An understanding of the time-varying mechanical impedance of the ankle during walking is fundamental in the design of active ankle-foot prostheses and lower extremity rehabilitation devices. This paper describes the estimation of the time-varying mechanical impedance of the human ankle in both dorsiflexion–plantarflexion (DP) and inversion–eversion (IE) during walking in a straight line. The impedance was estimated using a two degrees-of-freedom (DOF) vibrating platform and instrumented walkway. The perturbations were applied at eight different axes of rotation combining different amounts of DP and IE rotations of four male subjects. The observed stiffness and damping were low at heel strike, increased during the mid-stance, and decreases at push-off. At heel strike, it was observed that both the damping and stiffness were larger in IE than in DP. The maximum average ankle stiffness was 5.43 N·m/rad/kg at 31% of the stance length (SL) when combining plantarflexion and inversion and the minimum average was 1.14 N·m/rad/kg at 7% of the SL when combining dorsiflexion and eversion. The maximum average ankle damping was 0.080 Nms/rad/kg at 38% of the SL when combining plantarflexion and inversion, and the minimum average was 0.016 Nms/rad/kg at 7% of the SL when combining plantarflexion and eversion. From 23% to 93% of the SL, the largest ankle stiffness and damping occurred during the combination of plantarflexion and inversion or dorsiflexion and eversion. These rotations are the resulting motion of the ankle's subtalar joint, suggesting that the role of this joint and the muscles involved in the ankle rotation are significant in the impedance modulation in both DP and IE during gait.

## Introduction

The human ankle plays a major role in locomotion as it is the first major joint to transfer the ground reaction torques to the rest of the body and provide power for locomotion and stability. One of the main causes of the ankle impedance modulation is muscle activation [1,2], which can tune the ankle's stiffness and damping during the stance phase of gait. The ankle's time-varying impedance is also task dependent, meaning that different activities such as walking at different speeds, turning, and climbing/descending stairs would impose different profiles of time-varying impedance modulation.

The mechanical impedance correlates output force to input disturbance motion. Its inverse is the admittance, which correlates output motion to input disturbance force. For simplicity, mechanical impedance may be simply referred to as impedance in this paper. The ankle impedance at low frequencies has little effect of damping and inertia and is closely representative of the ankle's stiffness [3,4]. The ankle impedance in dorsiflexion–plantarflexion (DP) at low frequencies has been one of the primary parameters in the design of active ankle-foot prostheses and lower extremity rehabilitation devices. Powered prostheses that mimic the stiffness of the human ankle while providing adequate torque during the push-off can potentially reduce secondary injuries and metabolic cost. Some of the most notable prostheses include SPARKy, a tendon driven ankle-foot prosthesis capable of generating a net positive work during walking and running [5,6]. Sup et al. developed a powered ankle and knee prosthesis with impedance control of the knee and ankle in the sagittal plane [7,8]. BiOM [9] is an ankle-foot prosthesis capable of generating net positive work during plantarflexion. Ficanha et al. recently developed an ankle-foot prosthesis with two powered degrees-of-freedom (DOF) in DP and inversion–eversion (IE) capable of injecting energy throughout the gait in those two DOF [1013]. Active IE DOF has also been considered in an ankle-foot prosthesis emulator by Collins et al. [14].

Although these prostheses can improve gait, currently there is limited information about the time-varying impedance of the ankle during the stance phase of walking in DP and no direct estimation in IE (to the best of the author's knowledge). Additionally, the literature is limited on how the ankle impedance in the sagittal and frontal planes change during different gait scenarios, such as turning or walking on slopes, compared to walking on a straight path. The aforementioned prostheses could potentially benefit from implementing the time-varying impedance of the ankle to generate a natural gait. Recently, two research groups have explored the time-varying mechanical impedance of the human ankle during straight walking on level ground. A perturbation device capable of applying perturbations to the ankle in the sagittal plane, named Perturberator, was developed by Rouse et al. [15]. The device applied perturbations to the ankle in the DP direction at four distinct points of the stance phase from 100 ms to 475 ms after heel-strike, showing an increase in ankle stiffness from 1 N·m/rad/kg to 4.6 N·m/rad/kg, and an increase in damping from 0.012 Nms/rad/kg to 0.038 N·m/rad/kg in the same range [1]. Lee et al. estimated the time-varying mechanical impedance of the ankle in 2DOF during swing phase in DP and IE during walking on a treadmill. The ankle was continually perturbed by a wearable robot. Their experiment resulted in the estimation of the ankle impedance during the late stance (toe-off), swing, and early stance (heel strike) of the gait, but did not evaluate the impedance during the mid-stance. The results showed great variability of the ankle impedance in both DOF during the evaluated phases of straight path walking. Other related research includes a platform developed to apply horizontal translations to the angle to investigate the phase-dependent behavior in human locomotion [16], and extensive work has being done on system identification of human joints by Kearney et al. [17,18] and Van der Helm et al. [19].

This paper describes the results of experiments using the vibrating platform on four male subjects. For the first time, the time-varying impedance of the human ankle in both DP and IE during walking in a straight line is reported.

## Vibrating Platform

A vibrating platform and instrumented walkway was previously introduced by Ficanha et al. [20]. The device has two DOF to estimate the ankle's mechanical impedance in DP and IE during the stance of walking and consists of a vibrating platform installed in a walkway (Fig. 1). The vibrating platform consists of two independent modules connected by Bowden cables. It contains two voice coil actuators that can generate ±351.3 N of force at 10% duty cycle each, and have 63.5 mm stroke length. A force plate is mounted on the frame of the platform and it rotates using a universal joint, allowing the force plate to apply perturbations to the ankle in DP and IE. A motion capture system consisting of 8 Prime 17W OptiTrack cameras was mounted in a square formation covering a volume of about 9 m3 and an area of 9 m2. The camera system is used to record the foot and shin motion required for calculating the ankle angles and center of rotation. The ankle torque was calculated based on the center of pressure of the foot on the surface of the force plate, the ankle center of rotation, and the ground reaction forces in the foot coordinate system. A detailed description of the method used to calculate the ankle torques was presented in previous work [20]. A quasi-static experiment was conducted to validate the instrumented walkway and vibrating platform independently of the force plate or camera system measurements. This experiment showed relative errors of 1.68% in DP and 0.54% in IE [20].

## Methods

Four male volunteers with no self-reported neuromuscular or biomechanical disorder gave written consent to participate in the experiment, which was approved by the Michigan Tech's Institutional Review Board. The volunteers ranged from 25 to 31 years of age. The experiments consisted of applying ramp perturbations to the ankle with an average magnitude of 1.9 deg at a random instance of the stance length (SL) while the subjects walk on the vibrating platform. The magnitude of the perturbations is consistent with previous work, which used 2 deg perturbations [1]. Each perturbation was applied to 1 of 8 different axes of rotation, with 22.5 deg increments between axes, to produce rotation in both DP and IE directions. In addition, the direction of the perturbation was random, generating random combinations of positive and negative DP and IE angles, where positive perturbations were considered perturbations inducing rotation in dorsiflexion and inversion. This resulted in 16 possible types of perturbation. For each perturbation type, 100 steps were recorded, resulting in 1600 steps for each test subject. The experiments were conducted in blocks of 100 steps over two days, and the subjects could rest in between each block of experiments for as long as they needed.

During the tests, the subjects walked in a straight line for three steps, and placed the fourth step on top of the force plate module, where they received the random perturbation, and continued walking for three more steps. Next, they turned around and performed the same steps in the reverse direction. The ankle angles were calculated based on the motion of the shin and foot, which were recorded using the motion capture camera system. In addition, the ground reaction forces were recorded from the force plate module of the vibrating platform. All data were recorded at 250 Hz. Based on the measured forces and the ankle' center of rotation, the torques were calculated.

The estimated torques (τE) are a combination of the torques generated by the acceleration of the force plate (τFP) and the torques generated by the foot and ankle joint (τA). The torques due to the inertia of the force plate were calculated as the product of the force plate acceleration and its inertia, and the resultant torques were rotated to the foot coordinate system. The ankle torques were obtained by subtracting torques due to the inertia of the force plate from the estimated torques
$τA=τM−τFP$
(1)

The recorded data are the sum of ankle angles and torques due to normal walking and the ankle angles and torques due to the perturbations. In the impedance estimation, only the angles and torques due to the perturbations are required because the angles and torques due to walking are closely repeated at each step. By averaging all the recorded angles and torques for a specific axis of rotation and removing these averages from each perturbed step, the angles and torques due to the perturbations for each step were obtained. This is possible because the angles and torques due to the perturbations are uncorrelated, and therefore they average to zero.

Next, the data of each step were subdivided with a rolling window of 25 samples (100 ms window) with 20 samples overlap. This generated vectors of data for the ankle torque τ and ankle angles θ that are 25 samples long. A second-order parametric model used the vectors τ and θ to solve for the impedance parameters J (inertia), B (damping), and K (stiffness), as described in Eq. (2). The obtained impedance parameters are representative for that specific pair of vectors τ and θ and the first and second derivatives of the vector $θ$ ($θ˙$ and $θ¨$, respectively). These resulted in 36 estimated impedance parameters for each step, which were averaged among all the 100 steps for each perturbation type
$τ=θ¨J+θ˙B+θK$
(2)

## Results

The impedance parameters of the four human subjects and 16 perturbation types were normalized with respect to their body mass, and the results were averaged across subjects. From the obtained impedance, the data were segmented into 12 points in time from 7% of the SL to 93% of the SL. The plots were normalized with respect to the SL as each step showed small variations in length. Time normalization on similar experiments has been reported as less than 3% [2]. The contour plots of the averaged stiffens and damping are presented in Figs. 2 and 3 for each of the 12 points in time of the SL.

## Discussion

This paper describes a method for the estimation of the multidirectional time-varying ankle impedance in the frontal and sagittal planes during the stance phase of gait. As shown in Figs. 2 and 3, the ankle stiffness and damping changes continually during the gait cycle. Overall, the stiffness is low at heel strike, increases during the mid-stance, and decreases again at push-off. The maximum average ankle stiffness was 5.43 N·m/rad/kg at 31% of the SL when combining plantarflexion and inversion and the minimum average stiffness was 1.14 N·m/rad/kg at 7% of the SL when combining dorsiflexion and eversion. The ankle damping follows a similar pattern with low values at heel strike, increased values during the mid-stance, and decreases again at push-off. The maximum average ankle damping was 0.080 Nms/rad/kg at 38% of the SL when combining plantarflexion and inversion and the minimum average damping was 0.016 Nms/rad/kg at 7% of the SL when combining plantarflexion and eversion. Interestingly, after heel strike (7% SL), both the damping and stiffness are larger in IE than DP. This mechanism may be needed to stabilizing the gait in IE while remaining compliant to absorb the impact in DP. After heel strike, from 15% to 23% of the SL, the largest stiffness and damping progressively rotate to an axis of rotation combining plantarflexion and inversion and combining dorsiflexion and eversion, respectively. Afterward, specifically from 23% to 93% of the SL, the largest ankle stiffness and damping were at either a combined plantarflexion and inversion or at a combined dorsiflexion and eversion. For the ankle to generate this motion, it needs to rotate about an axis of rotation, which was not aligned with the anatomical axis. In fact, rotations combining plantarflexion and inversion or dorsiflexion and eversion are the results of the rotations of the ankle's subtalar joint [21], indicating the importance of this joint and the muscles involved in its rotation in the impedance modulation in both DP and IE during gait.

Previous literature has reported stiffness values for combined dorsiflexion and plantarflexion of 1.0 ± 0.6 N·m/rad/kg and 4.6 ± 1.3 N·m/rad/kg at 0.1 s and 0.48 s of the SL, respectively [1]. The work presented in this paper showed comparable stiffness values with 1.9 ± 0.4 N·m/rad/kg and 3.4 ± 0.7 N·m/rad/kg at the same instances of the SL when averaging dorsiflexion and plantarflexion stiffness values. The previously reported dampings were 0.01 ± 0.01 Nms/rad/kg at 0.1 s of the SL and 0.04 ± 0.02 Nms/rad/kg at 0.48 s of the SL [1]. The results presented in this paper, when averaging the damping in dorsiflexion and plantarflexion at the same instances of the SL, showed comparable values of 0.03 ± 0.01 Nms/rad/kg and 0.05 ± 0.01 Nms/rad/kg. Currently, the ankle impedance in IE, or the combination of DP and IE, during the stance phase has not been reported.

## Conclusions

This paper described the estimation of the time-varying impedance of the human ankle in both DP and IE during walking in a straight line. The observed stiffness and damping were low at heel strike, increased during the mid-stance, and decreases again at push-off. At heel strike, it was observed that both the damping and stiffness were larger in IE than DP, which may be needed to stabilize the gait in IE while remaining compliant to absorb the impact in DP. From 23% to 93% of the SL, the largest ankle stiffness and damping were at an axis combining plantarflexion and inversion or rotations combing dorsiflexion and eversion. These rotations are results of the rotations of the ankle's subtalar joint, suggesting that the role of this joint and the muscles involved in its rotation in the impedance modulation in both DP and IE during gait is significant. The presented results can be used in the control strategies of powered prosthesis with variable impedance in DP and/or IE. Future research will include different types of gait, including but not limited to different types of turning steps, walking on incline planes, stepping on different ground profiles, climbing/descending stairs, and walking while carrying different types of loads. In addition, the vibrating platform will be used to develop and tune ankle-foot prostheses. The vibrating platform and the developed method for time-varying impedance estimation can be used to test and tune powered prostheses to assure they can mimic the human ankle's time-varying angles, torques, and impedance.

## Funding Data

• National Science Foundation (Grant No. 1350154).

## Nomenclature

• B =

damping

•
• DP =

dorsiflexion–plantarflexion

•
• IE =

inversion–eversion

•
• J =

inertia

•
• K =

stiffness

•
• SL =

stance length

•
• θ =

vector of ankle angles

•
• $θ˙$ =

vector of ankle velocities

•
• $θ¨$ =

vector of ankle accelerations

•
• τ =

vector of ankle torques

•
• τA =

torque generated by the ankle joint

•
• τE =

estimated torques

•
• τFP =

torque generated by the acceleration of the force plate

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