Stability analysis and control for bipedal locomotion using energy methods

In this thesis, we investigate the stability of limit cycles of passive dynamic walking. The formation process of the limit cycles is approached from the view of energy interaction. We introduce for the first time the notion of the energy portrait analysis originated from the phase portrait. The energy plane is spanned by the total energy of the system and its derivative, and different energy trajectories represent the energy portrait in the plane. One of the advantages of this method is that the stability of the limit cycles can be easily shown in a 2D plane regardless of the dimension of the system. The energy portrait of passive dynamic walking reveals that the limit cycles are formed by the interaction between energy loss and energy gain during each cycle, and they are equal at equilibria in the energy plane. In addition, the energy portrait is exploited to examine the existence of semi-passive limit cycles generated using the energy supply only at the take-off phase. It is shown that the energy interaction at the ground contact compensates for the energy supply, which makes the total energy invariant yielding limit cycles. This result means that new limit cycles can be generated according to the energy supply without changing the ground slope, and level ground walking, whose energy gain at the contact phase is always zero, can be achieved without actuation during the swing phase. We design multiple switching controllers by virtue of this property to increase the basin of attraction. Multiple limit cycles are linearized using the Poincare map method, and the feedback gains are computed taking into account the robustness and actuator saturation. Once a trajectory diverges from a basin of attraction, we switch the current controller to one that includes the trajectory in its basin of attraction. Numerical simulations confirm that a set of limit cycles can be used to increase the basin of attraction further by switching the controllers one after another. To enhance our knowledge of the limit cycles, we performed sophisticated simulations and found all stable and unstable limit cycles from the various ground slopes not only for the symmetric legs but also for the unequal legs that cause gait asymmetries. As a result, we present a novel classification of the passive limit cycles showing six distinct groups that are consecutive and cyclical

Universal Dynamics of Damped-Driven Systems: The Logistic Map as a Normal Form for Energy Balance

Damped-driven systems are ubiquitous in engineering and science. Despite the diversity of physical processes observed in a broad range of applications, the underlying instabilities observed in practice have a universal characterization which is determined by the overall gain and loss curves of a given system. The universal behavior of damped-driven systems can be understood from a geometrical description of the energy balance with a minimal number of assumptions. The assumptions on the energy dynamics are as follows: the energy increases monotonically as a function of increasing gain, and the losses become increasingly larger with increasing energy, i.e. there are many routes for dissipation in the system for large input energy. The intersection of the gain and loss curves define an energy balanced solution. By constructing an iterative map between the loss and gain curves, the dynamics can be shown to be homeomorphic to the logistic map, which exhibits a period doubling cascade to chaos. Indeed, the loss and gain curves allow for a geometrical description of the dynamics through a simple Verhulst diagram (cobweb plot). Thus irrespective of the physics and its complexities, this simple geometrical description dictates the universal set of logistic map instabilities that arise in complex damped-driven systems. More broadly, damped-driven systems are a class of non-equilibrium pattern forming systems which have a canonical set of instabilities that are manifest in practice.Comment: 26 pages, 31 figure

Impulsive torque control of biped gait with power packets

Many strategies for an actuated biped gait generation have been proposed based on the passive dynamic gait. Among them, this study focuses on an impulsive excitation at the toe-off instance. The strategy offers advantages in its experimental implementation; for example, it is not required to measure and control the trajectory of the legs all the time. However, there has been no study on a realistic design of the impulsive torque itself. In this paper, we propose an impulsive actuation method based on a power packet dispatching system. Power packet is a unit of electric power transfer in a pulse shape with information tags attached in voltage waveforms. According to the tag, power packets are transferred from sources to loads. On the basis of the power packetization, the torque input is configured as a result of a power packet supply to electric motors in a realistic setup. The proposed scheme controls the supply in a digitized way, that is, by changing the number of power packets supplied in a gait step. We confirm the successful gait generation with the power packets through numerical simulations

Discrete Mechanics and Optimal Control Applied to the Compass Gait Biped

This paper presents a methodology for generating locally optimal control policies for simple hybrid mechanical systems, and illustrates the method on the compass gait biped. Principles from discrete mechanics are utilized to generate optimal control policies as solutions of constrained nonlinear optimization problems. In the context of bipedal walking, this procedure provides a comparative measure of the suboptimality of existing control policies. Furthermore, our methodology can be used as a control design tool; to demonstrate this, we minimize the specific cost of transport of periodic orbits for the compass gait biped, both in the fully actuated and underactuated case

Sharp changes in fractal basin of attraction in passive dynamic walking

The version of record of this article, first published in Nonlinear Dynamics, is available online at Publisher’s website: https://doi.org/10.1007/s11071-023-08913-wA passive dynamic walker is a mechanical system that walks down a slope without any control, and gives useful insights into the dynamic mechanism of stable walking. This system shows specific attractor characteristics depending on the slope angle due to nonlinear dynamics, such as period-doubling to chaos and its disappearance by a boundary crisis. However, it remains unclear what happens to the basin of attraction. In our previous studies, we showed that a fractal basin of attraction is generated using a simple model over a critical slope angle by iteratively applying the inverse image of the Poincaré map, which has stretching and bending effects. In the present study, we show that the size and fractality of the basin of attraction sharply change many times by changing the slope angle. Furthermore, we improved our previous analysis to clarify the mechanisms for these changes and the disappearance of the basin of attraction based on the stretching and bending deformation in the basin formation process. These findings will improve our understanding of the governing dynamics to generate the basin of attraction in walking

The basic mechanics of bipedal walking lead to asymmetric behavior

Abstract-This paper computationally investigates whether gait asymmetries can be attributed in part to basic bipedal mechanics independent of motor control. Using a symmetrical rigid-body model known as the compass-gait biped, we show that changes in environmental or physiological parameters can facilitate asymmetry in gait kinetics at fast walking speeds. In the environmental case, the asymmetric family of high-speed gaits is in fact more stable than the symmetric family of lowspeed gaits. These simulations suggest that lower extremity mechanics might play a direct role in functional and pathological asymmetries reported in human walking, where velocity may be a common variable in the emergence and growth of asymmetry