Asymptotically Stable Walking of a Five-Link Underactuated 3D Bipedal Robot
CNRS（法国国家科学研究中心）；IRCCyN（南特中央理工学院机器人研究中心）；Nantes Atlantic University（南特大西洋大学）；University of Michigan（密歇根大学）；National Taiwan University of Science and Technology（台湾科技大学）
Christine Chevallereau, Jessy W. Grizzle, Ching-Long Shih
This paper presents three feedback controllers that achieve an asymptotically stable, periodic, and fast walking gait for a 3D (spatial) bipedal robot consisting of a torso, two legs, and passive (unactuated) point feet. The contact between the robot and the walking surface is assumed to inhibit yaw rotation. The studied robot has 8 DOF in the single support phase and 6 actuators. The interest of studying robots with point feet is that the robot's natural dynamics must be explicitly taken into account to achieve balance while walking. We use an extension of the method of virtual constraints and hybrid zero dynamics, in order to simultaneously compute a periodic orbit and an autonomous feedback controller that realizes the orbit. This method allows the computations to be carried out on a 2-DOF subsystem of the 8-DOF robot model. The stability of the walking gait under closed-loop control is evaluated with the linearization of the restricted Poincaré map of the hybrid zero dynamics. Three strategies are explored. The first strategy consists of imposing a stability condition during the search of a periodic gait by optimization. The second strategy uses an event-based controller. In the third approach, the effect of output selection is discussed and a pertinent choice of outputs is proposed, leading to stabilization without the use of a supplemental event-based controller.
In this paper, we holistically present a Hybrid-Linear Inverted Pendulum (H-LIP) based approach for synthesizing and stabilizing 3D foot-underactuated bipedal walking, with an emphasis on thorough hardware realization. The H-LIP is proposed to capture the essential components of the underactuated and actuated part of the robotic walking. The robot walking gait is then directly synthesized based on the H-LIP. We comprehensively characterize the periodic orbits of the H-LIP and provably derive the stepping stabilization via its step-to-step (S2S) dynamics, which is then utilized to approximate t…
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 …
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…
Dynamic walking on bipedal robots has evolved from an idea in science fiction to a practical reality. This is due to continued progress in three key areas: a mathematical understanding of locomotion, the computational ability to encode this mathematics through optimization, and the hardware capable of realizing this understanding in practice. In this context, this review article outlines the end-to-end process of methods which have proven effective in the literature for achieving dynamic walking on bipedal robots. We begin by introducing mathematical models of locomotion, from reduced order mo…
We present a stepping stabilization control that addresses external push disturbances on bipedal walking robots. The stepping control is synthesized based on the step-to-step (S2S) dynamics of the robot that is controlled to have an approximately constant center of mass (COM) height. We first learn a linear S2S dynamics with bounded model discrepancy from the undisturbed walking behaviors of the robot, where the walking step size is taken as the control input to the S2S dynamics. External pushes are then considered as disturbances to the learned S2S (L-S2S) dynamics. We then apply the system-l…
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…
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…
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 …
Florida Institute for Human and Machine Cognition（佛罗里达人机认知研究所）；University of West Florida（西佛罗里达大学）
In trying to build humanoid robots that perform useful tasks in a world built for humans, we address the problem of autonomous locomotion. Humanoid robot planning and control algorithms for walking over rough terrain are becoming increasingly capable. At the same time, commercially available depth cameras have been getting more accurate and GPU computing has become a primary tool in AI research. In this paper, we present a newly constructed behavior control system for achieving fast, autonomous, bipedal walking, without pauses or deliberation. We achieve this using a recently published rapid p…
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…
Bipedal-Walking-Dynamics Model on Granular Terrains
Department of Mechanical and Aerospace Engineering, Rutgers University（罗格斯大学机械与航空航天工程系）；Bridgewater Raritan High School（布里奇沃特-拉里坦高中）；State Key Lab of Fluid Power Transmission and Control and the School of Mechanical Engineering, Zhejiang University（浙江大学流体动力传动与控制国家重点实验室和机械工程学院）
Real-world applications of bipedal robot walking require accurate, real-time state estimation. State estimation for locomotion over dynamic rigid surfaces (DRS), such as elevators, ships, public transport vehicles, and aircraft, remains under-explored, although state estimator designs for stationary rigid surfaces have been extensively studied. Addressing DRS locomotion in state estimation is a challenging problem mainly due to the nonlinear, hybrid nature of walking dynamics, the nonstationary surface-foot contact points, and hardware imperfections (e.g., limited availability, noise, and drif…
Not until recently, robust robot locomotion has been achieved by deep reinforcement learning (DRL). However, for efficient learning of parametrized bipedal walking, developed references are usually required, limiting the performance to that of the references. In this paper, we propose to design an adaptive reward function for imitation learning from the references. The agent is encouraged to mimic the references when its performance is low, while to pursue high performance when it reaches the limit of references. We further demonstrate that developed references can be replaced by low-quality r…
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）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
In this paper, we propose a novel framework capable of generating various walking and running gaits for bipedal robots. The main goal is to relax the fixed center of mass (CoM) height assumption of the linear inverted pendulum model (LIPM) and generate a wider range of walking and running motions, without a considerable increase in complexity. To do so, we use the concept of virtual constraints in the centroidal space which enables generating motions beyond walking while keeping the complexity at a minimum. By a proper choice of these virtual constraints, we show that we can generate different…
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…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
For humanoids to be deployed in demanding situations, such as search and rescue, highly intelligent decision making and proficient sensorimotor skill is expected. A promising solution is to leverage human prowess by interconnecting robot and human via teleoperation. Towards creating seamless operation, this paper presents a dynamic telelocomotion framework that synchronizes the gait of a human pilot with the walking of a bipedal robot. First, we introduce a method to generate a virtual human walking model from the stepping behavior of a human pilot which serves as a reference for the robot to …
The control of bipedal robotic walking remains a challenging problem in the domains of computation and experiment, due to the multi-body dynamics and various sources of uncertainty. In recent years, there has been a rising trend towards model reduction and the design of intuitive controllers to overcome the gap between assumed model and reality. Despite its viability in practical implementation, this local representation of true dynamics naturally indicate limited scalibility towards more dynamical behaviors. With the goal of moving towards increasingly dynamic behaviors, we leverage the detai…
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…
Bipedal robots are a prime example of systems which exhibit highly nonlinear dynamics, underactuation, and undergo complex dissipative impacts. This paper discusses methods used to overcome a wide variety of uncertainties, with the end result being stable bipedal walking. The principal contribution of this paper is to establish sufficiency conditions for yielding input to state stable (ISS) hybrid periodic orbits, i.e., stable walking gaits under model-based and phase-based uncertainties. In particular, it will be shown formally that exponential input to state stabilization (e-ISS) of the cont…
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…
This work presents Neural Gaits, a method for learning dynamic walking gaits through the enforcement of set invariance that can be refined episodically using experimental data from the robot. We frame walking as a set invariance problem enforceable via control barrier functions (CBFs) defined on the reduced-order dynamics quantifying the underactuated component of the robot: the zero dynamics. Our approach contains two learning modules: one for learning a policy that satisfies the CBF condition, and another for learning a residual dynamics model to refine imperfections of the nominal model. Im…
Indian Institute of Technology (BHU) Varanasi（瓦拉纳西印度理工学院（BHU））；Ohio State University（俄亥俄州立大学）；Indian Institute of Science（印度科学学院）
In this work, we demonstrate robust walking in the bipedal robot Digit on uneven terrains by just learning a single linear policy. In particular, we propose a new control pipeline, wherein the high-level trajectory modulator shapes the end-foot ellipsoidal trajectories, and the low-level gait controller regulates the torso and ankle orientation. The foot-trajectory modulator uses a linear policy and the regulator uses a linear PD control law. As opposed to neural network-based policies, the proposed linear policy has only 13 learnable parameters, thereby not only guaranteeing sample efficient …
Laboratoire des Sciences du Numérique de Nantes (LS2N)（南特数字科学实验室）；CNRS（法国国家科学研究中心）；Ecole Centrale de Nantes（南特中央理工学院）；Université de Nantes（南特大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；University of Michigan（密歇根大学）
This paper seeks insight into stabilization mechanisms for periodic walking gaits in 3D bipedal robots. Based on this insight, a control strategy based on virtual constraints, which imposes coordination between joints rather than a temporal evolution, will be proposed for achieving asymptotic convergence toward a periodic motion. For planar bipeds with one degree of underactuation, it is known that a vertical displacement of the center of mass---with downward velocity at the step transition---induces stability of a walking gait. This paper concerns the qualitative extension of this type of pro…
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…
Collaborative Robotics and Intelligent Systems Institute, Oregon State University（俄勒冈州立大学协作机器人与智能系统研究所）
In this work, we propose a learning approach for 3D dynamic bipedal walking when footsteps are constrained to stepping stones. While recent work has shown progress on this problem, real-world demonstrations have been limited to relatively simple open-loop, perception-free scenarios. Our main contribution is a more advanced learning approach that enables real-world demonstrations, using the Cassie robot, of closed-loop dynamic walking over moderately difficult stepping-stone patterns. Our approach first uses reinforcement learning (RL) in simulation to train a controller that maps footstep comm…
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…
Many methods exist for a bipedal robot to keep its balance while walking. In addition to step size and timing, other strategies are possible that influence the stability of the robot without interfering with the target direction and speed of locomotion. This paper introduces a multifaceted feedback controller that uses numerous different feedback mechanisms, collectively termed corrective actions, to stabilise a core keypoint-based gait. The feedback controller is experimentally effective, yet free of any physical model of the robot, very computationally inexpensive, and requires only a single…
University of Tehran（德黑兰大学）；K. N. Toosi University of Technology（图西理工大学）
Available possibilities to prevent a biped robot from falling down in the presence of severe disturbances are mainly Center of Pressure (CoP) modulation, step location and timing adjustment, and angular momentum regulation. In this paper, we aim at designing a walking pattern generator which employs an optimal combination of these tools to generate robust gaits. In this approach, first, the next step location and timing are decided consistent with the commanded walking velocity and based on the Divergent Component of Motion (DCM) measurement. This stage which is done by a very small-size Quadr…
Robust and energy-efficient bipedal locomotion in robotics is still a challenging topic. In order to address issues in this field, we can take inspiration from nature, by studying human locomotion. The Spring-Loaded Inverted Pendulum (SLIP) model has shown to be a good model for this purpose. However, the human musculoskeletal system enables us to actively modulate leg stiffness, for example when walking in rough terrain with irregular and unexpected height variations of the walking surface. This ability of varying leg stiffness is not considered in conventional SLIP-based models, and therefor…
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Technische Universität Ilmenau（伊尔默瑙工业大学）
The hybrid zero dynamics control concept for bipedal walking is extended to include a non-instantaneous double support phase. A symmetric robot that consists of five rigid body segments which are connected by four actuated revolute joints is considered. Periodic walking gaits with a constant average walking speed consists of alternating single (SSP) and double support phases (DSP). Hybrid zero dynamics control designs usually assume an instantaneous DSP, which is a severe limitation. The proposed controllers use continuous SSPs and DSPs. Transitions between both phases are modeled as instantan…
The ability to generate robust walking gaits on bipedal robots is key to their successful realization on hardware. To this end, this work extends the method of Hybrid Zero Dynamics (HZD) -- which traditionally only accounts for locomotive stability via periodicity constraints under perfect impact events -- through the inclusion of the saltation matrix with a view toward synthesizing robust walking gaits. By jointly minimizing the norm of the extended saltation matrix and the torque of the robot directly in the gait generation process, we demonstrate that the synthesized gaits are more robust t…
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 …
An omnidirectional closed-loop gait based on the direct feedback of orientation deviation estimates is presented in this paper. At the core of the gait is an open-loop central pattern generator. The orientation feedback is derived from a 3D nonlinear attitude estimator, and split into the relevant angular deviations in the sagittal and lateral planes using the concept of fused angles. These angular deviations from expected are used by a number of independent feedback mechanisms, including one that controls timing, to perform stabilising corrective actions. The tuning of the feedback mechanisms…
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…
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…
Collaborative Robotics and Intelligent Systems Institute, Oregon State University（俄勒冈州立大学协作机器人与智能系统研究所）；Agility Robotics（Agility Robotics（敏捷机器人公司））
Recently, work on reinforcement learning (RL) for bipedal robots has successfully learned controllers for a variety of dynamic gaits with robust sim-to-real demonstrations. In order to maintain balance, the learned controllers have full freedom of where to place the feet, resulting in highly robust gaits. In the real world however, the environment will often impose constraints on the feasible footstep locations, typically identified by perception systems. Unfortunately, most demonstrated RL controllers on bipedal robots do not allow for specifying and responding to such constraints. This missi…
This work presents algorithms for the feedback-stabilised walking of bipedal humanoid robotic platforms, along with the underlying theoretical and sensorimotor frameworks required to achieve it. Bipedal walking is inherently complex and difficult to control due to the high level of nonlinearity and significant number of degrees of freedom of the concerned robots, the limited observability and controllability of the corresponding states, and the combination of imperfect actuation with less-than-ideal sensing. The presented methods deal with these issues in a multitude of ways, ranging from the …
Machine learning algorithms have found several applications in the field of robotics and control systems. The control systems community has started to show interest towards several machine learning algorithms from the sub-domains such as supervised learning, imitation learning and reinforcement learning to achieve autonomous control and intelligent decision making. Amongst many complex control problems, stable bipedal walking has been the most challenging problem. In this paper, we present an architecture to design and simulate a planar bipedal walking robot(BWR) using a realistic robotics sim…
