This paper presents the design and validation of controlling hopping on the 3D bipedal robot Cassie. A spring-mass model is identified from the kinematics and compliance of the robot. The spring stiffness and damping are encapsulated by the leg length, thus actuating the leg length can create and control hopping behaviors. Trajectory optimization via direct collocation is performed on the spring-mass model to plan jumping and landing motions. The leg length trajectories are utilized as desired outputs to synthesize a control Lyapunov function based quadratic program (CLF-QP). Centroidal angula…
Nonlinear Model Predictive Control for Robust Bipedal Locomotion: Exploring Angular Momentum and CoM Height Changes
School of Power and Mechanical Engineering, Wuhan University（武汉大学动力与机械学院）；Shenzhen Institute of Artificial Intelligence and Robotics for Society（深圳人工智能与机器人研究院）；School of Mechanical Engineering, University of Leeds（利兹大学机械工程学院）；School of Informatics, The University of Edinburgh（爱丁堡大学信息学院）；Humanoid and Human Centered Mechatronics Research Line, Istituto Italiano di Tecnologia（意大利技术研究院人形与以人为中心的机电一体化研究线）
Task-Space Riccati Feedback based Whole Body Control for Underactuated Legged Locomotion
School of System Design and Intelligent Manufacturing, Southern University of Science and Technology（南方科技大学系统设计与智能制造学院）；Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology（香港科技大学土木与环境工程系）；Department of Aerospace and Mechanical Engineering, University of Notre Dame（圣母大学航空航天与机械工程系）
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Laban/Bartenieff Institute of Movement Studies（拉班/芭特妮芙动作研究学院）
Humans are efficient, yet expressive in their motion. Human walking behaviors can be used to walk across a great variety of surfaces without falling and to communicate internal state to other humans through variable gait styles. This provides inspiration for creating similarly expressive bipedal robots. To this end, a framework is presented for stylistic gait generation in a compass-like under-actuated planar biped model. The gait design is done using model-based trajectory optimization with variable constraints. For a finite range of optimization parameters, a large set of 360 gaits can be ge…
Walking motion planning based on Divergent Component of Motion (DCM) and Linear Inverted Pendulum Model (LIPM) is one of the alternatives that could be implemented to generate online humanoid robot gait trajectories. This algorithm requires different parameters to be adjusted. Herein, we developed a framework to attain optimal parameters to achieve a stable and energy-efficient trajectory for real robot's gait. To find the optimal trajectory, four cost functions representing energy consumption, the sum of joints velocity and applied torque at each lower limb joint of the robot, and a cost func…
The ability to realize nonlinear controllers with formal guarantees on dynamic robotic systems has the potential to enable more complex robotic behaviors -- yet, realizing these controllers is often practically challenging. To address this challenge, this paper presents the end-to-end realization of dynamic bipedal locomotion on an underactuated bipedal robot via hybrid zero dynamics and control Lyapunov functions. A compliant model of Cassie is represented as a hybrid system to set the stage for a trajectory optimization framework. With the goal of achieving a variety of walking speeds in all…
Locomotion of legged machines faces the problems of model complexity and computational costs. Algorithms based on complex models and/or reinforcement learning exist to solve the walking control task. In this project, we aim to develop a bipedal walking control system based on a simple model the Linear Inverted Pendulum model. In order to simplify the complex process of controlling legged locomotion, we make use of the technique of splitting the control into three parts as height control, forward velocity control and balance control. The forward velocity of the body has a linear relationship wi…
This paper presents an online walking synthesis methodology to enable dynamic and stable walking on constrained footholds for underactuated bipedal robots. Our approach modulates the change of angular momentum about the foot-ground contact pivot at discrete impact using pre-impact vertical center of mass (COM) velocity. To this end, we utilize the underactuated Linear Inverted Pendulum (LIP) model for approximating the underactuated walking dynamics to provide the desired post-impact angular momentum for each step. Desired outputs are constructed via online optimization combined with closed-fo…
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）
K. N. Toosi University of Technology（K. N. 图西理工大学）；Max-Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；New York University（纽约大学）
In this paper, we extend state of the art Model Predictive Control (MPC) approaches to generate safe bipedal walking on slippery surfaces. In this setting, we formulate walking as a trade off between realizing a desired walking velocity and preserving robust foot-ground contact. Exploiting this formulation inside MPC, we show that safe walking on various flat terrains can be achieved by compromising three main attributes, i. e. walking velocity tracking, the Zero Moment Point (ZMP) modulation, and the Required Coefficient of Friction (RCoF) regulation. Simulation results show that increasing t…
Hybrid systems, such as bipedal walkers, are challenging to control because of discontinuities in their nonlinear dynamics. Little can be predicted about the systems' evolution without modeling the guard conditions that govern transitions between hybrid modes, so even systems with reliable state sensing can be difficult to control. We propose an algorithm that allows for determining the hybrid mode of a system in real-time using data-driven analysis. The algorithm is used with data-driven dynamics identification to enable model predictive control based entirely on data. Two examples---a simula…
