Estimates of stability measures using data from nonaligned inertial sensors Sjoerd M. Bruijn1, Joost Haeck, Jaap H. van Dieen1, Warner R.Th. ten Kate2, Onno G. Meijer1 & Peter J. Beek1 Research Institute MOVE, Faculty of Human Movement Sciences, VU University Amsterdam, The Netherlands 2 Philips Research, Eindhoven, The Netherlands [email protected]

Abstract—We compared estimates of gait stability obtained using an optoelectronic measurement system with estimates obtained from non-aligned, co-moving inertial sensors. Estimates corresponded well for some, but not for all measures. Keywords: Gait stability, Lyapunov exponents, Floquet multipliers, inertial sensors

1. INTRODUCTION With their high incidence and associated costs, falls pose a large problem in modern society [1]. Consequently, there is a growing body of research focusing on (in) stability of posture and gait in both the elderly and various patient groups. Recently, two “dynamical systems” methods for quantifying stability have been applied to human locomotion; local dynamic stability [2-4], and orbital stability [5, 6]. Local dynamic stability is calculated by estimating the so called divergence coefficient, or maximal time-finite Lyapunov exponent, which quantifies how the system responds continuously to very small (i.e. “local”) perturbations [7] Orbital stability is calculated by estimating maximum Floquet multipliers, which quantify how the system responds to such perturbations discretely, from one cycle to the next [5]. In most literature, data from optoelectronic measurement systems are being used to calculate these measures. Using wireless inertial sensors may be more practical when using these measures in a clinical setting. Theoretically, using data from different sources should not be a problem when estimating these measures, as they rely on the global dynamics, rather than specifics. Thus, alignment of the sensor data to a global coordinate system seems unnecessary, making inertial sensors an even better candidate for use in the clinic. Although this sounds promising, a direct comparison between estimates of Lyapunov exponents and Floquet multipliers obtained from optoelectronic data and from non-aligned inertial sensor data has not been reported yet. 2. METHODS Nine healthy male volunteers participated in the study. Neoprene bands with clusters of 3 infrared LED’s were attached to the right calf, right femur, the pelvis and the thorax. Under these bands, wireless inertial sensor nodes [8] (which include 3D gyroscopes and accelerometers, see Fig. 1) were placed. The LED’s were used for movement registration with an optoelectronic system [Optotrak, NDI, Canada]. Subjects walked on a treadmill at 3 different velocities (2.0, 4.0 and 6.0 km/h) for a period of 5 minutes. Sample rate was 50 samples per second. For each sensor location (i.e. calf, thigh, pelvis and thorax) a 12 D state space was reconstructed from the data of

the inertial sensor and of the optoelectronically registered cluster marker. For the inertial sensor, state spaces consisted of the 3D angular velocities and their derivatives, as well as the 3D linear accelerations and their derivatives. For the cluster marker, state spaces consisted of 3D Figure 1: wireless rotations and rotational velocities, as inertial sensor node [8] well as 3D linear velocities and accelerations. From these state spaces, maximum Lyapunov exponents were estimated as the slope of the average logarithmic divergence of two initially nearest neighbors [1] (Fig. 2). This slope was estimated at 0-0.5 strides (λs) and at 410 strides (λL). Orbital stability was estimated using Floquet multipliers which were calculated as the largest (MaxFm) [5] and mean (MeanFm) [9] Eigenvalue(s) of the Jacobian mapping S*-Sk to S*-Sk+1 for each point in the stride cycle, where S* is the average state space at a certain percentage in the stride cycle, and Sk represents the values of the state space at stride k. (A) State space

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Figure 2: Calculation of maximum time finite Lyapunov exponents. (A) 3D state space (Note that calculations were carried out on a 12D state space, which cannot be visualized). (B): a section of A; for each point, the nearest neighbor was calculated, and divergence from this point was calculated as dj(i). (C): average logarithmic rate of divergence, from which, λs and λL, are calculated.

A 3 x 4 x 2 (velocity x sensor location x sensor type) repeated measures ANOVA was carried out for λs, λL, MaxFm and MeanFm. Since we were mainly interested in the question if

both sensor types would respond the same to a change in walking behaviour, we stratified to sensor placement if any velocity x placement interaction was found. P<0.05 was considered significant.

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Figure 3. Estimates of MeanFm. * is estimated using Optotrak and ∆ is estimated using the wireless inertial node sensors. Error bars represent standard errors.

3. RESULTS We found clear effects of speed, without interactions with sensor type or sensor placement for MeanFm (Fig. 3). For MaxFm, no velocity x sensor placement or velocity x sensor type interactions were found, but also no effect of velocity. When analysis of the other measures was stratified to sensor placement, λL showed a clear effect of velocity for all sensor placements, with velocity x sensor type interactions for the calf, thigh and pelvis, but not for the thorax, however visual inspection showed that these interactions did not lead to qualitatively different results (Fig. 4). For λs, interactions between sensor type and velocity were present for all sensor placements.

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Figure 4. Estimates of λL. * is estimated using Optotrak and ∆ is estimated using the wireless inertial node sensors. Error bars represent standard errors.

4. DISCUSSION The results of the current study show that non-aligned inertial sensors may give the same results for some, but not all,

stability measures. As all of the estimated measures rely on the global dynamics, the question arises as to why differences between different sensor types exist for some of them. A possible reason is the difference in the order of the derivatives used for the state space. For the optoelectronic data, we used a second order derivative, which may lead to some extra noise. This may especially lead to a problem for λs as there are indications that λs is correlated with variability [10]. The interactions between sensor type and velocity for λL may stem from this same problem, although the relation between λL and variability is much less clear. MaxFm did not show any velocity x sensor type interactions, but no effect of velocity either, which renders this measure less useful to pickup changes in walking behavior. It may however well be that it can detect more substantial changes in walking behavior as for instance caused by a pathology or ageing. Although the results of the current study show that some stability measures can be estimated using non-aligned inertial sensors, it should be kept in mind that the relationship between these measures and real-life perturbation resistance is yet to be established. Nonetheless, in a recent study [9] , it was shown that Floquet multipliers could adequately discriminate a group of elderly with a history of falling from a group of elderly without such a history. 5. 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: p. 848-863. [3] Dingwell, J.B., H. Gu Kang, and L.C. Marin, The effects of sensory loss and walking speed on the orbital dynamic stability of human walking. J Biomech, 2006. [4] Kang, H.G. and J.B. Dingwell, A direct comparison of local dynamic stability during unperturbed standing and walking. Exp Brain Res, 2006. 172: p. 35-48. [5] Hurmuzlu, Y. and C. Basdogan, On the measurement of dynamic stability of human locomotion. J Biomech Eng, 1994. 116: p. 30-6. [6] Hurmuzlu, Y., C. Basdogan, and D. Stoianovici, Kinematics and dynamic stability of the locomotion of post-polio patients. J Biomech Eng, 1996. 118: p. 405-11. [7] 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: p. 117-134. [8] van Acht, V., E. Bongers, N. Lambert and R. Verberne, Miniature Wireless Inertial Sensor for Measuring Human Motions. Proc. 29th I.C. IEEE EMBC, Lyon, France, 2007. [9] Granata, K.P. and T.E. Lockhart, Dynamic stability differences in fall-prone and healthy adults. J Electromyogr Kinesiol, 2007. [10]Su, J.L.S. and J.B. Dingwell, Dynamic stability of passive dynamic walking on an irregular surface. Journal of Biomechanical Engineering-Transactions of the Asme, 2007. 129: p. 802-810