Proceedings of USNCTAM2010 16th US National Congress of Theoretical and Applied Mechanics June 27 –July 2, 2010, University Park, Pennsylvania, USA
USNCTAM2010-828 THE VALIDITY OF STABILITY MEASURES: A MODELLING APPROACH Daan J. J. Bregman Research Institute MOVE VU University Amsterdam Amsterdam, The Netherlands
Sjoerd M. Bruijn Research Institute MOVE VU University Amsterdam Amsterdam, The Netherlands
[email protected]
Onno G. Meijer Research Institute MOVE VU University Amsterdam Amsterdam, The Netherlands
Peter J. Beek Research Institute MOVE VU University Amsterdam Amsterdam, The Netherlands
INTRODUCTION With their high incidence and associated costs, falls form a considerable problem in modern society [1]. Consequently, there is a rapidly growing body of research focusing on the (in)stability of posture and gait in the elderly and various patient groups. Recently, two methods to quantify stability have been borrowed from “dynamical systems theory” and applied in the study of human locomotion: local dynamic stability[2], and orbital stability[3]. Local dynamic stability is calculated by estimating the maximal finite time Lyapunov exponents (s and L), which quantify how a system responds continuously to very small (i.e. “local”) perturbations. Orbital stability assumes strict periodicity and is calculated by estimating maximum Floquet multipliers, which quantify how the system responds to very small perturbations in a discrete manner (i.e. from one cycle to the next). Although both measures have a sound mathematical foundation, it is still largely unknown how well they correlate with the real-life notion of stability—the probability of falling [2] . The relationship between stability measures and the probability of falling can be tested using (physical) models in which the probability of falling can be manipulated. Su and Dingwell[4] used the simplest walking model[5] to study the relationship between measures of dynamic gait stability and the probability of falling (as expressed by variability). They found that s correlated with the probability of falling, while L and MaxFm did not. In their study however, the probability of
Jaap H. van Dieën Research Institute MOVE VU University Amsterdam Amsterdam, The Netherlands
falling was only modified by having the model walk over a bumpier slope, which implied that the basin of attraction of the model remained the same and variability was a valid measure of stability. Thereby, the inherent stability (basin of attraction) of the model was not altered, and it remains unclear whether these measures still work if the basin of attraction is changed. Using an extended version of the simplest walking model, with arched feet, the inherent stability of the model can be altered via the foot radius[6]. However, changing the foot radius also results in changes in the basin of attraction, and thus variability cannot be used as an indicator of stability. The Gait Sensitivity Norm[6] (GSN) provides a valid alternative, as it correlates well with both the maximum perturbation a model can handle and the actual stability of the model, expressed by the bumpiness level for which the model can keep walking in 95% of 80-steps trials[6]. Using the Gait Sensitivity Norm as gold standard, we studied the relationship between maximum Lyapunov exponents and Floquet multipliers and actual stability in a passive dynamic walker when foot radius was changed. METHODS We performed simulations of a simple passive dynamic walker with arched feet, walking over a bumpy slope with different foot radii (ranging from 0 to 1). Step length was kept constant by adapting the slope at which the model walked downwards. To this aim, first, stable period-one solutions were found for each foot radius and slope combination. Then, for each foot radius and slope combination, 10 simulations of 100 strides were performed, with each simulation starting from the same stable solution. Eventually the gold standard of stability
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was calculated as the Gait Sensitivity Norm for each foot radius. The time series of the simulations were first time normalized, so that each time series consisted of 10,000 data points[7]. Then, 4D state spaces were constructed for the calculation of maximum Lyapunov exponents and Floquet multipliers, and these measures were calculated in accordance with the literature[2, 3, 8]. Note that we calculated s and L as the slopes of the log (divergence) curves for 0-0.5 and 4-10 strides, respectively. RESULTS Figure 1 shows the results of the simulations. Increasing foot radius while keeping step length constant resulted in an increase in the stability of the model, as indicated by increased values of 1/GSN. It was only 1/S that showed the same pattern as the Gait Sensitivity Norm, which suggests that L and MaxFm might not be related to actual stability.
REFERENCES 1. Rubenstein, L.Z., Falls in older people: epidemiology, risk factors and strategies for prevention. Age Ageing, 2006. 35 Suppl 2: p. ii37-ii41. 2. Dingwell, J.B. and J.P. Cusumano, Nonlinear time series analysis of normal and pathological human walking. Chaos, 2000. 10(4): p. 848-863. 3. Hurmuzlu, Y. and C. Basdogan, On the measurement of dynamic stability of human locomotion. J Biomech Eng, 1994. 116(1): p. 30-6. 4. Su, J.L.S. and J.B. Dingwell, Dynamic stability of passive dynamic walking on an irregular surface. J Biomech Eng, 2007. 129(6): p. 802-810. 5. Garcia, M., A. Chatterjee, A. Ruina, and M. Coleman, The simplest walking model: Stability, complexity, and scaling. Journal of Biomechanical EngineeringTransactions of the Asme, 1998. 120(2): p. 281-288. 6. Hobbelen, D.G.E. and M. Wisse, A disturbance rejection measure for limit cycle walkers: The Gait Sensitivity Norm. IEEE Transactions on Robotics, 2007. 23(6): p. 1213-1224. 7. Bruijn, S.M., J.H. van Dieën, O.G. Meijer, and P.J. Beek, Statistical precision and sensitivity of measures of dynamic gait stability. J Neurosci Methods, 2009. 178(2): p. 327-33. 8. Rosenstein, M.T., J.J. Collins, and C.J. Deluca, A practical method for calculating largest Lyapunov exponents from small data sets. Physica D, 1993. 65(1-2): p. 117-134.
Fig.1: Simulation results. For each foot radius the model was ran 10 times with a trial duration of 100 strides: results shown here represent the mean of those 10 trials. To facilitate the comparison with the results of Hobbelen and Wisse[6] we plotted 1/stability measure for all stability measures, and all variables were normalized.
DISCUSSION The results of the present study suggest that of all stability measures considered, only S relates to the probability of falling. Although L and MaxFm apparently do not reflect this real life notion of stability, they may capture other important aspects of movement control during walking. For now however, it seems safe to say that whatever those measures reflect, it is not the probability of falling. More research is needed to identify the physiological meaning of these stability measures.
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