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1.
Ahn J  Hogan N 《PloS one》2012,7(3):e31767
The control architecture underlying human reaching has been established, at least in broad outline. However, despite extensive research, the control architecture underlying human locomotion remains unclear. Some studies show evidence of high-level control focused on lower-limb trajectories; others suggest that nonlinear oscillators such as lower-level rhythmic central pattern generators (CPGs) play a significant role. To resolve this ambiguity, we reasoned that if a nonlinear oscillator contributes to locomotor control, human walking should exhibit dynamic entrainment to periodic mechanical perturbation; entrainment is a distinctive behavior of nonlinear oscillators. Here we present the first behavioral evidence that nonlinear neuro-mechanical oscillators contribute to the production of human walking, albeit weakly. As unimpaired human subjects walked at constant speed, we applied periodic torque pulses to the ankle at periods different from their preferred cadence. The gait period of 18 out of 19 subjects entrained to this mechanical perturbation, converging to match that of the perturbation. Significantly, entrainment occurred only if the perturbation period was close to subjects' preferred walking cadence: it exhibited a narrow basin of entrainment. Further, regardless of the phase within the walking cycle at which perturbation was initiated, subjects' gait synchronized or phase-locked with the mechanical perturbation at a phase of gait where it assisted propulsion. These results were affected neither by auditory feedback nor by a distractor task. However, the convergence to phase-locking was slow. These characteristics indicate that nonlinear neuro-mechanical oscillators make at most a modest contribution to human walking. Our results suggest that human locomotor control is not organized as in reaching to meet a predominantly kinematic specification, but is hierarchically organized with a semi-autonomous peripheral oscillator operating under episodic supervisory control.  相似文献   

2.
Phase reset and dynamic stability during human gait   总被引:1,自引:0,他引:1  
Yamasaki T  Nomura T  Sato S 《Bio Systems》2003,71(1-2):221-232
The human walking movement shows transient changes in response to single short-lived external perturbations, termed "stumbling reactions." During the stumbling reactions, the walking phase is reset. It has been considered that the reactions contribute to stabilizing the motion, but less evidence bridging between the rhythm reset and the dynamic stability of the gait has been provided. The present study tries to establish the relationship between them. To this end, we construct a simple dynamical system model of the human musculo-skeletal system interacting with the ground, whose joint kinematics during walking is constrained by a given periodic joint-angles-profile. We show first that the model can exhibit a stable limit cycle corresponding to the steady walking with no perturbations. The responses of the limit cycle oscillation are examined by applying a type of perturbations at various timings with various intensities, elucidating the stability of the model's walking when no phase reset is performed. We then observe that modifications of the periodic joint-angles-profile within a short time interval in response to the perturbation can alter the responses of the limit cycle oscillation and induce phase reset of the model's walking. It is shown that appropriate amounts of the phase reset can prevent the model from falling, even for the perturbation that induces falling in the case without the phase reset. This suggests that those phase resets can improve the dynamic stability of the gait. Moreover, the appropriate phase resets predicted by the model are compared with the experimentally observed phase resets during human stumbling reaction to show they share similar characteristics.  相似文献   

3.
A new principle of sensorimotor control of legged locomotion in an unpredictable environment is proposed on the basis of neurophysiological knowledge and a theory of nonlinear dynamics. Stable and flexible locomotion is realized as a global limit cycle generated by a global entrainment between the rhythmic activities of a nervous system composed of coupled neural oscillators and the rhythmic movements of a musculo-skeletal system including interaction with its environment. Coordinated movements are generated not by slaving to an explicit representation of the precise trajectories of the movement of each part but by dynamic interactions among the nervous system, the musculo-skeletal system and the environment. The performance of a bipedal model based on the above principle was investigated by computer simulation. Walking movements stable to mechanical perturbations and to environmental changes were obtained. Moreover, the model generated not only the walking movement but also the running movement by changing a single parameter nonspecific to the movement. The transitions between the gait patterns occurred with hysteresis.  相似文献   

4.
During human walking, plantar flexor activation in late stance helps to generate a stable and economical gait pattern. Because plantar flexor activation is highly mediated by proprioceptive feedback, the nervous system must modulate reflex pathways to meet the mechanical requirements of gait. The purpose of this study was to quantify ankle joint mechanical output of the plantar flexor stretch reflex response during a novel unexpected gait perturbation. We used a robotic ankle exoskeleton to mechanically amplify the ankle torque output resulting from soleus muscle activation. We recorded lower-body kinematics, ground reaction forces, and electromyography during steady-state walking and during randomly perturbed steps when the exoskeleton assistance was unexpectedly turned off. We also measured soleus Hoffmann- (H-) reflexes at late stance during the two conditions. Subjects reacted to the unexpectedly decreased exoskeleton assistance by greatly increasing soleus muscle activity about 60 ms after ankle angle deviated from the control condition (p<0.001). There were large differences in ankle kinematic and electromyography patterns for the perturbed and control steps, but the total ankle moment was almost identical for the two conditions (p=0.13). The ratio of soleus H-reflex amplitude to background electromyography was not significantly different between the two conditions (p=0.4). This is the first study to show that the nervous system chooses reflex responses during human walking such that invariant ankle joint moment patterns are maintained during perturbations. Our findings are particularly useful for the development of neuromusculoskeletal computer simulations of human walking that need to adjust reflex gains appropriately for biomechanical analyses.  相似文献   

5.
The human biped walking shows phase- dependent transient changes in gait trajectory in response to external brief force perturbations. Such responses, referred to as the stumbling reactions, are usually accompanied with phase reset of the walking rhythm. Our previous studies provided evidence, based on a human gait experiment and analyses of mathematical models of gait in the sagittal plane, that an appropriate amount of phase reset in response to a perturbation depended on the gait phase at the perturbation and could play an important role for preventing the walker from a fall, thus increasing gait stability. In this paper, we provide a further material that supports this evidence by a gait experiment on a biped humanoid. In the experiment, the impulsive force perturbations were applied using push-impacts by a pendulum-like hammer to the back of the robot during gait. The responses of the external perturbations were managed by resetting the gait phase with different delays or advancements. The results showed that appropriate amounts of phase resetting contributed to the avoidance of falling against the perturbation during the three-dimensional robot gait. A parallelism with human gait stumbling reactions was discussed.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.  相似文献   

6.
Falls during walking are a major contributor to accidental deaths and injuries that can result in debilitating hospitalization costs, lost productivity, and diminished quality of life. To reduce these losses, we must develop a more profound understanding of the characteristic responses to perturbations similar to those encountered in daily life. This study addresses this issue by building on our earlier studies that examined mechanical and visual perturbations in the same environment by applying the same continuous pseudo-random perturbations at multiple (3 mechanical, 5 visual) amplitudes. Walking variability during mechanical perturbations increased significantly with amplitude for all subjects and differences as measured by variabilities of step width, COM position, and COM velocity. These parameters were the only ones sensitive to the presence of visual perturbations, but none of them changed significantly with perturbation amplitude. Additionally, visual perturbation effects were far less consistent across participants, with several who were essentially unaffected by visual perturbations at any level. The homogeneity of the mechanical perturbation effects demonstrates that human responses to mechanical perturbations are similar because they are driven by kinetics that require similar corrections that must be made in order to maintain balance. Conversely, responses to visual perturbations are driven by the perceived need to make corrections and this perception is not accurate enough to produce amplitude-related corrections, even for a single participant, nor is this perception consistent across individuals. This latter finding is likely to be relevant to future visual perturbation studies and the diagnosis and rehabilitation of gait and balance disorders.  相似文献   

7.
Walking on uneven surfaces or while undergoing perturbations has been associated with increased gait variability in both modeling and human studies. Previous gait research involving continuous perturbations has focused on sinusoidal oscillations, which can result in individuals predicting the perturbation and/or entraining to it. Therefore, we examined the effects of continuous, pseudo-random support surface and visual field oscillations on 12 healthy, young participants. Participants walked in a virtual reality environment under no perturbation (NOP), anterior–posterior (AP) walking surface and visual oscillation and mediolateral (ML) walking surface and visual oscillation conditions. Participants exhibited shorter (p≤0.005), wider (p<0.001) and faster (p<0.001) steps relative to NOP during ML perturbations and shorter (p≤0.005) and wider (p<0.001) steps during AP perturbations. Step length variability and step width variability both increased relative to NOP during all perturbation conditions (p<0.001) but exhibited greater increases for the ML perturbations (p<0.001). Participants exhibited greater trunk position variability and trunk velocity variability in the ML direction than in the AP direction during ML perturbations relative to NOP (p<0.001). Significantly greater variability in the ML direction indicates that to maintain stability, participants needed to exert greater control in the ML direction. This observation is consistent with prior modeling predictions. The large and consistent responses observed during ML visual and walking surface perturbations suggest potential for application during gait training and patient assessment.  相似文献   

8.
In gait stability research, neither self-selected walking speeds, nor the same prescribed walking speed for all participants, guarantee equivalent gait stability among participants. Furthermore, these options may differentially affect the response to different gait perturbations, which is problematic when comparing groups with different capacities. We present a method for decreasing inter-individual differences in gait stability by adjusting walking speed to equivalent margins of stability (MoS). Eighteen healthy adults walked on a split-belt treadmill for two-minute bouts at 0.4 m/s up to 1.8 m/s in 0.2 m/s intervals. The stability-normalised walking speed (MoS = 0.05 m) was calculated using the mean MoS at touchdown of the final 10 steps of each speed. Participants then walked for three minutes at this speed and were subsequently exposed to a treadmill belt acceleration perturbation. A further 12 healthy adults were exposed to the same perturbation while walking at 1.3 m/s: the average of the previous group. Large ranges in MoS were observed during the prescribed speeds (6–10 cm across speeds) and walking speed significantly (P < 0.001) affected MoS. The stability-normalised walking speeds resulted in MoS equal or very close to the desired 0.05 m and reduced between-participant variability in MoS. The second group of participants walking at 1.3 m/s had greater inter-individual variation in MoS during both unperturbed and perturbed walking compared to 12 sex, height and leg length-matched participants from the stability-normalised walking speed group. The current method decreases inter-individual differences in gait stability which may benefit gait perturbation and stability research, in particular studies on populations with different locomotor capacities. [Preprint: https://doi.org/10.1101/314757]  相似文献   

9.
Perturbation training is an emerging approach to reduce fall risk in the elderly. This study examined potential differences in retention of improvements in reactive gait stability over 14 weeks resulting from unexpected trip-like gait perturbations. Twenty-four healthy middle-aged adults (41–62 years) were assigned randomly to either a single perturbation group (SINGLE, n = 9) or a group subjected to eight trip-like gait perturbations (MULTIPLE, n = 15). While participants walked on a treadmill a custom-built brake-and-release system was used to unexpectedly apply resistance during swing phase to the lower right limb via an ankle strap. The anteroposterior margin of stability (MoS) was calculated as the difference between the anterior boundary of the base of support and the extrapolated centre of mass at foot touchdown for the perturbed step and the first recovery step during the first and second (MULTIPLE group only) perturbation trials for the initial walking session and retention-test walking 14 weeks later. Group MULTIPLE retained the improvements in reactive gait stability to the perturbations (increased MoS at touchdown for perturbed and first recovery steps; p < 0.01). However, in group SINGLE no differences in MoS were detected after 14 weeks compared to the initial walking session. These findings provide evidence for the requirement of a threshold trip-perturbation dose if adaptive changes in the human neuromotor system over several months, aimed at the improvement in fall-resisting skills, are to occur.  相似文献   

10.
There is a growing body of evidence that the step-to-step variations present in human walking are related to the biomechanics of the locomotive system. However, we still have limited understanding of what biomechanical variables influence the observed nonlinear gait variations. It is necessary to develop reliable models that closely resemble the nonlinear gait dynamics in order to advance our knowledge in this scientific field. Previously, Goswami et al. [1998. A study of the passive gait of a compass-like biped robot: symmetry and chaos. International Journal of Robotic Research 17(12)] and Garcia et al. [1998. The simplest walking model: stability, complexity, and scaling. Journal of Biomechanical Engineering 120(2), 281-288] have demonstrated that passive dynamic walking computer models can exhibit a cascade of bifurcations in their gait pattern that lead to a deterministic nonlinear gait pattern. These computer models suggest that the intrinsic mechanical dynamics may be at least partially responsible for the deterministic nonlinear gait pattern; however, this has not been shown for a physical walking robot. Here we use the largest Laypunov exponent and a surrogation analysis method to confirm and extend Garcia et al.'s and Goswami et al.'s original results to a physical passive dynamic walking robot. Experimental outcomes from our walking robot further support the notion that the deterministic nonlinear step-to-step variations present in gait may be partly governed by the intrinsic mechanical dynamics of the locomotive system. Furthermore the nonlinear analysis techniques used in this investigation offer novel methods for quantifying the nature of the step-to-step variations found in human and robotic gait.  相似文献   

11.
Several methods derived from nonlinear time series analysis have been suggested to quantify stability in human gait kinematics. One of these methods is the definition of the maximum finite time Lyapunov exponent (λ) that quantifies how the system responds to infinitesimal perturbations. However, there are fundamental limitations to the conventional definition of λ for gait kinematics. First, exponential increase in initial perturbations cannot be assumed since real-life perturbations of gait kinematics are finite sized. Second, the transitions between single and double support phase within each stride cycle define two distinct dynamical regimes that may not be captured by a single λ. The present article presents a new method to quantify intra-stride changes λ(t) in local dynamical stability and employs the method to 3D lower extremity gait kinematics in 10 healthy adults walking on a treadmill at 3 different speeds. All participants showed an intra-stride change in λ(t) in the transition between single and double support phase. The intra-stride change reflected an both a increase and decrease in λ(t) at heel strike and toe off, respectively, with increased gait speed. Furthermore, a close relationship was found between the intra-stride change in standard deviation of foot velocity in the anterior-posterior direction and the intra-stride change of the initial perturbations. The present results indicate that local dynamical stability has gait phase-dependent changes that are not identified by conventional computation of a single λ.  相似文献   

12.
The present study scrutinizes the popular view that tremors of central origin but not those of peripheral origin are largely resistant to mechanical perturbations. We explore the effects of perturbations in a well-established model of peripheral tremor and document that (a) tremor frequency can remain unchanged when spring or weight loads are added, (b) entrainment by external drives can be limited to drives of similar frequency, and (c) resetting of tremor phase by torque pulses can remain fractional. This resistance to mechanical perturbations arises in the model because peripheral neuromuscular dynamics act as a limit-cycle oscillator which, by its very nature, will absorb moderate changes to signals and parameters. We conclude from our study that resistance to mechanical perturbations is not an exclusive property of central tremors, but rather may also be found in peripheral tremors. Other criteria are therefore needed to distinguish between different origins of tremor. Received: 8 April 1998 / Accepted in revised form: 6 October 1998  相似文献   

13.
The walking rhythm is known to show phase shift or "reset" in response to external impulsive perturbations. We tried to elucidate functional roles of the phase reset possibly used for the neural control of locomotion. To this end, a system with a double pendulum as a simplified model of the locomotor control and a model of bipedal locomotion were employed and analyzed in detail. In these models, a movement corresponding to the normal steady-state walking was realized as a stable limit cycle solution of the system. Unexpected external perturbations applied to the system can push the state point of the system away from its limit cycle, either outside or inside the basin of attraction of the limit cycle. Our mathematical analyses of the models suggested functional roles of the phase reset during walking as follows. Function 1: an appropriate amount of the phase reset for a given perturbation can contribute to relocating the system's state point outside the basin of attraction of the limit cycle back to the inside. Function 2: it can also be useful to reduce the convergence time (the time necessary for the state point to return to the limit cycle). In experimental studies during walking of animals and humans, the reset of walking rhythm induced by perturbations was investigated using the phase transition curve (PTC) or the phase resetting curve (PRC) representing phase-dependent responses of the walking. We showed, for the simple double-pendulum model, the existence of the optimal phase control and the corresponding PTC that could optimally realize the aforementioned functions in response to impulsive force perturbations. Moreover, possible forms of PRC that can avoid falling against the force perturbations were predicted by the biped model, and they were compared with the experimentally observed PRC during human walking. Finally, physiological implications of the results were discussed.  相似文献   

14.
Knee instability is a major problem in patients with anterior cruciate ligament injury or knee osteoarthritis. A valid and clinically meaningful measure for functional knee instability is lacking. The concept of the gait sensitivity norm, the normalized perturbation response of a walking system to external perturbations, could be a sensible way to quantify knee instability. The aim of this study is to explore the feasibility of this concept for measurement of knee responses, using controlled external perturbations during walking in healthy subjects.Nine young healthy participants walked on a treadmill, while three dimensional kinematics were measured. Sudden lateral translations of the treadmill were applied at five different intensities during stance. Right knee kinematic responses and spatio-temporal parameters were tracked for the perturbed stride and following four cycles, to calculate perturbation response and gait sensitivity norm values (i.e. response/perturbation) in various ways.The perturbation response values in terms of knee flexion and abduction increased with perturbation intensity and decreased with an increased number of steps after perturbation. For flexion and ab/adduction during midswing, the gait sensitivity norm values were shown to be constant over perturbation intensities, demonstrating the potential of the gait sensitivity norm as a robust measure of knee responses to perturbations.These results show the feasibility of using the gait sensitivity norm concept for certain gait indicators based on kinematics of the knee, as a measure of responses during perturbed gait. The current findings in healthy subjects could serve as reference-data to quantify pathological knee instability.  相似文献   

15.
Simulation of the double limb support phase of human gait   总被引:1,自引:0,他引:1  
Dynamic mechanical models of the double limb support phase of human gait were developed for both two-dimensional (sagittal plane) and three-dimensional motion. A "foot" model with a curved plantar surface was also developed such that the model foot motion was kinematically equivalent to that of a walking subject. This foot model was incorporated into the planar model for double limb support. The dynamic formulations were based on Kane's method and were implemented symbolically using MACSYMA. The development of the formulations for the constrained systems, application of these formulations to the study of normal gait, the sensitivity of the simulation to the frequency content of the input data, the sensitivity of limb displacements to changes in joint moments and the application of a nonlinear feedback controller to correct for perturbations in limb trajectories were investigated.  相似文献   

16.
Disinhibition of reflexes is a problem amongst spastic patients, for it limits a smooth and efficient execution of motor functions during gait. Treadmill belt accelerations may potentially be used to measure reflexes during walking, i.e. by dorsal flexing the ankle and stretching the calf muscles, while decelerations show the modulation of reflexes during a reduction of sensory feedback. The aim of the current study was to examine if belt accelerations and decelerations of different intensities applied during the stance phase of treadmill walking can evoke reflexes in the gastrocnemius, soleus and tibialis anterior in healthy subjects. Muscle electromyography and joint kinematics were measured in 10 subjects. To determine whether stretch reflexes occurred, we assessed modelled musculo-tendon length and stretch velocity, the amount of muscle activity, as well as the incidence of bursts or depressions in muscle activity with their time delays, and co-contraction between agonist and antagonist muscle. Although the effect on the ankle angle was small with 2.8±1.0°, the perturbations caused clear changes in muscle length and stretch velocity relative to unperturbed walking. Stretched muscles showed an increasing incidence of bursts in muscle activity, which occurred after a reasonable electrophysiological time delay (163–191 ms). Their amplitude was related to the muscle stretch velocity and not related to co-contraction of the antagonist muscle. These effects increased with perturbation intensity. Shortened muscles showed opposite effects, with a depression in muscle activity of the calf muscles. The perturbations only slightly affected the spatio-temporal parameters, indicating that normal walking was retained. Thus, our findings showed that treadmill perturbations can evoke reflexes in the calf muscles and tibialis anterior. This comprehensive study could form the basis for clinical implementation of treadmill perturbations to functionally measure reflexes during treadmill-based clinical gait analysis.  相似文献   

17.
Chaos is a central feature of human locomotion and has been suggested to be a window to the control mechanisms of locomotion. In this investigation, we explored how the principles of chaos can be used to control locomotion with a passive dynamic bipedal walking model that has a chaotic gait pattern. Our control scheme was based on the scientific evidence that slight perturbations to the unstable manifolds of points in a chaotic system will promote the transition to new stable behaviors embedded in the rich chaotic attractor. Here we demonstrate that hip joint actuations during the swing phase can provide such perturbations for the control of bifurcations and chaos in a locomotive pattern. Our simulations indicated that systematic alterations of the hip joint actuations resulted in rapid transitions to any stable locomotive pattern available in the chaotic locomotive attractor. Based on these insights, we further explored the benefits of having a chaotic gait with a biologically inspired artificial neural network (ANN) that employed this chaotic control scheme. Remarkably, the ANN was quite robust and capable of selecting a hip joint actuation that rapidly transitioned the passive dynamic bipedal model to a stable gait embedded in the chaotic attractor. Additionally, the ANN was capable of using hip joint actuations to accommodate unstable environments and to overcome unforeseen perturbations. Our simulations provide insight on the advantage of having a chaotic locomotive system and provide evidence as to how chaos can be used as an advantageous control scheme for the nervous system.  相似文献   

18.
After perturbation of the gait, feedback information may help regaining balance adequately, but it remains unknown whether adaptive feedback responses are possible after repetitive and unexpected perturbations during gait and if there are age-related differences. Prior experience may contribute to improved reactive behavior. Fourteen old (59-73 yrs) and fourteen young (22-31 yrs) males walked on a walkway which included one covered element. By exchanging this element participants either stepped on hard surface or unexpectedly on soft surface which caused a perturbation in gait. The gait protocol contained 5 unexpected soft trials to quantify the reactive adaptation. Each soft trial was followed by 4-8 hard trials to generate a wash-out effect. The dynamic stability was investigated by using the margin of stability (MoS), which was calculated as the difference between the anterior boundary of the base of support and the extrapolated position of the center of mass in the anterior-posterior direction. MoS at recovery leg touchdown were significantly lower in the unexpected soft trials compared to the baseline, indicating a less stable posture. However, MoS increased (p<0.05) in both groups within the disturbed trials, indicating feedback adaptive improvements. Young and old participants showed differences in the handling of the perturbation in the course of several trials. The magnitude of the reactive adaptation after the fifth unexpected perturbation was significantly different compared to the first unexpected perturbation (old: 49±30%; young: 77±40%), showing a tendency (p=0.065) for higher values in the young participants. Old individuals maintain the ability to adapt to feedback controlled perturbations. However, the locomotor behavior is more conservative compared to the young ones, leading to disadvantages in the reactive adaptation during disturbed walking.  相似文献   

19.
PurposeTo compare the responses in knee joint muscle activation patterns to different perturbations during gait in healthy subjects.ScopeNine healthy participants were subjected to perturbed walking on a split-belt treadmill. Four perturbation types were applied, each at five intensities. The activations of seven muscles surrounding the knee were measured using surface EMG. The responses in muscle activation were expressed by calculating mean, peak, co-contraction (CCI) and perturbation responses (PR) values. PR captures the responses relative to unperturbed gait. Statistical parametric mapping analysis was used to compare the muscle activation patterns between conditions.ResultsPerturbations evoked only small responses in muscle activation, though higher perturbation intensities yielded a higher mean activation in five muscles, as well as higher PR. Different types of perturbation led to different responses in the rectus femoris, medial gastrocnemius and lateral gastrocnemius. The participants had lower CCI just before perturbation compared to the same phase of unperturbed gait.ConclusionsHealthy participants respond to different perturbations during gait with small adaptations in their knee joint muscle activation patterns. This study provides insights in how the muscles are activated to stabilize the knee when challenged. Furthermore it could guide future studies in determining aberrant muscle activation in patients with knee disorders.  相似文献   

20.
External perturbations applied to the walking surface or visual field can challenge an individual's ability to maintain stability during walking. Accurately quantifying and predicting changes in stability during walking will further our understanding of how individuals respond to challenges encountered during daily life and guide the development of assessments and rehabilitation interventions for individuals at increased risk of falling. This study is the first to determine how orbital and local dynamic stability metrics, including maximum Floquet multipliers and local divergence exponents, change in response to continuous mediolateral visual and surface perturbations of different amplitudes. Eleven healthy individuals walked in a fully immersive virtual environment. Participants completed two 3-min walking trials each under the following nine conditions: no perturbations, surface perturbations at each of 3 amplitudes, and visual perturbations at each of 5 amplitudes. All perturbations were applied as continuous pseudo-random oscillations. During both surface and visual perturbations, individuals were significantly more orbitally and locally unstable compared to un-perturbed walking. As walking surface perturbation amplitudes increased, individuals were more orbitally (but not locally) unstable. As visual perturbation amplitudes increased, individuals were more locally (but not orbitally) unstable between lower and higher amplitudes. Overall, these dynamic stability metrics were much less sensitive to changes in perturbation amplitudes than to differences between un-perturbed and perturbed walking, or to differences between mechanical and visual perturbations. This suggests that the type of perturbation(s) applied has a far greater impact than the magnitude of those perturbations in determining the response that will be elicited.  相似文献   

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