Gait Entrainment to Torque Pulses From a Hip Exoskeleton Robot

—Robot-aided locomotor rehabilitation has proven challenging. To facilitate progress, it is important to ﬁrst understand the neuro-mechanical dynamics and con-trol of unimpaired human locomotion. Our previous studies found that human gait entrained to periodic torque pulses at theanklewhenthepulseperiodwasclosetopreferredstrideduration.Moreover,synchronizedgaitexhibitedaconstant phaserelationwiththepulsessothattherobotprovidedmechanicalassistance.Totestthegeneralityofmechan-ical gait entrainment, this study characterized unimpaired humansubjects’responsesto periodictorquepulsesduring overground walking. The intervention was applied by a hip exoskeleton robot, Samsung GEMS-H. Gait entrainment was assessed based on the time-course of the phase at which torque pulses occurred within each stride. Experiments were conducted for two consecutive days to evaluate whether the second day elicited more entrainment. Whether entrainment was affected by the difference between pulse period and preferred stride duration was also assessed. Results indicated that the intervention evoked gait entrainment that occurred more often when the period of perturbation was closer to subjects’ preferred stride duration, but the difference between consecutive days was insigniﬁcant. Entrainment was accompanied by convergence of pulse phase to a similar value across all conditions, where the robot maximized mechanical assistance. Clear evidence of motor adaptation indicated the potential of the intervention for rehabilitation. This study quantiﬁed important aspects of the nonlinear neuro-mechanical dynamics underlying unimpaired human walking, which will inform the development of effective approaches to robot-aided locomotor rehabilitation, exploiting natural dynamics in a minimally-encumbering way.


I. INTRODUCTION
T HE incidence of gait and balance disorders is expected to rise with the aging of the population and consequent increase in neurological injuries and disorders such as stroke [1]- [4].Thus, there is a great need for effective methods to aid and restore human locomotion.
Robot-aided locomotor therapy has emerged as a promising method to meet the enormous demand, as it allows repetitive, high-intensity, and task-oriented training in a safe and efficient manner [5], which is important for successful rehabilitation.There is evidence to suggest that complementing conventional gait therapy with robot-aided training can be beneficial [6].But on its own, robot-aided gait therapy does not outperform conventional physical therapy in clinical measures (such as walking speed, step length, and step frequency) [7].The recent rise of autonomously powered exoskeleton robots promises new avenues for delivering therapy in more ecological contexts, outside of a formal clinical setting [8]- [10].However, to live up to their promise and maximize gait recovery, we must better understand natural human locomotion and how humans react to exoskeletal robotic interventions.
Earlier robotic gait rehabilitation approaches have been criticized as they enforced repetition of preprogrammed kinematics which may have discouraged active engagement of participants and suppressed the natural rhythmic dynamics of walking [7], [11], [12].Our previous studies proposed a novel robotic intervention to modulate gait frequency-gait entrainmentthat respects the rhythmic dynamic nature of walking [13].It was shown that applying periodic torque pulses at the ankle, using an ankle exoskeleton robot, could induce subjects to increase their cadence to synchronize with the period of the pulses, when the pulse period was close to but shorter than subjects' preferred stride duration.This experimental observation suggests that a nonlinear neuro-mechanical limit cycle oscillator is a reasonable description of the dynamics of human walking.Moreover, subjects adapted their gait so that plantar-flexion torque pulses from the robot aligned with ankle push-off, which maximized their mechanical assistance.Subsequent studies showed that entrainment was observed more often, occurred earlier, and persisted longer during overground walking compared to treadmill walking [11].Motivated by this success with healthy individuals, Ahn et al. studied the feasibility of this intervention to treat locomotor deficits of neurologically-impaired patients and increase their cadence and walking speed [14].
This paradigm is similar to auditory entrainment but different in that it involves physical interaction and energy exchange.Gait entrainment to rhythmic, auditory signals is well-known and has been studied in depth [15]- [18], but synchronization to other forms of rhythmic stimuli is underexplored.Since the pioneering work of Ahn and Hogan that investigated gait entrainment to exoskeletal mechanical signals [13], there is a growing interest in gait entrainment to various types of stimuli, including an oscillating treadmill [19], [20], periodic vertical force [21], and electrical muscle stimulation [22], [23].These studies share the motivation that gait entrainment holds promise for locomotor rehabilitation, as well as providing a useful experimental paradigm to better inform neuro-motor control of human walking.However, results also show that while promising, gait entrainment is not straightforward to achieve.
The goal of the present study was to test whether humans also entrain to periodic torque pulses applied at the hip joints while walking overground.Torque pulses were applied by an autonomous hip exoskeleton robot, the Samsung Gait Enhancing and Motivating Systems for Hip (GEMS-H).We hypothesized that subjects would entrain their gait to periodic perturbations applied to the hip joint, consistent with prior results from ankle entrainment studies.A preliminary study showed that entrainment to hip torque pulses seemed promising [24].In the study reported here, experiments were conducted for two consecutive days to evaluate whether the second day elicited more entrainment.We also assessed how increasing the difference between the torque pulse period and subjects' preferred stride duration affected human responses.Assessment of gait entrainment requires accurate and reliable phase estimation.We developed an off-line stride segmentation algorithm and applied a phase estimation algorithm developed in [25], which does not require external sensors such as motion capture or foot-switches.Finally, we investigated whether subjects aligned the torque pulses to a specific phase of the gait cycle; and whether it was related to the mechanical power or work done by the hip exoskeleton robot, similar to the behavior observed in prior entrainment studies using an ankle exoskeleton [11], [26].

II. METHODS A. Subjects
A total of fifteen healthy young adults (gender: 4 females, 11 males; mean age: 25.5 ± 5.1 years old) participated in this study.The total number of subjects was chosen to be slightly larger than that which yielded statistically significant results in a related prior study of entrainment to periodic ankle torques [11].Subjects were divided into two groups: group-25ms (N = 7) and group-50ms (N = 8).All subjects gave informed written consent before the experiment.The experimental protocol was reviewed and approved by the Institutional Review Board of the Massachusetts Institute of Technology (protocol #: 1809534122; approval date: 10/18/2018).

B. Equipment: Samsung GEMS-H Exoskeleton
The Samsung Gait Enhancing and Motivating Systems for Hip (GEMS-H) developed by Samsung Advanced Fig. 1.A The Samsung GEMS-H exoskeleton applied torque pulses between the two thighs.Subjects were instructed to walk comfortably.B Subjects were divided into two groups.C Experimental protocol for each day.
Institute of Technology (Suwon, South Korea) was used in this study (Fig. 1A; [27]- [30]).This low-mass (2.1 kg) robot is worn around the waist and fastened to the thighs.A pair of actuators, one at each hip joint, applies torque in the sagittal plane (hip flexion and extension).Passive hinges allow unencumbered hip ab/adduction motion in the frontal plane.The torque output of each actuator is estimated and controlled by sensing electrical current in the respective motor.Encoders embedded in the actuator modules measure hip joint angles.All electronics, actuators, and power sources are located onboard the device, allowing for untethered operation.Unlike laboratory-based tethered exoskeleton testbeds [31]- [34], this autonomous exoskeleton allows experiments to be conducted overground, thereby enabling study of the effects of intervention in more ecological contexts.

C. Experimental Procedure
On two consecutive days, all subjects performed ten walking trials (two baseline trials followed by eight pulse trials) per day wearing the GEMS-H exoskeleton (Fig. 1A).Each trial consisted of 120 strides, and all trials were performed in a long corridor (approximately 250 m) with low foot traffic to simulate real-world walking conditions.Throughout the trials, subjects listened to white noise through wireless, over-theear headphones to mask the sound of the exoskeleton and environment.Subjects were instructed to walk at their comfortable pace, but they were neither informed that the torque pulses would be delivered periodically, nor asked to "entrain" to these perturbations.The experiment lasted approximately 45 minutes on each day.
1) Baseline Trials: In baseline trials, the hip exoskeleton was unpowered (i.e., zero motor current), and subjects were instructed to walk at a comfortable pace.On each day, preferred stride duration (T o ) was quantified as the average stride period from the middle 30 strides in the second baseline trial.
2) Pulse Trials: In the pulse trials, the hip exoskeleton was unpowered during the first 20 strides, powered during the subsequent 80 strides, and unpowered for the remaining 20 strides (Fig. 1B).When powered, the exoskeleton delivered periodic torque pulses to both legs simultaneously.The torque pulses were trapezoidal in shape with a duration (T d ) of 200 ms (Fig. 1A).The magnitude of peak torque (τ M ) was set to 0.1 times the subject's body mass in kg and was constant across trials for each subject.For safety, the torque magnitude was upper bounded by 8 Nm (group-25ms: median = 7.2 Nm, inter-quartile range (IQR) = 1.2 Nm; group-50ms: median = 8 Nm, IQR = 1.4 Nm).Note that the magnitude of torque applied was only ∼10% of the magnitude of the peak hip moment during natural gait [35].
The torque pulse applied to the right hip (τ R ) was always in the flexion direction (negative) and in the extension direction (positive) for the left hip (τ L ).The torques to the left and right hips were always of the same magnitude but opposite direction, i.e., τ R + τ L = 0 (Fig. 1A).For subjects in group-25ms, the pulse period (T p ) was set to be 25 ms faster than their preferred stride duration (T o − T p = 25 ms); for subjects in group-50ms, the pulse period was set to be 50 ms faster than their preferred stride duration (T o − T p = 50 ms).Subjects were instructed to walk in whatever way they found most comfortable.They were aware that the exoskeleton would alternate between powered and unpowered states during these trials, but they were not informed how the exoskeleton was controlled.

D. Hypotheses
To understand whether a nonlinear stable limit cycle oscillator is a reasonably competent descriptive model of the neuro-mechanical system controlling human walking, we assessed how the difference between torque pulse period and subjects' preferred stride duration affected entrainment.We hypothesized that entrainment would be observed more often when the difference was smaller, i.e., more often in group-25ms than in group-50ms (hypothesis 1).Synchronization itself does not specify the phase at which entrainment occurs.However, if entrainment occurs due to the dynamic structure of the human walking controller, a consistent phase when entrained was expected.Therefore, we hypothesized that the phase when entrained would be consistent across trials and subjects, i.e., for all entrained trials the distributions of the terminal pulse phases would not be different (hypothesis 2).
It is important to assess neural and/or biomechanical contributions.Experiments were conducted for two consecutive days to evaluate whether the second day elicited more entrainment.If the central nervous system (CNS) learned the adapted behavior, repeated bouts should evoke more frequent entrainment.
We hypothesized that entrainment would be observed more often on day 2 than on day 1 (hypothesis 3).Lastly, we tested whether mechanical energy/work done by the robot had any relation to entrainment, similar to the behavior observed in the prior entrainment studies using an ankle exoskeleton.We hypothesized that the locking phase would be commensurate with the phase where the torque pulse could provide maximal positive power (hypothesis 4).

E. Data Processing
1) Stride Segmentation: Each stride began with maximum extension (positive) of the left hip angle (θ L ), which approximately corresponds to left toe-off [36].Because torque pulses applied during the pulse trials affected position measurements, simple peak detection did not reliably separate each stride from the entire time-series data.To account for the inevitable relative motion between the exoskeleton and the wearer, we developed an off-line stride segmentation algorithm, as described in Appendix I.
2) Phase Estimation: Determining phase variables from experimental signals is not trivial, especially when the signals are non-stationary [37], [38].Previous studies [39], [40] proposed to compute phase from kinematic observations (kinematic phase) to understand neuromechanical control of animal locomotion subject to mechanical perturbation.Recent work in the robotics community exploited kinematic phase variables to design biped walking controllers, and this showed more robust performance than using a time-based phase variable [41].Inspired by these previous studies, Gregg and colleagues [25], [42]-[44] recently proposed reliable and robust methods to determine a phase variable for human walking, using hip angle.We adopted this method to estimate the gait phase of human subjects φ from the angular positions measured from the exoskeleton encoder of the left hip joint (θ L ).The estimated phase was used to assess entrainment, and to investigate to which phase of the gait cycle the mechanical perturbations converged.We briefly present the method in Appendix II, but readers are referred to the original work for details.

F. Dependent Measures
1) Pulse Phase: Pulse phase φ P (%) was defined as the phase at which the onset of the torque pulse occurred within the gait cycle.The initial pulse phase (φ P,initial ) of a given trial refers to the pulse phase of the first pulse and the terminal pulse phase ( φ P,terminal ) refers to average pulse phase of the last 10 pulses in that trial.
2) Pulse Phase Slope: Pulse phase slope (%/#) was defined as the average change of pulse phase (%) with respect to pulse number (#).Pulse phase slope was computed over the first 10 pulses (initial pulse phase slope) and the last 10 pulses (terminal pulse phase slope) of each trial by linear regression (MATLAB function fitlm).
3) Entrainment Criteria: Each trial was classified as entrained if the magnitude of the terminal pulse phase slope was < 0.5 (%/#); otherwise, it was classified as not-entrained.
4) Period Deviation: Period deviation T (ms) was defined as the difference between each stride duration and the torque pulse period for each respective trial.
5) Pulse Mechanical Energy: Normalized pulse mechanical energy E P (J/Nm) passed from the robot to the human was estimated for each torque pulse by integrating instantaneous power over time.To normalize the measure, the net energy was divided by peak commanded torque (τ M ) of the corresponding trial E p ≈ 1 τ M τ R θR + τ L θL dt.The joint torques were estimated from the current sensors.Joint position signals were filtered to estimate joint velocities off-line using an FIR filter (order: 50, passband: 20 Hz, stopband: 30 Hz) to approximate an ideal low-pass filtered differentiator.The group delay of the filter was used to compensate for the time shift of the signals due to the filtering process.Signal processing was performed using MATLAB R2018b (Mathworks, MA; function: designfilt).
6) Predicted Pulse Phase: Ignoring kinematic variation induced by the robot perturbation, one can estimate the dependence of pulse mechanical energy on pulse phase.Assuming an ideal pulse was applied to the wearers with magnitude τ M (= τ L = −τ R ), and using the kinematics obtained during baseline trials, the estimated pulse mechanical energy was calculated as where , and t on is the pulse onset time corresponding to φ P .Based on hypothesis 4, we predicted the pulse phase φP (%) would maximize the pulse mechanical energy, i.e. φP = argmax φ P Ê P (φ P ).

G. Statistical Analyses
Statistical analyses were conducted using the Statistics and Machine Learning Toolbox of MATLAB R2018b (Mathworks, MA).For all statistical tests, the significance level was set to α = 0.05.For each analysis of variance (ANOVA), the F-statistic and corresponding p-value are reported for all main effects and interactions.
A 2 (group: 25ms vs. 50ms) × 2 (day: 1 vs 2) ANOVA on the number of entrained trials was performed to test whether entrainment occurred more often when the difference between natural stride duration and pulse period was smaller (hypothesis 1) and whether the second day elicited more entrainment (hypothesis 3).Group was a between-subjects factor and day was a within-subject factor.
A 2 (group: 25ms vs. 50ms) × 2 (day: 1 vs. 2) ANOVA on terminal pulse phase (φ P,terminal ) of entrained trials tested whether the terminal pulse phase (i.e., locking phase) was influenced by the experimental conditions (hypothesis 2).Group was a between-subjects factor and day was a withinsubject factor.
A residual analysis was conducted to compare the agreement between the actual pulse phase and predicted pulse phase for maximizing mechanical work φP (%) during the last ten pulses of each trial (hypothesis 4).Fig. 2. A Representative entrained trial.B Representative not-entrained trial.Top: Left hip angle vs. phase, all strides.Black circles denote onsets of torque pulses.Middle: Pulse phase (φ P ) vs. pulse number.Bottom: Period deviation vs. stride number.Torque pulses were applied during strides 20-100 (black, solid).Pre-and post-pulse strides are also plotted (gray, dotted).In the top and middle rows, the initial pulse (green dot) and the last 10 pulses (red dots) are highlighted.

A. Representative Trials
Fig. 2 presents two representative trials: entrained (group-25ms, day 1, subject 1, trial 1) and not-entrained (group-50ms, day 1, subject 2, trial 1).In the entrained trial (Fig. 2A), the pulse phase converged by the end of the trial (M: 65.6%, SD: 0.54% in the last 10 pulse phases).The stride duration was approximately matched to the torque pulse period (i.e., period deviation close to zero).After the torque pulses ceased, stride duration slowly returned towards its pre-perturbation value (stride number 100 -120).However, in the not-entrained trial (Fig. 2B), the pulse phase drifted through all phases of the gait cycle and stride duration was little affected by the intervention (non-zero period deviation).

B. Group Results
Consistent with hypothesis 1, gait entrainment was observed more frequently in group-25ms (day1: 71%, 40 out of 56 trials; day2: 79%, 44 out of 56 trials) than in group-50ms (day1: 39%, 25 out of 64 trials; day2: 45%, 29 out of 64 trials), and this difference was statistically significant (F[1, 13] = 7.8, p = 0.015).Counter to hypothesis 3, there was no statistical difference between consecutive days of experiments (F[1, 13] = 0.37, p = 0.55).There was no statistically significant interaction between groups and Fig. 3. Pulse phase (φ P ) vs. pulse number for all entrained trials in all conditions.The number (n) of entrained trials, along with the percentage of entrained trials, for each condition is also presented.Different colors represent different subjects.The pulse phases are unwrapped such that the last value of each trial is between 0 % and 100 %.Pulse phase slope of initial and terminal segments of the entrained trials are also presented.

TABLE I PERCENTAGE OF ENTRAINED TRIALS PER SUBJECT
days (F[1, 13] = 0.002, p = 0.97).The percentages of the entrained trials for individual subjects are listed in Table I.
Fig. 3 presents the progression of pulse phase (φ P ) vs. pulse number for all entrained trials for each condition.In entrained trials, pulse phase converged to an approximately constant value before the perturbation ceased.Within each trial, the pulse phases tended to drift in one direction (either increasing or decreasing).Fig. 3 (bottom) presents the distribution of the initial pulse phase slopes and the terminal pulse phase slopes of the entrained trials.After initial transients, subjects adapted their gait and reached steady-state motion within 80 strides.

D. Mechanical Energy
Fig. 5 compares the normalized pulse mechanical energy E P and φ P values from the last 10 pulses of entrained and not-entrained trials in all conditions.The histograms of the entrained trials show that these two quantities were closely related.The unimodal distribution of the φ P values from the terminal pulses (i.e., terminal phase) from all entrained trials (M = 63.8%,S D = 10.4%) was centered about those at which subjects could gain the most mechanical benefit from the exoskeleton robot (M = 65.1%, S D = 1.57%).A representative trial exemplifies the similarity between the actual φ P,terminal and the predicted phase for maximal work done by the robot φP (Fig. 6A).The distribution of the error between predicted and actual locking phase (φ err = φP −φ P,terminal ) (Fig. 6B) for all entrained trials was centered around zero (M = −1.3%,S D = 10.2%).These results indicate a close relationship between the converged phase and positive mechanical work done by the robot confirming hypothesis 4.

E. Stride Period
Fig. 7 presents how the period deviation of entrained trials and not-entrained trials varied in each condition.In all entrained trials, stride duration adapted to match the torque pulse period, resulting in a mean period deviation close to zero.Upon removal of the robotic intervention, stride durations slowly returned to their pre-perturbation values.Conversely, stride duration did not converge to the torque pulse period in not-entrained trials.T-tests indicated that in each condition, the mean period deviation was not significantly different from zero during entrained trials (group-25ms × day1: p = 0.51, group-25ms × day 2: p = 0.77, group-50ms × day1: p = 0.61, group-50ms × day2: p = 0.11), but was significantly different from zero during not-entrained trials (group-25ms × day1: p < 0.001, group-25ms × day 2: p < 0.001, group-50ms × day1: p < 0.001, group-50ms × day2: p < 0.001).There was a constant offset in period deviation for not-entrained trials.While this offset was smaller on day 2 of not-entrained trials, in particular of group-25ms, it was larger than that of entrained trials.

A. Summary of Results
This study characterized unimpaired human subjects' responses to periodic torque pulses applied about the hip joints during overground walking.
We observed that human subjects entrained their gait to the periodic mechanical torque pulses applied by a hip exoskeleton robot.During this process, subjects synchronized their stride duration to match that of the external mechanical perturbation (Fig. 7).For all groups, by the end of entrained trials, pulse phase converged to a unimodal distribution centered around 63-65% (Fig. 3 and Fig. 4).Further analysis revealed that entrainment occurred such that the mechanical energy flow from the robot to the wearer was maximized (Figs. 5 and 6).
Gait entrainment was observed more often when the pulse period was closer to subjects' preferred stride duration, and Fig. 7. Period deviation vs. stride number of A entrained trials and B not-entrained trials.Pre-pulse strides and post-pulse strides are distinguished (cyan).The means and standard deviations of trials for each condition are presented as black, thick lines and red, thin lines, respectively.In computing means and standard deviations for each stride number, outliers were omitted using MATLAB function rmoutliers with its default setting.1142 out of 28,800 total strides (4%) were deemed outliers.
this difference was statistically significant ( p = 0.015).However, there was no statistical difference between the two days of experiments ( p = 0.55), suggesting that longer training periods might be required to elicit further changes in neuro-motor behavior or that the effect size of day was too small to detect with the number of subjects included in this study.

B. Limitation: Robot-Embedded Measurements
There are trade-offs between tightly-controlled lab experiments and experiments in real-world conditions.For example, walking overground enables a study of natural human behaviors in more ecological contexts, but the ability to measure human behavior is limited.On the other hand, walking on a treadmill in a lab enables reliable collection of various data (e.g., ground reaction forces using force plates or whole-body kinematics using a motion capture system).However, walking on a treadmill is mechanically different from walking overground.For instance, the treadmill belt speed changes periodically depending on the gait phase, and may require motor adaptation to a dynamic environment [45].The goal of this study was to investigate entrainment in real-world conditions, despite the associated limitations.
Entrainment was assessed based on robot-embedded joint position measurements.Because the exoskeleton robot did not perfectly conform to each individual, the hip joint angles measured by the robot-embedded encoders may have differed from the true human joint angles.The robot actuators transmitted power and/or torque through a thigh frame and belt assembly that was tightly coupled to the human subjects.However, torque transmission may have been imperfect for a myriad of reasons (e.g., friction in the actuator, elasticity in human tissue or the thigh frame, and relative motion between the exoskeleton robot and the wearer).The difference between torque estimated from on-board sensors and external force-sensitive resistor (FSR) sensors was presented in [46].Nonetheless, our conclusion that periodic torque pulses at the hip joints induced gait entrainment is still valid because it relies on the trend of behavior rather than exact values.Inaccurate kinematic measurement and inaccurate torque application cannot dismiss our results.
Measurements based on whole-body kinematics (e.g., stride length), or stride segmentation based on external sensors (e.g., foot-mounted sensors) would be useful to confirm and further illuminate our results.In addition, direct measurements of muscle activity (e.g., surface electromyography, sEMG) would be useful to further identify neuro-muscular mechanisms associated with gait entrainment.Measurements of metabolic cost would also add useful information to track consequential changes in locomotion economy following adaptation.

C. Gait Entrainment to Mechanical Perturbations
Periodic torque pulses applied by a hip exoskeleton evoked gait entrainment, accompanied by convergence of pulse phase to a constant value across all conditions.Entrainment occurred more often when the period of perturbation was closer to subjects' natural stride duration.The results are consistent with previous studies that showed entrainment to mechanical perturbations [11], [13], [19]- [21], [47].While a direct, withinsubject comparison of the basin of entrainment for ankle versus hip torque is still needed, the present results suggest that the basin of entrainment is smaller with a hip exoskeleton than with an ankle exoskeleton [11], [13], [47].For instance, the percentage of trials entrained to pulse periods 50ms shorter was higher for ankle perturbations (92% of overground trials in [11]) compared to hip perturbations (day1: 39%, 25 out of 64 trials; day2: 45%, 29 out of 64 trials).Assessment of this difference, and identification of the underlying cause, remain areas for future work.
While entrainment itself does not enforce convergence to a particular constant phase, subjects entrained their gait to maximize the work done by the hip exoskeleton robot, as illustrated in Fig. 5.This result is consistent with previous studies with different mechanical interventions, in which individuals adapted to leverage positive power from the devices [48], aligning the timing of robotic torques from an ankle exoskeleton with ankle propulsion [11], [13], [33].Subjects tended to get the most mechanical assistance during the gait phase when the ankle does the most positive work [49].
Adapting gait to maximize mechanical energy supply from the hip exoskeleton robot has biomechanical advantages.The hip joints produce 40 -45% of positive power during walking, and this is mostly done during the initial-and mid-swing phases [36].To get the swing leg out in front requires substantial effort and power, which is influenced by hip torque.The converged pulse phase (63 -65%) roughly corresponds to when the left leg is extending (from maximum flexion) and the right leg is flexing (See Fig. 2A).At this phase the robot drove the legs apart, as it always applied flexion torque pulses on the right and extension on the left, providing assistance to swing the right leg forward.Moreover, this phase also approximately coincides with activity bursts of the major hip flexor muscles of the right leg and hip extensor muscles of the left leg [36].Additional sEMG measurements would be useful to confirm this explanation.Assuming a major limitation to walking speed is the effort required to get the swing leg out in front, assisting that action may provide some useful therapeutic effect (e.g. for stroke survivors with hemiparesis).

D. Gait Entrainment: Neural or Mechanical?
Does gait entrainment involve any central neural process, or was it purely due to peripheral neuro-mechanics?The potential bio-mechanical benefits discussed in the previous section seem to suggest that biomechanical mechanisms at least played a significant role.In fact, a previous modeling study suggested that gait entrainment may not require any supra-spinal mechanisms [26].However, that model could not reproduce some experimental results (e.g., entrainment to periodic perturbations slower than preferred walking period [11], [50]), suggesting that some higher-level neural contribution may be required.Indeed, it is hard to dismiss the role of supra-spinal control in gait entrainment.
The intervention in this study was designed to either drive the legs apart or pull them together, depending on the phase at which the pulse occurred.Since the torque pulse period was similar to the stride duration not the step duration, this intervention influenced inter-leg coordination, breaking symmetry, which might be detected by the CNS as an error to be corrected [51].Mechanical perturbations affecting inter-leg coordination have been shown to evoke locomotor adaptation, e.g., split-belt treadmill walking [51] and many other studies involving unilateral perturbation [52]- [54].Conversely, our companion study [55] showed that an intervention that did not affect symmetry did not evoke motor adaptation.
Moreover, the timing of locomotor patterns is thought to be mediated at the spinal level (e.g., central pattern generators) but under supraspinal control (e.g., motor cortex, via brainstem centers) and afferent sensory feedback [56]- [59].While auditory signals cannot directly influence inter-leg coordination, substantial studies have shown that the rhythmic adaptation observed in auditory-motor synchronization involves central neural mechanisms (e.g.cortical areas, basal ganglia and the cerebellum) [18].There is also evidence that neuro-motor adaptation to a mechanical perturbation was predominantly due to descending drive from supra-spinal levels [60].For example, an intact cerebellum and motor cortex appear to be critical for motor adaptation [61].
In sum, it seems reasonable to suggest that both high-level (supra-spinal) control and low-level peripheral neuromechanical structures contributed to the observed gait entrainment to mechanical perturbation at the hip joints.

E. A Nonlinear Limit-Cycle Oscillator as a Descriptive Model of Human Walking
When exposed to external periodic forcing, a nonlinear limit cycle oscillator is entrained and synchronizes its frequency with that of the stimulus [62].This synchronization only occurs when the frequency and strength of the stimuli are in a finite region called the basin of entrainment1 ; outside this region in parameter space, entrainment does not occur.
The experiments reported here were designed based on a working hypothesis that a nonlinear, stable limit cycle oscillator is a reasonable descriptive model of human walking.The experimental observations of this study can be summarized by two characteristics: phase-locking to a unimodal phase distribution; and a finite basin of entrainment (more entrainment in group-25ms than in group-50ms).Inevitable noise in biological systems was also observed, e.g., in stride durations or pulse phase distributions.Despite the unquestionable complexity of the human neuromotor system, there is considerable practical value to describing human walking with an exoskeleton robot by a simplified mathematical model: a nonlinear limit-cycle oscillator with periodic forcing, subject to the presence of stochasticity.Note that a linear model (e.g. a second-order mass-spring-damper system) cannot exhibit a finite basin of entrainment.Despite the evident power of linear analysis, some phenomena require nonlinearity.Stochasticity is also required, though deterministic chaos may not be necessary [63].
This simple theoretical model provides useful insights to predict and interpret complicated experimental observations.First, the finite basin of entrainment is determined by two parameters: the period of perturbation and the coupling strength.From this point of view, it is natural to observe less entrainment in group-50ms than in group-25ms.When the parameters are outside the basin of entrainment, entrainment will not occur and the pulse phase will drift.However, when the parameters are close to the boundary, the pulse phase will drift slowly when it is near the converged phase (critical slowing [64]).In that case, stochasticity may 'push' the system into the basin of entrainment.We believe this is what we observed in some trials in Fig. 3.
Second, this simple theoretical model might potentially be useful to quantify individuals with a small number of parameters.For example, the model presented in [62] only requires three parameters: the convergence rate of the limit cycle (this may be a characteristic of individual subjects), the relative frequency between the stimulus (pulse period T p ) and the oscillator (subjects' preferred stride duration T o ), and the coupling strength (which may be related to pulse magnitude τ m or pulse duration T d or both).A model of this kind may serve to customize treatment protocols to individual patients.

F. Gait Entrainment: Clinical Implications
Mechanical gait entrainment may serve as a novel permissive locomotor rehabilitation therapy that does not disrupt the limit-cycle behavior underlying locomotion.Similar to the auditory gait entrainment which has shown promising therapeutic effects [17], mechanically entraining gait to increase cadence may result in increased walking speed [15], [65] or stride length [17], which might be important functional outcomes for post-stroke survivors [66], [67].Moreover, walking in recovering stroke patients (and in some healthy elders) looks a lot like a sporadic slow sequence of individual steps.Mechanical entrainment may promote more natural (rhythmic) action and hence be beneficial as demonstrated in [14].
To maximize the potential therapeutic effects of mechanical gait entrainment, different torque pulse profiles (including continuous rhythmic torque patterns) and performance-based protocols for gradual improvement [14], [47], [68] should be investigated.To understand underlying neuro-mechanical mechanisms of entrainment, a competent model would be insightful [69].Once the most effective method has been established, studies with patient populations should follow to evaluate whether neuromotor adaptation in healthy subjects translates to neuro-recovery in clinical settings.These matters are left for future studies.

APPENDIX I OFF-LINE STRIDE SEGMENTATION
For each pulse trial, the time-series of left hip angle θ L was post-processed to detect the peaks using a peak detection algorithm (MATLAB findpeaks).The indices of the peaks were stored i pk ∈ I P K and served as an initial seed for optimization for stride segmentation (Fig. 8A).The indices of the onset and offset pair of each torque pulse τ L were stored as tuples: (i on , i o f f ) ∈ I O N−O F F .If a peak index i pk was within any pair of indices [i on , i of f ], the corresponding stride segmentation was deemed affected by the torque pulse.Those peaks were removed from I P K and stored as decision variables to be updated, I DV (See Algorithm 1).return V 17: end procedure Assuming human subjects exhibited similarity across different strides within each trial, the decision variables were updated so as to minimize inter-stride variability V .To compute inter-stride variability V , the time-series of left θ L and right θ R hip angle were first segmented using I P K ∪ I DV .Each stride segment was interpolated and time-normalized as a length N vector (MATLAB interp1).The inter-stride variability was calculated as the sum of squared errors of each stride and the ensemble average; see Algorithm 2.
In this work, the genetic algorithm (MATLAB function ga) was used to find the integer-valued decision variables I DV that globally minimized V .Other methods may be developed for computational efficiency.To avoid over-correction, each of the corrected peak indices (i dv ) was bounded in time by ±T d (= ± 200 ms) from its initial value (i pk ).The optimal stride segments S were used to calculate dependent measures in this study.
The effect of the developed stride segmentation algorithm is clearly demonstrated in Fig. 8.When motion artifacts were significant as in this study (due to relative motion between the exoskeleton robot and the wearers), this method outperformed the näive peak detection algorithm to segment strides from long time-series data (e.g., MATLAB findpeaks).Of course, using external sensors such as a motion capture  system or a force plate to detect gait events would be more accurate, but may not be compatible with human studies outside the lab.

APPENDIX II PHASE ESTIMATION ALGORITHM
The phase estimation algorithm was adopted from [25], [43], [44].For the s-th stride θ (s) L ∈ S from Appendix I, the angle and its integral were shifted and scaled to obtain two signals s (s)  P and s (s) I .) ∈ [0, 1].(6)

Fig. 4 .
Fig.4.Histograms of initial and terminal pulse phases (φ P,initial and φ P,terminal , respectively) for all entrained trials across all conditions.

Fig. 5 .
Fig. 5. Histograms of pulse phase (φ P ) and mean normalized terminal work done by the robot (E P ) from the last 10 pulses (i.e., terminal pulses) of all (A) entrained and (B) not-entrained trials in each experimental condition.Color indicates normalized frequency (as does the z-axis); Greener bars denote higher occurrences of a particular combination of φ P and E P values.

Fig.
Fig. Stride segmentation using naive peak detection algorithm (A) and using the improved algorithm (B).Each segment of the data between squares (top, right of each panel) corresponds to a stride.

Fig. 9 .
Fig. 9. Representative entrained trial (A) and not entrained trial (B).Top: Hip angle (positive: extension; negative: flexion) vs. normalized time.All 80 strides of the representative trial.Bottom: trajectory of s P vs. s I of all strides on the phase plane.Black circles indicate the onsets of torque pulses.Green dot indicates the initial pulse.Red dots indicate the onsets of ten terminal pulses.

I
construct a closed orbit on the phase plane.The phase of each data point for each stride was computed as the angle of the data point on the phase plane as shown in Fig.9.

Algorithm 1
Off-Line Stride Segmentation Initialization 1: Given left hip angle peak indices I P K , torque pulse onset-offset index tuples I O N−O F F 2: Initialize decision variable I DV 3: for i pk ∈ I P K do 4:if ∃(i on , i of f ) ∈ I O N−O F F s.t.i pk ∈ [i on , i o f f ] then Inter-Stride Variability 1: procedure INTER-STRIDE VAR(θ L , θ R , I P K , I DV , N)Sort I S = I P K ∪ I DV in ascending order 5:for s-th index i s ∈ I S do 6: