Foot trajectory approximation using the pendulum model of walking

Generating a natural foot trajectory is an important objective in robotic systems for rehabilitation of walking. Human walking has pendular properties, so the pendulum model of walking has been used in bipedal robots which produce rhythmic gait patterns. Whether natural foot trajectories can be produced by the pendulum model needs to be addressed as a first step towards applying the pendulum concept in gait orthosis design. This study investigated circle approximation of the foot trajectories, with focus on the geometry of the pendulum model of walking. Three able-bodied subjects walked overground at various speeds, and foot trajectories relative to the hip were analysed. Four circle approximation approaches were developed, and best-fit circle algorithms were derived to fit the trajectories of the ankle, heel and toe. The study confirmed that the ankle and heel trajectories during stance and the toe trajectory in both the stance and the swing phases during walking at various speeds could be well modelled by a rigid pendulum. All the pendulum models were centred around the hip with pendular lengths approximately equal to the segment distances from the hip. This observation provides a new approach for using the pendulum model of walking in gait orthosis design.     

Control of foot trajectory in walking toddlers: adaptation to load changes

On earth, body weight is an inherent constraint, and accordingly, load-regulating mechanisms play an important role in terrestrial locomotion. How do toddlers deal with the effects of their full body weight when faced with the task of independent upright locomotion for the first time? Here we studied the effect of load variation on walking in 12 toddlers during their first unsupported steps, 15 older children (1.3-5 yr), and 10 adults. To simulate various levels of body weight, an experimenter held the trunk of the subject with both hands and supplied an approximately constant vertical force during stepping on a force platform. During unsupported stepping, the shape of the foot path in toddlers (typically single-peak toe trajectory) was different from that of adults and older children (double-peak trajectory). In contrast to adults and older children, who showed only limited changes in kinematic coordination, the "reduced-gravity" condition considerably affected the shape of the foot path in toddlers: they tended to make a high lift and forward foot overshoot at the end of swing. In addition, stepping at high levels of body unloading was characterized by a significant change in the initial direction of foot motion during early swing. Intermediate walkers (1.5-5 mo after walking onset) showed only partial improvement in foot trajectory characteristics. The results suggest that, at the onset of walking, changes in vertical body loads are not compensated accurately by the kinematic controllers; compensation necessitates a few months of independent walking experience.     

The healing balm effect: using a walking group to feel better

This qualitative study combines methods from grounded theory and ethnography, exploring members' experience of a mental health day service walking group, with regards psychological benefits of the physical activity, the outdoor environment and the social setting. Data were collected via participant observation, interviews and a group discussion, and analysed using a grounded theory approach. An overall model, the Healing Balm Effect, brings together seven categories, which describe healing properties of the experience. The categories are: Closer to What is More Natural; Feeling Safe; Being Part; Striving; Getting Away; Being Me; Finding Meaning. Clinical relevance and methodological issues are discussed.     

Quasi-steady state approximation to a fungal growth model

In a previous paper, we proposed a fungal growth model (Lamouret al., 2001 IMA J. Math. Appl. Med. Biol., 17, 329–346),describing the colonization and decomposition of substrate,subsequent uptake of nutrients, and incorporation into fungalbiomass, and performed an overall-steady-state analysis. Inthis paper we assume that where nutrient dynamics are much fasterthan the dynamics of fungal biomass and substrate, the systemwill reach a quasi-steady-state relatively quickly. We showhow the quasi-steady-state approximation is a simplificationof the full fungal growth model. We then derive an explicitfungal invasion criterion, which was not possible for the fullmodel, and characterize parameter domains for invasion and extinction.Importantly, the fungal invasion criterion takes two forms:one for systems where carbon is limiting, another for systemswhere nitrogen is limiting. We focus attention on what happensin the short term immediately following the introduction ofa fungus to a fungal-free system by analysing the stabilityof the trivial steady state, and then check numerically whetherthe fungus is able to persist. The derived invasion criterionwas found to be valid also for the full model. Knowledge ofthe factors that determine invasion is essential to an understandingof fungal dynamics. The simplified model allows the invasioncriterion to be tested with experimental data.     

Foot contact timing and the effect of walking speed in normal childhood and adult gait

Gait foot contact timing patterns are analysed to determine their relationship with walking speed and the effects of age. The effects on these relationships of normalising times by stride time and normalising speeds by stature are also determined. A total of 272 foot contact patterns from 47 normal subjects with an age range of 4 to 39 years are used in this study. The onset and offset foot contact times and walking speed comprise the data set. Two mathematical models are developed, an exponential one for the actual times and a linear one for the normalised times, using a minimum mean square error criterion. A statistical test using the F-ratio criterion is presented and used to compare the characteristic regressions of different age groups. The tests determine if the model regressions are either parallel or coincident. The results indicate that normalising by stride time is the most effective in reducing age differences. A trend is observed which differs from previous investigators' belief that mature gait is developed by age four. Similarities in all ages are revealed by the parallelism of the regressions but a distinct difference exists between the children's and adult's groups.     

Light dosimetry using the P3 approximation

In earlier work, we demonstrated that radiance, calculated using the P3 approximation in a plane wave geometry, could be used to accurately predict the optical parameters of an Intralipid/methylene blue phantom. Plane wave geometry is impractical for clinical use but the results of this work encouraged us to further develop the P3 approximation for a spherical geometry, described in this paper. Radiance predicted by this model for a defined Intralipid/methylene blue phantom was compared with radiance measured in this phantom. The results demonstrate that the spherical derivation of the P3 approximation will reproducibly predict optical parameters of a tissue phantom as effectively as the slab geometry derivation of the P3 approximation. In a similar protocol, the P3 approximation was used to estimate the optical parameters of ex vivo human prostate. Radiance in this case was measured in the prostate samples using an after loading technique. Three prostate samples tested were found to be surprisingly optically homogeneous. The after loading protocol described in this paper could form the basis of a minimally invasive and effective clinical method to optically characterize human prostate.     

The dynamics of postural sway cannot be captured using a one-segment inverted pendulum model: a PCA on segment rotations during unperturbed stance

Research on unperturbed stance is largely based on a one-segment inverted pendulum model. Recently, an increasing number of studies report a contribution of other major joints to postural control. Therefore this study evaluates whether the conclusions originating from the research based on a one-segment model adequately capture postural sway during unperturbed stance. High-pass filtered kinematic data (cutoff frequency 1/30 Hz) obtained over 3 min of unperturbed stance were analyzed in different ways. Variance of joint angles was analyzed. Principal-component analysis (PCA) was performed on the variance of lower leg, upper leg, and head-arms-trunk (HAT) angles, as well as on lower leg and COM angle (the orientation of the line from ankle joint to center of mass). It was found that the variance in knee and hip joint angles did not differ from the variance found in the ankle angle. The first PCA component indicated that, generally, the upper leg and HAT segments move in the same direction as the lower leg with a somewhat larger amplitude. The first PCA component relating ankle angle variance and COM angle variance indicated that the ankle joint angle displacement gives a good estimate of the COM angle displacement. The second PCA component on the segment angles partly explains the apparent discrepancy between these findings because this component points to a countermovement of the HAT relative to the ankle joint angle. It is concluded that postural control during unperturbed stance should be analyzed in terms of a multiple inverted pendulum model.     

Adaptation of the walking pattern to uphill walking in normal and spinal-cord injured subjects

 Lower-limb movements and muscle-activity patterns were assessed from seven normal and seven ambulatory subjects with incomplete spinal-cord injury (SCI) during level and uphill treadmill walking (5, 10 and 15°). Increasing the treadmill grade from 0° to 15° induced an increasingly flexed posture of the hip, knee and ankle during initial contact in all normal subjects, resulting in a larger excursion throughout stance. This adaptation process actually began in mid-swing with a graded increase in hip flexion and ankle dorsiflexion as well as a gradual decrease in knee extension. In SCI subjects, a similar trend was found at the hip joint for both swing and stance phases, whereas the knee angle showed very limited changes and the ankle angle showed large variations with grade throughout the walking cycle. A distinct coordination pattern between the hip and knee was observed in normal subjects, but not in SCI subjects during level walking. The same coordination pattern was preserved in all normal subjects and in five of seven SCI subjects during uphill walking. The duration of electromyographic (EMG) activity of thigh muscles was progressively increased during uphill walking, whereas no significant changes occurred in leg muscles. In SCI subjects, EMG durations of both thigh and leg muscles, which were already active throughout stance during level walking, were not significantly affected by uphill walking. The peak amplitude of EMG activity of the vastus lateralis, medial hamstrings, soleus, medial gastrocnemius and tibialis anterior was progressively increased during uphill walking in normal subjects. In SCI subjects, the peak amplitude of EMG activity of the medial hamstrings was adapted in a similar fashion, whereas the vastus lateralis, soleus and medial gastrocnemius showed very limited adaptation during uphill walking. We conclude that SCI subjects can adapt to uphill treadmill walking within certain limits, but they use different strategies to adapt to the changing locomotor demands. Received: 10 March 1998 / Accepted: 29 December 1998     

Three-dimensional kinematics and dynamics of the foot during walking: a model of central control mechanisms

The foot is a critical interface between the body and supporting surface during walking, but there is no coherent framework on which to model the dynamics of the stance and swing phases. To establish this framework, we studied the rotational and translational dynamics of foot movement in three dimensions with a motion detection system (OPTOTRAK), while subjects walked on a treadmill. Positions, velocities, and durations were normalized to leg-length and gravity. Foot position and rotation at toe-off were closely related to walking velocity. Foot pitch at toe clearance increased with walking velocity, but the medial–lateral and vertical toe positions were unaltered. Phase–plane trajectories along the fore-aft direction, i.e., plots of toe velocity versus position, were circular during the swing phases, with radii proportional to walking velocity. Peak forward, lateral, and upward velocities were linearly related to corresponding excursions, forming main sequences. A second order model predicted the changes in toe position and velocity, and the approximately hyperbolic decrements in duration as a function of walking velocity. The model indicates that the foot is controlled in an overdamped manner during the stance phase and as a feedback-controlled undamped pendulum during the swing. The data and model suggest that the state of the foot at toe-off, set by walking velocity during the stance phase, determines the dynamics of the swing phase. Thus, in addition to determining locomotion kinematics, walking velocity plays a critical role in determining the phase–plane trajectories and main sequence relationships of foot movements during the swing phases.     

Erratum to: Building a mouse model hallmarking the congenital human cytomegalovirus infection in central nervous system

Archives of Virology -