Leg stiffness increases with speed to modulate gait frequency and propulsion energy

Bipedal walking models with compliant legs have been employed to represent the ground reaction forces (GRFs) observed in human subjects. Quantification of the leg stiffness at varying gait speeds, therefore, would improve our understanding of the contributions of spring-like leg behavior to gait dynamics. In this study, we tuned a model of bipedal walking with damped compliant legs to match human GRFs at different gait speeds. Eight subjects walked at four different gait speeds, ranging from their self-selected speed to their maximum speed, in a random order. To examine the correlation between leg stiffness and the oscillatory behavior of the center of mass (CoM) during the single support phase, the damped natural frequency of the single compliant leg was compared with the duration of the single support phase. We observed that leg stiffness increased with speed and that the damping ratio was low and increased slightly with speed. The duration of the single support phase correlated well with the oscillation period of the damped complaint walking model, suggesting that CoM oscillations during single support may take advantage of resonance characteristics of the spring-like leg. The theoretical leg stiffness that maximizes the elastic energy stored in the compliant leg at the end of the single support phase is approximated by the empirical leg stiffness used to match model GRFs to human GRFs. This result implies that the CoM momentum change during the double support phase requires maximum forward propulsion and that an increase in leg stiffness with speed would beneficially increase the propulsion energy. Our results suggest that humans emulate, and may benefit from, spring-like leg mechanics.     

A bipedal compliant walking model generates periodic gait cycles with realistic swing dynamics

A simple spring mechanics model can capture the dynamics of the center of mass (CoM) during human walking, which is coordinated by multiple joints. This simple spring model, however, only describes the CoM during the stance phase, and the mechanics involved in the bipedality of the human gait are limited. In this study, a bipedal spring walking model was proposed to demonstrate the dynamics of bipedal walking, including swing dynamics followed by the step-to-step transition. The model consists of two springs with different stiffnesses and rest lengths representing the stance leg and swing leg. One end of each spring has a foot mass, and the other end is attached to the body mass. To induce a forward swing that matches the gait phase, a torsional hip joint spring was introduced at each leg. To reflect the active knee flexion for foot clearance, the rest length of the swing leg was set shorter than that of the stance leg, generating a discrete elastic restoring force. The number of model parameters was reduced by introducing dependencies among stiffness parameters. The proposed model generates periodic gaits with dynamics-driven step-to-step transitions and realistic swing dynamics. While preserving the mimicry of the CoM and ground reaction force (GRF) data at various gait speeds, the proposed model emulated the kinematics of the swing leg. This result implies that the dynamics of human walking generated by the actuations of multiple body segments is describable by a simple spring mechanics.     

Resonance-based oscillations could describe human gait mechanics under various loading conditions

The oscillatory behavior of the center of mass (CoM) and the corresponding ground reaction force (GRF) of human gait for various gait speeds can be accurately described in terms of resonance using a spring–mass bipedal model. Resonance is a mechanical phenomenon that reflects the maximum responsiveness and energetic efficiency of a system. To use resonance to describe human gait, we need to investigate whether resonant mechanics is a common property under multiple walking conditions. Body mass and leg stiffness are determinants of resonance; thus, in this study, we investigated the following questions: (1) whether the estimated leg stiffness increased with inertia, (2) whether a resonance-based CoM oscillation could be sustained during a change in the stiffness, and (3) whether these relationships were consistently observed for different walking speeds. Seven healthy young subjects participated in over-ground walking trials at three different gait speeds with and without a 25-kg backpack. We measured the GRFs and the joint kinematics using three force platforms and a motion capture system. The leg stiffness was incorporated using a stiffness parameter in a compliant bipedal model that best fitted the empirical GRF data. The results showed that the leg stiffness increased with the load such that the resonance-based oscillatory behavior of the CoM was maintained for a given gait speed. The results imply that the resonance-based oscillation of the CoM is a consistent gait property and that resonant mechanics may be useful for modeling human gait.     

Compliant bipedal model with the center of pressure excursion associated with oscillatory behavior of the center of mass reproduces the human gait dynamics

Although the compliant bipedal model could reproduce qualitative ground reaction force (GRF) of human walking, the model with a fixed pivot showed overestimations in stance leg rotation and the ratio of horizontal to vertical GRF. The human walking data showed a continuous forward progression of the center of pressure (CoP) during the stance phase and the suspension of the CoP near the forefoot before the onset of step transition. To better describe human gait dynamics with a minimal expense of model complexity, we proposed a compliant bipedal model with the accelerated pivot which associated the CoP excursion with the oscillatory behavior of the center of mass (CoM) with the existing simulation parameter and leg stiffness. Owing to the pivot acceleration defined to emulate human CoP profile, the arrival of the CoP at the limit of the stance foot over the single stance duration initiated the step-to-step transition. The proposed model showed an improved match of walking data. As the forward motion of CoM during single stance was partly accounted by forward pivot translation, the previously overestimated rotation of the stance leg was reduced and the corresponding horizontal GRF became closer to human data. The walking solutions of the model ranged over higher speed ranges (~1.7 m/s) than those of the fixed pivoted compliant bipedal model (~1.5 m/s) and exhibited other gait parameters, such as touchdown angle, step length and step frequency, comparable to the experimental observations. The good matches between the model and experimental GRF data imply that the continuous pivot acceleration associated with CoM oscillatory behavior could serve as a useful framework of bipedal model.     

Compliant leg behaviour explains basic dynamics of walking and running

The basic mechanics of human locomotion are associated with vaulting over stiff legs in walking and rebounding on compliant legs in running. However, while rebounding legs well explain the stance dynamics of running, stiff legs cannot reproduce that of walking. With a simple bipedal spring-mass model, we show that not stiff but compliant legs are essential to obtain the basic walking mechanics; incorporating the double support as an essential part of the walking motion, the model reproduces the characteristic stance dynamics that result in the observed small vertical oscillation of the body and the observed out-of-phase changes in forward kinetic and gravitational potential energies. Exploring the parameter space of this model, we further show that it not only combines the basic dynamics of walking and running in one mechanical system, but also reveals these gaits to be just two out of the many solutions to legged locomotion offered by compliant leg behaviour and accessed by energy or speed.     

Passive bipedal walking with phasic muscle contraction

The existence of self-organizing walking patterns is often considered the result of a mechanical system interacting with the environment and a (neural) oscillating unit. The pattern generators might be thought of as an indispensable component for the existence of limit cycle behavior. This paper shows that this is not a necessity for the existence of a self-organizing bipedal walking pattern. Stable walking cycles emerge from a simple passive bipedal structure, with an energy source inevitably present to sustain the oscillation. In this work the energy source is chosen to be phasic muscle contraction. A two-dimensional model is composed of two legs and a hip mass, symbolizing the trunk. The stance leg stiffness is generated by two muscles. The hip stiffness is generated by four muscles. Muscle activation is caused by two reflex-like trigger signals, without feedback control. Human equivalent model parameters such as geometry and mass distribution were assumed. With return map analysis, the model is analyzed on periodic behavior. Stable walking cycles were found and could be manipulated during walking by varying the muscle or reflex parameters, forcing the oscillation to converge to a new attractor. Received: 5 November 1998 / Accepted in revised form: 26 March 1999     

Stable walking with asymmetric legs

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 such as 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 a bipedal spring-mass model for simulating walking with compliant legs. With the help of the model, 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.     

Energy efficient walking with central pattern generators: from passive dynamic walking to biologically inspired control

Like human walking, passive dynamic walking—i.e. walking down a slope with no actuation except gravity—is energy efficient by exploiting the natural dynamics. In the animal world, neural oscillators termed central pattern generators (CPGs) provide the basic rhythm for muscular activity in locomotion. We present a CPG model, which automatically tunes into the resonance frequency of the passive dynamics of a bipedal walker, i.e. the CPG model exhibits resonance tuning behavior. Each leg is coupled to its own CPG, controlling the hip moment of force. Resonance tuning above the endogenous frequency of the CPG—i.e. the CPG’s eigenfrequency—is achieved by feedback of both limb angles to their corresponding CPG, while integration of the limb angles provides resonance tuning at and below the endogenous frequency of the CPG. Feedback of the angular velocity of both limbs to their corresponding CPG compensates for the time delay in the loop coupling each limb to its CPG. The resonance tuning behavior of the CPG model allows the gait velocity to be controlled by a single parameter, while retaining the energy efficiency of passive dynamic walking.     

Bipedal walking and running with spring-like biarticular muscles

Compliant elements in the leg musculoskeletal system appear to be important not only for running but also for walking in human locomotion as shown in the energetics and kinematics studies of spring-mass model. While the spring-mass model assumes a whole leg as a linear spring, it is still not clear how the compliant elements of muscle-tendon systems behave in a human-like segmented leg structure. This study presents a minimalistic model of compliant leg structure that exploits dynamics of biarticular tension springs. In the proposed bipedal model, each leg consists of three leg segments with passive knee and ankle joints that are constrained by four linear tension springs. We found that biarticular arrangements of the springs that correspond to rectus femoris, biceps femoris and gastrocnemius in human legs provide self-stabilizing characteristics for both walking and running gaits. Through the experiments in simulation and a real-world robotic platform, we show how behavioral characteristics of the proposed model agree with basic patterns of human locomotion including joint kinematics and ground reaction force, which could not be explained in the previous models.     

Effects of hip torque during step-to-step transition on center-of-mass dynamics during human walking examined with numerical simulation

Besides the leg force actuator, humans also use a hip torque actuator during the step-to-step transition to redirect the velocity of CoM (Center of Mass). Although the leg force actuator has been widely studied, few researches analyze the hip torque actuator during the step-to-step transition. In this paper, we build a powered walking model which consists of a point mass linked with two compliant legs. Each leg has a spring and a damper in parallel. Two types of active actuators, the force actuator on the leg and the torque actuator at the hip, are added to simulate the leg force and hip torque actuator during the step-to-step transition. The cycle walk is solved by numerical simulations under different hip torque strength, and the energetics and stability are evaluated. The simulation results show that the hip torque actuator can reduce the energy cost and improve the stability of walking. Further analysis shows that the hip torque actuator can reduce mechanical works of both legs with small extra energy cost. To understand the principle of hip torque actuator, the CoM dynamics is analyzed. It is shown that the hip torque actuator is efficient on the redirection of CoM. Thus, it can improve the stability and reduce required forces of both legs, which decreases the energy cost. Our work provides a fundamental understanding of the hip torque during the step-to-step transition, and may help improve the design of bipedal robots and prosthesis.     

Stance leg control: variation of leg parameters supports stable hopping

The spring-loaded inverted pendulum describes the planar center-of-mass dynamics of legged locomotion. This model features linear springs with constant parameters as legs. In biological systems, however, spring-like properties of limbs can change over time. Therefore, in this study, it is asked how variation of spring parameters during ground contact would affect the dynamics of the spring-mass model. Neglecting damping initially, it is found that decreasing stiffness and increasing rest length of the leg during a stance phase are required for orbitally stable hopping. With damping, stable hopping is found for a larger region of rest-length rates and stiffness rates. Here, also increasing stiffness and decreasing rest length can result in stable hopping. Within the predicted range of leg parameter variations for stable hopping, there is no need for precise parameter tuning. Since hopping gaits form a subset of the running gaits (with vanishing horizontal velocity), these results may help to improve leg design in robots and prostheses.     

Does a crouched leg posture enhance running stability and robustness?

Humans and birds both walk and run bipedally on compliant legs. However, differences in leg architecture may result in species-specific leg control strategies as indicated by the observed gait patterns. In this work, control strategies for stable running are derived based on a conceptual model and compared with experimental data on running humans and pheasants (Phasianus colchicus). From a model perspective, running with compliant legs can be represented by the planar spring mass model and stabilized by applying swing leg control. Here, linear adaptations of the three leg parameters, leg angle, leg length and leg stiffness during late swing phase are assumed. Experimentally observed kinematic control parameters (leg rotation and leg length change) of human and avian running are compared, and interpreted within the context of this model, with specific focus on stability and robustness characteristics. The results suggest differences in stability characteristics and applied control strategies of human and avian running, which may relate to differences in leg posture (straight leg posture in humans, and crouched leg posture in birds). It has been suggested that crouched leg postures may improve stability. However, as the system of control strategies is overdetermined, our model findings suggest that a crouched leg posture does not necessarily enhance running stability. The model also predicts different leg stiffness adaptation rates for human and avian running, and suggests that a crouched avian leg posture, which is capable of both leg shortening and lengthening, allows for stable running without adjusting leg stiffness. In contrast, in straight-legged human running, the preparation of the ground contact seems to be more critical, requiring leg stiffness adjustment to remain stable. Finally, analysis of a simple robustness measure, the normalized maximum drop, suggests that the crouched leg posture may provide greater robustness to changes in terrain height.     

Dynamic Stability of Passive Bipedal Walking on Rough Terrain:A Preliminary Simulation Study

A simplified 2D passive dynamic model was simulated to walk down on a rough slope surface defined by deterministic profiles to investigate how the walking stability changes with increasing surface roughness.Our results show that the passive walker can walk on rough surfaces subject to surface roughness up to approximately 0.1％ of its leg length.This indicates that bipedal walkers based on passive dynamics may possess some intrinsic stability to adapt to rough terrains although the maximum roughness they can tolerate is small.Orbital stability method was used to quantify the walking stability before the walker started to fall over.It was found that the average maximum Floquet multiplier increases with surface roughness in a non-linear form.Although the passive walker remained orbitally stable for all the simulation cases,the results suggest that the possibility of the bipedal model moving away from its limit cycle increases with the surface roughness if subjected to additional perturbations.The number of consecutive steps before falling was used to measure the walking stability after the passive walker started to fall over.The results show that the number of steps before falling decreases exponentially with the increase in surface roughness.When the roughness magnitude approached to 0.73％ of the walker's leg length,it fell down to the ground as soon as it entered into the uneven terrain.It was also found that shifting the phase angle of the surface profile has apparent affect on the system stability.This is probably because point contact was used to simulate the heel strikes and the resulted variations in system states at heel strikes may have pronounced impact on the passive gaits,which have narrow basins of attraction.These results would provide insight into how the dynamic stability of passive bipedal walkers evolves with increasing surface roughness.     

Effect of Flexible Spine Motion on Energy Efficiency in Quadruped Running

Energy efficiency is important in the performance of quadruped robots and mammals.Flexible spine motion generally exists in quadruped mammals.This paper mainly explores the effect of flexible spinal motion on energy efficiency.Firstly,a planar simplified model of the quadruped robot with flexible spine motion is introduced and two simulation experiments are carried out.The results of simulation experiments demonstrate that both spine motion and spinal flexibility can indeed increase energy efficiency,and the curve of energy efficiency change along with spinal stiffness is acquired.So,in order to obtain higher energy efficiency,quadruped robots should have flexible spine motion.In a certain speed,there is an optimal spinal stiffness which can make energy efficiency to be the best.Secondly,a planar quadruped robot with flexible spine motion is designed and the conclusions drawn in the two simulation experiments are verified.Lastly,the third simulation experiment is carried out to explore the relationship between the optimal spinal stiffness,speed and total mass.The optimal spinal stiffness increases with both speed and total mass,which has important guiding significance for adjusting the spinal stiffness of quadruped robots to make them reach the best energy efficiency.     

Evaluating functional roles of phase resetting in generation of adaptive human bipedal walking with a physiologically based model of the spinal pattern generator

The central pattern generators (CPGs) in the spinal cord strongly contribute to locomotor behavior. To achieve adaptive locomotion, locomotor rhythm generated by the CPGs is suggested to be functionally modulated by phase resetting based on sensory afferent or perturbations. Although phase resetting has been investigated during fictive locomotion in cats, its functional roles in actual locomotion have not been clarified. Recently, simulation studies have been conducted to examine the roles of phase resetting during human bipedal walking, assuming that locomotion is generated based on prescribed kinematics and feedback control. However, such kinematically based modeling cannot be used to fully elucidate the mechanisms of adaptation. In this article we proposed a more physiologically based mathematical model of the neural system for locomotion and investigated the functional roles of phase resetting. We constructed a locomotor CPG model based on a two-layered hierarchical network model of the rhythm generator (RG) and pattern formation (PF) networks. The RG model produces rhythm information using phase oscillators and regulates it by phase resetting based on foot-contact information. The PF model creates feedforward command signals based on rhythm information, which consists of the combination of five rectangular pulses based on previous analyses of muscle synergy. Simulation results showed that our model establishes adaptive walking against perturbing forces and variations in the environment, with phase resetting playing important roles in increasing the robustness of responses, suggesting that this mechanism of regulation may contribute to the generation of adaptive human bipedal locomotion.     

Do highly trained monkeys walk like humans? A kinematic study of bipedal locomotion in bipedally trained Japanese macaques

In this study, we examined the kinematics of bipedal walking in macaque monkeys that have been highly trained to stand and walk bipedally, and compared them to the kinematics of bipedal walking in ordinary macaques. The results revealed that the trained macaques walked with longer and less frequent strides than ordinary subjects. In addition, they appear to have used inverted pendulum mechanics during bipedal walking, which resulted in an efficient exchange of potential and kinetic energy. These gait characteristics resulted from the relatively more extended hindlimb joints of the trained macaques. By contrast, the body of the ordinary macaques translated downward during the single-limb stance phase due to more flexed hindlimb joints. This resulted in almost in-phase fluctuations of potential and kinetic energy, which indicated that energy transformation was less efficient in the ordinary macaques. The findings provide two insights into the early stage of the evolution of human bipedalism. First, the finding that training considerably improved bipedal walking a posteriori may explain why the very first bipeds that might not yet have been morphologically adapted to bipedal walking continued to walk bipedally. The evolutionary transition from quadrupedalism to bipedalism might not be as difficult as has been envisioned. In addition, the finding that macaques, which are phylogenetically distant from humans and in which bipedal walking is unlike human walking, could develop humanlike gait characteristics with training, provides strong support for the commonly held but unproven idea that the characteristics of the human gait are advantageous to human bipedalism.     

Theories of bipedal walking: an odyssey

In this paper six theories of bipedal walking, and the evidence in support of the theories, are reviewed. They include: evolution, minimising energy consumption, maturation in children, central pattern generators, linking control and effect, and robots on two legs. Specifically, the six theories posit that: (1) bipedalism is the fundamental evolutionary adaptation that sets hominids--and therefore humans--apart from other primates; (2) locomotion is the translation of the centre of gravity along a pathway requiring the least expenditure of energy; (3) when a young child takes its first few halting steps, his or her biomechanical strategy is to minimise the risk of falling; (4) a dedicated network of interneurons in the spinal cord generates the rhythm and cyclic pattern of electromyographic signals that give rise to bipedal gait; (5) bipedal locomotion is generated through global entrainment of the neural system on the one hand, and the musculoskeletal system plus environment on the other; and (6) powered dynamic gait in a bipedal robot can be realised only through a strategy which is based on stability and real-time feedback control. The published record suggests that each of the theories has some measure of support. However, it is important to note that there are other important theories of locomotion which have not been covered in this review. Despite such omissions, this odyssey has explored the wide spectrum of bipedal walking, from its origins through to the integration of the nervous, muscular and skeletal systems.     

Biomechanical analysis of primate bipedal walking by computer simulation

A computer simulation technique was applied to make clear the mechanical characteristics of primate bipedal walking. A primate body and the walking mechanism were modeled mathematically with a set of dynamic equations. Using a digital computer, the following were calculated from these equations by substituting measured displacements and morphological data of each segment of the primate: the acceleration, joint angle, center of gravity, foot force, joint moment, muscular force, transmitted force at the joint, electric activity of the muscle, generated power by the leg and energy expenditure in walking.The model was evaluated by comparing some of the calculated results with the experimental results such as foot force and electromyographic data, and improved in order to obtain the agreement between them.The level bipedal walking of man, chimpanzee and Japanese monkey and several types of synthesized walking were analyzed from the viewpoint of biomechanics.It is concluded that the bipedal walking of chimpanzee is nearer to that of man than to that of the Japanese monkey because of its propulsive mechanism, but it requires large muscular force for supporting the body weight.     

Forward dynamic simulation of bipedal walking in the Japanese macaque: investigation of causal relationships among limb kinematics, speed, and energetics of bipedal locomotion in a nonhuman primate

Japanese macaques that have been trained for monkey performances exhibit a remarkable ability to walk bipedally. In this study, we dynamically reconstructed bipedal walking of the Japanese macaque to investigate causal relationships among limb kinematics, speed, and energetics, with a view to understanding the mechanisms underlying the evolution of human bipedalism. We constructed a two-dimensional macaque musculoskeletal model consisting of nine rigid links and eight principal muscles. To generate locomotion, we used a trajectory-tracking control law, the reference trajectories of which were obtained experimentally. Using this framework, we evaluated the effects of changes in cycle duration and gait kinematics on locomotor efficiency. The energetic cost of locomotion was estimated based on the calculation of mechanical energy generated by muscles. Our results demonstrated that the mass-specific metabolic cost of transport decreased as speed increased in bipedal walking of the Japanese macaque. Furthermore, the cost of transport in bipedal walking was reduced when vertical displacement of the hip joint was virtually modified in the simulation to be more humanlike. Human vertical fluctuations in the body's center of mass actually contributed to energy savings via an inverted pendulum mechanism.