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

Energy Conservative Limit Cycle Oscillations

This paper shows how globally attractive limit cycle oscillations can be induced in a system with a nonlinear feedback element. Based on the same principle as the Van der Pol oscillator, the feedback behaves as a negative damping for low velocities but as an ordinary damper for high velocities. This nonlinear damper can be physically implemented with a continuous variable transmission and a spring, storing energy in the spring when the damping is positive and reusing it when the damping is negative. The resulting mechanism has a natural limit cycle oscillation that is energy conservative and can be used for the development of robust, dynamic walking robots

Beyond Basins of Attraction: Quantifying Robustness of Natural Dynamics

Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the basins of attraction of passive limit-cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond basins of attraction. We show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.Comment: 15 pages. This work has been accepted to IEEE Transactions on Robotics (2019

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 which 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.Comment: Note: only corrections so far have been fixing typo's in these comments. 3 pages, 2 eps figures, uses epsf.tex, revtex.sty, amsfonts.sty, aps.sty, aps10.sty, prabib.sty; Accepted for publication in Phys. Rev. E. 4/9/2001 ; information about Andy Ruina's lab (including Coleman's, Garcia's and Ruina's other publications and associated video clips) can be found at: http://www.tam.cornell.edu/~ruina/hplab/index.html and more about Georg Bock's Simulation Group with whom Katja Mombaur is affiliated can be found at http://www.iwr.uni-heidelberg.de/~agboc

Acceleration feedback control of human-induced floor vibrations

Active vibration control (AVC) via a proof-mass actuator is considered to be a suitable technique for the mitigation of vibrations caused by human motions in floor structures. It has been observed that actuator dynamics strongly influence structure dynamics despite considering collocated actuator/sensor control. The well-known property of the interlacing of poles and zeros of a collocated control system is no longer accomplished. Therefore, velocity-based feedback control, which has been previously used by other researchers, might not be a good solution. This work presents a design process for a control scheme based on acceleration feedback control with a phase-lag compensator, which will generally be different from an integrator circuit. This first-order compensator is applied to the output (acceleration) in such a way that the relative stability and potential damping to be introduced are significantly increased accounting for the interaction between floor and actuator dynamics. Additionally, a high-pass filter designed to avoid stroke saturation is applied to the control signal. The AVC system designed according to this procedure has been assessed in simulation and successfully implemented in an in-service open-plan office floor. The actual vibration reductions achieved have been approximately 60% for walking tests and over 90% for a whole-day vibration monitoring. (C) 2009 Elsevier Ltd. All rights reserved

Walking dynamics are symmetric (enough)

Many biological phenomena such as locomotion, circadian cycles, and breathing are rhythmic in nature and can be modeled as rhythmic dynamical systems. Dynamical systems modeling often involves neglecting certain characteristics of a physical system as a modeling convenience. For example, human locomotion is frequently treated as symmetric about the sagittal plane. In this work, we test this assumption by examining human walking dynamics around the steady-state (limit-cycle). Here we adapt statistical cross validation in order to examine whether there are statistically significant asymmetries, and even if so, test the consequences of assuming bilateral symmetry anyway. Indeed, we identify significant asymmetries in the dynamics of human walking, but nevertheless show that ignoring these asymmetries results in a more consistent and predictive model. In general, neglecting evident characteristics of a system can be more than a modeling convenience---it can produce a better model.Comment: Draft submitted to Journal of the Royal Society Interfac

A geometric approach to three-dimensional hipped bipedal robotic walking

This paper presents a control law that results in stable walking for a three-dimensional bipedal robot with a hip. To obtain this control law, we utilize techniques from geometric reduction, and specifically a variant of Routhian reduction termed functional Routhian reduction, to effectively decouple the dynamics of the three-dimensional biped into its sagittal and lateral components. Motivated by the decoupling afforded by functional Routhian reduction, the control law we present is obtained by combining three separate control laws: the first shapes the potential energy of the sagittal dynamics of the biped to obtain stable walking gaits when it is constrained to the sagittal plane, the second shapes the total energy of the walker so that functional Routhian reduction can be applied to decoupling the dynamics of the walker for certain initial conditions, and the third utilizes an output zeroing controller to stabilize to the surface defining these initial conditions. We numerically verify that this method results in stable walking, and we discuss certain attributes of this walking gait