Periodic Walking Gaits Research Articles

A three-dimensional spring-loaded inverted pendulum walking model considering human movement speed and frequency

The spring-loaded inverted pendulum (SLIP) model is an effective model to capture the essential dynamics during human walking and/or running. However, most of the existing three-dimensional (3D) SLIP model does not explicitly account for human movement speed and frequency. To address this knowledge gap, this paper develops a new SLIP model, which includes a roller foot, massless spring, and concentrated mass. The governing equations-of-motion for the SLIP model during its double support phase are derived. It is noted that in the current formulation, the motion of the roller foot is prescribed; therefore, only the equations for the concentrated mass need to be solved. To yield model parameters leading to a periodic walking gait, a constrained optimization problem is formulated and solved using a gradient-based approach with a global search strategy. The optimization results show that when the attack angle ranges from 68° to 74°, the 3D SLIP model can yield a periodic walking gait with walking speeds varying from 0.5 to 2.0 m s−1. The predicted human walking data are also compared with published experimental data, showing reasonable accuracy.

Human-in-the-Loop Cooperative Control of a Walking Exoskeleton for Following Time-Variable Human Intention.

This article presents a human-in-the-loop cooperative control of a walking exoskeleton to provide assistance to the user and enhance human mobility. First, a dynamic mathematical model of the human-exoskeleton system is derived, and then, a human-in-the-loop cooperative control framework is proposed in two ways: separable cooperative control (SCC) and interactive cooperative control (ICC), respectively. The SCC introduces a space division of the human and the exoskeleton, while the ICC allows the robot to perceive the human intention and follows human motor, thereby improving the collaborative performance. The ICC formulates the impedance connection between the human and the exoskeleton in the divided orthogonal subspaces of walking, such that the robot is able to modify its position of center of mass (COM) when its motor trajectory deviates the one of the human. In addition, a novel adaptation control is proposed to deal with the unmodeled dynamics and trajectory tracking. Finally, to validate the effectiveness of our proposed controller, a series of experiments are conducted in three adults in gait at different speeds. It shows that the proposed controller can preserve the periodic walking gait and inherit the robustness of dealing with perturbations during walking.

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.

Dynamic Walking of Bipedal Robots on Uneven Stepping Stones via Adaptive-Frequency MPC

This letter presents a novel Adaptive-frequency MPC framework for bipedal locomotion over terrain with uneven stepping stones. In detail, we intend to achieve adaptive gait periods with variable MPC frequency for bipedal periodic walking gait to traverse terrain with discontinuities without slowing down. We pair this adaptive-frequency MPC with kino-dynamics trajectory optimization to obtain MPC adaptive frequencies (in terms of sampling times), 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 offline optimization. In numerical validations, our adaptive-frequency optimization and MPC framework have shown advantages over fixed-frequency MPC. The proposed framework can control the bipedal robot to traverse through uneven stepping stone terrains with perturbed stone heights, widths, and surface shapes while maintaining an average speed of 1.5 m/s.

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.

From standing balance to walking: A single control structure for a continuum of gaits

This article presents a control algorithm framework with which a bipedal robot can perform a variety of gaits by only modifying a small set of control parameters. The controller drives a number of variables, called non-emergent variables, to their desired trajectories resulting in a desired emergent walking gait. While the non-emergent variables remain the same independent of the gait, their desired trajectories are functions of a small set of control parameters that change as a function of the desired gait. This control algorithm has been tested on the humanoid robot COMAN, where different gaits including standing balance, stepping in place, periodic walking gaits with different velocities, as well as gait switching are demonstrated in experiments.

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.

An essential model for generating walking motions for humanoid robots

The modeling of humanoid robots with many degrees-of-freedom (DoF) can be done via the complete dynamic model. However, the complexity of the model can hide the essential factor of the walking, i.e. the equilibrium of the robot. One alternative is to simplify the model by neglecting some dynamical effects like in the 3D Linear Inverted Pendulum (LIP) model. Nonetheless, the assumption that the ZMP will be at the base of the pendulum is not ensured and the resulting walking gaits can make the Zero Moment Point (ZMP) evolve outside of the convex hull of support when they are replicated on the complete model of any humanoid robot. The objective of this paper is to propose a new model for walking that has the same dimension as the 3D LIP model but considers the complete dynamics of the humanoid. The proposed model is called essential model and it can be written based on the internal states of the robot and possible external information, thereby generating models for different purposes. The main advantage of the essential model is that it allows to generate walking gaits that ensure that the Zero Moment Point (ZMP) is kept in a desired position or it follows a desired path while the gait is performed. Furthermore, impacts of the swing foot with the ground can be considered to compute periodic walking gaits. In order to show the advantages of the proposed model, numerical studies are performed to design periodic walking gaits for the humanoid robot ROMEO.

Self-synchronization and self-stabilization of 3D bipedal walking gaits

Abstract 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 – inducesstability of a walking gait. This paper concerns the qualitative extension of this type of property to 3D walking with two degrees of underactuation. It is shown that a condition on the position of the center of mass in the horizontal plane at the transition between steps induces synchronization between the motions in the sagittal and frontal planes. A combination of the conditions for self-synchronization and vertical oscillations leads to stable gaits. The algorithm for self-stabilization of 3D walking gaits is first developed for a simplified model of a walking robot (an inverted pendulum with variable length legs), and then it is extended to a complex model of the humanoid robot Romeo using the notion of Hybrid Zero Dynamics. Simulations of the model of the robot illustrate the efficacy of the method and its robustness.

First Steps Towards Translating HZD Control of Bipedal Robots to Decentralized Control of Exoskeletons

This paper presents preliminary results toward translating gait and control design for bipedal robots to decentralized control of an exoskeleton aimed at restoring mobility to patients with lower limb paralysis, without the need for crutches. A mathematical hybrid dynamical model of the human-exoskeleton system is developed and a library of dynamically feasible periodic walking gaits for different walking speeds is found through nonlinear constrained optimization using the full-order dynamical system. These walking gaits are stabilized using a centralized (i.e., full-state information) hybrid zero dynamics-based controller, which is then decentralized (i.e., control actions use partial state information) so as to be implementable on the exoskeleton subsystem. A control architecture is then developed so as to allow the user to actively control the exoskeleton speed through his/her upper body posture. Numerical simulations are carried out to compare the two controllers. It is found that the proposed decentralized controller not only preserves the periodic walking gaits but also inherits the robustness to perturbations present in the centralized controller. Moreover, the proposed velocity regulation scheme is able to reach a steady state and track desired walking speeds under both, centralized, and decentralized schemes.

Investigation of optimal bipedal walking gaits subject to different energy‐based objective functions

AbstractOptimal bipedal walking gaits subject to different energy‐based objective functions are investigated using a simple planar rigid body model of a bipedal robot with upper body, thighs and shanks. The robot's segments are connected by revolute joints actuated by electric motors. The actuators' torques are generated by a trajectory tracking controller to produce periodic walking gaits. A numerical optimization routine is used to find optimal reference trajectories for average speeds in the range of 0.3 – 2.3 m/s to investigate the influence of different objective functions. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Predicting human walking gaits with a simple planar model

Models of human walking with moderate complexity have the potential to accurately capture both joint kinematics and whole body energetics, thereby offering more simultaneous information than very simple models and less computational cost than very complex models. This work examines four- and six-link planar biped models with knees and rigid circular feet. The two differ in that the six-link model includes ankle joints. Stable periodic walking gaits are generated for both models using a hybrid zero dynamics-based control approach. To establish a baseline of how well the models can approximate normal human walking, gaits were optimized to match experimental human walking data, ranging in speed from very slow to very fast. The six-link model well matched the experimental step length, speed, and mean absolute power, while the four-link model did not, indicating that ankle work is a critical element in human walking models of this type. Beyond simply matching human data, the six-link model can be used in an optimization framework to predict normal human walking using a torque-squared objective function. The model well predicted experimental step length, joint motions, and mean absolute power over the full range of speeds.

Power Consumption Optimization for a Hexapod Walking Robot

Power consumption is one of the main operational restrictions on autonomous walking robots. In this paper, an energy efficiency analysis is performed for a hexapod walking robot to reduce these energy costs. To meet the power-saving demands of legged robots, the torque distribution algorithm required to minimize the system's energy costs was established with an energy-consumption model formulated. In contrast to the force distribution method, where the objective function is related to the tip-point force components, the torque distribution scheme is based on minimization of the mechanical energy cost and heat loss power. The simulation results show that this scheme could reduce the system energy costs with use of the appropriate walking velocities and duty factors for the robot. The paper also discusses the effects of the gait patterns and the mechanical structure on the system energy costs. For this purpose, the prescribed periodic walking gait of the robot is described in terms of several parameters, including the duty factor, the stride length, the body height, and the foot trajectory lateral offset. The numerical results indicate some analogies between the characteristics of the simulated walking robot and those of animals in nature. The optimized parameters derived here are intended for robot platform development applications.

Planning and Control of Stable Walking for a 3D Bipedal Robot

This paper presents a time-invariant feedback controller that simultaneously regulates the ZMP (zero-moment point) position and the joint configuration of a 3D biped in order to achieve an asymptotically, periodic walking gait for a 3D bipedal robot with feet. The cyclic walking gait is composed of a successive single-support phase and an impulsive impact with full plane-contact between the feet and the ground. The biped robot has 10 DOFs (degrees of freedom) in the single-support phase and 10 actuators. In order to avoid the unexpected rotation of the supporting foot, the position of the ZMP in the horizontal plane has to be controlled. It is also desired that the feedback controller tracks a parameterized reference trajectory to achieve walking stability. We use the method of virtual constraints previously implemented for controlling point-feet bipedal robots to create a set of parameterized reference walking trajectories. By creating the hybrid zero dynamics, an orbital stability study with Poincaré map is evaluated in a reduced space. We then design a supplemental event-based feedback controller to enhance walking stability. The walking gait has an average walking speed of 0.76m/sec (or 0.72 body lengths per second) in the simulation study.

Gait generation and control for biped robots with underactuation degree one

The paper develops a unified feedback control law for n degree-of-freedom biped robots with one degree of underactuation so as to generate periodic orbits on different slopes. The periodic orbits on different slopes are produced from an original periodic orbit, which is either a natural passive limit cycle on a specific slope or a stable periodic walking gait on level ground generated with active control. First, inspired by the controlled symmetries approach, a general result on gait generation on different slopes based on a periodic orbit on a specific slope is obtained. Second, the time-scaling control approach is integrated to reproduce geometrically same periodic orbits for biped robots with one degree of underactuation. The degree of underactuation is compensated by one degree-of-freedom in the temporal evolution that scales the original periodic orbit. Necessary and sufficient conditions are investigated for the existence and stability properties of periodic orbits on different slopes with the proposed control law. Finally, the proposed approach is illustrated by two kinds of underactuated biped robots: one has a passive gait on a specific ground slope and the other does not have a natural passive gait.

Stability analysis and time-varying walking control for an under-actuated planar biped robot

This paper presents two walking controllers for a planar biped robot with unactuated point feet. The control is based on the tracking of reference motions expressed as a function of time. First, the reference motions are adapted at each step in order to create a hybrid zero dynamic (HZD) system. Next, the stability of the walking gait under closed-loop control is evaluated with the linearization of the restricted Poincaré map of the HZD. When the controlled outputs are selected to be the actuated coordinates, most periodic walking gaits for this robot are unstable, that is, the eigenvalues of the linearized Poincaré map (ELPM) is larger than one. Therefore, two control strategies are explored to produce stable walking. The first strategy uses an event-based feedback controller to modify the ELPM and the second one is based on the choice of controlled outputs. The stability analysis show that, for the same robot and for the same reference trajectory, the stability of the walking (or ELPM) can be modified by some pertinent choices of controlled outputs. Moreover, by studying some walking characteristics of many stable cases, a necessary condition for stable walking is proposed. It is that the height of swing foot is nearly zero at the desired moment of impact. Based on this condition, the duration of the step is almost constant in presence of initial error, so a method for choosing controlled outputs for the second controller is given. By using this method, two stable domains for the controlled outputs selection are obtained.

Asymptotically Stable Walking of a Five-Link Underactuated 3-D Bipedal Robot

This paper presents three feedback controllers that achieve an asymptotically stable, periodic, and fast walking gait for a 3-D bipedal robot consisting of a torso, revolute knees, and passive (unactuated) point feet. The walking surface is assumed to be rigid and flat; 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 six actuators. In addition to the reduced number of actuators, the interest of studying robots with point feet is that the feedback control solution must explicitly account for the robot's natural dynamics in order to achieve balance while walking. We use an extension of the method of virtual constraints and hybrid zero dynamics (HZD), a very successful method for planar bipeds, in order to simultaneously compute a periodic orbit and an autonomous feedback controller that realizes the orbit, for a 3-D (spatial) bipedal walking robot. This method allows the computations for the controller design and the periodic orbit 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 Poincare map of the HZD. Most periodic walking gaits for this robot are unstable when the controlled outputs are selected to be the actuated coordinates. Three strategies are explored to produce stable walking. 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 to modify the eigenvalues of the (linearized) Poincare map. In the third approach, the effect of output selection on the zero dynamics is discussed and a pertinent choice of outputs is proposed, leading to stabilization without the use of a supplemental event-based controller.