1.1. Fall Risk Assessment

According to the World Health Organization, falls are the second most common cause of unintentional injury and death worldwide, after traffic accidents, killing approximately 684,000 people annually ^a. Falls are a major problem for society because they lead to a decline in activities of living and increase medical costs ¹. It has been reported that falls often occur when mechanical or kinematic disturbances are applied to a person, such as slipping or tripping ^2,3,4.

To evaluate the fall risk, the Get Up and Go test ⁵ and the time required to complete it, the Timed Up and Go test ⁶ are used as simple fall risk assessment indicators. Other commonly used instruments include the Berg baklance scale ⁷, which measures balance maintenance ability, and the Performance-Oriented Mobility Assessment ⁸, which comprehensively evaluates gait quality and balance. Furthermore, MoS, defined as the distance between the extrapolated center of gravity \(X_{\mathrm{CoM}}\), which is the future position of the center of gravity taking into account velocity \(V_{\mathrm{CoM}}\), and the boundary of the base of support (BoS), is used as a measure of postural stability margin ^{9,10,11,12,13}. However, MoS, for example, indicates the stable limit distance at which a fall does not occur during a specific gait cycle, and does not represent the inherent postural stabilization performance of that individual.

Since falls are common during walking, the ability to return to normal walking posture when it is disrupted by various external disturbances during walking is thought to be important in assessing the risk of falls. Therefore, when assessing the risk of tipping over when an external disturbance is applied, it is important not only to look at the resulting stability margin, but also to evaluate the postural control dynamics, which shows how the postural control system responds. Furthermore, Miyazaki et al. ¹⁴ are discussing strategies to improve posture control ability itself.

1.2. Quantifying Postural Recovery Ability

Previous research by Togoe et al. has shown a fall risk assessment method that focuses on postural control dynamics ¹⁵. They apply an impact force to the lower limbs from a standing position and obtain postural change output by measuring the center of pressure (COP). Using these input/output response results, the posture control dynamics is experimentally obtained as a transfer function through system identification. By evaluating the poles of this transfer function, they quantitatively assess the ability to recover posture from an upright position.

This method makes it possible to quantitatively evaluate postural control ability during static standing. However, since most falls occur while walking ^3,4, it is important to evaluate postural control ability while walking. During walking, center of gravity sway occurs, depending on walking period and walking speed. This is equivalent to being constantly affected by mechanical disturbances during stationary standing, making it difficult to evaluate postural recovery ability.

This paper extends the previous method for assessing the risk of falling during stationary standing to the case of walking in order to evaluate the risk of falling while normal walking. There are two challenges for this problem. The first one is that it is difficult to distinguish between COP fluctuations due to normal walking and those due to postural control in response to disturbances, making identification using only the latter difficult.

The second problem is that it is not so easy to apply disturbances that excite the postural control dynamics during walking. Several studies have investigated the effects and variability of walking motion by applying some kind of disturbance during walking. However, few of these studies have focused on postural control dynamics, and it is unclear what kind of disturbances excite postural control dynamics.

Regarding the first point, this paper proposes a method for identifying the postural recovery dynamics system by using the results of time-frequency analysis to remove, to some extent, the COP fluctuation component caused by normal walking. In a time-frequency analysis of COP fluctuations that include elements of normal walking, we show that frequency characteristics are observed immediately after the application of a disturbance. This suggests that it is possible to identify postural control dynamics even without completely eliminating COP fluctuations during normal walking.

To address the second issue, we use a split treadmill and suddenly reduce the speed of only one of the belts. Using the techniques, we propose a method to derive a mathematical model of postural control dynamics, which represents the postural response to disturbances during walking, using a system identification method. We then propose a method to quantitatively evaluate postural stabilization performance during walking based on the pole placement of the mathematical model.

2.1. Postural Response System Output

Physical quantities that represent human postural responses during walking include the center of mass (CoM), the future position of the center of gravity (\(X_{\mathrm{CoM}}\)) ¹², walking speed ¹⁶, the acceleration of the head and trunk ¹⁷, and the variance of step length and step width ¹⁸. The CoM, \(X_{\mathrm{CoM}}\), walking speed, and head and trunk acceleration are control targets in postural control and appear as results of postural control.

On the other hand, the COP ¹⁹ is something that humans actively change during postural control. Because this includes the influence of acceleration, it is thought to be an index that more directly reflects human postural control, taking dynamics into account (Fig. 2). Furthermore, unlike stride length and step width, COP can be obtained as a continuous value, making it suitable for system identification. Therefore, in this paper, we measure COP as a postural response to external disturbances.

2.2. Postural Response System Input

To date, various disturbance methods have been used to obtain postural responses during walking. For example, Martelli et al. applied disturbances by independently slowing down the belts of a treadmill ⁹.

The disturbance application method for identifying postural response dynamics during human walking must be able to continue walking without falling, be quantitatively measurable, and be reproducibly applied during a specific walking cycle. The study uses a mechanical disturbance to a walking human by decelerating one side of a separated treadmill, taking into account the variable magnitude of the disturbance input and ease of measurement. The method for decelerating one side of the belt is shown in Fig. 3. In the experiment, the speed of the belt is controlled by LabVIEW program.

First, during normal walking before the application of a disturbance, the same value is commanded to both the left and right sides. Next, shortly after the decelerating leg touches the ground, the post-deceleration value is commanded to only the decelerating leg. This acts as a mechanical disturbance on the supporting leg. The speed difference was determined to be small enough to allow the subject to continue walking without falling, but large enough to induce a postural response to the disturbance. The time from touch down to the start of deceleration is determined as the time when the large COP fluctuations caused by the impact of the legs with the ground at touchdown converged as shown in Fig. 4. In addition, the belt speed indication during deceleration is set to a step-like manner.

2.3. Posture Recovery Identification Method

This section explains a method for system identification of human postural control ability during walking as a mathematical model. The section derives a mathematical model based on COP fluctuations, which represent the relatively small postural response caused by disturbances due to treadmill belt deceleration.

The postural response dynamics to large mechanical disturbances that can lead to falls during walking is inherently considered to be nonlinear dynamics and time-varying systems. However, previous research (citing Togoe et al.’s paper ¹⁵) has shown that, with relatively small disturbance inputs, if appropriate COP input data for a short period representing the attitude change corresponding to that dynamic disturbance is available, the attitude correction capability can be fairly well represented by a linear system approximation. Therefore, we assume a linear input–output system and consider a discrete-time linear time-invariant system that generally represents this. We apply the least squares method using the Auto Regression eXogenous (ARX) model, which is commonly used for identifying discrete-time linear time-invariant systems.

The discrete-time state equation of the ARX model is expressed as follows:

First, we find the coefficients of the numerator and denominator polynomials.

From Eqs. (3) and (7), \(\boldsymbol{A}\) is composed of output data \(y\) and input data \(u\), and from Eq. (6), \(\boldsymbol{y}\) is the output data itself. Therefore, the coefficient vector \(\boldsymbol{c}\) can be calculated from Eq. (12). Then, we obtain a transfer function, which is a dynamics model, using the coefficient vector \(\boldsymbol{c}\). Assuming \(r_t=0\), and performing the \(\mathcal{Z}\) transformation for \(y_k\),

2.4. How to Select the Order of the System

To obtain the transfer function of the system to be identified, the order of the transfer function \(N, M\) in Eq. (14) must be selected in advance. It is difficult to determine the order that can adequately represent the attitude control dynamics before identification. Furthermore, if the order is too small, the attitude control dynamics cannot be adequately represented, which may result in insufficient identification accuracy. On the other hand, if the order is too large, overfitting may occur to various errors, such as COP measurement errors, which may result in insufficient identification accuracy.

Therefore, while changing the order \(N, M\), the error between the experimental COP response \(y_{i, \mathrm{actual}}\) and the response \(y_{i, \mathrm{simulation}}\) calculated from the identification model using the same input data is calculated using the following formula:

2.5. Evaluation of Posture Recovery Ability Using Poles of the Transfer Function

As shown in Section 2.3, the method allows the numerator and denominator of the transfer function to be obtained as rational polynomials, and the roots, i.e., the poles and zeros, can be calculated using numerical solutions. When this transfer function is proper,

In a real response, poles close to the origin decay rapidly as the time step progresses, and therefore have less effect on the output. On the other hand, poles close to the unit circle are less damped as the time step advances and become dominant in the output. Therefore, the following index \(S\), which takes into account the poles after \(K\) steps as a weighting for poles close to the unit circle, is considered as an index of posture recovery ability in this study.

As mentioned in Section 2.3, this method is thought to simultaneously estimate the poles of the dynamics specific to normal walking. The poles related to the dynamics corresponding to posture recovery from mechanical disturbances during walking are considered important. Therefore, as shown in Fig. 6, only poles whose argument \(\theta\) is within a specific range are used to calculate the index. The pole angle \(\theta\) corresponds to the frequency component of the output \(y\), and we select poles with angles corresponding to the frequency bands where the characteristics of posture recovery dynamics have been confirmed through frequency analysis, which means frequency filtering for poles. The pole angle \(\theta\) [deg] and the output frequency \(f\) [Hz] are related as follows:

3.1. Experiment and Measurement

Participants walk at a constant, self-selected comfortable walking speed on the treadmill. As a disturbance during walking, only one side of the belt is suddenly decelerated. At that point, the participant must suddenly decelerate the speed of the leg that is standing on that belt, which becomes a mechanical disturbance. A schematic diagram of treadmill belt deceleration is shown in Fig. 3. The time from the start of the stance phase to the start of deceleration is determined based on preliminary experiments, which is explained in Section 2.2. The participants are asked to walk at a comfortable speed, and COP fluctuations during normal walking were measured. An example is shown in Fig. 4. Sampling time \(T=1\) ms is used in the experiment. These results allowed us to estimate the point at which the large COP fluctuations caused by normal walking during heel strike converge to some extent, and we set the timing of applying artificial disturbance input due to treadmill speed differences to 0.15 s after this point.

This allows for some degree of separation of the influence of postural response dynamics caused by normal walking. Furthermore, even when using COP response data that include some dynamics for normal walking, it is believed that the proposed method can still be used to evaluate postural correction ability.

3.2. COP Measurement with Different Postual Recovery Ability Conditions

To confirm whether the proposed method can quantitatively evaluate the apparent decline in postural control ability alone, we conducted experiments by four participants under conditions in which postural recovery ability was artificially reduced and under normal walking conditions for each.

During walking, the sole of the foot is the only part of contact with the environment, and postural control via the sole is essential for maintaining stable walking ²⁰. Therefore, we set two conditions to evaluate posture recovery ability: one in which a marble (sphere) is held between the toes and one in which it is not held. Fig. 9 shows the results of measurements of plantar pressure distribution and COP trajectory for participant A when walking without holding a marble and when walking with a marble held between his toes. The green area in the figure represents the part where the sole of the foot touches the ground during walking, and the white line segment shows the COP trajectory. By comparing the cases with and without a marble clamp, it can be seen that when a marble is clamped, the contact with the toes, especially the area around the big toe, disappears. These results suggest that the action of holding the marble between the toes inhibits the toes from stepping on or pushing off the ground. Therefore, the use of marbles in this experiment is considered to be a valid condition for intentionally reducing posture recovery ability.

3.3. Time-Frequency Analysis of COP Fluctuations

Figure 10 shows the time-frequency analysis of COP fluctuations for a representative trial of participant A. The figure shows frequency on the vertical axis, time on the horizontal axis, and gain as color intensity. Note that the time window length was set to 0.128 s, so the 0.064 s at both ends are blank. In the figures, (i) shows the time-frequency distribution during a normal walking cycle without disturbances due to belt speed differences, and (ii) shows the time-frequency distribution when disturbances due to belt speed differences are applied from 0.15 s after ground contact.

The gain is particularly different in the areas marked with black squares, from 0 Hz to 40 Hz and from 0.15 s to 0.5 s, indicating that the frequency characteristics are different. From the result, it is predicted that different dynamics are excited before and during belt deceleration. In particular, for this part, the dynamics during belt deceleration (ii) was considered as the attitude control dynamics in response to disturbances, and system identification was performed.

The postural control dynamics in response to belt deceleration, which differs from the dynamics of normal walking before belt deceleration, was identified as a transfer function using the method described in the previous section, where the system order \(N=315\) (\(=M\)) is selected by the optimization in Section 2.4. In this case, the time interval for the input/output data used for identification is set to 350 ms from the start of belt deceleration, based on the results of Fig. 10 in this section.

An example of the pole distribution of the identified transfer function is shown in Fig. 11. Note that in discrete linear models, stable poles lie within the unit circle, and the distance from the origin indicates the speed of convergence, i.e., how quickly the orientation converges.

3.4. Calculation of Posture Recovery Performance Evaluation Index

For each participant, we confirmed that there was a difference in the evaluation index of postural control ability defined by Eq. (22) between the experimental conditions in which the marble was sandwiched and the experimental conditions in which it was not sandwiched. Based on the results in Section 2.5, the poles used for evaluation were determined to have angles between 0.1 Hz and 40 Hz. Therefore, from Eq. (23), \(\theta_\mathrm{min}=0.036°\), \(\theta_\mathrm{max}=14.4°\). The values of the evaluation indices for each participant are shown in Figs. 12–15. A significant difference in the value of the evaluation index \(S\) was confirmed between the two experimental conditions for all four participants, demonstrating that this method can quantitatively determine postural recovery ability.