Haptic Human-Human Interaction During an Ankle Tracking Task: Effects of Virtual Connection Stiffness

While treating sensorimotor impairments, a therapist may provide physical assistance by guiding their patient’s limb to teach a desired movement. In this scenario, a key aspect is the compliance of the interaction, as the therapist can provide subtle cues or impose a movement as demonstration. One approach to studying these interactions involves haptically connecting two individuals through robotic interfaces. Upper-limb studies have shown that pairs of connected individuals estimate one another’s goals during tracking tasks by exchanging haptic information, resulting in improved performance dependent on the ability of one’s partner and the stiffness of the virtual connection. In this study, our goal was to investigate whether these findings generalize to the lower limb during an ankle tracking task. Pairs of healthy participants (i.e., dyads) independently tracked target trajectories with and without connections rendered between two ankle robots. We tested the effects of connection stiffness as well as visual noise to manipulate the correlation of tracking errors between partners. In our analysis, we compared changes in task performance across conditions while tracking with and without the connection. We found that tracking improvements while connected increased with connection stiffness, favoring the worse partner in the dyad during hard connections. We modeled the interaction as three springs in series, considering the stiffness of the connection and each partners’ ankle, to show that improvements were likely due to a cancellation of random tracking errors between partners. These results suggest a simplified mechanism of improvements compared to what has been reported during upper-limb dyadic tracking.

individuals through robotic interfaces.Upper-limb studies have shown that pairs of connected individuals estimate one another's goals during tracking tasks by exchanging haptic information, resulting in improved performance dependent on the ability of one's partner and the stiffness of the virtual connection.In this study, our goal was to investigate whether these findings generalize to the lower limb during an ankle tracking task.Pairs of healthy participants (i.e., dyads) independently tracked target trajectories with and without connections rendered between two ankle robots.We tested the effects of connection stiffness as well as visual noise to manipulate the correlation of tracking errors between partners.In our analysis, we compared changes in task performance across conditions while tracking with and without the connection.We found that tracking improvements while connected increased with connection stiffness, favoring the worse partner in the dyad during hard connections.We modeled the interaction as three springs in series, considering the stiffness of the connection and each partners' ankle, to show that improvements were likely due to a cancellation of random tracking errors between partners.These results suggest a simplified mechanism of improvements compared to what has been reported during upper-limb dyadic tracking.Index Terms-Stiffness, tracking performance, haptic rendering.

I. INTRODUCTION
P HYSICAL interaction is an important aspect of daily life, allowing humans to assist and learn from one another through the exchange of haptic information [1].During rehabilitation, for instance, a physiotherapist may interact with their patient to stimulate (re)learning of motor tasks through physical assistance or cues (e.g., supporting paretic limb during swing phase of walking).This interaction allows the therapist to both feel and respond to their patient's movements, adjusting the amount of guidance based on the intended outcome.
To understand these interactions and their benefits, it is necessary to monitor the forces exchanged between individuals and corresponding changes in motor performance.To this end, various aspects of human-human physical interaction have been studied using virtual haptic connections between robotic devices with one or more degrees of freedom (DoF), typically implemented as spring and damper elements between joints [2].With these systems, two individuals interface with separate manipulators and perform motor tasks independently while experiencing forces due to their partner's movements.
One scenario where human-human interaction can be beneficial is during upper-limb reaching.Studies have reported that pairs of individuals (i.e., dyads) track sinusoidal trajectories more accurately while haptically connected compared to tracking alone, and that these improvements depend on the abilities of each partner [3], [4], [5], [6].Another important factor is the stiffness of the connection.Takagi et al. showed that the worse partner in a dyad improves more as the connection stiffness increases, at the expense of greater effort exerted by the better partner [5].Through dynamic modeling, these improvements have been attributed to individuals mutually estimating their partners' tracking strategies to improve their own prediction of a target's movement [4], [5], [7].Studies have associated this mutual estimation with faster individual motor learning after the connection is removed, demonstrated during 1-DoF position tracking [8] as well as multi-DoF adaptations to visuomotor rotations [3] and force fields [9].
While many studies have reported on upper-limb physical interaction, few have focused on lower-limb tasks.It remains unclear whether changes in task performance and individual learning obtained for upper-limb movements can be generalized to the lower limb, given differences in physiology and functional tasks in which each are involved [10], [11].As a first study in healthy individuals, our group investigated a 1-DoF ankle motor task, comparing changes in tracking performance in response to compliant haptic connections with a partner [12].Our results showed no difference in shortterm, individual learning when training alone or with a partner.During connected trials, we found that tracking performance depends on the ability of one's partner, such that collaborative tracking with a better partner results in greater improvements.Simulating the connection as a system of springs in series suggested that improvements in tracking could be explained by the averaging of random errors within each dyad, depending on the stiffness of the connection and the relative ankle stiffness of each partner.One potential explanation for these results is the relatively soft stiffness of the connection used in our previous work [12], possibly limiting the communication of tracking goals as described by Takagi et al. [5].
If the performance improvements during haptic connections were simply due to error averaging within dyads, then we would expect that these improvements are sensitive to the stiffness of the connection between individuals and the amount of random error each individual generates while tracking the target.For instance, connected partners with highly correlated tracking strategies (i.e., low random error) would improve less than partners with uncorrelated strategies (i.e., high random error), due to less attenuation of random errors.To evaluate this assumption, we can manipulate the magnitude and type of errors (i.e., random or bias) between partners during tracking tasks by adding visual noise to the displayed target trajectory, creating perceptual differences in the target's location and movement [5], [7], [13].
Understanding the mechanism of improvements during dyadic tracking would inform the development of natural-istic controllers and improved rehabilitation strategies for individuals with lower-limb motor impairments (e.g., stroke) [14].We believe that the literature lacks a clear evaluation of the response to various connection stiffnesses and visual disturbances during dyadic target tracking to elucidate these mechanisms in the lower limb.Therefore, the goal of our study was to investigate the effects of virtual stiffness on task performance during a 1-DoF ankle tracking task with and without random noise added to a visually-displayed target.Based on similarities between upper-and lower-limb tracking in our initial findings [12], we hypothesized that dyadic improvements increase with connection stiffness, benefiting the worse partner in the dyad, and that these improvements are explained by the averaging of random errors across partners.We also expected that less correlated tracking errors across partners would correspond to greater improvements due to a larger amount of random error corrected by an elastic connection.To test these hypotheses, we had pairs of participants, connected to commercial ankle robots, track sinusoidal targets with and without haptic connections rendered between their devices.We analyzed task performance during tracking, then performed simulations to suggest a mechanism of the dyadic tracking behaviors in the lower limb.

A. Description of the Ankle Robots
Two ankle rehabilitation robots (M1-AnkleMotus, Fourier Intelligence, Singapore) were used in this study.These robots are designed for 1-DoF ankle exercises (dorsi-and plantarflexion) and come equipped with a sensor to measure the interaction torque between the ankle joint and robot.A custom interaction torque controller was developed to allow transparent motion (i.e., near zero interaction torque) for each device and to render virtual haptic environments between multiple devices [15].Virtual connections were implemented using rotational spring and damper elements.In connected trials, the desired interaction torque between users A and B was calculated as where λ int is the interaction torque applied to either user, θ is the ankle angular position measured by each robot, θ is the ankle angular velocity, K virt is the virtual stiffness that is applied between the ankle angles of the two users, and C virt is the virtual damping that is applied between the ankle velocities.Desired torque commands and sensor measurements including joint position, velocity, and torque data were updated at 400 Hz.

B. Experimental Protocol
We recruited 30 healthy individuals (20 females and 10 males; 27.0 ± 4.2 years) to participate in this study and paired them into age-and sex-matched dyads.Participants gave informed consent for their participation.The study protocol (registered on clinicaltrials.govas NCT04578665) was conducted in accordance with the Declaration of Helsinki and approved by the institutional review board (IRB) of Northwestern University (STU00212684).
During the experiment, each participant was seated with their right leg strapped into the ankle robot using foot and shank braces (Fig. 1A).Foot placement was adjusted so that the heel was in contact with the heel support of the pedal.The distance between the chair and the ankle robot was adjusted to allow each participant to reach the upper and lower limits of the target trajectory during the tracking trials.Dyads were seated side by side, and a physical divider was placed between them to prevent any social interaction.
To investigate changes in effort during the tracking task, we analyzed muscle activation of the ankle dorsi-and plantarflexors.Electromyography (EMG) sensors (Bagnoli, Delsys Inc., USA) were placed on the Tibialis Anterior (TA), Medial Gastrocnemius (MG), Lateral Gastrocnemius (LG), and Soleus (SOL) muscles.Recordings were sampled at 1500 Hz and synchronized across devices via analog trigger using a data acquisition board (USB-6218, National Instruments, USA) and a custom Python script.After preparation of EMG sensors, participants performed isometric maximum voluntary contractions (MVC) with the pedal of each robot fixed at an angle of 45 deg relative to the home position (i.e., foot pedal parallel to the ground as shown in Fig. 1A).This angle was consistent with the bias of our target trajectory described in the section below.Due to differences in seated posture, the ankle angle (i.e., angle between fifth metatarsal and fibula) during MVC trials varied across participants, but all participants were slightly plantarflexed while strapped to the pedal in this fixed configuration.Participants were instructed to push (plantarflexion) or pull (dorsiflexion) on the pedal as hard as possible for a period of five seconds while receiving visual feedback of their applied torque.Trials were repeated three times in each direction with 30 seconds of rest between each period of pushing or pulling.
The main tracking experiment was divided into four blocks, featuring different combinations of visual noise and connection stiffness during solo and dyad tracking trials as shown in Fig. 1B.In each block, participants performed 15 tracking trials simultaneously with their partner.The type of tracking trial alternated between solo, where the ankle robots operated in transparent mode (λ int = 0), and dyad, where a virtual connection was rendered between the two participants according to Eq. 1.In these tracking trials, participants were provided visual feedback of a target angle (red point[s]) and their own instantaneous ankle angle (blue bar) as shown in Fig. 1A.Participants were instructed to match their ankle angle to the target angle as accurately as possible while the target moved along a grey arc defining the bounds of its trajectory.The target varied according to a multi-sine function defined as where θ des is the target angle in deg at a given time point.Phase shifts (φ 1 , φ 2 , φ 3 ) were added so that the trajectory was different in each block.A time shift (t r ) was added to change the starting point of the trajectory from trial to trial.These time and phase shifts prevented memorization of the target trajectory [3].Consistent with our previous study [12], the amplitude and frequency of the three multi-sine components were chosen to emulate ankle kinematics during healthy gait [16].In each trial, participants tracked the target trajectory for 26 seconds, then rested for 10 seconds.A longer rest of 60 seconds was provided after eight trials.Participants were told that they would experience forces from their robot during some trials, but they were blinded to the nature of these forces (i.e., the virtual connection).
To evaluate the effects of visual noise, the target angle was represented in one of two ways (Fig. 1A).( 1) In blocks without Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
visual noise, the target was a single point (10 mm square) indicating the true desired angle.(2) In blocks with visual noise, the target was a "cloud" of 10 points (3 mm squares) randomly distributed about the true desired angle.Similar to [5], points in the cloud spawned every 400 ms in a circular region (radius = 10 mm) around the target.After spawning, each point travelled in a randomized direction at a constant velocity (0.1 m/s) before disappearing and respawning.The visual noise characteristics in this condition were the same for all participants, however, the randomized starting location and movement direction of each point was different across partners in a dyad, creating perceptual differences throughout the tracking trials.
During the dyad trials, we varied the connection properties depending on the block (Fig. 1A).( 1) In blocks with soft connections, the stiffness (K virt ) and damping (C virt ) were set to 20 Nm/rad and 2 Nm•s/rad, respectively, consistent with our previous study [12].(2) In blocks with hard connections, the stiffness and damping were set to 200 Nm/rad and 6.3 Nm•s/rad, respectively.The damping constants (C virt ) were selected such that the damping ratio (ζ ∝ C virt / √ K virt ) of each connection was the same.This ensured that oscillations decayed at the same speed in all conditions.
The four block types (soft without visual noise, hard without visual noise, soft with visual noise, hard with visual noise) were performed in a randomized order, always starting with a soft block (soft without visual noise or soft with visual noise).Consistent with Takagi et al. [5], we chose the soft block as the first condition to demonstrate the type of torques experienced during the dyad trials at lower magnitude.One possible block order is shown in Fig. 1B.In total, eight dyads experienced the soft block with visual noise first and seven dyads experienced the soft block without visual noise first.

C. Data Analysis
1) Task Performance: Performance during the task was evaluated using tracking error (E).This was quantified for each trial as the root-mean-square error between the target and actual ankle trajectories, where θ des is the target trajectory, θ is the actual trajectory, and N is the total number of time points.In our analysis, the first two seconds of each 26-second trial were removed to exclude errors due to initiation of the tracking task.Additionally, because we did not provide an explicit familiarization block, we excluded the first two trials of each block to account for adjustment to the dynamics of the ankle robot.We then took the average of the errors in the remaining trials to compute the mean solo (E S ) and dyad (E D ) tracking errors for each block and participant.E S was similar across the two blocks of tracking without noise (soft vs. hard block: δ = 0.12 ± 0.15 deg, t 116 = 0.6, p =0.6) and the two blocks of tracking with noise (soft vs. hard block: δ = 0.24 ± 0.15 deg, t 116 = 1.1, p = 0.3).This indicates that block order did not have a significant effect on task performance, despite all dyads starting with a soft block (with or without noise).Within each block, relative partner performances ( E P S ) were computed by taking the difference between each participant's solo tracking error (E S ) and their partner's tracking error (E P S ), normalized by E S : In Eq. 4, negative values of E P S indicate that a participant was a better performer than their partner (i.e., more accurate) while positive values indicate the participant was worse than their partner (i.e., less accurate).Dyadic improvements were computed by taking the difference between each participant's solo tracking error (E S ) and their dyad tracking error (E D ), normalized by E S : In Eq. 5, positive values of E D indicate better task performance as a result of the haptic connection while negative values indicate a degradation in performance due to the connection.Measures of relative partner performance ( E P S ) and dyadic improvement ( E D ) were computed within each block, therefore we obtained one value per condition for each participant.
2) Error Correlation: To quantify the amount of random error generated by participants when tracking with and without visual noise, we computed the correlation between tracking errors within each dyad: In this equation, r represents Pearson's correlation coefficient between a dyad's error profiles (i.e., the time series difference between target and actual trajectories).We then took an average of the error correlations during solo trials in each block to obtain a single value per condition for each dyad (r S ).
3) Effort Exerted: Effort (M) was quantified as the percentage of muscle activation relative to maximum output during MVC trials.Raw EMG data from the MVC and tracking trials were band-pass filtered to remove low-frequency motion artifacts and high-frequency noise (sixth-order Butterworth filter with a passband from 20 to 450 Hz), then notch filtered to remove power line interference (second-order IIR at 60 Hz) [17].Filtered data were rectified and smoothed using a moving average filter (1-second window).After these preprocessing steps, we used the maximum EMG value during dorsiflexion (TA) and plantarflexion (MG, LG, SOL) in the MVC trials to normalize the EMG values in the tracking trials.Finally, to obtain a single value for effort exerted per trial, we summed the normalized values across a dorsi-and plantarflexor pair: where E M G d f is the normalized value of TA activity and E M G p f is the normalized value of the most active plantarflexor (MG, LG, or SOL).The most active plantarflexor was specific to each participant and was determined by averaging the mean normalized values across all trials to find the muscle Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
with the most activity throughout the experiment.This step was performed to account for differences in plantarflexor activity which depend on the posture (i.e., knee angle) and muscle activation strategy of each participant.The paired sum of EMG values was averaged across each trial, excluding the first two seconds of each trial and first two trials per block as described in the task performance section.Within each block, changes in effort ( M D ) were computed by taking the difference between each participant's mean effort during dyad (M D ) and solo (M S ) trials, normalized by M S : In Eq. 8, positive values of M D indicate increased muscle activation (i.e., more effort) as a result of the haptic connection while negative values indicate decreased muscle activation (i.e., more relaxed).
4) Simulated Dyadic Connection: Exploring the mechanism of the changes in performance across connection stiffnesses, we simulated dyad trials by modeling the haptic interaction as three springs in series [12], where each participant's simulated angle was influenced by the physiological stiffness of their own ankle, the stiffness of the virtual spring (K virt ), and the physiological stiffness of their partner's ankle.We compute the relative displacement of one spring to another, simulating the mechanical effect of the virtual connection on each user's intended motion.Similarities between our simulated and experimental results would suggest that changes in dyadic performance were due to the elastic properties of the virtual connection as opposed to visuo-haptic sensing of partner goals [5].We used each dyad's actual trajectories during solo trials (i.e., no connection) to represent their intended motion in the three-spring system and balanced the torques across each spring: where θ sim is the simulated trajectory of each partner, K is the physiological ankle stiffness of each partner, and θ is each partner's actual trajectory during solo tracking trials.Simulated trajectories were calculated by solving for θ sim in the following: Because we did not measure instantaneous ankle stiffness during the tracking trials, we estimated values of K based on the finding that stiffness increases linearly with torque production during isometric ankle contractions [18], [19].Specifically, we used the mean model coefficients from Kearney et al. [18] which describe the linear relationship between torque and ankle stiffness during dorsi-and plantarflexion (slope = 10.8 rad −1 , intercept = 35.3Nm/rad).To obtain estimates of ankle stiffness for each participant, we used the mean absolute interaction torque during dyad trials as an input to the linear relationship described above.For each block, subject-specific estimates of K (range: [38.4 to 54.0 Nm/rad]), virtual connection stiffness (soft: K virt = 20 Nm/rad; hard: K virt = 200 Nm/rad), and the solo trajectories within each dyad were used in Eq. 10 to generate the simulated trajectories.For the simulated dyad trials, tracking errors between the target and simulated trajectories were calculated using Eq.3; changes in performance during simulated trials ( E A ) were computed relative to solo trials as defined in Eq. 5.
5) Statistical Analysis: The goal of this study was to assess how task performance improvements and exerted effort change during dyadic ankle tracking under different connection stiffnesses and visual noise conditions.We focused on three hypotheses for our experimental results: (1) task performance improvements will be dependent on partner ability and increase with connection stiffness, (2) improvements will be greater when partners track targets with visual noise, due to less correlated errors caused by perceptual differences, and (3) better partners will exert increased effort compared to worse partners as a response to perceived disturbances during dyad trials.Additionally, we used simulated connected trials to determine whether task performance improvements can be explained by the averaging of random errors between partners.
To test our hypotheses related to changes in task performance, we used multiple linear regression models with the tracking improvement in the dyad trials ( E D or E A ) as a dependent variable, visual noise and connection stiffness as categorical variables, and difference in partner performances ( E P S ) as a continuous variable, as well as the interaction between each of these predictors.A three-way ANOVA was used to evaluate the changes in exerted effort by better ( E P S < 0%) and worse partners ( E P S > 0%) across each of the visual noise and connection stiffness conditions.Onesample t-tests were conducted to determine whether changes in effort ( M D ) were different from zero (i.e., exerted effort in dyad trials different than solo) for each condition.The DoF of the multiple regression models were estimated using a Satterthwaite approximation [20].Significance was set to 0.05 for all hypotheses related to dyadic improvements.A Holm-Bonferroni correction [21] with a significance level of 0.05 was applied for the one-sample t-tests related to changes in exerted effort.Results are presented as mean ± SE, unless otherwise specified.

A. Dyadic Improvements Scale With Connection Stiffness
Improvements were linearly related to partner ability when tracking across connection stiffnesses and visual noise conditions (R 2 = 0.69; Fig. 2).This means that tracking with a better partner resulted in lower tracking errors during the dyad trials compared to solo trials.There was a significant difference in slopes between dyadic improvement rates comparing soft and hard connections without (δ = 0.13 ± 0.05, t 112 = 2.1, p < 0.05) and with (δ = 0.20 ± 0.05, t 112 = 2.6, p < 0.05) visual noise.This increase in slopes during hard connections resulted in additional improvements for the worse partner compared to soft connections regardless of the visual noise condition.Comparing results across visual noise conditions (Fig. 3), we found that the magnitude of dyadic improvements was lower when tracking with visual noise.There was a significant decrease in intercepts of dyadic improvements comparing without and with visual noise during soft (δ = −3.0 ± 0.7%, t 112 = −3.0,p < 0.01) and hard (δ = −4.1 ± 0.7%, t 112 = −4.0,p < 0.001) connections.

B. No Change in Correlation During Visual Noise Tracking
Tracking targets with visual noise increased the magnitude of tracking errors, but not the correlation of errors across partners.Comparing tracking performance during solo trials (E S ), we found that the mean error was 5.7 ± 0.8 deg (mean ± SD) when tracking without visual noise and 7.8 ± 0.8 deg with visual noise, indicating worse performance with visual noise (t 29 = 23.1, p < 0.001).However, there was no significant difference in mean error correlation (r S ) between visual conditions (t 14 = −0.4,p = 0.7).Pearson's correlation coefficients were 0.59 ± 0.07 without noise and 0.58 ± 0.07 with noise, suggesting that the change in random error when tracking with visual noise did not substantially influence the correlation of partner errors.Therefore, we cannot attribute the changes in dyadic performance with and without visual noise, reported in the previous section, to differences in correlated strategies.

C. Increase in Muscle Activation for the Better Partner
During dyad trials, we observed more effort exerted by the better partner in each dyad, indicated by higher levels of muscle activation compared to the worse partner (Fig. 4).The three-way ANOVA which quantified the effects of partner ability (better: E P S < 0%, worse: E P S > 0%), connection stiffness, and visual noise on exerted effort determined that there was a significant effect of partner ability (F 1,112 = 11.3,p < 0.01).On average, the change in effort for better partners during dyad trials was 9.9 ± 1.8%, compared to −0.9 ± 2.7% for worse partners.This increase in effort for better partners was significantly different from solo trials for all conditions except for tracking without visual noise during soft connections.When tracking with visual noise, mean effort of the better partners increased during soft (10.3 ± 2.7%, t 14 = 3.8, p < 0.01) and hard connections (13.5 ± 4.2%, t 14 = 3.2, p < 0.01).When tracking without visual noise, mean effort of the better partner also increased during hard connections (12.7 ± 4.4%, t 14 = 2.9, t 14 < 0.05), however these changes did not reach significance for soft connections (3.3 ± 2.1%, t 14 = 1.6, p = 0.14).For the worse partner, changes in effort were not significantly different from zero across all conditions, indicating that, on average, the worse partner in the dyad did not change their muscle activation strategy in response to the haptic connection.

D. Improvements Explained by Error Averaging
To explain the mechanism of the improvements during dyadic ankle tracking across connection stiffnesses, we modeled the interaction between partners as a system of springs described in our previous work [12].We found similarities when comparing relative partner performances ( E P S ) to improvements observed as a result of simulating the haptic interaction within each dyad ( E A ) and the experimental dyadic improvements ( E D ) (Fig. 5).Simulated and experimental improvements were described by linear fits with E P S , connection stiffness, and visual noise as the predictors (R 2 = 0.75).In the simulated case, consistent with our experimental results, there was a significant increase in the slope of dyadic improvement rates comparing soft and hard connections without (δ = 0.19 ± 0.04, t 224 = 3.7, p < 0.001) and with (δ = 0.19 ± 0.04, t 224 = 3.0, p < 0.01) visual noise.
Though we did not find a difference in error correlations when tracking without and with visual noise, we observed a distribution of solo error correlations for both visual conditions (range: [0.4 to 0.8]).As our simulated results used solo trials from which these correlations were calculated, we performed an additional analysis to determine how correlated errors across partners affect simulated dyadic improvements ( E A ) (Fig. 6).We found the error correlation had a significant negative relationship with the simulated dyadic improvements (−20.6 ± 0.6%/r S , t 57 = −32.3,p < 0.001), such that partners who had less correlated errors experienced greater improvements than more correlated partners.At the extremes of our dataset, this implies that partners who had a mean error correlation of 0.4 would improve during soft connected trials by an additional 4.6% (added to their changes due to partner ability) while partners with a correlation of 0.8 would get worse by 3.7%.

IV. DISCUSSION
In this study, we investigated how the stiffness of haptic connections between pairs of healthy individuals influences dyadic tracking performance.To the best of our knowledge, this was the first study to investigate such behaviors in the lower limb, comparing changes in performance and exerted effort across two stiffnesses (soft and hard) and visual feedback conditions (without and with visual noise) during unconnected and connected trials of 1-DoF ankle tracking.We found that the better partner in each dyad exerted increased effort during the interaction, independent of the connection stiffness or visual condition.Furthermore, we showed that changes in task performance depended on the ability of one's partner, and that stronger (i.e., hard) connections increased the slope of these changes.Finally, we performed simulations using unconnected tracking trials to suggest that changes in tracking performance can be explained by the averaging of random errors across partners, relative to the virtual connection stiffness, and the physiological stiffness of each partner's ankle.
In agreement with upper-and lower-limb findings, we found that improvements in dyadic tracking performance depend on the ability of one's partner [3], [4], [5], [6], [12].While it is still unclear whether haptic dyadic training can improve the rate of individual learning [6], this finding of improved dyadic performance is well-established during trajectory tracking tasks, regardless of differences in robotic devices, virtual stiffnesses, and task parameters used by each group.Consistent with the findings of Takagi et al. during 1-DoF wrist tracking  [5] and Beckers et al. during 2-DoF upper-limb tracking [6], we observed that changes in task performance scale with the stiffness of the virtual connection between individuals, favoring the worse partner in each dyad during stiffer connections.
Exploring the mechanism of changes in dyadic task performance, we found that improvements during both soft and hard connected trials were similar to those obtained by simulating the interaction as a system of three springs in series.Our simulation was analogous to taking a weighted average of two partners' trajectories during unconnected trials, suggesting that performance improvements were due to the cancellation of random errors caused by the mechanics of an elastic connection.During hard connections, the virtual stiffness was three to four times larger than the assumed ankle stiffness of each partner.This likely resulted in greater attenuation of random errors for the worse partner in the dyad, while introducing random disturbances to the better partner.These findings confirm our previous work on dyadic ankle tracking during soft, compliant connections [12] and show that the same mechanism can be used to explain interactions where dyads are more rigidly connected.
Despite the agreement between our experimental and simulated results, these comparisons rely on the assumption that ankle stiffness remains constant during the tracking task and can be approximated by the stiffness of the joint during isometric, sub-maximal contractions [18].For our tracking task, we argue this assumption is more appropriate than assuming the passive stiffness of the ankle [12], as our participants produced and experienced significant torques during connected trials.However, we acknowledge this approximation of ankle stiffness is limited as joint stiffness can be lower during dynamic movement compared to postural tasks [22].It is also possible that unaccounted changes in stiffness due to antagonist and agonist muscle activity [23] may explain some variability between our simulated and experimental fits.To address this limitation, our model could be improved by estimating ankle stiffness through kinematic responses to torque perturbations during the tracking task [24].According to our simulated connection, the ankle stiffness of each partner can influence the effect of error averaging, such that improvements can be augmented or reduced if one or both partners are stiffer or more relaxed.A continuous estimate of stiffness would allow us to characterize these behaviors during tracking, as well as explore changes in ankle stiffness in response to experimental variables (i.e., connection stiffness, partner ability).
Overall, similarities between our experimental and simulated results suggest a mechanism for dyadic improvements that is fundamentally different from the "interpersonal goal integration" or "neuromechanical goal sharing" models proposed for upper-limb dyadic behaviors [4], [5], [7].These models assume that partners mutually estimate one another's strategies while haptically connected to improve their estimation of a target's movement.Our simulation suggests that partners track targets independently whether they are connected or not; any tracking improvements are due to the corrective forces from the virtual connection.Further work is needed to determine how well these upper-limb models explain our lower-limb experimental results, and whether the error averaging mechanism proposed here generalizes to upper-limb tasks.
If there are two distinct mechanisms which explain dyadic improvements during tracking tasks, this could reflect differences in the functional role of upper-and lower-limb movements.While the upper limb is typically involved in goal-directed activities like reaching and grasping, the lower limb performs semi-automatic, rhythmic behaviors like stepping.These differences in behavior are supported by differences in neural control.Motor neurons of the forelimb in non-human primates receive greater monosynaptic input compared to the hindlimb [10], contributing to higher resolution descending control.This is consistent with findings in humans that the cortical control of wrist movements is more lateralized than for movements at the ankle [11].In contrast to the high-resolution cortical control of wrist and hand function, it is well known that spinal networks are essential for locomotion [25], and that their function tends to be less mutable over short-term periods than cortical networks [26], [27].In the context of dyadic wrist tracking, rigidly connected partners exhibit lower levels of muscle co-contraction when tracking targets with visual noise, suggesting that less accurate visual information causes increased reliance on the haptic guidance of a partner [13].In our ankle study, we found no difference in co-contraction levels across visual noise conditions during hard connections, regardless of partner ability (Fig. 7).In addition, we found less distinct changes in exerted effort across different connection stiffnesses compared to Takagi et al., as they reported effort was linearly related to partner ability and significantly increased from soft to hard connections during wrist tracking [5].A comparative study with identical protocols for the wrist and ankle could elucidate whether these findings are due to physiology or simply experimental discrepancies.
In our previous work, we found no difference in the rate of individual learning when participants trained with and without compliant connections to a partner [12].We posited that the lack of difference in learning rates could be due to the relatively soft connection stiffness implemented, potentially limiting the amount of haptic information that could be communicated between partners [5].Our current findings suggest that increasing the connection stiffness during dyadic ankle tracking does not change the mechanism of improvements via error averaging; we expect similar learning rates whether training alone or with a partner under soft or hard connections.
Studies on robot-mediated training in healthy individuals have suggested that reducing tracking errors through haptic guidance is not an effective strategy to accelerate learning of motor tasks [28], given the importance of error adaptation in motor learning [29].Though haptic guidance may not be effective for healthy individuals, it could be beneficial in the early stages of robotic therapy for individuals with sensorimotor impairments (e.g., stroke).In the context of dyadic interactions, reduction of random errors through virtual haptic connections can have a similar effect to assist-as-needed strategies [30], allowing patients to learn tasks with reduced noise in their motor output due to impaired motor control.Targeting motor control of the ankle is one approach to lower-limb rehabilitation [31], [32], but it is also interesting to consider the effects of dyadic interaction on more complex, multi-joint tasks such as walking or balancing.These tasks are particularly relevant in locomotor recovery, and it is an open question how individuals perceive feedback from a partner when additional loads are present (e.g., contact between the foot and ground).

V. CONCLUSION
During a continuous ankle motor task, we found that tracking performance while haptically connected to a partner depends on the stiffness of the virtual connection and the relative abilities of each partner.Performance improvements were likely dictated by the mechanics of the interaction, including the stiffness of the virtual connection and each individuals' ankle.The resulting mechanism averages and attenuates random errors across partners, resulting in improvements in tracking performance unless paired with a considerably worse partner.Future work should investigate individual estimates of ankle stiffness during dyadic tracking to accurately characterize the proposed mechanism, as well as comparisons with upper-limb models which suggest that partners improve through the haptic exchange of goals as opposed to error Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
averaging.If the mechanism of error averaging is the primary source of improvements during dyadic tracking, this is not a recommended strategy for enhancing learning of 1-DoF lowerlimb tasks in healthy individuals.However, different responses could be obtained in individuals with lower-limb sensorimotor impairments who may benefit from assistive interventions early in rehabilitation.

APPENDIX
In addition to changes in effort across conditions, we computed changes in co-contraction across antagonist-agonist muscle pairs as a supplementary measure of tracking strategies (Fig. 7).This was calculated for each trial using the normalized activation of the dorsi-and plantarflexor pair, where E M G low and E M G high were chosen based on which muscle of the pair was more active [33].This distinction was determined at each time point, comparing the normalized activation of the TA and most active plantarflexor (MG, LG or SOL).For each participant, co-contraction values were averaged across solo (CC I S ) and dyad (CC I D ) trials within blocks.

Fig. 1 .
Fig. 1.Experimental setup and block design.(A) Representation of visual feedback (multi-sine trajectories without and with visual noise added to the target) and virtual connection (stiffness varying between soft and hard) conditions during the tracking task.(B) Example order of the four experimental blocks of tracking trials.The first block presented was always soft (either with or without noise), while the order of the remaining blocks was randomized.Each block consisted of 15 trials (26 second duration) alternating between solo (transparent) and dyad (virtual connection).

Fig. 2 .
Fig. 2. The slope of dyadic improvements relative to partner ability increased from soft to hard connections.Left panels show the mean improvements (∆E D = 100 × (E S − E D )/E S ) plot relative to partner performances (∆E P S = 100 × (E S − E P S )/E S ).Each point indicates the average of a single participant's improvements during the associated condition.Right panels show the linear model results for dyadic improvements relative to partner performance, connection stiffness, and visual noise.Top panels compare soft and hard connections without visual noise (Soft and Hard) while bottom panels compare soft and hard connections with visual noise (Soft-Noise and Hard-Noise).

Fig. 3 .
Fig. 3.The magnitude of dyadic improvements relative to partner ability decreased when tracking with visual noise.Left panels show the mean improvements (∆E D = 100 × (E S − E D )/E S ) plot relative to partner performances (∆E P S = 100 × (E S − E P S )/E S ).Each point indicates the average of a single participant's improvements during the associated condition.Right panels show the linear model results for dyadic improvements relative to partner performance, connection stiffness, and visual noise.Top panels compare soft connections without and with visual noise (Soft and Soft-Noise) while bottom panels compare hard connections without and with visual noise (Hard and Hard-Noise).

Fig. 4 .
Fig. 4. Better partners in the dyad exerted greater effort during dyad trials compared to worse partners.Top panel shows changes in effort during dyad trials (∆M D = 100 × (M D − M S )/M S ) separated between better (∆E P S < 0%) and worse partners (∆E P S > 0%) and averaged across all participants (mean ± SE). * indicates significant effect of partner ability on exerted effort obtained from ANOVA results.Bottom panel shows boxplots of individual changes in effort separated by partner ability, connection stiffness, and visual noise condition.∧ indicates significant difference after Holm-Bonferroni correction for one-sample t-test.Crosses represent outliers.

Fig. 5 .
Fig. 5.Improvements were attributed to error averaging due to virtual spring connection.Left panels show the mean improvements plot relative to partner performances (∆E P S ) for experimental (filled points: ∆E D ) and simulated trials (unfilled points:∆E A = 100 × (E S − E A )/E S ).Each point indicates the average of a single participant's experimental or simulated improvements during the associated condition.Right panels show the linear model results for dyadic improvements relative to partner performance, connection stiffness, and visual noise for experimental (solid lines) and simulated improvements (dashed lines).Top panels compare simulated and experimental results during soft and hard connections without visual noise while bottom panels show soft and hard connections with visual noise.

Fig. 6 .
Fig. 6.Simulated dyadic improvements decreased as errors across partners became more correlated.Left panel shows the residuals of the model used to describe simulated dyadic improvements (∆E A ) as a function of partner performance (∆E P S ) for soft connections.Residuals are plot relative to mean error correlations for each dyad (r S ).Visual noise conditions were grouped in this model as noise did not have a significant effect on simulated dyadic improvements.Right panel shows the linear model results for simulated improvements explained by r S .Predictions and confidence intervals were obtained by subtracting the predictions of two models with and without error correlation as a predictor.

Fig. 7 .
Fig. 7. Better partners increased their co-contraction index (CCI) more than worse partners.Top panel shows changes in CCI during dyad trials (∆CCI D = 100 × (CCI D − CCI S )/CCI S ) separated between better (∆E P S < 0%) and worse partners (∆E P S > 0%) and averaged across all participants (mean ± SE). * indicates significant effect of partner ability on CCI obtained from ANOVA results.Bottom panel shows boxplots of individual changes in CCI separated by partner ability, connection stiffness, and visual noise condition.∧ indicates significant difference after Holm-Bonferroni correction for one-sample t-test.Crosses represent outliers.