Dynamic Stability of Human Walking in Response to Sudden Speed Changes

Abstract

Abrupt changes in gait speed can interfere with the symmetry of the overall gait apparatus and result in unstable joint movement patterns. Because unstable joint movements may cause slips, trips, and falls, it is necessary to quantitatively characterize the changes in joint movement patterns in response to sudden speed changes. The purpose of this study is to examine how abrupt changes in gait speed affect gait dynamics. Twenty-two healthy young subjects walked for four minutes, including a warm-up period, under three different speed conditions. Utilizing nonlinear dynamics tools, including the maximum Lyapunov exponent, Sample Entropy, and Detrended Fluctuation Analysis, we quantitatively assessed gait dynamics for the different speed conditions. Our findings highlight how different speed change patterns impact joint instability, notably within the knee joint during gait (p < 0.05). Furthermore, introducing a resting phase during random speed changes exhibited the potential to restore gait symmetry and control movement patterns. This research offers valuable insights into human gait stability dynamics, especially concerning sudden speed changes. Understanding how controlled speed variations affect gait and joint instability informs fall prevention and rehabilitation strategies, emphasizing speed management to improve gait symmetry and reduce joint instability.

1. Introduction

Minor changes in lower-extremity joint movements during walking can have profound effects on overall gait patterns [1,2]. The intricate coordination of body movements during gait is influenced by various factors including time, posture, and dynamic stability [3]. Dynamic stability, characterized by the ability to maintain desired movement states despite internal and external perturbations, is crucial for effective walking [4]. It enables individuals to swiftly recognize and respond to environmental changes, thus sustaining gait trajectory and preventing falls. Given the implications of lower-extremity instability for falls and injuries, evaluating dynamic stability under conditions of repetitive perturbations is paramount.

Gait speed, a fundamental parameter of walking, has been shown to serve as an adaptive mechanism to respond to perturbations and enhance dynamic stability [5,6,7,8]. Individuals typically select a preferred walking speed, optimizing energy expenditure while ensuring efficient locomotion [6,7]. This preference shapes various kinematic parameters, impacting joint angles, body segments, and stride lengths, thereby contributing to gait pattern variability [9]. While studies have explored gait characteristics across varying speeds, including slow, fast, and fixed paces, the investigation of dynamic stability during sudden speed changes remains limited [10,11]. Despite the natural variability in gait parameters, particularly in limb joint kinematics, the stability implications of abrupt speed changes remain relatively unexplored.

The intricate interplay between joint movement control and spatiotemporal adjustments during walking contributes to stable gait patterns [12,13]. Notably, different age groups and pathological walkers exhibit distinct strategies. Older adults and individuals with pathologies tend to prioritize reduced variability and increased stability through slower walking, while healthy young adults exhibit greater variability but higher stability across various speeds [14,15,16]. However, understanding dynamic stability when encountering sudden speed changes remains an understudied area. The correlation between low gait stability and increased variability is not straightforward, necessitating a comprehensive approach to assessing stability within the context of variable speeds.

To address these gaps, this study employs nonlinear dynamics methods, including the Maximum Lyapunov Exponent (MLE), Sample Entropy (SaEn), and Detrended Fluctuation Analysis (DFA), to quantitatively analyze the relationship between gait pattern characteristics and dynamic stability. These methods offer a holistic perspective on gait behavior, capturing intricate dynamics throughout the entire gait cycle [10,17,18,19,20,21,22]. Unlike previous studies focusing on individual variables, the utilization of nonlinear dynamics tools allows us to delve into the complex interactions that shape gait stability during sudden speed changes [23,24].

The application of nonlinear dynamics methods within the context of gait evaluation offers a novel approach to understand the complexities of human locomotion. These metrics provide a deeper insight into the dynamic interactions and variability inherent in gait patterns. The Maximum Lyapunov Exponent (MLE) measures the rate of divergence of trajectories in a reconstructed state space, reflecting the sensitivity to initial conditions and quantifying the predictability of movement dynamics during gait. Sample Entropy (SaEn) quantifies the irregularity and complexity of gait patterns, reflecting the system’s unpredictability and adaptability in response to perturbations. Meanwhile, Detrended Fluctuation Analysis (DFA) assesses the long-range correlations within gait time series, indicating the degree of self-similarity and persistence in gait patterns across various timescales. These nonlinear metrics serve as powerful tools to capture the subtle intricacies of gait dynamics, enabling a comprehensive understanding of how sudden speed changes affect gait stability and movement patterns.

In this investigation, we quantify gait pattern characteristics during speed changes using nonlinear dynamics methods and explore their interplay with dynamic stability. Our experiment encompasses three walking speed conditions: (1) maintaining the preferred walking speed; (2) proportional speed increase or decrease; and (3) random speed changes. We hypothesize that the highest stability will emerge during random speed changes due to the adaptive responses required. Thus, the study aims to quantitatively assess gait stability under variable speed conditions, shedding light on its impact on gait patterns and providing insights into the effects of sudden speed changes.

2. Materials and Methods

2.1. Data Collection

Twenty-two healthy young subjects (eleven males and eleven females) participated in this study. The average age was 24.36 ± 1.6 years, average height was 1.67 ± 3.45 m, and average weight was 63.03 ± 6.56 kg. All the subjects had no difficulty with gait and were accustomed to walking on a treadmill. The subjects had not been diagnosed with any neurological disease or musculoskeletal injury. As the aim of the experiment was to evaluate changes in gait stability on a treadmill, only subjects who had no walking problems and could walk naturally on a treadmill were included. In a similar gait analysis study, a statistically significant study required at least 10 subjects per group. With a confidence level of 95%, the sample size was estimated to be 15 and was increased to 22 participants for this study. This study was approved by the Institutional Review Board of Incheon National University (7007971-202003-006-01).

The subjects completed a two-minute warm-up trial to become acquainted with treadmill walking before data collection began. The subjects were allowed to experience all speeds based on a fixed preferred speed during warm-up time. Subjects walked on a horizontal motorized treadmill (Xiaomi Walking Pad A1 PRO, Beijing, China) to obtain sufficient time-series data for dynamic analysis while controlling the walking speed.

For each of the three conditions, every participant engaged in a 4 min trial, which included preparation time, based on the specific speed condition assigned. To determine their preferred speed, participants began at a slower pace and gradually increased the treadmill’s speed until they indicated that their current pace was faster than their preferred speed. The treadmill featured 7 speed modes, initiating from a complete stop and incrementing by 0.5 units up to a maximum speed of 3.5.

The proportional speed change condition followed the pattern of a speed increase every 5 s according to the participant’s preferred speed. When the maximum speed was attained, the speed increments ceased, followed by a gradual decrease until complete stop. On the other hand, the random speed change condition commenced at the participant’s preferred speed and randomly adjusted within the 7 speed modes every five seconds. This random speed condition varied for each participant.

Each speed condition trial lasted four minutes, during which kinematic data were collected for a continuous two-minute walking period. One subject completed a total of nine trials, performing three trials per speed condition. Participants were encouraged to walk comfortably and refrain from actions that might disrupt walking, such as excessive body movement or coughing. To mitigate gait alterations due to fatigue, participants rested for a minimum of two minutes prior to each trial.

Eight Optitrack (Prime 17W model, Corvallis, OR, USA) motion capture devices were used to record kinematic data at 100 Hz. A total of 39 infrared reflective markings (14 mm in diameter) were applied to the whole body. Based on the kinematic data of the markers, 12,000 joint motion time-series data were generated. The average number of gait cycles considered for each speed condition was 80, and these were analyzed using three methods. It is theoretically possible to capture the complexity of the entire system by studying the dynamics of a single section over a sufficiently long period of time [10].

2.2. Maximum Lyapunov Exponent Implementation

The MLE has been reported as a valid method for estimating fall risk by quantifying dynamic stability during walking [25,26]. The MLE was calculated by measuring the exponential ratio of the mean log divergence in a state space reconstructed from trajectories obtained during walking. It can quantify a person’s ability to respond to changes in movement and small perturbations during gait [27,28,29]. The MLE quantifies the change in distance between two adjacent points in a continuous trajectory [27,28]. The maximum MLE and dynamic stability of the gait trajectory were inversely proportional.

Time-series data of joint angles are required to reconstruct an attractor in the m-dimensional state space to implement perturbations (Figure 1a). A time-delay method for averaging mutual information was selected to reconstruct the attractor’s state space [30]. An attractor essentially represents the set of states that a dynamic system tends to gravitate toward over time, irrespective of its initial conditions. In this method, mutual information serves as a measure to quantify the statistical dependence or shared information between two variables within a dataset [31,32]. It exhibits robustness against inherent noise in time-series data and is particularly effective for studying nonlinear dynamical systems susceptible to chaotic behavior. Specifically, in time-series data, it helps gauge the association or predictability between different data points. Understanding the degree to which knowing one point in the series reduces uncertainty about another is crucial.

The determination of the ‘embedding dimension’ was pivotal in this study [31,32]. This dimension refers to the number of components utilized to reconstruct the phase space from a single scalar time series. In complex systems, a single scalar series might not adequately represent the underlying dynamics. By embedding the time series into a higher-dimensional space, via time-delay embedding, a more comprehensive representation of the system’s behavior becomes possible. All kinematic data were quantitatively estimated under the same conditions, with fixed time delays and embedded dimensions [32]. Fixed time delays and the included dimension values increased the reliability of calculating the MLE using fixed values as the overall mean of individual trial T and m values for all participants (T = 12 and m = 7) (Figure 1b). From the phase trajectories of the reconstructed attractor according to the Rosenstein algorithm, it is necessary to identify neighboring trajectories with the closest points, excluding identical trajectories (Figure 1c).

$$P\left(i\right)=\frac{1}{i∆t}ln{d}_j\left(i\right),$$

(1)

The MLE value was identified as the slope of the average divergence log of the trajectory (Equation (1)) (Figure 1d). A positive MLE value indicates the tendency of the reconstructed attractor to deviate from the mean trajectory, indicating an unstable pattern. The stability of the gait pattern decreased as the MLE value increased. When the MLE was less than zero, deviation was assumed to be absent and the motion was considered stable.

2.3. Sample Entropy

Sample Entropy (SaEn) serves as a statistical metric that provides a quantitative assessment of the level of predictability or repeatable pattern characteristics within time-series data. An advantage of SaEn lies in its ability to identify nonlinearities by utilizing a deterministic probabilistic system. Unlike traditional Approximate Entropy, SaEn demonstrates robustness by yielding consistent results and being less sensitive to varying data lengths. The calculation of SaEn involves determining the natural logarithm of the probability estimate of length ‘m’ matched point-by-point within tolerance ‘r’ and then repeated for points ‘m + 1’ (Figure 1e). In our analysis, we employed specific values of ‘m = 2’ and ‘r = 0.2*standard deviation’ for this purpose. In the process of estimating m values ranging from 2 to 10 and r values of 0.2, 0.25, and 0.3, consistent results were observed despite changes in m values, so fixed values were used for the specified parameters [33,34]. Smaller SaEn values correspond to higher regularity and predictability in the time series, while larger SaEn values signify lower repeatability. Notably, an SaEn value of 0 within a periodic time series suggests a highly prescriptive and predictable pattern.

2.4. Detrended Fluctuation Analysis

The DFA exponent was used to quantify the mapping of self-similarity in the long-range correlated time series. Long-distance dependence was used as an indicator of the system adaptability in the physiological time series (Figure 2a). Long-distance correlations characterize the variability structure of time-series joint data, allowing a quantitative indication of the dependence of the next stride on the preceding one. Each time-series data point represents the change in variation $F\left(n\right)$ with n (number of strides) in different observation windows, where $F\left(n\right)$ increases with n [35].

$$F\left(n\right)\approx n^{\alpha },$$

(2)

The linear relationship in the double-log graph is represented by Equation (2), and the slope was determined using the trended practical exponent (Figure 2b). The magnitude of α is ≤0.5, which indicates a semi-permanent correlation, whereas 0.5 < α < 1 indicates a long-term correlation.

2.5. Statistical Analysis

Two-way repeated measures were used to test the significant difference between different speed conditions. ANOVA using SPSS (Statistical Package for the Social Sciences, Chicago, IL, USA) was performed to compare the quantitative values of dynamic stability during walking, and p < 0.05 was considered significant. In the post hoc analysis, Bonferroni’s test was used to assess the main effects.

3. Results

Dynamic stability during gait was quantitatively expressed using the MLE, SaEn, and DFE values for each speed condition using kinematic data and then determined. In the proportional condition, the MLE value for joint movement was highest at 2.36 ± 0.12. A larger MLE value means greater volatility, and a significant difference from the random condition with an MLE value of 2.29 ± 0.03 was confirmed (Figure 3) (p < 0.05). In all the speed conditions, it was confirmed that the knee joint movement had a high MLE value. The lowest MLE value among the lower-limb joints was determined in the ankle joint, and the same results were verified in all cases in

Table 1. MLE for lower-extremity joint movement in each speed condition.

Condition	Joint	Max. Lyapunov Exponent	SD	p-Value
Normal	Ankle	1.66	0.16	<0.001
	Hip	1.97	0.18
	Knee	2.11	0.18
Proportional	Ankle	2.11	0.13	<0.001
	Hip	2.44	0.17
	Knee	2.53	0.15
Random	Ankle	2.03	0.14	<0.001
	Hip	2.37	0.15
	Knee	2.48	0.17

. The SaEn value in the proportional condition was the highest at 1.67 ± 0.13, and in the random condition, there was no significant difference at 1.54 ± 0.13. In both conditions, a significant difference from the preferred speed with no change in speed was confirmed (Figure 4) (p > 0.05). For all the speed settings, it was determined that the knee joint movement had a high SaEn value in

Table 2. SaEn for lower extremity joint movement in each speed condition.

Condition	Joint	Sample Entropy	SD	p-Value
Normal	Ankle	1.15	0.29	<0.001
	Hip	1.01	0.28
	Knee	1.73	0.41
Proportional	Ankle	1.55	0.29	<0.001
	Hip	1.57	0.40
	Knee	1.90	0.47
Random	Ankle	1.46	0.35	<0.001
	Hip	1.54	0.37
	Knee	1.64	0.49

. There was a significant difference between the Normal and other conditions when the DFA exponent value was greatest (Figure 5) (p < 0.05).

4. Discussion

This study quantitatively investigated the movement patterns and variability of lower-extremity joints under time-varying speed conditions. Walking was performed at three different speeds, and the movement of the lower-extremity joints was measured. In this study, dynamic stability during gait was calculated using nonlinear dynamics tools, namely MLE, SaEn, and DFA. We hypothesized that a random change in velocity would be the most critical change in the pattern of the lower-extremity joints. The main finding was that the ability to respond to an unexpected environment (change in speed) was highly dependent on the pattern of change in speed. Specifically, it suggests that the change in gait dynamics according to speed conditions can be characterized by various aspects according to the nonlinear dynamics tools.

Interestingly, the MLE showed significantly higher exponential values for proportional pattern changes than for random pattern changes. The movement pattern of the lower-extremity joint was found to be the most unstable under proportionally changing speed conditions and this was confirmed by quantitative MLE values. Contrary to the initial hypothesis, there was no significant decrease in the mobility stability under the random change condition. In healthy young adults identified in a previous study, the rapid response to unexpected situations may support our results [34]. Overall, this study strongly suggests that younger, healthier individuals have higher predictive abilities, which may be a result of choosing gait patterns in response to unexpected perturbations.

The second hypothesis of this study was that the regularity of the gait pattern also increases when the gait speed is proportionally changed. The largest MLE value was obtained when the gait speed condition that changed with time had a proportional pattern, and no relationship was found between the gait pattern and gait speed pattern. The SaEn value, which quantifies the regularity of the gait pattern, showed a lower difference between the gait speed conditions that changed with time. From the pedestrian point of view, it is estimated that there is a high probability that no significant difference was found because all the speed change conditions were perceived as unexpected environments. Notably, changes in speed conditions alter the regularity of gait patterns compared to a fixed speed; however, this may be independent of the speed change patterns.

The fractal scaling exponent, which quantifies the long-distance correlation between the stride length, width, and time, showed clear differences under all the speed change conditions. This suggests that it is difficult to predict gait patterns under randomly changing speed conditions. This refutes the hypothesis of previous studies that gait trajectory is more consistent at a fixed speed [35]. In random speed change, long-distance correlation of gait variables was found, but the continuity of the gait pattern was confirmed to be low. It is reasonable to infer that a person controls the gait pattern even when the speed changes; if the gait pattern is changed, self-similarity is detected even within an arbitrary pattern and compensates for the next movement.

MLE acknowledges the possibility of a relationship between different methods, and as MLE increases, SaEn increases and DFA decreases. Our findings allow us to investigate the relationship between MLE, SaEn, and DFA. Rapid changes in speed affect the dynamics of joints during gait and simultaneously have practical implications in terms of local dynamic stability [30,36]. An interpretation of the stability of gait patterns suggested by this study is that young and healthy adult subjects may have the cognitive abilities to handle the changed demands of their coordination. Therefore, further studies are needed to confirm the association between biomechanical parameters, gait parameters, and cognition.

The knee joint movement pattern over time was consistently observed to be the most unstable of the lower-extremity joints. MLE and SaEn, which quantified predictability and regularity, respectively, demonstrated that the movement of the knee joint was clearly unstable in all speed conditions compared with other lower-extremity joints. Contrary to previous results, the movement of the ankle joint with severe instability has high regularity, predictability, and fast response speed under speed change conditions, so it is well prepared for the next trajectory [37,38,39,40]. Because the ankle’s movement pattern is closest to the walking ground during gait, it can be inferred that this is the result of recognizing and adapting to the change in the treadmill faster than other joints.

Our findings suggest that the pattern of changes in gait speed over time affects the gait dynamics in young and healthy adults. Changes in gait speed had a distinct effect on the variability of lower-extremity joint movement and the long-distance correlation quantified by MLE and DFA, respectively. Although speed change had a clear effect on the complexity and regularity of the lower-extremity joints quantified by SaEn, there were no significant differences between the patterns of speed change. As our study was limited to healthy young adults, further investigation of gait stability is required for older adults at high risk of falls and for those with gait difficulties. In addition to walking speed, speed change over time can provide useful clinical information regarding movement instability in patients with Parkinson’s disease. Further studies should be conducted to assess the ability of gait tasks to identify, prevent, and respond to rehabilitative treatment of speed change over time.

5. Conclusions

This study delved into the nuanced dynamics of lower-extremity joint movements under varying speed conditions during walking. By employing nonlinear dynamics tools (MLE, SaEn, and DFA), we quantified dynamic stability and its relationship with gait pattern changes. Our investigation centered on three distinct walking speed conditions: maintaining preferred speed, proportional speed changes, and random speed changes.

Our findings unveil the intricate interplay between gait dynamics and speed change patterns. Contrary to initial expectations, the knee joint emerged as the most vulnerable point, consistently demonstrating instability across all speed conditions. The MLE, SaEn, and DFA metrics collectively revealed that knee joint movements exhibited significant unpredictability and irregularity, potentially reflecting the complex adjustments required to respond to speed changes.

Interestingly, our results indicated that proportional speed changes induced higher instability in knee joint movements than random speed changes. This emphasizes that the specific pattern of speed change substantially influences the ability to adapt to unexpected environments. Young and healthy individuals demonstrated remarkable predictive capabilities, suggesting an adaptive cognitive component to gait stability. The ankle joint’s swift response to speed changes, despite its inherent instability, further highlights the importance of recognizing and adapting to dynamic shifts.

While our study offers comprehensive insights into the relationship between gait dynamics and speed change patterns in healthy young adults, it carries certain limitations. Firstly, our research was confined to a specific demographic of healthy young individuals, limiting the generalization of the findings to broader age groups or clinical populations. Future studies should encompass diverse age brackets, particularly older adults susceptible to falls and individuals with movement disorders, to provide a more comprehensive understanding of gait stability under varying speed conditions.

Secondly, our analysis focused on a select set of nonlinear dynamics metrics (MLE, SaEn, and DFA) to assess gait stability. Incorporating additional metrics or complementary methodologies might offer a more nuanced evaluation of gait dynamics and their response to speed changes.

In line with our title and research objectives, this study underscores that sudden speed changes indeed disrupt gait stability. Our quantification of gait patterns using nonlinear dynamics methods further demonstrates the influence of speed change patterns on joint behavior. The implications of these findings extend to fall-prevention strategies and rehabilitation efforts.

Moving forward, studies that investigate real-time interventions or training regimens aimed at enhancing gait stability in response to abrupt speed changes could significantly contribute to fall-prevention strategies and rehabilitation practices. Furthermore, exploring the applicability of our findings in clinical settings, such as devising personalized treatments for individuals with movement disorders like Parkinson’s disease, remains an intriguing avenue for future research. It is worth noting that our study focuses on healthy young adults, warranting further exploration in older populations prone to falls and individuals with movement disorders like Parkinson’s disease.

In conclusion, while our study elucidates critical insights into the impact of speed change patterns on gait dynamics and joint behavior in healthy young adults, further investigations encompassing diverse populations and additional methodologies are necessary to comprehensively delineate the complexities of gait stability and its implications for clinical applications.