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1 Elsevier Editorial System(tm) for Journal of Biomechanics Manuscript Draft Manuscript Number: BM-D Title: Floquet Multipliers Do Not Correlate with Gross Mechanical Stability in a Neuromusculoskeletal Model Article Type: Full Length Article (max 3000 words) Keywords: walking; stability; dynamic; simulation Corresponding Author: Dr. Mukul Talaty, Corresponding Author's Institution: First Author: Mukul C Talaty, Ph.D. Order of Authors: Mukul C Talaty, Ph.D.; Antonie J van den Bogert, Ph.D.

2 Title Page & Abstract Page 1 of 2 Floquet Multipliers Do Not Correlate with Gross Mechanical Stability in a Neuromusculoskeletal Model ORIGINAL ARTICLE Mukul Talaty 1 Antonie J. van den Bogert 2 1 Gait & Motion Analysis Laboratory; MossRehab; Elkins Park, PA 19027; Phone (215) , Fax (215) ; mctalaty@einstein.edu 2 Department of Biomedical Engineering; Lerner Research Institute; 9500 Euclid Avenue; Cleveland, Ohio Keywords : walking, stability, dynamic, simulation Word Count: 2855

3 Page 2 of 2 Abstract A practical method to quantify dynamic stability of a biped during walking can assist with clinical movement analysis and the evaluation of computational models designed to explore clinical scenarios. Floquet analysis of cyclic movement has been used towards the former goal, but its meaning with respect to risk of falling is unclear. We compared Floquet and empirical stability estimates for a neuromusculoskeletal model with human-like muscles and a neural controller. The model demonstrated cycle to cycle variability as humans do and did not obtain a true limit cycle, though it did get quite close. We used the cycle to cycle variability to estimate the Floquet multipliers of a family of three models that differed only in their trunk center of mass; Empirical stability was estimated by applying forces at varying directions, body segments, and instants of the gait cycle, until the model fell over. The maximum Floquet multiplier did not correlate with the empirical stability and thus falls risk in this moderately complex model. This is consistent with previous findings from simpler models, however seems contrary to human subject research using Floquet analyses. The dissociation between falls risk and maximum Floquet multiplier raises the question to what extent Floquet analysis is a useful tool in clinical movement analysis and musculoskeletal modeling.

4 *Manuscript Page 1 of 14 Introduction Bipedal walking in inherently unstable. Nevertheless, humans can walk without much apparent control effort. This is generally thought to be a consequence of sensory feedback and the organization of the nervous system. Neuromusculoskeletal models have been developed to study the basic mechanisms of locomotion and its stability (Hase, 1999, Hase and Yamazaki, 1998, Taga, 1994, Taga, 1995a, Taga, 1995b, Taga, et al., 1991). These and other models can capture many of the salient observable features of human gait dynamics. They show that stable locomotion can be achieved with specific neural control architectures. Those with explicit and physiologically inspired neural control schemes may be ideally useful for exploring clinical scenarios. Such models must, however, be able to respond in ways similar humans to realize this potential. Having commensurate stability to the real (i.e. human) system is an aspect of model validity that has not been well addressed. Reports of model stability and the contribution of model properties to overall stability were important steps in this area. Taga quantitatively evaluated stability by applying backward directed impulsive forces of varying magnitudes and at varying times of gait cycle, applying masses to various segments and changing the ground slope (Taga, 1995b). For the first approach, he reported maximum loads that could be tolerated while walking maintained. Hase evaluated the contribution of muscle properties, trunk degrees of freedom and time delays on stability in a neutrally controlled model using only tolerance to an external force as the stability measure (Hase, 2002). Gerritsen et al. demonstrated the contribution of muscle properties to stability by similarly applying loads until models fell over (Gerritsen, et al., 1998). They developed a family of models with varying inclusion of forcelength and force-velocity muscle properties. By and large, model stability has been demonstrated

5 Page 2 of 14 qualitatively and anecdotally. Ironically, formal estimation of stability on human data has developed and been reported more. Floquet analysis may be a step in the direction towards a concise conceptual framework and efficient means to apply it to evaluate stability. Hurmuzlu suggested the relevance of Floquet to gross stability by inference when reporting that Polio subjects had higher overall range of maximum Floquet multiplier ( max ) than healthy normal subjects(hurmuzlu, et al., 1996). Similarly Granata showed statistical differences in max for healthy and fall prone adults (Granata and Lockhart, 2008). Su et al. measured max on a passive dynamic walker over varying bumpiness slopes. However, they found, as they expected, that the max did not vary as the model itself (and thus its basin of attraction) was not changed across conditions (Su and Dingwell, 2007). Other work too has objectively quantified walking stability in humans and simple models (Dingwell and Kang, 2007, Garcia, et al., 1998, Hurmuzlu, et al., 1996, Poulakakis, et al., 2006, Su and Dingwell, 2007) but the relationship between the Floquet results and more functional measures of stability has not been well described. True stability limits are difficult if not impossible, not to mention likely unethical, to assess in human subjects, but models are ideal for this type of experimenting. The current work aims to clarify the interpretation of Floquet analysis for estimating stability in a neuromusculoskeletal model by comparing Floquet results to those from empirical stability testing. We quantify empirical stability by tolerance to applied external force before falling over. Despite the suggested relationship from human subject studies, it is unclear whether Floquet multipliers do relate to empirical measures of gross stability. Floquet analysis linearizes the responses of a system to small perturbations while gross stability necessitates relatively large forces to cause a fall. Considerable nonlinearities in musculoskeletal dynamics and neural

6 Page 3 of 14 control exist, which may undermine the relationship between linearized response and actual falls. The purpose of this paper is to quantify stability using Floquet analysis as well as large perturbations in the same model system. Clarifying the interpretation of Floquet multipliers will help to establish its role as part of an overall assessment of stability in humans as well as in neuromusculoskeletal models. Methods The Neuromusculoskeletal Model. The model contains 8 segments and is constrained to move in the sagittal plane under the control of a central pattern generator (CPG) coupled to 16 Hill-type muscle models with first order activation and fixed muscle moment arms. The muscle models have been used extensively in simulation studies for over a decade (Gerritsen, et al., 1998). The neural circuits originated from those described by Taga (Taga, 1995a). Seven neural oscillators generate control signals to the muscles. Each oscillator consists of a coupled pair of flexor and extensor neurons. Each neuron is self-inhibiting as well as reciprocally inhibiting to its antagonist neuron and are known to have stable limit cycles (Matsuoka, 1985). The joint between the pelvis and trunk was heavily damped resulting in only approximately 2.2 degrees of motion over each gait cycle. The ground contact model consists of 10 visco-elastic elements per foot and was of the form described previously (Gerritsen, et al., 1995). A family of 3 models was created by shifting the center of mass of the trunk up and down 75mm from the base model. The models were simulated using Matlab (The Mathworks, Natick MA) ode23 variable step integrator using 1e-6 absolute and 1e- 3 relative error tolerances.

7 Page 4 of 14 The model performance captures many of the salient features of walking (Figure 1) and can walk indefinitely. It does not appear to have a fixed point limit cycle based on observation of a large number of cycles (900) even though the gross performance appears to have settled into a steady state pattern. This cycle to cycle variability was the basis for using Floquet techniques to estimate orbital stability. [Figure 1] Floquet Theory. Well established methods for estimating Floquet multipliers were used (Dingwell and Kang, 2007, Granata and Lockhart, 2008, Hurmuzlu, et al., 1996). In short, multiple cycles of walking kinematics are used to estimate the relationship between one stride and the next for each state. Each state is referenced to a steady state value which is estimated as the average of all steady state cycles. Because the neuromusculoskeletal model demonstrates step to step variability similar to that of a human, we are able to apply the methods similarly to how they have been applied to date. Figures 2 and 3 below illustrate the variability present in the model performance. Shown are phase plots of one of the more variable and one of the less variable states. The blue traces show the transient variability in the first approximately 5 cycles (first 15 are shown). The red traces demonstrate the model nearly, but not quite, achieving a fixed point limit cycle during the next 695 cycles. In compiling state data from the point of maximum knee flexion during each cycle, we have estimated an equivalent system to the one we have actually simulated; stability assessments were performed on this equivalent system.

8 Page 5 of 14 [Figures 2,3] Floquet Calculations. Each model was simulated continuously for 900 seconds of continuous walking or about 700 gait cycles. For stability analysis, a subset of sixteen state variables was used, consisting of angular orientations and angular velocities of pelvis, trunk, and bilateral hip, knee and ankle. Poincare sections were produced by sampling these variables at the time of maximum knee flexion in each gait cycle. The estimation of Floquet multipliers was performed as described previously (Hurmuzlu, et al., 1996), except that all data were obtained from a single long simulation. In brief, cycle to cycle change in state matrices, da, db, were compiled : da A( k) X ss [1] db A( k 1) X ss [2] where A was an mx16 matrix of state values at the selected Poincare section (m=number of cycles evaluated), k was cycle number and X ss was the steady state value : X ss n 1 X [3] n k 1 k The Jacobian J was defined as the linear relationship between step k and the next step k+1: db = J da [4] and estimated from da and db using linear least squares (Matlab, The Mathworks Inc.).

9 Page 6 of 14 The eigenvalue (of this Jacobian) having the largest magnitude was the maximum Floquet multiplier, max. Floquet analysis was performed on subsets of 150 gait cycles (Bruijn, et al., 2009) allowing max estimates to be obtained from four non-overlapping data sets for each model. Empirical Stability Testing. 100ms horizontal or vertical impulses were applied to the pelvis or foot of the models during midswing and during double support of steady state walking. The exact time of the gait cycle at which the perturbation was applied was selected by the similarity of the joint angles and ground reaction forces as shown below in Figure 4a,b. [Figure 4a,b] Statistical Analysis of Stability Measures. The mean and standard deviations of max were calculated for each model and max were compared to the baseline model values using simple t-test (Microsoft Excel, assuming two tailed distribution and homoscedasticity). The Wilcoxon Signed Rank Test (SPSSv10.0.5, SPSS Inc.) was used to compare the empirical results. The outcomes of Walk and Fall were first transformed into numerical values of 1 and 0, respectively. Two tailed asymptotic significance is reported. Results

10 Page 7 of 14 Floquet results for the three models are reported in Table 1. [Table 1] The horizontal and vertical perturbation forces and result (fall or walk) are presented in Table 2 as the empirical measure of stability. [Table 2] Discussion Maximum Floquet multipliers ( max ) and empirical testing results were not consistent with each other suggesting orbital stability and falls risk are not related for this model. For the purposes of this study, we assumed that being able to tolerate greater forces without falling over was equivalent to having a lower falls risk. Statistical analysis of max indicated that all three models were equally stable, though the high CM model trended towards being less stable (p=0.10). From empirical testing, the base and the high models were similarly stable (p=0.414) and the low model was less stable (p=0.008). The latter result was surprising, because low center of mass is generally thought to lead to increased stability. However, the neural control system was designed for a model with normal mass distribution, and might cause counterintuitive responses to large perturbations in the altered model.. This is the first such assessment using Floquet techniques on data from a complex model with human-like muscle properties under closed loop neural control. These results are consistent with two other recent findings that suggest orbital stability does not correlate with falls risk (Su and Dingwell, 2007, Verdaasdonk, et al., 2007). Both of these studies were also on simulated data

11 Page 8 of 14 from a model. In the Su work, a passive dynamic walker model was perturbed at varying levels and the max calculated. max did not vary with perturbation levels despite the model performing closer to the edges of the basin of attraction with larger perturbations. Verdaasdonk et al. used a very simple (2 rigid legs) but actively controlled model and applied varying levels of a moment to block swing leg progression and reported that max did not correlate with the varying levels of risk. Floquet estimation of orbital stability in the conditions tested may not correlate to empirical (gross) stability because Floquet analysis is based on measuring the system response to small perturbations whereas the empirical stability involved large perturbations. The perturbations used to calculate max were the inherently small cycle to cycle variations during steady state (Figure2,3) However the model responses to perturbations near the levels that cause it to fall are considerably greater (c.f. perturbation magnitudes in Table 2 and Figure 5 below) than the cycle to cycle variability exhibited by the model when unperturbed. [Figure 5] Because the perturbations are large, it is possible that the model response is outside of the linear response region estimated by the least-squares best-fit Jacobian. It has been suggested but not rigorously demonstrated that the asymptotic stability of a nonlinear system can be estimated from the linearized map (Granata and Lockhart, 2008, Mombaur, et al., 2001). It is certainly conceivable that there is a neurophysiological or systems level relationship between the small and large perturbation response, but that relationship if it exists is not known to the authors at this time.

12 Page 9 of 14 The current study results and the two other model-based ones cited have a more direct quantification of falls risk than recent human studies which suggested that max may be related to gross stability. Hurmuzlu (Hurmuzlu, 1994, Hurmuzlu, et al., 1996) tacitly linked Floquet to gross stability by comparison of healthy normal Floquet values to those of Polio survivors who generally have gait instabilities that predispose them to falling. Kang et al. (Kang and Dingwell, 2008) more directly related the measures stating that Floquet results might be more sensitive indicators of locomotor impairment and potential future risk of falls. Granata et al. (Granata and Lockhart, 2008) used self-reported history of falls during overground community walking, and found that this was correlated with Floquet estimates from controlled treadmill walking. They provided a strong statistical basis for concluding max could identify fall prone individuals. However, the self-report falls risk and discrepancy between venue of reported falls and laboratory testing are sources of potential error. The accuracy of self-reported falls and falls risk is not without debate (Hauer, et al., 2009, Kanten, et al., 1993, Srygley, et al., 2009). Differences along many dimensions between treadmill and overground walking have been noted to exist for several decades and continue to be reported (Brouwer, et al., 2009, Puh and Baer, 2009, van Ingen Schenau, 1980, Vogt, et al., 2002). How well the empirical testing adequately sampled all the relevant aspects of the model s stability responses is not known. It is always a question with empirical testing whether all modes by which the model may fall and thus all aspects of the models relevant performance space have been evaluated. This is central to the motivation to discover a formal stability analysis method, such as Floquet, that can be practically implemented. It is possible, but hopefully unlikely, that applying other load configurations (timing, location, magnitude, etc.) would produce empirical results more consistent with the Floquet analysis. We applied perturbation forces during both

13 Page 10 of 14 single and double support, to proximal as well as distal segments, and in vertical as well as horizontal directions to sample many aspects of the performance space and minimize this possibility. However, we admit our basis for selecting these variations were not guided by a theoretical knowledge of the overall performance space. A similar limitation may affect the Floquet analysis. Bruijn et al. (Bruijn, et al., 2009) used a bootstrapping technique to demonstrate that approximately 150 cycles was sufficient to produce statistically precise estimates of max. Our informal sensitivity analysis revealed that max did not appear to change appreciably with increasing cycles beyond about 150 but varied considerably when fewer than about 130 cycles were used to estimate max. We have observed varying amounts of variability (precision) in max estimates (see standard deviations in Table 1) that raise the question of whether 150 or even 600 cycles is enough to adequately sample a high dimensional space or the lower dimensional dynamics that space represents. Variability in this and other aspects of Floquet methodology suggests its use is not yet fully mature. Granata et al. used only ~35 cycles in estimating max, while our own work (unpublished) as well as that of Bruijn (Bruijn, et al., 2009) suggests at least 150 cycles are needed for a reliable estimate of max. Still others have suggested 85 cycles may be enough (Scheiner, et al., 1995). One other possible contributor to the discrepancy between model and human results may stem from fundamental differences in the source of cycle to cycle variability in these two systems. It has been argued that Natural variance is attributed to mechanical disturbances or neuromotor control errors. (Granata and Lockhart, 2008) And yet it is the neural controllers and the musculoskeletal system that attenuate these disturbances. So there should be a relationship between this variability and the system s ability to attenuate disturbances. The cause of the variability in our neuromusculoskeletal model is not clear. That a deterministic model shows

14 Page 11 of 14 human-like variability in steady state is in and of itself interesting. We have ruled out (results not reported) that technical issues such as appropriateness of computation time step, event selection, and concluded that our system has slightly chaotic behavior. The similarity in magnitude of our Floquet results to those from human studies perhaps suggests that some fundamental characteristics of the model that contribute to this variability are in accord with actual human characteristics. This unexpected finding could be a useful and potentially novel adjunct in establishing model validity. Conclusions We present a case of bipedal, powered, human-like gait in which the maximum Floquet multiplier does not correlate with falls risk. This is similar to what a few other recent studies using simpler models have suggested, but contrary to what others have reported using the same methods applied to human walking. To what practical aspect of stability does max relate is still unclear. Further clarification is needed for both this particular neuromusculoskeletal model as well as for human walking. Floquet multipliers do seem to have a role in understanding walking stability but are not the holy grail of stability and need to be interpreted with caution and in the context of not yet well-understood limitations. Acknowledgements Professor John Guckenheimer offered many helpful insights and stimulating discussions on Floquet theory and its application to simulated data. Thomas Coulter was instrumental in coding early versions of the analyses routines.

15 Page 12 of 14 References Brouwer, B., Parvataneni, K. and Olney, S., A comparison of gait biomechanics and metabolic requirements of overground and treadmill walking in people with stroke. Clin Biomech (Bristol, Avon) 24, Bruijn, S. M., Dieën, J. H. v., Meijer, O. G. and Beek, P. J., Statistical precision and sensitivity of measures of dynamic gait stability. Journal of Neuroscience Methods 22, Dingwell, J. B. and Kang, H. G., Differences Between Local and Orbital Dynamic Stability During Human Walking. Journal of Biomechanical Engineering 129, Garcia, M., Chatterjee, A., Ruina, A. and Coleman, M., The Simplest Walking Model : Stability, Complexity, and Scaling. Journal of Biomechanical Engineering Gerritsen, K. G. M., van den Bogert, A. J., Hulliger, M. and Zernicke, D. F., Intrinsic muscle properties facilitate locomotor control -- a computer simulation study. Motor Control Gerritsen, K. G. M., van den Bogert, A. J. and Nigg, B., Direct dynamics simulation of the impact phase in heel-toe running. Journal of Biomechanics 28, Granata, K. and Lockhart, T. E., Dynamic stability differences in fall-prone and healthy adults. Journal of Electromyography and Kinesiology 18, Hase, K., A Model of Human Walking with a Three-Dimensional Musculoskeletal System and a Hierarchical Neuronal System. In VIIth International Symposium On Computer Simulation in Biomechanics. University of Calgary, Calgary, Canada. Hase, K., Computer Simulation Study of Human Locomotion with a Three-Dimensional Entire-Body Neuro-Musculo-Skeletal Model (II. Biomechanical Relationship between Walking Stability and Neuro-Musculo-Skeletal System). JSME International Journal Series C (Special Issue on Bioengineering) 45, Hase, K. and Yamazaki, N., Computer simulation of the ontogeny of bipedal walking. Anthropological Science 106, Hauer, K., Yardley, L., Beyer, N., Kempen, G., Dias, N., Campbell, M., Becker, C. and Todd, C., Validation of the Falls Efficacy Scale and Falls Efficacy Scale International in Geriatric Patients with and without Cognitive Impairment: Results of Self-Report and Interview- Based Questionnaires. Gerontology [epub ahead of print]. Hurmuzlu, Y., and Basdogan, C., On the Measurement of Dynamic Stability of Human Locomotion. ASME J. Biomech. Eng. 116,

16 Page 13 of 14 Hurmuzlu, Y., Basdogan, C. and Stoianovici, D., Kinematics and Dynamic Stability of the Locomotion of Polio Patients. ASME Journal of Biomechanical Engineering 118, Kang, H. G. and Dingwell, J. B., Effects of walking speed, strength and range of motion on gait stability in healthy older adults. J Biomech 41, Kanten, D., CD., M., Gerety, M., Lichtenstein, M., Aguilar, C. and Cornell, J., Falls: an examination of three reporting methods in nursing homes. J Am Geriatr Soc. 41, Matsuoka, K., Sustained oscillations generated by mutually inhibiting neurons with adaptation. Biol Cybern 52, Mombaur, K. D., Hans Georg Bock, Schloder, J. P. and Longman, R. W., Human-Like Actuated Walking that is Asymptotically Stable Without Feedback. In International Conference on Robotics & Automation. Seoul, Korea. Poulakakis, I., Papadopoulos, E. and Buehler, M., On the Stability of the Passive Dynamics of Quadrupedal Running with a Bounding Gait. IJRR 25, Puh, U. and Baer, G., A comparison of treadmill walking and overground walking in independently ambulant stroke patients: a pilot study. Disabil Rehabil. 31, Scheiner, A., Ferencz, D. and Chizeck, H., Quantitative Measurement of Stability in Human Gait through Computer Simulation and Floquet Analysis. In IEEE-EMBC and CMBEC. Srygley, J., Herman, T., Giladi, N. and Hausdorff, J., Self-report of missteps in older adults: a valid proxy of fall risk? Arch Phys Med Rehabil.): 90, Su, J. L. and Dingwell, J. B., Dynamic stability of passive dynamic walking on an irregular surface. J Biomech Eng 129, Taga, G., Emergence of bipedal locomotion through entrainment among the neuromusculo-skeletal system and the environment. Physica D 70, Taga, G., 1995a. A Model of the Neuro-Musculo-Skeletal System for Human Locomotion I. Emergence of basic gait. Biological Cybernetics 73, Taga, G., 1995b. A model of the neuro-musculo-skeletal system for human locomotion II. Realtime adaptability under various constraints. Biol. Cyber. 73, Taga, G., Yamaguchi, Y. and Shimizu, H., Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biological Cybernetics 651, van Ingen Schenau, G., Some fundamental aspects of the biomechanics of overground versus treadmill locomotion. Med Sci Sports Exerc. 12,

17 Page 14 of 14 Verdaasdonk, B., van den Broeke, A., Koopman, H. and van der Helm, F., Achieving Energy Efficient and Robust Bipedal Gait with a CPG-Controlled Bipedal Walker: Tuning the Neural Coupling Gains. In T. O. Williams (Eds), Biological Cybernetics Research Trends. Nova Science Publishers, Inc., New York, Vogt, L., Pfeifer, K. and Banzer, W., Comparison of angular lumbar spine and pelvis kinematics during treadmill and overground locomotion. Clin Biomech (Bristol, Avon) 17,

18 Figure Legends Figure Captions Figure 1. The basic walking pattern for the base model in early steady state for approximately one cycle. Figure 2. Phase plot of one of the more variable system states (Pelvis). Figure 3. Phase plot of one of the less variable system states (Knee). Figure 4. Illustration of the times during the gait cycle at which perturbations were applied. (a) MidSwing (b) Double Support Figure 5. The phase plot of a perturbed walking trial superimposed on steady state cycles for the base model gives a reference for the magnitude of the perturbation relative to the cycle to cycle variability. A 50N downward perturbation was applied to the foot during midswing phase.

19 Figures Figure 1.

20 Figure 2

21 Figure 3

22 Figure 4 (a) (b)

23 Figure 5.

24 Tables Table 1. Maximum Floquet Multipliers ( max ) for the Models. Model Base High Low Mean Standard deviation p value (relative to Base model)

25 Table 2. Empirical Testing Results Magni- Model Segment Direction Time tude (N) Base High Low MS 5250 Walk Walk Fall Vert Fall Fall Fall DS 700 Walk Walk Fall Pelvis 800 Fall Fall Fall MS 150 Walk Walk Fall Horiz. 250 Walk Fall Fall DS 150 Walk Walk Fall 250 Walk Fall Fall MS 50 Walk Walk Walk Foot Vert. 75 Walk Fall Walk DS 900 Walk Walk Walk 1500 Walk Walk Walk MS 350 Fall Fall Fall Horiz. 150 Fall Fall Fall DS 150 Walk Walk Fall 250 Fall Fall Fall MS 900 Fall Fall Fall Thigh Vert Fall Fall Fall DS 1200 Fall Fall Fall 900 Fall Walk Fall MS 150 Fall Walk Fall Horiz. 250 Fall Fall Fall DS 250 Walk Walk Walk 350 Walk Fall Walk Total # of Falls Wilcoxon Rank Sum Test significance Notes. MS=MidSwing; DS=Double Support; Bold indicate differences relative to the base model; Italic indicate changes contrary to the Floquet results. Data were transformed to numerical values (0=Fall,1=Walk) and the Wilcoxon Signed Rank Test performed. Two tailed asymptotic significance is reported.

26 *Conflict of Interest Statement Conflict of Interest Statement Neither author has any personal or financial relationships that would constitute a real or apparent conflict of interest.

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