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Movement Termination and Stability in Standing

Pai, Yi-Chung

Exercise and Sport Sciences Reviews: January 2003 - Volume 31 - Issue 1 - p 19-25
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PAI, Y.-C. Movement termination and stability in standing. Exerc. Sport Sci. Rev., Vol. 31, No. 1, pp. 19–25, 2003. This brief review develops a new conceptual framework for describing the influence of center-of-mass position and velocity on ability to terminate movement. Limitations of this dynamic conceptual framework and its applications in physical rehabilitation are explored.

Department of Physical Therapy, University of Illinois at Chicago

Accepted for publication: July 22, 2002.

Address for correspondence: Yi-Chung Pai, PhD, MPT, Department of Physical Therapy, University of Illinois at Chicago, 1919 W. Taylor St. 4th floor, M/C 898, Chicago, IL 60612-7251 (E-mail: cpai@uic.edu).

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INTRODUCTION

Upright standing posture of humans is inherently unstable. A primary objective for the central nervous system (CNS) must then be the maintenance of standing stability, which may be characterized by the control of the relative motion between the body’s center of mass (COM) and its base of support (BOS). Loss of balance occurs when the motion state of the COM (i.e., its instantaneous position and velocity with respect to the BOS) exceeds certain stability limits, resulting in the initiation of a fall, whereby upright posture can only be restored by establishing a new BOS. It is only logical to assume that the CNS requires afferent input to monitor and frequently update the current motion state between the COM and the BOS, which can then be readily compared with internal representations of these stability limits. With the aid of this information, the CNS can select appropriate motor responses to achieve the biomechanical objectives of avoiding a fall.

A wide range of motor responses is available for fall prevention. In standing, the control of balance can be achieved by employing ankle/hip movement strategies that keep the COM motion state within its associated stability limits without a loss of balance. Alternatively, grasping and stepping strategies can be used to alter the BOS to recover balance. The corrections generated by ankle/hip movement are typically insufficient to protect against a fall when the disturbance is of large magnitude. One obvious limitation of the grasping strategy is the potential lack of any graspable fixtures where the fall occurs. Large disturbances of balance can seldom be restored without the subject taking a step. The stepping response thus has a unique and irreplaceable importance in the recovery of balance.

In locomotion, the control of the COM and BOS motion state consists of seemingly contradictory objectives of achieving both mobility and stability. For instance, high mobility in walking is accomplished through periods of controlled loss of balance with a constantly moving COM. Each such loss of balance must be interrupted before it proceeds to an actual fall. Periodic stability is accomplished through stepping, whereby the previously established BOS is abandoned and replaced with a new BOS to avert a fall.

Two types of physical limits must commonly exist in standing and in locomotion. One can quantitatively define a loss of balance, and the other, a failure in recovery, resulted in a fall (Fig. 1). First, there must exist certain stability limits between the COM and the BOS, beyond which movement related perturbation (e.g., movements caused by muscle forces or resulted from externally applied forces) cannot be terminated without a loss of balance. Most frequently, a balance loss can be recovered with a change of the BOS. These physical limits correspond to the threshold values that divide Figure 1a and 1b. Nevertheless, there must also exist a recovery limits (e.g., threshold of the COM state and/or threshold of knee moment that set a recovery apart from limb collapse), beyond which a loss of balance will no longer be recoverable, leading to an actual fall. These physical limits correspond to the threshold values that differentiate Figure 1a and 1b from Figure 1c. From the perspective of the sequence of events, however, a robust stability is the first line of defense against a fall. Therefore, to avoid a fall, the CNS must obey physical laws, selecting and scaling a motor action appropriately within the biomechanical stability limits imposed on the motion between the COM and the BOS. An important question thus emerges:what are the stability limits beyond which a loss of balance must occur?

Figure 1

Figure 1

This brief review is intended to focus on a line of ideas and concepts that relates movement termination to the quantification of the stability limits. Specifically, the purposes are to 1) develop a new conceptual framework for describing the influence of center-of-mass position and velocity on ability to terminate movement in the context of stability limits and 2) explore the limitations of this framework and its potential applications in assessing and improving stability. As a first step, this review will focus on maintaining or restoring upright standing posture, rather than the control of the COM and BOS motion state during locomotion. Furthermore, this review will focus on the conditions that lead to a loss of balance without exploring the recovery limits beyond which an actual fall (not merely a loss of balance) will occur.

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STABILITY IN QUIET STANDING

Stability is “the property of a body that causes it when disturbed from a condition of equilibrium or steady motion to develop forces or moments that restore the original condition” (Webster’s 3rd International Dictionary, Unabridged). Stability can be defined in terms of a person’s ability to resist perturbation and to maintain or restore equilibrium state of upright stance, without changing the BOS. Traditionally, the boundary of stability is quantified by limits pertaining to the relative position between the COM and the BOS. As stated by Borelli, who is often considered to be the founding father of contemporary biomechanics, “…the line of support must fall vertically from the center of gravity of the whole body to between the plantar soles or through one sole. Otherwise, the body could not stand and would fall to the side toward which the line joining its center of gravity and the support is inclined” (1). In other words, a forward loss of balance will be initiated when the COM is located anterior to the BOS, and vice versa for a backward loss of balance. This concept of stability is simple yet elegant and has been widely adopted for the evaluation of a person’s ability to control balance. Stability, within this traditional concept, can be conveniently assessed by the amount of body sway (i.e., the displacement of the COM with respect to the BOS), often approximated by the displacement of the center of pressure of the ground reaction forces in standing. It is generally believed that the greater the body sway measured during standing, the closer the COM will approach the limits of the BOS, hence the less stable a person will be. As it becomes likely that the COM will exceed these limits, a loss of balance is expected. However, an apparent limitation with this approach can be demonstrated among patients who have very little body sway during quiet stance but who are also known to be very unstable, as in individuals with Parkinson’s disease (2).

This concept reflects primarily a static view of stability and, arguably, is suitable only when one’s COM velocity is negligible. It is possible that body sway evaluated during quiet standing will not be an appropriate indicator of stability in daily living, when the velocity of the COM is neither small nor negligible. In comparison with static stability, the concept of dynamic stability is more elusive and is neither easy to define nor measure (3). It is possible, however, that in addition to the relative position between the COM and the BOS, other global level variables (i.e., variables describing total body motion versus segmental motion) must also be tightly controlled by the CNS to achieve stability. Identifying such variables will be a necessary first step toward the understanding of dynamic stability.

The horizontal component of the COM velocity appears to be one of these variables. Recent studies of a sit-to-stand task have revealed that the peak magnitude of the horizontal component of the COM velocity remained invariant as movement speed increased from a person’s preferred speed to “as fast as possible” (7,9,10,11). The change in movement speed was primarily accomplished by an increase in the vertical component of the COM velocity (10). This tight control over the horizontal component of the COM velocity, accomplished primarily by limiting the variability in the motion of the upper body (11), was not caused by the subjects’ inability to increase the propulsive impulse. Rather, it resulted from the constraint associated with terminating the movement in an upright posture (i.e., with the COM within the BOS) (13). When the constraint was removed by allowing the subjects to fall forward onto an anteriorly placed protective bar, the peak horizontal COM velocity continued to increase with movement speed, comparable with its vertical counterpart (13). These findings, verifiable across different age groups (7), suggest that the CNS has to control not only the relative position (displacement) between the COM and the BOS, but also the relative velocity to maintain stability during movement. A dynamic concept of stability must therefore simultaneously take into account both of these variables.

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STABILITY IN MOVEMENT TERMINATION

There are only three general outcomes possible after the occurrence of a balance disturbance during standing (Fig. 1). One outcome is the termination of the movement in an upright, stable posture without the need to change the BOS (Fig. 1a). Alternatively, a loss of balance occurs during falling or after a movement termination failure (curved arrow in Fig. 1) forcing a change in BOS (e.g., stepping) (Fig. 1b), but balance can be regained following a successful movement termination (curved arrow). Finally, a recovery attempt may either be absent or fail, resulting in an actual fall (Fig. 1c).

Although falling is a prerequisite for the fall to occur, it is quite different from an actual fall (Fig. 1c). Unfortunately, there is a common confusion in nomenclature, in which the distinction between falling and a fall is blurred. The word “fall” is frequently misused to describe both “falling” (or loss of balance) and “fall.” A fall is a sudden, unintended change in position (i.e., displacement) causing an individual to land inadvertently at a lower level on an object, the floor, or the ground. In contrast, falling merely indicates the initiation and the process rather than the consequence of a loss of balance. The falling does not always lead to a fall because once a fall is initiated, a protective action is often used to avert an actual fall. Only when such an action fails will a fall occur.

Given these possible outcomes, we operationally define stability as a person’s ability to resist perturbation by terminating movement while maintaining or restoring the original equilibrium state of upright stance (8). Loss of balance occurs when stability can no longer be restored without changing the BOS. The feasible stability region can then be defined as the set of all combinations of COM position and velocity for which a loss of balance is preventable. The boundaries of the feasible stability region thus outlines the threshold values that discriminates between the “fixed BOS” and the other “changing BOS” outcomes described in Figure 1a and 1b, rather than a fall (Fig. 1c).

The boundaries of the feasible stability region are shaped by a person’s functional and physiological capacities, as well as by anatomical and environmental constraints. These boundaries can be numerically determined with the aid of the computer simulation of movement termination and an optimization routine (Appendix) (4,9). For instance, at a given COM position, the feasible stability region for a forward-traveling body ranges between the maximum COM velocity that can be reduced to zero and the minimum COM velocity that is required to carry the COM forward into the BOS (Fig. 2). The regions of forward and backward loss of balance lie anterior and posterior to the feasible stability region, respectively (Fig. 2). When the horizontal velocity of the COM is either above or below this range, a respective forward or backward loss of balance must occur and a change in the BOS must take place to protect the falling body. A similar quantification of the feasible stability region can be made for situations where the body is traveling backward (Fig. 2) or laterally. It is readily apparent that the static stability limits and the feasible stability region differ substantially because the static concept fails to consider the relative velocity between the COM and the BOS (Fig. 2). An important question is, therefore, can this dynamic conceptualization of the stability better predict the loss of balance than relying on the static approach?

Figure 2

Figure 2

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EVALUATION OF THE PREDICTIONS

The static stability region and the feasible stability region derived for a two-link (feet and rest-of-body) sagittal model have been evaluated against the experimentally derived COM position (alone) or state (position-velocity) trajectories during volitional movements that included a sit-to-stand (8) and a standing bimanual pull (14). At time of liftoff from a seated position, the COM was often outside of the BOS but still inside the feasible stability region, usually near the boundary for a backward loss of balance (8). Similarly, in 99.8% of the 2367 volitional bimanual pull trials, the COM state trajectory stayed within the feasible stability region during balance recovery immediately after the pulling force receded (14). Contradictory to the static predictions, the trajectories of both movements clearly indicate that the COM position can be outside of the BOS without losing balance, lending credence to the dynamic theory of stability.

Nevertheless, a major limitation of such a validation approach is that a loss of balance is generally absent in these volitional movements. When the objective is to identify the stability limits and to evaluate the conditions under which a loss of balance occurs inadvertently, one must be able to experimentally reproduce these boundary conditions. Applying a mechanical perturbation becomes an obvious necessity to achieve this objective. This approach has been adopted in two studies.

In one attempt (12), the responses to forward pulls externally applied at the waist were recorded among 13 young and 36 older subjects. Half of the older subjects had a history of falls. The subjects stood quietly before the onset of perturbation and were instructed to “react naturally” in response to pulls delivered at waist level near their body COM at three levels of peak acceleration, ranging from 1.8 m·s−2 to 5.4 m·s−2. In 428 analyzable trials, there were 313 trials where stepping had occurred. When the COM position-velocity trajectory exceeded the feasible stability region, subjects consistently stepped. In all trials where stepping did not occur, the COM state trajectory remained entirely within the feasible stability region. The dynamic (feasible stability region) concept was substantially better than the static one in predicting stability. The dynamic predictions were correct in 65% of stepping trials versus only 4% for the static predictions (Table 1).

TABLE 1

TABLE 1

In another attempt (6), the responses to BOS translations caused by a forward or backward translation of the supporting surface were recorded in 10 healthy young adults. The subjects stood quietly before the onset of perturbation. They were instructed to try to maintain balance without stepping after the onset of the BOS translation at each of three levels of peak acceleration, ranging from 0.7 m·s−2 to 1.5 m·s−2 in forward translation and from 2.0 m·s−2 to 3.0 m·s−2 in backward translation. Stepping occurred in nearly half (222) of the 480 trials analyzed. When predictions of forward and backward loss of balance were compared with these outcomes, the dynamic concept again provided better prediction of loss of balance than does the static one. The dynamic predictions were correct for 71% of all stepping responses versus only 11% of static predictions (Table 1).

The verification approach used in these studies rests upon an essential assumption: a step taken after the perturbation must reflect a loss of balance. A false classification of unstable condition will result if a subject does not step after his or her COM position or state trajectory has traveled outside of the corresponding stability region. Alternately, a false classification of stable condition will result if a subject takes a step that the predictive model considers to be “unnecessary” because the COM position or state trajectory remained within the corresponding stability region for the entire period before step liftoff. This latter aspect of the basic assumption may introduce certain bias for the verification of model predications.

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LIMITATIONS

The above-described experimental verification results revealed that the predictive accuracy for “step necessary” was excellent, but the predictive accuracy for “step unnecessary” was only reasonably good (Table 1). To understand why this has happened, one must examine the associated methodological limitations. One limitation, related to the verification approach per se, is that a loss of balance is not a prerequisite for initiating a step. Thus stepping (changing the BOS) may not always indicate a loss of balance, although by definition, a loss of balance must be accompanied by a change in the BOS. A step might be initiated defensively because of a fear of falling or a perception of danger rather than an actual loss of balance. The fact that older adults step more frequently than young adults when facing a similar perturbation (12) and that likely false predictions of “stable” occur more frequently for backward than forward stepping (6) lends support to this possibility. An inability to exert visual control over foot placement, as well as an increased risk of serious injury (e.g., because of an inability to protect the head), could explain a heightened perception of instability when facing the threat of losing balance in a backward direction. As seen in young adults (5), it is possible that some older adults would prefer to take a step when balance is minimally perturbed even after they are explicitly told not to. A change in BOS can occur volitionally (before loss of balance) or unintentionally (after loss of balance) in such situations. Thus this verification approach faces a fundamental challenge resulting from our limited ability to distinguish between these two scenarios.

An additional limitation may be inherent to the methodology used to derive the feasible stability region. The region is established by ruling out conditions for loss of balance that require a step, given the assumed anatomical and physiological characteristics of the simulation model, the assumed constraints, and control that is optimal. This last assumption means an optimal performance on the part of the CNS and the musculoskeletal systems. Errors in any or all of these assumptions or a failure to consider one or more critical constraints could cause the size of the feasible stability region to be overestimated. For example, a fear of falling and heightened concerns for safety may be considered as constraints pertaining to psychomotor limitations on movement termination and stability. The role of vision upon the threat of a fall may also be important to consider, especially with respect to the perceptual difference between backward and forward falling. These psychomotor limitations, if their constraints could be formulated in mathematical terms, might act to fine tune the feasible stability region by eliminating otherwise existing movement options. Additional as-yet neglected factors, such as limitations in the rate of resultant joint moment rise or delays in reaction time, might further affect the feasible stability region and thus require consideration.

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POTENTIAL APPLICATIONS

Despite the aforementioned limitations, the proposed dynamic conceptual framework may provide solutions for challenging problems in physical rehabilitation. For example, clinical assessments in physical rehabilitation can routinely quantify regional limitations in a joint or body segment (e.g., deficits in joint range of motion, muscle strength, or joint position sensation). However, such quantification generally does not present clinicians with a global evaluation of a patient’s functional status because of its lack of apparent connection to the patient’s ability to perform daily tasks (e.g., impairment in the control of balance during rising from a chair). The connection between regional deficits and global functional impairment may be established based on the computational methods associated with the proposed conceptual framework. In fact, this approach has been applied to estimating the impacts of reductions in ankle strength or in functional BOS on balance and stability (Fig. 3) (8). For example, a deficit in strength will have a negligible effect on the feasible stability region until it exceeds 51% from the normal value for the dorsiflexor muscles or 35% for the plantar flexors (8).

Figure 3

Figure 3

The computational methods can also be applied to the search for movement strategies that are optimal for the control of balance. For instance, the model predicts that if a person increases the COM velocity and/or shifts the COM anteriorly before the onset of a slip, the initiation of a backward fall can be avoided (Fig. 4) (9). Optimal movement strategies can be identified that serve the dual purposes of resisting a loss of balance upon a slip while still permitting the performance of regular movement functions under nonslipping conditions (the shaded area in Fig. 4) (9). A similar process can be applied to the search for optimal strategies for movement termination and balance recovery. It has been predicted that knee motion can enhance the effect of “hip/ankle” strategies for balance recovery under certain conditions (4). These predictions can be verified by properly designed experiments and potentially used to guide physical rehabilitation aimed at improving stability.

Figure 4

Figure 4

Finally, it is possible that the proposed conceptual framework can provide us with insights for the control of stability during steady locomotion. For instance, walking consists of cycles of balance loss and recovery similar to that for achieving stability during movement termination in standing. Without the movement-termination–like interruption, an actual fall would occur after the voluntarily initiated loss of balance in gait. The initial stance phase of a gait is stable, similar to the “fixed BOS” component of the standing model presented in Figure 1. This period is followed by the controlled loss of balance, which continues through the entire swing phase, comparable with the “changing BOS” component of that model. The quantitative relationship between these two components for gait (similar to the circular arrows between Fig. 1a and 1b) has been developed for backward stepping (15). At the touch down of each step, the boundaries of the feasible stability region can be used to predict three distinct outcomes of the stance phase (i.e., a backward loss of balance, movement termination, or a forward loss of balance). Such predictions may provide guidance for improving stability when perturbed such as a slip (9) or trip. Strengthening one’s own neuromotor protective mechanisms should be an effective physical rehabilitation strategy against falls, and improving one’s movement stability will then be a worthy contender for the first line of defense. After all, an improved stability can reduce or even eliminate the need for a protective reaction, thus reducing the likelihood of failure in the recovery response, which is prone to have serious limitations from neural circuitry loop-gain and significant delays from receptor transduction, neural conduction, and central processing.

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CONCLUSIONS

The control of balance is essential to the successful execution of activities of daily living. If a loss of balance or a fall is to be avoided, the CNS must rely on internal representations of the corresponding stability and recovery limits, as can be deduced following two basic concepts for quantifying the stability limits. The static concept places its emphasis on the position of the COM with respect to the BOS. The dynamic concept simultaneously considers both the relative position and velocity between the COM and the BOS. While the static representation is perhaps sufficiently accurate in low movement velocity and low perturbation conditions (open star and triangle in Fig. 4), the resulting differences can be drastic in predicting loss of balance with moderate level of perturbation where the dynamic concept can improve the predictability by sixfold or more (12). If the COM position is an essential control variable in the CNS’ internal representations of the stability limits, arguably the relative velocity between the COM and the BOS must be another control variable to properly predict the threats and protect this person against unintended consequences resulting from a loss of balance.

The predictive ability of the dynamic model might be further improved by identifying and including other important physiological and psychomotor variables. It has been suggested that the new, dynamic, conceptual framework may become a practical tool in physical rehabilitation, such as establishing a quantitative connection between local assessments of joints/segments and the control of whole body balance during functional performance and identifying optimal movement strategies for the improvement of movement stability. Future research may also extend the conceptual framework to the study of locomotion. A conceptual distinction has been drawn between a loss of balance and an actual fall, which is presumably associated with a set of yet-to-be-determined thresholds. Finally, a process similar to that presented in this review should be carried out to identify the essential control variables and the corresponding threshold values beyond which a fall is inevitable.

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Acknowledgments

The author thanks Drs. Michael J. Pavol, James L. Patton, and Suzann K. Campbell for their important inquiries, suggestions, comments, and editing of the text. This work was supported by NIH AG16727 and the Whitaker Foundation.

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References

1. Borelli, G. A. De Motu Animalium. Translated by P. Maguet. Berlin: Springer-Verlag, 1989, p. 130.
2. Frank, J. S., F. B. Horak, and J. Nutt. Centrally initiated postural adjustments in parkinsonian patients on and off levodopa. J. Neurophysiol. 84: 2440–2448, 2000.
3. Grabiner, M. D., and R. M. Enoka. Changes in movement capabilities with aging. Exerc. Sport Sci. Rev. 23: 65–104, 1995.
4. Iqbal, K., and Y.-C. Pai. Predicted region of stability for balance recovery: motion at the knee joint can improve termination of forward movement. J. Biomech. 33: 1619–1627, 2000.
5. McIlroy, W. E., and B. E. Maki. Task constraints on foot movement and the incidence of compensatory stepping following perturbation of upright stance. Brain Res. 616: 30–38, 1993.
6. Pai, Y.-C., B. E. Maki, K. Iqbal, W. E. McIlroy, and S. D. Perry. Thresholds for step initiation induced by support-surface translation: a dynamic center-of-mass model provides much better prediction than a static model. J. Biomech. 33: 387–392, 2000.
7. Pai, Y.-C., B. J. Naughton, R. W. Chang, and M. W. Rogers. Control of body center of mass momentum during sit-to-stand among young and elderly adults. Gait Posture. 2: 109–116, 1994.
8. Pai, Y.-C., and J. L. Patton. Center of mass velocity-position predictions for balance control. J. Biomech. 30: 347–354, 1997.
9. Pai, Y.-C., and K. Iqbal. Simulated movement termination for balance recovery: can movement strategies be sought to maintain stability even in the presence of slipping or forced sliding? J. Biomech. 32: 779–786, 1999.
10. Pai, Y.-C., and M. W. Rogers. Control of body mass transfer as a function of speed of ascent in sit-to-stand. Med. Sci. Sports Exerc. 22: 378–384, 1990.
11. Pai, Y.-C., and M. W. Rogers. Segmental contributions to total body momentum in sit-to-stand. Med. Sci. Sports Exerc. 22: 225–230, 1991.
12. Pai, Y.-C., M. W. Rogers, J. Patton, T. D. Cain, and T. A. Hanke. Static versus dynamic predictions of protective stepping following waist-pull perturbations in young and older adults. J. Biomech. 30: 347–354, 1998.
13. Pai, Y.-C., and W. A. Lee. Effect of a terminal constraint on control of balance during sit-to-stand. J. Mot. Behav. 26: 247–256, 1994.
14. Patton, J. L., Y. Pai, and W. A. Lee. Evaluation of a model that determines the stability limits of dynamic balance. Gait Posture. 9: 38–49, 1999.
15. Pavol, M. J., and Y.-C. Pai. Center of mass state at step contact influences the ability to restore balance with a backward step. Paper presented at:The 25th Annual Meeting of the American Society of Biomechanics. San Diego, CA, August 8–11, 2001, p. 205–206.
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APPENDIX

Simulation of Human Movement

Movement simulation requires a model of the dynamics of the human body. A simple model on stability in the sagittal plane may include only two links with one segment representing the symmetrical placement of the feet and the other segment for the rest of the body. The equations of motion for a two-link model with two-degrees-of-freedom under slipping condition (9) can be given as follows:EQUATION

where m = mass of the body minus feet, mf = mass of the feet, r = length of the body segment, θ = angular position of the pendulum, x = displacement of the BOS, τ = joint moment, Fx = Horizontal component of the ground reaction force, Fy = Vertical component of the ground reaction force, μ = coefficient of static friction, and g = acceleration due to gravity.

A four-link model may include a representation of symmetrical placement of feet, lower leg, upper leg, and the rest of the body (head, arms, and trunk). The equations of motion for the four-link model are complex, nonlinear, and highly coupled (4). The equations often do not have analytical solutions. Rather, they will be iteratively solved by using a forward dynamics approach and numerical integration methods. Forward dynamic solutions require several inputs to simulate movement termination. These inputs include the initial body state (i.e., initial segment positions and velocities) and joint moment estimates. The model is controlled through joint moments, and the joint moment histories may be parameterized by mathematical functions as model input. The outputs of the simulation may include time histories of segment positions, velocities, and accelerations, the horizontal and vertical components of the ground reaction force, and the COM position and velocity (4).

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Search for the Stability Boundary

Optimizations will entail a cyclic process of movement simulation, evaluation of the cost function from the simulation results, then an update of the model inputs based on an optimization algorithm. The task objectives of a successful balance recovery and termination of movement are quantified through a cost function. It incorporates mathematical expressions representing the desired final state of the model, the anatomical (e.g., joint range of motion) and physiological (e.g., muscle strength) limitations, and the environmental constraints (e.g., characteristics of the ground reaction force). For example, this cost function (FCOST) can have the form:EQUATION

The first term, f0, represents the contribution of the initial velocity. The second term, Σki, represents balance equilibrium and movement termination criteria that include COM position error, residual COM velocity, segment position error, and residual segment velocity and acceleration. The third term, Σgi, represents the cost due to violations of continuous constraints on vertical ground reaction forces, frictional forces, and center of pressure location. This term also includes costs due to violations of the joint range of motion limits and the limits on the parameters that define the joint moment profiles.

The Simulated Annealing algorithm has been used to solve this problem (4). It involves an iterative, constrained random search through the space of all possible model input values; a statistical criterion based on the corresponding change in the cost function value will be used to accept or reject each step. Regular periodic adjustments to the search space and a gradual tightening of the statistical criterion for step acceptance result in a refinement and convergence to the “optimal” solution.

The solution derived from the simulation and optimization process will determine, for a given COM position, the minimum initial COM velocity below which threshold a backward loss of balance will be initiated. This process will be repeated at other COM positions and for forward loss of balance. Polynomial interpolation between solutions will be used to outline the boundary of the feasible stability region.

Keywords:

fall; loss of balance; locomotion; motor control; model simulation; optimization; physical rehabilitation

©2003 The American College of Sports Medicine