Formation mechanism of a basin of attraction for passive dynamic walking induced by intrinsic hyperbolicity

Fractal mechanism of basin of attraction in passive dynamic walking

Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how humans walk with low energy consumption and provides design principles for energy-efficient biped robots. However, the basin of attraction is very small and thin and has a fractal-like complicated shape, which makes producing stable walking difficult. In our previous study, we used the simplest walking model and investigated the fractal-like basin of attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics caused by the impact upon foot contact. We clarified that the fractal-like basin of attraction is generated through iterative stretching and bending deformations of the domain of the Poincaré map by sequential inverse images. However, whether the fractal-like basin of attraction is actually fractal, i.e., whether infinitely many self-similar patterns are embedded in the basin of attraction, is dependent on the slope angle, and the mechanism remains unclear. In the present study, we improved our previous analysis in order to clarify this mechanism. In particular, we newly focused on the range of the Poincaré map and specified the regions that are stretched and bent by the sequential inverse images of the Poincaré map. Through the analysis of the specified regions, we clarified the conditions and mechanism required for the basin of attraction to be fractal.

Task-level regulation enhances global stability of the simplest dynamic walker

Much remains unknown about how considerations such as stability and energy minimization shape the way humans walk. While active neuromotor control keeps humans upright, they also need to choose from multiple stepping regulation strategies to achieve one or more task goals, such as maintaining a desired speed or direction. Experiments on human treadmill walking motivate an important question: why do humans prefer one task-level regulation strategy over another-perhaps to enhance their ability to reject large disturbances? Here, we study the relationship between task-level regulation and global stability in a powered compass walker on a treadmill, with added step-to-step speed and position regulators. For treadmill walking, we find that speed regulation greatly enlarges and regularizes the unregulated walker's stability region, i.e. its basin of attraction, much more than position regulation. Thus, our results suggest a possible explanation for the experimental finding that humans strongly prioritize regulating speed from one stride to the next, even as they walk economically on average. Furthermore, our work suggests a functional connection between task-level motor regulation and global stability-and, thus, perhaps even fall risk.

Existence and stability of limit cycles in the model of a planar passive biped walking down a slope

We consider the simplest model of a passive biped walking down a slope given by the equations of switched coupled pendula (McGeer T. 1990 Passive dynamic walking. Int. J. Robot. Res. 9, 62-82. (doi:10.1177/027836499000900206)). Following the fundamental work by Garcia (Garcia et al. 1998 J. Biomech. Eng. 120, 281. (doi:10.1115/1.2798313)), we view the slope of the ground as a small parameter γ ≥ 0. When γ = 0, the system can be solved in closed form and the existence of a family of cycles (i.e. potential walking cycles) can be computed in closed form. As observed in Garcia et al. (Garcia et al. 1998 J. Biomech. Eng. 120, 281. (doi:10.1115/1.2798313)), the family of cycles disappears when γ increases and only isolated asymptotically stable cycles (walking cycles) persist. However, no mathematically complete proofs of the existence and stability of walking cycles have been reported in the literature to date. The present paper proves the existence and stability of a walking cycle (long-period gait cycle, as termed by McGeer) by using the methods of perturbation theory for maps. In particular, we derive a perturbation theorem for the occurrence of stable fixed points from 1-parameter families in two-dimensional maps that can be of independent interest in applied sciences.

The bipedal saddle space: modelling and validation

Better understanding of human balance control is pivotal for applications such as bipedal robots and medical technologies/therapies targeting human locomotion. Despite the inverted pendulum model being popular for describing bipedal locomotion, it does not properly capture the step-to-step transition dynamics. The major drawback has been the requirement for both feet to be on the ground which generates a discontinuity along the intersection of the potential energy surfaces produced by the two legs. To overcome this problem, we propose a generalized inverted pendulum-based model that can describe both single and double support phases. The full characterization of the system's potential energy allows the proposed model to drop the main limitation. This framework also enables optimal strategies to be designed for the transition between the two feet without the optimization algorithms. The proposed theory has been validated by comparing the human locomotor strategies output of our planner with real data from multiple experimental studies. The results show that our model generates trajectories consistent with human variability and performs better than existing well-known methods.

Asymptotic Realization of Desired Control Performance by Body Adaptation of Passive Dynamic Walker

[abstFig src='/00290003/03.jpg' width='300' text='Schematic of the proposed design method' ] This article proposes a design method of legged walking robot hardware capable of performing passive dynamic walking with its desirable characteristics. Passive dynamic walking has a relatively good energy efficiency, and is said to be similar to the walking style of animals. However, most legged robot hardware capable of passive dynamic walking is designed through trial and error on the basis of experience. One of the major problems of designing through trial and error is the difficulty of verifying walking for the legged robot hardware that has many degree of freedom. It is relatively easy to determine the initial condition for compass-type robot hardware. However, it often takes long time to determine the appropriate initial conditions and slope angles for complicated robots such as legged robots with knees. We proposed and verified a method to design a legged robot with knees that has a desired leg length and leg mass from a compass-type legged robot. In this article, we propose a method to design a passive dynamic walker that has a desired leg angle, step length, leg mass, etc., and verify the resulting design. More specifically, the physical parameters, such as the leg length, leg mass, and joint friction, are defined as “physical parameters” and the parameters acquired as the result of walking, such as the leg angle, step length, and walking cycle, are defined as “variable parameters.” By observing variable parameters while the robot is walking and by changing the physical parameters according to the observed variable parameters, the variable parameters are indirectly changed to desired values.

Dynamically and Biologically Inspired Legged Locomotion: A Review

[abstFig src='/00290003/01.jpg' width='300' text='Passive dynamic walking: RW03 and Jenkka III' ] Legged locomotion, such as walking, running, turning, and jumping depends strongly on the dynamics and the biological characteristics of the body involved. Gait patterns and energy efficiency, for instance, are known to be greatly affected, not only by travel speed and ground contact conditions but also by body structure such as joint stiffness and coordination, and foot sole shape. To understand legged locomotion principles, we must elucidate how the body’s dynamic and biological characteristics affect locomotion. Efforts should also be made to incorporate these characteristics inventively in order to improve locomotion performance with regard to robustness, adaptability, and efficiency, which realize more refined legged locomotion. For this special issue, we invited our readers to submit papers with approaches to achieving legged locomotion based on dynamic and biological characteristics and studies investigating the effects of these characteristics. In this paper, we review studies on dynamically and biologically inspired legged locomotion.