Journal of Biomechanics

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1 Journal of Biomechanics 44 (2011) Contents lists available at ScienceDirect Journal of Biomechanics journal homepage: Multivariable static ankle mechanical impedance with relaxed muscles Hyunglae Lee a,n, Patrick Ho a, Mohammad A. Rastgaar a, Hermano I. Krebs a,b, Neville Hogan a,c a Department of Mechanical Engineering, Massachusetts Institute of Technology, Room 3-147, 77 Massachusetts Avenue, Cambridge, MA 02139, USA b Department of Neurology, University of Maryland School of Medicine, Baltimore, MD 21201, USA c Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA article info Article history: Accepted 16 April 2011 Keywords: Ankle joint Ankle mechanical impedance Ankle joint stiffness Human ankle Multivariable stiffness abstract Quantitative characterization of ankle mechanical impedance is important to understand how the ankle supports lower-extremity functions during interaction with the environment. This paper reports a novel procedure to characterize static multivariable ankle mechanical impedance. An experimental protocol using a wearable therapeutic robot, Anklebot, enabled reliable measurement of torque and angle data in multiple degrees of freedom simultaneously, a combination of inversion eversion and dorsiflexion plantarflexion. The measured multivariable torque angle relation was represented as a vector field, and approximated using a method based on thin-plate spline smoothing with generalized cross validation. The vector field enabled assessment of several important characteristics of static ankle mechanical impedance, which are not available from prior single degree of freedom studies: the directional variation of ankle mechanical impedance, the extent to which the ankle behaves as a spring, and evidence of uniquely neural contributions. The method was validated by testing a simple physical mock-up consisting of passive elements. Experiments with young unimpaired subjects quantified the behavior of the maximally relaxed human ankle, showing that ankle mechanical impedance is springlike but strongly direction-dependent, being weakest in inversion. Remarkably, the analysis was sufficiently sensitive to detect a subtle but statistically significant deviation from spring-like behavior if subjects were not fully relaxed. This method may provide new insight about the function of the ankle, both unimpaired and after biomechanical or neurological injury. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction While all joints in the kinematic chain between foot and trunk (ankle, knee, and hip) participate in lower-extremity functions, the contribution of the ankle is significant (DeVita et al., 2007). Ankle mechanical impedance plays an important role in natural interaction of the lower extremity with the environment, including postural stabilization during standing as well as propulsion, energy-absorption, and lower limb joint coordination during locomotion. Sinkjaer et al. (1998) measured intrinsic and reflex contributions to the stiffness of the ankle dorsiflexors. Lamontagne et al. (1997) studied the viscoelastic behavior of the plantarflexors at rest. Other research measured the relaxed stiffness of the ankle plantarflexors and/or dorsiflexors to compare unimpaired and neurologically impaired subjects (Singer et al., 2002; Harlaar et al., 2000; Rydahl and Brouwer, 2004; Chung et al., 2004; Kobayashi et al., 2010). Stochastic system identification methods were used by Hunter and Kearney (1982), Kearney and Hunter (1982), and Weiss et al. (1986a,b). Other work measured ankle stiffness during quiet standing (Winter et al., 2001; Morasso and n Corresponding author. Tel.: þ address: hyunglae@mit.edu (H. Lee). Sanguineti, 2002; Loram and Lakie 2002a,b) and quasi-stiffness during locomotion (Davis and DeLuca 1996; Palmer, 2002; Hansen et al., 2004). Mizrahi et al. (1990) studied the passive dynamics of the ankle in sudden inversion. Saripalli and Wilson (2005) examined the variation of frontal plane ankle stiffness under different weight bearing conditions. Zinder et al. (2007) measured inversion/eversion ankle stiffness using a medial/lateral swaying cradle. All of the above studies considered only a single degree of freedom (DOF) either in the sagittal plane (dorsiflexion plantarflexion (DP)) or the frontal plane (inversion eversion (IE)). Recently, Roy et al. (2009) measured static ankle impedance for IE and DP directions separately. However, we found no prior work characterizing ankle mechanical impedance in coupled multiple DOFs. Unlike the ankle, multivariable upper-extremity mechanical impedance has been studied extensively. Multivariable analysis allows a distinction between the contributions of neural feedback and intrinsic muscle properties to static mechanical impedance (Hogan, 1985; Mussa-Ivaldi et al., 1985; Lacquaniti et al., 1993). As the ankle also has multiple DOFs and single DOF movements are rare under natural conditions, characterization of multivariable ankle mechanical impedance in multiple DOFs promises deeper understanding of its roles in lower-extremity function. In electrical engineering usage impedance maps current onto voltage but the analogy between electrical and mechanical /$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi: /j.jbiomech

2 1902 H. Lee et al. / Journal of Biomechanics 44 (2011) systems has been debated for decades (Hogan and Breedveld, 2008). Mechanical impedance could be defined to map force onto velocity or vice-versa but neither definition would reflect the profound role of geometry and kinematics in all aspects of mechanical system dynamics. We define the term mechanical impedance to be a dynamic operator that maps a time-history of displacement onto the corresponding force time-history, Z : xðtþ-f ðtþ, where the force and displacement vectors are energetically conjugate such that they may define mechanical work, dw ¼ f t dx. In this study we identified the static component of multivariable ankle mechanical impedance. We report an experimental protocol and analysis methods using a wearable robot and a vector field approximation method. The methods were validated on a simple physical mock-up consisting of passive elements. Experiments with human subjects provided a precise and sensitive characterization of the multivariable torque angle relation at the ankle in IE DP space as a vector field. This vector field enabled assessment of several important ankle characteristics: the spatial structure of ankle impedance, the extent to which the ankle behaves as a spring, and evidence of uniquely neural contributions. This information may be valuable in orthopedic and neurological applications, providing a detailed quantitative characterization of ankle mechanics in health for comparison with biomechanical or neurological disorders. 2. Methods 2.1. Subjects Eight unimpaired young human subjects with no reported history of biomechanical or neuromuscular disorders (4 males, 4 females; age 22 31; height m; weight kg) were recruited for this study. Participants gave written informed consent to participate as approved by MIT s Committee on the Use of Humans as Experimental Subjects Experimental setup The static torque angle relation was measured for IE and DP simultaneously using a wearable robot, Anklebot (Interactive Motion Technologies, Inc.) It is highly back-drivable and provides actively controllable torques up to 15 Nm and 23 Nm in IE and DP, respectively, (Roy et al., 2009) and allows the maximum range of motion required for the typical gait of healthy or pathological subjects (Perry, 1992). A PD controller for joint angle with proportional gain 100 Nm/rad and derivative gain 2 Nm-s/rad guaranteed safe, stable, and highly compliant operations in human robot interactions. Subjects wore a modified shoe and a knee brace, to which the robot was connected. Subjects were seated and the knee brace was securely fastened to the chair to support the weight of the robot and the leg and to ensure that measurements were made in a repeatable posture (Fig. 1). To monitor muscle activation levels electromyographic (EMG) signals were recorded using surface electrodes with a bandpass between 20 and 450 Hz (Delsys Inc.) placed on the bellies of major muscles related to ankle movement: tibialis anterior (TA), soleus (SOL), gastrocnemius (GAS), and peroneus longus (PL). Because aliasing due to under-sampling is expected to affect the shape of the sampled EMG spectrum but not its total variance (Sadhukhan et al., 1994), and to ensure synchronization with other measurements, EMG was sampled at 200 Hz and its magnitude was estimated using a root-mean-square filter with a sliding window of 0.2 s (Clancy and Hogan, 1997) Experimental protocol Subjects were seated with their ankle held by the robot clear of the ground in a neutral position with the sole at a right angle to the tibia. Subjects were instructed to relax while Anklebot applied terminated ramp perturbations to the ankle with a slow velocity (51/s), selected to avoid evoking spindle-mediated stretch reflexes and maintain quasi-static conditions. Anklebot moved the ankle along a commanded trajectory, and held the foot for 0.1 s at the starting and ending positions. The complete protocol consisted of 48 movements total along 24 equally spaced directions in IE DP space, once outbound and once inbound per direction, with a nominal displacement amplitude of 201 in each direction. Perturbations began with pure eversion (01), rotated by 151 on each subsequent perturbation (451 corresponded to equal perturbations in eversion and dorsiflexion) and ended at Applied torque and actual angular displacement in both DOF as well as EMG data were sampled at 200 Hz. The complete set of measurements took a little over 3 min Analysis methods Fig. 1. Subject wearing Anklebot in a seated position. Any torque components required to overcome the inertia and friction of the actuators were measured by 10 repetitions of the experimental protocol explained above but with no human subject. The average of these measurements was expressed as a multivariable relation between torque and angle and subtracted from the recorded torque. A general angular mechanical impedance is a time-varying, multidimensional, nonlinear, dynamic operator that maps an angular motion time-history onto a corresponding torque time-history. The time-invariant, static components of ankle impedance we measured may be represented as a vector field (Eq. (1)) ðt IE,t DP Þ¼Vðy IE,y DP Þ ð1þ where y IE and y IE are angular displacements in the IE and DP directions, respectively, and y IE and y IE are the corresponding applied torques. We decomposed the vector field approximation (V) problem into two scalar function estimation ðf 1,f 2 Þ problems (Eq. (2)) t IE ¼ f 1 ðy IE,y DP Þ, t DP ¼ f 2 ðy IE,y DP Þ For scalar function estimation, we used Thin-Plate Spline (TPS) smoothing with Generalized Cross Validation (GCV). TPS smoothing provides a surface approximation for each scalar function. In addition, the best fit can be obtained from GCV, which determines an optimal smoothing parameter to compromise between fidelity to the data and roughness of the surface. Thus, with the application of GCV, each scalar function ðf 1,f 2 Þ is uniquely determined in the form of a TPS, and the total vector field (V) can be defined accordingly. Details of TPS smoothing with GCV are provided in Appendix A. Inbound and outbound data were treated separately. To represent the directional variation of static ankle mechanical impedance, the torque opposing displacement in different directions was calculated from the friction-compensated nonlinear vector field approximation as follows. For each of the 48 movements described above (24 outbound, 24 inbound) displacements up to 201 were imposed and the projection of the torque field directed towards the origin (co-linear with the displacement) was identified. The straight line (not constrained to intersect the origin) that best fits this data in a least squares sense was determined. Its slope represented the effective stiffness opposing displacement in that direction. ð2þ

3 H. Lee et al. / Journal of Biomechanics 44 (2011) The nonlinear vector field approximation was also decomposed into a conservative component with zero curl and a rotational component with zero divergence. The conservative component is spring-like. Here, a set of muscles is defined as spring-like if the torque vector is an integrable function of displacement so that a potential function analogous to the elastic energy stored in a spring may be determined (Hogan, 1985). In that case, the torque vector is the gradient of a potential function and, as a result, the curl of the torque field must be identically zero. Spring-like behavior is passive, since no energy is generated or removed by cyclic displacements. On the other hand, the rotational component is active because cyclic displacements may add or remove energy. Thus we can quantify how spring-like the ankle is by comparing the size of the rotational (curl) component of the field with its conservative component. A detailed definition of spring-like and the decomposition method are provided in Appendix B and C, respectively Physical mock-up To validate our methods, the procedure was tested using a simple physical mock-up. It consisted of two wooden blocks joined by a flexible steel plate and connected to the Anklebot in the same way as the shoe and knee brace (Fig. 2 left). To allow for the robot torque limits and prevent plastic deformation of the steel plate, the nominal angular displacement of this mock-up was set as 31, and the torque angle data were recorded for movements in 24 directions. Although the mock-up did not display the same torque angle relation as the human ankle, it provided a challenging test of our methods because (i) by design it had strongly direction-dependent behavior and (ii) its torque angle relation had zero curl, because we know from physical principles that any collection of passive elements (elastic, inertial or frictional), however connected, cannot generate non-zero curl. 3. Results 3.1. Validation of analysis methods The total vector field measured from the physical mock-up is shown in Fig. 2. As expected from the geometry of the thin steel plate connecting the blocks, it was substantially stiffer in IE than in DP, and this was reflected in the pattern of the vector field. In addition, whether the torque angle relation is linear or nonlinear, because the mock-up was a passive mechanical structure, the vector field must be conservative. Decomposition of the measured vector field showed that its curl components were not significantly different from zero (Fig. 2). If curl is zero in one coordinate frame, it should be zero in all coordinate frames. We verified this by computing the curl in joint coordinates and in actuator coordinates, first transforming the raw data between the two frames through the nonlinear kinematic relations due to the mechanical connection of the Anklebot to the limb (detailed in Appendix D) then applying the vector field fitting procedure. Although the shape of the total vector field was changed by this coordinate transformation, the curl value remained zero (Fig. 2). These results showed that our analysis method correctly identified curl when it was absent. To test whether the method correctly identified curl when it was present, we simulated a torque field with non-zero curl and noise comparable to our experimental data and applied our method to analyze it. The results (detailed in Appendix E) verified that curl was also correctly identified when it was present Relaxed static ankle mechanical impedance Outbound and inbound torque and angular displacement data were approximated separately (Fig. 3). The data points distributed in 24 directions are the friction-compensated measurements. The surfaces are estimates of the two torque components obtained by TPS approximation with an optimal smoothing parameter determined by GCV. The actual displacement amplitude was about 151 (mean value across 24 directions of all subjects data is with standard deviation 1.541), and all data except points near the starting position (0 11) of each movement direction were used for scalar function estimation. The mean deviation between measurements and estimates was less than Nm with standard deviation (SD) Nm, from which we defined zero deviation as Nm with 95% confidence. Because of the substantial smoothing effect of the surface fitting procedure used to identify the vector field, this error was substantially smaller than the apparatus measurement error range, 71 Nm. (Roy et al., 2009) Representative 2D slices of the vector field in 4 major directions (inversion, eversion, dorsiflexion, and plantarflexion) show how well the field fit the observed data (Fig. 4). For all subjects, the effective ankle stiffness was calculated from the estimated vector field in 48 movement directions (24 outbound, 24 inbound) as detailed above. To check the validity of this stiffness estimate, we quantified how well this linearized approximation fit the nonlinear torque field by Fig. 2. Verification of the vector field approximation using a physical mock-up consisting of passive elements. (Left: physical mock-up setup, Middle: total vector field, Right: Rotational field with zero curl. Top panels: joint coordinates; Bottom panels: actuator coordinates).

4 1904 H. Lee et al. / Journal of Biomechanics 44 (2011) Fig. 3. Representative friction-compensated measurements (depicted as point sets in 24 directions) and the corresponding vector field (consisting of two TPS surface estimates ( 1, 2 Þ). Both (a) outbound and (b) inbound data are depicted. calculating an R 2 value. The lowest R 2 value was 0.88 and most subjects showed R 2 values higher than 0.90, indicating that the linearized stiffness accounted for at least 90% of the variance. The averaged stiffness estimation results of all subjects but one (subject #6) are plotted in polar coordinates in (Fig. 5(a)). Ankle stiffness varied significantly with direction in IE DP space and was consistently lower in the IE direction than in the DP direction, resulting in a peanut shape, pinched in the IE direction. Subject #6, who was excluded from the means presented in Fig. 5(a) had an unusually low dorsiflexion impedance (9.86 Nm/rad) and an unusually high plantarflexion impedance (33.15 Nm/rad) compared to the mean values of all other subjects (16.99 Nm/rad for dorsiflexion and Nm/rad for plantarflexion). The torque required for outbound (loading) movements was consistently greater than for inbound (unloading) movements, even after static torques due to Anklebot friction were subtracted from the measurements. In all 24 directions, this torque was statistically different from zero (po0.05). The resulting energy dissipation was quantified by the area enclosed within the hysteresis loop formed by the compensated outbound and inbound curves of the raw data. The result is graphically depicted in IE DP space in Fig. 5(b). Hysteresis was highly correlated with stiffness (R 2 ¼0.88) and larger in DP than IE, exhibiting a pinched peanut shape. Decomposition of the vector field into a conservative field (with zero curl) and a rotational field (with zero divergence) showed that, in general, the rotational components were much smaller than the conservative components (Fig. 6). The mean magnitude of curl and the ratio of the determinants of the anti-symmetric and symmetric parts of the stiffness matrix (K ratio ¼det(K anti-sym )/det(k sym )) were computed (Hogan, 1985). Excluding one subject (subject #3, discussed below), the average magnitude of curl was 0.22 Nm with standard deviation 0.19 Nm, which was statistically indistinguishable from zero. The mean value of K ratio for the same 7 subjects was (see Table 1 for details). The ankle joint is predominantly spring-like in the fully relaxed condition. One subject (subject #3) exhibited undesired muscle activation during the protocol, apparently being unable to relax fully. This subject significantly activated tibialis anterior (evident as a large magnitude of EMG, 0.02 mv) during movements in directions between 2551 and For comparison, the mean EMG magnitude for each of four fully relaxed muscles was lower than mv. This subject s muscle activation was accompanied by non-zero rotational components, notably in the 2551 and 2701 directions (the highlighted region in the right panel of Fig. 7). Another interesting subject (subject #2) who fell asleep during the measurement protocol exhibited an average magnitude curl of 0.13 Nm with standard deviation 0.1 Nm. These were the smallest values recorded among all subjects. 4. Discussion Given its importance during interaction with the environment, an accurate characterization of ankle mechanical impedance would be a valuable asset to understand the role of the ankle in lower-extremity function in health or with biomechanical or neurological disorders. To overcome a limitation of previous single DOF studies, we developed a novel multivariable

5 H. Lee et al. / Journal of Biomechanics 44 (2011) Fig. 4. Representative 2D slices of the vector field in 4 major directions (inversion, eversion, dorsiflexion and plantarflexion): measurements in 4 major directions and the fitted field (top), projected 2D data in inversion eversion (mid) and dorsiflexion plantarflexion (bottom). Fig. 5. Directional variation of (a) ankle stiffness (Dark solid band: Mean value of outbound and inbound data, Light solid bands: Mean7SE, Dotted band: Mean of outbound data, Dashed band: Mean of inbound data) and (b) hysteresis (Dark solid band: Mean value, Light solid bands: Mean7SE) in IE DP space. experimental protocol using a wearable therapeutic robot that enabled fast and reliable measurements of the static ankle torque angle relation in IE DP space. We used a robust function-approximation method (TPS with optimal smoothing determined by GCV) to identify the vector torque angle relation. This provided a reliable estimate of static

6 1906 H. Lee et al. / Journal of Biomechanics 44 (2011) Fig. 6. Decomposition of a representative vector field. The torque vector is represented by an arrow drawn with its tail at the tip of the angular displacement vector (Red: positive curl, Black: negative curl). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Rotational components and effective stiffness in major directions for all subjects. OUTBOUND subject Mean curl magnitude (Nm) Curl std. dev. (Nm) K_ratio Stiffness in major directions (Nm/rad) Inversion Eversion Dorsiflexion Plantarflexion INBOUND subject Mean curl magnitude (Nm) Curl std. dev. (Nm) K_ratio Stiffness in major directions (Nm/rad) Inversion Eversion Dorsiflexion Plantarflexion Fig. 7. Non-zero curl components (right panel, color code as in Fig. 6) due to undesired muscle activation (left panel). (Red: Tibialis Anterior, Green: Peroneus Longus, Blue: Soleus, Pink: Gastrocnemius). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) ankle mechanical impedance even in the (inevitable) presence of noisy data. The mean deviation between the estimated vector field and the raw data was much less than the imprecision of the measurement system. Identifying a vector field by treating each vector component as a scalar field has been challenged by (Mussa-Ivaldi, 1992) who pointed out that the details of a scalar approximation (e.g., using the Gaussian radial basis functions) vary with the coordinates

7 H. Lee et al. / Journal of Biomechanics 44 (2011) chosen by the experimenter, potentially rendering the structure of the estimated vector field sensitive to an investigator s arbitrary choices. We validated our method experimentally using a simple physical mock-up consisting of wooden blocks and a steel plate. From physical considerations we know that the rotational component (curl) of the vector torque angle relation for the mock-up must be identically zero, and our measurements confirmed this. In addition, we showed that the conclusion that curl was zero did not depend on the coordinate frame used to represent the data. Although the detailed shape of the total vector field may change when it is represented in different coordinates (Fig. 2), certain physical properties (such as whether the field is energetically conservative, with zero curl) must be independent of coordinates. We verified that the observation of zero curl was invariant even under nonlinear coordinate transformations. We quantified the multivariable static ankle mechanical impedance of healthy young human subjects previously unavailable from single DOF studies. Torque values ðt IE,t DP Þ at any position ðy IE,y DP Þ were determined from the approximated vector field and a single effective stiffness opposing displacement in any direction in IE DP space was calculated. We found that ankle stiffness varied substantially with direction. A plot of stiffness vs. direction exhibited a characteristic peanut shape, pinched in the IE direction. Although ankle stiffness is highly variable even within a group of young healthy subjects, on average, ankle stiffness in the DP direction was higher than in the IE direction by about a factor of 2. The orientation of this peanut shape was not precisely aligned with the anatomical axes, the maximumstiffness direction being consistently tilted slightly from the DP axis in a counterclockwise direction. These are clinically meaningful results and indicate the direction of rotation in which the ankle is most vulnerable. Lower frontal plane stiffness indicates a greater tendency for the foot to roll, especially in inversion. Perturbations, e.g., from uneven ground, may evoke excessive displacement and possibly injury of the ankle. This is consistent with the observation that most ankle-related injuries occur in the frontal plane rather than in the sagittal plane (O Donoghue, 1984; Baumhauer et al., 1995). We also found that the torque evoked by displacement away from the equilibrium posture (outbound) was typically greater than the torque corresponding to the same displacement as the foot returned towards equilibrium (inbound). As positive network was required to displace the ankle and return it to equilibrium, the presence of some dissipative phenomenon is indicated, similarly observed in the wrist (Rijnveld and Krebs, 2007). As forces due to static friction or other non-ideal behavior of the Anklebot were measured prior to the experiments and subtracted from the recorded data, and the phenomenon varied with direction similarly to ankle stiffness, it appears to be a characteristic of muscle. Further study seems warranted. Analysis of the estimated multivariable vector field provided a precise quantification of the extent to which the ankle is springlike. Across almost all subjects, curl was indistinguishable from zero; the ankle was spring-like. Because muscles were relaxed, this may seem unsurprising. The force length relation for each individual muscle resembles a spring (albeit possibly nonlinear) and any arbitrary connection of spring-like muscles to the skeleton will yield a spring-like joint behavior (again possibly nonlinear). However, neural feedback (e.g., from spindles and Golgi tendon organs) may also contribute to static ankle mechanical impedance. If it does, then inter-muscular feedback between muscles that act on different degrees of freedom may introduce a deviation from spring-like behavior (Hogan, 1985). If curl is zero while inter-muscular feedback is non-zero, then the feedback gains must be exactly balanced; a dorsi-plantar flexion torque evoked by an inversion eversion displacement must be identical to the inversion eversion torque evoked by a comparable dorsiplantar flexion displacement. Conversely, with constant muscle activity, non-zero curl can only be due to unbalanced intermuscular feedback. Spring-like behavior (zero curl) in the presence of non-zero inter-muscular feedback is important because it would indicate that although the neuromuscular system is internally complex, it apparently exhibits an externally simple behavior. Spring-like behavior is consistent with dynamically passive ankle mechanical impedance, which would help to ensure stable interaction with a dynamically passive environment (Hogan, 1986, 1988). Remarkably, we observed non-zero curl in one alert subject. This subject showed a small but significantly non-zero activation of TA and a small but significantly non-zero curl when the foot was displaced towards plantar flexion, stretching the TA. Movement in other directions evoked zero curl and EMG was silent, showing that this subject was able to relax fully under our experimental manipulation. The activation of TA when it was stretched suggests an involuntary action of neural feedback. If so, the observation of non-zero curl may be due to unbalanced intermuscular feedback. Work in progress to estimate ankle impedance when muscles are active may clarify the relation between muscle activation and spring-like behavior. Conversely, our most relaxed subject (who fell asleep during the measurement procedure) exhibited the smallest curl measured in all subjects. Taken together, these observations show that vector field approximation based on scalar methods using TPS smoothing with GCV works well sufficiently precise to detect the subtle differences in structure due to small changes in muscle activation (Fig. 7). Quantifying the rotational components of static ankle mechanical impedance is important because they are fundamentally nonpassive: unlike the conservative components, under certain conditions, the rotational components may function as an energy source. Consequently, they may affect the stability of the musculo-skeletal system during posture and locomotion. This may be especially important in persons with neurological injury (e.g., stroke or spinal cord injury). The prevalence of abnormal muscle tone and spasticity indicates that the peripheral neural networks may be disordered following neurological injury. This will affect the static ankle mechanical impedance and, if inter-muscular feedback is unbalanced, it will manifest as a non-zero rotational component. We anticipate that this study will serve as a baseline to explore the ankle mechanical impedance characteristics of older and neurologically impaired subjects. Conflict of interest statement The authors declare that there is no conflict of interest in this study. Acknowledgments This work was funded by Toyota Motor Corporation s Partner Robot Division. H. Lee was supported by a Samsung Scholarship. Drs. N. Hogan and H. I. Krebs are co-inventors of the MIT patents for the robotic devices used in this study. They hold equity positions in Interactive Motion Technologies, Inc., the company that manufactures this type of technology under license to MIT. Appendix. Supporting information Supplementary data associated with this article can be found in the online version at doi: /j.jbiomech

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