3LP: a linear 3D-walking model including torso and swing dynamics

In this paper, we present a new model of biped locomotion which is composed of three linear pendulums (one per leg and one for the whole upper body) to describe stance, swing and torso dynamics. In addition to double support, this model has different actuation possibilities in the swing hip and stance ankle which could be widely used to produce different walking gaits. Without the need for numerical time-integration, closed-form solutions help finding periodic gaits which could be simply scaled in certain dimensions to modulate the motion online. Thanks to linearity properties, the proposed model can provide a computationally fast platform for model predictive controllers to predict the future and consider meaningful inequality constraints to ensure feasibility of the motion. Such property is coming from describing dynamics with joint torques directly and therefore, reflecting hardware limitations more precisely, even in the very abstract high level template space. The proposed model produces human-like torque and ground reaction force profiles and thus, compared to point-mass models, it is more promising for precise control of humanoid robots. Despite being linear and lacking many other features of human walking like CoM excursion, knee flexion and ground clearance, we show that the proposed model can predict one of the main optimality trends in human walking, i.e. nonlinear speed-frequency relationship. In this paper, we mainly focus on describing the model and its capabilities, comparing it with human data and calculating optimal human gait variables. Setting up control problems and advanced biomechanical analysis still remain for future works.Comment: Journal paper under revie

How Crouch Gait Can Dynamically Induce Stiff-Knee Gait

Children with cerebral palsy frequently experience foot dragging and tripping during walking due to a lack of adequate knee flexion in swing (stiff-knee gait). Stiff-knee gait is often accompanied by an overly flexed knee during stance (crouch gait). Studies on stiff-knee gait have mostly focused on excessive knee muscle activity during (pre)swing, but the passive dynamics of the limbs may also have an important effect. To examine the effects of a crouched posture on swing knee flexion, we developed a forward-dynamic model of human walking with a passive swing knee, capable of stable cyclic walking for a range of stance knee crouch angles. As crouch angle during stance was increased, the knee naturally flexed much less during swing, resulting in a 'stiff-knee' gait pattern and reduced foot clearance. Reduced swing knee flexion was primarily due to altered gravitational moments around the joints during initial swing. We also considered the effects of increased push-off strength and swing hip flexion torque, which both increased swing knee flexion, but the effect of crouch angle was dominant. These findings demonstrate that decreased knee flexion during swing can occur purely as the dynamical result of crouch, rather than from altered muscle function or pathoneurological control alone. © 2010 The Author(s)

Passive dynamic walking with knees : a point foot model

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 57-59).In this thesis, a hybrid model for a passive 2D walker with knees and point feet is presented. The step cycle of the model has two phases of continuous dynamics: one with an unlocked knee configuration and a second one with a locked knee configuration. These stages are modeled as three-link and two-link pendulums correspondingly. The model switches between the two stages at knee-strike and heel-strike, which are both discrete events modeled as instantaneous inelastic collisions. The dynamics of this model were fully derived analytically. Furthermore, a stable gait was found given a set of physical parameters and initial conditions. A basic stability analysis of this stable limit cycle was performed around the fixed point of the Poincar6 return map examined right after heel-strike. This thesis also presents the design and construction of a planar robot based on this kneed walker model, as well as a careful examination of its correspondence to the motion predicted by the model simulation. The goal is to be able to study the nonlinear dynamics of simplified dynamic models which are also physically realizable, in order to build robots based on them in a more rigorous and reproducible manner. The work presented here aims to bridge the gap between existing theoretical models and successful experiments in passive dynamic walking.by Vanessa F. Hsu Chen.M.Eng

Master of Science

thesisThis thesis focuses on the design, modeling, fabrication, and testing of a ?ying and walking robot, called the Dynamic Underactuated Flying-Walking (DUCK) robot. The DUCK robot combines a high-mobility ?ying platform, such as a quadcopter (quadrotor helicopter), with passive-dynamic legs to create a versatile system that can ?y and walk. One of the advantages of using passive-dynamic legs for walking is that additional actuators are not needed for terrestrial locomotion, therefore simplifying the design, reducing overall weight, and decreasing power consumption. First, a mathematical model is developed for the DUCK robot, where the modeling combines the passive-dynamic walking mechanism with the swinging mass of the aerial platform. Second, simulations based on the model are used to help guide the design of two prototype robots, speci?cally to tailor the shape of the feet and the dimensions of the passive-dynamic walking mechanism. Third, an energy analysis is performed to compare the performances between ?ying and walking. More specifically, simulation results show that continuous active walking has a comparable energy efficiency to that of flying for the two prototype designs. For design Version 1, it is estimated that the robot is able to walk up to 1600 meters on a 30kJ battery (standard Li-Po battery) with a cost of transport of 1.0, while the robot can potentially fly up to 1800 meters horizontally with the weight of its legs and up to 2300 meters without the weight of its legs. Design Version 2 is estimated to be able to walk up to 4600 meters on a 30kJ battery with a cost of transport of .50, while it could fly up to 2600 meters with the weight of its legs or 4300 meters without its legs. The cost of transport of flying is estimated to be .89 in all scenarios. Finally, experimental results demonstrate the feasibility of combining an aerial platform with passive-dynamic legs to create an effective flying and walking robot. Two modes of walking are experimentally demonstrated: (1) passive walking down inclined surfaces for low-energy terrestrial locomotion and (2) active (powered) walking leveraging the capabilities of the flying platform, where thrust from the quadcopter's rotors enables the DUCK robot to walk on flat surfaces or up inclined surfaces

Dynamics of the Inertia Coupled Rimless Wheel with Frictional Losses and Actuation

The Inertia Coupled Rimless (ICR) wheel is a mechanically simple walking device capable of energy efficient motion. Typically, walking robots that are capable of level ground transport are extremely energy inefficient. To address this performance issue, the ICR wheel was examined while considering real-world frictional losses. The ICR wheel has been previously shown to be capable of collisionless, periodic motion, but until now, the ICR wheel had only been examined as an ideal, theoretical model. The inertia device within the system was tested to determine both the magnitude of energy loss due to damping and a suitable model for its motion. Fitting friction models to the experimental results showed that the a visciously damped model most accurately represented the system\u27s motion. Simulations revealed that the ICR wheel with friction would be capable of walking passively on a ramp with half stable, periodic walking, but the collisionless motion was lost. An actuation scheme was designed in simulation to allow an ICR wheel with damping to achieve collisionless motion on level ground. Experimental testing of a passive ICR wheel on a $3^o$ ramp showed that a cost of transport of at least 0.052 is possible with this system. Simulations suggest that, with the inclusion of an actuation scheme, the cost of transport for the same system on level ground could be as low as 0.024. Understanding how to overcome frictional losses lays the foundation for the creation of a walking robot capable of level ground transport with significantly less energy use than current models are capable of achieving

Dynamical Analysis Of A Passive Dynamic Walking Biped Robot

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2012Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2012Bu tezde, Uzaysal Operatör Cebri (UOC) kullanılarak pasif dinamik yürüme yapan bir robotun dinamik incelemesi yapılmıştır. Bu yöntem robotik bir sistemin eklem ve bağlantı elemanlarındaki kuvvet ve hız dağılımlarını vermekle birlikte, özyinemeli yapısı sayesinde hızlı hesaplama yapmaya olanak sağlamaktadır. Bir diğer yandan pasif dinamik yürüme ise, genellikle iki bacaklı olarak seçilen bir sistemin eğik bir düzlemde sadece yerçekimi kuvveti etksininde yürüme hareketini gerçekleştirdiği durumdur. İki bacaklı sistem yerçekimi etkisi altında tam bir yürüme hareketini tekrarlar. Bu fikir genellikle iki bacaklı robotların verimlerinin arttırılmasında kullanılmaktadır. Tezde tasarlanan beş serbestlik dereceli iki bacaklı sistem iki boyutlu uzayda tanımlanmıştır. İnsan bacağının doğası gereği sistem diz ve diz kitleme mekanizmaları ile modellenmiş ve bu mekanizmalar sözde eklem yöntemi yardımıyla oluşturulmuştur. Dizlerdeki ve ayaklardaki sınır kuvvetlerinin hesaplanması da bu çalışma dahilinde incelenmiştir. Sonuçlar pasif dinamik yürüme yapan bir robotun iç dinamiklerini göz önüne sermektedir.In this thesis using the Spatial Operator Algebra (SOA) the dynamic analysis of a passive dynamic walking biped robot is obtained. This method reveals the force and velocity distributions of links and joints of a robotic system, and also computationally efficient because of its recursive manner. On the other hand, passive dynamic walking is a case, where a mechanism, mostly biped, walks only by the gravitational forces on a shallow slope ground. Two legged system repeats full walking cycle under the influence of the gravitational force. This idea is mostly being used for efficiency problem of bipeds. In this thesis, a 5 degree of freedom biped robot is designed in 2D space. Due to the nature of human legs, the biped is modeled with knees and knee locking system using pseudo joint technique. The constraint forces due to the boundary conditions on knees and feet have also been computed as a result of this thesis. The results provided a deep insight of a passive dynamic walking biped robot.Yüksek LisansM.Sc

Energy Shaping of Mechanical Systems via Control Lyapunov Functions with Applications to Bipedal Locomotion

This dissertation presents a method which attempts to improve the stability properties of periodic orbits in hybrid dynamical systems by shaping the energy. By taking advantage of conservation of energy and the existence of invariant level sets of a conserved quantity of energy corresponding to periodic orbits, energy shaping drives a system to a desired level set. This energy shaping method is similar to existing methods but improves upon them by utilizing control Lyapunov functions, allowing for formal results on stability. The main theoretical result, Theorem 1, states that, given an exponentially-stable limit cycle in a hybrid dynamical system, application of the presented energy shaping controller results in a closed-loop system which is exponentially stable. The method can be applied to a wide class of problems including bipedal locomotion; because the optimization problem can be formulated as a quadratic program operating on a convex set, existing methods can be used to rapidly obtain the optimal solution. As illustrated through numerical simulations, this method turns out to be useful in practice, taking an existing behavior which corresponds to a periodic orbit of a hybrid system, such as steady state locomotion, and providing an improvement in convergence properties and robustness with respect to perturbations in initial conditions without destabilizing the behavior. The method is even shown to work on complex multi-domain hybrid systems; an example is provided of bipedal locomotion for a robot with non-trivial foot contact which results in a multi-phase gait

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

From bipedal locomotion to prosthetic walking: A hybrid system and nonlinear control approach

When modeled after the human form, humanoid robots more easily garner societal acceptance and gain increased dexterity in human environments. During this process of humanoid robot design, research on simulated bodies also yields a better understanding of the original biological system. Such advantages make humanoid robots ideal for use in areas such as elderly assistance, physical rehabilitation, assistive exoskeletons, and prosthetic devices. In these applications specifically, an understanding of human-like bipedal robotic locomotion is requisite for practical purposes. However, compared to mobile robots with wheels, humanoid walking robots are complex to design, difficult to balance, and hard to control, resulting in humanoid robots which walk slowly and unnaturally. Despite emerging research and technologies on humanoid robotic locomotion in recent decades, there still lacks a systematic method for obtaining truly kinematic and fluid walking. In this dissertation, we propose a formal optimization framework for achieving stable, human-like robotic walking with natural heel and toe behavior. Importantly, the mathematical construction allows us to directly realize natural walking on the custom-designed physical robot, AMBER2, resulting in a sustainable and robust multi-contact walking gait. As one of the ultimate goals of studying human-like robotic locomotion, the proposed systematic methodology is then translated to achieve prosthetic walking that is both human-like and energy-efficient, with reduced need for parameter tuning. We evaluate this method on two custom, powered transfemoral prostheses in both 2D (AMPRO1) and 3D (AMPRO3) cases. Finally, this dissertation concludes with future research opportunities.Ph.D