Investigation of the functional stability limits while squatting

This study investigates the functional stability limits (FSLs) in the squatting positions. Eleven male participants leaned and moved their pelvis horizontally in the clockwise and counter‐clockwise directions while squatting at 11 depth levels. The depth was controlled by changing the hip height from 100% to 0% of the upright position. The FSLs and the center of pressure excursion lengths were calculated from the force‐plate data, and the musculoskeletal loads on the lower limbs were estimated from the joint torques and surface electromyograms. As the hip height reduced, the area of the FSLs narrowed by up to 20% of the base of support (BOS) area at the deepest squatting position. The narrowing was affected by the decreasing FSLs in the forward direction, which also decreased by up to 20% of BOS. These quantitative data accurately evaluate the postural stability, suggesting a considerable fall risk during tasks requiring the squatting position.

. Therefore, the human posture is balanced and the biped-robot gait is statically stable if the gravity line from its COM falls within the convex hull of the foot-support area (Goswami, 1999).
The foot-support area, called the base of support (BOS), defines the theoretical maximum motion of the COP (Holbein & Redfern, 1997) and is commonly used as an indicator of whether STF injuries will follow a loss of balance during static standing. Several researchers have studied the voluntary movable range of the COP within the BOS. The portion of the BOS within which individuals are willing to extend their COP is defined as the functional stability region (FSR; Holbein & Chaffin, 1997;Holbein & Redfern, 1997) or the functional stability boundary (FSB; Bagchee, Bhattacharya, Succop, & Emerich, 1998). The functional limit of the BOS (FBOS), which is defined as the effectively utilized area for the COP movement, is a concept that is similar to FSR (Fujimoto, Hsu, Woollacott, & Chou, 2013;King, Judge, & Wolfson, 1994). The reduced FSR or FBOS limits an individual's ability to maintain balance because the COP-COM distance is proportional to the COM acceleration (Winter, Patla, Prince, Ishac, & Gielo-Perczak, 1998). The functional stability limits (FSLs) are defined as the distance that an individual will displace their COP from the center of the BOS to the FBOS in a particular direction (Holbein, McDermott, Shaw, & Demchak, 2007).
FSLs and FSR are smaller than the theoretical maximum because they are limited by muscle strength, internal postural control, external load control, and other factors (Holbein & Chaffin, 1997). Moreover, the effects of various factors on FSLs and FSRs have been revealed; among these factors are postural muscle strength, reaction time, feet placement, ability to coordinate body-segment movements, somatotopic differences, age, psychological factors, and load-carrying (Bagchee et al., 1998;Fujimoto, Bair, & Rogers, 2015;Holbein & Redfern, 1997;Qu & Nussbaum, 2009).
FSLs have been used in biomechanical models to predict the horizontal pushing/pulling forces and working postures or movements and when the compensatory stepping is required (Holbein et al., 2007). Few studies, however, have investigated the FSLs while squatting. Although squatting postures are expected to reduce the FSLs, the postural balance of squatting postures is poorly understood. The risk of falls, which is often calculated as the ratio of actual COM or COP movement to FSLs, may be underestimated relative to the actual state. Therefore, to investigate the effect of squatting on the postural balance, this research measures the FSLs of squatting postures at multi-level hip heights (HHs). Furthermore, the musculoskeletal loads on lower limbs are then evaluated to examine the biomechanical constraints while squatting.

| Participants
Eleven men aged 22-27 years participated in the study. The sample size was determined by a statistical power calculation based on pilot study data . All participants were in good health with no medical history of musculoskeletal injuries within the previous 12 months. The means and standard deviations of the age, height, and weight variables were 22.8 ± 0.9 years, 171 ± 4.0 cm, and 65.0 ± 5.3 kg, respectively. When standing upright, the vertical height from the floor to the greater trochanter was 88.6 ± 4.1 cm. All participants wore the same model of safety shoes (MZ010J; Midori Anzen Co. Ltd., Japan).
The average foot length of the participants (International Organization for Standardization, 2017) was 26.0 ± 0.7 cm, and the shoe size ranged from 25.0 to 27.0 cm. After an oral overview of the experiment, each participant gave their informed consent to participate. This experiment was approved by the ethics committee of Tokyo Metropolitan University, Hino Campus (approval number: 238).

| Experimental apparatus and procedure
While FSLs or the limits of stability (LOS) have been measured by various inspection methods, inconsistencies in techniques may have limited the accuracy and reproducibility of the estimates due to leaning speed, allowing loss of foot contact or constructing ellipses from limited data samples (Forth, Fiedler, & Paloski, 2011). Therefore, this study adopted the verified functional limits testing procedure using a controlled lowspeed voluntary leaning protocol that requires feet to remain in continuous contact with the ground (Forth et al., 2011;Juras, Słomka, Fredyk, Sobota, & Bacik, 2008). The participants of this study were required to lean forward as far as possible while maintaining heel contact with a force plate (9286AA; Kistler Instrument Corp., Amherst, NY). After reaching this forward stability limit, they moved their hip horizontally while maintaining a squatting posture for 20 s. Figure 1 shows the flow of their movements. Within the first 5 s, the participants squatted and adjusted their knee height until the hip marker overlapped the level line of the feedback system. They then leaned and moved their pelvis, drawing an imaginary circle in the horizontal plane as widely as possible without changing the HH. Next, they moved in the clockwise direction for 10 s, followed by the counter-clockwise direction for 10 s.
Before the experiments, the participants had determined their comfortable foot position for squatting on a force plate. The outline of the standing position was marked on the plate by 10 mm-wide masking tape, and the trajectory was electronically recorded by pressing the tape with a finger. Using the recorded coordinates, the inner area was calculated as the BOS by summing the infinitesimal triangular areas formed by the two neighboring extracted points and the central coordinates. In all trials, participants were asked to maintain the same foot position.
The HH was defined as the vertical height from the floor to the greater trochanter during upright standing. A reflective marker The height of the camera was adjusted such that the marker on the greater trochanter was always captured directly from the side.
In this study, the HH relative to the body height was controlled as an experimental factor. The vertical heights of the greater trochanter in the upright position and in the squatting position with maximum knee flexion were defined as 100%HH and 0%HH, respectively ( Figure 2). The experimental conditions were 11 levels of HH (0%, 10%, 20%, …, 100%HH). The experimental conditions were completely randomized, and measurements were duplicated under each condition.

| Center of pressure and functional stability limits
While the participants squatted and leaned, the force components (F x , F y , and F z ) and moment components (M x , M y , and M z ) were sampled at 100 Hz by the single force-plate system with an analog-to-digital data converter (PH-703; DKH Co. Ltd, Japan) and analysis software (TRIAS2; DKH Co. Ltd). The signals were low-pass filtered by the moving-average method (1-s average block). As the fast Fourier transformed signals peaked near 0.1 Hz, the cutoff frequency was set to 0.443 Hz. The position-time trajectory of the COP in the horizontal plane was determined through standard transformations (Winter, 2009).  (Bagchee et al., 1998). The MP was defined as follows: Here ⃗ a and ⃗ b represents the position vector of FSLs and BOS, respectively, from the average point of the COP perturbation for the case that the distance between FSLs and BOS ( ⃗ ⃗ = − b a) has a minimum value.

| Lower-limb loads
Muscles transmit forces and create torque through the insertion into the skeletal structure around the joints. The torque is a function of the moment arm of the insertion and the angle at which the muscle is applying the force (Redfern, 1992). Therefore, it is common to see electromyogram (EMG) signals related to the generated torque instead of the internal muscle force. Therefore, this study measured the joint torque and muscle activities during the squatting tasks.
Before calculating the joint angles, the entire body posture was captured by a motion capture system (Perception Neuron, Noitom Ltd., China) using a software (Axis Neuron Ver. based on the BVH format data and estimated the joint torques at the knee and ankle joints using the three-dimensional biomechanical analysis method (Chaffin, 1997;Winter, 2009). The estimated joint torques were then normalized by the maximum voluntary torque on each joint (Chaffin, Andersson, & Martin, 2006).
The resulting values are called the joint torque ratios. The physical loads on the knee and ankle joints during the rotational motion were then determined from the maximum and 1-s-averaged joint torque ratios. The maximum joint torque ratios in the forward, backward, right, and left directions were calculated, as described for the FSLs above.
The activities of six muscles (R and L rectus femoris, R and L gastrocnemius, and R and L tibialis anterior) were recorded by   (Criswell, 2008). Before the trials, the sEMG signals during maximum voluntary contraction (MVC) of each muscle were recorded in a manual muscle test (Hislop, Avers, & Brown, 2013).
The raw signals were sampled at 1000 Hz and lowpass-filtered through a Butterworth filter with a 2-Hz cutoff frequency.
Finally, to eliminate the individual differences among the participants, the raw signals were converted to relative MVC values (%MVC).

| Statistical analysis
The effects of the experimental factors on the measured indices were compared by analysis of variance (ANOVA) with a two-way factorial design (viz., the hip-height conditions and participants) and a post hoc Tukey's test. Sphericity was checked by the Mauchly sphericity test.
The statistical significance level of all tests was set to 5%. Data analyses were carried out using BellCurve for Excel version 3.10 (Social Survey Research Information Co., Ltd., Japan).

| Functional stability limits
The COP trajectories at HHs of 100%, 60%, 30%, and 0% are shown in Figure 4. These trajectories are the average COP coordinates of all participants in each 6°range of azimuthal angles. According to these results, squatting narrowed the area of the FSLs as the HH lowered.
The reduction was notable in the forward, left, and right directions, but was slightly in the backward direction (near the line connecting the lateral malleoli).
ANOVA indicated that the FSL area ratios have the main effect of HH. The bar chart in Figure 5 shows 3.2 | Joint torque ratio Figure 6 plots the joint torque ratios at the different HHs. The average and maximum knee-torque ratios at each HH are shown in Figure 6a.
Both values increased with decreasing HH. Under the 0% HH condition, the maximum and average knee-torque ratios exceeded 50% and 30%, respectively. On the other hand, the maximum ankle-joint torque ratios were below 30% under all HH conditions and did not significantly depend on HH (Figure 6b).

| Electromyography
The mean muscular activities on the left and right legs during the LOS tests are shown in Figure 7a- When the participants leaned and moved their hips in a circular motion, the COP shifted by up to 60% of the BOS boundary in the left and right directions. This trend concurs with the literature (Holbein & Redfern, 1997), in which the center of gravity shifted by 60.3 ± 9.2% to the right and 58.6 ± 11.5% to the left as participants performed similar motions with no load.
The FSL area decreased with lowering of the HH, being minimized at 19% at the deepest squat position (0%HH). As the participants which is the minimum distance between COP and BOS, was greater for fallers than nonfallers. The reduced FSLs predispose people to more precarious stability conditions by limiting their ability to regulate COM momentum induced by perturbations (Fujimoto et al., 2015).
At work sites, the use of the human simulation software, called the digital human model, helps to accurately estimate the risk of falls (Kawano et al., 2003(Kawano et al., , 2009. However, few models use the models of

| Relation between hip-height and lower-limb loads
With a decrease in the HH, the flexion of the knee joint becomes deeper, and the activity of the tibialis anterior muscle approximately linearly increases. We evaluated the correlation between muscle activities and joint torque ratios by Pearson's correlation coefficient.
The R and L tibialis anterior muscles highly correlate with the kneejoint torque ratios (R: r = 0.967; L: r = 0.955), while no significant correlation was observed between other joints and muscles.
During the squatting tasks, the participant needs to maintain the angles of ankle joints within an appropriate range to prevent a fall. The dorsiflexion angles of the ankles were close to the maximum range of motion (ROM) when the HH is lower than 90%. Therefore, the tibialis anterior muscles require the stronger contractions compared to the normal positions because the joint position around the maximum ROM generates a passive torque in the opposite rotational direction (Chaffin et al., 2006). Thus, the tibialis anterior muscles were significantly correlated with the knee-joint torque. However, the activity of the rectus femoris was gradually reduced when the HH was lower than 40%. These phenomena are attributable to the contact between the thighs and lower legs. When the thigh is strongly contacted with the posterior surface of the lower leg in the deepest squatting position, the applied reaction forces generate a passive torque, which offsets the joint torque on the knee joints (Fukunaga, Ayaka, Ito, & Morimoto, 2016;Zelle, Barink, Loeffen, De Waal Malefijt, & Verdonschot, 2007).
Consequently, the activity of the rectus femoris differs from the trend of the knee-joint torque.
The joint torque estimation in this study was based on the inverse dynamics approach, where several factors can influence the results.
Hence, only particular muscles were significantly correlated with the joint torque. We assumed the torque on body joints as the external mechanical moments around the joints, maintaining equilibrium with internal forces generated by muscles, tendons, or articular capsule.
Then the angle of pull, the moment arm to the center of rotation, and even the structure of the particular muscle significantly vary during normal movements (Redfern, 1992). Therefore, sEMG values do not necessarily reflect the mechanical loads on the biomechanical system during complex and dynamic movements. To estimate the torques exerted by the muscles, the biomechanical model with a more precise inner structure is needed.

| Relevance of lower-limb loads on the functional stability limits during squatting
We now discuss the effect of hip height on the FSL based on the joint torque ratio and sEMG values. For this purpose, we divide the 11 experimental conditions into three groups. In the first group (100-70% HHs), the FSLs remained constant. In this group, the muscular activities of the rectus femoris and tibialis anterior were below 30% MVC, and the knee-joint torque ratio was at most 30%.
These results indicate that while leaning and moving their hips, the participants sufficiently shifted their COP despite the increased physical loads on the lower limbs.
In the second group (60-40% HHs), the FSLs narrowed toward the inner edge of the feet.  (Jensen, 2008). Therefore, as the torque increased in their knee joints during deep squatting, the participants dominantly shifted their COP by moving their ankle joints. This interpretation is supported by the high muscular activity of the tibialis anterior.

| Limitations
This study has several limitations. First, the FSLs of the feet positions were measured only during static squatting postures.
The participants were instructed to maintain their hips at constant height while moving their pelvis in a circular motion. In any practical work environment, squatting motions are accompanied by a more complex series of movements, and some work tasks require different positions of the feet. These movements were not considered here. Further studies will examine more realistic motions on participants over a wider age range, enabling a comprehensive quantification of the FSL indicator.
Especially important is the dynamic property of postural balance, which relates to the fall risk on real work sites. For the dynamic balance of a biped robot, several models have been introduced as indicators of whether the current state is balanced or falling. The prevalent approaches are based on the virtual foot-rotation point on the ground (Goswami, 1999), the ability of a biped system to come to a stop after taking maximum N steps (Koolen, De Boer, Rebula, Goswami, & Pratt, 2012), or a contact-specific partition of the COM state space (position and velocity) (Mummolo, Peng, Gonzalez, & Kim, 2018). Moreover, we may be able to apply dynamic mechanical models and evaluation indices such as virtual time-to-contact approaches, which quantify the temporal proximity to the stability boundary (Dutt-Mazumder, Challis, & Newell, 2016;Kilby et al., 2014).

| CONCLUSIONS
The present study investigated the effect of HH on the area of the