(Soumis à Journal of Neurophysiology)

M. Simoneau* and P. Corbeil

Faculté de Médecine, Division de Kinésiologie, Université Laval, Québec, Québec, Canada, G1K7P4.

**Running head: **
Feasible forward balance safety margins

Corresponding author:

Martin Simoneau

Division de Kinésiologie, PEPS, Université Laval, Québec, Canada, G1K 7P4

Phone : 418-656-2131 ext. 7788 ; Fax : 418-656-2441

Email : Martin.Simoneau@kin.msp.ulaval.ca

**RÉSUMÉ**

Récemment, certains auteurs ont proposé qu’il existe un ensemble de combinaisons vitesse et position du centre de masse qui garantit la stabilité posturale en station debout (limite d’équilibre). Les objectifs de cette étude sont : (1) de montrer que, en utilisant un modèle mathématique, cette limite d’équilibre est dépendante du temps nécessaire pour développer un moment de force aux chevilles, et (2) de valider le modèle à l’aide d’une expérimentation. Dans cette expérimentation, huit sujets devaient initier une déstabilisation vers l’avant et enclencher une réponse correctrice afin de rétablir l’équilibre en utilisant une stratégie impliquant uniquement les muscles des chevilles (nombre d’essais égal à 120). Le modèle mathématique présenté dans cette étude a prédit 73.3 % des chutes et 73.3 % des stabilisations observées dans l’expérimentation. Cette étude montre clairement que le temps de développement du moment de force aux chevilles contraint la capacité d’une personne à retrouver l’équilibre en station debout suite à une déstabilisation vers l’avant.

**ABSTRACT**

Pai and Patton (1997) determined a set of feasible center of mass (CM) velocity-position combinations (safety margin) that guarantees upright stability. Recent studies demonstrated that the ability to maintain a stable posture depended not only on the magnitude of the restoring torque but also the time to peak of the restoring torque. The objectives of the present study were: (1) to determine the effect of ankle time to peak torque on safety margin using a mathematical model and (2) to determine the capacity of the model to predict experimental fall and balance recovery. A single-link-plus-foot biomechanical model was used to determine a set of safety margins, computed from the combination of maximum CM velocity and related CM position, for various ankle times to peak torque. In a validation experiment, the participants self-initiated a forward destabilization and were asked to regain balance using an ankle strategy. The outcomes of the mathematical model showed that the feasible combinations of CM velocity-position wewere reduced when ankle time to peak torque increased, whereas constant ankle torque led to a more tolerant safety margin. The mathematical model predicted 73.3% of the experimental fall trials when safety margins were built upon time varying ankle torque (5.6% for constant ankle torque input). Also, the model predicted 73.3% of the balance recovery trials for the time varying ankle torque (100% for constant ankle torque input). This study provides evidence that ankle time to peak torque drastically constrains the ability of a person to regain balance following forward destabilization.

**Keywords**
: Fall prediction, Balance, Speed of ankle torque

**INTRODUCTION**

One of the most debilitating consequences of falling is the injuries that follow. About 1% of falls cause hip fracture and about 5% result in any type of fracture. It is now recognized that about 90% of hip fractures are due to an impact of the hip on the floor following a fall. Altogether, falls and fall-related injuries cost nearly 10$ billion annually (U.S. Department of Health and Human Service 1993) and lead to serious medical problems, especially for elderly persons (about 300,000 cases in the United States yearly).

Balance is particularly challenging for people. Whole body center of mass (CM) is precariously balanced by several limbs, which are under the control of multiple muscles. Multiple sensory inputs are combined and mapped onto appropriate spatiotemporal muscle actions, which are necessary to counteract perceived destabilization and maintain balance of the whole body. A person’s ability to restore balance following destabilization largely depends on how that person negotiates physiological, mechanical and environmental constraints. Pai and Patton (1997) recently determined a set of feasible CM velocity-position combinations that guarantees stable upright posture. In the present paper, our aim was to build upon the efforts of Pai and Patton by integrating one physiological characteristic of human motor responses that could constrain the feasible safety margin: ankle time to peak torque.

Recently, there have been suggestions that inappropriate ankle time to peak torque could result in an inefficient balance recovery, during either locomotion (e.g.,van den Bogert et al. 2002) or upright standing (Corbeil et al. 2001; Robinovitch et al. 2002). An upright position is a complex task both mechanically and neurologically. If a person is falling forward, then the appropriate change of ankle torque must be greater than the gravitational torque to avoid a fall. Several studies have indicated that balance recovery depends on peak magnitude of lower limb joint torques and the rate of development of these torques (Lord et al. 1999; Luchies et al. 1994; McIlroy and Maki 1996; Tang and Woollacott 1998; Thelen et al. 1997). It is likely that balance recovery will fail to occur, even among young healthy adults, if the ankle time to peak torque is too long to counteract the destabilizing momentum. Considering that ankle torque amplitude is constrained by the length of the base of support (e.g., Kuo 1995) and the ankle torque onset delay is fixed, ankle time to peak torque is likely to be a crucial parameter to produce an adequate postural response to recover balance without stepping. Ankle time to peak torque should reduce the feasible combinations of CM velocity-position that permit balance recovery.

Consequently, in the present study we developed a mathematical model constrained by physiological ankle time to peak torque and mechanical constraints to determine the set of feasible safety margins leading to forward balance recovery. Safety margins were computed from the combination of maximum CM velocity and related CM position permitting balance recovery for various initial CM positions. The input of the mathematical model was the mean experimental ankle torque profile, whereas the outputs were the CM position and velocity. The predictive capability of the model was evaluated using experimental data. In our experiment, following a first auditory signal, participants self-initiated a forward fall. Shortly after, a second unpredictable auditory signal indicated to the participants that they must initiate their postural recovery response using ankle torque strategy. From the experimental data, we measured ankle torque using inverse dynamics and CM velocity and position using whole body kinematics.

The objectives of the present study were: (1) to determine the effect of ankle time to peak torque on forward balance recovery safety margins using a mathematical model and (2) to determine the predictive capability of the mathematical model using experimental data.

**METHODS**

**Mathematical model**

An inverted pendulum rotating about a base of support was used to model the postural system. Although this model has known limitations, it provides a good approximation of upright standing behavior (e.g., Johansson et al., 1988) even during large whole body sway (Lee and Patton 1998; Patton et al. 1999).

Dynamic equation of angular motion and mechanical constraints similar to that of Pai and Patton (1997) were used to predict the forward safety margin leading to balance recovery for different ankle time to peak torque.

All simulations were run using Matlab/Simulink (MathWorks, Inc., Natick, MA, USA). A total of 2.5 s was allowed for each computer simulation. A five-order variable-step Dormand-Prince method was used to integrate numerically the following equation of motion

where
*θ *
and
are the pendulum angular position (rad) and acceleration (rad/s2),
*J *
and
*m*
are the moment of inertia (kgm2) and mass (kg),
*l *
is the distance between the ankle joint and the CM of the pendulum,
*g*
is the gravitational acceleration (9.81 m/s2), and
*TA(t)*
is the ankle torque profile (Nm) that stabilizes the pendulum. To determine the ability of the model to predict experimental fall and balance recovery, we set
*m*
,
*l*
and
*J*
equal to 78 kg, 0.95 m (pendulum height times 0.536i) and 68.9 kgm2. The input of the model was the time varying ankle torque and the outputs of the model were the pendulum CM position and velocity. Linear CM position was derived from pendulum angular position. The linear CM position was differentiated numerically with a central finite difference technique to obtain the linear CM velocity.

The ankle torque profile
*TA(t)*
used in the mathematical model was based on the mean experimental ankle torque of all participants. To do so, we first normalized both the amplitude and temporal aspects of the experimental ankle torque profiles of each participant. The normalization was performed between two specific instances: the beginning of the stabilizing ankle torque and the first relative ankle torque maximum. The period between these two instances is referred to as ankle time to peak torque throughout this manuscript. Then, we averaged the normalized mean ankle torque profiles across all participants. Since, all human movements are characterized by an intrinsic motor and neural variability; we modified the temporal aspect of the normalized experimental ankle torque to evaluate the effect of various ankle times to peak torque on the balance recovery safety margins. Maximum ankle torque amplitude is constrained by the maximum anterior excursion of the center of pressure (CP). On average, the maximum anterior CP distance for all participants was equal to 74.05 % of the distance between the ankle joint and the toe (BOSii). Hence, in our mathematical model, we used this reduced BOS to calculate the maximum amplitude of the ankle torque profile (
*Tmax*
, equal 117.4 Nmiii).
*TA(t) *
reaches maximum ankle torque amplitude at time to peak torque. At first sight, this reduced BOS may seem conservative, however, several studies (Maki and McIlroy 1997; Pai et al. 1998) have revealed that stepping reactions or rising heels are triggered well before the center of pressure has reached the anterior edge of the feet, suggesting that the sensorimotor BOS is smaller than the biomechanical BOS.

The following equation is the elaboration of ankle torque (
*TA(t)*
) presented in the equation of motion. Muscle tone (
*Ttonus*
) was added to ensure stability before the initiation of the forward destabilization.
*Ttonus *
was equal to
where
represents the first step of the computer simulation.
is the normalized experimental ankle torque (explained above).

When
*t*
(time) was greater than ankle time to peak torque, the amplitude of the ankle torque was equal to
*T*
max.

The solid line is for ankle time to peak torque of 0.4 s, the dashed line is for ankle time to peak torque of 0.6 s, the dotted line is for ankle time to peak torque of 0.8 s. The dashed-dotted line represents the constant ankle torque simulated.

Figure 1 presents the ankle torque profile for four ankle times to peak torque (0, 0.4, 0.6 and 0.8 s) computed from the normalized mean ankle torque profile. To examine the effect of ankle time to peak torque on the forward balance recovery safety margins, ankle time to peak torque varied from 0 to 1.6 s by step of 0.1 s.

Balance recovery was identified when the CM position of the model stopped inside the BOS and reversed motion (anterior CM velocity changed direction). For a given initial CM position (varying from 0 to 0.7405 times BOS by step of 0.01 m; 0 represents a perfect alignment of the CM and ankle joint positions), the initial CM velocity was increased iteratively until the mechanical constraints were violated. Then, the maximum anterior CM velocity and CM positioniv were identified. For each ankle time to peak torque, the balance recovery safety margin was computed from the linear fit between the CM position and maximum anterior CM velocity for various initial CM positions. The maximum anterior CM velocity was normalized to body height, whereas CM position was normalized to foot length.

**Subjects**

Eight subjects (four males, age: 24.5.± 2.6 years, mass: 72.1 ± 3.9 kg, height: 1.77 ± 0.03 m and four females, age: 24.0 ± 4.8 years, mass: 56.7 ± 7.5 kg, height: 1.64 ± 0.07 m) with no known neurological or musculoskeletal pathologies participated. The experimental methods were reviewed and approved institutionally and each participant provided written informed consent prior to their participation.

**Experimental protocol**

An experimental protocol was developed to validate the outcomes of our mathematical model. Participants were standing upright barefoot with their feet 10 cm apart. Following an initial auditory signal, they self-initiated a forward fall keeping their body straight; they were encouraged not to bend their knee and hip joints or to raise their heels. A second auditory signal indicated that they had to initiate, as fast as possible, a recovery response in order to return to their initial position. The time elapsed between both auditory signals was varied to ascertain ankle time to peak torque variability, to increase the likelihood of forward fall and to acquire various CM position-velocity profiles. Participants were encouraged not to initiate balance recovery before the second auditory signal. When it happened, the trial was repeated. This occurred for less than 5 % of the trials. Each participant was submitted to 120 trials. An experimenter was located in front of the participants to secure them. When the participants raised their heels (the angle between the foot and the ground greater than 5 degrees) the trial was discarded. Less than 24 % of the trials were removed from the subsequent analysis. When the participants took a step, the trial was identified as a fall. Hence, trials were classified into two categories: balance recovery or forward fall.

**Materials**

Participants stood on an AMTI force platform, which was used to measure the displacement of the CP. The CP displacement was calculated from ground reaction forces and moments. The force platform signals were sampled at 1000 Hz using a 12-bit A/D converter. The 3D motion of the body was recorded using an opto-electronic motion analysis system (Selspot II) at a sampling frequency of 250 Hz. Eight hemispherical infra-red diodes were taped to the right side of the body over the temple (outher canthus), the ear (tragus), the neck (over C7), the shoulder (acromion), the hip (greater trochanter of the femur), the knee (lateral femoral epicondyle), the ankle (lateral malleolus), and the foot (head of the fifth metatarsal). Kinematics and CP data were filtered using a dual-pass fourth-order Butterworth filter having 7 Hz cut-off frequency. CM motion was calculated using a six-segment rigid body model (foot, shank, thigh, pelvis, trunk with arms incorporated, and head) based on Dempster’s estimates of the segment weight and segment mass-center location (Dempster 1955).

**Data analysis**

For each subject and for each trial, we defined ankle time to peak torque (Fig. 2A), maximum CM velocity (Fig. 2B) and CM position (Fig. 2C). The ankle torque first decreased (negative slope) until it reached a minimum relative permitting the CM to free fall forward. Then, the ankle torque increased (positive slope) to counteract the forward destabilization. Ankle time to peak torque was defined as time elapsed between the initiation of the stabilizing ankle torque (minimum relative indicated as the first vertical dotted line on Fig. 2A) and first relative maximum ankle torque (second vertical dotted line on Fig. 2A). The experimental velocity and position of the CM were identified at the instant when the CM reached maximum anterior velocity (vertical dotted lines on Fig. 2B and 2C). Trials where the CM position was outside the BOS were discarded (less than 3 % of all trials).

A) Typical experimental ankle torque profile. The first vertical line indicates the beginning of the ankle torque response and the second vertical line represents the first relative maximum of the ankle torque profile. B) Typical experimental center of mass velocity profile. The vertical line identifies the maximum anterior CM velocity. C) Typical experimental center of mass position. The vertical line indicates the position of the CM at maximum anterior CM velocity. On all panels, the thick lines and the thin lines are for a fall trial and for a balance recovery trial respectively.

To evaluate the predictive capability of the model, we checked to see if the experimental maximum anterior CM velocity and the CM position for the balance recovery trials were inside the safety margins and if the experimental fall trials were outside the model safety margins. In order to emphasize the effect of ankle time to peak torque on the predictive capacity of our mathematical model, we compared the model performance for safety margins computed from time varying ankle torque to a safety margin calculated from constant ankle torque (ankle time to peak torque equals to 0 s). For all trials, experimental ankle time to peak torque was rounded to the tenth of a second. This allowed us to group trials with similar ankle time to peak torque and compare them to the proper safety margin (i.e., model ankle time to peak torque identical to experimental ankle time to peak torque).

**Statistics **

Based on the rational behind the mathematical model, we hypothesized that ankle time to peak torque for the fall trials should be longer than for the balance recovery trials. In addition, the experimental maximum anterior CM velocity and CM position for the fall trials should be greater than for the balance recovery trials. Therefore, we used paired t-test to detect whether average values of ankle time to peak torque and CM position and maximum anterior CM velocity differed from the experimental recovery and fall trials. The level of significance was set as
*P*
< 0.05.

**RESULTS**

The balance recovery safety margin, defined as the combination of CM position and maximum anterior CM velocity, decreased with longer ankle time to peak torque (Fig. 3A). Moreover, the reduction of the safety margin slope as the ankle time to peak torque becomes longer supports this result (Fig. 3B). It is noteworthy that for ankle time to peak torque from 0 to 0.5 s, the reduction of the variation of the safety margin slope is much greater than for longer ankle time to peak torque (0.6 to 1.6). Overall, this demonstrates that ankle time to peak torque drastically reduced the forward balance recovery safety margins.

When we look at the CM anterior velocity-position time series for a representative experimental fall trial (Fig. 4 thick line), the maximum anterior CM velocity is greater than the lower safety margin (solid line); however, it is lower than the upper safety margin (dashed line). Lower and upper safety margins are for ankle time to peak torque equal to 0.9 s (time varying ankle torque input) and 0 s (constant ankle torque input), respectively and both experimental trials presented are for ankle time to peak torque equal to 0.9 s. The experimental fall trial is correctly identified as a fall (maximum anterior CM velocity and CM position outside the lower safety margin) by the lower safety margin. In contrast, it is considered as a recovery trial (maximum anterior CM velocity and CM position inside the upper safety margin) by the upper safety margin. On the other hand, the maximum anterior CM velocity and CM position of the recovery trial (Fig. 4 thin line) are inside both safety margins and well identified by both safety margins. When the safety margins were computed using time varying ankle torque input, the mathematical model predicted 73.3 % of the experimental fall trials, compared to 5.6 % for constant ankle torque input. On the other hand, the model’s capability to predict balance recovery was 73.3 % and 100 % for time varying ankle torque and constant ankle torque input respectively.

A) Forward balance recovery safety margins for ankle time to peak torque starting from 0 s to 1.6 s by step 0.1 s. These safety margins are bounded by the maximum anterior center of mass velocity normalized to body height and by the maximum center of mass position normalized to foot length. B) Safety margin slope variation across ankle time to peak torque.

The experimental trials in Fig. 5 are for ankle time to peak torque of 0.7 s, 0.8 s, and 0.9 s (upper, middle and lower panel respectively). The upper safety margin (dashed line) is for the constant ankle torque input, whereas the lower safety margin (solid line) is for the time varying ankle torque input. The experimental recovery trials (crossed) are inside the lower safety margin, whereas the experimental fall trials (open circle) are outside. In contrast, most of the experimental fall trials are inside the upper safety margin but outside the lower safety margin. Some experimental fall trials were outside both (upper and lower) safety margins. Finally, all experimental recovery trials were correctly identified by both safety margins.

Mathematical model safety margins for constant ankle torque (upper dashed line) and for time varying ankle torque (lower solid line). Experimental fall trials and balance recovery trials are illustrated by open circle and cross respectively. On all panels, upper safety margin is for constant ankle torque whereas lower safety margin is for ankle time to peak torque similar to the experimental trials. A) Experimental fall and balance recovery data for ankle time to peak torque of 0.7 s. B) Experimental fall and balance recovery data for ankle time to peak torque of 0.8 s. C) Experimental fall and balance recovery trials for ankle to peak ankle torque of 0.9 s.

As hypothesized, we found that experimental fall trials had longer ankle time to peak torque compared to experimental recovery trials. On average, ankle time to peak torque was 0.89 s for trials where balance was regained, compared to 0.94 s when participants failed to recover their balance (Fig. 6A). This difference was significantly different (
*t*
= -2.43, df = 7,
*P*
< 0.05). Moreover, participant’s maximum anterior CM velocity was much faster (Fig. 6B) and CM position closer to the anterior limit of their base of support (Fig. 6C) for the fall trials compared to the recovery trials (
*t*
= -2.97, df = 7,
*P*
< 0.05 and
*t*
= -5.38, df = 7,
*P*
< 0.01, for maximum CM anterior velocity and CM position, respectively).

A) Mean ankle time to peak torque for the experimental recovery and fall trials. B) Mean maximum anterior center of mass velocity normalized to body height for the experimental recovery and fall trials. C) Mean maximum center of mass position normalized to foot length at maximum anterior center of mass velocity for experimental fall and balance recovery trials.

**DISCUSSION**

The main outcomes of the present study clearly suggest that feasible balance recovery safety margin is constrained by ankle time to peak torque. The model predicts that long ankle time to peak torque drastically decreases the maximum anterior CM velocity that the postural system can tolerate during a forward destabilization. Hence, it is likely that persons who are unable to generate ankle torque quickly (e.g., frail elderly) are much more constrained than healthy people. The results of the mathematical model confirm what several authors have recently suggested (Lord et al. 1999; Luchies et al. 1994; McIlroy and Maki 1996; Tang and Woollacott 1998; Thelen et al. 1997; van den Bogert et al. 2002), that in addition to muscle strength, slow ankle torque development drastically reduces the capacity of a person to regain balance. Pai and Patton (1997), using mechanical constraints, presented two types of feasible limits of stability: torque and state boundaries. Results of the present mathematical model illustrate that safety margins are constrained as well by ankle time to peak torque. We suggest that the central nervous system organizes postural responses by selecting an operational subset of feasible postural responses from the entire family of feasible responses that are limited by ankle torque amplitude as well as ankle time to peak torque. Therefore, speed of ankle torque development appears to be an important determinant of balance recovery following a forward destabilization.

Several limitations could have influenced the predictive capability of the model. The mathematical model was derived from several simple assumptions. First of all, a single segment was assumed above the ankle joint and the pendulum motion was restricted to the sagittal plane. Although we tried to minimize knee and hip strategy to recover balance, it is possible that participants did rely on this kind of postural strategy to regain balance. However, on average, the range of motion of the trunk angle was relatively low for recovery trials (5.31 ± 2.46°). The model also neglected the possibility that heel rise could occur without causing forward fall. Experimental trials where subjects raised on their heels were discarded. The results of the model were computed using constant intrinsic properties of the pendulum (mass 78 kg and height 1.78 m). It is likely that fixing these parameters yielded small discrepancies in the safety margins between the actual model and an anthropometric adjusted model (see sensitive analysis in Appendix A). However, sensitivity analyses revealed that the model is very robust to pendulum length and mass variation. Moreover, the model was robust as well to the reduced BOS. The model prediction capability was not improved when the reduced BOS was adjusted to each participant (76% for balance recovery and 73% for fall prediction). Another assumption that could have reduced the predictive capability of the model is that all simulations were performed using the ankle torque profile computed by averaging normalized mean ankle torque profiles of all participants. However, this procedure permitted to have only one free input parameter (i.e., ankle time to peak torque). One avenue we pursued with the present mathematical model was the development of a model that could predict balance recovery and forward fall for a wide range of individuals. Finally, the safety margins were constrained by ankle time to peak torque and computed from the maximum anterior CM velocity and CM position for various initial CM positions. It is conceivable that the use of only three discrete measures in order to predict fall and balance recovery may limit the predictive capability of the mathematical model, since balance control is a very complex task. Nevertheless, the significant statistical differences observed for these three discrete parameters between the experimental balance recovery and fall trials (see Fig. 6) suggest that at least these parameters may be reliable to discriminate balance recovery from fall.

For ankle time to peak torque equal to zero, the safety margin was identical to that of Pai and Patton (1997). Based upon Pai and Patton’s modeling efforts, we added ankle time to peak torque to mimic physiological ankle torque, which allowed us to compute a set of safety margins that were constrained by speed of ankle torque development. The present safety margins, compared to Pai and Patton’s feasible stability region, were bounded by slower maximum anterior CM velocity regardless of CM position.

Previous studies have demonstrated that sometimes people step before their CM velocity-position has reached the anterior stability limit (e.g., Pai et al., 1998). As recently suggested by Pai (2003), ”unnecessary” steps might be initiated defensively because of a fear of falling or a perception of danger. On the other hand, the present model’s outcomes may suggest another explanation. Indeed, it is possible that the CM velocity-position may have actually reached the safety margin constrained by the person’s capacity to rapidly generate ankle torque.

Despite all the assumptions, we are confident that these limitations do not considerably affect the predictive capability of the model. The model predicted 73.3 % of the experimental fall trials where safety margins were built upon time varying ankle torque. In contrast, only 5.6 % of the fall trials were predicted when ankle torque was constant. It is clear that the CM position-velocity profile for the experimental fall trials ended outside the safety margin computed with both types of ankle torque (see Fig. 4). The present mathematical model, based on maximum anterior CM velocity and CM position, attempted to predict forward fall
*before*
it actually reached the anterior mechanical limit of stability proposed by Pai and Patton (1997). On the other hand, the predictive ability of the model to identify balance recovery may seem low (73.3 %) for time varying ankle torque compared to constant ankle torque input (100 %). The greater capability of our mathematical model to predict balance recovery when ankle torque was constant is not surprising. Constant ankle torque led to balance recovery safety margin that can tolerate much greater maximum anterior CM velocity. However, constant ankle torque is not physiologically possible. Moreover, the capability of the mathematical model to predict forward fall, was less than 6 %. The consequences of a false detection in identifying forward fall are certainly much more damaging than not being able to detect balance recovery. Our mathematical model included biomechanical (reduce BOS) and physiological (ankle time to peak torque) parameters that had been previously neglected. In addition, the mathematical model and the experimental data integrated a real-life loss-of-balance aspect, initial anterior CM velocity different from zero.

In conclusion, based on the outcomes of a mathematical model and experimental validation, this study provides evidences that speed of ankle torque development drastically constrains the ability of a person to regain balance following forward destabilization. It is likely that exercise could be an interesting paradigm for improving someone’s ankle time to peak torque since it has already been showed that exercise improves strength, reaction time, balance and walking velocity (Gardner et al. 2000; Lord et al. 1996; Lord et al. 1995). However, further research using different exercises and experimental designs are needed to determine whether improvement of ankle time to peak torque alone in a task not involving balance recovery would carry over to the recovery of balance. Altogether, the model and experimental protocol may serve as assessment tools to examine and hopefully to better understand the neuromechanical parameters that must be investigated to direct specific intervention towards improving a patient’s speed of ankle torque development and to possibly reduce the risk of falling.

**APPENDIX A**

Sensitivity analyses have been conducted to determine the robustness of the forward balance recovery safety margin with respect to the mass and height parameters of the pendulum and the length of the reduced BOS. A summary of the sensitivity analysis with respect to the height parameter is showed in Table I. For this sensitivity analysis, the BOS was set at 74.05 % of the distance between ankle joint and toe. In general, for short and tall participants (1.5 to 1.9 m, respectively) and various ankle times to peak torque (0 to 1.6 s), safety margins were less than 10 % which is different from the safety margins computed using the standard parameters (e.g. pendulum height = 1.78 m, mass = 78 kg and BOS equal to 74.05 % of the distance between the ankle joint and the toe). The slopes of the safety margins underestimated the safety margins that generate ankle torque in a short period of time (see Table I ) for a short participant (height 1.5 m). On the other hand, for a tall participant (height equal to 1.9 m), the mathematical model overestimated less than 5 % the forward balance recovery safety margin.

In order to evaluate the effect of the mass of the model on the safety margin slope variation, sensitivity analysis was performed. To do so, the height (1.78 m) and the length of the reduced BOS (74.05%) of the model were kept constant, but the mass varied from 50 kg to 90 kg. The sensitivity analysis failed to demonstrate any safety margin slope variation.

For safety margin sensitivity, with respect to the length of the reduced BOS, additional computer simulations were executed. The performance of the model using a smaller (64.05%) and a larger (84.05%) reduced BOS was evaluated. The height and the mass parameters were kept constant. The slope of the safety margins did not vary, but the area bounded by the safety margin decreased for the smaller or increased for the larger reduced BOS. Although the area bounded by the safety margin is related to the length of the BOS, the predictive capability of the model did not increase much when the length of the reduced BOS was adjusted to each participant (see Discussion).

** Tableau 1 : Sensitivity analyses of the model’s height parameter**

Height (m) |
||

TTP (s) |
1.5 |
1.9 |

0 |
8.3 |
-3.3 |

0.4 |
7.3 |
-2.4 |

0.8 |
5.6 |
-1.7 |

1.2 |
4.1 |
-1.5 |

1.6 |
-3.2 |
-3.2 |

Percentage of variation (%) of the safety margin slope for various ankle times to peak torque (TTP) and height parameters. For different combination of height parameter and ankle time to peak torque, keeping the mass parameter constant (78 kg), the percentage of variation is calculated in function of the safety margin slope used throughout this study: mass and height parameters equal to 78 kg and 1.78 m and the base of support length equal to 0.7405 times foot length.

**TEXT FOOTNOTES**

i CM position is equal to 0.575 times body height minus 0.039 times body height (vertical ankle height). Body proportions are derived from Dempters (1955).

ii Generally, the base of support (BOS) is the contact area between the feet and the ground, in which the resultant ground reaction force can be applied. In this manuscript, the BOS represents the distance between the ankle joint and the toe.

iii Maximum ankle torque was equal to (0.7405 x BOS) x
*mg*
. BOS was derived from Pai and Patton (1997), which equals 0.1231 times the height of the subject.

iv The CM position represents the position of the CM with respect to the BOS at maximum anterior CM velocity. For the sake of clarity, CM position has been used throughout the manuscript.

**ACKNOWLEDGEMENTS**

The authors thank Dr. Normand Teasdale for useful comments on previous versions of this paper and Julie Marcotte for her assistance during data collection. This study was supported by a grant from FQRNT (Fonds Québecois de la Recherche sur la Nature et les Technologies) young investigator award (MS). PC is supported by a Doctoral scholarship from NSERC (Natural Sciences and Engineering Research Council of Canada).

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