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

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

Dynamic Walking Models to Understand Asymmetric Gait Characteristics

Passive dynamic walking models remarkably predict gait behaviour such as walk-run transition speeds, preferred step length, stride frequencies and - with the inclusion of springs - ground reaction forces. Muscular or neurological conditions may lead to asymmetric walking characteristics that, in turn, come with long term health risks. Gait analysis may be used to understand an individual patient’s conditions to help rehabilitate them. However, people adapt their kinematic and kinetic walking patterns so it can be hard to distinguish the effects of gait alterations such as inertial imbalance or injury. In this thesis a compass walking model with no active controllers is explored to understand the dynamics of gait. To help us interpret the effects of mass imbalance with a prosthetic foot or orthotic device, asymmetric loading conditions are investigated. A simple spring-mass walking model is used to explore the effects of altered touch-down angles and effective leg stiffness to see if these are used as strategies to alter the characteristics of gait. Results show that an asymmetric touch-down angle alters step length while retaining a symmetric stance time. A hybrid model is then derived with springs to emulate human-like ground reaction forces and asymmetric inertial loading of the legs. Results support previous research that push-off from the trailing leg propels the leg mass more than the body mass. Higher rates of joint forces, larger step lengths and a longer stance time on the residual limb may be due to the prosthetic leg stiffness or the higher location of centre-of-mass. These results help us understand how the dynamic components affect gait characteristics such as step length, stance time and walking speeds. This work is motivated by the needs of persons with disabilities and by the desire to understand human walking

Numerical bifurcation analysis of the asymmetric spring-mass model

In this thesis, an approach is presented that is based on the calculation of bifurcations in the spring-mass model. The new approach consists of the transformation of the series of initial value problems on different intervals into a single boundary value problem. Using this technique, discontinuities can be avoided and sophisticated numerical methods for studying parameterized nonlinear boundary value problems can be applied. Thus, appropriate extended systems are used to compute transcritical and period-doubling bifurcation points as well as turning points. We show that the resulting boundary value problems can be solved by the simple shooting method with sufficient accuracy, making the application of the more extensive multiple shooting superfluous. The proposed approach is fast, robust to numerical perturbations and allows to determine highly unstable periodic solutions of the original problem. Asymmetric leg function is often an undesired side-effect in artificial legged systems and may reflect functional deficits or variations in the mechanical construction. It can also be found in legged locomotion in humans and animals, for example after an accident or in specific gait patterns. So far, it is not clear to what extent differences in the leg function of contralateral limbs can be tolerated during walking or running. Here, we address this issue using the spring-mass model for simulating walking and running with compliant legs. With the help of the original realization of the model and the boundary value problem approach, we show that considerable differences between contralateral legs can be tolerated and may even provide advantages to the robustness of the system dynamics. A better understanding of the mechanisms and potential benefits of asymmetric leg operation may help to guide the development of artificial limbs or the design novel therapeutic concepts and rehabilitation strategies

Metastable legged-robot locomotion

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 195-215).A variety of impressive approaches to legged locomotion exist; however, the science of legged robotics is still far from demonstrating a solution which performs with a level of flexibility, reliability and careful foot placement that would enable practical locomotion on the variety of rough and intermittent terrain humans negotiate with ease on a regular basis. In this thesis, we strive toward this particular goal by developing a methodology for designing control algorithms for moving a legged robot across such terrain in a qualitatively satisfying manner, without falling down very often. We feel the definition of a meaningful metric for legged locomotion is a useful goal in and of itself. Specifically, the mean first-passage time (MFPT), also called the mean time to failure (MTTF), is an intuitively practical cost function to optimize for a legged robot, and we present the reader with a systematic, mathematical process for obtaining estimates of this MFPT metric. Of particular significance, our models of walking on stochastically rough terrain generally result in dynamics with a fast mixing time, where initial conditions are largely "forgotten" within 1 to 3 steps. Additionally, we can often find a near-optimal solution for motion planning using only a short time-horizon look-ahead. Although we openly recognize that there are important classes of optimization problems for which long-term planning is required to avoid "running into a dead end" (or off of a cliff!), we demonstrate that many classes of rough terrain can in fact be successfully negotiated with a surprisingly high level of long-term reliability by selecting the short-sighted motion with the greatest probability of success. The methods used throughout have direct relevance to machine learning, providing a physics-based approach to reduce state space dimensionality and mathematical tools to obtain a scalar metric quantifying performance of the resulting reduced-order system.by Katie Byl.Ph.D

Augmented Linear Inverted Pendulum Model for Bipedal Gait Planning

Ph.DDOCTOR OF PHILOSOPH

Using MapReduce Streaming for Distributed Life Simulation on the Cloud

Distributed software simulations are indispensable in the study of large-scale life models but often require the use of technically complex lower-level distributed computing frameworks, such as MPI. We propose to overcome the complexity challenge by applying the emerging MapReduce (MR) model to distributed life simulations and by running such simulations on the cloud. Technically, we design optimized MR streaming algorithms for discrete and continuous versions of Conway’s life according to a general MR streaming pattern. We chose life because it is simple enough as a testbed for MR’s applicability to a-life simulations and general enough to make our results applicable to various lattice-based a-life models. We implement and empirically evaluate our algorithms’ performance on Amazon’s Elastic MR cloud. Our experiments demonstrate that a single MR optimization technique called strip partitioning can reduce the execution time of continuous life simulations by 64%. To the best of our knowledge, we are the first to propose and evaluate MR streaming algorithms for lattice-based simulations. Our algorithms can serve as prototypes in the development of novel MR simulation algorithms for large-scale lattice-based a-life models.https://digitalcommons.chapman.edu/scs_books/1014/thumbnail.jp