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.     

Leg stiffness adjustment for a range of hopping frequencies in humans

The purpose of the present study was to determine how humans adjust leg stiffness over a range of hopping frequencies. Ten male subjects performed in place hopping on two legs, at three frequencies (1.5, 2.2, and 3.0 Hz). Leg stiffness, joint stiffness and touchdown joint angles were calculated from kinetic and/or kinematics data. Electromyographic activity (EMG) was recorded from six leg muscles. Leg stiffness increased with an increase in hopping frequency. Hip and knee stiffnesses were significantly greater at 3.0 Hz than at 1.5 Hz. There was no significant difference in ankle stiffness among the three hopping frequencies. Although there were significant differences in EMG activity among the three hopping frequencies, the largest was the 1.5 Hz, followed by the 2.2 Hz and then 3.0 Hz. The subjects landed with a straighter leg (both hip and knee were extended more) with increased hopping frequency. These results suggest that over the range of hopping frequencies we evaluated, humans adjust leg stiffness by altering hip and knee stiffness. This is accomplished by extending the touchdown joint angles rather than by altering neural activity.     

Knee stiffness is a major determinant of leg stiffness during maximal hopping

Understanding stiffness of the lower extremities during human movement may provide important information for developing more effective training methods during sports activities. It has been reported that leg stiffness during submaximal hopping depends primarily on ankle stiffness, but the way stiffness is regulated in maximal hopping is unknown. The goal of this study was to examine the hypothesis that knee stiffness is a major determinant of leg stiffness during the maximal hopping. Ten well-trained male athletes performed two-legged hopping in place with a maximal effort. We determined leg and joint stiffness of the hip, knee, and ankle from kinetic and kinematic data. Knee stiffness was significantly higher than ankle and hip stiffness. Further, the regression model revealed that only knee stiffness was significantly correlated with leg stiffness. The results of the present study suggest that the knee stiffness, rather than those of the ankle or hip, is the major determinant of leg stiffness during maximal hopping.     

Influence of ligament stiffness on the mechanical behavior of a functional spinal unit

Data on the stiffnesses of spinal ligaments are required for analytical studies on the mechanical behavior of spinal segments. Values obtained experimentally vary widely in the literature. A finite element model of an L3/L4 functional spinal unit was used to determine the influence of ligament stiffness on intersegmental rotation and forces in the ligaments. The lowest values for ligament stiffness selected from the literature were used in one set of calculations, and the highest values were simulated in a second set. The nonlinear model was loaded with pure moments of 7.5 and 15 Nm in the three main anatomical planes. The mechanical behavior of the functional spinal unit was strongly influenced by ligament stiffness. In some cases, a ligament with low stiffness does not carry any load, while the same ligament with high stiffness has to carry a high load. This indicates that finite element models of spinal segments have to be validated and that a realistic quantitative prediction of ligament forces is extremely difficult.     

Resonant hopping of a robot controlled by an artificial neural oscillator

The bouncing gaits of terrestrial animals (hopping, running, trotting) can be modeled as a hybrid dynamic system, with spring-mass dynamics during stance and ballistic motion during the aerial phase. We used a simple hopping robot controlled by an artificial neural oscillator to test the ability of the neural oscillator to adaptively drive this hybrid dynamic system. The robot had a single joint, actuated by an artificial pneumatic muscle in series with a tendon spring. We examined how the oscillator-robot system responded to variation in two neural control parameters: descending neural drive and neuromuscular gain. We also tested the ability of the oscillator-robot system to adapt to variations in mechanical properties by changing the series and parallel spring stiffnesses. Across a 100-fold variation in both supraspinal gain and muscle gain, hopping frequency changed by less than 10%. The neural oscillator consistently drove the system at the resonant half-period for the stance phase, and adapted to a new resonant half-period when the muscle series and parallel stiffnesses were altered. Passive cycling of elastic energy in the tendon accounted for 70-79% of the mechanical work done during each hop cycle. Our results demonstrate that hopping dynamics were largely determined by the intrinsic properties of the mechanical system, not the specific choice of neural oscillator parameters. The findings provide the first evidence that an artificial neural oscillator will drive a hybrid dynamic system at partial resonance.     

Leg stiffness primarily depends on ankle stiffness during human hopping

When humans hop in place or run forward, they adjust leg stiffness to accommodate changes in stride frequency or surface stiffness. The goal of the present study was to determine the mechanisms by which humans adjust leg stiffness during hopping in place. Five subjects hopped in place at 2.2 Hz while we collected force platform and kinematic data. Each subject completed trials in which they hopped to whatever height they chose ("preferred height hopping") and trials in which they hopped as high as possible ("maximum height hopping"). Leg stiffness was approximately twice as great for maximum height hopping as for preferred height hopping. Ankle torsional stiffness was 1.9-times greater while knee torsional stiffness was 1.7-times greater in maximum height hopping than in preferred height hopping. We used a computer simulation to examine the sensitivity of leg stiffness to the observed changes in ankle and knee stiffness. Our model consisted of four segments (foot, shank, thigh, head-arms-trunk) interconnected by three torsional springs (ankle, knee, hip). In the model, increasing ankle stiffness by 1.9-fold, as observed in the subjects, caused leg stiffness to increase by 2.0-fold. Increasing knee stiffness by 1.7-fold had virtually no effect on leg stiffness. Thus, we conclude that the primary mechanism for leg stiffness adjustment is the adjustment of ankle stiffness.     

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.     

Postural differences in shoulder dynamics during pushing and pulling

Assessments of shoulder dynamics (e.g. the inertial, viscous, and stiffness properties of the joint) can provide important insights into the stability of the joint at rest and during volitional contraction. The purpose of this study was to investigate how arm posture influences shoulder dynamics while generating pushing or pulling torques in the horizontal plane. Sixteen healthy participants were examined in seven postures encompassing a large workspace of the shoulder. At each posture, the participant’s shoulder was rapidly perturbed while measuring the resultant change in shoulder torque about the glenohumeral axis. Participants were examined both at rest and while producing horizontal flexion and extension torques scaled to 15% of a maximum voluntary contraction. Shoulder stiffness, viscosity, and damping ratio were estimated using impedance-based matching, and changes in these outcome measures with torque level, elevation angle, and plane of elevation angle were explored with a linear mixed effects model. Shoulder stiffness was found to decrease with increasing elevation angles (p < 0.001) without subsequent changes in viscosity, leading to a greater damping ratios at higher elevation angles (p < 0.001). Shoulder stiffness, viscosity, and damping ratio (all p < 0.05) were all found to significantly increase as the plane of elevation of the arm was increased. The relationship between the viscosity, stiffness and the damping ratio of the shoulder is one that the central nervous system must regulate in order to maintain stability, protect against injury, and control the shoulder joint as the inertial and muscle contributions change across different arm postures.     

Neuromechanical adaptation to hopping with an elastic ankle-foot orthosis.

When humans hop or run on different surfaces, they adjust their effective leg stiffness to offset changes in surface stiffness. As a result, the overall stiffness of the leg-surface series combination remains independent of surface stiffness. The purpose of this study was to determine whether humans make a similar adjustment when springs are placed in parallel with the leg via a lower limb orthosis. We studied seven human subjects hopping in place on one leg while wearing an ankle-foot orthosis. We used an ankle-foot orthosis because the ankle joint is primarily responsible for leg stiffness during hopping. A spring was added to the ankle-foot orthosis so that it increased orthosis stiffness by providing plantar flexor torque during ankle dorsiflexion. We hypothesized that subjects would decrease their biological ankle stiffness when the spring was added to the orthosis, keeping total ankle stiffness constant. We collected kinematic, kinetic, and electromyographic data during hopping with and without the spring on the orthosis. We found that total ankle stiffness and leg stiffness did not change across the two orthosis conditions (ANOVA, P > 0.05). This was possible because subjects decreased their biological ankle stiffness to offset the orthosis spring stiffness (P < 0.0001). The reduction in biological ankle stiffness was accompanied by decreases in soleus, medial gastrocnemius, and lateral gastrocnemius muscle activation (P < 0.0002). These results suggest that an elastic exoskeleton might improve human running performance by reducing muscle recruitment.     

Effect of flexural and torsional wing flexibility on lift generation in hoverfly flight

The effect of wing flexibility in hoverflies was investigated using an at-scale mechanical model. Unlike dynamically-scaled models, an at-scale model can include all phenomena related to motion and deformation of the wing during flapping. For this purpose, an at-scale polymer wing mimicking a hoverfly was fabricated using a custom micromolding process. The wing has venation and corrugation profiles which mimic those of a hoverfly wing and the measured flexural stiffness of the artificial wing is comparable to that of the natural wing. To emulate the torsional flexibility at the wing-body joint, a discrete flexure hinge was created. A range of flexure stiffnesses was chosen to match the torsional stiffness of pronation and supination in a hoverfly wing. The polymer wing was compared with a rigid, flat, carbon-fiber wing using a flapping mechanism driven by a piezoelectric actuator. Both wings exhibited passive rotation around the wing hinge; however, these rotations were reduced in the case of the compliant polymer wing due to chordwise deformations during flapping which caused a reduced effective angle of attack. Maximum lift was achieved when the stiffness of the hinge was similar to that of a hoverfly in both wing cases and the magnitude of measured lift is sufficient for hovering; the maximum lift achieved by the single polymer and carbon-fiber wings was 5.9?×?10(2)(?)μN and 6.9?×?10(2)(?)μN, respectively. These results suggest that hoverflies could exploit intrinsic compliances to generate desired motions of the wing and that, for the same flapping motions, a rigid wing could be more suitable for producing large lift.     

Linear center-of-mass dynamics emerge from non-linear leg-spring properties in human hopping

Given the almost linear relationship between ground-reaction force and leg length, bouncy gaits are commonly described using spring–mass models with constant leg-spring parameters. In biological systems, however, spring-like properties of limbs may change over time. Therefore, it was investigated how much variation of leg-spring parameters is present during vertical human hopping. In order to do so, rest-length and stiffness profiles were estimated from ground-reaction forces and center-of-mass dynamics measured in human hopping. Trials included five hopping frequencies ranging from 1.2 to 3.6 Hz. Results show that, even though stiffness and rest length vary during stance, for most frequencies the center-of-mass dynamics still resemble those of a linear spring–mass hopper. Rest-length and stiffness profiles differ for slow and fast hopping. Furthermore, at 1.2 Hz two distinct control schemes were observed.     

Determinants of difference in leg stiffness between endurance- and power-trained athletes

Understanding the leg and joint stiffness during human movement would provide important information that could be utilized for evaluating sports performance and for injury prevention. In the present study, we examined the determinants of the difference in the leg stiffness between the endurance-trained and power-trained athletes. Seven distance runners and seven power-trained athletes performed in-place hopping, matching metronome beats at 3.0 and 1.5Hz. Leg and joint stiffness were calculated from kinetic and kinematics data. Electromyographic activity (EMG) was recorded from six leg muscles. At both hopping frequencies, the power-trained athletes demonstrated significantly higher leg stiffness than the distance runners. Hip, knee, and ankle joints were analyzed for stiffness and touchdown angles. Ankle stiffness was significantly greater in the power-trained athletes than the distance runners at 3.0Hz as was knee stiffness at 1.5Hz. There was no significant difference in touchdown angle between the DR and PT groups at either hopping frequencies. When significant difference in EMG activity existed between two groups, it was always greater in the distance runners than the power-trained athletes. These results suggest that (1) the difference in leg stiffness between endurance-trained and power-trained athletes is best attributed to increased joint stiffness, and (2) the difference in joint stiffness between the two groups may be attributed to a lack of similarity in the intrinsic stiffness of the muscle-tendon complex rather than in altered neural activity.     

Gender differences in active musculoskeletal stiffness. Part II. Quantification of leg stiffness during functional hopping tasks.

Leg stiffness was compared between age-matched males and females during hopping at preferred and controlled frequencies. Stiffness was defined as the linear regression slope between the vertical center of mass (COM) displacement and ground-reaction forces recorded from a force plate during the stance phase of the hopping task. Results demonstrate that subjects modulated the vertical displacement of the COM during ground contact in relation to the square of hopping frequency. This supports the accuracy of the spring-mass oscillator as a representative model of hopping. It also maintained peak vertical ground-reaction load at approximately three times body weight. Leg stiffness values in males (33.9+/-8.7 kN/m) were significantly (p<0.01) greater than in females (26.3+/-6.5 kN/m) at each of three hopping frequencies, 3.0, 2.5 Hz, and a preferred hopping rate. In the spring-mass oscillator model leg stiffness and body mass are related to the frequency of motion. Thus male subjects necessarily recruited greater leg stiffness to drive their heavier body mass at the same frequency as the lighter female subjects during the controlled frequency trials. However, in the preferred hopping condition the stiffness was not constrained by the task because frequency was self-selected. Nonetheless, both male and female subjects hopped at statistically similar preferred frequencies (2.34+/-0.22 Hz), therefore, the females continued to demonstrate less leg stiffness. Recognizing the active muscle stiffness contributes to biomechanical stability as well as leg stiffness, these results may provide insight into the gender bias in risk of musculoskeletal knee injury.     

Evaluation of a cruciate ligament model: sensitivity to the parameters during drawer test simulation

The knowledge of how cruciate ligaments stabilize the knee joint could be very useful during the execution of daily living activities for the development of clinical procedures. The objective of this study was to evaluate a cruciate ligament model that could achieve this knowledge while avoiding any destructive measurements in living healthy subjects. Subject-specific geometries and kinematic data, acquired from a living subject, were the foundations of the devised model. Each cruciate ligament was modeled with 25 linear-elastic elements and their geometrical properties were subject specific. The anteroposterior drawer test was simulated, and the sensitivity to the reference length and the elastic modulus was performed. Laxity, anterior, and posterior stiffness were calculated and compared with the literature. The laxity was most sensitive to reference length but fitted the literature well considering the reference length estimated from the subject. Both stiffnesses were most sensitive to elastic modulus variations. At full extension, anterior stiffness overestimated the literature, but at 90 degrees good comparisons with the literature were obtained. Posterior stiffness showed smaller overestimations. The devised model, when properly improved, could evaluate the role of the cruciate ligaments of a living subject during the execution of daily living activities.     

Identification and experimental validation of damping ratios of different human body segments through anthropometric vibratory model in standing posture

A 15 degrees of freedom lumped parameter vibratory model of human body is developed, for vertical mode vibrations, using anthropometric data of the 50th percentile US male. The mass and stiffness of various segments are determined from the elastic modulii of bones and tissues and from the anthropometric data available, assuming the shape of all the segments is ellipsoidal. The damping ratio of each segment is estimated on the basis of the physical structure of the body in a particular posture. Damping constants of various segments are calculated from these damping ratios. The human body is modeled as a linear spring-mass-damper system. The optimal values of the damping ratios of the body segments are estimated, for the 15 degrees of freedom model of the 50th percentile US male, by comparing the response of the model with the experimental response. Formulating a similar vibratory model of the 50th percentile Indian male and comparing the frequency response of the model with the experimental response of the same group of subjects validate the modeling procedure. A range of damping ratios has been considered to develop a vibratory model, which can predict the vertical harmonic response of the human body.     

Dynamic and static control of the human knee joint in abduction-adduction

It is unclear whether humans can voluntarily control dynamic and static properties in knee abduction-adduction, which may be important in performing functional tasks and preventing injuries, whether the main load is about the abduction axis or not. A joint-driving device was used to perturb the knee in abduction-adduction at full knee extension under both passive (muscle relaxed) and active (muscle contracted in abduction or adduction) conditions. Dynamic control properties in knee abduction-adduction were characterized by joint stiffness, viscosity, and limb inertia, and quasi-static knee torque-angle relationship was characterized by knee abduction-adduction laxity and quasi-static stiffness (at a 20Nm moment). It was found that the subjects were capable of generating net abduction and adduction moment through differential co-contraction of muscles crossing the medial and lateral sides of the knee, which helped to reduce the abduction-adduction joint laxity (p< or =0.01) and increase stiffness (p<0.027) and viscous damping. Knee abduction laxity was significantly lower than adduction laxity (p=0.043) and the quasi-static abduction stiffness was significantly higher than adduction stiffness (p<0.001). The knee joint showed significantly higher stiffness and viscosity in abduction-adduction than their counterparts in knee flexion-extension at comparable levels of joint torque (p<0.05). Similar to dynamic flexion-extension properties, the system damping ratio remained constant over different levels of contraction, indicating simplified control tasks for the central nervous system; while the natural undamped frequency increased considerably with abduction-adduction muscle contraction, presumably making the knee a quicker system during strenuous tasks involving strong muscle contraction.     

Modeling of control and learning in a stepping motion

In a previous study (Beuter et al. 1986) the authors modeled a stepping motion using a three-body linkage with four degrees of freedom. Stepping was simulated by using three task parameters (i.e., step height, length, and duration) and sinusoidal joint angular velocity profiles. The results supported the concept of a hierarchical control structure with open-loop control during normal operation. In this study we refine the dynamic model and improve the simulation technique by incorporating the dynamics of the leg after landing, adding a foot segment to the model, and preprogramming the complete step motion using cycloids. The equations of the forces and torques developed on the ground by the foot during the landing phase are derived using the Lagrangian method. Simulation results are compared to experimental data collected on a subject stepping four times over an obstacle using a Selspot motion analysis system. A hierarchical control model that incorporates a learning process is proposed. The model allows an efficient combination of open and closed loop control strategies and involves hardwired movement segments. We also test the hypothesis of cycloidal velocity profiles in the joint programs against experimental data using a novel curve-fitting procedure based on analytical rather than numerical differentiation. The results suggest multiob-jective optimization of the joint's motion. The control and learning model proposed here will help the understanding of the mechanisms responsible for assembling selected movement segments into goaldirected movement sequences in humans.     

Identification of a frequency response model of joint rotation

A second order, linear oscillator transfer function model is fit to the measured transfer function relating the abduction-adduction rotation of the first finger to the applied moment. Nearly constant isometric contractions of the first palmar and dorsal interossei are maintained by the subjects during the measurements. The stiffness and damping components of the identified models increase significantly with increasing isometric contraction when compared to those recorded under relaxed contraction. Muscle fatigue causes the natural frequency, damping ratio and stiffness of the joint rotation to decrease under full isometric contraction, and it causes the natural frequency and stiffness to increase when the muscles are relaxed.     

Mechanism of leg stiffness adjustment for hopping on surfaces of different stiffnesses

When humans hopin place or run forward, leg stiffness is increased to offsetreductions in surface stiffness, allowing the global kinematics andmechanics to remain the same on all surfaces. The purpose of thepresent study was to determine the mechanism for adjusting legstiffness. Seven subjects hopped in place on surfaces of differentstiffnesses (23-35,000 kN/m) while force platform, kinematic, andelectromyographic data were collected. Leg stiffness approximatelydoubled between the most stiff surface and the least stiff surface.Over the same range of surfaces, ankle torsional stiffness increased1.75-fold, and the knee became more extended at the time of touchdown(2.81 vs. 2.65 rad). We used a computer simulation to examine thesensitivity of leg stiffness to the observed changes in ankle stiffnessand touchdown knee angle. Our model consisted of four segments (foot,shank, thigh, head-arms-trunk) interconnected by three torsionalsprings (ankle, knee, hip). In the model, an increase in anklestiffness 1.75-fold caused leg stiffness to increase 1.7-fold. A changein touchdown knee angle as observed in the subjects caused legstiffness to increase 1.3-fold. Thus both joint stiffness and limbgeometry adjustments are important in adjusting leg stiffness to allow similar hopping on different surfaces.

     

Vertical stiffness during one-legged hopping with and without using a running-specific prosthesis

Although athletes with unilateral below-the-knee amputations (BKAs) generally use their affected leg, including their prosthesis, as their take-off leg for the long jump, little is known about the spring-like leg behavior and stiffness regulation of the affected leg. The purpose of this study was to investigate vertical stiffness during one-legged hopping in an elite-level long jump athlete with a unilateral BKA. We used the spring-mass model to calculate vertical stiffness, which equals the ratio of maximum vertical ground reaction force to maximum center of mass displacement, while the athlete with a BKA hopped on one leg at a range of frequencies. Then, we compared the vertical stiffness of this athlete to seven non-amputee elite-level long-jumpers. We found that from 1.8 to 3.4 Hz, the vertical stiffness of the unaffected leg for an athlete with a BKA increases with faster hopping frequencies, but the vertical stiffness of the affected leg remains nearly constant across frequencies. The athlete with a BKA attained the desired hopping frequencies at 2.2 and 2.6 Hz, but was unable to match the lowest (1.8 Hz) and two highest frequencies (3.0 and 3.4 Hz) using his affected leg. We also found that at 2.5 Hz, unaffected leg vertical stiffness was 15% greater than affected leg vertical stiffness, and the vertical stiffness of non-amputee long-jumpers was 32% greater than the affected leg vertical stiffness of an athlete with a BKA. The results of the present study suggest that the vertical stiffness regulation strategy of an athlete with a unilateral BKA is not the same in the unaffected versus affected legs, and compared to non-amputees.