A mathematical model of the stability control of human thorax and pelvis movements during walking

A mathematical model is developed to study the human thorax and pelvis movements in the frontal plane during normal walking. The model comprises of two-link base-excited inverted pendulums with one-degree of rotational freedom for each link. Since the linear motion of the pelvis has a significant effect on the upper body stability, this effect is included in the model by having a base point moving in the frontal plane in a general way. Furthermore, because the postural stability is the primary requirement of normal human walking, the control law is developed based on Lyapunov's stability theory, which guarantees the stability of the pendulum system around the up-right position. To evaluate the model, the simulation results, including the angular displacement of each link and the torque applied on each link, are compared with those from gait measurements. It is shown that the simulation results match those from gait measurements closely. These results suggest that the proposed model can provide a useful framework for analysis of postural control mechanisms.     

Evaluation of a generalized model of human postural dynamics and control in the sagittal plane

A two-step identification method is used to evaluate a generalized model of human postural control in the sagittal plane. Postural dynamics are represented as a planar open-chain linkage system supported by a triangular foot. The control mechanism is modeled as a state feedback element in which the torque acting at a given link is an arbitrary function of the state variables — angles and angular velocities. To validate the approach, six normal subjects underwent two series of experiments. The first series were used to determine an appropriate model of the system dynamics. The second series were used to estimate the parameters of the feedback model. A computer simulation of the complete system shows that the model predictions closely match the observed responses. These results suggest that the proposed model provides a useful framework for analysis of postural control mechanisms.This work was supported by the National Institutes of Health under Grant NS 21363     

Modular control of human walking: A simulation study

Recent evidence suggests that performance of complex locomotor tasks such as walking may be accomplished using a simple underlying organization of co-active muscles, or “modules”, which have been assumed to be structured to perform task-specific biomechanical functions. However, no study has explicitly tested whether the modules would actually produce the biomechanical functions associated with them or even produce a well-coordinated movement. In this study, we generated muscle-actuated forward dynamics simulations of normal walking using muscle activation modules (identified using non-negative matrix factorization) as the muscle control inputs to identify the contributions of each module to the biomechanical sub-tasks of walking (i.e., body support, forward propulsion, and leg swing). The simulation analysis showed that a simple neural control strategy involving five muscle activation modules was sufficient to perform the basic sub-tasks of walking. Module 1 (gluteus medius, vasti, and rectus femoris) primarily contributed to body support in early stance while Module 2 (soleus and gastrocnemius) contributed to both body support and propulsion in late stance. Module 3 (rectus femoris and tibialis anterior) acted to decelerate the leg in early and late swing while generating energy to the trunk throughout swing. Module 4 (hamstrings) acted to absorb leg energy (i.e., decelerate it) in late swing while increasing the leg energy in early stance. Post-hoc analysis revealed an additional module (Module 5: iliopsoas) acted to accelerate the leg forward in pre- and early swing. These results provide evidence that the identified modules can act as basic neural control elements that generate task-specific biomechanical functions to produce well-coordinated walking.     

A mathematical model of hind-limb control in cats when walking backward

The “walking backward” mode was achieved within a single model of cat hind-limb locomotion with the balance maintenance only due to a change in the controlling actions (in addition to the “forward walking” mode). The skeletal part of the model contains the spine, pelvis, and two limbs consisting of the thigh, shin, and foot. The hip joint and spine mount in the thoracic region have three degrees of freedom; the knee and ankle joints have one degree of freedom. The pelvis is rigidly connected to the spine. Control is performed by model muscles (flexors and extensors of the thigh, shin, and foot). The muscle activation is performed by the effects that are typical for motoneurons that control the muscles. The feet in the support phase touch the treadmill, which moves at a constant speed. The model qualitatively reproduces multiple characteristics of feline movements during forward and backward walking (supporting its validity).     

Three-dimensional modular control of human walking

Recent studies have suggested that complex muscle activity during walking may be controlled using a reduced neural control strategy organized around the co-excitation of multiple muscles, or modules. Previous computer simulation studies have shown that five modules satisfy the sagittal-plane biomechanical sub-tasks of 2D walking. The present study shows that a sixth module, which contributes primarily to mediolateral balance control and contralateral leg swing, is needed to satisfy the additional non-sagittal plane demands of 3D walking. Body support was provided by Module 1 (hip and knee extensors, hip abductors) in early stance and Module 2 (plantarflexors) in late stance. In early stance, forward propulsion was provided by Module 4 (hamstrings), but net braking occurred due to Modules 1 and 2. Forward propulsion was provided by Module 2 in late stance. Module 1 accelerated the body medially throughout stance, dominating the lateral acceleration in early stance provided by Modules 4 and 6 (adductor magnus) and in late stance by Module 2, except near toe-off. Modules 3 (ankle dorsiflexors, rectus femoris) and 5 (hip flexors and adductors except adductor magnus) accelerated the ipsilateral leg forward in early swing whereas Module 4 decelerated the ipsilateral leg prior to heel-strike. Finally, Modules 1, 4 and 6 accelerated the contralateral leg forward prior to and during contralateral swing. Since the modules were based on experimentally measured muscle activity, these results provide further evidence that a simple neural control strategy involving muscle activation modules organized around task-specific biomechanical functions may be used to control complex human movements.     

Active control of lateral balance in human walking

We measured variability of foot placement during gait to test whether lateral balance must be actively controlled against dynamic instability. The hypothesis was developed using a simple dynamical model that can walk down a slight incline with a periodic gait resembling that of humans. This gait is entirely passive except that it requires active control for a single unstable mode, confined mainly to lateral motion. An especially efficient means of controlling this instability is to adjust lateral foot placement. We hypothesized that similar active feedback control is performed by humans, with fore-aft dynamics stabilized either passively or by very low-level control. The model predicts that uncertainty within the active feedback loop should result in variability in foot placement that is larger laterally than fore-aft. In addition, loss of sensory information such as by closing the eyes should result in larger increases in lateral variability. The control model also predicts a slight coupling between step width and length. We tested 15 young normal human subjects and found that lateral variability was 79% larger than fore-aft variability with eyes open, and a larger increase in lateral variability (53% vs. 21%) with eyes closed, consistent with the model's predictions. We also found that the coupling between lateral and fore-aft foot placements was consistent with a value of 0.13 predicted by the control model. Our results imply that humans may harness passive dynamic properties of the limbs in the sagittal plane, but must provide significant active control in order to stabilize lateral motion.     

A model of human walking energetics with an elastically-suspended load

Elastically-suspended loads have been shown to reduce the peak forces acting on the body while walking with a load when the suspension stiffness and damping are minimized. However, it is not well understood how elastically-suspended loads can affect the energetic cost of walking. Prior work shows that elastically suspending a load can yield either an increase or decrease in the energetic cost of human walking, depending primarily on the suspension stiffness, load, and walking speed. It would be useful to have a simple explanation that reconciles apparent differences in existing data. The objective of this paper is to help explain different energetic outcomes found with experimental load suspension backpacks and to systematically investigate the effect of load suspension parameters on the energetic cost of human walking. A simple two-degree-of-freedom model is used to approximate the energetic cost of human walking with a suspended load. The energetic predictions of the model are consistent with existing experimental data and show how the suspension parameters, load mass, and walking speed can affect the energetic cost of walking. In general, the energetic cost of walking with a load is decreased compared to that of a stiffly-attached load when the natural frequency of a load suspension is tuned significantly below the resonant walking frequency. The model also shows that a compliant load suspension is more effective in reducing the energetic cost of walking with low suspension damping, high load mass, and fast walking speed. This simple model could improve our understanding of how elastic load-carrying devices affect the energetic cost of walking with a load.     

Ankle torque control that shifts the center of pressure from heel to toe contributes non-zero sagittal plane angular momentum during human walking

A principle objective of human walking is controlling angular motion of the body as a whole to remain upright. The force of the ground on each foot (F) reflects that control, and recent studies show that in the sagittal plane F exhibits a specific coordination between F direction and center-of-pressure (CP) that is conducive to remaining upright. Typical walking involves the CP shifting relative to the body due to two factors: posterior motion of the foot with respect to the hip (stepping) and motion of the CP relative to the foot (foot roll-over). Recent research has also shown how adjusting ankle torque alone to shift CP relative to the foot systematically alters the direction of F, and thus, could play a key role in upright posture and the F measured during walking. This study explores how the CP shifts due to stepping and foot roll-over contribute to the observed F and its role in maintaining upright posture. Experimental walking kinetics and kinematics were combined with a mechanical model of the human to show that variation in F that was not attributable to foot roll-over had systematic correlation between direction and CP that could be described by an intersection point located near the center-of-mass. The findings characterize a component of walking motor control, describe how typical foot roll-over contributes to postural control, and provide a rationale for the increased fall risk observed in individuals with atypical ankle muscle function.     

Muscle-tendon interaction and elastic energy usage in human walking.

The present study was designed to explore how the interaction between the fascicles and tendinous tissues is involved in storage and utilization of elastic energy during human walking. Eight male subjects walked with a natural cadence (1.4 +/- 0.1 m/s) on a 10-m-long force plate system. In vivo techniques were employed to record the Achilles tendon force and to scan real-time fascicle lengths for two muscles (medial gastrocnemius and soleus). The results showed that tendinous tissues of both medial gastrocnemius and soleus muscles lengthened slowly throughout the single-stance phase and then recoiled rapidly close to the end of the ground contact. However, the fascicle length changes demonstrated different patterns and amplitudes between two muscles. The medial gastrocnemius fascicles were stretched during the early single-stance phase and then remained isometrically during the late-stance phase. In contrast, the soleus fascicles were lengthened until the end of the single-stance phase. These findings suggest that the elastic recoil takes place not as a spring-like bouncing but as a catapult action in natural human walking. The interaction between the muscle fascicles and tendinous tissues plays an important role in the process of release of elastic energy, although the leg muscles, which are commonly accepted as synergists, do not have similar mechanical behavior of fascicles in this catapult action.     

Synthesis of human walking: A planar model for single support

A mathematical model for the single support phase of normal, level, human walking is formulated. The motion of the lower extremity is synthesized using a preprogrammed set of inputs, recognized by the model as a simple collection of applied joint moments.

Two mechanisms are forwarded as candidates for producing the observed peaks in the vertical ground reaction. The first, stance knee flexion-extension, generates the necessary level of whole-body vertical acceleration during the initial region of single support (opposite toe-off to heel-off). A model accounting for the determinants of foot and knee interaction then predicts the second peak to be the result of an increasing ankle moment in the region from heel-off to opposite heel-strike.     

A negative control model of hormone receptor interaction.

An original model of hormone-receptor interaction is proposed: the hormone would interact with its receptor by dissociating a regulatory unit from a catalytic unit, which, freed from the negative constraint exerted by the regulatory unit, would become active. It is shown that such a model shares some features with negatively cooperative models: 1. Scatchard plot has a hyperbolic shape; 2. Dissociation of bound labeled hormone, promoted by chemical dilution is enhanced by an excess of cold hormone. A new graph, in which the ratio between the square of bound hormone concentration and the free hormone concentration is plotted against the bound hormone concentration, allows a discrimination between both models. Moreover, the behavior of a negatively controlled receptor unlike a negatively cooperative one, is critically dependent on its physical state (soluble or particulate). The possible application of this model to hormone responsive adenylate cyclases and other biochemical systems, such as cAMP dependent protein kinases, is mentioned.     

A three-dimensional whole-body model to predict human walking on level ground

Predictive simulation of human walking has great potential in clinical motion analysis and rehabilitation engineering assessment, but large computational cost and reliance on measurement data to provide initial guess have limited its wide use. We developed a computationally efficient model combining optimization and inverse dynamics to predict three-dimensional whole-body motions and forces during human walking without relying on measurement data. Using the model, we explored two different optimization objectives, mechanical energy expenditure and the time integral of normalized joint torque. Of the two criteria, the sum of the time integrals of the normalized joint torques produced a more realistic walking gait. The reason for this difference is that most of the mechanical energy expenditure is in the sagittal plane (based on measurement data) and this leads to difficulty in prediction in the other two planes. We conclude that mechanical energy may only account for part of the complex performance criteria driving human walking in three dimensions.

     

A biomechanical model of sagittal tongue bending

The human tongue is a structurally complex and extremely flexible organ. In order to better understand the mechanical basis for lingual deformations, we modeled a primitive movement of the tongue, sagittal tongue bending. We hypothesized that sagittal bending is a synergistic deformation derived from co-contraction of the longitudinalis and transversus muscles. Our model of tongue bending was based on classical bimetal strip theory, in which curvature is produced when one muscle layer contracts more so than another. Contraction was modulated via mismatched thermal expansion coefficients and temperature change (to simulate muscular contraction). Our results demonstrated that synergistic contraction produced curvature and strain results which were in better correspondence to empirical results derived from tagging MRI than were the results of contraction of the longitudinalis muscle alone. This fundamental reliance of tongue bending on the synergistic contraction of its intrinsic fibers supports the muscular hydrostat theory of tongue function.     

Robust control of CPG-based 3D neuromusculoskeletal walking model

This paper proposes a method for enhancing the robustness of the central pattern generator (CPG)-based three-dimensional (3D) neuromusculoskeletal walking controller. The CPG has been successfully applied to walking controllers and controllers for walking robots. However, the robustness of walking motion with the CPG-based controller is not sufficient, especially when subjected to external forces or environmental variations. To achieve a realistic and stable walking motion of the controller, we propose the use of an attracting controller in parallel with the CPG-based controller. The robustness of the proposed controller is confirmed through simulation results.     

A manipulability analysis of human walking

From the literature of biomechanics, it is now clear that humans use elevating, lowering and delayed-lowering strategies in order to maintain stability during perturbed walking. The main purpose of this study is to provide insights into the role of manipulability in selection of these strategies. A 37 degrees of freedom (DoFs) model of the human body is developed to evaluate the manipulability indices during walking. The model is considered as a tree-like structure and its forward kinematics equations and the Jacobian are derived based on the Denavit-Hartenberg (DH) convention. A hybrid genetic algorithm (HGA) is then employed to map the experimental kinematics of a human to the model. The kinematic and dynamic manipulability indices of the swing phase of walking are evaluated concentrating on early, mid and late swing phases. The results indicate that the manipulability indices can characterize well the selection of elevating, lowering and delayed-lowering strategies at different stages of the swing phase. The results kinematically describe the reason of selecting delayed-lowering strategy at mid-swing phase that was not obvious in previous studies. Moreover, the results show that at mid-swing phase of walking the kinematic maneuverability is lower than that of the early and late swing phases.     

A control model of human tongue movements in speech

Tongue movements during speech production have been investigated by means of a simple yet realistic biomechanical model, based on a finite elements modeling of soft tissues, in the framework of the equilibrium point hypothesis (-model) of motor control. In particular, the model has been applied to the estimation of the “central” control commands issued to the muscles, for a data set of mid-sagittal digitized tracings of vocal tract shape, r ecorded by means of low-intensity X-ray cineradiographies during speech. In spite of the highly non-linear mapping between the shape of the oral cavity and its acoustic consequences, the organization of control commands preserves the peculiar spatial organization of vowel phonemes in acoustic space. A factor analysis of control commands, which have been decomposed into independent or “orthogonal” muscle groups, has shown that, in spite of the great mobility of the tongue and the highly complex arrangement of tongue muscles, its movements can be explained in terms of the activation of a small number of independent muscle groups, each corresponding to an elementary or “primitive” movement. These results are consistent with the hypothesis that the tongue is controlled by a small number of independent “articulators”, for which a precise biomechanical substrate is provided. The influence of the effect of jaw and hyoid movements on tongue equilibrium has also bee n evaluated, suggesting that the bony structures cannot be considered as a moving frame of reference, but, indeed, there may be a substantial interaction between them and the tongue, that may only be accounted for by a “global” model. The reported results also define a simple control model for the tongue and, in analogy with similar modelling studies, they suggest that, because of the peculiar geometrical arrangement of tongue muscles, the central nervous system (CNS) may not need a de tailed representation of tongue mechanics but rather may make use of a relatively small number of muscle synergies, that are invariant over the whole space of tongue configurations. Received: 27 August 1996 / Accepted in revised form: 25 February 1997     

A linear canal-otolith interaction model to describe the human vestibulo-ocular reflex

A control systems model of the vestibulo-ocular reflex (VOR) originally derived for yaw rotation about an eccentric axis (Crane et al. 1997) was applied to data collected during ambulation and dynamic posturography. The model incorporates a linear summation of an otolith response due to head translation scaled by target distance, adding to a semi-circular canal response that depends only on angular head rotation. The results of the model were compared with human experimental data by supplying head angular velocity as determined by magnetic search coil recording as the input for the canal branch of the model and supplying linear acceleration as determined by flux gate magnetometer measurements of otolith position. The model was fit to data by determining otolith weighting that enabled the model to best fit the data. We fit to the model experimental data from normal subjects who were: standing quietly, walking, running, or making active sinusoidal head movements. We also fit data obtained during dynamic posturography tasks of: standing on a platform sliding in a horizontal plane at 0.2 Hz, standing directly on a platform tilting at 0.1 Hz, and standing on the tilting platform buffered by a 5-cm thick foam rubber cushion. Each task was done with the subject attending a target approximately 500, 100, or 50 cm distant, both in light and darkness. The model accurately predicted the observed VOR response during each test. Greater otolith weighting was required for near targets for nearly all activities, consistent with weights for the otolith component found in previous studies employing imposed rotations. The only exceptions were for vertical axis motion during standing, sliding, and tilting when the platform was buffered with foam rubber. In the horizontal axis, the model always fit near target data better with a higher otolith component. Otolith weights were similar with the target visible and in darkness. The model predicts eye movement during both passive whole-body rotation and free head movement in space implying that the VOR is controlled by a similar mechanism during both situations. Factors such as vision, proprioception, and efference copy that are available during head free motion but not during whole-body rotation are probably not important to gaze stabilization during ambulation and postural stabilizing movement. The linearity of the canal-otolith interaction was tested by re-analysis of the whole body rotation data on which the model is based (Crane et al. 1997). Normalized otolith-mediated gain enhancement was determined for each axis of rotation. This analysis uncovered minor non-linearities in the canal-otolith interaction at frequencies above 1.6 Hz and when the axis of rotation was posterior to the head. Received: 11 March 1998 / Received in revised form: 1 March 1999     

A multisensory integration model of human stance control

A model is presented to study and quantify the contribution of all available sensory information to human standing based on optimal estimation theory. In the model, delayed sensory information is integrated in such a way that a best estimate of body orientation is obtained. The model approach agrees with the present theory of the goal of human balance control. The model is not based on purely inverted pendulum body dynamics, but rather on a three-link segment model of a standing human on a movable support base. In addition, the model is non-linear and explicitly addresses the problem of multisensory integration and neural time delays. A predictive element is included in the controller to compensate for time delays, necessary to maintain erect body orientation. Model results of sensory perturbations on total body sway closely resemble experimental results. Despite internal and external perturbations, the controller is able to stabilise the model of an inherently unstable standing human with neural time delays of 100 ms. It is concluded, that the model is capable of studying and quantifying multisensory integration in human stance control. We aim to apply the model in (1) the design and development of prostheses and orthoses and (2) the diagnosis of neurological balance disorders. Received: 25 August 1997 / Accepted in revised form: 8 December 1998