. feedback control may actually increase rather than decrease instability due to the inherent time - delay in the neural close - loop control .
we focused on measuring the mechanical impedance of the ankle during walking because the ankle is critical for propulsion , shock absorption , and balance during standing and walking .
the ankle plantarflexors contribute as much as 50% of positive mechanical work in a single stride to enable forward propulsion . in pre - swing
, ankle plantarflexors also act to advance the leg into swing phase while promoting knee flexion at toe - off .
additionally , the ankle helps maintain body - weight support during gait and balance , .
finally , the ankle dorsiflexor musculature helps absorb impact forces during foot strike to enable controlled landing and foot - floor swing clearance at toe off .
extensive previous work estimating ankle impedance has provided a rich characterization of these properties during stationary , non - moving conditions . when investigated in the sagittal plane , ankle joint impedance has been shown to vary with many factors , including mean ankle torque , ankle position , perturbation amplitude , and muscle fatigue .
the research showed that modulation of these factors can vary the stiffness component of impedance by up to three orders of magnitude .
few studies have focused on characterizing ankle impedance during non - stationary conditions , which are more relevant to locomotion .
these studies use changes in ankle torque or position in combination with time - varying system identification techniques to estimate ankle impedance . rather than using a perturbation , researchers have instead focused on measuring the instantaneous slope of the ankle s torque - angle profile during locomotion , known as the  quasi - stiffness  , .
these studies has provided important insight into how torque and angle co - vary at a joint during locomotion . however , because the agonist and antagonist muscles that cross human joints are capable of producing net - positive mechanical work , joint torque and angle can be varied independently of joint stiffness . thus , there is no causal relationship between quasi - stiffness and joint impedance , and these concepts are decoupled .
ultimately , a perturbation is required to assess the joint impedance that characterizes the underlying system dynamics .
the mechanical impedance of the human ankle plays a critical role in locomotion , and has broad applications in assistive technologies , such as robotic prostheses and exoskeletons , as well as humanoid robotics . for the first time , the previous studies have been merged to provide a comprehensive representation of how joint impedance varies throughout the entire gait cycle .
specifically , we have examined the work of lee and hogan   and rouse et al . 
  to characterize ankle impedance from pre - swing to terminal stance .
the purpose of this study is to show how ankle impedance varies throughout the gait cycle , and discuss the implications of the results in the context of the whole gait cycle .
this study motivates many changes in the development of future assistive technologies , as well as a new understanding of neuromuscular pathologies , and the neural mechanisms that underlie these disorders .
two different approaches were used to construct a trajectory of ankle impedance modulation during walking . a mechatronic platform and a wearable ankle robot
were utilized to quantify ankle impedance from late - loading response to terminal stance phase and from pre - swing phase to early - loading response , respectively ( fig .
a : from late - loading response to terminal stance phase , i.e , when the foot was flat to the ground , ankle impedance was estimated by a mechatronic platform , recessed into a walkway ( b ) . from pre - swing phase through the entire swing phase to early - loading response , i.e , when the toe and/or heel were off the ground , ankle impedance was estimated by a wearable ankle robot ( c ) .
b : the perturberator robot was comprised of a force platform coupled to a gear - motor that was controlled by a servodrive and microcontroller .
c : the anklebot was mounted onto the knee brace and its end - effectors were connected to a rigid u - shaped bracket attached to the bottom of a shoe ( left ) .
the robot was properly attached to the subjects right leg , and subjects were instructed to walk on a treadmill with the robot ( right ) .
a : from late - loading response to terminal stance phase , i.e , when the foot was flat to the ground , ankle impedance was estimated by a mechatronic platform , recessed into a walkway ( b ) . from pre - swing phase through the entire swing phase to early - loading response , i.e , when the toe and/or heel were off the ground , ankle impedance was estimated by a wearable ankle robot ( c ) .
b : the perturberator robot was comprised of a force platform coupled to a gear - motor that was controlled by a servodrive and microcontroller .
c : the anklebot was mounted onto the knee brace and its end - effectors were connected to a rigid u - shaped bracket attached to the bottom of a shoe ( left ) .
the robot was properly attached to the subjects right leg , and subjects were instructed to walk on a treadmill with the robot ( right ) .
joint impedance was estimated using perturbations and data recorded from a mechatronic platform , termed the perturberator robot ( fig .
the mechatronic platform was comprised of a multi - axis force platform coupled to a gear - motor that received position commands from a servodrive and microcontroller .
the mechatronic platform was recessed into a walkway and the platform section of the robot was flush with the walkway surface .
the total length of the walkway was 5.25 m. ten healthy , able - bodied subjects participated in this experimental protocol .
subjects walked at a self - selected pace that ranged from 85  90 steps per minute across the walkway that included the recessed mechatronic platform .
when subjects stepped on the mechatronic platform , a 2.0 ramp perturbation was randomly applied with a probability of 50% in either the dorsiflexion or plantarflexion directions .
one hundred perturbation trials were recorded at each timing point , and approximately 400 trials were recorded where no perturbation occurred , totaling approximately 800 walking trials .
perturbations were applied at four timing points , ranging from 2070% of the stance phase of walking , chosen randomly during each trial .
the mid - stance region of stance phase was chosen because it is a critical part of the gait cycle when the body is supported by a single leg . additionally , during mid - stance , the perturbation caused changes in the ankle s angle , rather than deformations of the mid - foot , improving the accuracy of our estimation .
data were low - pass filtered at 20 hz and segmented to include the 100 ms window beginning with the perturbation onset . to obtain ankle torque ,
ground reaction force data were resolved to the equivalent force - torque couple at the ankle s center of rotation .
the ankle s center of rotation was identified from high - definition video obtained during the experiment . a second - order model consisting of stiffness , damping , and
the stiffness , damping , and inertia describe the position , velocity , and acceleration dependent components of response torque , respectively .
prior to estimation , a bootstrapping procedure was used to provide a more reliable estimate of the parameter variability .
the bootstrapping procedure involved included the following steps : a random selection of 60% of the torque and angle profiles from perturbed trials were chosen and averaged .
2a black lines ) was then subtracted from the mean of the perturbed torque and angle profiles ( fig .
the joint impedance was then estimated from these resultant profiles using the optimal least squares estimate , with a constraint that the estimated parameters be greater than zero .
this process was repeated one hundred times for each perturbation direction and time point . in summary ,
the purpose of the bootstrapping procedure was to remove the torque and angle changes that occurred simply as a result of walking , while preserving the variability of the parameter estimates that were subsequently obtained . 

figure 2.samples of inputs and outputs for the estimation of ankle impedance . a : data during the analysis window ( 100 ms ) from the perturberator robot .
random torque inputs ( top ) and the corresponding angles ( black curve - blue curve ; bottom ) .
the blue curve is the nominal angle trajectory . only the data in the shaded region ( from pre - swing phase to early - loading response )
a : data during the analysis window ( 100 ms ) from the perturberator robot .
random torque inputs ( top ) and the corresponding angles ( black curve - blue curve ; bottom ) .
only the data in the shaded region ( from pre - swing phase to early - loading response ) were used .
the aforementioned methods that were used to estimate ankle impedance during locomotion were previously validated and shown to provide highly accurate estimates of ankle stiffness .
the stiffness of a passive prosthetic foot was estimated during locomotion at four time points and compared to values obtained independently using a materials testing machine .
the stiffness component of impedance was shown to be accurate within 5% on average during walking .
a correction has been applied to the data from , which updated the inertia estimates .
the stiffness and damping values did not change significantly , however , the data presented here reflect the updated values .
a wearable ankle robot , anklebot , was used to estimate ankle impedance from pre - swing phase before toe - off to early - loading response right after heel - strike .
the robot was designed to have very low intrinsic mechanical impedance at the interaction port , i.e , the ankle .
this made possible highly back - drivable operations , allowing subjects to have movements with minimal resistance .
the robot was mounted onto the knee brace and its end - effectors were connected to a rigid u - shaped bracket attached to the bottom of a shoe .
the robot was properly attached to the subjects dominant leg , and subjects were instructed to walk on a treadmill to familiarize themselves with the experimental setup and select their preferred walking speed . in a main experiment , subjects walked with the anklebot on a treadmill for 13 minutes while the robot continuously applied mild random torque perturbations ( band - limited white noise with stop frequency 100 hz ) to the ankle .
the magnitude of perturbations was determined low enough not to disturb normal walking but strong enough to perturb the ankle .
the resulting ankle displacements due to the mild perturbations were 1.6 ( root - mean - square value ) and 4.1 3.9 ( peak - to - peak value ; negative and positive denote plantarflexion and dorsiflexion , respectively ) when averaged across subjects .
ankle impedance was estimated based on ensemble - based linear time - varying system identification .
this is an effective and robust system identification technique when repetitive and periodic data are available . in this study ,
more than 500 realizations for ensemble sets were generated from human walking , where each realization was defined by a gait cycle that begins with heel - strike of one foot and ends with another heel - strike of the same foot .
2b bottom ) , respectively . by applying correlation - based system identification approach   to the ensemble sets ,
measured dynamics were estimated along the time axis ( at 2 ms intervals ) in the form of finite impulse response functions ( irf ) .
the irf estimates were further approximated by second order models consisting of inertia , damping , and stiffness .
finally , ankle parameters ( \documentclass[12pt]{minimal } 
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 \end{document } ) were obtained by compensating actuator dynamics . as the ankle and the robot share the same displacement , ankle impedance was calculated by subtracting impedance of the robot ( also approximated as a second order model consisting of inertia , damping , and stiffness ) from the measured impedance .
while impedance was estimated every 2 ms , representative ankle parameters were calculated at five sub - gait phases , ranging from 010% ( early - loading response : elr ) , 5060% ( pre - swing phase : psw ) , 6073% ( initial swing phase : isw ) , 7486% ( mid - swing phase : msw ) , and 87100% ( terminal swing phase : tsw ) of the gait cycle by averaging estimated parameters within each sub - gait phase into single values .
based on the aforementioned two methods , ankle impedance was estimated for the whole gait cycle .
time - varying ankle stiffness and damping were summarized at nine timing points ( fig .
3 ) . first , ankle stiffness linearly increased from loading response to terminal stance ; the correlation coefficient ( \documentclass[12pt]{minimal } 
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 \end{document } between the gait phase and the corresponding ankle stiffness was 0.998 .
ankle stiffness substantially decreased at the end of the stance phase , remained relatively constant through the entire swing phase , and increased again around heel - strike ( fig .
stiffness in the swing phase was comparable with results from a previous steady - state study of matched muscle activation .
the change in damping during the gait cycle is much less than the changes observed in stiffness ( fig .
inertia estimates by the anklebot were rather invariant throughout the gait phase around 0.008 kgm , consistent with previous findings , .
although also generally invariant throughout the gait cycle , inertia estimates by the perturberator robot were approximately 0.027 kgm , differing by a factor of three .
this is likely a result of residual inertia of the mechanism structure and other coupled body segments . 

figure 3.time-varying ankle mechanical impedance during walking.ankle stiffness and damping were summarized at nine timing points .
the stance and swing phases account for approximately 60% and 40% of the gait cycle .
time - varying ankle mechanical impedance during walking.ankle stiffness and damping were summarized at nine timing points .
the stance and swing phases account for approximately 60% and 40% of the gait cycle .
red and blue colors denote results from the perturberator robot and anklebot , respectively . dashed black lines indicate the regions where no results were available .
the quality of estimation by second order models was demonstrated by high percentage variance accounted for ( % vaf ) between measurements and estimates by the models . when averaged across subjects and timing points from late - loading response to terminal stance phase
, the % vaf between torque measurements and estimates by second order models was 926.3% ( meanstandard deviation ) . for timing points from pre - swing phase to early - loading response , the % vaf between angle measurements and predicted outputs by input torques and estimated second order models was 85.74.5% .
for the first time , we were able to provide an estimate of the ankle impedance in the sagittal plane during the complete gait cycle .
especially , stiffness increases from heel strike to terminal stance , remaining low during the swing phase .
this modulation of ankle mechanical impedance parallels functional need to prevent foot slap following heel strike , increase stability demands during stance and metabolic savings during the swing phase .
it goes beyond coarse correlations between anthropometric measures with intrinsic ankle   or prior studies showing that humans adjust leg stiffness to accommodate surface changes .
we speculate that proper modulation of ankle impedance is part of the learning process of human biped walking and it is an important aspect when assessing neurological deficits , aging , and determining proper therapeutic interventions as well as designing prosthetics .
this study highlights the need to incorporate joint impedance in the design of future wearable robotic technologies , including prostheses and exoskeletons .
historically , the development of wearable robotic technologies has only used the torques and angles of lower limb joints to define the design criteria , and accompanying metrics of success .
however , the present study demonstrates that joint impedance is also regulated during gait , and must be included in the future design and control of wearable robotic systems that can truly mimic the system dynamics of the healthy neuromuscular system .
the impedance control law proposed in rouse et al . has been implemented in a robotic prosthetic leg , which led to more natural locomotor behavior in amputees .
additionally , a quasi - passive prosthetic ankle has been developed that can provide the natural joint impedance relationship during gait , which will be soon be tested in amputees . because impedance is at the interface between the neural controller and the world ,
one implication of our working model is that restoring appropriate impedance occurs concurrently with recovery . in a preliminary study krebs and
colleagues found that the ankle quasi - stiffness of neurologically impaired subjects was significantly different from that of age - matched healthy subjects , .
they also found that seated (  open - chain ) robot - aided therapy successfully resolved this abnormality and  most important  this therapy resulted in a 20% improvement of over - ground walking speed .
the identification techniques described here will allow rehabilitation therapy to be targeted toward the specific deficits in the mechanical impedance and its modulation that each individual exhibits .
we will alter the adaptive algorithm developed in 2003 to compensate also for impedance abnormalities .
it will include time - varying impedance that depends on arm reaching or gait phase and aims to normalize the patient s impedance with respect to unimpaired impedance ( suitably scaled ) .
while the stiffness and quasi - stiffness are distinct entities in general , linearly increasing stiffness from loading response to terminal stance in this study are remarkably similar to those from the slope of the ankle s torque - angle profile , which was not previously known .
however , despite their similar magnitudes , the joint stiffness and quasi - stiffness have different meanings and , therefore also different interpretations .
moreover , we do not yet know if these properties are similar for other joints or regions of the gait cycle .
most notably , the identification procedures assume a linear , time - invariant behavior of the ankle within a short period of time ; the procedure with the perturberator robot used data within the 100 ms window beginning with the perturbation onset , and the procedure with the anklebot used data within the 40 ms window for irf estimation .
furthermore , the system structure was assumed to be second order , consisting of stiffness , damping and inertia .
the rationale for this assumption was based on previous findings that the human ankle was accurately described as a second order system over a wide range of ankle positions and muscle activation , . even with these assumptions ,
the methods were valid under the given experimental conditions , supported by the high % vaf between measurements and predicted outputs across the gait cycle .
this study used two distinct approaches to complete a trajectory of ankle impedance modulation throughout the gait cycle . to estimate ankle impedance
, we should apply external energy to the ankle and analyze the corresponding responses . in the entire swing phase and when the toe or heel were off the ground , a wearable device is proper to provide accurately controlled perturbations to the ankle .
however , when weight bearing is substantial during stance phase , i.e. , when the foot was flat to the ground , a wearable device is less desirable . due to high weight bearing , mild perturbations are not strong enough for the purpose of identification .
strong perturbations from the wearable device are not desirable either ; they will significantly change the normal gait or may cause slippage at the interface of the lower - limb and the device . on the other hand ,
a robotic platform recessed into a walkway can provide strong perturbations to the ankle without disturbing the normal gait .
however , this approach can be used only when solid contact exists between the foot and the device .
thus , we would argue that the current best practice to estimate ankle impedance across the full gait - cycle could be achieved through the combination of two distinct experimental setups : a wearable ankle device and an instrumented walkway .
however it is important to note that the robot is not mounted at the ankle , but mounted proximally to the leg and anterior to the shank to minimize perception of loading .
we further examined the effects of asymmetric or unilateral loading of the limb during task - oriented gait in adults both healthy and stroke ( 9 chronic stroke patients ) .
specifically , we sought to assess the effects of the added inertia and friction of unpowered ankle robot on gait parameters , interlimb symmetry , and lower extremity joint kinematics . in summary , our results demonstrated that the added inertia anterior to the shank ( above knee ) had no statistically significant effect on spatiotemporal parameters of gait , including paretic and nonparetic step time and stance percentage , in both overground and treadmill conditions .
noteworthy , interlimb symmetry as characterized by relative stance duration was greater on the treadmill than overground regardless of loading conditions .
the current study utilized a treadmill in estimating ankle impedance from pre - swing phase to early - loading response .
the primary reason for this is to minimize the duration of the experiment in a limited space .
one recent study has demonstrated that the anklebot could be well used for overground walking . since dynamics of treadmill walking
may be significantly different from those of overground walking , it is important to compare the modulation of ankle impedance in these two conditions . as a limitation of this study ,
the analysis methods presented do not attempt to separate intrinsic and reflex components of joint impedance .
however , it is presumed that impedance measurements by both systems are dominated by the intrinsic components of the ankle joint , supported by the time course of short latency reflexes that occur approximately 40 ms following the onset of an imposed movement ,   and an additional 60 ms delay in subsequent force production , .
it was also demonstrated that the influence of random perturbations by the anklebot on muscle activity is small ; the change of muscle activation due to perturbations was less than 0.5% mvc of each of the major ankle muscles .
together , it is presumed that reflex contribution to ankle impedance measurements is minimal in this study .
the current results are limited to the characterization of ankle impedance in the dorsi - plantarflexion direction .
we plan to expand this work and measure healthy young subjects ankle mechanical impedance in all 2-d directions during walking , including dorsi - plantarflexion and inversion - eversion directions .
we will employ the anklebot described above and the mit - skywalker   or a new multi - axis robotic platform   during stance to implement in 2-d what the perturberator robot did in 1-d .
in addition , it is possible to extend the current study to characterize ankle impedance in different walking conditions , such as walking in different speeds and slopes .
 the human ankle joint plays a critical role during walking and understanding the biomechanical factors that govern ankle behavior and provides fundamental insight into normal and pathologically altered gait . 
 previous researchers have comprehensively studied ankle joint kinetics and kinematics during many biomechanical tasks , including locomotion ; however , only recently have researchers been able to quantify how the mechanical impedance of the ankle varies during walking . 
 the mechanical impedance describes the dynamic relationship between the joint position and the joint torque during perturbation , and is often represented in terms of stiffness , damping , and inertia . 
 the purpose of this short communication is to unify the results of the first two studies measuring ankle mechanical impedance in the sagittal plane during walking , where each study investigated differing regions of the gait cycle . 
 rouse et al . 
 measured ankle impedance from late loading response to terminal stance , where lee et al . quantified ankle impedance from pre - swing to early loading response . 
 while stiffness component of impedance increases significantly as the stance phase of walking progressed , the change in damping during the gait cycle is much less than the changes observed in stiffness . 
 in addition , both stiffness and damping remained low during the swing phase of walking . 
 future work will focus on quantifying impedance during the  push off  region of stance phase , as well as measurement of these properties in the coronal plane . 
Introduction
Methods
Results
Discussion