Mahdiar Edraki
email: edraki.m@northeastern.edu

Pauline Maurice
email: pauline.maurice@loria.fr

Dagmar Sternad
email: d.sternad@northeastern.edu

Humans Need Augmented Feedback to Physically Track Non-Biological Robot Movements

Keywords: Physical Human-Robot Interaction, Human Factors and Human-in-the-Loop, Human-Centered Robotics

An important component for the effective collaboration of humans with robots is the compatibility of their movements, especially when humans physically collaborate with a robot partner. Following previous findings that humans interact more seamlessly with a robot that moves with humanlike or biological velocity profiles, this study examined whether humans can adapt to a robot that violates human signatures. The specific focus was on the role of extensive practice and realtime augmented feedback. Six groups of participants physically tracked a robot tracing an ellipse with profiles where velocity scaled with the curvature of the path in biological and nonbiological ways, while instructed to minimize the interaction force with the robot. Three of the 6 groups received real-time visual feedback about their force error. Results showed that with 3 daily practice sessions, when given feedback about their force errors, humans could decrease their interaction forces when the robot's trajectory violated human-like velocity patterns. Conversely, when augmented feedback was not provided, there were no improvements despite this extensive practice. The biological profile showed no improvements, even with feedback, indicating that the (non-zero) force had already reached a floor level. These findings highlight the importance of biological robot trajectories and augmented feedback to guide humans to adapt to non-biological movements in physical human-robot interaction. These results have implications on various fields of robotics, such as surgical applications and collaborative robots for industry.

I. INTRODUCTION

Robots are transitioning from operating in isolated rooms to working in close collaboration with humans [START_REF] Abou Allaban | A systematic review of robotics research in support of in-home care for older adults[END_REF] [START_REF] Pervez | Safe physical human robot interactionpast, present and future[END_REF]. Humanrobot interaction (HRI) introduces unique challenges for planning and control of robots to ensure the safety and comfort of the human partner while enhancing the overall task efficiency [START_REF] Admoni | Editorial: Special issue on human-robot interaction[END_REF] [START_REF] Agostino De Santis | An atlas of physical human-robot interaction[END_REF]. Optimal action planning [START_REF] Kelsey | Probabilistic human action prediction and wait-sensitive planning for responsive human-robot collaboration[END_REF] [START_REF] Sadrfaridpour | Collaborative assembly in hybrid manufacturing cells: An integrated framework for human-robot interaction[END_REF], human intent recognition [START_REF] Przemyslaw | Analyzing the effects of human-aware motion planning on close-proximity human-robot collaboration[END_REF][8] [START_REF] Teja | Bi-directional navigation intent communication using spatial augmented reality and eye-tracking glasses for improved safety in human-robot interaction[END_REF], and collision avoidance [START_REF] Woo | A temporal bayesian network with application to design of a proactive robotic assistant[END_REF] [START_REF] Ding | With proximity servoing towards safe human-robotinteraction[END_REF] [START_REF] Nascimento | Collision avoidance in human-robot interaction using kinect vision system combined with robot's model and data[END_REF] are among the important challenges that need to be addressed. Physical human-robot interaction (pHRI), where the human and the robot work in direct contact with one another, adds even more complexity because it tightly couples the heterogeneous motor abilities of the two partners.

In pHRI, the interaction can be limited to a few seconds as in object handover [START_REF] Rosenberger | Object-independent human-to-robot handovers using real time robotic vision[END_REF] or it can be present throughout the entire task, e.g., collaboratively carrying an object [START_REF] Ikeura | Optimal variable impedance control for a robot and its application to lifting an object with a human[END_REF]. In either case, the movements of the human and the robot directly impact the task execution. The interaction can be enhanced either by relying on the learning capabilities of humans to adjust their behavior to the robot, or by adapting the robot to move in ways that are intuitive for the human [START_REF] Wesley P Chan | A human-inspired object handover controller[END_REF].

In the robotics literature, intuitiveness of a movement is often tied to its human-like aspects. To characterize trajectories as human-like, one must turn to human motor control studies that have tried to identify the core signatures present in human movement. Examining hand or end-effector trajectories, some robust features have been revealed: most notably the speedaccuracy trade-off in pointing tasks [START_REF] Paul | The information capacity of the human motor system in controlling the amplitude of movement[END_REF], [START_REF] Plamondon | Speed/accuracy tradeoffs in target-directed movements[END_REF] and the velocitycurvature relationship (two-thirds power law) in continuous trajectories [START_REF] Lacquaniti | The law relating the kinematic and figural aspects of drawing movements[END_REF], [START_REF] Viviani | A developmental study of the relationship between geometry and kinematics in drawing movements[END_REF], [START_REF] Schaal | Origins and violations of the 2/3 power law in rhythmic three-dimensional arm movements[END_REF]. These kinematic features have been replicated using optimal control with a variety of cost functions, such as minimum jerk [START_REF] Flash | The coordination of arm movements: an experimentally confirmed mathematical model[END_REF], minimum endpoint variance [START_REF] Harris | Signal-dependent noise determines motor planning[END_REF], and minimum torque change [START_REF] Uno | Formation and control of optimal trajectory in human multijoint arm movement[END_REF].

Robot movements that are intuitive to humans have been explored in HRI tasks that do not involve physical contact. Bisio et al. [START_REF] Bisio | Motor contagion during human-human and human-robot interaction[END_REF] demonstrated that humans modulate their movements based on the robot movements as long as the kinematics is consistent with movements recorded from real humans. Conversely, when the robot violates the human-like features, the human does not adjust their own movement to those of the robot. A study by Kupferberg et al. [START_REF] Kupferberg | Biological movement increases acceptance of humanoid robots as human partners in motor interaction[END_REF] reinforced this finding in a contact-free interaction with a humanoid robot, showing that humans tended to perceive the robot as interaction partners as long as it exhibited minimum-jerk profiles, i.e., maximally smooth profiles. These results suggest that humans dissociate from their robot partner and do not execute the task collaboratively, if the robot displays non-human-like behavior.

However in physical interactions humans no longer have the option to dissociate themselves from their robot partner. This raises the question, how do humans react to non-human-like movements of a robot? Previous work showed that untrained subjects prefer interacting with robots that move in more biological patterns during collaborative tasks, such as pointto-point reaching [START_REF] Corteville | Human-inspired robot assistant for fast point-to-point movements[END_REF], object handover [START_REF] Huber | Human-robot interaction in handing-over tasks[END_REF], and exoskeleton applications [START_REF] Zhao | Bio-inspired balance control assistance can reduce metabolic energy consumption in human walking[END_REF]. A previous study of our group on a humanrobot tracking task also showed that humans tended to follow the robot's movements if its kinematics obeyed the two-thirds power law. Specifically, humans exerted less force on the robot if its tangential velocity varied with the path's curvature with a specific power relation identified in humans [START_REF] Maurice | Velocity-curvature patterns limit human-robot physical interaction[END_REF].

Programming robots to elicit human-like movement patterns therefore seems the best approach to achieve seamless interaction and increase the usability and acceptance of the robot. But this is not always an option, as external constraints may be imposed by the task or the environment. In such cases, one could expect that humans are able to adapt if they are given enough practice. Indeed, humans have been shown to adjust to the robot behavior to increase the task efficiency [START_REF] Nikolaidis | Humanrobot mutual adaptation in collaborative tasks: Models and experiments[END_REF] and they could learn to predict non-biological robot movements during visual interaction [START_REF] Dragan | Familiarization to robot motion[END_REF]. And, after all, humans seem to demonstrate a sheer boundless ability to learn new and complex skills in everyday and leisure time activities. But is this adaptability really without limits?

Maurice et al. started to examine human abilities in physically tracking a robot, and has reported first evidence that humans can adapt to robot movements that violated the velocitycurvature relation of the two-thirds power law [START_REF] Maurice | Velocity-curvature patterns limit human-robot physical interaction[END_REF]. Specifically, participants were instructed to hold a robot as it traced out an elliptical path with a constant (non-biological) velocity profile, while minimizing the force exerted against the robot. After 1.5-hour practice, participants who received visual feedback about their exerted force revealed some reductions in force, while those without feedback did not improve. But interestingly, despite this improvement, the performance reached was still less optimal compared to when the robot followed the power law. Performance in the biological condition remained superior even when subjects had no practice and no visual feedback. This suggested that the untrained performance of the biological profile is the best humans can do. However, the practice comprised only a single session and performance did not reach a clear plateau, hence real performance limits could not be assessed.

For the development of physical human-robot interaction, it is crucial to understand to what extent humans can learn and overcome the challenges posed by non-biological movement patterns. A related question is what is needed to facilitate learning. Therefore, the present study investigated to what extent humans can adapt to and learn the robot motion when violating human movement signatures. Using a paradigm similar to our previous study [START_REF] Maurice | Velocity-curvature patterns limit human-robot physical interaction[END_REF], we extended the practice over 3 days to assess whether performance with a biological profile without training remains a hard limit. Participants underwent practice with the same constant velocity profiles as in [START_REF] Maurice | Velocity-curvature patterns limit human-robot physical interaction[END_REF], but also with another non-biological profile to assess the generalizability of the findings. Extended practice with the biological profile aimed to scrutinize whether further improvement was possible even in this familiar condition.

Since our previous study suggested that visual feedback facilitated learning, half of the participants in the present study were provided with augmented feedback. In addition, to provide the best opportunity to learn, we modified the visual feedback to better match the elliptical hand-robot movements than the previously used display.

II. METHODS

A. Participants

A total of 41 healthy college students (22 females and 19 males, aged 18-35 yrs) participated in the study. All participants were right-handed, did not report any biomechanical problems in their upper extremity, and were naive to the purpose of the study. All participants signed an informed consent form approved by Northeastern University Institutional Review Board prior to the start of the experiment (#10-06-19).

B. Two-Thirds Power Law in Velocity Profiles

Previous research in human motor control showed that human endpoint trajectories exhibit the so-called two-thirds power law, a systematic relation between the kinematic characteristics of the hand movement and the curvature of the associated path [START_REF] Lacquaniti | The law relating the kinematic and figural aspects of drawing movements[END_REF][19] [START_REF] Schaal | Origins and violations of the 2/3 power law in rhythmic three-dimensional arm movements[END_REF]. For trajectories with no inflection points, the power law can be written as,

v(t) = Kr(t) β (1)
with β = 1/3, where v is the tangential hand velocity, r is the radius of curvature of the path, and K is a gain factor that determines the tempo of the overall trajectory 1 . Hence, the kinematics scales with the geometric features of the endpoint movement. Essentially, this means that the movement slows down in more curved portions of the path and speeds up in straighter portions. Thus, this power relation can be used to characterize biological 'power law' profiles and non-biological profiles that violate this relation in robot movements.

C. Experimental Conditions and Procedure

A robotic manipulandum was programmed to trace out an elliptic path in a horizontal plane (major axis = 30 cm, minor axis = 10 cm) with its end-effector, moving in counterclockwise direction (Fig. 1A). The tangential velocity of the traversal was modulated according to the two-thirds power law. By varying the exponent β of the power law in (1), we created 3 different velocity profiles: biological (β = 1/3), constant (β = 0, profile used in the learning experiment of [START_REF] Maurice | Velocity-curvature patterns limit human-robot physical interaction[END_REF]), and exaggerated (β = 2/3). For the biological profile, the robot followed the power law, where the movement velocity decreased as the path curvature increased, and vice versa. The constant condition enforced a constant tangential velocity on the robot, similar to a control strategy commonly used for robot movements. The exaggerated velocity profile was an amplified version of the power law profile, where the robot significantly slowed down around curves and moved faster in the straighter sections of the ellipse. The K parameter in (1) was adjusted for each velocity profile to hold the ellipse period at 3 s across all conditions.

In each trial, the robot traced the ellipse 6 times without pause, with each ellipse lasting 3 s (18 s per trial). A 5 s break between successive trials allowed for rest to avoid fatigue. After 5 s, the robot automatically began the next trial. Each trial began and ended with a short sound. After a block of 10 trials, participants could rest for 2 to 3 minutes.

Participants were instructed to firmly hold the robot handle and move with the robot as it traced the ellipses, while exerting as little force as possible on the robot handle (Fig. 1B). Participants used their right-dominant hand to interact with the robot. The robot handle was free to rotate about its vertical axis which decoupled its orientation from the participants' wrist orientation (Fig. 1C). The elliptic robot and hand movements were confined to the 2-dimensional horizontal plane. The height of the robot was adjusted for each participant to ensure their forearm was approximately horizontal when holding the robot end-effector. Before starting the experiment, participants could find their most comfortable orientation and distance relative to the robot, but were instructed to maintain this position during the trials. A video of the experiment is available here: https://youtu.be/t6VRMFxMxfs. D. Experimental Apparatus 1) Robot: Participants interacted with the HapticMaster, a 3-DOF robotic manipulandum (Motek Medical, The Netherlands) [START_REF] Van Der Linde | Hapticmaster -a generic force controlled robot for human interaction[END_REF] (Fig. 1B). The robot was programmed through a custom C++ API and controlled using an impedance controller with high stiffness. Given the high stiffness, the robot continued to trace the predefined path even if the human applied resistive or assistive forces to its end-effector. The robot's desired position was updated at 120Hz. A 3-axis force sensor embedded at the robot end-effector measured the force exerted by participants on the robot handle. The force data and the position of the robot's end-effector were recorded at 120Hz.

An additional force sensitive resistor (FSR) afixed to the handle measured grip force (Fig. 1C) to test whether subjects firmly held the robot handle and were actively engaged in the task. The grip force sensor would trigger a buzzing sound, if participants loosened their grip on the handle beyond a threshold value. Without this check, participants may apply less force on the robot simply by loosening their grip on the handle during the experiments. This could be misinterpreted as compliance with the task. Note that the FSR was only used to ensure active participation, and not for analysis purpose.

2) Augmented Feedback: Since our previous study [START_REF] Maurice | Velocity-curvature patterns limit human-robot physical interaction[END_REF] suggested that additional feedback may facilitate learning, visual feedback about online performance was provided to half of the participants to give them the best opportunity to learn. This online feedback was shown on a projector screen in front of the participant with a cursor tracing the ellipse as the robot moved along its predefined elliptical path (Fig. 1B). The color of the cursor changed on-line to indicate the error in the force applied. Its shade changed between green (good) to red (bad) depending on the real-time error. Real-time error was quantified as the root mean square (RMS) of the magnitude of force in the horizontal plane that the participant exerted on the robot over a sliding window of 80 ms. The range of color was adjusted for each participant, based on their average force exerted in their baseline trials (see section II-E). The cursor was green if the force error was less than half of their baseline, and it was red if it was more than double of their force at baseline. In order to ensure subjects were continuously challenged to reduce their force on the robot handle, the reference force was updated after every 4 blocks of 10 trials. The reference force was updated to be the average of the most recent 10 trials. Participants who received visual feedback were instructed to ensure that the cursor color remained green.

For comparison, the other half of the participants were blindfolded and saw neither visual feedback, nor their hands or the robot. Hence, they had to rely on their proprioceptive and haptic feedback to smoothly follow the robot's motion without exerting additional forces against the robot.

E. Experimental Design

Participants were randomly assigned to 1 of 6 groups, with 6 to 8 participants per group (Fig. 2). Three groups received visual feedback (with-FB) of their real-time force error throughout the training, as shown in Fig. 1B; 3 groups were blindfolded (no-FB). All participants received haptic and proprioceptive feedback through grip contact with the robot handle.

Prior to the training, all participants performed 2 blocks of 4 trials each, in which all groups were blindfolded: 4 trials of the biological profile determined reference performance; the second block of 4 trials presented the assigned velocity profile to determine baseline performance before introducing feedback. In the experiment proper, participants practiced their assigned velocity profile in 3 one-hour-long training sessions on 3 consecutive days. Every training session involved 8 blocks, with each block containing 10 successive trials. Hence participants practiced 24 blocks with 240 trials, for a grand total of 1440 elliptical movements.

F. Data Analysis

1) Data Filtering: The recorded data consisted of continuous forces in X and Y horizontal directions that a participant exerted on the robot end-effector during the interaction. The magnitude of the force at every point of the ellipse was calculated. In order to eliminate transient data, the first and last ellipse of every trial were excluded from the data analysis.

The human-robot interaction force data were filtered forward and backward (to avoid phase-shifts) through a 4th-order 6 Hz low-pass Butterworth filter. Over 90% of the original signal's power was maintained post filtering. Code and data are available here: https://gitfront.io/r/mahdiaredraki/ YqQmHMJPnq1t/ICRA2023/.

2) Performance Metrics: The explicitly instructed goal in all conditions was to minimize the magnitude of the force exerted against the robot, which therefore served as first performance metric. The force magnitude metrics F-RMS was calculated as the root mean square of force applied across the 40 ellipses of one block (10 trials per block and 4 ellipses per trial, since the first and last ellipse of each trial were excluded). As human performance is always variable and reduction of this variability has been shown a reliable characteristic of learning [START_REF] Sternad | It's not (only) the mean that matters: Variability, noise and exploration in skill learning[END_REF], the standard deviation of force F-SD was added as a second metric. This metric was calculated as follows: for each ellipse in a block (4 x 10 trials), force magnitude was computed at each spatial location across the ellipse (360 bins). At each spatial location, the standard deviation of the force magnitude was computed over the 40 ellipses. Finally, all location-dependent values were averaged.

3) Statistical Analysis: Performance across the training sessions was evaluated by fitting linear regressions to both the F-RMS and the F-SD values over the 24 blocks of the 3 days of training. Blocks rather than trials were used as regressors, because the calculation of standard deviations required aggregating trials of one block. The confidence intervals of the regression slopes served to determine whether each participant's performance significantly changed across 3 practice days. If the zero slope was not within the confidence intervals, the change was considered statistically significant (Fig. 3).

In the groups where improvements were detected, additional analyses were conducted to assess whether the performance reached after training was better than the reference performance (biological profile without visual feedback before any practice). This was done by comparing the 4 last trials of the training phase with the 4 reference trials, using a paired t-test. Fig. 3. Linear regressions for two representative participants. Black dots show block means across the 24 blocks. The shaded areas show the 95% confidence interval of the regression slopes. A: Linear fit for a participant that improved through practice as the confidence intervals did not include zero slope. B: Linear fit for a participant that did not improve as the confidence intervals did embrace zero slope.

III. RESULTS

Based on the measured forces, this experiment aimed to evaluate whether extensive practice over multiple days with compatible and updating feedback could elicit performance improvements both in biological and non-biological conditions. We also assessed whether practice could take participants beyond baseline performance with the biological robot profile.

A. Elimination of Trials

Prior to data analysis, trials were scrutinized to assure that participants maintained a solid grip of the robot to prevent that low interaction forces only resulted from a loose grip of the robot. Trials where the buzzer was triggered for more than 10% of the trial duration (>1.8 s out of 18 s) were excluded from subsequent analysis. With this criterion, the data for 2 participants had to be excluded as they failed to follow the task instructions (reducing number of participants to 6 in 2 groups as shown in Fig. 2). In the remaining 39 participants, ∼4% of the trials across all participants were excluded.

B. Force Patterns Across the Ellipse

Fig. 4 illustrates the spatial and temporal pattern of the interaction force exerted by representative participants in the 3 velocity profiles. Different force patterns were elicited depending on the profile, highlighting how different segments of the ellipse posed difficulties to minimize the interaction force. Most notably, the forces were high at the highly curved segments in the constant condition, while the forces in the biological conditions were lowest and more distributed along the ellipse. In the exaggerated condition two peaks occurred at the linear portions where the speed was faster than the biological pattern. Except for short moments, the magnitude of force did not reach zero Newtons.

C. Force Magnitude and Variability Across Blocks

This analysis focused on how the force magnitude and variability changed across the 3-day-long practice in all 6 conditions. To this end, the values of F-RMS and F-SD of all participants over the 24 blocks were plotted for each condition in Fig. 5A andC. The linear regressions are also shown as lines. The R2 values for all participants ranged from 0.19 to 0.77 with a median of 0.38 (see Fig. 3). 2 Starting with the feedback conditions, the individual participants' regression slopes of F-RMS in the biological condition tended to be variable and without any visible trend to decline across blocks. This differed from the constant and exaggerated condition where the majority of participants exhibited negative slopes. This pattern was reinforced in the F-SD results, although less clear. In the no-feedback conditions, the individual regression slopes did not signal any consistent change with practice, neither for F-RMS, nor for F-SD. The slopes and their 95% confidence intervals for each group and participant are summarized in Fig. 5B andD, where the error bars indicate the boundaries of the confidence intervals. number of participants in each condition that show regression slopes different from zero, i.e., significant change across 3 days of practice. Overall, the data showed that participants learnt the non-biological patterns, but only when visual feedback was provided. Performance in the biological profile did not change, even when feedback was given. Hence, performance in the biological profile seemed to be at a 'hard limit'.

Given these results, the next question was whether performance in the non-biological conditions had also reached that limit. Using paired t-tests, the mean F-RMS in the last 4 trials was compared with the mean performance in the reference trials. These comparisons were only conducted for participants that received feedback in the constant and exaggerated conditions. Neither of the two comparisons was significant (p=0.068 for constant, p=0.265 for exaggerated condition). Therefore, even with practice, participants were not able to reduce their interaction forces beyond their untrained biological performance. When F-SD values were compared, the exaggerated condition showed no difference (p-value = 0.107), only the constant conditions showed a small difference (p=0.029).

Given the high initial force values in some of the feedback conditions, additional control tests were conducted to rule out that the introduction of visual feedback presented an initial difficulty. To this end, the first 4 trials of the training phase of all participants who received visual feedback were compared with the 4 trials at baseline with the same velocity profile, but blindfolded. For these 3 comparisons, both F-RMS and F-SD of all participants were compared by paired t-tests. None of these comparisons revealed a significant difference.

IV. DISCUSSION This study investigated to what extent humans can adapt to a robot motion in a pHRI task that violates human movement signatures, specifically the two-thirds power law, a humanpreferred speed-curvature relation. Results showed that humans could improve their interaction with a robot moving in two nonbiological velocity profiles, but only if augmented feedback was provided. This finding generalizes our previous findings [START_REF] Maurice | Velocity-curvature patterns limit human-robot physical interaction[END_REF]. However, even when given extensive 3-day-long practice with dedicated visual feedback of their performance, participants did not show any improvement in the biological pro- file. Additionally, improved performance with a non-biological profile never exceeded their untrained biological performance. This suggests that the untrained biological performance is a limit that humans cannot surpass.

A. Role of Augmented Feedback

It is noteworthy that participants who received continuous force feedback but were deprived of augmented visual feedback about their task error did not show any improvements in practiced velocity profiles, even after 3 daily practice sessions. This suggests that haptic feedback about the forces alone was insufficient to fine-tune their interaction with the robot. This inferior performance with haptic information alone suggests that humans may not have enough sensitivity to perceive the modulations of the relatively small forces. Visualization of force error, i.e., providing augmented real-time feedback, was evidently necessary.

These results were based on two main performance metrics that were regarded as two main indicators of performance and learning: While F-RMS expresses the error from the instructed zero force, variability F-SD is an independent metric of performance [START_REF] Schmidt | Motor control and learning: A behavioral emphasis[END_REF]. With one exception where variability in the constant profile was better than the biological condition, the current results showed parallel declines in the two metrics, therefore reinforcing the observations. Note that both Maurice et al. [START_REF] Maurice | Velocity-curvature patterns limit human-robot physical interaction[END_REF] and a recent study by West et al. [START_REF] Michael | Dynamic primitives limit human force regulation during motion[END_REF] also provided online visual feedback, but with mixed results. Not only was the practice duration in both studies much shorter, they also presented a different design of visual feedback. Their visualization of error was a horizontal bar deviating from a baseline representing the target force. Anecdotally, this vertical graphic arrangement proved difficult to map onto the horizontal elliptical pattern, i.e., showed little compatibility. Therefore, the present study presented the cursor moving online on an elliptical path and used cursor color to indicate force error. Note that augmented feedback can be of different degrees of 'compatibility' and therefore careful design of how feedback is provided is necessary [START_REF] Proctor | Stimulus-response compatibility: An integrated perspective[END_REF].

In fact, feedback can also be detrimental if it is 'too complicated' because it can create an additional cognitive load. An increase in cognitive load due to feedback has been reported in surgical robotics applications, where novice surgeons could not benefit from additional haptic information and actually deteriorated in their performance accuracy [START_REF] Zhou | Role of haptic feedback and cognitive load in surgical skill acquisition[END_REF]. Similarly, a study on object manipulation with a phantom robot showed that the effect of haptic feedback on performance was dependent on the context [START_REF] Stepp | Contextual effects on robotic experiments of sensory feedback for object manipulation[END_REF]. To probe whether the introduction of feedback in our study did not require undesired attentional resources, we compared the blindfolded baseline trials with the first feedback trials. Results confirmed that this feedback did not introduce any noticeable extra load and, hence, the subsequent improvements were not a side-effect of the initial integration of feedback.

B. Limits of Human Performance in Biological Condition

The biological profiles did not show any improvements, even after 3 days of practice with enhanced feedback and regardless of whether augmented visual feedback was given. Note that the experiment also adjusted the visual force feedback to the current level of force error in order to maintain the challenge for participants. These efforts reinforced that participants have reached their maximum performance in the biological condition, even though the force errors were not reduced to zero. This is in apparent contrast with a recent paper by West et al. [START_REF] Michael | Dynamic primitives limit human force regulation during motion[END_REF] on a similar ellipse-tracking task that reported improvements in a biological profile. However, an important difference is that participants were instructed to apply a constant tangential force of 5 N against the robot. Further, the research focus was on force-motion or hybrid control. Hence, trial blocks with visual feedback alternated with blocks of no feedback, precluding direct inferences about learning.

C. Implications for Robotic Applications

The straightforward conclusion of our results is to program robots to move with human-like features to facilitate the interaction without requiring extensive practice. However, this may not be feasible in all pHRI applications. Some pHRI tasks have pre-set constraints on the types of motion that the robot can display. For example, Glogowski et al. developed a trajectory planning algorithm that allowed for the online adaptation of the robot velocity to satisfy constraints such as collision avoidance [START_REF] Glogowski | Robot speed adaption in multiple trajectory planning and integration in a simulation tool for human-robot interaction[END_REF]. This may have traded off velocity with collision avoidance. In the same vein, robot control needs to avoid actuator torque saturation and prioritize stability and inertia compensation, all issues that may compromise the independent control of trajectory velocity [START_REF] Buerger | Relaxing passivity for humanrobot interaction[END_REF]. For such situations where task constraints may prevent the robot to adopt humanlike trajectories, the current study showed that humans do have the ability to adapt, at least in a limited way and if appropriate feedback about the interaction is provided.

V. CONCLUSIONS

This study examined the conditions in which humans could optimally track robot movements and learn to minimize undesired interaction forces. With extensive practice and real-time augmented feedback about the force error, humans could adapt to robot trajectories that violated human signatures. However, humans were unable to improve their performance when no additional feedback was provided. These results show that when humans have to interact with robots that move with nonbiological profiles, they need proper guidance to master the interaction. Thus, training modalities have to be considered carefully when deploying such robots. Further, even in the most human-like trajectories, humans could not perfectly follow robot movements. Hence, future work should further probe into human preferences and their limited ability to interact with and adapt to robot motions. Feedback should be carefully designed to minimize additional cognitive load and to provide the best possible guidance to potentially push these human limits.

Fig. 1 .

 1 Fig. 1. A. Experimental setup. B. Participant performing the task with visual feedback of their performance. The cursor on the screen traced the ellipse simultaneously as the robot (HapticMaster) moved around its elliptical path. C. Grip sensor attached to the robot end-effector to ensure participants' engagement in the task. D. Tangential velocity across the ellipse for the three velocity profiles used in the experiment.

Fig. 2 .

 2 Fig.2. Layout of the experimental design. Six groups of participants performed the 3 velocity conditions, either with or without visual feedback (n is the number of participants per group). The experiment began with recording reference and baseline performance (4 trials each, blind-folded). This was followed by 3 days of training, each day consisting of 8 blocks of 80 trials total.

Fig. 4 .

 4 Fig. 4. Top row: Vectors representing the interaction force (direction and magnitude) exerted on the robot across one ellipse in the three velocity profiles, taken from 3 representative participants of the respective groups. The red dot marks the starting point of the ellipse. The color code indicates the progression of time matching the time series below. Bottom row: Time series of the magnitude of the interaction force around the ellipse of the same trial as shown above.

Fig. 5 .

 5 Fig. 5. Overview of participants' performance across the 3 practice days. A. Force magnitude F-RMS across blocks. Each point is the average value per block for each participant. The 6 panels correspond to the 6 conditions (3 velocity profiles and 2 feedback conditions). Within each condition, color shading within each condition denote different participants (consistently with panel B). The solid lines represent the regression slopes for each participant. B. Slopes (point) and their corresponding 95% confidence intervals (bars) of the linear regressions of F-RMS for each participant in the biological (Bio), constant (Con), and exaggerated (Exa) velocity profiles, with and without visual feedback. C. Same as panel A for variability of force F-SD. D. Same as panel B for variability of force F-SD.

TABLE I NUMBER

 I OF PARTICIPANTS IN EACH CONDITION THAT IMPROVED VERSUS DID NOT IMPROVE THROUGH PRACTICE.

	Performance Metric	F-RMS	F-SD
		Condition	(improved / did not improve)
		Biological	1/7	1/7
	with-FB Constant	5/2	5/2
		Exaggerated	4/3	4/3
		Biological	1/5	1/5
	no-FB	Constant	1/5	1/5
		Exaggerated	0/7	2/5

The name two-thirds power law comes from the original formulation with the angular velocity and curvature (instead of tangential velocity and radius of curvature), for which the value of the exponent is

2/3.

The R 2 metric evaluates the amount of variance explained by the linear fit in determining change in performance compared to mean performance for each participant. This makes it meaningless to report the R 2 values for regressions where the 95% confidence interval includes zero slope. Thus, only the R 2 of regressions with non-zero slopes are reported.

Dagmar Sternad and Mahdiar Edraki were supported by NIH-R37-HD087089 and NSF-M3X-1825942. Pauline Maurice was supported by French Research Agency, ANR-20-CE33-0004 (project ROOIBOS).


Benoit Dompierre
email: benoit.dompierre@thalesgroup.com

Véronique Aubin

Eric Charkaluk

Wilson Maia Filho

Michel Brizoux

W C Maia Filho

Cyclic mechanical behavior of Sn3.0Ag0.5Cu alloy under high temperature isothermal ageing

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Introduction

In the electronics industry, components are assembled on the printed circuit board using solder joints, which ensure the electric, thermal and mechanical connection between the board and the components. Before July 2006, most electronic products were assembled with eutectic SnPb alloy. Due to the July 2006 RoHS (Restriction of Hazardous Substances) directive, the mass market turned to lead-free processes for electronic assemblies. Most electronic manufacturers chose SnAgCu alloys.

Thermal ageing induces changes in the SnAgCu alloy's microstructure. The growth kinetics of the interface between the solder and substrates are well known [START_REF] Siewert | Proceedings of the APEX Conference on Electronics Manufacturing[END_REF][START_REF] Choi | [END_REF][3]: for example, Siewert et al. [START_REF] Siewert | Proceedings of the APEX Conference on Electronics Manufacturing[END_REF] determined the equations and activation energies of the growth of intermetallic layers for four various lead-free alloys. In the bulk solder, two phenomena can be encountered. First, Wiese and Rzepka [4] observed the coalescence of intermetallic phases after thermal storage for 1500 h at 150 • C. Second, Telang et al. [5] demonstrated a grain size growth phenomenon for the Sn3.8Ag0.7Cu alloy. After thermal ageing for 200 h at 150 • C, the increase in the grain size was greater than 100%. Other studies [START_REF] Xu | Proceedings of the 56th Electronic Components and Technology Conference (ECTC 2006)[END_REF][START_REF] Dompierre | Proceedings of the 10th International Conference on Thermal Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Micro-Systems (EuroSimE)[END_REF] demonstrated that the same phenomenon is observed after thermal ageing at other high temperatures.

For thermal ageing of the interface layers, various studies have demonstrated the relation between the changes in the microstructure and in certain mechanical properties [START_REF] Zhang | [END_REF][9][10][11]. In these cases, ageing laws have been defined for various chemical compositions. Regarding the bulk alloy, Ma et al. [START_REF] Ma | Proceedings of the 9th International Conference on Thermal Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Micro-Systems (EuroSimE)[END_REF] suggested that high-temperature thermal ageing causes a significant deterioration in mechanical properties such as hardness and yield stress after storage for hundreds of hours at 100 • C, but the relation between changes in the microstructure and in mechanical properties remains to be proved.

Moreover, until now, changes in bulk solder mechanical properties have been demonstrated using monotonic tests [START_REF] Ma | Proceedings of the 9th International Conference on Thermal Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Micro-Systems (EuroSimE)[END_REF][START_REF] Wiese | Proceedings of the 55th Electronic Components and Technology Conference (ECTC 2005)[END_REF][START_REF] Anderson | [END_REF][15][16], which is a good but insufficient indicator of the influence of thermal ageing on the cyclic mechanical behaviour of SnAgCu alloys. The objective of the present study was to evaluate the influence of thermal ageing on the cyclic mechanical behaviour of SnAgCu alloys. This study is original in that it simultaneously analyses microstructure and cyclic mechanical behaviour changes.

This study was divided into three parts. First, the influence of thermal ageing on monotonic mechanical properties of the Sn3.0Ag0.5Cu alloy was evaluated. The Vickers hardness was measured on as-received samples (specimens before ageing) and regularly during thermal ageing up to 6500 h at various temperatures making it possible to evaluate ageing kinetics and to define the thermal ageing conditions for the second phase of the study. Simultaneously, cross-polarised light microscopy was carried out to analyse the microstructure changes after thermal ageing. These results related the decrease in hardness to the changes in the microstructure. Finally, the effects of thermal ageing on mechanical properties were characterised using monotonic tensile tests and reversed cyclic tests. This paper is organised as follows. First, it presents the material studied and the experimental methodology. Then the results of hardness tests and polarised light microscopy observations are described and followed by the results of monotonic and cyclic tests. Finally, these results are summarised and the conclusions are presented.

Material

Many chemical compositions have been suggested to replace SnPb alloys. In large part because of their relatively low price and low melting point, SnAgCu alloys emerged and the majority of new brazing alloys are SnAgCu-based. For electronic products, Sn3.0Ag0.5Cu is a very widely used composition because it is a compromise between low melting point (near eutectic composition) and price (low Ag rate). For this reason, only the Sn3.0Ag0.5Cu composition was investigated in this study.

Because of the low melting point of SnAgCu alloys, around 220 • C, room temperature corresponds to 0.6 Tm (in Kelvin). Consequently, SnAgCu alloys present a time-dependent behaviour at room temperature and the effects of strain rate and stress relaxation cannot be neglected [17][START_REF] Poh | Proceedings of the 9th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Micro-Systems[END_REF][START_REF] Jun | Proceedings of the 8th International Conference on Thermal, Mechanical and Multi-Physics Simulation Experiments in Microelectronics and Micro-Systems (EuroSimE)[END_REF][START_REF] Pei | Proceedings of the International Symposium on Advanced Packaging Materials: Processes, Properties and Interfaces[END_REF][START_REF] Bai | [END_REF][START_REF] Metasch | Proceedings of the 10th International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Micro-Systems (EuroSimE)[END_REF].

Regarding the microstructure, pure Sn presents three allotropic forms: ␣ (diamond cubic) at 13 • C, ␤ (body-centered tetragonal) between 13 • C and 161 • C and ␥ (orthorhombic) between 161 • C and 232 • C (Tm). In SnAgCu alloys, Sn accounts for more than 95% of the composition. The alloy's microstructure is therefore very similar to the microstructure of pure Sn: pure ␤Sn dendrites surrounded by ␤Sn. Ag can be found in Ag 3 Sn needles or plates, while Cu 6 Sn 5 precipitates are generally rod-like [START_REF] Park | [END_REF].

Experimental procedure

This section first presents the fabrication of specimens, then the hardness tests and the microstructural observations, and finally the reversed cyclic tests.

Fabrication of samples

For reproducibility purposes, all samples were extracted from the same batch of solder bars used for wave soldering. This ensures that the initial microstructure is the same for both the hardness and tensile tests. For hardness tests, the samples were extracted from solder bars by cutting with a band saw. The samples measured 10 mm × 10 mm × 30 mm, as shown in Fig. 1a. Band saw cutting generally warms the specimens. Great care was taken to achieve good lubrication of the solder bars while sawing so as to limit heating. In addition to this precaution, samples were prepared by polishing with the grit size ranging from 120 to 600 to remove the external layer of material, which could be affected by the cutting process.

Two main fabrication methods of the tensile specimens were compared: casting and machining. Casting problems have been encountered in some studies [START_REF] Park | [END_REF]24] in which polarised light microscopy demonstrated that only a few grains constituted the cross-section because the cooling rate was too low. A sample cast- ing test demonstrated the same results. However, controlling grain size is crucial in this type of test. The grain size has to be homogeneous within all the specimens and small enough to consider the specimens as a representative elementary volume of the material. To ensure this condition, it is estimated that a specimen's cross-section must contain a minimum of 100 grains. Polarised light microscopy observations of the solder bars proved that the grain size in these bars was about 10 m, which ensures thousands of grains in the specimen's cross-section. As the specimen obtained by casting did not encounter this condition, the specimens were extracted from solder bars from the batch used for hardness samples by high-pressure water cutting, as illustrated in Fig. 1b. Microstructure observations on the edges of the specimens showed that the impact of the water cutting on the microstructure was negligible. The specimen's dimensions and geometry are illustrated in Fig. 1b. The gauge part of the specimen was 10 mm in length, 10 mm in width and 10 mm in height.

Isothermal ageing

For the former SnPb alloys, the influence of thermal ageing on the mechanical properties stabilises in a very short time. Miyazawa and Ariga [START_REF] Miyazawa | Proceedings of the First International Symposium On Environmentally Conscious Design and Inverse Manufacturing (EcoDesign'99)[END_REF] demonstrated that thermal storage for only 250 h at room temperature is enough to reach the thermally aged condition. For new SnAgCu alloys, previous studies seem to prove that there is a stabilisation state for thermal ageing for hundreds of hours at high temperature, but the relation with room temperature has not yet been clarified.

To evaluate this, bulk samples were thermally aged at three different temperatures (75 • C, 125 • C and 150 • C) up to 6500 h in order to characterise the isothermal acceleration of the phenomenon using hardness tests, as shown in Table 1. Each ageing condition was associated with at least one sample.

Thermal ageing at 125 • C was chosen in order to be coherent with the IPC9701A specification [26], which describes "use environments for surface mounted electronics". In this specification, the highest storage temperature for military aircraft is 125 • C. The second storage temperature, 150 • C, was chosen to create a higher acceleration phenomenon, which corresponds to the automotive (under the hood) maximum storage temperature. Finally, the third temperature, 75 • C, corresponds to a lower acceleration phenomenon and is half-way between room temperature (25 • C) and 125 • C.

Cross-polarised light observations

The ␤Sn structure is body-centred tetragonal. Directions [1 0 0] and [0 1 0] present the same properties but the direction [0 0 1] differs, so that ␤Sn has a birefringent behaviour. Using cross-polarised light, each grain orientation shows a different colour, so that the grains can be counted and measured.

Before the observation, one side of the samples was prepared using polishing paper with grit sizes ranging from 120 to 4000, then 3-, 1-and 0.25-m Struers diamond pastes were used. For each observation, all grains within a circle of a defined area were identified. The results obtained were presented in a previous study [START_REF] Dompierre | [END_REF]. Using ImageJ software, the area of each grain was measured. Then the mean diameter of the grain was calculated from the measured area assuming a circular grain. Finally, the grain size distribution was evaluated with these data.

While cooling the solder bar, the temperature drop is probably not the same in all locations. The highest difference in the cooling rate could be between the edge and the centre of the bars and can therefore influence grain size. Ingots were first observed before thermal ageing in order to study the possible influence of the testing site on the results. Two areas were analysed for comparison: the centre and the edge of the sample. Grain size was determined using cross-polarised microscopy observations as presented above. There was no significant difference between the centre of the sample and its edge. On these two locations, the mean grain size was about 4 m for the as-received material.

Secondly, to establish whether a relation existed between the microstructure and thermal ageing, grain size distributions were estimated for various ageing conditions. For all samples, the distribution was fitted with a lognormal law.

Hardness tests

The results of hardness tests could be affected by the indented grain's crystal orientation. However, the bulk samples' mean grain size (d m ) was up to 10 m (for the aged material) and the indenter size was about 1 mm 2 . As the indenter size was much larger than the grain size, this ensured that a mean hardness value would be obtained.

The hardness tests were divided into two steps. First, in order to check whether the test location had an impact on the result, the scattering of the hardness was characterised over the entire surface. Consequently, 21 hardness tests were performed following a grid. As the relative standard deviation was 3%, it can be concluded that there is no significant influence of the test location on the results.

Secondly, the tests were performed with various ageing conditions. As described above, at room temperature, SnAgCu alloys present a time-dependent behaviour. Thus, to prevent scattering of the values measured, the duration of each test was carefully controlled. All tests were carried out at room temperature with a 30-N load applied for 10 s. For each sample, the hardness value was obtained by averaging at least nine values to reduce uncertainty. The results presented in the following section are the mean values of all the data measured in each test condition.

hour Strain Time

Monotonic and reversed cyclic tests

All monotonic and cyclic tensile tests were carried out at room temperature on a 100-kN Instron servo-hydraulic tensile test machine. Strain was checked by means of an Instron contact extensometer with a 10-mm gauge length.

As previously specified, at room temperature, SnAgCu alloys present a time-dependent behaviour; the effects of the strain rate and stress relaxation have to be taken into account. Therefore, two types of tests were performed. First, monotonic tensile tests were carried out in order to determine the influence of the strain rate, the reproducibility between various specimens and relaxation characteristics. Fig. 2 presents the strain control shape for monotonic tensile tests.

Then reversed cyclic tests were carried out to analyse the changes of the mechanical behaviour during mechanical cycling for various strain amplitudes and strain rates. Fig. 3a presents the cyclic strain control without dwell time (triangular shape) and Fig. 3b presents the same cyclic control with dwell time (trapezoid shape).

All the parameters used for monotonic and cyclic tests are presented in Table 2. Strain amplitudes and strain rates were chosen to be coherent with the strain amplitudes that can be encountered in solder joints during thermal cycling tests. The dwell time for monotonic tests was chosen to ensure that stress stabilisation was reached. The dwell time for cyclic tests was chosen as a compromise between relaxation behaviour and test duration. These results suggest that the stabilisation level could be the same for both temperatures, which is consistent with the hypothesis of a single stabilisation state.

Time

Results

Hardness tests

Regarding the results presented in Fig. 4, the decrease in hardness is related to ageing temperature and time. Eq. ( 1) was chosen to represent the changes observed. This equation corresponds first to an exponential decay to represent the time dependence with three parameters (A, B and ) where the time parameter () follows an Arrhenius law to represent the temperature dependence. The parameters of this equation were determined with data at 75 • C and 150 • C with a mean error of 1.9%. Data at 25 • C and 125 • C were then computed, and the comparison with experimental data gave a relative error of approximately 2.6% (Table 3).

H v = A + Be -(t/) with = 0 e (Ea/kT ) (1)
According to Fig. 4 and Eq. ( 1), above 125 • C, 95% of the stabilisation was reached after approximately 500 h. Complementary tests have to be made at 75 • C to confirm that there is only one steady state for all temperatures. If so, according to these results, thermal ageing for 500 h at 125 stabilised state for all temperatures. The ageing time at room temperature is highly inaccurate because of various uncertainties, one of which is activation energy (E a ). A 10% variation in E a leads to a variation in stabilisation time at room temperature by a factor of 2. This means that the ageing time required before stabilisation at room temperature seems to be at least 20 years. These results are in good agreement with the results of Ma et al. [START_REF] Ma | Proceedings of the 9th International Conference on Thermal Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Micro-Systems (EuroSimE)[END_REF], who focused on the characterisation of the changes in monotonic mechanical properties using tensile tests on Sn4.0Ag0.5Cu. They demonstrated that the decrease in the Young modulus and the yield stress of this alloy seems to stabilise after thermal ageing for less than 500 h at 125 • C.

Cross-polarised light observations

Fig. 5 compares the microstructures obtained initially and after thermal ageing at 125 • C for 384 h. The initial microstructure is very fine, with the grain size around 4 m, whereas the aged microstructure presents very large grains. Comparing these two images demonstrates substantial grain growth.

Fig. 6 shows the distribution of grain sizes for Sn3.0Ag0.5Cu samples for the as-received condition. This distribution was esti- mated with about 300 grains. Distributions corresponding to other ageing conditions are presented in a previous article [START_REF] Dompierre | [END_REF]. From these results, two observations can be made: first, the mean grain size increased with ageing time; second, very large grains were more frequent when ageing time increased. Both observations can be made for the microstructures between 0 h, 72 h and 192 h, but no significant difference was observed between the microstructures after 192 h and 384 h. Differences between these two ageing conditions were probably so small that they were hidden by the uncertainties induced by the measurement method. For this reason, no additional observations were made after 384 h of thermal ageing at 125 • C.

Influence of isothermal ageing on mechanical behaviour

The objective of the following mechanical tests was to analyse the cyclic behaviour of the Sn3.0Ag0.5Cu alloy before and after ageing, when the stabilisation level has been reached. Two ageing conditions were chosen: specimens before ageing (asreceived material) and specimens aged 500 h at 125 • C (aged material).

Fig. 7a shows the mechanical responses to monotonic tensile tests at two strain rates (2.10 -3 s -1 and 2.10 -4 s -1 ) with specimens before and after ageing. Each condition was tested at least three times. Good reproducibility between specimens was observed so that only one test per condition during cyclic test analysis was necessary. Fig. 7b shows the relaxation behaviour during the dwell time. Stabilisation was reached about 1 h after dwell time began.

The influence of the strain rate on the maximum stress is shown in Fig. 8, in which the influence of the strain rate is characterised by an 18-22% decrease in the maximum stress for a decrease in the strain rate by a factor of 10. These results are in good agreement with Wang et al. [START_REF] Wang | Proceedings of the 7th International Conference on Electronic Packaging Technology (ICEPT'06)[END_REF], who determined that for two strain rates, 2.10 -3 s -1 and 2.10 -5 s -1 , the yield stress differs by 30%. In the present study, this was valid before and after ageing.

The influence of thermal ageing can be seen in Fig. 8. For each strain rate, the maximum stress was 35-40% lower for the aged condition than for as-received specimens. Moreover, for all the loading conditions considered, after the dwell time, the specimens reached the same stabilisation level, depending only on the ageing condition (as-received or aged material). The stress after the dwell time obtained on aged material was half the stress of the as-received material.

Fig. 9a shows the behaviour of the material during stress relaxation. The data were normalised as follows: maximum stress, the value referred to as "start of dwell time" in Fig. 7b, was converted to "1" and the stress value after dwell time, referred as "stabilisation level" in Fig. 7b, was converted to "0". Fig. 9a illustrates that aged and as-received materials presented similar stress relaxation behaviours, which suggests that the viscous behaviour is not modified by thermal ageing. Fig. 9b shows that more than 90% of the stress relaxation was achieved before 30 min. The relaxation time depends on the strain rate and was the same for as-received and aged specimens. The higher the strain rate was, the faster the relaxation was, as shown in Fig. 9b. Fig. 10 shows softening curves without normalisation for asreceived and aged specimens under various conditions. These curves present a softening time, which leads to stabilisation or a quasi-stabilisation of the stress amplitude. These results are quite similar to those reported by Kanchanomai et al. [START_REF] Kanchanomai | [END_REF], who obtained a softening cyclic behaviour for four lead-free alloys, including Sn3.0Ag0.5Cu. Two points should be underlined:

• As for monotonic tests, thermal ageing induced a nearly 40% drop in stress amplitude. This direct comparison between as-received and aged conditions is possible because a good reproducibility between specimens had been observed previously; • Even if the results of tests with and without the dwell time are not exactly the same, no clear effect of dwell time on stress amplitude was found. For a test condition, the dotted line (reversed cyclic test with dwell time) can be above or below the solid line (reversed cyclic tests without dwell time) corresponding to the same test condition.

In the following figures, all the data were normalised to make the comparison clearer. For each curve, all data were divided by the stress amplitude value of the first cycle, so that the first value for each curve was "1". Fig. 11 shows the cyclic hardening/softening on as-received specimens. For the strain amplitude of 0.2%, shown in Fig. 11b, the first five cycles resulted in the material hardening (first stage) followed by softening (second stage), which finally provided quasistabilisation. The same general behaviour can be seen in Fig. 11a, for the strain amplitude of 1%. However, in that case, the hardening phase did not exist. It can also be noted that the stabilised stress amplitude at a strain amplitude of 1% was lower than the strain amplitude at 0.2%. For aged specimens, the cyclic behaviour was only a softening behaviour, as shown in Fig. 10. After 200 cycles, the drop in stress amplitude was the same as the as-received specimens (about 20%).

Discussion

In this study, hardness tests, monotonic tensile tests and cyclic reversed tests were carried out on the Sn3.0Ag0.5Cu alloy. In all these tests, the material was highly deformed (1% for tensile tests and more than 10% for hardness tests). The as-received and the aged materials differed in terms of hardness by about 30%. In tensile tests, maximum stress and the stress reached after relaxation differed by 40% and 50%, respectively. These deviations were also observed during cyclic tests with or without dwell time. These similarities are not surprising given that both tests measure the elastoplastic behaviour of the material and it is well known that hardness and yield stress are often linked by empirical relationships [START_REF] Fischer-Cripps | Nanoindentation[END_REF], which is even more accurate when the behaviour is perfectly plastic, as is Sn3.0Ag0.5Cu. The results also show that thermal ageing induces an increase in grain size and a decrease in hardness. For metallic materials, the yield stress is often related to the grain size through the Hall-Petch law [START_REF] Hansen | [END_REF], presented in Eq. ( 2). Fig. 12 represents the decrease in the hardness value and the grain size growth over time during isothermal ageing at 125 • C. The grain growth is correlated to the decrease of 1/ √ d. The hardness and the grain size follow analogous variations according to time and temperature, which confirms that grain growth and changes in mechanical properties may be related. This type of relationship is generally observed in metallic materials [START_REF] Moniz | Metallurgy[END_REF].

y = 0 + k √ d (2)

Conclusions

The aim of this paper was to improve the understanding of the microstructural and mechanical changes during thermal ageing of the Sn3.0Ag0.5Cu alloy and their impact on the cyclic behaviour of the alloy.

Assessing ageing through hardness tests and polarised light microscopy demonstrated that thermal ageing induces an increase in grain size and a decrease in hardness. These two phenomena are strongly related through a Hall-Petch-type relationship. Ageing ends (95% achieved) after about 500 h at 125 • C and this ageing condition is equivalent to more than 20 years at room temperature. Therefore, for systems intended to have a long lifespan, it appears that ageing may occur before or during the operational phase and evaluating the influence of thermal ageing on the reliability of electronic assemblies is required.

The mechanical behaviour of Sn3.0Ag0.5Cu was evaluated by monotonic and cyclic tensile tests on as-received and aged materials. The results of these tests show common points between these two materials. First, the influence of the strain rate is a well-known phenomenon for Sn3.0Ag0.5Cu and remains the same after thermal ageing for 500 h at 125 • C. Then the relaxation behaviour remains the same, and depends only on the strain rate. Finally, the softening behaviour seems to be the same with a 20% decrease in stress amplitude after 200 cycles with or without dwell time. However, major differences can be noted between as-received and aged materials. First, the stress amplitude during monotonic or cyclic tests is about 40% lower after thermal ageing. Secondly, after dwell time, stress is halved in specimens after compared to before isothermal ageing.

All these observations lead to the conclusion that thermal ageing must be fully understood in the reliability analysis of solder joints on electronic assemblies. Complementary studies have to be conducted first in solder joint scaled specimens to validate the influence of thermal ageing at the joint scale and then on the assembled board. These studies are in progress [16,[START_REF] Dompierre | [END_REF].

Fig. 1 .

 1 Fig. 1. (a) Fabrication of hardness test samples; (b) fabrication of tensile test specimens.

Fig. 2 .

 2 Fig. 2. Monotonic tensile test strain control.

Fig. 3 .

 3 Fig. 3. (a) Reversed cyclic test strain control without dwell time. (b) Reversed cyclic test strain control with dwell time.

Fig. 4

 4 Fig. 4 shows the normalised values of hardness for four tested temperatures: 25 • C, 75 • C, 125 • C and 150 • C. In this figure, all the hardness values were divided by the initial value before ageing. This figure shows an approximately 30% decrease in hardness, followed by stabilisation for 125 • C and 150 • C. The stabilisation level obtained with samples aged at 125 • C was approximately 0.72, very close to the stabilisation level obtained with samples aged at 150 • C.These results suggest that the stabilisation level could be the same for both temperatures, which is consistent with the hypothesis of a single stabilisation state.Regarding the results presented in Fig.4, the decrease in hardness is related to ageing temperature and time. Eq. (1) was chosen to represent the changes observed. This equation corresponds first to an exponential decay to represent the time dependence with three parameters (A, B and ) where the time parameter () follows an Arrhenius law to represent the temperature dependence. The parameters of this equation were determined with data at 75 • C and 150 • C with a mean error of 1.9%. Data at 25 • C and 125 • C were then computed, and the comparison with experimental data gave a relative error of approximately 2.6% (Table3).

Fig. 4 .

 4 Fig.4. Influence of ageing temperature and time on hardness with exponential fit[START_REF] Dompierre | [END_REF].

Fig. 5 .

 5 Fig. 5. (a) Initial microstructure and (b) microstructure after thermal ageing at 125 • C for 384 h.

Fig. 6 .

 6 Fig.6. Grain size distribution depending on the ageing condition.

Fig. 7 .

 7 Fig. 7. (a) Strain-stress curves in various loading conditions. (b) Example of stress relaxation during dwell time.

Fig. 8 .

 8 Fig.8. Influence of the strain rate on the stress obtained when strain = 1%.

Fig. 9 .

 9 Fig. 9. (a) Influence of strain rate on relaxation behaviour. (b) Influence of strain rate on relaxation time.

Fig. 10 .

 10 Fig.10. Cyclic softening curve -strain amplitude = 1% -strain rate = 2.10 -3 s -1 .

Fig. 11 .

 11 Fig. 11. As-received specimens -cyclic hardening/softening curve with a strain amplitude of (a) 1% and (b) 0.2%.

Fig. 12 .

 12 Fig. 12. Decrease in hardness and increase in grain size over time at 125 • C.

Table 1

 1 Ageing conditions.

	Condition ( • C)	Maximum ageing time tested (h)
	75	6500
	125	3000
	150	1350

Table 2

 2 Monotonic and cyclic testing conditions.

	Monotonic tests	Strain rate (s -1 )	2.10 -3	2.10 -4	2.10 -5
		Strain amplitude		0.2-2%
		Dwell time		1 h
	Cyclic tests	Strain rate (s -1 )	2.10 -3		2.10 -4
		Strain amplitude	0.2%	0.6%	1%
		Dwell time		0 or 5 min

Table 3

 3 Parameters of Eq. (1).

	Parameter	Value
	A	0.72
	B	0.23
	0 Ea (activation energy)	3.31 × 10 -8 h 0.77 eV
	k (Boltzmann constant)	8.617 × 10 -5 eV K -1