No abstract

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 stateof-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 stepdown of 5% of its leg length, while walking efficiently at a mechanical cost of transport of 0.09.

Abstract-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 "passive dynamic walking," 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.

The concept of 'Limit Cycle Walking' in bipedal robots removes the constraint of dynamic balance at every instance during gait. We hypothesize that this is crucial for the development of increasingly versatile and energy-effective humanoid robots. It allows the application of a wide range of gaits and it allows a robot to utilize its natural dynamics in order to reduce energy use. This paper presents the design and experimental results of our latest walking robot 'Flame' and the design of our next robot in line 'TUlip'. The focus is on the mechanical implementation of series elastic actuation, which is ideal for Limit Cycle Walkers since it offers high controllability without having the actuator dominating the system dynamics. Walking experiments show the potential of our robots, showing good walking performance, though using simple control.

"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 11 (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-tostep 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.

Passive dynamic walking is a promising idea for the development of simple and efficient two-legged walking robots. One of the difficulties with this concept is the addition of a stable upper body; on the one hand, a passive swing leg motion must be possible, whereas on the other hand, the upper body (an inverted pendulum) must be stabilized via the stance leg. This paper presents a solution to the problem in the form of a bisecting hip mechanism. The mechanism is studied with a simulation model and a prototype based on the concept of passive dynamic walking. The successful walking results of the prototype show that the bisecting hip mechanism forms a powerful ingredient for stable, simple, and efficient bipeds.

No abstract

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.