The hybrid mass-spring pendulum model of human leg swinging: stiffness in the control of cycle period

Human leg swinging is modeled as the harmonic motion of a hybrid mass-spring pendulum. The cycle period is determined by a gravitational component and an elastic component, which is provided by the attachment of a soft-tissue/muscular spring of variable stiffness. To confirm that the stiffness of the spring changes with alterations in the inertial properties of the oscillator and that stiffness is relevant for the control of cycle period, we conducted this study in which the simple pendulum equivalent length was experimentally manipulated by adding mass to the ankle of a comfortably swinging leg. Twenty-four young, healthy adults were videotaped as they swung their right leg under four conditions: no added mass and with masses of 2.27, 4.55, and 6.82kg added to the ankle. Strong, linear relationships between the acceleration and displacement of the swinging leg within subjects and conditions were found, confirming the motion's harmonic nature. Cycle period significantly increased with the added mass. However, the observed increases were not as large as would be predicted by the induced changes in the gravitational component alone. These differences were interpreted as being due to increases in the active muscular stiffness. Significant linear increases in the elastic component (and hence stiffness) were demonstrated with increases in the simple pendulum equivalent length in 20 of the individual subjects, with r ² values ranging between 0.89 and 0.99. Significant linear relationships were also demonstrated between the elastic and gravitational components in 22 subjects, with individual r ² values between 0.90 and 0.99. This finding suggests stiffness is varied concomitantly with alterations in the inertial properties of the leg pendulum in a simplified mechanism of control.The views expressed in this article are those of the authors and do not reflect the official policy or position of the Department of the Army, Department of Defense, or the US Government     

A balance control model of quiet upright stance based on an optimal control strategy

Models of balance control can aid in understanding the mechanisms by which humans maintain balance. A balance control model of quiet upright stance based on an optimal control strategy is presented here. In this model, the human body was represented by a simple single-segment inverted pendulum during upright stance, and the neural controller was assumed to be an optimal controller that generates ankle control torques according to a certain performance criterion. This performance criterion was defined by several physical quantities relevant to sway. In order to accurately simulate existing experimental data, an optimization procedure was used to specify the set of model parameters to minimize the scalar error between experimental and simulated sway measures. Thirty-two independent simulations were performed for both younger and older adults. The model's capabilities, in terms of reflecting sway behaviors and identifying aging effects, were then analyzed based on the simulation results. The model was able to accurately predict center-of-pressure-based sway measures, and identify potential changes in balance control mechanisms caused by aging. Correlations between sway measures and model parameters are also discussed.     

Active stiffness of the ankle in response to inertial and elastic loads.

Effective stiffness of the musculoskeletal system was examined as a function of the characteristics of an external load. Thirteen healthy subjects provided active contraction of the ankle plantarflexion musculature in a neutral ankle posture to support an external load. Musculoskeletal stiffness was computed from kinetic data recorded in response to dorsiflexion/plantarflexion perturbations. Ankle dynamics were recorded while supporting external loads of 19 and 38 kg with and without antagonistic co-contraction. External loads were applied using pure gravitational mass. In separate trials external loads were applied from stretch of steel springs in parallel with the plantarflexion musculature that also provided added parallel stiffness to the system. Adding external stiffness of 4.9 and 8.1 kN/m surprisingly failed to significantly change the stiffness of the ankle-plus-spring system. This suggests contributions from intrinsic muscle stiffness and reflex stiffness declined in response to added external stiffness. This could not be explained by load magnitudes, ankle postures, or co-activation as these were similar between the inertial and elastic loading conditions. However, non-linear parametric analyses suggest mean intrinsic stiffness of 35.5 kN/m and reflex gain of 11.6 kN/m with a constant reflex delay of 70 ms accurately described the empirical results. The phase response between the mechanical dynamics of the musculoskeletal system and delayed neuromotor feedback combine to provide robust control of system behavior.     

Ankle intrinsic stiffness changes with postural sway

In standing, the human body is inherently unstable and its stabilization requires constant regulation of ankle torque, generated by a combination of ankle intrinsic properties, peripheral reflexes, and central contributions. Ankle intrinsic stiffness, which quantifies the joint intrinsic properties, has been usually assumed constant in standing; however, there is strong evidence that it is highly dependent on the joint torque, which changes significantly with sway in stance. In this study, we examined how ankle intrinsic stiffness changes with postural sway during standing. Ten subjects stood on a standing apparatus, while subjected to pulse perturbations of ankle position. The mean torque of a short period before the start of each pulse was used as a measure of background torque. Responses with similar background torques were grouped together and used to estimate the parameters of an intrinsic stiffness model. Stiffness estimates were normalized to the critical stiffness and the background torque was transformed to the center of pressure location. We found that in most subjects, the normalized stiffness increased linearly with the movement of center of pressure towards the toes, with an average slope of 2.11 ± 0.80 1/m·rad. This modulation of ankle intrinsic stiffness seems functionally appropriate, since the intrinsic stiffness increases quickly, as the center of pressure moves toward the toes and the limits of stability. These large changes of ankle intrinsic stiffness with postural sway must be incorporated in any model of stance control.     

Intralimb compensation strategy depends on the nature of joint perturbation in human hopping

Due to the well-described spring-mass dynamics of bouncing gaits, human hopping is a tractable model for elucidating basic neuromuscular compensation principles. We tested whether subjects would employ a multi-joint or single-joint response to stabilize leg stiffness while wearing a spring-loaded ankle-foot orthosis (AFO) that applied localized resistive and assistive torques to the ankle. We analyzed kinematics and kinetics data from nine subjects hopping in place on one leg, at three frequencies (2.2, 2.4, and 2.8Hz) and three orthosis conditions (freely articulating AFO, AFO with plantarflexion resistance, and AFO with plantarflexion assistance). Leg stiffness was invariant across AFO conditions, however, compensation strategy depended upon the nature of the applied load. Biological ankle stiffness increased in response to a resistive load at twice the rate that it decreased with an assitive load. Ankle adjustments alone fully compensated for an assistive load with no net change in combined (biological plus applied) total ankle stiffness (p > or =0.133). In contrast, a resistive load resulted in a 7.4-9.0% increase in total ankle stiffness across frequencies and a concomitant 10-15% increase in knee joint stiffness at each frequency (p< or =0.037). The increased knee joint stiffness in response to resistive ankle load allowed subjects to maintain a more flexed knee at mid-stance, which attenuated the effect of the increased total ankle joint stiffness to preserve leg stiffness and whole limb biomechanical performance. Our findings suggest humans maintain invariant leg stiffness in bouncing gaits through different intralimb compensation strategies that are specific to the nature of the joint loading.     

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.     

The inverted pendulum model of bipedal standing cannot be stabilized through direct feedback of force and contractile element length and velocity at realistic series elastic element stiffness

Control of bipedal standing is typically analyzed in the context of a single-segment inverted pendulum model. The stiffness K (SE) of the series elastic element that transmits the force generated by the contractile elements of the ankle plantarflexors to the skeletal system has been reported to be smaller in magnitude than the destabilizing gravitational stiffness K ( g ). In this study, we assess, in case K (SE) + K ( g ) < 0, if bipedal standing can be locally stable under direct feedback of contractile element length, contractile element velocity (both sensed by muscle spindles) and muscle force (sensed by Golgi tendon organs) to alpha-motoneuron activity. A theoretical analysis reveals that even though positive feedback of force may increase the stiffness of the muscle-tendon complex to values well over the destabilizing gravitational stiffness, dynamic instability makes it impossible to obtain locally stable standing under the conditions assumed.     

Prestress revealed by passive co-tension at the ankle joint

This study was designed to test the assumption that elastic tissues of the ankle are prestressed, by investigating the presence of simultaneous opposite passive elastic moments and thus, passive co-tension, at the ankle joint. A prestressed two-spring model used to generate qualitative predictions of the effects of stretching the posterior elastic structures of the ankle on the net passive moment of this joint was used. Twenty-seven healthy individuals were subjected to passive evaluation of the net elastic moment of the ankle in the sagittal plane, with the knee positioned at 90°, 60°, 30° and 0° of flexion, in order to change the length of the posterior biarticular elastic structures. The placement of the knee in the more extended positions caused changes in the net passive moment as predicted by the prestressed model. The ankle position in which the net passive moment was equal to zero was shifted to more plantar flexed positions (p<0.001) and there was a global increase in ankle stiffness since both passive dorsiflexion stiffness (p≤0.037) and passive plantar flexion stiffness (p≤0.029) increased. The normalized terminal plantar flexion stiffness also increased (p≤0.047), suggesting that biarticular posterior elastic structures are pre-strained and still under tension when the ankle is maximally plantar flexed and the knee is positioned at 60° of flexion. Resting positions were indicative of equilibrium between opposite passive elastic moments. The results revealed that there is passive co-tension at the ankle, demonstrating the existence of prestress in elastic structures of this joint.     

The effect of speed on leg stiffness and joint kinetics in human running

The goals of this study were to examine the following hypotheses: (a) there is a difference between the theoretically calculated (McMahon and Cheng, 1990. Journal of Biomechanics 23, 65-78) and the kinematically measured length changes of the spring-mass model and (b) the leg spring stiffness, the ankle spring stiffness and the knee spring stiffness are influenced by running speed. Thirteen athletes took part in this study. Force was measured using a "Kistler" force plate (1000 Hz). Kinematic data were recorded using two high-speed (120 Hz) video cameras. Each athlete completed trials running at five different velocities (approx. 2.5, 3.5, 4.5, 5.5 and 6.5 m/s). Running velocity influences the leg spring stiffness, the effective vertical spring stiffness and the spring stiffness at the knee joint. The spring stiffness at the ankle joint showed no statistical difference (p < 0.05) for the five velocities. The theoretically calculated length change of the spring-mass model significantly (p < 0.05) overestimated the actual length change. For running velocities up to 6.5 m/s the leg spring stiffness is influenced mostly by changes in stiffness at the knee joint.     

Joint stiffness of the ankle and the knee in running

The spring-mass model is a valid fundament to understand global dynamics of fast legged locomotion under gravity. The underlying concept of elasticity, implying leg stiffness as a crucial parameter, is also found on lower motor control levels, i.e. in muscle-reflex and muscle-tendon systems. Therefore, it seems reasonable that global leg stiffness emerges from local elasticity established by appropriate joint torques. A recently published model of an elastically operating, segmented leg predicts that proper adjustment of joint elasticities to the leg geometry and initial conditions of ground contact provides internal leg stability. Another recent study suggests that in turn the leg segmentation and the initial conditions may be a consequence of metabolic and bone stress constraints. In this study, the theoretical predictions were verified experimentally with respect to initial conditions and elastic joint characteristics in human running. Kinematics and kinetics were measured and the joint torques were estimated by inverse dynamics. Stiffnesses and elastic nonlinearities describing the resulting joint characteristics were extracted from parameter fits. Our results clearly support the theoretical predictions: the knee joint is always stiffer and more extended than the ankle joint. Moreover, the knee torque characteristic on the average shows the higher nonlinearity. According to literature, the leg geometry is a consequence of metabolic and material stress limitations. Adapted to this given geometry, the initial joint angle conditions in fast locomotion are a compromise between metabolic and control effort minimisation. Based on this adaptation, an appropriate joint stiffness ratio between ankle and knee passively safeguards the internal leg stability. The identified joint nonlinearities contribute to the linearisation of the leg spring.     

The effect of Knee-Ankle-Foot orthosis stiffness on the parameters of walking

The purpose of this simulation study was to analyze the effect of variation in Knee-Ankle-Foot-Orthosis stiffness on the joint power and the energy cost of walking. The effect of contractile tissue was simulated using linear elastic spring and viscous dampers in knee and ankle joints. Then, joint angles, ground reaction force, were collected from Twenty chronic hemiparesis subjects (15 males and 5 females) and twenty control subjects (14 males and 6 females), and spring stiffness were considered as the inputs. In this new study, the generated muscle torques were optimized by changing the stiffness as the desired output in the mathematical model attained by the MATLAB SimMechanics toolbox. Finally, the simulated mathematical model was introduced as an appropriate substitute in obtaining the optimized stiffness with a more convenient and efficient designed orthosis.     

Spectral analysis of the human body sway during standing on firm and compliant surfaces under different visual conditions

Effects of different visual conditions on the vertical posture maintenance were compared in subjects standing on a firm or compliant surface. These visual conditions included a motionless visual environment (MVE), eyes-closed condition (EC), and a virtual visual environment (VVE). The VVE consisted of two planes: the foreground and background. The foreground displayed a room window with adjacent walls, and the background was represented by an aqueduct with the adjacent landscape. The VVE was destabilized by inducing either the cophased or the antiphased relation between the foreground of the visual scene and the body sway. We evaluated changes in the amplitude spectra of two elementary variables calculated from the trajectories of the plantar center of pressure (CoP) displacements in the anteroposterior and lateral directions, namely, the trajectories for the center of gravity projections on the support (the CG variable) and the differences between the CoP and CG trajectories (the CoP–CG variable).The CG trajectory was considered as a controlled variable, and the difference between the CoP and CG trajectories were considered as a variable related to the body acceleration and reflecting changes in the resultant stiffness in ankle joints. The rootmean-square (RMS) values for the spectra of both variables calculated from the body sway in the anteroposterior direction in standing on a firm support decreased proportionately with antiphased relation between the foreground and the body sway and increased with the cophased relation, compared with the RMS calculated for the MVE conditions. RMS for the spectra of the CG variable in the cophased relation were nearly the same, as in standing with eyes closed (EC), while the RMS for the spectra of the CoP–CG variable were significantly less than with EC. The body sway during standing on a compliant support significantly increased in both the anteroposterior and the lateral directions under all visual conditions. RMS for the spectra of both variables with EC increased considerably higher than in the cophased relation. Furthermore, the RMS for the spectra of the CG variable calculated from the body sway in the lateral direction on a compliant support was substantially higher in the antiphased relation than in the cophased relation, whereas the RMS for the spectra of the CoP–CG variable under both conditions had similar values. The analysis of body sway and the results under some visual conditions have shown that the amplitude characteristics of the CG and CoP–CG variables changed not always proportionately with the passage from standing on a firm support to a compliant support. It is suggested that the found disproportion of changes in these two variables is probably associated with the contribution of another additional factor to the process of postural control, the passive elastic component of musculo-articular stiffness generated by fascial-tendon tissues.     

An efficient robotic tendon for gait assistance

A robotic tendon is a spring based, linear actuator in which the stiffness of the spring is crucial for its successful use in a lightweight, energy efficient, powered ankle orthosis. Like its human analog, the robotic tendon uses its inherent elastic nature to reduce both peak power and energy requirements for its motor. In the ideal example, peak power required of the motor for ankle gait is reduced from 250 W to just 77 W. In addition, ideal energy requirements are reduced from nearly 36 J to just 21 J. Using this approach, an initial prototype has provided 100% of the power and energy necessary for ankle gait in a compact 0.95 kg package, seven times less than an equivalent motor/gearbox system.     

An active balance board system with real-time control of stiffness and time-delay to assess mechanisms of postural stability

Increased time-delay in the neuromuscular system caused by neurological disorders, concussions, or advancing age is an important factor contributing to balance loss (Chagdes et al., 2013, 2016a,b). We present the design and fabrication of an active balance board system that allows for a systematic study of stiffness and time-delay induced instabilities in standing posture. Although current commercial balance boards allow for variable stiffness, they do not allow for manipulation of time-delay. Having two controllable parameters can more accurately determine the cause of balance deficiencies, and allows us to induce instabilities even in healthy populations. An inverted pendulum model of human posture on such an active balance board predicts that reduced board rotational stiffness destabilizes upright posture through board tipping, and limit cycle oscillations about the upright position emerge as feedback time-delay is increased. We validate these two mechanisms of instability on the designed balance board, showing that rotational stiffness and board time-delay induced the predicted postural instabilities in healthy, young adults. Although current commercial balance boards utilize control of rotational stiffness, real-time control of both stiffness and time-delay on an active balance board is a novel and innovative manipulation to reveal balance deficiencies and potentially improve individualized balance training by targeting multiple dimensions contributing to standing balance.     

Identification of passive elastic joint moment-angle relationships in the lower extremity

The purpose of this study was to develop a method for identifying subject-specific passive elastic joint moment-angle relationships in the lower extremity, which could subsequently be used to estimate passive contributions to joint kinetics during gait. Twenty healthy young adults participated in the study. Subjects were positioned side-lying with their dominant limb supported on a table via low-friction carts. A physical therapist slowly manipulated the limb through full sagittal hip, knee, and ankle ranges of motion using two hand-held 3D load cells. Lower extremity kinematics, measured with a passive marker motion capture system, and load cell readings were used to compute joint angles and associated passive joint moments. We formulated a passive joint moment-angle model that included eight exponential functions to account for forces generated via the passive stretch of uni-articular structures and bi-articular muscles. Model parameters were estimated for individual subjects by minimizing the sum of squared errors between model predicted and experimentally measured moments. The model predictions closely replicated measured joint moments with average root-mean-squared errors of 2.5, 1.4, and 0.7 Nm about the hip, knee, and ankle respectively. We show that the models can be coupled with gait kinematics to estimate passive joint moments during walking. Passive hip moments were substantial from terminal stance through initial swing, with energy being stored as the hip extended and subsequently returned during pre- and initial swing. We conclude that the proposed methodology could provide quantitative insights into the potentially important role that passive mechanisms play in both normal and abnormal gait.     

持续性军事职业活动对平衡能力的影响*

目的:评定持续性军事职业活动对平衡能力的影响及视觉系统在其中的作用,为精准军事运动训练提供依据。方法:54名健康男性受试者,年龄(20.28±3.72)岁;身高(173.21±5.67)cm;体重(64.29±5.12) kg,在36 h的时间内完成多项军事科目,记录全程运动负荷(随机抽取11人),军事活动结束后进行睁眼平衡能力测试(54人)和闭眼平衡能力测试(随机抽取27人)。结果:内部负荷,与安静状态相比,军事职业活动中的心率(HR)、运动后的过氧消耗(EPOC)和运动冲量(TRIMP)值均显著增加(P＜0.05);平衡能力,与睁眼安静值相比,军事职业活动后睁眼状态下整体、前后和左右方向的重心动摇距离和速度均显著增加(P＜0.05),而整体、前后和左右动摇面积,前后和左右方向的最大动摇径,以及运动椭圆面积无显著变化(P＞0.05);与闭眼安静值相比,军事活动后闭眼状态下平衡能力所有指标均显著增加(P＜0.05)。结果表明视觉可控制持续性军事职业活动后人体重心动摇的幅度和面积。结论:长时间军事职业活动会损害人体的平衡能力,持续性的军事职业活动后,闭眼状态下平衡能力的破坏程度较睁眼状态大,表现为重心动摇幅度和范围的增加,表明视觉系统在控制姿态稳定方面起一定作用。     

Invariance of ankle dynamic stiffness during fatiguing muscle contractions

Our objective was to determine the effect of muscle fatigue on the dynamic stiffness of the human ankle. Four subjects were required to maintain constant-force contractions of tibialis anterior until the required force could no longer be maintained. Repeated pseudo-random displacements of ankle angular position were applied throughout each contraction. The dynamic relation between ankle angular position and ankle torque was identified by determining non-parametric compliance impulse response functions (CIRFs). The CIRFs were redetermined every 2.55s throughout the sustained contractions to provide a quantitative measure of changes in ankle stiffness dynamics. Inspection of these CIRFs revealed little change in shape or magnitude throughout the contractions, despite large increases in tibialis anterior EMG. The dynamics were further quantified by estimating the equivalent joint inertia, viscosity and elasticity associated with each CIRF. As each contraction progressed, the inertial and elastic terms remained constant whereas the viscous term decreased slightly. These findings demonstrate that fatigue of tibialis anterior during sustained constant mean force contractions results in little change in the mechanical dynamics of the human ankle.     

Validity and reliability of a new in vivo ankle stiffness measurement device

This investigation was designed to test the validity and reliability of a new measure of inversion/eversion ankle stiffness on a unique medial/lateral swaying cradle device utilizing a test/retest with comparison to a known standard. Ankle stiffness is essential to maintaining joint stability. Most ankle injuries occur via an inversion mechanism. To date, very little information is available regarding stiffness of the evertor muscles in the prevention of excessive inversion joint rotation. Transient oscillation data representing inversion/eversion stiffness was obtained in a bipedal weight-bearing stance with an upright posture. Using commercially available springs with stiffness of 4.80N/cm the measured value recorded by the cradle was 4.87N/cm. Mean active stiffness values of the ankle were 35.70Nm/cm (SD 9.45). The trial-to-trial reliability ICC (2,1) coefficient was 0.96 with an SEM of 2.05Nm/rad, and the day-to-day reliability ICC (2,k) coefficient was 0.93 and an SEM of 3.00Nm/rad. The results demonstrate that inversion/eversion ankle stiffness measures on this device are a valid, repeatable and consistent measure. This is relevant because the ability to accurately quantify inversion/eversion ankle stiffness will improve our understanding of biomechanical stability and factors that influence it. It will also enable identification of ankle injury risk factors that will lead to more efficient rehabilitation programs and injury prevention strategies.     

Age-related changes in the behavior of the muscle-tendon unit of the gastrocnemius medialis during upright stance

Mechanical properties of the muscle-tendon unit change with aging, but it is not known how these modifications influence the control of lower leg muscles during upright stance. In this study, young and elderly adults stood upright on a force platform with and without vision while muscle architecture and myotendinous junction movements (expressed relative to the change in the moment on the x-axis of the force platform) were recorded by ultrasonography and muscle activity by electromyography. The results show that the maximal amplitude of the sway in the antero-posterior direction was greater in elderly adults (age effect, P < 0.05) and was accompanied by an increase in lower leg muscle activity compared with young adults. Moreover, the data highlight that fascicles shorten during forward sway and lengthen during backward sways but more so for young (-4 ± 3 and -4 ± 3 mm/Nm, respectively) than elderly adults (-0.7 ± 3 and 0.8 ± 3 mm/Nm, respectively; age × sway, P < 0.001). Concurrently, the pennation angle increased and decreased during forward and backward sways, respectively, with greater changes in young than elderly adults (age × sway, P < 0.001). In contrast, no significant differences were observed between age groups for tendon lengthening and shortening during sways. The results indicate that, compared with young, elderly adults increase the stiffness of the muscular portion of the muscle-tendon unit during upright stance that may compensate for the age-related decrease in tendon stiffness. These observations suggest a shift in the control strategy used to maintain balance.