Chapter 3 The Simplest 2-D Walker

Lecture Notes in Computer Science, 2010

Stability of bipedal locomotion is analyzed using a model of a planar biped written in the framework of systems with unilateral constraints. Based on this model, two different stable walking gaits are derived: one which fulfills the widely used criterion of the Zero Moment Point (ZMP) and another one violating this criterion. Both gaits are determined using systematic model-based designs. The model and the two gaits are used in simulations to illustrate conservatisms of two commonly used methods for stability analysis of bipedal walking: the ZMP criterion and Poincaré return map method. We show that none of these two methods can give us a general qualification of bipedal walking stability.

Intelligent Autonomous Systems 15, 2018

The aim of this paper is to investigate the role of some mechanical quantities in the challenging task to make a robot walking or running. Because the upright posture of an humanoid is the main source of instability, the maintenance of the equilibrium during locomotion requires the gait-controller to deal with a number of constraints, such as ZMP, whose dynamical satisfactions prevent the humanoid from an harmful fall. Walking humanoids are open systems heavily interacting with a perturbing environment and the rapid loss of mechanical energy could be an hallmark of instability. In this paper we shall show how certain dimensionless parameters could be useful to design the walking gait of a bipedal robot.

Robotica, 2008

This paper presents a passive dynamic walking model with toed feet that can walk down a gentle slope under the action of gravity alone. The model is the simplest of its kind with a point mass at the hip and two rigid legs each hinged at the hip on the one end and equipped with toed foot on the other end. We investigate two cases of the model, one with massless legs and another with infinitesimal leg masses. Rotation of the stance foot about the toe joint is initiated by ankle-strike, which is caused by the inelastic collision of the stance leg with a stop mounted on the stance foot. Numerical simulations of walking show that larger step lengths, higher speeds, stability, and energy efficiency can be achieved than what is achievable by a point-feet walker of same hip mass and leg lengths. Period-two gait of a pointfeet walker is compared with period-one gait of the toed-feet walker and the mechanism responsible for achieving longer step lengths is described. It is shown that the advantage of the proposed walker comes from its relation to arc-feet walker. The characteristics of deterministic gait with infinitesimal leg masses is compared with that of nondeterministic gait with zero leg masses. It is shown that deterministic gait does not give maximum speed and efficiency compared to nondeterministic gait with swing leg control. Finally, active dynamic walking of the proposed walker is discussed.

2010 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2010

This paper presents a seven-link dynamic walking model that is more close to human beings. We add hip actuation, upper body, flat feet and compliant ankle joints to the model. Walking sequence of the flat-foot walker has several sub-streams that form bipedal walking with dynamic series of phases, which is different with the motion of roundfoot and point-foot models. We investigate the characteristics of three different walking gaits with different step lengths. Comparison of these walking gaits in walking velocity, efficiency and stability reveals the relation between step length and walking performance. Experimental results indicate that the gait which is more close to human normal walking achieves higher stability and energetic efficiency.

A popular hypothesis regarding legged locomotion is that humans and other large animals walk and run in a manner that minimizes the metabolic energy expenditure for locomotion. Here, using numerical optimization and supporting analytical arguments, I obtain the energy-minimizing gaits of many different simple biped models. I consider bipeds with point-mass bodies and massless legs, with or without a knee, with or without a springy tendon in series with the leg muscle and minimizing one of many different 'metabolic cost' models—correlated with muscle work, muscle force raised to some power, the Minetti – Alex-ander quasi-steady approximation to empirical muscle metabolic rate (from heat and ATPase activity), a new cost function called the 'generalized work cost' C g having some positivity and convexity properties (and includes the Minetti – Alexander cost and the work cost as special cases), and generalizations thereof. For many of these models, walking-like gaits are optimal at low speeds and running-like gaits at higher speeds, so a gait transition is optimal. Minimizing the generalized work cost C g appears mostly indistinguishable from minimizing muscle work for all the models. Inverted pendulum walking and impulsive running gaits minimize the work cost, generalized work costs C g and a few other costs for the springless bipeds; in particular, a knee-torque-squared cost, appropriate as a simplified model for electric motor power for a kneed robot biped. Many optimal gaits had symmetry properties; for instance, the left stance phase was identical to the right stance phases. Muscle force – velocity relations and legs with masses have predictable qualitative effects, if any, on the optima. For bipeds with compliant tendons, the muscle work-minimizing strategies have close to zero muscle work (isometric muscles), with the springs performing all the leg work. These zero work gaits also minimize the generalized work costs C g with substantial additive force or force rate costs, indicating that a running animal's metabolic cost could be dominated by the cost of producing isometric force, even though performing muscle work is usually expensive. I also catalogue the many differences between the optimal gaits of the various models. These differences contain information that might help us develop models that better predict loco-motion data. In particular, for some biologically plausible cost functions, the presence or absence of springs in series with muscles has a large effect on both the coordination strategy and the absolute cost; the absence of springs results in more impulsive (collisional) optimal gaits and the presence of springs leads to more compliant optimal gaits. Most results are obtained for specific speed and stride length combinations close to preferred human behaviour , but limited numerical experiments show that some qualitative results extend to other speed-stride length combinations as well.

2011

The paper deals with the historical development of human body dynamics, and it presents results received for simple models by parameter optimization. The scientific research on human walking dynamics started already in the 19 th century and was later promoted by the physicist and physiologist Otto Fischer who published in 1906 his fundamental book. Fischer used the mechanism theory for modeling and analysis of human walking. His research was based on the barycenters of the corresponding reduced mechanisms. By the end of the 20 th century computational multibody dynamics provided more complex models which were applied to human body dynamics, too. More recently parameter optimization has been used to deal with the overactuation of biomechanical systems still a very active research topic in biomechanics. As an example a gait disorder simulation is presented showing that even today mechanism models, muscle group selection, inverse dynamics approaches and parameter optimization techniques using energy and aesthetics criteria are essential tools.

Mechanism and Machine Theory, 2009

The mechanical analysis of bipedal walking is a fundamental subject of research in biomechanics. Such analysis is useful to better understand the principles underlying human locomotion, as well as to improve the design and control of bipedal robotic prototypes. Modelling the dynamics of walking involves the analysis of its two phases of motion: (1) the single support phase, which represents finite motion; and (2) the impulsive motion of the impact that occurs at the end of each step (heel strike). The latter is an important event since it is the main cause of energy loss during motion and, in turn, it makes the topology of the system change. In this paper, we present a unified method to analyze the dynamics of both phases of walking. Emphasis is given to the heel strike event, for which we introduce a novel method that gives a complete decomposition of the dynamic equations and the kinetic energy of the system at topology change. As an application example, the presented approach is applied to a compass-gait biped with point feet. Based on this, the work includes a thorough analysis and discussions about the effect of the biped configuration and its inertial parameters on the dynamics and energetics of heel strike.

… and Automation (ICRA), …

The implementation of bipedal gaits in legged robots is still a challenge in state-of-the-art engineering. Human gaits could be realized by imitating human leg dynamics where a spring-like leg behavior is found as represented in the bipedal spring-mass model. In this study we explore the gap between walking and running by investigating periodic gait patterns. We found an almost continuous morphing of gait patterns between walking and running. The technical feasibility of this transition is, however, restricted by the duration of swing phase. In practice, this requires an abrupt gait transition between both gaits, while a change of speed is not necessary.

Bipedal locomotion is a phenomenon that still eludes a fundamental and concise mathematical understanding. Conceptual models that capture some relevant aspects of the process exist but their full explanatory power is not yet exhausted. In the current study, we introduce the robustness criterion which defines the conditions for stable 1 arXiv:1403.0879v1 [cs.RO] 4 Mar 2014 locomotion when steps are taken with imprecise angle of attack. Intuitively, the necessity of a higher precision indicates the difficulty to continue moving with a given gait. We show that the spring-loaded inverted pendulum model, under the robustness criterion, is consistent with previously reported findings on attentional demand during human locomotion. This criterion allows transitions between running and walking, many of which conserve forward speed. Simulations of transitions predict Froude numbers below the ones observed in humans, nevertheless the model satisfactorily reproduces several biomechanical indicators such as hip excursion, gait duty factor and vertical ground reaction force profiles. Furthermore, we identify reversible robust walk-run transitions, which allow the system to execute a robust version of the hopping gait. These findings foster the spring-loaded inverted pendulum model as the unifying framework for the understanding of bipedal locomotion.

1) IUGG, centre de recherches.

Journal of Engineering Science and Technology Review

In this article we present a mathematical model of the bipedal gait as well as the respective simulation of the joint movement that makes it possible. The data acquisition was carried out with the Tracker® software, and the dynamic simulation is presented in MATLAB® based on the obtained data. From this procedure, a perspective of the displacement generated by the lower train of the digitized human body is generated, which allows a study of the most concrete and successful movement for medical studies, since it allows the implementation of specialized designs and that meet specific characteristics in structures to reproduce the movement of the different joints.

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Handbook of Human Motion, 2018

Th e c o nv e n tio n al g ai t m o d els u c c e s s a n d li mit a tio n s

2001

A 3-dimensional computer model of sustainedbipedal walking is presented. It is intended be used as adevelopment tool for walking controllers. The directdynamic simulation has 8 segments, 19 degrees offreedom and is driven by prescribed joint moment andstiffness trajectories. Limited feedback in the form of aproportional-derivative controller provides upper bodystability and allows walking to be sustained indefinitely.The joint moment and stiffness trajectories are specifiedin coarse block segments. By changing the intensity of hipextensor activity during terminal stance the walking stridelength is modulated.

2004

This paper studies periodic gaits of multi-legged locomotion systems based on dynamic models. The purpose is to determine the system performance during walking and the best set of locomotion variables. For that objective the prescribed motion of the robot is completely characterized in terms of several locomotion variables such as gait, duty factor, body height, step length, stroke pitch, foot clearance, legs link lengths, foot-hip offset, body and legs mass and cycle time. In this perspective, we formulate three performance measures of the walking robot namely, the mean absolute energy, the mean power dispersion and the mean power lost in the joint actuators per walking distance. A set of model-based experiments reveals the influence of the locomotion variables in the proposed indices.