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European Journal of Sport Science Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tejs20
Effect of speed on local dynamic stability of locomotion under different task constraints in running a
a
bc
Sina Mehdizadeh , Ahmed Reza Arshi & Keith Davids a
Biomechanics and Sports Engineering Group, Faculty of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran b
Centre for Sports Engineering Research, Sheffield Hallam University, Sheffield, UK
c
FiDiPro Programme, University of Jyväskylä, Jyväskylä, Finland Published online: 10 Apr 2014.
Click for updates To cite this article: Sina Mehdizadeh, Ahmed Reza Arshi & Keith Davids (2014) Effect of speed on local dynamic stability of locomotion under different task constraints in running, European Journal of Sport Science, 14:8, 791-798, DOI: 10.1080/17461391.2014.905986 To link to this article: http://dx.doi.org/10.1080/17461391.2014.905986
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European Journal of Sport Science, 2014 Vol. 14, No. 8, 791–798, http://dx.doi.org/10.1080/17461391.2014.905986
ORIGINAL ARTICLE
Effect of speed on local dynamic stability of locomotion under different task constraints in running
SINA MEHDIZADEH1, AHMED REZA ARSHI1, & KEITH DAVIDS2,3 1
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Biomechanics and Sports Engineering Group, Faculty of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran, 2Centre for Sports Engineering Research, Sheffield Hallam University, Sheffield, UK, 3FiDiPro Programme, University of Jyväskylä, Jyväskylä, Finland
Abstract A number of studies have investigated effects of speed on local dynamic stability of walking, although this relationship has been rarely investigated under changing task constraints, such as during forward and backward running. To rectify this gap in the literature, the aim of this study was to investigate the effect of running speed on local dynamic stability of forward and backward running on a treadmill. Fifteen healthy male participants took part in this study. Participants ran in forward and backward directions at speeds of 80%, 100% and 120% of their preferred running speed. The three-dimensional motion of a C7 marker was recorded using a motion capture system. Local dynamic stability of the marker was quantified using shortand long-term largest finite-time Lyapunov exponents (LyE). Results showed that short-term largest finite-time LyE values increased with participant speed meaning that local dynamic stability decreased with increasing speed. Long-term largest finite-time LyEs, however, remained unaffected as speed increased. Results of this study indicated that, as in walking, slow running is more stable than fast running. These findings improve understanding of how stability is regulated when constraints on the speed of movements is altered. Implications for the design of rehabilitation or sport practice programmes suggest how task constraints could be manipulated to facilitate adaptations in locomotion stability during athletic training. Keywords: Running, task constraint, local dynamic stability, treadmill
Introduction Stability is an important factor for functional human locomotion strategies (England & Granata, 2007). It has been defined as the capacity of a movement system to preserve its functional state of organisation in the presence of sudden kinematic perturbations due to changing task constraints (Davids, Button, & Bennett, 2008; England & Granata, 2007). One aspect of maintaining stability in human locomotion is the capacity to attenuate inherent stride-to-stride perturbations through, for example, visual control strategies (England & Granata, 2007; Scott, Li, & Davids, 1997). To quantify the ability to maintain stability, a number of studies have measured rate of divergence of trajectories in the state-space constructed from kinematic data, using largest finite-time Lyapunov exponents (LyE; Bruijn, van Dieen, Meijer, & Beek, 2009;
Dingwell, Cusumano, Cavanagh, & Sternad, 2001; Dingwell & Marin, 2006; England & Granata, 2007; Stergiou, Moraiti, Giakas, Ristanis, & Georgoulis, 2004). However, these efforts have tended to focus solely on examining walking behaviours, and system stability under different task constraints, such as during forward and backward running, has rarely been investigated. The present study, therefore, aimed at investigating the stability of forward and backward running in order to complement existing research. The LyE measures the exponential rate of divergence of neighbouring trajectories of the state-space constructed by kinematic data obtained from a movement system during performance (Dingwell & Marin, 2006; Rosenstein, Collins, & De Luca,
Correspondence: A. R. Arshi, Biomechanics and Sports Engineering Group, Faculty of Biomedical Engineering, Amirkabir University of Technology, Hafez Ave. Tehran, Iran. E-mail: [email protected] © 2014 European College of Sport Science
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1993). This type of system stability is considered as local dynamic stability since LyE quantifies the ability to respond to small local perturbations during movement performance (Dingwell & Marin, 2006). In addition, Since LyE measures the rate of divergence of the trajectories, a greater LyE value indicates a lower level of local dynamic stability in the system. Using the LyE method, previous research has indicated that stability of walking is speed-dependent (Bruijn et al., 2009; Dingwell et al., 2001; Dingwell & Marin, 2006; England & Granata, 2007; Stergiou et al., 2004). Dingwell and Marin (2006), and England and Granata (2007) in separate studies, for example, have demonstrated that the value of LyE increased (i.e. local dynamic stability decreased) with increments in walking speed. These observations suggest that slow walking is inherently more stable than fast walking. The results of a study by Bruijn et al. (2009), on the other hand, showed that slow walking was not necessarily more stable than fast walking. Their findings led Bruijn et al. (2009) to argue that the relationship between local dynamic stability and system speed depends on the plane of motion. Regardless, all these studies were conducted under the task constraints of walking, and characterisation of the effect of speed on local dynamic stability in running has rarely been approached. In one study, Jordan, Challis, Cusumano, and Newell (2009) investigated the local dynamic stability of running patterns compared to walking patterns. The range of speeds used in that study, however, was limited to speeds very close to walk–run transition speeds in participants and no relation between speed and stability in running was established. Furthermore, to the best of our knowledge, no study has investigated the effect of speed on local dynamic stability in backward running. Backward running is a common running pattern in many sports such as soccer, basketball and football (Zampeli et al., 2010). In addition, backward running has been considered as a common rehabilitation technique by sport rehabilitation specialists (Cipriani, Armstrong, & Gaul, 1995; Threlkeld, Horn, Wojtowicz, Rooney, & Shapiro, 1989). Research studies have suggested that a single neuro-motor mechanism might be responsible for movement system regulation in locomotion in forward and backward directions (van Deursen, Flynn, McCrory, & Morag, 1998). Therefore, it could be argued that the effect of speed on the ability to maintain stability is identical in both forward and backward running. The aim of this study was therefore to investigate the effect of running speed on local dynamic stability of forward and backward running on a treadmill. It is hypothesised that speed will affect the local dynamic
stability of participants under the task constraints of both forward and backward running.
Methods Participants Fifteen male participants took part in this study. They were 24.1, s = 1.0 years old with average mass and height of 68.8, s = 3.9 kg and 1.76, s = 0.04 m, respectively. Participants were asked to report any musculoskeletal injuries that prevent them from running in forward and backward directions. None of them reported any musculoskeletal injuries at the time of experiments. All participants provided written informed consent before participation in the study. The ethics committee of Amirkabir University of Technology approved the experimental procedure. Marker placement Since this study was a part of a broader study on the stability of the movement, 17 passive reflective markers (14 mm diameter) were attached to the skin of each participant at the right and left bony landmark on the second metatarsal head (toe), calcaneus (heel), lateral malleolus (ankle), mid-tibia, lateral epicondyle of knee (knee), mid-thigh, anterior superior iliac spine and also on the sacrum, midway between posterior superior iliac spines, 10th thoracic vertebrae (T10) and 7th cervical vertebrae (C7). For this particular study, right toe and C7 markers were used for further analyses. Task Before starting the experiments, participants had enough time to familiarise themselves with running forward and backward on the treadmill. During the actual tests, all participants ran in forward and backward directions on a motorised treadmill (Cosmed® T150, Rome, Italy) at 80%, 100% and 120% of their preferred forward and backward running speeds. Preferred running speed (PRS) was obtained following a top-down and bottom-up approach similar to protocols described in Dingwell and Marin (2006) and Jordan et al. (2009). Participants began by running on the treadmill at a slow speed (as defined by participant perceptions) followed by a gradual increase of 0.1 km/h increments until each participant declared that he was running at his PRS. This speed value was recorded, and increments were introduced until each participant declared that the speed was fast. From this point on, the speed value was gradually decreased until once again the participant reported that he was running at his PRS. This speed was also recorded. The average of these
Effect of speed on local dynamic stability of locomotion
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two recorded PRSs was consequently determined. This procedure was repeated three times, and the mean value of three average PRSs was considered as each participant’s actual PRS. This procedure was followed under the task constraints of both forward and backward running. During acquisition of the PRS data, participants were not allowed to view the speed at which they were running on the treadmill (Jordan, Challis, & Newell, 2007). Each participant was then asked to run for two minutes in every one of the six test trials. The trials were conducted at speeds of 80%, 100% and 120% of actual PRS values for each participant, under both forward and backward running task constraints in random order (Dingwell & Marin, 2006). Sufficient rest periods were allocated between the tests to allow participants to recover.
Data recording The three-dimensional coordinate data of the markers were recorded using five Vicon® VCAM motion capture calibrated cameras (Oxford Metrics, Oxford, UK) at the sampling frequency of 100 samples/ second. Reconstruction and labelling were performed using Vicon® Workstation software (Oxford Metrics, Oxford, UK).
Data analysis Data associated with x (Anterior–Posterior: AP), y (Medio–Lateral: ML) and z (Vertical: VT) components of the C7 marker motion were analysed. The necessity of a stationary characteristic for nonlinear analysis of the time series data resulted in the adoption of velocity profiles for further ana‐ lyses (Dingwell & Marin, 2006; Kantz & Schreiber, 2004). Although the common approach at this stage of analysis is to filter the signals obtained from movement system, possible loss of information at critical points prohibited implementation of any filtering algorithms (Kantz & Schreiber, 2004). For each trial, a total number of 100 consecutive strides were analysed. In addition, all time series were timenormalised to an equal length of 10,000 points that in turn signified that each individual stride comprised approximately 100 data points. To quantify local dynamic stability, first, an appropriate state space was reconstructed using Equation (1): X ðtÞ ¼ ½xðtÞ, xðt þ sÞ, xðt þ 2sÞ, . . . xðt þ ðdE 1ÞsÞT ð1Þ where X ðtÞ is the reconstructed state vector, xðtÞ is the original velocity time vector, s is the time delay and dE is the dimension of the state space, i.e. the
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embedding dimension (Kantz & Schreiber, 2004; Takens, 1981). Time delay s is determined as the first local minimum of Average Mutual information (AMI) function (Fraser, 1986). The AMI is a statistical measure from theory of information that shows how much information about a random variable can be obtained from the information of another random variable (Shelhamer, 2006). In this method, mutual information between a time series xðtÞ and its time shift xðt þ sÞ is calculated for different values of s until the mutual information is minimised (Shelhamer, 2006). A time delay of 10 samples was found to be appropriate for data associated with the three AP, ML and VT directions. This value of time delay has also been determined in other studies (Bruijn et al., 2009; England & Granata, 2007). In addition, a Global False Nearest Neighbours (GFNN) measure was used to determine embedding dimension dE (Kennel, Brown, & Abarbanel, 1992). False Nearest Neighbours are points which are close together in dimension dE but not in dimension dE þ 1 (Shelhamer, 2006). According to this method, the dimension is gradually increased until the number of false nearest neighbours is reduced to zero. For the purpose of this study, an embedding dimension of dE ¼ 5 was calculated for data associated with AP, ML and VT directions. Similar embedding dimension values have also been determined in previous studies (Dingwell & Marin, 2006; England & Granata, 2007; Jordan, et al., 2009). In order to quantify local dynamic stability, the largest finite-time LyE was determined from kinematic data (Figure 1). The LyE measures the exponential rate of divergence of neighbouring trajectories in the state space (Rosenstein et al., 1993). Since LyE measures the rate of divergence of the trajectories, a greater LyE value indicates lower levels of local dynamic stability of a system. The approach implemented in this study was introduced by Rosenstein et al. (1993), which is most suitable for a finite time series. Here, the largest finite-time LyE ðk1 Þ could be determined using Equation (2): dðtÞ ¼ C expðk1 tÞ
ð2Þ
where dðtÞ is the average distance between neighbouring points at time t, and the initial separation of the neighbouring points is represented by C. According to Expression (2), for the jth pairs of neighbouring points in state space, we have: dj ðiÞ Cj expðk1 ðiDtÞÞ
ð3Þ
Taking the logarithm from both sides of Equation (3) results in: ln½dj ðiÞ ln½Cj þ k1 ðiDtÞ
ð4Þ
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S. Mehdizadeh et al. slope of a linear fit to Equation (5) in the range ðiDtÞ of 0 to 0.5 stride (approximately 0–50 samples; Bruijn et al., 2009), and 4–10 strides (approximately 400–1000 samples), respectively. In the present study, all LyE values were presented as the rate of divergence/stride. Before analysing the experimental data, the LyE algorithms were validated by feeding a Lorenz system with typical inputs to the algorithms and comparing the outcomes with the published results (Gates & Dingwell, 2009; Graham, Sadler, & Stevenson, 2011; Rosenstein et al., 1993).
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Statistical analysis Separate one-way repeated measure analysis of variance (ANOVA) tests were applied to determine whether speed affected LyE values. Here, k1S and k1L in different planes of motion and in forward and backward running were considered as dependent variables. A linear fit was used to determine the relationship between speed and LyE where statistically significant effects of speed were found. Statistical significance levels were set at P < 0.05. Results As shown in Table I, speed had a significant effect (P < 0.05) on the value of k1S in all planes of both forward and backward running direction. k1L , on the other hand remained unaffected as speed increased in forward running. During backward running, however, speed did provide a significant effect on k1L in the AP direction only. In addition, trend analyses indicated that the relationship between speed and k1S was well described by a positive slope Figure 1. State-space reconstruction and local dynamic stability analysis of time series obtained from experiment. (A) Original velocity time series x(t), (B) 3D representation of state-space reconstructed from time series x(t) and its time copies xðt þ sÞ, and xðt þ 2sÞ. Note that the dimension of state space might be greater than three but cannot be visually observed. The expanded view is a region of state space in which it is schematically shown that initial separation of jth pairs of points, dj ð0Þ, diverge after i time steps shown by dj ðiÞ. (C) Average logarithmic divergence of all pairs of neighbouring points plotted over time (shown as stride number) to calculate k1S and k1L . See text for further information.
k1 is determined using a linear fit to the following curve:
yðiÞ/ð1/DtÞ dj ðiÞ
ð5Þ
where dj ðiÞ denotes the average over all pairs of j. For this study, values of a short-term (k1S ) and a long-term ðk1L Þ LyE were calculated (Dingwell & Marin, 2006). k1S and k1L were calculated as the
Table I. Results of ANOVA tests for short- ðk1S Þ and long-term ðk1L Þ largest finite-time LyEs in forward and backward running. The results of a linear fit to the data are also presented.
Forward running k1S AP ML VT k1L AP ML VT Backward running k1S AP ML VT k1L AP ML VT
F
P-value
η2
Linear fit P-value
3.90 3.47 5.20 0.094 2.11 1.32
0.03 0.04 0.01 0.89 0.15 0.28
0.21 0.19 0.27 0.007 0.13 0.08
0.07 0.08 0.02 NA NA NA
6.07 4.15 5.12 6.08 0.30 0.63
0.01 0.03 0.03 0.008 0.59 0.48
0.30 0.22 0.26 0.30 0.02 0.04
0.03 0.03 0.03 0.50 NA NA
η2 = effect size (partial eta-squared). AP = anterior-posterior, ML = medial-lateral, VT = vertical. NA = not applicable.
Effect of speed on local dynamic stability of locomotion
It should, however, be pointed out that although the relationship between speed and k1S in AP and ML directions of forward running did not reach levels of statistical significance, they were interestingly
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straight line fit (P < 0.05), except in AP and ML directions during forward running. This was a clear indication that k1S increased linearly with increasing speed (Table I and Figure 2).
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Figure 2. Results of short- (k1S , left) and long-term (k1L , right) largest finite-time LyE in forward and backward running and in anterior– posterior (AP, top), medial–lateral (ML, middle) and vertical (VT, down) planes of motion. Speed is presented as the percent of preferred running speed (% PRS). The linear fit lines are also shown in the figure. FR = forward running, BR = backward running. Error bars are standard deviations of the mean.
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close to significance (P = 0.07 and 0.08 for AP and ML directions, respectively). Furthermore, a linear fit did not appear to be statistically significant for the relationship between speed and k1L in AP direction on backwards running (P = 0.50).
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Discussion Running in different directions is a vital component of performance during many sports and physical activities. Studying local dynamic stability of running patterns could play a key role in understanding stability in such human movements during performance. Running speed, on the other hand, seems to provide an overriding effect upon local dynamic stability in movement systems. Although many studies have investigated the effect of speed on local dynamic stability in walking (Bruijn et al., 2009; Dingwell et al., 2001; Dingwell & Marin, 2006; England & Granata, 2007; Stergiou et al., 2004), to the best of our knowledge, no study as yet has characterised the effect of speed on local dynamic stability of participants during running. This study investigated the effect of speed on local dynamic stability in both forward and backward running patterns. It was hypothesised that speed could affect local dynamic stability in both forward and backward running patterns. The short- ðk1S Þ and long-term ðk1L Þ finite-time LyEs were consequently adopted as the measures of local dynamic stability. Results showed that, while incrementing speed resulted in an increase in k1S (i.e. reducing local dynamic stability) in both forward and backward running, it did not lead to any significant changes in k1L . Effect of speed on local dynamic stability in running compared to walking Dingwell and Marin (2006) and England and Granata (2007) in separate studies showed that local dynamic stability decreased with increasing walking speed. The results presented in our study, supported these observations, indicating that k1S changed in direct proportionality with running speed (i.e. local dynamic stability reduced as speed increased). Several factors might contribute to the observed relationship between speed of running and changes in local dynamic stability. First, as in walking (England & Granata, 2007), fast running is associated with shorter stride times. The reduced duration of strides in running tends to limit the capacity of the neuromuscular control system to respond properly to kinematic perturbations. This in turn leads to greater divergence of trajectories in the statespace constructed from kinematic data and thus greater values of LyE. In addition, as speed
increases, the heel (toe in backward running) contacts the ground with greater force which might result in greater kinematic perturbations and consequently the emergence of a less-stable running pattern. In the study of Dingwell and Marin (2006), it was also shown that k1L increased with increments in walking speed. Our results, however, showed that running speed did not have an effect on k1L . This observation, in turn, indicates that running speed did not affect the divergence of state-space trajectories after the first stride. It should be noted that in studies where both k1S and k1L were measured, k1L exhibited much smaller changes compared to k1S (Bruijn et al., 2009; Dingwell & Marin, 2006). Similar results were also obtained in this paper.
Effect of speed on local dynamic stability in backward running From another point of view, the results of the present study demonstrated that effect of speed on local dynamic stability is identical in both forward and backward running. That is, increasing speed led to a reduction of local dynamic stability in both running patterns. This finding could be interpreted by observations from other studies suggesting that a single neuro-motor mechanism might be responsible for system regulation of locomotion in forward and backward directions (van Deursen et al.,1998).
Methodological issues in calculating LyE There are two important methodological issues which should be considered when calculating LyE, that is, the number of cycles being analysed and the average stride time (Bruijn et al., 2009). As discussed in Bruijn et al. (2009), a fixed number of cycles in addition to approximately fixed stride times should be used when calculating LyE for different speeds. In their study, Dingwell and Marin (2006) calculated the LyE values based on a different number of cycles. In addition, the LyE values calculated by England and Granata (2007) were dependent on the stride times. Therefore, it is possible that their results might have been influenced by these two factors (Bruijn et al., 2009). Considering these issues in their study, Bruijn et al. (2009) indicated that LyE did not necessarily increase with increasing speed and that they considered the relationship between walking speed and local dynamic stability might depend upon the plane of motion. Since our methodology in calculating LyE resembled that of Bruijn et al. (2009), our results could be reliably construed to suggest that fast running is less stable than slow running.
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Limitation of the study
References
There are some issues associated with methods adopted in the study described here. First, local dynamic stability was quantified under the task constraints of treadmill running to enhance experimental control of locomotion speed in participants. Research studies on stability of walking patterns have demonstrated that using a treadmill might affect levels of local dynamic system stability observed (Dingwell et al., 2001). It is important to note that no published results have so far been elicited in studies comparing local dynamic stability of running under the specific task constraints of both treadmill and overground running, and future research should take this distinction into consideration. However, it could be argued that the treadmill effect on local dynamic stability of running exists across all speeds. This implies that the trend of relationship between speed and local dynamic stability might not be influenced by treadmill. Therefore, although overground running might have different values of local dynamic stability, similar trend could be expected (i.e. decreasing local dynamic stability with increasing speed) for overground running as well. A second issue in our study is that the local dynamic stability of running patterns was investigated over three low-to-high speeds (i.e. 80%, 100% and 120% of PRS). Previous studies on walking have tended to investigate system stability tendencies over a range of five different speeds from low to high (60%–140% of preferred walking speed with 20% increments). In the current study, however, lowering the constrained running speed to below 80% PRS (e.g. 60% PRS) led to transitions from running to walking patterns in all participants. This was not desirable in our work, and it was not considered viable to observe performance under a broader range of speeds.
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Conclusions This study demonstrated that the effect of speed on local dynamic stability is identical in both forward and backward running patterns. That is, speed and local dynamic stability were inversely proportional in both running patterns. The results of this study are in accordance with those of previous studies, where it has been suggested that a single neuro-motor mechanism might be responsible for control of both forward and backward motions. Further implementation of this study could be manifested in design of rehabilitation or sport exercise programmes with the objective of manipulating task constraint to facilitate and enhance adaptations in stability in forward and backward running.
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Threlkeld, A. J., Horn, T. S.,Wojtowicz, G., Rooney, J. G., Shapiro, R. (1989). Kinematics, ground reaction force, and muscle balance produced by backward running. Journal of Orthopaedic & Sports Physical Therapy, 11, 56–63. doi:10.2519/ jospt.1989.11.2.56 van Deursen, R. W. M., Flynn, T. W., McCrory, J. L., & Morag, E. (1998). Does a single control mechanism exist for both forward and backward walking? Gait & Posture, 7, 214–224. doi:10.1016/S0966-6362(98)00007-1
Zampeli, F., Moraiti, C. O., Xergia, S., Tsiaras, V. A., Stergiou, N., Georgoulis, A. D. (2010). Stride-to-stride variability is altered during backward walking in anterior cruciate ligament deficient patients. Clinical Biomechanics, 25, 1037–1041. doi:10.1016/j. clinbiomech.2010.07.015
European Journal of Sport Science Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tejs20
Effect of speed on local dynamic stability of locomotion under different task constraints in running a
a
bc
Sina Mehdizadeh , Ahmed Reza Arshi & Keith Davids a
Biomechanics and Sports Engineering Group, Faculty of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran b
Centre for Sports Engineering Research, Sheffield Hallam University, Sheffield, UK
c
FiDiPro Programme, University of Jyväskylä, Jyväskylä, Finland Published online: 10 Apr 2014.
Click for updates To cite this article: Sina Mehdizadeh, Ahmed Reza Arshi & Keith Davids (2014) Effect of speed on local dynamic stability of locomotion under different task constraints in running, European Journal of Sport Science, 14:8, 791-798, DOI: 10.1080/17461391.2014.905986 To link to this article: http://dx.doi.org/10.1080/17461391.2014.905986
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European Journal of Sport Science, 2014 Vol. 14, No. 8, 791–798, http://dx.doi.org/10.1080/17461391.2014.905986
ORIGINAL ARTICLE
Effect of speed on local dynamic stability of locomotion under different task constraints in running
SINA MEHDIZADEH1, AHMED REZA ARSHI1, & KEITH DAVIDS2,3 1
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Biomechanics and Sports Engineering Group, Faculty of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran, 2Centre for Sports Engineering Research, Sheffield Hallam University, Sheffield, UK, 3FiDiPro Programme, University of Jyväskylä, Jyväskylä, Finland
Abstract A number of studies have investigated effects of speed on local dynamic stability of walking, although this relationship has been rarely investigated under changing task constraints, such as during forward and backward running. To rectify this gap in the literature, the aim of this study was to investigate the effect of running speed on local dynamic stability of forward and backward running on a treadmill. Fifteen healthy male participants took part in this study. Participants ran in forward and backward directions at speeds of 80%, 100% and 120% of their preferred running speed. The three-dimensional motion of a C7 marker was recorded using a motion capture system. Local dynamic stability of the marker was quantified using shortand long-term largest finite-time Lyapunov exponents (LyE). Results showed that short-term largest finite-time LyE values increased with participant speed meaning that local dynamic stability decreased with increasing speed. Long-term largest finite-time LyEs, however, remained unaffected as speed increased. Results of this study indicated that, as in walking, slow running is more stable than fast running. These findings improve understanding of how stability is regulated when constraints on the speed of movements is altered. Implications for the design of rehabilitation or sport practice programmes suggest how task constraints could be manipulated to facilitate adaptations in locomotion stability during athletic training. Keywords: Running, task constraint, local dynamic stability, treadmill
Introduction Stability is an important factor for functional human locomotion strategies (England & Granata, 2007). It has been defined as the capacity of a movement system to preserve its functional state of organisation in the presence of sudden kinematic perturbations due to changing task constraints (Davids, Button, & Bennett, 2008; England & Granata, 2007). One aspect of maintaining stability in human locomotion is the capacity to attenuate inherent stride-to-stride perturbations through, for example, visual control strategies (England & Granata, 2007; Scott, Li, & Davids, 1997). To quantify the ability to maintain stability, a number of studies have measured rate of divergence of trajectories in the state-space constructed from kinematic data, using largest finite-time Lyapunov exponents (LyE; Bruijn, van Dieen, Meijer, & Beek, 2009;
Dingwell, Cusumano, Cavanagh, & Sternad, 2001; Dingwell & Marin, 2006; England & Granata, 2007; Stergiou, Moraiti, Giakas, Ristanis, & Georgoulis, 2004). However, these efforts have tended to focus solely on examining walking behaviours, and system stability under different task constraints, such as during forward and backward running, has rarely been investigated. The present study, therefore, aimed at investigating the stability of forward and backward running in order to complement existing research. The LyE measures the exponential rate of divergence of neighbouring trajectories of the state-space constructed by kinematic data obtained from a movement system during performance (Dingwell & Marin, 2006; Rosenstein, Collins, & De Luca,
Correspondence: A. R. Arshi, Biomechanics and Sports Engineering Group, Faculty of Biomedical Engineering, Amirkabir University of Technology, Hafez Ave. Tehran, Iran. E-mail: [email protected] © 2014 European College of Sport Science
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1993). This type of system stability is considered as local dynamic stability since LyE quantifies the ability to respond to small local perturbations during movement performance (Dingwell & Marin, 2006). In addition, Since LyE measures the rate of divergence of the trajectories, a greater LyE value indicates a lower level of local dynamic stability in the system. Using the LyE method, previous research has indicated that stability of walking is speed-dependent (Bruijn et al., 2009; Dingwell et al., 2001; Dingwell & Marin, 2006; England & Granata, 2007; Stergiou et al., 2004). Dingwell and Marin (2006), and England and Granata (2007) in separate studies, for example, have demonstrated that the value of LyE increased (i.e. local dynamic stability decreased) with increments in walking speed. These observations suggest that slow walking is inherently more stable than fast walking. The results of a study by Bruijn et al. (2009), on the other hand, showed that slow walking was not necessarily more stable than fast walking. Their findings led Bruijn et al. (2009) to argue that the relationship between local dynamic stability and system speed depends on the plane of motion. Regardless, all these studies were conducted under the task constraints of walking, and characterisation of the effect of speed on local dynamic stability in running has rarely been approached. In one study, Jordan, Challis, Cusumano, and Newell (2009) investigated the local dynamic stability of running patterns compared to walking patterns. The range of speeds used in that study, however, was limited to speeds very close to walk–run transition speeds in participants and no relation between speed and stability in running was established. Furthermore, to the best of our knowledge, no study has investigated the effect of speed on local dynamic stability in backward running. Backward running is a common running pattern in many sports such as soccer, basketball and football (Zampeli et al., 2010). In addition, backward running has been considered as a common rehabilitation technique by sport rehabilitation specialists (Cipriani, Armstrong, & Gaul, 1995; Threlkeld, Horn, Wojtowicz, Rooney, & Shapiro, 1989). Research studies have suggested that a single neuro-motor mechanism might be responsible for movement system regulation in locomotion in forward and backward directions (van Deursen, Flynn, McCrory, & Morag, 1998). Therefore, it could be argued that the effect of speed on the ability to maintain stability is identical in both forward and backward running. The aim of this study was therefore to investigate the effect of running speed on local dynamic stability of forward and backward running on a treadmill. It is hypothesised that speed will affect the local dynamic
stability of participants under the task constraints of both forward and backward running.
Methods Participants Fifteen male participants took part in this study. They were 24.1, s = 1.0 years old with average mass and height of 68.8, s = 3.9 kg and 1.76, s = 0.04 m, respectively. Participants were asked to report any musculoskeletal injuries that prevent them from running in forward and backward directions. None of them reported any musculoskeletal injuries at the time of experiments. All participants provided written informed consent before participation in the study. The ethics committee of Amirkabir University of Technology approved the experimental procedure. Marker placement Since this study was a part of a broader study on the stability of the movement, 17 passive reflective markers (14 mm diameter) were attached to the skin of each participant at the right and left bony landmark on the second metatarsal head (toe), calcaneus (heel), lateral malleolus (ankle), mid-tibia, lateral epicondyle of knee (knee), mid-thigh, anterior superior iliac spine and also on the sacrum, midway between posterior superior iliac spines, 10th thoracic vertebrae (T10) and 7th cervical vertebrae (C7). For this particular study, right toe and C7 markers were used for further analyses. Task Before starting the experiments, participants had enough time to familiarise themselves with running forward and backward on the treadmill. During the actual tests, all participants ran in forward and backward directions on a motorised treadmill (Cosmed® T150, Rome, Italy) at 80%, 100% and 120% of their preferred forward and backward running speeds. Preferred running speed (PRS) was obtained following a top-down and bottom-up approach similar to protocols described in Dingwell and Marin (2006) and Jordan et al. (2009). Participants began by running on the treadmill at a slow speed (as defined by participant perceptions) followed by a gradual increase of 0.1 km/h increments until each participant declared that he was running at his PRS. This speed value was recorded, and increments were introduced until each participant declared that the speed was fast. From this point on, the speed value was gradually decreased until once again the participant reported that he was running at his PRS. This speed was also recorded. The average of these
Effect of speed on local dynamic stability of locomotion
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two recorded PRSs was consequently determined. This procedure was repeated three times, and the mean value of three average PRSs was considered as each participant’s actual PRS. This procedure was followed under the task constraints of both forward and backward running. During acquisition of the PRS data, participants were not allowed to view the speed at which they were running on the treadmill (Jordan, Challis, & Newell, 2007). Each participant was then asked to run for two minutes in every one of the six test trials. The trials were conducted at speeds of 80%, 100% and 120% of actual PRS values for each participant, under both forward and backward running task constraints in random order (Dingwell & Marin, 2006). Sufficient rest periods were allocated between the tests to allow participants to recover.
Data recording The three-dimensional coordinate data of the markers were recorded using five Vicon® VCAM motion capture calibrated cameras (Oxford Metrics, Oxford, UK) at the sampling frequency of 100 samples/ second. Reconstruction and labelling were performed using Vicon® Workstation software (Oxford Metrics, Oxford, UK).
Data analysis Data associated with x (Anterior–Posterior: AP), y (Medio–Lateral: ML) and z (Vertical: VT) components of the C7 marker motion were analysed. The necessity of a stationary characteristic for nonlinear analysis of the time series data resulted in the adoption of velocity profiles for further ana‐ lyses (Dingwell & Marin, 2006; Kantz & Schreiber, 2004). Although the common approach at this stage of analysis is to filter the signals obtained from movement system, possible loss of information at critical points prohibited implementation of any filtering algorithms (Kantz & Schreiber, 2004). For each trial, a total number of 100 consecutive strides were analysed. In addition, all time series were timenormalised to an equal length of 10,000 points that in turn signified that each individual stride comprised approximately 100 data points. To quantify local dynamic stability, first, an appropriate state space was reconstructed using Equation (1): X ðtÞ ¼ ½xðtÞ, xðt þ sÞ, xðt þ 2sÞ, . . . xðt þ ðdE 1ÞsÞT ð1Þ where X ðtÞ is the reconstructed state vector, xðtÞ is the original velocity time vector, s is the time delay and dE is the dimension of the state space, i.e. the
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embedding dimension (Kantz & Schreiber, 2004; Takens, 1981). Time delay s is determined as the first local minimum of Average Mutual information (AMI) function (Fraser, 1986). The AMI is a statistical measure from theory of information that shows how much information about a random variable can be obtained from the information of another random variable (Shelhamer, 2006). In this method, mutual information between a time series xðtÞ and its time shift xðt þ sÞ is calculated for different values of s until the mutual information is minimised (Shelhamer, 2006). A time delay of 10 samples was found to be appropriate for data associated with the three AP, ML and VT directions. This value of time delay has also been determined in other studies (Bruijn et al., 2009; England & Granata, 2007). In addition, a Global False Nearest Neighbours (GFNN) measure was used to determine embedding dimension dE (Kennel, Brown, & Abarbanel, 1992). False Nearest Neighbours are points which are close together in dimension dE but not in dimension dE þ 1 (Shelhamer, 2006). According to this method, the dimension is gradually increased until the number of false nearest neighbours is reduced to zero. For the purpose of this study, an embedding dimension of dE ¼ 5 was calculated for data associated with AP, ML and VT directions. Similar embedding dimension values have also been determined in previous studies (Dingwell & Marin, 2006; England & Granata, 2007; Jordan, et al., 2009). In order to quantify local dynamic stability, the largest finite-time LyE was determined from kinematic data (Figure 1). The LyE measures the exponential rate of divergence of neighbouring trajectories in the state space (Rosenstein et al., 1993). Since LyE measures the rate of divergence of the trajectories, a greater LyE value indicates lower levels of local dynamic stability of a system. The approach implemented in this study was introduced by Rosenstein et al. (1993), which is most suitable for a finite time series. Here, the largest finite-time LyE ðk1 Þ could be determined using Equation (2): dðtÞ ¼ C expðk1 tÞ
ð2Þ
where dðtÞ is the average distance between neighbouring points at time t, and the initial separation of the neighbouring points is represented by C. According to Expression (2), for the jth pairs of neighbouring points in state space, we have: dj ðiÞ Cj expðk1 ðiDtÞÞ
ð3Þ
Taking the logarithm from both sides of Equation (3) results in: ln½dj ðiÞ ln½Cj þ k1 ðiDtÞ
ð4Þ
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S. Mehdizadeh et al. slope of a linear fit to Equation (5) in the range ðiDtÞ of 0 to 0.5 stride (approximately 0–50 samples; Bruijn et al., 2009), and 4–10 strides (approximately 400–1000 samples), respectively. In the present study, all LyE values were presented as the rate of divergence/stride. Before analysing the experimental data, the LyE algorithms were validated by feeding a Lorenz system with typical inputs to the algorithms and comparing the outcomes with the published results (Gates & Dingwell, 2009; Graham, Sadler, & Stevenson, 2011; Rosenstein et al., 1993).
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Statistical analysis Separate one-way repeated measure analysis of variance (ANOVA) tests were applied to determine whether speed affected LyE values. Here, k1S and k1L in different planes of motion and in forward and backward running were considered as dependent variables. A linear fit was used to determine the relationship between speed and LyE where statistically significant effects of speed were found. Statistical significance levels were set at P < 0.05. Results As shown in Table I, speed had a significant effect (P < 0.05) on the value of k1S in all planes of both forward and backward running direction. k1L , on the other hand remained unaffected as speed increased in forward running. During backward running, however, speed did provide a significant effect on k1L in the AP direction only. In addition, trend analyses indicated that the relationship between speed and k1S was well described by a positive slope Figure 1. State-space reconstruction and local dynamic stability analysis of time series obtained from experiment. (A) Original velocity time series x(t), (B) 3D representation of state-space reconstructed from time series x(t) and its time copies xðt þ sÞ, and xðt þ 2sÞ. Note that the dimension of state space might be greater than three but cannot be visually observed. The expanded view is a region of state space in which it is schematically shown that initial separation of jth pairs of points, dj ð0Þ, diverge after i time steps shown by dj ðiÞ. (C) Average logarithmic divergence of all pairs of neighbouring points plotted over time (shown as stride number) to calculate k1S and k1L . See text for further information.
k1 is determined using a linear fit to the following curve:
yðiÞ/ð1/DtÞ dj ðiÞ
ð5Þ
where dj ðiÞ denotes the average over all pairs of j. For this study, values of a short-term (k1S ) and a long-term ðk1L Þ LyE were calculated (Dingwell & Marin, 2006). k1S and k1L were calculated as the
Table I. Results of ANOVA tests for short- ðk1S Þ and long-term ðk1L Þ largest finite-time LyEs in forward and backward running. The results of a linear fit to the data are also presented.
Forward running k1S AP ML VT k1L AP ML VT Backward running k1S AP ML VT k1L AP ML VT
F
P-value
η2
Linear fit P-value
3.90 3.47 5.20 0.094 2.11 1.32
0.03 0.04 0.01 0.89 0.15 0.28
0.21 0.19 0.27 0.007 0.13 0.08
0.07 0.08 0.02 NA NA NA
6.07 4.15 5.12 6.08 0.30 0.63
0.01 0.03 0.03 0.008 0.59 0.48
0.30 0.22 0.26 0.30 0.02 0.04
0.03 0.03 0.03 0.50 NA NA
η2 = effect size (partial eta-squared). AP = anterior-posterior, ML = medial-lateral, VT = vertical. NA = not applicable.
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It should, however, be pointed out that although the relationship between speed and k1S in AP and ML directions of forward running did not reach levels of statistical significance, they were interestingly
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straight line fit (P < 0.05), except in AP and ML directions during forward running. This was a clear indication that k1S increased linearly with increasing speed (Table I and Figure 2).
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Figure 2. Results of short- (k1S , left) and long-term (k1L , right) largest finite-time LyE in forward and backward running and in anterior– posterior (AP, top), medial–lateral (ML, middle) and vertical (VT, down) planes of motion. Speed is presented as the percent of preferred running speed (% PRS). The linear fit lines are also shown in the figure. FR = forward running, BR = backward running. Error bars are standard deviations of the mean.
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close to significance (P = 0.07 and 0.08 for AP and ML directions, respectively). Furthermore, a linear fit did not appear to be statistically significant for the relationship between speed and k1L in AP direction on backwards running (P = 0.50).
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Discussion Running in different directions is a vital component of performance during many sports and physical activities. Studying local dynamic stability of running patterns could play a key role in understanding stability in such human movements during performance. Running speed, on the other hand, seems to provide an overriding effect upon local dynamic stability in movement systems. Although many studies have investigated the effect of speed on local dynamic stability in walking (Bruijn et al., 2009; Dingwell et al., 2001; Dingwell & Marin, 2006; England & Granata, 2007; Stergiou et al., 2004), to the best of our knowledge, no study as yet has characterised the effect of speed on local dynamic stability of participants during running. This study investigated the effect of speed on local dynamic stability in both forward and backward running patterns. It was hypothesised that speed could affect local dynamic stability in both forward and backward running patterns. The short- ðk1S Þ and long-term ðk1L Þ finite-time LyEs were consequently adopted as the measures of local dynamic stability. Results showed that, while incrementing speed resulted in an increase in k1S (i.e. reducing local dynamic stability) in both forward and backward running, it did not lead to any significant changes in k1L . Effect of speed on local dynamic stability in running compared to walking Dingwell and Marin (2006) and England and Granata (2007) in separate studies showed that local dynamic stability decreased with increasing walking speed. The results presented in our study, supported these observations, indicating that k1S changed in direct proportionality with running speed (i.e. local dynamic stability reduced as speed increased). Several factors might contribute to the observed relationship between speed of running and changes in local dynamic stability. First, as in walking (England & Granata, 2007), fast running is associated with shorter stride times. The reduced duration of strides in running tends to limit the capacity of the neuromuscular control system to respond properly to kinematic perturbations. This in turn leads to greater divergence of trajectories in the statespace constructed from kinematic data and thus greater values of LyE. In addition, as speed
increases, the heel (toe in backward running) contacts the ground with greater force which might result in greater kinematic perturbations and consequently the emergence of a less-stable running pattern. In the study of Dingwell and Marin (2006), it was also shown that k1L increased with increments in walking speed. Our results, however, showed that running speed did not have an effect on k1L . This observation, in turn, indicates that running speed did not affect the divergence of state-space trajectories after the first stride. It should be noted that in studies where both k1S and k1L were measured, k1L exhibited much smaller changes compared to k1S (Bruijn et al., 2009; Dingwell & Marin, 2006). Similar results were also obtained in this paper.
Effect of speed on local dynamic stability in backward running From another point of view, the results of the present study demonstrated that effect of speed on local dynamic stability is identical in both forward and backward running. That is, increasing speed led to a reduction of local dynamic stability in both running patterns. This finding could be interpreted by observations from other studies suggesting that a single neuro-motor mechanism might be responsible for system regulation of locomotion in forward and backward directions (van Deursen et al.,1998).
Methodological issues in calculating LyE There are two important methodological issues which should be considered when calculating LyE, that is, the number of cycles being analysed and the average stride time (Bruijn et al., 2009). As discussed in Bruijn et al. (2009), a fixed number of cycles in addition to approximately fixed stride times should be used when calculating LyE for different speeds. In their study, Dingwell and Marin (2006) calculated the LyE values based on a different number of cycles. In addition, the LyE values calculated by England and Granata (2007) were dependent on the stride times. Therefore, it is possible that their results might have been influenced by these two factors (Bruijn et al., 2009). Considering these issues in their study, Bruijn et al. (2009) indicated that LyE did not necessarily increase with increasing speed and that they considered the relationship between walking speed and local dynamic stability might depend upon the plane of motion. Since our methodology in calculating LyE resembled that of Bruijn et al. (2009), our results could be reliably construed to suggest that fast running is less stable than slow running.
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Limitation of the study
References
There are some issues associated with methods adopted in the study described here. First, local dynamic stability was quantified under the task constraints of treadmill running to enhance experimental control of locomotion speed in participants. Research studies on stability of walking patterns have demonstrated that using a treadmill might affect levels of local dynamic system stability observed (Dingwell et al., 2001). It is important to note that no published results have so far been elicited in studies comparing local dynamic stability of running under the specific task constraints of both treadmill and overground running, and future research should take this distinction into consideration. However, it could be argued that the treadmill effect on local dynamic stability of running exists across all speeds. This implies that the trend of relationship between speed and local dynamic stability might not be influenced by treadmill. Therefore, although overground running might have different values of local dynamic stability, similar trend could be expected (i.e. decreasing local dynamic stability with increasing speed) for overground running as well. A second issue in our study is that the local dynamic stability of running patterns was investigated over three low-to-high speeds (i.e. 80%, 100% and 120% of PRS). Previous studies on walking have tended to investigate system stability tendencies over a range of five different speeds from low to high (60%–140% of preferred walking speed with 20% increments). In the current study, however, lowering the constrained running speed to below 80% PRS (e.g. 60% PRS) led to transitions from running to walking patterns in all participants. This was not desirable in our work, and it was not considered viable to observe performance under a broader range of speeds.
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Conclusions This study demonstrated that the effect of speed on local dynamic stability is identical in both forward and backward running patterns. That is, speed and local dynamic stability were inversely proportional in both running patterns. The results of this study are in accordance with those of previous studies, where it has been suggested that a single neuro-motor mechanism might be responsible for control of both forward and backward motions. Further implementation of this study could be manifested in design of rehabilitation or sport exercise programmes with the objective of manipulating task constraint to facilitate and enhance adaptations in stability in forward and backward running.
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Threlkeld, A. J., Horn, T. S.,Wojtowicz, G., Rooney, J. G., Shapiro, R. (1989). Kinematics, ground reaction force, and muscle balance produced by backward running. Journal of Orthopaedic & Sports Physical Therapy, 11, 56–63. doi:10.2519/ jospt.1989.11.2.56 van Deursen, R. W. M., Flynn, T. W., McCrory, J. L., & Morag, E. (1998). Does a single control mechanism exist for both forward and backward walking? Gait & Posture, 7, 214–224. doi:10.1016/S0966-6362(98)00007-1
Zampeli, F., Moraiti, C. O., Xergia, S., Tsiaras, V. A., Stergiou, N., Georgoulis, A. D. (2010). Stride-to-stride variability is altered during backward walking in anterior cruciate ligament deficient patients. Clinical Biomechanics, 25, 1037–1041. doi:10.1016/j. clinbiomech.2010.07.015
