Underactuated Biped Robot Research Articles

Autonomous navigation of underactuated bipedal robots in height-constrained environments

Navigating a large-scaled robot in unknown and cluttered height-constrained environments is challenging. Not only is a fast and reliable planning algorithm required to go around obstacles, the robot should also be able to change its intrinsic dimension by crouching in order to travel underneath height-constrained regions. There are few mobile robots that are capable of handling such a challenge, and bipedal robots provide a solution. However, as bipedal robots have nonlinear and hybrid dynamics, trajectory planning while ensuring dynamic feasibility and safety on these robots is challenging. This paper presents an end-to-end autonomous navigation framework which leverages three layers of planners and a variable walking height controller to enable bipedal robots to safely explore height-constrained environments. A vertically actuated spring-loaded inverted pendulum (vSLIP) model is introduced to capture the robot’s coupled dynamics of planar walking and vertical walking height. This reduced-order model is utilized to optimize for long-term and short-term safe trajectory plans. A variable walking height controller is leveraged to enable the bipedal robot to maintain stable periodic walking gaits while following the planned trajectory. The entire framework is tested and experimentally validated using a bipedal robot Cassie. This demonstrates reliable autonomy to drive the robot to safely avoid obstacles while walking to the goal location in various kinds of height-constrained cluttered environments.

Multimodal bipedal locomotion generation with passive dynamics via deep reinforcement learning.

Generating multimodal locomotion in underactuated bipedal robots requires control solutions that can facilitate motion patterns for drastically different dynamical modes, which is an extremely challenging problem in locomotion-learning tasks. Also, in such multimodal locomotion, utilizing body morphology is important because it leads to energy-efficient locomotion. This study provides a framework that reproduces multimodal bipedal locomotion using passive dynamics through deep reinforcement learning (DRL). An underactuated bipedal model was developed based on a passive walker, and a controller was designed using DRL. By carefully planning the weight parameter settings of the DRL reward function during the learning process based on a curriculum learning method, the bipedal model successfully learned to walk, run, and perform gait transitions by adjusting only one command input. These results indicate that DRL can be applied to generate various gaits with the effective use of passive dynamics.

Velocity-Based Gait Planning for Underactuated Bipedal Robot on Uneven and Compliant Terrain

This article develops a gait planning method for underactuated bipedal robot on uneven and compliant terrain. First, we employ a linear spring-damper model to describe foot-ground compliant contact, and establish a decoupled robot–ground three-dimensional dynamic mode. Second, based on a velocity-based bipedal stability definition and human walking characteristics, we propose a gait planning method to realize underactuated bipedal walking on uneven and compliant terrain. We decouple bipedal gait planning into sagittal and lateral master-slave ones. By planning the motion state of Center-of-Mass (CoM) of a robot, we make the movement of lateral and sagittal coincident such that bipedal walking is realized. Finally, underactuated bipedal walking with an average walking speed of 0.216 m/s and a step length of 183.9 mm is realized on uneven terrain where the maximum height of unevenness is 32 mm. The experimental results show that underactuated bipedal walking can be realized on uneven and compliant terrain by using the proposed method to control robot CoM and track its desired velocity.

3-D Underactuated Bipedal Walking via H-LIP Based Gait Synthesis and Stepping Stabilization

In this article, we holistically present a hybrid-linear inverted pendulum (H-LIP) based approach for synthesizing and stabilizing 3-D 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">directly</i> 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 the S2S dynamics of the horizontal state of the center of mass of the robotic walking. The approximation facilities a H-LIP based stepping controller to provide desired step sizes to stabilize the robotic walking. By realizing the desired step sizes, the robot achieves dynamic and stable walking. The approach is fully evaluated in both simulation and experiment on the 3-D underactuated bipedal robot Cassie, which demonstrates dynamic walking behaviors with both high versatility and robustness.

Modeling, Analysis and Control of the Compass Gait Biped Robot and Extensions: A Review

Abstract This paper gives a review of recent results in the field of modeling and control of hybrid walking systems with a particular focus on the Compass Gait model. Since dynamic models of biped robots are hybrid in nature, the paper presents an overview of modeling the continuous and discrete dynamics of a hybrid system representing a walking robot. The paper briefly describes the design evolution in underactuated bipedal robots and presents several prominent prototypes, explains a number of related concepts and lists a variety of suitable control techniques.

Dynamic Walking Analysis of an Underactuated Biped Robot with Asymmetric Structure

Owing to their nonlinear structures and dynamics, bipedal walking robots are commonly used as appropriate case studies for nonlinear modeling and control. In this study, the dynamics of a point-feet 4-link biped robot having asymmetric structure is studied. This asymmetry appears on the robot’s legs such that one leg of the robot does have an active knee while the other is knee-less. In this way, the style and analysis of each step depends on which leg is the stance leg. Although the stable steady state behavior of the system is purely periodic, the gait cycle does consist of two sequential steps. Since each step includes a continuous phase followed by an impact phase, hence, we need to model the system as a multiphase (4-phase) hybrid system. The main purpose of this research is to find stable gating pattern and employ appropriate controller to make sure that the gating is accomplished in an asymptotically stable manner. A combination of feedback linearization and finite-time controllers is used to control the walking posture, and the stability of the whole behavior is investigated by analysis of a one-dimensional Poincaré map. Simulation results successfully support the modeling and control approach.

Strict stealth walking gait generation for 3-link underactuated biped robots

In this paper, we propose a method for generating a stable stealth gait on frictionless road surface without including the double-limb support phase for biped robots. First, we introduce a model of a biped robot with an upper body, and derive the conditions to maintain the horizontal ground reaction force at zero. Second, we derive an approximate analytical solution for the target initial state using the linearized equation of motion. Third, to generate a stealth walking gait of the nonlinear model, we propose a numerical method to identify the target initial state based on the result of the linearized model, and evaluate the mathematical results obtained through numerical simulations. Furthermore, we discuss the effect of semicircular feet on the improvement of energy efficiency.

Feedforward control for underactuated bipedal walking on compliant continuous steps with varying height

This study develops a feedforward control strategy based on the motion state of center-of-mass (CoM) of a robot for underactuated biped robot stable walking on compliant continuous steps with a known varying height. First, considering ground deformation, a compliant contact model is employed to characterize foot-ground interaction, and a robot–step coupling dynamic model of sagittal and lateral planes are established through decoupling modelling. Second, based on the gait characteristics of human variable-step walking, a feedforward control strategy based on the motion state of CoM is proposed. Varying height step is equivalent to varying slope, an equivalent slope angle and a desired step length can be calculated for each step according to their height. Underactuated bipedal walking control is decoupled into sagittal and lateral master-slave control. The velocity of robot CoM is considered as a system output. It is controlled through the displacement of CoM in a single walking cycle, and thus walking is stabilized. By the proposed method, the walking system is modelled as a polynomial with definite number of degrees and the controlled input is derived through a simple inverse operation on it. Its effectiveness is validated through simulations in an environment with a step varying height of less than 0.032 m. Simulation results show that the proposed method can improve the tracking performance of robot CoM velocity on varying height steps, as compared to a hybrid zero dynamic (HZD)-based controller.

Control of ATRIAS in three dimensions: Walking as a forced-oscillation problem

In this article, we present a new controller for stable and robust walking control of ATRIAS, an underactuated bipedal robot designed based on the spring-loaded inverted pendulum (SLIP) model. We propose a forced-oscillation scheme for control of vertical motion, which we prove to be stable and contractive. Moreover, we prove that, through some mild assumptions, the dynamics of the system can be written in a hierarchical form that decouples the stability analyses of the horizontal and vertical directions. We leverage these properties to find a stabilizing class of functions for foot placement. The torso control is also proved to be decoupled using singular perturbation theory and is stabilized through a feedback linearization controller. We also take advantage of the proposed framework’s flexibility and extend it to include a new reflex-based uneven-terrain walking control scheme. We test the controller for various desired walking speeds (0 to 2.5 m/s), for stepping up and down unexpected obstacles (15 cm), and for high-speed walking on a random uneven terrain (up to 10 cm of step-ups and step-downs and up to 1.8 m/s). The results show successful performance of the controller and its stability and robustness against various perturbations.

Impulsive control and stability analysis of biped robot based on virtual constraint and adaptive optimization

This article is concerned with the problem of asymptotically stable periodic walking for a five‐link underactuated biped robot. Because of instantaneous change of two legs and complex dynamics during the walking process, the robot can be regarded as a nonlinear impulsive system. In order to make impulsive control of the robotic system, it is modeled as a rigid kinematic chain with Lagrange equations, which is strong coupling and hybrid. A novel state feedback controller is proposed to generate a closed‐loop periodic walking gait and track the desired trajectory of each joint. This controller is designed based on virtual constraint and hybrid zero dynamics. When the holonomic constraint of each actuator is zeroed by the controller, it can be regarded as an output. By using this method, the computation of the standard 5 degree of freedom (DOF) robot model can be reduced to 1 DOF. In addition, an adaptive optimization scheme is further investigated, which solves the problem of model parameter uncertainty. Finally, simulations are presented to illustrate the effectiveness of the proposed method.

Finite-time stabilization of periodic orbits for under-actuated biped walking with hybrid zero dynamics

This paper is concerned with the problem of finite-time stable walking for a 5-link under-actuated biped robot. Due to instantaneous change of 2 legs and complex dynamics during the walking process, the robot can be regarded as a nonlinear impulsive system. In order to make impulsive control of the robotic system, it is modeled as a rigid kinematic chain with Lagrange equations, which is strong coupling and hybrid. A novel finite-time feedback controller is proposed to realize finite-time stability of the nonlinear impulsive system. The controller is designed based on finite-time stabilizing control Lyapunov function (FTS-CLF) and hybrid zero dynamics (HZD). By establishing a finite-time stable periodic orbit, the controller can make the output of virtual constraints converge to zero rapidly. Restricted Poincare return map is then utilized to analyze the finite-time stability of nonlinear impulsive system. It ensures that the flow of the continuous subsystem can pass through the impact cross section. Additionally, a periodic walking gait planning is further investigated, and the existence of the gait is proved, which satisfy the joint trajectory tracking. Finally, the effectiveness of the mentioned method is illustrated by simulations.

Control strategy for gait transition of an underactuated 3D bipedal robot

Significant research interest has recently been attracted to the study of bipedal robots due to the wide variety of their potential applications. In reality, bipedal robots are often required to perform gait transitions to achieve flexible walking. In this paper, we consider the gait transition of a five-link underactuated three-dimensional (3D) bipedal robot, and propose a two-layer control strategy. The strategy consists of a unique, event-based, feedback controller whose feedback gain in each step is updated by an adaptive control law, and a transition controller that guides the robot from the current gait to a neighboring point of the target gait so that the state trajectory can smoothly converge to the target gait. Compared with previous works, the transition controller is parameterized and its control parameters are obtained by solving an optimization problem to guarantee the physical constraints in the transition process. Finally, the effectiveness of the control strategy is illustrated on the underactuated 3D bipedal robot.

Exponentially stable periodic walking of under-actuated biped robot

The biped robots have characteristics of strong environmental adaptability, complex structure and difficult motion control. This paper addresses the problem of exponential stable walking of a five-link under-actuated biped robot (RABBIT) through control Lyapunov function and hybrid zero dynamics. The walking system is input-output linearized after the full-order dynamic model is described with Lagrange dynamic equations. Based on hybrid zero dynamics, a state feedback controller is then utilized to realize exponential stability of periodic orbit by designing control Lyapunov function, which is proved to be exponentially stable by restricted Poincare Return Map. The effectiveness of the mentioned method is illustrated by simulations.

Hybrid Nonlinear Disturbance Observer Design for Underactuated Bipedal Robots.

Existence of disturbances in unknown environments is a pervasive challenge in robotic locomotion control. Disturbance observers are a class of unknown input observers that have been extensively used for disturbance rejection in numerous robotics applications. In this paper, we extend a class of widely-used nonlinear disturbance observers to underactuated bipedal robots, which are controlled using hybrid zero dynamics-based control schemes. The proposed hybrid nonlinear disturbance observer provides the autonomous biped robot control system with disturbance rejection capabilities, while the underlying hybrid zero-dynamics based control law remains intact.

Efficiency analysis of telescopic-legged bipedal robots

This paper investigates the stability of underactuated bipedal walking incorporating telescopic-leg actuation. In human walking, knee joints of swing and support legs are bent and stretched. The telescopic legs mimic the motion of the center of mass of human legs via their telescopic motion during the stance phase. First, underactuated telescopic-legged biped robot models are introduced. Second, an output-following control law is applied to the linearized equation of motion of the robot, and the controlled robot’s equation is then specified as a linear time-varying system. The error transition equation is developed to evaluate the stability during the stance phase. Numerical calculations are performed to show the influences of leg telescopic motion on the stability.

Decentralized control of rhythmic activities in fully-actuated/under-actuated robots

Rhythmic activities such as swimming stroke in the human body are learnable through conscious trainings. Inspiringly, the main objective of this study is to develop a control framework to reproduce the described functionality in the imitating robots. To do so, a two layer supervisory controller is proposed. The high-level controller, which acts as the conscious controller during trainings, is a supervisory dynamic-based controller and uses all system sensory data to generate stable rhythmic movements. On the other hand, the low-level controller in this structure is a distributed trajectory-based controller network. Each node in this network is an oscillatory dynamical system which has the ability to learn and reproduce the desired trajectory. Also, each node has a critic agent which evaluates the control eligibility of the low-level controllers for controlling the system. Then, based on the evaluation, these agents decide to assign the control of the system to the high-level controller or the low-level controllers. By using this structure, the system controller will act as simple and computing efficient as trajectory-based controllers and will perform as stably and robustly as dynamic-based controllers. At last, the applicability of this framework is demonstrated on a fully actuated robot and on an under-actuated biped robot.

Discrete sliding mode control to stabilize running of a biped robot with compliant kneed legs

This paper investigates performance of two event-based controllers applied to an underactuated biped robot to stabilize its running gait in presence of uncertainties. Mechanism of the biped robot includes four links leg, one point mass at the hip, point feet, and three motors parallel to rotational springs. So it has one degree of underactuation during stance phase and three degrees of underactuation during flight phase. A discrete sliding mode controller (DSMC) in comparison with a discrete linear-quadratic regulator (DLQR) is examined in order to stabilize the fixed point of the corresponding Poincare map. Using numerical simulations, it is concluded that DSMC has a better performance regarding basin of attraction and convergence speed compared to DLQR, especially in presence of disturbances.

Removing Phase Variables from Biped Robot Parametric Gaits.

Hybrid zero dynamics-based control is a promising framework for controlling underactuated biped robots and powered prosthetic legs. In this control paradigm, stable walking gaits are implicitly encoded in polynomial output functions of the robot configuration variables, which are to be zeroed via feedback. The biped output functions are parameterized by a suitable mechanical phasing variable whose evolution determines the biped gait progression during each step. Determining a proper phase variable, however, might not always be a trivial task. In this paper, we present a method for generating output functions from given parametric walking gaits without any explicit knowledge of the phase variables. Our elimination method is based on computing the resultant of polynomials, an algebraic tool widely used in computer algebra.

A brief review of dynamics and control of underactuated biped robots

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Orbital stabilization of an underactuated bipedal gait via nonlinear $${{\mathcal H}}_{\infty }$$ H ∞ -control using measurement feedback

The primary concern of the work is robust control of hybrid mechanical systems under unilateral constraints with underactuation degree one. Nonlinear \({{\mathcal H}}_{\infty }\) output feedback synthesis is developed in the hybrid setting, covering collision phenomena. Sufficient conditions are presented to ensure internal asymptotic stability while also attenuating external disturbances and plant uncertainties. The developed synthesis is applied to the orbital stabilization of an underactuated bipedal robot periodically touching the ground. Good performance of the closed-loop system is obtained not only in the presence of measurement noise and external disturbances, affecting the gait of the biped between collision time instants, but also under uncertainties at the velocity restitution when the ground collision occurs.