Spring-loaded inverted pendulum goes through two contraction-extension cycles during the single-support phase of walking

Quantifying changes in individual-specific template-based representations of center-of-mass dynamics during walking with ankle exoskeletons using Hybrid-SINDy

Ankle exoskeletons alter whole-body walking mechanics, energetics, and stability by altering center-of-mass (CoM) motion. Controlling the dynamics governing CoM motion is, therefore, critical for maintaining efficient and stable gait. However, how CoM dynamics change with ankle exoskeletons is unknown, and how to optimally model individual-specific CoM dynamics, especially in individuals with neurological injuries, remains a challenge. Here, we evaluated individual-specific changes in CoM dynamics in unimpaired adults and one individual with post-stroke hemiparesis while walking in shoes-only and with zero-stiffness and high-stiffness passive ankle exoskeletons. To identify optimal sets of physically interpretable mechanisms describing CoM dynamics, termed template signatures, we leveraged hybrid sparse identification of nonlinear dynamics (Hybrid-SINDy), an equation-free data-driven method for inferring sparse hybrid dynamics from a library of candidate functional forms. In unimpaired adults, Hybrid-SINDy automatically identified spring-loaded inverted pendulum-like template signatures, which did not change with exoskeletons (p > 0.16), except for small changes in leg resting length (p < 0.001). Conversely, post-stroke paretic-leg rotary stiffness mechanisms increased by 37-50% with zero-stiffness exoskeletons. While unimpaired CoM dynamics appear robust to passive ankle exoskeletons, how neurological injuries alter exoskeleton impacts on CoM dynamics merits further investigation. Our findings support Hybrid-SINDy's potential to discover mechanisms describing individual-specific CoM dynamics with assistive devices.

A simple model of human walking

Aim. We investigate Alexander’s inverted pendulum model, the simplest mathematical model of human walking. Although it successfully explains some kinematic features of human walking, such as the velocity of the body's centre of mass, it does not account for others, like the vertical reaction force and the maximum walking speed. This paper aims to minimally extend Alexander’s model in such a way as to make it a viable and quantitative model of human walking for clinical biomechanics.Material and methods. In order to compare the predictions of Alexander’s model with experimental data on walking, we incorporate in it a robust phenomenological relation between stride frequency and stride length derived in the literature, and we introduce a step-angle dependent muscle force along the pendulum. We then analytically solve the pendulum's motion equation and find the corresponding analytical expression for the average walking speed.Results. The values of the average walking speed for different heights predicted by our model are in excellent agreement with the ones obtained in treadmill experiments. Moreover, it successfully predicts the observed walking-running transition speed, which occurs when the stride length equals the height of an individual. Finally, our extended model satisfactorily reproduces the experimentally observed ground reaction forces in the midstance and terminal stance phases. Consequently, the predicted value of the (height-dependent) maximum walking speed is in reasonable agreement with the one obtained in more sophisticated models of human walking.Conclusions. Augmented with our minimal extensions, Alexander’s model becomes an effective and realistic model of human walking applicable in clinical investigations of the human gate.

Development of a gait speed estimation model for healthy older adults using a single inertial measurement unit

Although gait speed changes are associated with various geriatric conditions, standard gait analysis systems, such as laboratory-based motion capture systems or instrumented walkways, are too expensive, spatially limited, and difficult to access. A wearable inertia sensor is cheap and easy to access; however, its accuracy in estimating gait speed is limited. In this study, we developed a model for accurately estimating the gait speed of healthy older adults using the data captured by an inertia sensor placed at their center of body mass (CoM). We enrolled 759 healthy older adults from two population-based cohort studies and asked them to walk on a 14 m long walkway thrice at comfortable paces with an inertia sensor attached to their CoM. In the middle of the walkway, we placed GAITRite™ to obtain the gold standard of gait speed. We then divided the participants into three subgroups using the normalized step length and developed a linear regression model for estimating the gold standard gait speed using age, foot length, and the features obtained from an inertia sensor, including cadence, vertical height displacement, yaw angle, and role angle of CoM. Our model exhibited excellent accuracy in estimating the gold standard gait speed (mean absolute error = 3.74%; root mean square error = 5.30 cm/s; intraclass correlation coefficient = 0.954). Our model may contribute to the early detection and monitoring of gait disorders and other geriatric conditions by making gait assessment easier, cheaper, and more ambulatory while remaining as accurate as other standard gait analysis systems.

Kinematics of sea star legged locomotion

Sea stars have slower crawling and faster bouncing gaits. Both speed and oscillation amplitude increase during the transition from crawling to oscillating. In the bouncy gait, oscillating vertical velocities precede oscillating horizontal velocities by 90 deg, as reflected by clockwise circular hodographs. Potential energy precedes horizontal kinetic energy by 9.6 deg and so they are nearly in phase. These phase relationships resemble terrestrial running gaits, except that podia are always on the ground. Kinetic and potential energy scale with body mass as Mb   1.1, with the change in kinetic energy consistently two orders of magnitude less, indicating that efficient exchange is not feasible. Frequency of the bouncy gait scales with Mb-0.14, which is similar to continuously running vertebrates and indicates that gravitational forces are important. This scaling differs from the Hill model, in which scaling of muscle forces determine frequency. We propose a simple torque-stabilized inverted pendulum (TS-IP) model to conceptualize the dynamics of this gait. The TS-IP model incorporates mathematics equivalent to an angular spring, but implemented by a nearly constant upward force generated by the podia in each step. That upward force is just larger than the force required to sustain the underwater weight of the sea star. Even though the bouncy gait is the rapid gait for these sea stars, the pace of movement is still very slow. In fact, the observed Froude numbers (10-2 to 10-3) are much lower than those typical of vertebrate locomotion and are as low or lower than those reported for slow-walking fruit flies, which are the lowest values for pedestrian Froude numbers of which we are aware.

Simple models highlight differences in the walking biomechanics of young children and adults

Adults conserve metabolic energy during walking by minimizing the step-to-step transition work performed by the legs during double support and by utilizing spring-like mechanisms in their legs, but little is known as to whether children utilize these same mechanisms. To gain a better understanding, we studied how children (5-6 years) and adults modulate the mechanical and metabolic demands of walking at their preferred speed, across slow (75%), preferred (100%) and fast (125%) step frequencies. We quantified (1) the positive mass-specific work done by the trailing leg during step-to-step transitions and (2) the leg's spring-like behavior during single support. On average, children walked with a 36% greater net cost of transport (COT; J kg-1 m-1) than adults (P=0.03), yet both groups increased their net COT at varying step frequencies. After scaling for speed, children generated ∼2-fold less trailing limb positive scaled mechanical work during the step-to-step transition (P=0.02). Unlike adults, children did not modulate their trailing limb positive work to meet the demands of walking at 75% and 125% of their preferred step frequency. In single support, young children operated their stance limb with much greater compliance than adults ( versus 11.35; P=0.023). Our observations suggest that the mechanics of walking in children aged 5-6 years are fundamentally distinct from the mechanics of walking in adults and may help to explain a child's higher net COT. These insights have implications for the design of assistive devices for children and suggest that children cannot be simply treated as scaled-down versions of adults.

Drosophila uses a tripod gait across all walking speeds, and the geometry of the tripod is important for speed control

Changes in walking speed are characterized by changes in both the animal's gait and the mechanics of its interaction with the ground. Here we study these changes in walking Drosophila. We measured the fly's center of mass movement with high spatial resolution and the position of its footprints. Flies predominantly employ a modified tripod gait that only changes marginally with speed. The mechanics of a tripod gait can be approximated with a simple model - angular and radial spring-loaded inverted pendulum (ARSLIP) - which is characterized by two springs of an effective leg that become stiffer as the speed increases. Surprisingly, the change in the stiffness of the spring is mediated by the change in tripod shape rather than a change in stiffness of individual legs. The effect of tripod shape on mechanics can also explain the large variation in kinematics among insects, and ARSLIP can model these variations.

Change in telescoping leg strategy with varying walking speed to modulate force advantage

Human walking consists of two major sequential events (i.e., single- and double-support phases). Although there have been many studies relating to basic principles of the each stage, how the two distinct but continuous phases interact with each other remains to be clarified. We examined the change in walking strategy with varying walking speed on a local reference frame with telescoping and tangential axes; we expect that the telescoping directional dynamics at the end of a single-support phase change with walking speed to facilitate the modulation of the push-off work during a double-support phase. The telescoping directional force and power are calculated using two methods: model simulation and kinematic configuration. The empirical walking data for eight healthy young subjects and the corresponding model parameters obtained from a data-fit optimization were used to investigate the changing trend of each factor (i.e., force and power) with the increase in speed. The resulting force at the end of the single-support phase significantly increased with the walking speed for both methods, whereas the resulting power remained nearly unchanged and was close to zero for the entire range of walking speeds. This result implies that the positive amount of the telescoping directional force at the end of the single-support phase may be a certain type of preparation for the double-support phase, which can contribute to a larger push-off.