The validity of stability measures: A modelling approach

The condition for dynamic stability in humans walking with feedback control

The walking human body is mechanically unstable. Loss of stability and falling is more likely in certain groups of people, such as older adults or people with neuromotor impairments, as well as in certain situations, such as when experiencing conflicting or distracting sensory inputs. Stability during walking is often characterized biomechanically, by measures based on body dynamics and the base of support. Neural control of upright stability, on the other hand, does not factor into commonly used stability measures. Here we analyze stability of human walking accounting for both biomechanics and neural control, using a modeling approach. We define a walking system as a combination of biomechanics, using the well known inverted pendulum model, and neural control, using a proportional-derivative controller for foot placement based on the state of the center of mass at midstance. We analyze this system formally and show that for any choice of system parameters there is always one periodic orbit. We then determine when this periodic orbit is stable, i.e. how the neural control gain values have to be chosen for stable walking. Following the formal analysis, we use this model to make predictions about neural control gains and compare these predictions with the literature and existing experimental data. The model predicts that control gains should increase with decreasing cadence. This finding appears in agreement with literature showing stronger effects of visual or vestibular manipulations at different walking speeds.

Predicting fall risk using multiple mechanics-based metrics for a planar biped model

For both humans and robots, falls are undesirable, motivating the development of fall prediction models. Many mechanics-based fall risk metrics have been proposed and validated to varying degrees, including the extrapolated center of mass, the foot rotation index, Lyapunov exponents, joint and spatiotemporal variability, and mean spatiotemporal parameters. To obtain a best-case estimate of how well these metrics can predict fall risk both individually and in combination, this work used a planar six-link hip-knee-ankle biped model with curved feet walking at speeds ranging from 0.8 m/s to 1.2 m/s. The true number of steps to fall was determined using the mean first passage times from a Markov chain describing the gaits. In addition, each metric was estimated using the Markov chain of the gait. Because calculating the fall risk metrics from the Markov chain had not been done before, the results were validated using brute force simulations. Except for the short-term Lyapunov exponents, the Markov chains could accurately calculate the metrics. Using the Markov chain data, quadratic fall prediction models were created and evaluated. The models were further evaluated using differing length brute force simulations. None of the 49 tested fall risk metrics could accurately predict the number of steps to fall by themselves. However, when all the fall risk metrics except the Lyapunov exponents were combined into a single model, the accuracy increased substantially. These results suggest that multiple fall risk metrics must be combined to obtain a useful measure of stability. As expected, as the number of steps used to calculate the fall risk metrics increased, the accuracy and precision increased. This led to a corresponding increase in the accuracy and precision of the combined fall risk model. 300 step simulations seemed to provide the best tradeoff between accuracy and using as few steps as possible.

Contribution of Phase Resetting to Statistical Persistence in Stride Intervals: A Modeling Study

Stride intervals in human walking fluctuate from one stride to the next, exhibiting statistical persistence. This statistical property is changed by aging, neural disorders, and experimental interventions. It has been hypothesized that the central nervous system is responsible for the statistical persistence. Human walking is a complex phenomenon generated through the dynamic interactions between the central nervous system and the biomechanical system. It has also been hypothesized that the statistical persistence emerges through the dynamic interactions during walking. In particular, a previous study integrated a biomechanical model composed of seven rigid links with a central pattern generator (CPG) model, which incorporated a phase resetting mechanism as sensory feedback as well as feedforward, trajectory tracking, and intermittent feedback controllers, and suggested that phase resetting contributes to the statistical persistence in stride intervals. However, the essential mechanisms remain largely unclear due to the complexity of the neuromechanical model. In this study, we reproduced the statistical persistence in stride intervals using a simplified neuromechanical model composed of a simple compass-type biomechanical model and a simple CPG model that incorporates only phase resetting and a feedforward controller. A lack of phase resetting induced a loss of statistical persistence, as observed for aging, neural disorders, and experimental interventions. These mechanisms were clarified based on the phase response characteristics of our model. These findings provide useful insight into the mechanisms responsible for the statistical persistence of stride intervals in human walking.

The validation of new phase-dependent gait stability measures: a modelling approach

Identification of individuals at risk of falling is important when designing fall prevention methods. Current measures that estimate gait stability and robustness appear limited in predicting falls in older adults. Inspired by recent findings on changes in phase-dependent local stability within a gait cycle, we devised several phase-dependent stability measures and tested for their usefulness to predict gait robustness in compass walker models. These measures are closely related to the often-employed maximum finite-time Lyapunov exponent and maximum Floquet multiplier that both assess a system's response to infinitesimal perturbations. As such, they entail linearizing the system, but this is realized in a rotating hypersurface orthogonal to the period-one solution followed by estimating the trajectory-normal divergence rate of the swing phases and the foot strikes. We correlated the measures with gait robustness, i.e. the largest perturbation a walker can handle, in two compass walker models with either point or circular feet to estimate their prediction accuracy. To also test for the dependence of the measures under state space transform, we represented the point feet walker in both Euler-Lagrange and Hamiltonian canonical form. Our simulations revealed that for most of the measures their correlation with gait robustness differs between models and between different state space forms. In particular, the latter may jeopardize many stability measures' predictive capacity for gait robustness. The only exception that consistently displayed strong correlations is the divergence of foot strike. Our results admit challenges of using phase-dependent stability measures as objective means to estimate the risk of falling.

Fractal mechanism of basin of attraction in passive dynamic walking

Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how humans walk with low energy consumption and provides design principles for energy-efficient biped robots. However, the basin of attraction is very small and thin and has a fractal-like complicated shape, which makes producing stable walking difficult. In our previous study, we used the simplest walking model and investigated the fractal-like basin of attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics caused by the impact upon foot contact. We clarified that the fractal-like basin of attraction is generated through iterative stretching and bending deformations of the domain of the Poincaré map by sequential inverse images. However, whether the fractal-like basin of attraction is actually fractal, i.e., whether infinitely many self-similar patterns are embedded in the basin of attraction, is dependent on the slope angle, and the mechanism remains unclear. In the present study, we improved our previous analysis in order to clarify this mechanism. In particular, we newly focused on the range of the Poincaré map and specified the regions that are stretched and bent by the sequential inverse images of the Poincaré map. Through the analysis of the specified regions, we clarified the conditions and mechanism required for the basin of attraction to be fractal.

Effects of Self-Selected Step Length and Trunk Position on Joint Kinetics in Highly Physically Fit Older Adults

The causes of age-related differences in lower-extremity joint moments and powers are unknown. The purpose of this study was to determine the effects of highly physically active older adults walking with (1) a step length similar to young adults and (2) an upright trunk posture, on hip and ankle joint kinetics. The authors hypothesized that, compared with their self-selected walking mechanics, older adults would exhibit decreased hip kinetics and increased ankle kinetics when prescribed a young adult step length, and would exhibit decreased hip extension moments when maintaining an upright trunk posture during walking. A total of 12 active older adults (67 [5] y) and 13 active young adults (21 [3] y) walked at 1.3 m/s. The older adults also walked at 1.3 m/s with step lengths prescribed from height-matched young adults and, in a separate condition, walked with an upright trunk. The older adults did not display larger ankle kinetics or smaller hip kinetics in either condition compared to walking with a self-selected step length. These findings indicate that step length and trunk position do not primarily contribute to age-related differences in kinetics in highly active older adults and should serve as a starting point for investigating alternative explanations.

Lower Local Dynamic Stability and Invariable Orbital Stability in the Activation of Muscle Synergies in Response to Accelerated Walking Speeds

In order to achieve flexible and smooth walking, we must accomplish subtasks (e. g., loading response, forward propulsion or swing initiation) within a gait cycle. To evaluate subtasks within a gait cycle, the analysis of muscle synergies may be effective. In the case of walking, extracted sets of muscle synergies characterize muscle patterns that relate to the subtasks within a gait cycle. Although previous studies have reported that the muscle synergies of individuals with disorders reflect impairments, a way to investigate the instability in the activations of muscle synergies themselves has not been proposed. Thus, we investigated the local dynamic stability and orbital stability of activations of muscle synergies across various walking speeds using maximum Lyapunov exponents and maximum Floquet multipliers. We revealed that the local dynamic stability in the activations decreased with accelerated walking speeds. Contrary to the local dynamic stability, the orbital stability of the activations was almost constant across walking speeds. In addition, the increasing rates of maximum Lyapunov exponents were different among the muscle synergies. Therefore, the local dynamic stability in the activations might depend on the requirement of motor output related to the subtasks within a gait cycle. We concluded that the local dynamic stability in the activation of muscle synergies decrease as walking speed accelerates. On the other hand, the orbital stability is sustained across broad walking speeds.

Effects of backpack load and positioning on nonlinear gait features in young adults

Overloaded backpacks can cause changes in posture and gait dynamic balance. Therefore, the aim of this study was to assess gait regularity and local dynamic stability in young adults as they carried a backpack in different positions, and with different loads. Twenty-one healthy young adults participated in the study, carrying a backpack that was loaded with 10 and 20% of their body weight (BW). The participants walked on a level treadmill at their preferred walking speeds for 4 min under different conditions of backpack load and position (i.e. with backpack positioned back bilaterally, back unilaterally, frontally or without a backpack). Results indicate that backpack load and positioning significantly influence gait stability and regularity, with the exception of the 10% BW bilateral back position. Therefore, the recommended safe load for school-age children and adolescents (10% of BW) should also be considered for young adults. Practitioner summary: Increase in load results in changes in posture, muscle activity and gait parameters, so we investigated the gait adaptations related to regularity and stability. Conditions with high backpack loads significantly influenced gait stability and regularity in a position-dependent manner, except for 10% body weight bilateral back position.

Treadmill walking reduces pre-frontal activation in patients with Parkinson's disease

BACKGROUND

Among patients with Parkinson's disease (PD), gait is typically disturbed and less automatic. These gait changes are associated with impaired rhythmicity and increased prefrontal activation, presumably in an attempt to compensate for reduced automaticity.

RESEARCH QUESTION

We investigated whether during treadmill walking, when the pace is determined and fixed, prefrontal activation in patients with PD is lower, as compared to over-ground walking.

METHODS

Twenty patients with PD (age: 69.8 ± 6.5 yrs.; MoCA: 26.9 ± 2.4; disease duration: 7.9 ± 4.2 yrs) walked at a self-selected walking speed over-ground and on a treadmill. A wireless functional near infrared spectroscopy (fNIRS) system measured prefrontal lobe activation, i.e., oxygenated hemoglobin (Hb02) in the pre-frontal area. Gait was evaluated using 3D-accelerometers attached to the lower back and ankles (Opal™, APDM). Dynamic gait stability was assessed using the maximum Lyapunov exponent to investigate automaticity of the walking pattern.

RESULTS

Hb02 was lower during treadmill walking than during over-ground walking (p = 0.001). Gait stability was greater on the treadmill, compared to over-ground walking, in both the anteroposterior and medio-lateral axes (p < 0.001).

SIGNIFICANCE

These findings support the notion that when gait is externally paced, prefrontal lobe activation is reduced in patients with PD, perhaps reflecting a reduced need for compensatory cognitive mechanisms.

Formation mechanism of a basin of attraction for passive dynamic walking induced by intrinsic hyperbolicity

Passive dynamic walking is a useful model for investigating the mechanical functions of the body that produce energy-efficient walking. The basin of attraction is very small and thin, and it has a fractal-like shape; this explains the difficulty in producing stable passive dynamic walking. The underlying mechanism that produces these geometric characteristics was not known. In this paper, we consider this from the viewpoint of dynamical systems theory, and we use the simplest walking model to clarify the mechanism that forms the basin of attraction for passive dynamic walking. We show that the intrinsic saddle-type hyperbolicity of the upright equilibrium point in the governing dynamics plays an important role in the geometrical characteristics of the basin of attraction; this contributes to our understanding of the stability mechanism of bipedal walking.

Can stability really predict an impending slip-related fall among older adults?

The primary purpose of this study was to systematically evaluate and compare the predictive power of falls for a battery of stability indices, obtained during normal walking among community-dwelling older adults. One hundred and eighty seven community-dwelling older adults participated in the study. After walking regularly for 20 strides on a walkway, participants were subjected to an unannounced slip during gait under the protection of a safety harness. Full body kinematics and kinetics were monitored during walking using a motion capture system synchronized with force plates. Stability variables, including feasible-stability-region measurement, margin of stability, the maximum Floquet multiplier, the Lyapunov exponents (short- and long-term), and the variability of gait parameters (including the step length, step width, and step time), were calculated for each subject. Sensitivity of predicting slip outcome (fall vs. recovery) was examined for each stability variable using logistic regression. Results showed that the feasible-stability-region measurement predicted fall incidence among these subjects with the highest sensitivity (68.4%). Except for the step width (with an sensitivity of 60.2%), no other stability variables could differentiate fallers from those who did not fall for the sample included in this study. The findings from the present study could provide guidance to identify individuals at increased risk of falling using the feasible-stability-region measurement or variability of the step width.

Influence of input parameters on dynamic orbital stability of walking: in-silico and experimental evaluation

Many measures aiming to assess the stability of human motion have been proposed in the literature, but still there is no commonly accepted way to define or quantify locomotor stability. Among these measures, orbital stability analysis via Floquet multipliers is still under debate. Some of the controversies concerning the use of this technique could lie in the absence of a standard implementation. The aim of this study was to analyse the influence of i) experimental measurement noise, ii) variables selected for the construction of the state space, and iii) number of analysed cycles on the outputs of orbital stability applied to walking. The analysis was performed on a 2-dimensional 5-link walking model and on a sample of 10 subjects performing long over-ground walks. Noise resulting from stereophotogrammetric and accelerometric measurement systems was simulated in the in-silico analysis. Maximum Floquet multipliers resulted to be affected by both number of analysed strides and state space composition. The effect of experimental noise was found to be slightly more potentially critical when analysing stereophotogrammetric data then when dealing with acceleration data. Experimental and model results were comparable in terms of overall trend, but a difference was found in the influence of the number of analysed cycles.

Movement variability near goal equivalent manifolds: fluctuations, control, and model-based analysis

Fluctuations in the repeated performance of human movements have been the subject of intense scrutiny because they are generally believed to contain important information about the function and health of the neuromotor system. A variety of approaches has been brought to bear to study these fluctuations. However it is frequently difficult to understand how to synthesize different perspectives to give a coherent picture. Here, we describe a conceptual framework for the experimental study of motor variability that helps to unify geometrical methods, which focus on the role of motor redundancy, with dynamical methods that characterize the error-correcting processes regulating the performance of skilled tasks. We describe how goal functions, which mathematically specify the task strategy being employed, together with ideas from the control of redundant systems, allow one to formulate simple, experimentally testable dynamical models of inter-trial fluctuations. After reviewing the basic theory, we present a list of five general hypotheses on the structure of fluctuations that can be expected in repeated trials of goal-directed tasks. We review recent experimental applications of this general approach, and show how it can be used to precisely characterize the error-correcting control used by human subjects.

Using dynamic walking models to identify factors that contribute to increased risk of falling in older adults

Falls are common in older adults. The most common cause of falls is tripping while walking. Simulation studies demonstrated that older adults may be restricted by lower limb strength and movement speed to regain balance after a trip. This review examines how modeling approaches can be used to determine how different measures predict actual fall risk and what some of the causal mechanisms of fall risk are. Although increased gait variability predicts increased fall risk experimentally, it is not clear which variability measures could best be used, or what magnitude of change corresponded with increased fall risk. With a simulation study we showed that the increase in fall risk with a certain increase in gait variability was greatly influenced by the initial level of variability. Gait variability can therefore not easily be used to predict fall risk. We therefore explored other measures that may be related to fall risk and investigated the relationship between stability measures such as Floquet multipliers and local divergence exponents and actual fall risk in a dynamic walking model. We demonstrated that short-term local divergence exponents were a good early predictor for fall risk. Neuronal noise increases with age. It has however not been fully understood if increased neuronal noise would cause an increased fall risk. With our dynamic walking model we showed that increased neuronal noise caused increased fall risk. Although people who are at increased risk of falling reduce their walking speed it had been questioned whether this slower speed would actually cause a reduced fall risk. With our model we demonstrated that a reduced walking speed caused a reduction in fall risk. This may be due to the decreased kinematic variability as a result of the reduced signal-dependent noise of the smaller muscle forces that are required for slower. These insights may be used in the development of fall prevention programs in order to better identify those at increased risk of falling and to target those factors that influence fall risk most.

Assessment of gait sensitivity norm as a predictor of risk of falling during walking in a neuromusculoskeletal model

Quantifying the risk of falling (falls risk) would be helpful in treating people with gait disorders. The gait sensitivity norm (GSN) is a stability measure that correlates well to risk of falling in passive dynamic walkers but has not been evaluated on humans or human-like walking models. We assessed the correlation of GSN to risk of falling in a neuromusculoskeletal (NMS) walking model. Specifically, we evaluated the correlation of GSN to the actual disturbance rejection (ADR) of the model and the sensitivity of this relationship to gait parameter, Poincaré section selection and steady state variability correction. Statistically significant results at p<0.05 were obtained for some of the gait indicators evaluated at the point in the gait cycle where they were most variable. The correlation between GSN and ADR was sensitive to gait indicator and Poincaré sections evaluated but not to steady state variability correction. The current work suggests some simple steps to reduce the sensitivity of GSN to arbitrary and subjective factors. Overall, the findings support the potential of GSN to be a clinically applicable measure of falls risk. Further study is required to identify methods to more definitively select the various factors within the GSN calculation and to confirm its ability to predict falls risk in human subjects.

Influence of neuromuscular noise and walking speed on fall risk and dynamic stability in a 3D dynamic walking model

Older adults and those with increased fall risk tend to walk slower. They may do this voluntarily to reduce their fall risk. However, both slower and faster walking speeds can predict increased risk of different types of falls. The mechanisms that contribute to fall risk across speeds are not well known. Faster walking requires greater forward propulsion, generated by larger muscle forces. However, greater muscle activation induces increased signal-dependent neuromuscular noise. These speed-related increases in neuromuscular noise may contribute to the increased fall risk observed at faster walking speeds. Using a 3D dynamic walking model, we systematically varied walking speed without and with physiologically-appropriate neuromuscular noise. We quantified how actual fall risk changed with gait speed, how neuromuscular noise affected speed-related changes in fall risk, and how well orbital and local dynamic stability measures predicted changes in fall risk across speeds. When we included physiologically-appropriate noise to the 'push-off' force in our model, fall risk increased with increasing walking speed. Changes in kinematic variability, orbital, and local dynamic stability did not predict these speed-related changes in fall risk. Thus, the increased neuromuscular variability that results from increased signal-dependent noise that is necessitated by the greater muscular force requirements of faster walking may contribute to the increased fall risk observed at faster walking speeds. The lower fall risk observed at slower speeds supports experimental evidence that slowing down can be an effective strategy to reduce fall risk. This may help explain the slower walking speeds observed in older adults and others.

Gait stability in children with Cerebral Palsy

Children with unilateral Cerebral Palsy (CP) have several gait impairments, amongst which impaired gait stability may be one. We tested whether a newly developed stability measure (the foot placement estimator, FPE) which does not require long data series, can be used to asses gait stability in typically developing (TD) children as well as children with CP. In doing so, we tested the FPE's sensitivity to the assumptions needed to calculate this measure, as well as the ability of the FPE to detect differences in stability between children with CP and TD children, and differences in walking speed. Participants were asked to walk at two different speeds, while gait kinematics were recorded. From these data, the FPE, as well as the error that violations of assumptions of the FPE could have caused were calculated. The results showed that children with CP walked with marked instabilities in anterior-posterior and mediolateral directions. Furthermore, errors caused by violations of assumptions in calculation of FPE were only small (≈ 1.5 cm), while effects of walking speed (≈ 20 cm per m/s increase in walking speed) and group (≈ 5 cm) were much larger. These results suggest that the FPE may be used to quantify gait stability in TD children and children with CP.

Assessing the stability of human locomotion: a review of current measures

Falling poses a major threat to the steadily growing population of the elderly in modern-day society. A major challenge in the prevention of falls is the identification of individuals who are at risk of falling owing to an unstable gait. At present, several methods are available for estimating gait stability, each with its own advantages and disadvantages. In this paper, we review the currently available measures: the maximum Lyapunov exponent (λS and λL), the maximum Floquet multiplier, variability measures, long-range correlations, extrapolated centre of mass, stabilizing and destabilizing forces, foot placement estimator, gait sensitivity norm and maximum allowable perturbation. We explain what these measures represent and how they are calculated, and we assess their validity, divided up into construct validity, predictive validity in simple models, convergent validity in experimental studies, and predictive validity in observational studies. We conclude that (i) the validity of variability measures and λS is best supported across all levels, (ii) the maximum Floquet multiplier and λL have good construct validity, but negative predictive validity in models, negative convergent validity and (for λL) negative predictive validity in observational studies, (iii) long-range correlations lack construct validity and predictive validity in models and have negative convergent validity, and (iv) measures derived from perturbation experiments have good construct validity, but data are lacking on convergent validity in experimental studies and predictive validity in observational studies. In closing, directions for future research on dynamic gait stability are discussed.

Local dynamic stability and variability of gait are associated with fall history in elderly subjects

Gait parameters that can be measured with simple instrumentation may hold promise for identifying individuals at risk of falling. Increased variability of gait is associated with increased risk of falling, but research on additional parameters indicates that local dynamic stability (LDS) of gait may also be a predictor of fall risk. The objective of the present study was to assess the association between gait variability, LDS of gait and fall history in a large sample of elderly subjects. Subjects were recruited and tested at a large national fair. One hundred and thirty four elderly, aged 50-75, who were able to walk without aids on a treadmill, agreed to participate. After subjects walked on a treadmill, LDS (higher values indicate more instability) and variability parameters were calculated from accelerometer signals (trunk worn). Fall history was obtained by self-report of falls in the past 12 months. Gait variability and short-term LDS were, individually and combined, positively associated with fall history. In conclusion, both increased gait variability and increased short-term LDS are possible risk factors for falling in the elderly.