Limit Cycle Walking Research Articles

Limit cycle walking, running, and skipping of telescopic-legged rimless wheel

SUMMARYThis paper investigates the efficiency and properties of limit cycle walking, running, and skipping of a planar, active, telescopic-legged rimless wheel. First, we develop the robot equations of motion and design an output following control for the telescopic-legs' action. We then numerically show that a stable walking gait can be generated by asymmetrizing the impact posture. Second, we numerically show that a stable running gait can be generated by employing a simple feedback control of the control period, and compare the properties of the generated running gait with those of the walking gait. Furthermore, we find out another underlying gait called skipping that emerges as an extension of the walking gait. Through numerical analysis, we show that the generated skipping gaits are inherently stable and are less efficient than the other two gaits.

Stable Walking Gaits for a Three-Link Planar Biped Robot With One Actuator

We consider a benchmark example of a three-link planar biped walker with torso, which is actuated in between the legs. The torso is thought to be kept upright by two identical torsional springs. The mathematical model reflects a three-degree-of-freedom mechanical system with impulse effects, which describe the impacts of the swing leg with the ground, and the aim is to induce stable limit-cycle walking on level ground. The main contribution is a novel systematic trajectory planning procedure for solving the problem of gait synthesis. The key idea is to find a system of ordinary differential equations for the functions describing a synchronization pattern for the time evolution of the generalized coordinates along a periodic motion. These functions, which are known as virtual holonomic constraints, are also used to compute an impulsive linear system that approximates the time evolution of the subset of coordinates that are transverse to the orbit of the continuous part of the periodic solution. This auxiliary system, which is known as transverse linearization, is used to design a nonlinear exponentially orbitally stabilizing feedback controller. The performance of the closed-loop system and its robustness with respect to various perturbations and uncertainties are illustrated via numerical simulations.

Fully interconnected, linear control for limit cycle walking

Limit cycle walkers are a class of bipeds that achieve stable locomotion without enforcing full controllability throughout the gait cycle. Although limit cycle walkers produce more natural-looking and efficient gaits than bipeds that are based on other control principles such as zero moment point walking, they cannot yet achieve the stability and versatility of human locomotion. One open question is the degree of complexity required in the control algorithm to ensure reliable terrain adaptation and disturbance rejection. The present study applies a fully interconnected, linear controller to a two-dimensional, five-link walking model, achieving stable and efficient locomotion over unpredictable terrain (slopes varying between 2° and 7° and step-downs varying between 0 and 25% leg length). The results indicate that elaborate control principles are not necessarily required for stable bipedal walking.

Analysis of limit-cycle walking for a compass-like biped robot

We consider an impulsive mechanical system modeling dynamics of a planar two-link walker commonly known as a compass-gait biped. It is assumed that there is actuation in the hip but that the desired periodic trajectory describes an unstable passive walking gait. We recall and apply a recently developed technique for design of orbitally stabilizing feedback controllers. After that, we illustrate how to assess various properties of the closed-loop system. In particular, sensitivity to perturbations of the slope of the walking surface is analyzed and an estimate for possible deviations from the nominal trajectory is computed analytically.

ACTIVE LATERAL FOOT PLACEMENT FOR 3D STABILIZATION OF A LIMIT CYCLE WALKER PROTOTYPE

This study focuses on the application of active lateral foot placement for 3D stabilization of bipedal walkers. Within the paradigm of "limit cycle walking" foot placement is an important strategy as it can provide cyclic stability for walkers that are locally unstable. Moreover, human gait analysis studies suggest that the stability of human walking depends highly on lateral foot placement. Various simulation studies have already successfully implemented lateral foot placement in walking models, but this study demonstrates that an active lateral foot placement strategy can actually (cyclically) stabilize a physical walking robot that is locally unstable. In order to come to this result, first a study is performed on a simple 3D point mass walking model. This study establishes that, for a model with fixed step length, cyclic stability can already be obtained with a simple linear lateral foot placement strategy that only uses lateral state information (lateral position and velocity) of the center of mass. Moreover, it is found that increasing the walking speed and increasing the ankle roll stiffness enlarges the range of stable feedback gains. With this knowledge of stable feedback gains and parameter sensitivities, the same foot placement strategy is applied to the physical 3D walking prototype called Flame. Similar to the model, this prototype is shown to be unstable without foot placement and stable with the application of the simple, linear lateral foot placement strategy.

Controlling the Walking Speed in Limit Cycle Walking

“Limit Cycle Walking” is a relatively new paradigm for the design and control of two-legged walking robots. It states that achieving stable periodic gait is possible without locally stabilizing the walking trajectory at every instant in time, as is traditionally done in most walking robots. Well-known examples of Limit Cycle Walkers are the Passive Dynamic Walkers, but recently there are also many actuated Limit Cycle Walkers. Limit Cycle Walkers generally use less energy than other existing bipeds, but thus far they have not been as versatile. This paper focuses on one aspect of versatility: walking speed. We study how walking speed can be varied, which way is energetically beneficial and how walking speed affects a walker's ability to handle disturbances (that is, disturbance rejection). The study is performed using one prototype and one simulation model. The speed of these two walkers is adapted by changing three parameters: the amount of ankle push-off, upper body pitch and step length. The study has resulted in four conclusions. (1) Steady-state speeds between 0.24 and 0.68 m s -1 (for a 0.6 m leg length) were obtained, with loss of stability determining the lower limit and actuation limits determining the upper limit. This result shows the applicability of Limit Cycle Walking for versatile walking machines. (2) For any speed, powering the gait by leaning the body forward costs less energy than using ankle push-off. (3) In contrast to the apparent tradeoff between speed and stability in traditional walking robots, in Limit Cycle Walking we find that increasing the walking speed, independent of how this is done, automatically results in an increasing disturbance rejection. (4) A combination of feedforward actuation adjustment and step-to-step feedback from walking speed shows that it is possible to change walking speed in only a few steps and maintain a desired speed when performing tasks such as carrying loads and walking on slopes. In particular, this fourth conclusion underlines the applicability of the concept of Limit Cycle Walking for versatile two-legged walking machines.

Ankle Actuation for Limit Cycle Walkers

Limit Cycle Walkers are bipeds that exhibit a stable cyclic gait without requiring local controllability at all times during gait. Well-known example are McGeer's “Passive Dynamic Walkers”, but the concept expands to actuated bipeds as involved in this study. Current state-of-the-art Limit Cycle Walkers excel in being very energy efficient, but their ability to handle disturbances (i.e. disturbance rejection) is still limited. A way to improve this ability while maintaining low energy consumption is the use of ankle actuation, which has so far seen few applications in this type of walker. In this paper we study the effect of (1) applying (passive) stiffness in the ankle joint, (2) applying control in the stance ankle based only on local sensor information and (3) modulating ankle push-off. For all three strategies the paper shows how they influence energy use and disturbance rejection of a simple point mass walking model, a more realistic model and a physical prototype. We find that applying a passive ankle spring that results in premature heel rise is energetically optimal and gives an actuation pattern that largely resembles that of humans. Local stance ankle control and ankle push-off modulation can improve the disturbance rejection of a Limit Cycle Walker by at least 60%, without increasing its energy use. These findings are substantiated by showing that our prototype is able to handle large disturbances such as a step-down of 5% of its leg length, while walking efficiently at a mechanical cost of transport of 0.09.

Swing-Leg Retraction for Limit Cycle Walkers Improves Disturbance Rejection

Limit cycle walkers are bipeds that exhibit a stable cyclic gait without requiring local controllability at all times during gait. A well-known example of limit cycle walking is McGeer's ldquopassive dynamic walking,rdquo but the concept expands to actuated bipeds as involved in this study. One of the stabilizing effects in limit cycle walkers is the dissipation of energy that occurs when the swing foot hits the ground. We hypothesize that this effect can be enhanced with a negative relation between the step length and step time. This relation is implemented through an open-loop strategy called swing-leg retraction; a predefined time trajectory for the swing leg makes the swing leg move backwards just prior to foot impact. In this paper, we study the effect of swing-leg retraction through three bipeds; a simple point mass simulation model, a realistic simulation model, and a physical prototype. Their stability is analyzed using Floquet multipliers, followed by an evaluation of how well disturbances are handled using the Gait Sensitivity Norm. We find that mild swing-leg retraction is optimal for the disturbance rejection of a limit cycle walker, as it results in a system response that is close to critically damped, rejecting the disturbance in the fewest steps. Slower retraction results in an overdamped response, characterized by a positive dominant Floquet multiplier. Likewise, faster retraction results in an underdamped response, characterized by a negative Floquet multiplier.

1P1-B13 三次元二足ロボットのリミットサイクル歩行の腕振りによる安定化(受動歩行ロボット)

1P1-B13 三次元二足ロボットのリミットサイクル歩行の腕振りによる安定化(受動歩行ロボット)

A Disturbance Rejection Measure for Limit Cycle Walkers: The Gait Sensitivity Norm

The construction of more capable bipedal robots highly depends on the ability to measure their performance. This performance is often measured in terms of speed or energy efficiency, but these properties are secondary to the robot's ability to prevent falling given the inevitable presence of disturbances, i.e., its disturbance rejection. Existing disturbance rejection measures (zero moment point, basin of attraction, Floquet multipliers) are unsatisfactory due to conservative assumptions, long computation times, or bad correlation to actual disturbance rejection. This paper introduces a new measure called the Gait Sensitivity Norm that combines a short calculation time with good correlation to actual disturbance rejection. It is especially suitable for implementation on limit cycle walkers, a class of bipeds that currently excels in terms of energy efficiency, but still has limited disturbance rejection capabilities. The paper contains an explanation of the Gait Sensitivity Norm and a validation of its value on a simple walking model as well as on a real bipedal robot. The disturbance rejection of the simple model is studied for variations of floor slope, foot radius, and hip spring stiffness. We show that the calculation speed is as fast as the standard Floquet multiplier analysis, while the actual disturbance rejection is correctly predicted with 93% correlation on average.

Prediction of stable walking for a toy that cannot stand.

Previous experiments [M. J. Coleman and A. Ruina, Phys. Rev. Lett. 80, 3658 (1998)] showed that a gravity-powered toy with no control and that has no statically stable near-standing configurations can walk stably. We show here that a simple rigid-body statically unstable mathematical model based loosely on the physical toy can predict stable limit-cycle walking motions. These calculations add to the repertoire of rigid-body mechanism behaviors as well as further implicating passive dynamics as a possible contributor to stability of animal motions.