Force-driven Harmonic Oscillator Research Articles

Walking dynamics in preadolescents with and without Down syndrome.

A force-driven harmonic oscillator (FDHO) model reveals the elastic property of general muscular activity during walking. This study aimed to investigate whether children with Down syndrome (DS) have a lower K/G ratio, a primary variable derived from the FDHO model, compared with children with typical development during overground and treadmill walking and whether children with DS can adapt the K/G ratio to walking speeds, external ankle load, and a treadmill setting. A cross-sectional study design was used that included 26 children with and without DS, aged 7 to 10 years, for overground walking and 20 of them for treadmill walking in a laboratory setting. During overground walking, participants walked at 2 speeds: normal and fastest speed. During treadmill walking, participants walked at 75% and 100% of their preferred overground speed. Two load conditions were manipulated for both overground and treadmill walking: no load and an ankle load that was equal to 2% of body mass on each side. Children with DS showed a K/G ratio similar to that of their healthy peers and increased this ratio with walking speed regardless of ankle load during overground walking. Children with DS produced a lower K/G ratio at the fast speed of treadmill walking without ankle load, but ankle load helped them produce a K/G ratio similar to that of their healthy peers. The FDHO model cannot specify what muscles are used or how muscles are coordinated for a given motor task. Children with DS show elastic property of general muscular activity similar to their healthy peers during overground walking. External ankle load helps children with DS increase general muscular activity and match their healthy peers while walking fast on a treadmill.

Effect of stride frequency on the energy cost of walking in obese teenagers

The aim of this study was to compare the energy cost of obese and non-obese teenagers while walking at their preferred speed and different stride frequencies. Twelve obese and twelve non-obese teenagers walked continuously on the treadmill at their most comfortable speed for 6 periods of 4 min each. Each period corresponded to a specific stride frequency: preferred (PSF), force-driven harmonic oscillator (FDHO), PSF + 10%, PSF + 20%, PSF − 10% and PSF − 20%. Cardiorespiratory parameters were collected between the 3rd and 4th minute of each stage, and used to calculate the energy cost of walking (EC). The main results showed a significantly higher cost of walking expressed relative to lean body mass. In addition, a U-shaped relationship between EC and stride frequency was shown in both groups, with PSF and FDHO leading to a significantly lower value compared to all other frequencies. This showed first, that FDHO is a good predictor of PSF and minimal energy cost of walking in both groups, and second, that excess body fat does not affect the relationship between energy expenditure and stride frequency. Walking at lower or higher than preferred frequencies could be used as an exercise mode to promote weight loss in obese teenagers.

Development of walking in preterm and term infants: Age of onset, qualitative features and sensitivity to resonance

An increasing number of studies have examined the development of walking in preterm infants; however, the results concerning those who had no major neonatal disease were inconclusive. This study was therefore aimed to examine the age of onset, the quality of early walking movement, and the sensitivity to resonant period of the force-driven harmonic oscillator (FDHO) model in preterm infants who had no major neonatal disease and normal term infants. Twenty-nine preterm infants and 29 term infants were prospectively examined for their age of onset of independent walking and were subsequently assessed the qualitative features of walking at 18 months of corrected age using kinematic analysis. Kinematic variables examined included spatio-temporal organization, inter-joint coordination, and inter-limb coordination. The anthropometric data were used to calculate the resonant period. The results demonstrated that the preterm infants attained independent walking at significantly older ages than the term infants when corrected for prematurity. The preterm infants manifested similar walking characteristics, except for shorter stride lengths, at 18 months of corrected age compared with the term infants. Furthermore, the stride periods of both groups were accurately predicted by the resonant period of the FDHO model. We conclude that preterm birth without accompanied major neonatal disease may affect infants’ age of onset and spatial organization but not their sensitivity to resonance during the early stage of walking development.

Locomotor-respiratory coordination dynamics and metabolic cost during walking at different stride frequencies

Human locomotion has been modeled as a force-driven harmonic oscillator (FDHO). The minimum forcing function in locomotion has been shown to occur at the resonant frequency of the FDHO and results in the suggestion that oxygen cost may be considered an optimality criterion for locomotion. The purposes of this study were twofold: first, to determine the relationship between stride frequency and shock attenuation, and second, to determine whether shock attenuation may also be considered an optimality criterion. Ten healthy young adult males served as subjects in this study. Each subject's preferred running speed and preferred stride frequency (PSF) were determined. In addition to the PSF, they ran at stride frequencies corresponding to −20%, −10%, +10%, and +20% of the PSF at the preferred running speed. Metabolic data as well as leg and head acceleration data were collected during a steady state run at each of the stride frequency conditions. The metabolic data produced a U-shaped curve hypothesized by the FDHO model. Spectral analysis on the leg and head acceleration data were used to develop transfer functions for each of the stride frequency conditions. Analysis of the transfer function indicated that there was a gain at the low frequencies and an attenuation at the higher frequencies. The transfer function at the higher frequencies indicated that the impact shock signal was attenuated as it passed through the body. However, the transfer functions appeared to vary according to the amount of shock input to the system with the result that the head accelerations remained constant. It would appear that impact (high frequency) shock attenuation increases with stride frequency and thus does not fit the FDHO model as an optimization criterion. At all stride frequencies, regardless of the impact shock, head accelerations were maintained at a constant level.

Gait retraining after anterior cruciate ligament reconstruction

Decker MJ, Torry MR, Noonan TJ, Sterett WI, Steadman JR. Gait retraining after anterior cruciate ligament reconstruction. Arch Phys Med Rehabil 2004;85:848-56. Objectives To examine the effects of 2 gait retraining protocols on the gait patterns of patients with bone-patellar tendon-bone anterior cruciate ligament (ACL) reconstruction. Design Randomized control, repeated-measures design. Setting Private orthopedic center and research facility. Participants Sixteen patients with bone-patellar tendon-bone ACL reconstruction, randomly subdivided into 2 groups (group 1, n=8; group 2, n=8), and a healthy control group of 8 subjects. Intervention The 16 subjects with ACL reconstruction were randomly assigned to 2 different gait retraining protocols over a 6-week training interval: (1) a protocol using a predicted stride frequency calculated from the resonant frequency of a force-driven harmonic oscillator (FDHO) model or (2) a protocol using the preferred stride frequency (PSF). Main outcome measures Gait analyses examining the lower-extremity kinematic, kinetic, and energetic gait patterns of each group. Results Gait retraining with the FDHO model showed improvements in lower-extremity positions, hip and knee extensor angular impulse, and work parameters. Gait retraining with the PSF demonstrated no statistical improvements. The FDHO training protocol facilitated a greater midstance knee range of motion (ROM) and greater rates of improvement for midstance ROM, hip extensor angular impulse, and concentric hip extensor work. Conclusions Gait retraining with the resonant frequency of an FDHO model facilitated a greater recovery of gait function compared with training with the PSF.

The force-driven harmonic oscillator model for energy-efficient locomotion in individuals with transtibial amputation

The force-driven harmonic oscillator (FDHO) model states that the driving force is minimum at the resonant period of an oscillator. By manipulating prosthetic mass, this study explored the compromise of resonant periods between the two legs in persons with unilateral traumatic transtibial amputation (TTA) at self-selected walking velocity (SSWV), with an aim to better understand the energy minimization mechanisms of walking. It was hypothesized that (1) SSWV was the most energy-efficient walking velocity (MEWV), (2) the stride period at SSWV ( T s) is a compromise between the resonant periods of the normal leg ( T n) and the prosthetic leg ( T p) when they are dissimilar. Eight subjects completed multiple-speed treadmill walking tests (at 53, 67, 80, 93, and 107 m/min) according to three mass conditions (60%, 80%, and 100% of the normal leg below-knee mass) in a random order. Oxygen consumption and stride period were measured, and SSWV was empirically determined. The MEWV, the speed with minimum energy expenditure per distance traveled, was derived from quadratic regression, and its stride period ( T m) was estimated. A theoretical compromise period ( T v) between T n and T p was predicted by a virtual single pendulum system based on Huygens' Law. Across different mass conditions, comparisons were made among: T s, T m, T v, T n, and T p. Results showed that: (1) T s was significantly different from T m; (2) T s was greater than both T n and T p; (3) no significant difference was found between T m and T n. Implications for amputee rehabilitation in terms of thigh muscle training and prosthesis development were discussed.

THE EFFECT OF STRIDE PERIOD ON THE VERTICAL GROUND REACTIONFORCE DURING WALKING

1916 At self selected walking speeds, humans naturally employ a stride period consistent with the resonant period predicted by a modified force driven harmonic oscillator (FDHO) model. Metabolic cost is also minimized at self selected walking speeds, thus resonance has been suggested as the mechanism responsible for the self-optimization of gait. It is unclear whether other performance advantages accompany the use of this biomechanical strategy, such as minimization of ground reaction force loads. The purpose of this study was to determine the relationship between stride period and the vertical ground reaction force (VGRF) during locomotion. It was hypothesized that the first active vertical peak force (F1) would be minimized at the FDHO predicted stride period. Fourteen subjects (73.9 kg, 1.7 m, 28.4 y) were required to walk at a constant treadmill velocity during four stride period conditions: preferred, FDHO predicted and ± 15% of predicted. Stride period and the VGRF data were collected (1200 Hz) from a Gaitway treadmill equipped with two Kistler force plates. A paired t-test (p<0.05) was used to contrast preferred and predicted stride periods. A repeated measures ANOVA (p<0.05) and a trend analysis were implemented to investigate the effect of stride period on the F1 VGRF data. TableTableNo differences were found between preferred and predicted stride periods. The F1 VGRF demonstrated a significant quadratic (U-shaped) curve with a minimum at the predicted stride period condition. Post-hoc pairwise comparisons (Bonferroni) of the F1 VGRF revealed a significant difference between predicted and - 15% stride period conditions. Collectively, these data indicate that individuals optimize the F1 VGRF when they naturally adopt a stride period predicted by a modified FDHO model.

The force driven harmonic oscillator model accurately predicts the preferred stride frequency for backward walking

It has been established that when gait speed is freely selected, there is a strong natural tendency for a stride rate and length combination to be utilized that results in minimized metabolic cost. A possible mechanism for this self-optimizing behavior is that a biomechanically optimal movement pattern is adopted, which should reduce or minimize metabolic cost. When the legs are modeled as pendular force driven harmonic oscillators (FDHO) the stride frequency during freely selected gait is predictable as the pendulum's resonant frequency — the state in which force inputs to maintain oscillations are minimal. Thus, resonance has been proposed as a mechanism responsible for gait optimization. To further examine the durability of this mechanism, the FDHO model was applied to predict stride rates during both free forward and backward walking for two separate test sessions. Stride measures were stable across sessions. Forward walking resulted in 25% greater stride lengths than backward, however, stride rates were statistically equal. Also, the FDHO model successfully predicted the preferred stride rate for both backward and forward walking. The resulting invariance of the stride frequency and adjustment of stride length were interpreted as offering additional support for the resonance mechanism.

Optimization of walking in children.

Previous work demonstrated that adults naturally adopt a walking frequency to optimize physiological cost, symmetry, and stability. Furthermore, the optimal frequency is predictable using the force-driven harmonic oscillator (FDHO) model. However, no studies have established the developmental processes of optimization in children. Thus, the purposes of this study were to examine the predictability of the preferred stride frequency (PSF) and optimization features of 3- to 12-yr-old children using the FDHO model. Forty-five children and nine adults were measured for anthropometric data to calculate the predicted frequency. They later walked at three frequencies (PSF, PSF +25%, and PSF -25%) at a constant speed on a treadmill. The results indicated that the FDHO model was accurate in predicting the preferred frequency of children (prediction error < 0.07 s). We identified three stages of learning in the development of optimization: an early manifestation of sensitivity to resonant frequency, the subsequent development of ability to modulate walking frequency, and the final establishment of an adult optimization form at age seven. Our findings suggest that walking development may be determined by the dynamic cooperation of physiological, neural, and musculoskeletal systems with respect to the environmental context.

Self-Optimization of Walking in Nondisabled Children and Children With Spastic Hemiplegic Cerebral Palsy.

Children voluntarily adopt a frequency and movement pattern for walking. The force-driven harmonic oscillator (FDHO) model was used in this study for accurate prediction of the preferred walking frequency of nondisabled children and children with spastic hemiplegic cerebral palsy. Four potential optimality criteria with which the preferred walking pattern was forced to comply were examined: minimization of physiological costs, maximization of mechanical energy conservation, minimization of asymmetry in lower limb movements and minimization of variability of interlimb and intralimb coordination. Age and gender-matched nondisabled children (n = 6) and children with spastic hemiplegic cerebral palsy (n = 6) were tested under six frequency conditions of walking at a constant speed on a treadmill. For the nondisabled children, the results indicated that their preferred walking frequency could be accurately predicted by the FDHO model. They freely adopted a walking pattern that minimized physiological costs, asymmetry, and variability of inter- and intralimb coordination. For the children with spastic hemiplegic cerebral palsy, the prediction of preferred overground walking frequency required that the FDHO model be modified to account for muscle mass and leg length discrepancies between limbs and increased stiffness. Most of the children achieved the same optimality goals as the nondisabled when walking at the preferred frequency. However, the children were found to use different mechanisms to attain these goals: for example, a steeper increase observed in physiological cost at higher frequencies; a lowered center of gravity of the body, which allowed for angular symmetry; and greater variability of between-joint coordination in the nonaffected limb and less variability in the affected limb.

Shock attenuation and stride frequency during running

Human locomotion has been modeled as a force-driven harmonic oscillator (FDHO). The minimum forcing function in locomotion has been shown to occur at the resonant frequency of the FDHO and results in the suggestion that oxygen cost may be considered an optimality criterion for locomotion. The purposes of this study were twofold: first, to determine the relationship between stride frequency and shock attenuation, and second, to determine whether shock attenuation may also be considered an optimality criterion. Ten healthy young adult males served as subjects in this study. Each subject's preferred running speed and preferred stride frequency (PSF) were determined. In addition to the PSF, they ran at stride frequencies corresponding to −20%, −10%, +10%, and +20% of the PSF at the preferred running speed. Metabolic data as well as leg and head acceleration data were collected during a steady state run at each of the stride frequency conditions. The metabolic data produced a U-shaped curve hypothesized by the FDHO model. Spectral analysis on the leg and head acceleration data were used to develop transfer functions for each of the stride frequency conditions. Analysis of the transfer function indicated that there was a gain at the low frequencies and an attenuation at the higher frequencies. The transfer function at the higher frequencies indicated that the impact shock signal was attenuated as it passed through the body. However, the transfer functions appeared to vary according to the amount of shock input to the system with the result that the head accelerations remained constant. It would appear that impact (high frequency) shock attenuation increases with stride frequency and thus does not fit the FDHO model as an optimization criterion. At all stride frequencies, regardless of the impact shock, head accelerations were maintained at a constant level.

Evaluation of in-line skating for rehabilitation: Impact shock considerations

The study was conducted to determine whether the preferred frequency of locomotion was predictable as the least amount of energy required to drive a harmonic oscillator. Subjects were instructed to walk at a preferred rate under conditions where their ankles were unloaded and bilaterally loaded. These results were compared to the frequency which was predicted from the formula for a force-driven harmonic oscillator. The length of a simple pendulum equivalent of the lower extremity (thigh, shank, foot, and added mass) was used to approximate the length of the oscillator for prediction purposes. Results indicated that a constant of 2 applied to the gravitational constant of the period prediction formula provided a more accurate representation of the actual frequency. Similar results have been found in the walking gait of quadrupeds of widely varying sizes (Kugler and Turvey 1987).

Walking Cadence of 9-Year-Olds Predictable as Resonant Frequency of a Force Driven Harmonic Oscillator

Preferred stride frequency (PSF) of adult human walking has been shown to be predictable as the resonant frequency of a force driven harmonic oscillator (FDHO). The purpose of this study was to determine whether the PSF of 9-year-old children was predictable using the same resonance formula as that of adults. Subjects walked around a gymnasium at a rate at which they felt comfortable. Stride frequency was measured as the time for 20 strides and the stride period was calculated. The best-fit prediction based on resonance was then calculated using the overall center of mass of three segments (foot, shank, thigh) to determine the simple pendulum equivalent (SPE) length. Results indicated that a constant of 2 applied to the gravitational constant of the resonance formula, the same formulation used for adults, can be used to predict the cadence of children.

The force-driven harmonic oscillator as a model for human locomotion

The study was conducted to determine whether the preferred frequency of locomotion was predictable as the least amount of energy required to drive a harmonic oscillator. Subjects were instructed to walk at a preferred rate under conditions where their ankles were unloaded and bilaterally loaded. These results were compared to the frequency which was predicted from the formula for a force-driven harmonic oscillator. The length of a simple pendulum equivalent of the lower extremity (thigh, shank, foot, and added mass) was used to approximate the length of the oscillator for prediction purposes. Results indicated that a constant of 2 applied to the gravitational constant of the period prediction formula provided a more accurate representation of the actual frequency. Similar results have been found in the walking gait of quadrupeds of widely varying sizes (Kugler and Turvey 1987).