# Interpersonal coordination in a ball-and-beam paradigm: Transfer of skill from solo action to joint action

Marijn S.J. Hafkamp, Remy Casanova, Reinoud J Bootsma

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Marijn S.J. Hafkamp, Remy Casanova, Reinoud J Bootsma. Interpersonal coordination in a ball-and-beam paradigm: Transfer of skill from solo action to joint action. Human Movement Science, 2025, 101, pp.103365. 10.1016/j.humov.2025.103365. hal-05189960

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Full Length Article

# Interpersonal coordination in a ball-and-beam paradigm: Transfer of skill from solo action to joint action

Marijn S.J. Hafkamp, Remy Casanova, Reinoud J. Bootsma*

*Institut des Sciences du Mouvement, Aix-Marseille Université, CNRS, Marseille, France*

## ARTICLE INFO

**Keywords:**

Interpersonal coordination

Joint action

Cooperation

Motor control

Motor learning

## ABSTRACT

Interpersonal coordination is paramount to the success of a joint action. While the pattern formation process of interpersonal coordination is relatively well understood, models like HKB lack an ecological dimension that relates the motor pattern to the environment. We investigated this dimension in a two-step study using the ball-and-beam paradigm. Participants rolled a ball back-and-forth between two targets on a beam, by manipulating the beam inclination either individually or dyadically. In an (earlier reported) first step, 16 participants performed two solo action sessions of the task, allowing us to identify pertinent performance characteristics. Here, those participants were paired into 8 dyads to perform a joint action session, allowing us to assess the solo-to-joint transfer of those characteristics. Over blocks, dyads improved their performance by increasing the ball speed and accuracy. While the relative variability of the beam's inclination angle decreased, the range and the timing of the beam motion remained unchanged. Variables (indirectly) related to ball speed, such as the range and timing of the beam motion, were strongly propagated from solo to joint action, while the variables related to ball accuracy, like the beam variability, were only moderately transferred. Most dyads established an anti-phase mode of coordination, with a significant decrease in phase variability over blocks. We also observed significant asymmetries in the coordination. Dyad members with a better solo-action performance were more likely to lead the interaction. We concluded that interpersonal coordination in the ball-and-beam paradigm emerged from the interaction, while being constrained by the goal of the task.

## 1. Introduction

Many of our daily life activities cannot be accomplished without some form of coordination between individuals. Consequently, the interpersonal coordination of movements is at the heart of the human behavioral repertoire. Without much effort, we are able to tune into the actions of others and to establish a spatiotemporal coupling between our own movements and those of our co-actors. Think, for instance, about the ease with which we do the dishes together. Or consider the apparent fluency with which we carry a heavy table to the other side of the room together, often without even talking to one another. According to Interpersonal Synergies Theory (IST, Riley et al., 2011), we can achieve such spatiotemporal coupling of movements by linking the degrees of freedom (DFs) of our own perceptuomotor system with those of other individuals, thereby establishing a single coordinative structure. This structure is known as an interpersonal synergy. The core claim of IST is that an interpersonal synergy is established and sustained by the informational and/or

<footnote>* Corresponding author.
E-mail address: reinoud.bootsma@univ-amu.fr (R.J. Bootsma).</footnote>
mechanical coupling between the two or more interacting perception-action systems. With this position, IST falls within a broader
theoretical framework which holds that structures (or patterns) of coordination in complex systems emerge from an autonomous
process of pattern formation (e.g., Cross & Hohenberg, 1993; Haken, 1975). Within human movement sciences, this pattern formation
approach leans heavily on the HKB model of coupled oscillators (Haken et al., 1985). Originally developed in the domain of intra-
individual coordination, HKB models rhythmically moving limbs as coupled and self-sustained oscillators that are dynamically con-
strained to display only two stable patterns of coordination: in-phase (relative phase 0 degrees) or anti-phase (relative phase 180
degrees). Much like metronomes, human limbs can synchronize spontaneously in their swinging upon being coupled (Kelso, 1984). In
a paradigmatic study, Schmidt et al. (1990) showed that such pattern formation can also occur in inter-individual coordination. Dyads
of seated and visually coupled participants could only stably oscillate their lower limbs in in-phase and anti-phase with respect to each
other. In line with the HKB model, in-phase coordination was found to be more stable than anti-phase coordination and an involuntary
transition from the anti-phase mode to the in-phase mode occurred when the frequency of the limb oscillations was stepwise increased.
Although the stability of interpersonal coordination was found to be somewhat lower than that of intrapersonal coordination, the work
of Schmidt et al. (1990) formed the basis for IST as a new perspective on human motor interactions (Schmidt et al., 2011).

In the decades following the work of Schmidt et al. (1990), the finding of spontaneous inter-individual synchronization was
replicated in various other settings in- and outside the laboratory (see Schmidt and Richardson, 2008). From side-to-side pendulum
swinging (Schmidt & O'Brien, 1997), chair rocking (Richardson, Marsh, Isenhower, et al., 2007) and treadmill walking (Van Ulzen
et al., 2008) to elite level sprinting at the World Championships (Varlet & Richardson, 2015): rhythmic interpersonal coordination
consistently turned out to abide by the laws of non-linear dynamics. Moreover, extended versions of the coupled oscillator model
provided new predictions with respect to human interactions (Fuchs & Kelso, 1994). One of these predictions concerned a systematic
deviation from synchrony, also known as detuning. When the intrinsic (eigen)frequencies of two oscillators differ from each other, the
stability of their coordination will be lower and the phase locking will occur on a relative phase that slightly deviates from anti-phase
or in-phase. In particular, the oscillator with the higher intrinsic frequency will take the lead over the other, with a phase lag between
the two that is proportional to the difference in the eigenfrequencies. In a series of experiments with hand-held pendulums of different
lengths, these predictions were shown to hold in the case of inter-individual coordination (Amazeen et al., 1995; Schmidt et al., 1994;
Schmidt & Turvey, 1994). That is, the participant swinging the shorter pendulum -with the higher intrinsic frequency- systematically
led the interaction with a proportional phase lag. These experiments not only provided further support for the conceptualization of
interpersonal coordination in terms of frequency entrainment and phase locking of coupled oscillators (De Poel, 2016). They also
suggested that key phenomena such as who will lead and who will follow are (at least partially) dictated by the dynamical constraints
governing the interaction.

Despite its elegant account for motor stability and social synchronicity, we argue that the pattern formation approach alone has its
limitations in explaining interpersonal coordination. Most prominently, it lacks an ecological dimension that is intrinsic to any joint
action (Fusaroli et al., 2014). As intentional agents living in a meaningful and populated environment (Gibson, 1979; Reed, 1996), we
rarely (if ever) coordinate our movements with others for the sake of the synchronization itself. On the contrary, we coordinate with
others in order to achieve goals (Huys et al., 2018; Lagarde, 2013; Richardson et al., 2015). These goals can be diverse, but they are
directed at our environment and almost always go beyond the dynamics of the motor pattern. Indeed, interpersonal coordination often
allows us to achieve goals that we could not have achieved individually. Together, we can carry heavier and larger pieces of furniture
than we can carry alone. Most often, our action capabilities increase when we decide to coordinate our movements with other in-
dividuals, making available a greater range of affordances in the environment (Baron, 2007; Richardson, Marsh, & Baron, 2007). From
the ecological perspective of agents encountering their environment together (Reed, 1996), we therefore argue that an interpersonal
synergy is more than only a stable mode of coordination constrained by a pattern formation process. It is also a social tool—a collective
effectivity— of individuals actualizing an affordance together (Marsh et al., 2006, 2009). To do so, each individual contributes
something to the synergy, while also leaving something behind. Or, in the words of Baron (2007, p. 186), cooperation involves “a
symmetry-breaking process whereby organisms shift from individual to collective action (...) an ‘unfreezing’ process or qualitative
change”. Given that models like HKB lack a layer of affordances and effectivities, it seems unlikely that the transition from individual to
collective action, as necessitated in joint actions, can be entirely understood through those models alone. As such, we suggest that a
functional level of description is required to conceptualize interpersonal synergies. To appreciate the relevance of this level, we briefly
turn to some of the more recent applications of the HKB model to the domain of joint action.

Cuijpers (2019) investigated rhythmic interpersonal coordination in crew rowing. Rowers are coupled not only visually but also
mechanically, through the boat in which they are seated. This suggests that the motor coordination between them can be understood as
a pattern formation process. Indeed, Cuijpers et al. (2015) found that crew rowers can stably produce both in-phase and anti-phase,
akin to HKB. At the same time, the authors found some notable deviations from the classical model. For instance, both phase patterns
were found to be equally stable, although the anti-phase coordination deviated somewhat from the intended phase. Furthermore,
increases in the stroke frequency resulted in a higher consistency of the crew synchronization, instead of a breakdown in the stability of
coordination (Cuijpers et al., 2017). The main reason for these deviations was the mechanical coupling provided by the boat. This
induced lower fluctuations in the boat velocity and less detrimental boat movements in anti-phase than in in-phase. It is important to
note, here, that the coordination between the strokes is subordinate to the goal of rowing as fast as possible from start to finish. From an
ecological point of view, it is the movement of the boat in relation to the water that matters. The crew coordination constitutes a
collective effectivity—an interpersonal synergy—that allows crew members to row faster than they could have done individually (in a
so-called skiff). Synchronization of strokes is thus a necessary but not sufficient condition for successfully performing the joint action. In
another study, Huys et al. (2018) examined a dyadic rope-turning task. Participants were to turn a rope regularly and at a comfortable
pace, either individually or as a dyad. Although the interpersonal coordination (as captured by the relative phase) between dyad
members was more variable than the within-individual coordination, the regularity of the rope's motion was generally higher in joint action than in solo action. Similar to the crew rowing studies of Cuijpers (2019), the reason for this finding was the presence of a mechanical coupling. Given that the regularity of the rope improved when the turning frequency increased and that the possible turning frequency was higher in joint action than in solo action, the highest rope regularity could be actualized in joint action. Again, dyadic synchronization was subordinate to the goal of turning the rope regularly (and thus at a fast pace).

The work of Cuijpers (2019) and Huys et al. (2018) highlights the necessity for including and distinguishing at least two levels of description in a joint action. One level pertaining to the pattern formation process of interpersonal coordination, which may or may not conceptualize the agents as a pair of coupled oscillators. And another level pertaining to the affordance being actualized in the joint activity, which entails the team's or dyad's interaction with its environment. It is only with inclusion of this second level that joint action, as a social phenomenon embedded in a populated environment, can be fully understood. To further develop this ecological dimension of joint action while remaining in line with the pattern formation approach of IST, we recently introduced a new experimental paradigm: the manual ball-and-beam task (Hafkamp et al., 2023, 2024). In this task, participants have to control a ball rolling on a long and hand-held beam. By continuously tilting the beam up and down, participants aim to roll the ball back and forth between two indicated targets on the beam. More specifically, their goal is to hit the targets as often as possible within the timespan of a two-minute trial, by reversing the direction of the ball motion in the target zones. While the task can be performed individually (i.e., in a solo action setting, Fig. 1A), the current contribution focused on dyadic task performances (i.e., in a joint action setting, see Fig. 1B). In joint action setting, each of the two beam extremities is handled by a different participant such that dyads have to collaborate to control the ball. In doing so, they are not allowed to communicate with each other in our protocol. With this set-up, the ball-and-beam paradigm constitutes a clear example of a purposeful joint action that nevertheless necessitates the emergence of a stable (rhythmic) pattern of interpersonal coordination.

In a first experiment exploring this paradigm (Hafkamp et al., 2023), we found that the task's goal of hitting the targets could be achieved through at least two different patterns of interpersonal coordination. In this experiment, one group of participants first performed a joint action session and then a solo action session (J/S), while another group started with a solo action session, followed by a joint action session (S/J). In the J/S group, most of the dyads moved the beam handles sequentially and in a turn-taking fashion – each dyad member freezing their motion while the other moved. In the S/J group, most of the dyads moved the beam handles simultaneously and in opposite directions – one moving up while the other moved down. Even though the first pattern was superior in terms of coordinative stability, the latter pattern enhanced the action capabilities of the two individuals in comparison to the solo action. That is, only by moving the beam handles simultaneously instead of sequentially, dyads could produce larger beam inclination angles than what they could have produced individually, implying the possibility of more target hits as a dyad than as a soloist. These results illustrate the necessity of an ecological perspective to understand the performance of a joint action. Pattern formation theory alone cannot bring out the crucial fact that simultaneous action of the dyad members affords them to meet their goal of acquiring as many target hits as possible. This fact can only be brought out by a functional analysis of the performance. The question that remains, however, is why individual task experience in the solo action setting was necessary for the simultaneous pattern to emerge. It suggests

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Schematic illustration of the ball-and-beam task configuration used in the solo action setting (A) and in the joint action setting (B).</figcaption></figure>
a particular transfer of skill from solo action to the joint action, since two task settings are equivalent in terms of their goal. In a two-step study, of which the current contribution formed the second step, we set out to investigate this transfer.

In the first step (Hafkamp et al., 2024) we focused on the individual task execution in a solo action setting, with the goal to identify the pertinent characteristics of the individual performances. A group of 16 participants performed two solo action sessions, including trials with different target sizes to vary the accuracy demands of the task. In capturing the participants’ performances, we distinguished between the level of the ball motion and the level of the beam motion. On the level of the ball, performances were characterized by the number of target hits on each trial, but also by the number of (ball) half cycles and by the ratio between the number of hits and the number of half cycles. In other words, the ball level performances featured speed as well as accuracy, which both improved over practice. On the level of the beam, the performances were characterized by three aspects of the inclination angle motion. First of all, the range of motion of the beam. Over practice, individual participants moved the beam extremity up and down with increasingly large excursions so as to increase the speed of the ball. In second place, the consistency of the beam behavior. From trial to trial, participants enhanced the consistency of the beam movements so as to roll the ball more accurately and hit the targets more often. Third and last, the timing of the beam. A within-cycle analysis of the beam motion revealed two different timing (i.e. phase progression) strategies to drive the ball in motion with the beam. While most participants tended to maintain the beam at peak inclination angles for brief, but noticeable, durations (denoted extreme-stop strategy), some were more inclined to briefly halt the beam movement between two peaks, at inclination angle zero (denoted middle-stop strategy). Although the extreme-stop strategy exploits the force of gravity more efficiently than the middle-stop strategy, and thus may lead to more target hits, the timing strategies were found to remain invariant over practice.

In the current contribution we investigated to what extent the identified performance characteristics were propagated from the solo action setting into a joint action setting. To this end, we paired all individuals from Hafkamp et al. (2024) into dyads and had them perform a subsequent joint action session of the ball-and-beam task. Similar to the solo action, the joint action session consisted of trials with different target sizes, again to vary the accuracy demands of the task. Our first research question concerned the level of the ball and the level of the beam, which pertained directly to the purpose of the activity. To what extent can the rolling of the ball and the tilting of the beam in the joint action setting be explained by the dyad members’ performances in solo action? Given the functional equivalence of solo and joint action, we hypothesized a significant behavioral transfer from one setting to the other. That is, we expected the ball’s speed, accuracy and target hits as well as the beam’s range, timing and consistency in the joint action setting to be predictable from the dyad members’ solo action performances. Our second research question concerned the level of interpersonal coordination. Based on earlier research within this paradigm (Hafkamp et al., 2023), we expected simultaneous motion of the two beam extremities, each held by another dyad member. With respect to the relative phase of the interpersonal coordination, we hypothesized that the solo-to-joint transfer of individual participants’ timing strategy would prevent a stable phase locking to occur. That is, we expected each dyad member to retain a unique phase progression profile in their timing, implying a variable phase relationship between the two extremities. Moreover, we expected that the mean relative phase would be affected by inter-participant differences on the levels of the ball and the beam. In analogy to detuning in coupled oscillators (Amazeen et al., 1995; Schmidt & Turvey, 1994), we tentatively hypothesized that the dyad member with the larger number of target hits or half cycles in solo action would take the lead in joint action.

## 2. Methods

Although the research methods deployed in this two-step study were already partially reported in Hafkamp et al. (2024), for reasons of stand-alone readability of the present contribution we here provide a full description of all relevant methodological aspects.

### 2.1. Participants

A group of 16 students and junior staff members from Aix Marseille University (7 women, 9 men, with an average age of $M \pm SD$: 23.3 $\pm$ 2.9 years) participated voluntarily in these studies. None of them had any experience on the task. All participants were free from known motor impairments and reported normal or corrected-to-normal vision. Before the start of the first experimental session the participants were informed about the aim and procedure of the experiment. All participants provided written informed consent before participating in the study. The study was approved by the French National Ethics Committee for Research in Sports and Movement Sciences (CERSTAPS 15/12/2022 IRB00012476-2022-15-12-213) and conducted according to University regulations and the Declaration of Helsinki.

### 2.2. Task and procedure

The task equipment consisted of a standard golf ball (diameter 4.3 cm, weight 46 g) that could roll over a 2-m long, 3 by 3 cm V-shaped iron beam, with tubular handles (length 54 cm, diameter 3 cm) perpendicularly attached to its extremities. Participants’ task was to roll the ball as fast and accurately as possible back-and-forth between two targets clearly visible on the beam. Participants received one point for every time that the ball reversed its direction within a target’s boundaries, with the objective of achieving as many points as possible within every 2-min duration trial. With both target centers located 50 cm from the beam’s midpoint, the center-to-center inter-target distance was 100 cm in all trials. The size of both targets varied from trial to trial and was either small (6 cm), medium (12 cm) or large (24 cm). Bimanually holding one handle allowed a participant to lift or lower a beam extremity and thereby change its inclination angle so as to accelerate or decelerate the ball (see Fig. 1). In the solo action setting, the participant
manipulated one handle while the other was attached to a hinge on a static pole, the height of which was adjusted to the participants' hip height. In the joint action setting, each handle was manipulated by a different participant. In order to compensate for more than 10 cm height differences between participants within a dyad, the smaller of the two participants was placed on a wide and stable wooden platform that allowed 2-cm stepwise height adjustments.

The first part of this two-step study consisted of two sessions of solo action practice on the ball-and-beam task that were completed on two different days, with one or two days between the sessions. A detailed description of findings obtained during these solo action sessions is provided in Hafkamp et al. (2024). The relevant results are recapitulated and provided with the necessary context in the Results section of the current contribution. For the second part of the study, the 16 participants were subsequently paired into 8 dyads (referred to as D1 to D8). Each dyad then performed a session of joint practice on the ball-and-beam task. The present contribution focusses on this joint action session, which was performed three to four weeks after the second solo action session. Pairing of the participants into dyads was based on the average number of target hits of each individual in the second solo action session, such that in each dyad a high performer (HP) was paired to low a performer (LP). Similar to each of the two solo action sessions, the joint action session comprised six blocks of practice. Each block consisted of three trials: one trial with small targets, one with medium targets and one with large targets. The three conditions within a block were presented in a pseudo-randomized order, such that all possible orders of conditions were presented once during a session. Each dyad thus performed a total of 18 trials. After every two blocks of trials, participants were given a short break of 2–3 min. During the trials, dyads were not allowed to talk or communicate otherwise. Talking was allowed during the breaks but could not concern the task.

At the start of each 2-min trial, the ball was placed in the middle of the beam. Verbal signals from the experimenter signaled the beginning ('Go') and the end ('Stop') of a trial. After each trial, participants received verbal feedback on their score, based on experimenter observation. Participants were regularly encouraged to try to improve their score on each new trial, in order to reach the highest score possible in each target size condition. In both the solo and the joint settings, participants/dyads competed indirectly with the others via a leaderboard that expressed the high scores of the other participants or dyads on each target size condition. All of this was in function of motivating the participants to perform optimally on each new trial.

## 2.3. Data acquisition

All sessions took place in a darkened room. To capture the motion of beam, small reflective markers were attached to each handle through a thin, rigid and 10-cm long aluminum rod. In order to capture the motion of the ball, we used a bright yellow colored golf ball. Participants wore dark clothes and black gloves to ensure sufficient contrast with the ball and the two beam markers. All trials were filmed with a GoPro9 camera that was positioned at a 2-m perpendicular distance from the midpoint of the beam. The camera, set in linear lens mode so as to obtain minimal image deformation, was placed on a tripod at a 1-m height. All trials were recorded with a (recalculated) sample frequency of 29.97 Hz and a resolution of 2.7 K.

Video recordings of all trials were imported into Kinovea (open-source software https://www.kinovea.org/), which was used to track the two-dimensional (X, Y) motion of the ball and the two reflective beam markers. X and Y position time series were then analyzed using MATLAB. All timeseries were first filtered using a fourth-order low pass Butterworth filter (cut off frequency of 3 Hz) and trimmed to the 120-s trial duration. The ball's trajectory was subsequently transformed from motion in a two-dimensional space to one-dimensional motion along the beam's longitudinal axis. To do this, we calculated the ball's motion along the virtual vector between the two beam markers, with the inter-marker distance used for calibration. From the trajectories of the two beam markers we derived a single beam inclination angle at each timestep by taking the inverse tangent (atan) of the X and Y coordinates of both markers. The one-dimensional timeseries of the ball position on the beam, on the one hand, and the timeseries of beam's inclination angle, on the other hand, formed the basis of all further analyses of ball and beam behavior in this contribution. The timeseries of the vertical motions of the two hand-held beam extremities were used to analyze interpersonal coordination.

## 2.4. Dependent variables

The dependent variables in this study were related to three distinct levels of the ball-and-beam system. First of all, we addressed the level of the ball's motion between the two targets (Section 2.4.1). In second place, we analyzed the level of the beam's inclination angle (Section 2.4.2). In the third place, we investigated the level of coordination between the individuals in each dyad (Section 2.4.3).

### 2.4.1. Ball motion

Using a peak finding algorithm that split the ball position timeseries into separate half cycles, we computed three variables to assess the ball's motion between the targets. Firstly, we counted the number of target hits (TH) on each trial, which we defined as the number of ball motion reversals within the (recalculated) target zones of each task condition. Secondly, we counted the total number of ball half cycles (HC) on each trial. Given that the trials were time-constrained, this number was proportionally related to the average ball speed on a trial. Thirdly, we divided the number of target hits by the number of half cycles on a trial. Since the TH/HC-ratio indicated the accuracy on a trial, we refer to this variable as the ball's accuracy ratio (AR).

### 2.4.2. Beam motion

From the timeseries of the beam's inclination angle, we derived three variables to capture the relevant characteristics of the beam behavior. Firstly, we computed the beam's range of motion (ROM), which we defined as the average peak-to-peak amplitude of the beam inclination timeseries. Using a peak-finding algorithm, we determined the mean angular distance between the maximum and
minimum beam inclination angles in each half cycle. Secondly, we assessed the relative variability of the beam's angular motion using the coefficient of variation (CV). To compute the CV, we divided the inclination angle timeseries into separate cycles on the basis of the ball position peaks. We used the ball's cycles because these were more consistent in comparison to the beam's cycles. We then time-normalized all cycles to 100 timepoints and calculated the standard deviation (std) of beam inclination across cycles at each timepoint. We averaged all std.'s to come to one std., which we then divided by the ROM on that trial to compute the trial's CV. Thirdly, we quantified the shape of the beam's inclination angle motion using the Shape ratio (SR). This measure was inspired by the shape of the inclination angle frequency distributions and provided an indication of the time structure of the beam's motion pattern (see Hafkamp et al., 2024, for examples). We computed SR as the ratio of the average absolute inclination angle over half of the ROM (i.e., peak-to-peak amplitude) of the beam, with higher shape ratios thus indicating longer time spent at large inclination angle magnitudes.

### 2.4.3. Interpersonal coordination

As a first assessment of the dyadic interpersonal coordination, we analyzed the relative direction of the vertical motions of the two beam extremities (each extremity being controlled by a different participant). More specifically, we calculated the proportion of the total trial duration (120 s) during which both participants moved in the opposite direction (complementarity ratio, CompR), in the same direction (supplementarity ratio, SuppR) or that at least one of the two individuals moved slower than a threshold value of 0.05 m/s (freezing ratio, FreezR) (see Hafkamp et al., 2023). To investigate the spatiotemporal characteristics of this interpersonal coordination in more detail, we computed the continuous relative phase (CRP) between the vertical movements of the two beam extremities. To this end, we first centered the two vertical extremity position timeseries, removed any linear trends and normalized the signals by dividing them by their standard deviation. We then transformed them into analytic signals using the Hilbert transform (Lamb & Stöckl, 2014) and derived the continuous phase angles (in degrees) from that. As a last step, we subtracted the phase angle of the LP from the phase angle of the HP to arrive at the CRP. Given this definition, a CRP > 180 deg. indicated a leading role for HP, while a CRP < 180 deg. indicated that a leading role for LP. In addition to computing the mean CRP on each trial, we also assessed the within-trial variability of the coordination by calculating the standard deviation of the CRP on each trial (CRP<sub>STD</sub>) and we assessed the mean absolute deviation of the CRP from anti-phase (CRP<sub>DEV</sub>).

## 2.5. Statistical analyses

In order to compare the dyad's performances across target size conditions and over practice blocks, we performed 3 × 6 repeated measures ANOVAs on all dependent variables. Each ANOVA contained the within-participant factors Condition (3 target sizes) and Practice (6 blocks). The dependent variables to which these analyses were applied included TH, HC, AR, ROM, CV, SR, CompR, SuppR, FreezR, CRP<sub>STD</sub> and CRP<sub>DEV</sub>. For all analyses the significance level was set to $\alpha = 0.05$. Greenhouse-Geisser corrections were applied when violations of the sphericity assumption were detected. Post-hoc tests were performed using Bonferroni corrections. Where appropriate means and standard deviations are reported as $M \pm SD$.

Next, we analyzed the relationship between the dyadic performances in the joint action setting and the dyad members' individual performances in the (second) solo action setting. For each performance variable related to the ball motion (TH, HC, AR) and the beam motion (ROM, CV and SR), we did this via a Pearson correlation analysis between the joint action performances of each dyad and the sum of the corresponding individual's solo action performances. Per correlation analysis, we took the mean of all 6 blocks in the (solo/joint action) session, and we performed the analyses separately for each target size condition. In our exploratory analyses, we also conducted the same analysis by only taking the first three or last three blocks of either session, but this did not lead to significantly different results. Hence, we decided to report only the analysis in which we took the mean of all 6 blocks per (solo/joint action) session. We used the proportion of explained variance captured by $R^2$ as the measure of strength of the solo-joint relation. Given that each Pearson correlation was based on 8 datapoints (dyads) and $p = 0.05$ for $r(6) = 0.707$, we interpreted a $R^2 > 0.50$ (i.e., explaining more than 50 % of the variance) as indicative of a significant explanatory value of that variable in the transfer from solo to joint task performance.

Finally, at the level of interpersonal coordination, we analyzed (again using Pearson correlation analyses) the relation between the mean relative phase observed for each dyad and the mean performance differences (in TH, HC, AR, ROM, CV and SR) of the dyad members in the solo action setting, again performing these analyses separately for each target size condition. In doing so, we assessed whether any potential leader-follower dynamics could be explained from inter-individual differences in the preceding solo action setting.

## 3. Results

In presenting the results of this study, we progressively move along the three above-defined levels of the ball-and-beam system: the level of the ball motion (Section 3.1), the level of the beam motion (Section 3.2) and the level of interpersonal coordination (Section 3.3). At each level, we first report the results from the Condition (3 target sizes) x Practice (6 blocks) repeated measures ANOVAs on the dependent variables. Thereafter, we relate the dyad's performances in the joint action session to the individual's performances in the preceding solo action sessions. On the levels of the ball motion and the beam motion, we did this via correlational analyses between the indicator scores observed in the joint action session (JAS) and the (sum of the) indicator scores of the corresponding individual in the second solo action session (SAS). On the level of the interpersonal coordination, we related the dyad's mean relative phase -indicative of a leader-follower dynamics- to differences in the dyad members' performances in the solo action session.
## 3.1. Ball motion

### 3.1.1. Target hits

As shown in the upper left panel of Fig. 2, the number of target hits (TH) increased significantly over practice blocks ($F(5, 35) = 9.95, p < 0.001, \eta_p^2 = 0.59$). A post-hoc test revealed that the number of target hits was higher in the last (B6) than in the first (B1) block ($p < 0.001$). The TH was significantly different between target sizes ($F(2, 14) = 102.97, p < 0.001, \eta_p^2 = 0.94$), such that larger targets allowed for more target hits per trial. The increase in TH over blocks was not significantly different per target size ($F(10, 70) = 1.17, p = 0.329, \eta_p^2 = 0.14$).

### 3.1.2. Half cycles

The number of half cycles (HC) increased significantly over practice blocks (upper middle panel of Fig. 2, $F(2,20, 15.38) = 7.04, p = 0.006, \eta_p^2 = 0.50$, B1-B6 post hoc: $p < 0.001$). The HC was significantly higher in the larger target size conditions ($F(2, 14) = 170.57, p < 0.001, \eta_p^2 = 0.96$), while the increase in the HC over blocks was not significantly different per target size ($F(10, 70) = 1.67, p = 0.105, \eta_p^2 = 0.19$).

### 3.1.3. Accuracy ratio

The accuracy ratio (AR) increased significantly over practice (upper right panel Fig. 2, $F(5, 35) = 4.76, p = 0.002, \eta_p^2 = 0.41$, B1-B5 post-hoc $p = 0.003$), while revealing a slight decrease in block 6 (B1-B6 post hoc: $p = 0.333$). The AR was significantly higher for larger target sizes ($F(2, 14) = 43.24, p < 0.001, \eta_p^2 = 0.86$). The changes in the AR over practice were not significantly different per target size ($F(10, 70) = 0.94, p = 0.560, \eta_p^2 = 0.12$).

In summary, all ball-level variables (i.e., TH, HC and AR) increased significantly over blocks and were higher when the target sizes were larger. None of these variables showed an interaction effect between target size conditions and practice blocks.

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Ball motion variables: number of target hits (TH, left column), number of ball half cycles (HC, middle column) and accuracy ratio (AR, right column). The upper row of panels shows the evolution of each dependent variable across practice blocks, split per target size condition (small: green, medium: blue and large: red). Error bars represent between-participant standard deviations. The lower row of panels displays the dyad's joint action session (JAS) performances as a function of the sum of the low and high performing individual's solo action session ($SAS_{LP} + SAS_{HP}$) performances. Separate regression lines are fitted through the data points for each target size condition (small: green, medium: blue and large: red), whereas the dotted line represents JAS performance equal to the mean of $SAS_{LP}$ and $SAS_{HP}$ performances. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
### 3.1.4. Solo-to-joint transfer

Next, we investigated how the dyad's joint performances were related to the preceding solo performances of the two dyad members. Using correlation analyses we evaluated how much of the variance in the dyad's performances could be explained from the sum of the corresponding individuals' performances on the second (and last) solo action session. For each dyad Table 1 reports the average performances of the low and high performing individual in the (second) solo action session (SAS<sub>LP</sub> and SAS<sub>HP</sub>), together with the dyad's average joint action session performance (JAS). The results are reported for dependent variables TH, HC and AR.

In order to visualize the relations between dyadic and summed individual performance scores, we plotted for each indicator the JAS performance of a dyad against the sum of the corresponding SAS<sub>LP</sub> and SAS<sub>HP</sub> performances (lower row of panels in Fig. 2). As can be appreciated from these graphs, the TH and the HC were generally higher in the joint performances than in the solo performances, while the reverse was true for the AR, in particular for dyads with a low accuracy ratio. Yet, for all three dependent variables the joint performances were positively correlated with the sum of the solo performances. Table 2 reports the proportion of explained variance of each regression analysis, split per target size (L: large; M: medium, S: small). On average, the majority (R<sup>2</sup> = 0.60) of the variance in the dyads' number of target hits TH could be explained by the solo action performances, with an R<sup>2</sup> > 0.50 in two of the three target size conditions. The solo action performances had an even higher explanatory value with respect to the number of half cycles HC (average R<sup>2</sup> = 0.73, with R<sup>2</sup> > 0.50 in two of the three conditions). The variance in the dyads' accuracy ratio, in contrast, was only moderately explained by the solo sessions (average R<sup>2</sup> = 0.46), with an R<sup>2</sup> > 0.50 in only one condition.

Taken together, we conclude that the variance in the dyads' ball-level performances could be rather well explained by the (sum of) the corresponding individual's solo performances. In particular, it appeared that it was the speed of the ball rather than its accuracy that was propagated from the solo to the joint action setting.

## 3.2. Beam motion

### 3.2.1. Range of motion

As can be seen in the upper left panel of Fig. 3, the beam's range of motion (ROM) did not change significantly over blocks (F(1.78, 12.42) = 1.49, *p* = 0.262, *η*<sub>p</sub><sup>2</sup> = 0.18). The ROM was significantly different over target size conditions (F(2, 14) = 40.58, *p* < 0.001, *η*<sub>p</sub><sup>2</sup> = 0.85), such that the ROM was higher when the target size was larger. The Condition x Practice interaction was not significant (F(10, 70) = 1.66, *p* = 0.109, *η*<sub>p</sub><sup>2</sup> = 0.19).

### 3.2.2. Coefficient of variation

The beam's motion coefficient of variation (CV) decreased significantly over blocks, as shown in the upper middle panel of Fig. 3 (F(2.19, 15.31) = 4.87, *p* = 0.021, *η*<sub>p</sub><sup>2</sup> = 0.41). A post-hoc analysis demonstrated that the CV was significantly lower in B6 than in B1 (*p* = 0.013). The CV was different over target size conditions, with the CV being lower when the target size was larger (F(2, 14) = 18.20, *p* <

**Table 1**

The average ball-level performances of each dyad (D1-D8) split per target size condition (Large: L, Medium: M, Small: S). Column 3 to 5 display the number of target hits (TH), column 6 to 8 the number of half cycles (HC) and column 9 to 11 the accuracy ratio (AR). For each dependent variable, the table shows the mean of the 2nd solo action session of the low performer (SAS<sub>LP</sub>), the mean of the 2nd solo action session of the high performer (SAS<sub>HP</sub>) and the mean of the joint action session of the dyads (JAS).

<table><thead><tr><th rowspan="2">Dyad</th><th rowspan="2">Cond</th><th colspan="3">TH (n)</th><th colspan="3">HC (n)</th><th colspan="3">AR (%)</th></tr><tr><th>SAS<sub>LP</sub></th><th>SAS<sub>HP</sub></th><th>JAS</th><th>SAS<sub>LP</sub></th><th>SAS<sub>HP</sub></th><th>JAS</th><th>SAS<sub>LP</sub></th><th>SAS<sub>HP</sub></th><th>JAS</th></tr></thead><tbody><tr><td rowspan="3">D1</td><td>L</td><td>48.2</td><td>57.2</td><td>58.3</td><td>50.7</td><td>60.5</td><td>69.0</td><td>95.1</td><td>94.5</td><td>84.4</td></tr><tr><td>M</td><td>39.8</td><td>44.8</td><td>44.8</td><td>48.0</td><td>54.8</td><td>62.7</td><td>83.0</td><td>81.3</td><td>71.8</td></tr><tr><td>S</td><td>28.8</td><td>32.7</td><td>34.5</td><td>44.2</td><td>49.3</td><td>52.5</td><td>65.4</td><td>66.1</td><td>65.7</td></tr><tr><td rowspan="3">D2</td><td>L</td><td>42.0</td><td>55.2</td><td>53.7</td><td>47.7</td><td>68.0</td><td>72.8</td><td>88.0</td><td>80.9</td><td>73.9</td></tr><tr><td>M</td><td>29.5</td><td>42.7</td><td>33.5</td><td>41.8</td><td>59.8</td><td>64.7</td><td>70.6</td><td>71.3</td><td>51.9</td></tr><tr><td>S</td><td>20.8</td><td>28.7</td><td>25.3</td><td>33.2</td><td>50.0</td><td>59.2</td><td>63.0</td><td>57.2</td><td>42.8</td></tr><tr><td rowspan="3">D3</td><td>L</td><td>49.5</td><td>53.7</td><td>61.7</td><td>49.7</td><td>59.7</td><td>68.2</td><td>99.7</td><td>90.4</td><td>90.3</td></tr><tr><td>M</td><td>40.0</td><td>46.2</td><td>47.0</td><td>46.3</td><td>54.7</td><td>63.3</td><td>86.4</td><td>84.6</td><td>74.1</td></tr><tr><td>S</td><td>34.3</td><td>35.2</td><td>36.5</td><td>42.7</td><td>43.3</td><td>55.7</td><td>80.6</td><td>81.9</td><td>65.7</td></tr><tr><td rowspan="3">D4</td><td>L</td><td>51.3</td><td>58.7</td><td>61.2</td><td>54.0</td><td>62.0</td><td>72.7</td><td>95.1</td><td>94.6</td><td>84.2</td></tr><tr><td>M</td><td>41.2</td><td>43.0</td><td>50.2</td><td>44.8</td><td>54.7</td><td>68.8</td><td>91.9</td><td>78.3</td><td>72.9</td></tr><tr><td>S</td><td>31.5</td><td>32.5</td><td>36.0</td><td>37.0</td><td>48.2</td><td>63.8</td><td>85.3</td><td>67.4</td><td>56.4</td></tr><tr><td rowspan="3">D5</td><td>L</td><td>32.0</td><td>47.5</td><td>47.8</td><td>38.2</td><td>47.7</td><td>53.7</td><td>85.0</td><td>99.6</td><td>88.9</td></tr><tr><td>M</td><td>27.2</td><td>39.8</td><td>39.5</td><td>33.3</td><td>43.2</td><td>48.7</td><td>82.0</td><td>92.4</td><td>81.5</td></tr><tr><td>S</td><td>21.0</td><td>31.8</td><td>32.0</td><td>27.3</td><td>37.8</td><td>41.8</td><td>76.9</td><td>84.2</td><td>76.3</td></tr><tr><td rowspan="3">D6</td><td>L</td><td>43.5</td><td>50.5</td><td>57.3</td><td>47.8</td><td>53.8</td><td>61.0</td><td>91.0</td><td>93.8</td><td>93.9</td></tr><tr><td>M</td><td>33.0</td><td>42.2</td><td>45.7</td><td>43.2</td><td>50.0</td><td>51.3</td><td>76.6</td><td>84.3</td><td>88.9</td></tr><tr><td>S</td><td>27.2</td><td>35.2</td><td>37.5</td><td>37.5</td><td>43.0</td><td>45.3</td><td>72.5</td><td>81.7</td><td>82.7</td></tr><tr><td rowspan="3">D7</td><td>L</td><td>37.2</td><td>49.5</td><td>57.7</td><td>41.7</td><td>61.8</td><td>66.0</td><td>89.3</td><td>80.6</td><td>87.3</td></tr><tr><td>M</td><td>31.5</td><td>41.3</td><td>41.8</td><td>39.2</td><td>50.7</td><td>60.2</td><td>80.5</td><td>81.9</td><td>69.5</td></tr><tr><td>S</td><td>19.7</td><td>30.3</td><td>26.2</td><td>39.2</td><td>36.3</td><td>51.2</td><td>50.2</td><td>83.4</td><td>50.5</td></tr><tr><td rowspan="3">D8</td><td>L</td><td>51.3</td><td>59.2</td><td>65.3</td><td>61.0</td><td>68.5</td><td>83.0</td><td>84.4</td><td>86.6</td><td>78.7</td></tr><tr><td>M</td><td>36.5</td><td>45.7</td><td>42.2</td><td>54.8</td><td>55.0</td><td>75.8</td><td>67.0</td><td>82.8</td><td>54.9</td></tr><tr><td>S</td><td>29.3</td><td>32.7</td><td>28.0</td><td>48.2</td><td>41.2</td><td>67.8</td><td>61.7</td><td>80.8</td><td>41.1</td></tr></tbody></table>
<caption>**Table 2**</caption>

The proportions of explained variance (R²) derived from the correlation analyses between the dyad's joint action performances and the sums of the corresponding individual's solo action performances at the ball motion level. The rows are split per target size condition (L: Large, M: Medium, S: Small). The lower row reports the average over target size conditions, while the columns show respectively the number of target hits (TH), the number of half cycles (HC) and the accuracy ratio (TH/HC). Asterisks (*) denote R² values larger than 0.50.

<table><thead><tr><th>Condition</th><th>TH</th><th>HC</th><th>AR</th></tr></thead><tbody><tr><td>L</td><td>0.70*</td><td>0.98*</td><td>0.36</td></tr><tr><td>M</td><td>0.49</td><td>0.75*</td><td>0.59*</td></tr><tr><td>S</td><td>0.62*</td><td>0.45</td><td>0.42</td></tr><tr><td>Average</td><td>0.60*</td><td>0.73*</td><td>0.46</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Beam motion variables: range of motion (ROM, left column), coefficient of variation (CV, middle column) and shape ratio (SR, right column). The upper row shows the evolution of each dependent variable across practice blocks, split per target size condition (small: green, medium: blue and large: red). Error bars represent between-participant standard deviations. The lower row displays the dyad's joint action session (JAS) performances as a function of the sum of the low and high performing individual's solo action session (SAS<sub>LP</sub> + SAS<sub>HP</sub>) performances. Separate regression lines are fitted through the data points for each target size condition (small: green, medium: blue and large: red), whereas the dotted line represents JAS performance equal to the mean of SAS<sub>LP</sub> and SAS<sub>HP</sub>) performances. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

0.001, $\eta_p^2 = 0.72$). The Condition x Practice interaction was not significant ($F(10, 70) = 0.45, p = 0.917, \eta_p^2 = 0.06$).

### 3.2.3. Shape ratio

The beam's motion shape ratio (SR) did not change significantly over blocks (Fig. 3, upper right panel, $F(1.75, 12.25) = 0.58, p = 0.553, \eta_p^2 = 0.08$). The SR was significantly higher in larger target sizes ($F(1.16, 8.11) = 24.05, p < 0.001, \eta_p^2 = 0.76$). The Condition x Practice interaction was not significant ($F(10, 70) = 0.65, p = 0.768, \eta_p^2 = 0.09$).

All in all, only the CV decreased somewhat over blocks, while the ROM and the SR did not change significantly with practice. Thus, only the relative variability of the beam's inclination decreased over practice. All beam-related variables were different across target
sizes, without statistically significant evolutions of these differences over practice.

### 3.2.4. Solo-to-joint transfer

For each beam-level dependent variable, we subsequently assessed how much of the variance in the joint performances could be explained by the sum to the dyad members' performances on the last solo action session. Table 3 reports the average performances of the low and high performing individual in the (second) solo action session (SAS<sub>LP</sub> and SAS<sub>HP</sub>), together with the dyad's average joint action session performance (JAS). The results are reported for variables ROM, SR and CV.

In the lower row of panels in Fig. 3, we plotted the JAS beam-level performance indicators of each dyad against the sum of the corresponding SAS<sub>LP</sub> and SAS<sub>HP</sub> indicators. While the ROM was consistently larger in the joint performances than in the solo performances, the CV and SR were similar in size for both sessions. For all three dependent variables, the joint performances were positively correlated with the sum of the solo performances. Table 4 reports the explained variance from each correlation analysis, split per target size. The ROM of the dyads could be well explained by the solo action sessions (average R<sup>2</sup> = 0.79), with an R<sup>2</sup> > 0.50 in all three target size conditions. The beam's variability as captured by CV was only moderately (average R<sup>2</sup> = 0.52) explained by the solo performances, with an R<sup>2</sup> > 0.50 in two of the three conditions. The beam's timing, in the form of the shape ratio SR, could be well explained (average R<sup>2</sup> = 0.67) from the solo conditions, with R<sup>2</sup> > 0.50 in all three conditions.

In summary, the variance in the dyad's beam-level performances could be well explained by the sum of the solo action performances. The solo-to-joint transfer of beam behavior was the largest for the range (ROM) and the timing (SR) of the movement. The transfer was notably smaller for the relative variability (CV) of the beam movement, with exception of the smallest target size condition.

### 3.3. Interpersonal coordination

For some dependent variables, notably those (indirectly) related to the ball's speed, like TH and HC, but also ROM and SR, the joint performances were quite well accounted for by the corresponding individual's solo performances. For other variables, like those (indirectly) associated with the ball's accuracy, the AR and the CV, the joint performances were less well accounted for by the solo performances. However, for none of the indicators could the solo action performances *fully* predict the outcome of the joint action session. Thus, an important part of the dyadic performances emerged from the interaction between the two individuals. To capture this interaction, we proceeded to the level of the interpersonal coordination. All analyses on this level were based on the timeseries of the vertical motions of the two hand-held beam extremities.

#### 3.3.1. The beam extremities

In Fig. 4 the time evolutions of the two beam extremities (i.e., the two handles) are visualized for block 6 of the joint action session.

<caption>**Table 3**</caption>

The average beam-level performances of each dyad (D1-D8) split per target size condition (Large: L, Medium: M, Small: S). Column 3 to 5 display the range of motion (ROM), column 6 to 8 the coefficient of variation (CV) and column 9 to 11 the shape ratio (SR). For each dependent variable, the table shows the mean of the 2nd solo action session of the low performer (SAS<sub>LP</sub>), the mean of the 2nd solo action session of the high performer (SAS<sub>HP</sub>) and the mean of the joint action session of the dyads (JAS).

<table><thead><tr><th rowspan="2">Dyad</th><th rowspan="2">Cond</th><th colspan="3">ROM (deg)</th><th colspan="3">CV (prop.)</th><th colspan="3">SR (%)</th></tr><tr><th>SAS<sub>LP</sub></th><th>SAS<sub>HP</sub></th><th>JAS</th><th>SAS<sub>LP</sub></th><th>SAS<sub>HP</sub></th><th>JAS</th><th>SAS<sub>LP</sub></th><th>SAS<sub>HP</sub></th><th>JAS</th></tr></thead><tbody><tr><td rowspan="3">D1</td><td>L</td><td>17.5</td><td>23.1</td><td>36.8</td><td>0.066</td><td>0.047</td><td>0.091</td><td>70.5</td><td>78.3</td><td>67.7</td></tr><tr><td>M</td><td>16.9</td><td>19.7</td><td>30.8</td><td>0.062</td><td>0.050</td><td>0.086</td><td>67.1</td><td>78.2</td><td>65.6</td></tr><tr><td>S</td><td>15.1</td><td>16.2</td><td>23.2</td><td>0.077</td><td>0.081</td><td>0.129</td><td>63.8</td><td>76.4</td><td>62.5</td></tr><tr><td rowspan="3">D2</td><td>L</td><td>16.6</td><td>35.6</td><td>42.0</td><td>0.077</td><td>0.088</td><td>0.087</td><td>71.5</td><td>64.8</td><td>67.9</td></tr><tr><td>M</td><td>14.1</td><td>27.3</td><td>33.3</td><td>0.111</td><td>0.096</td><td>0.101</td><td>65.9</td><td>64.7</td><td>66.7</td></tr><tr><td>S</td><td>10.7</td><td>20.0</td><td>29.7</td><td>0.138</td><td>0.118</td><td>0.131</td><td>56.1</td><td>63.8</td><td>63.8</td></tr><tr><td rowspan="3">D3</td><td>L</td><td>16.2</td><td>23.8</td><td>31.3</td><td>0.111</td><td>0.085</td><td>0.071</td><td>76.4</td><td>77.8</td><td>81.3</td></tr><tr><td>M</td><td>15.2</td><td>21.5</td><td>29.4</td><td>0.106</td><td>0.119</td><td>0.115</td><td>76.2</td><td>71.7</td><td>76.2</td></tr><tr><td>S</td><td>13.8</td><td>16.6</td><td>25.3</td><td>0.150</td><td>0.173</td><td>0.163</td><td>73.0</td><td>62.4</td><td>68.4</td></tr><tr><td rowspan="3">D4</td><td>L</td><td>21.8</td><td>25.6</td><td>43.3</td><td>0.094</td><td>0.045</td><td>0.096</td><td>63.7</td><td>71.7</td><td>66.4</td></tr><tr><td>M</td><td>18.2</td><td>22.1</td><td>41.6</td><td>0.125</td><td>0.076</td><td>0.118</td><td>52.1</td><td>69.5</td><td>62.2</td></tr><tr><td>S</td><td>14.2</td><td>18.2</td><td>37.2</td><td>0.132</td><td>0.123</td><td>0.120</td><td>47.0</td><td>67.0</td><td>60.5</td></tr><tr><td rowspan="3">D5</td><td>L</td><td>12.8</td><td>15.9</td><td>22.2</td><td>0.178</td><td>0.113</td><td>0.127</td><td>68.5</td><td>72.6</td><td>67.2</td></tr><tr><td>M</td><td>10.2</td><td>14.8</td><td>22.6</td><td>0.179</td><td>0.118</td><td>0.156</td><td>64.8</td><td>65.4</td><td>58.4</td></tr><tr><td>S</td><td>7.8</td><td>13.8</td><td>19.0</td><td>0.136</td><td>0.151</td><td>0.158</td><td>55.9</td><td>57.1</td><td>51.2</td></tr><tr><td rowspan="3">D6</td><td>L</td><td>17.4</td><td>26.0</td><td>30.1</td><td>0.080</td><td>0.108</td><td>0.094</td><td>66.5</td><td>59.5</td><td>60.2</td></tr><tr><td>M</td><td>15.1</td><td>22.4</td><td>24.7</td><td>0.117</td><td>0.131</td><td>0.110</td><td>62.4</td><td>57.8</td><td>53.4</td></tr><tr><td>S</td><td>13.0</td><td>17.9</td><td>21.6</td><td>0.142</td><td>0.145</td><td>0.129</td><td>55.0</td><td>55.3</td><td>50.0</td></tr><tr><td rowspan="3">D7</td><td>L</td><td>11.9</td><td>30.9</td><td>31.7</td><td>0.087</td><td>0.100</td><td>0.082</td><td>81.8</td><td>61.8</td><td>69.6</td></tr><tr><td>M</td><td>10.5</td><td>22.5</td><td>27.0</td><td>0.095</td><td>0.133</td><td>0.101</td><td>83.0</td><td>55.0</td><td>68.8</td></tr><tr><td>S</td><td>10.7</td><td>15.4</td><td>22.1</td><td>0.117</td><td>0.174</td><td>0.171</td><td>82.0</td><td>46.5</td><td>65.8</td></tr><tr><td rowspan="3">D8</td><td>L</td><td>30.1</td><td>41.2</td><td>54.5</td><td>0.103</td><td>0.094</td><td>0.100</td><td>65.3</td><td>57.2</td><td>62.8</td></tr><tr><td>M</td><td>27.5</td><td>27.8</td><td>50.5</td><td>0.138</td><td>0.109</td><td>0.111</td><td>61.9</td><td>52.2</td><td>58.9</td></tr><tr><td>S</td><td>22.1</td><td>17.4</td><td>44.2</td><td>0.157</td><td>0.136</td><td>0.127</td><td>59.4</td><td>49.6</td><td>56.0</td></tr></tbody></table>
<caption>**Table 4**</caption>

The proportions of explained variance (R²) derived from the correlation analyses between the dyad's joint action performances and the sums of the corresponding individual's solo action performances at the ball motion level. The rows are split per target size condition (L: Large, M: Medium, S: Small). The lower row reports the average over target size conditions, while the columns show respectively the range of motion (ROM), the coefficient of variation (CV) and the shape ratio (SR). Asterisks (*) denote R² values larger than 0.50.

<table><thead><tr><td>Condition</td><td>Range of motion</td><td>Coeff. of variation</td><td>Shape ratio</td></tr></thead><tbody><tr><td>L</td><td>0.88*</td><td>0.27</td><td>0.74*</td></tr><tr><td>M</td><td>0.80*</td><td>0.59*</td><td>0.65*</td></tr><tr><td>S</td><td>0.70*</td><td>0.70*</td><td>0.61*</td></tr><tr><td>Average</td><td>0.79*</td><td>0.52*</td><td>0.67*</td></tr></tbody></table>

More specifically, the graphs display the vertical positions (i.e., the height) of the two beam extremities as a function of the standardized position of the ball on the beam. These graphs reveal that the two agents generally moved the beam extremities in opposite directions as a function of the ball's motion, leading to X-shaped evolutions. At the same time, the graphs show notable differences between the eight dyads. While some dyads moved the beam handles in a symmetrical fashion (e.g., D2 and D3), others moved asymmetrically (e.g., D6), implying that one of the individuals adopted a larger handle range of motion than the other. What remains invisible in Fig. 4, however, is how both individual's contributions related to each other in a spatiotemporal sense. To unravel this, we employed two different analyses. First, we computed the relative amount of motion of the two handles and thereafter we assessed the relative phase between them.

### 3.3.2. Freezing and freeing

We calculated the proportional amount of time that both beam extremities moved in the opposite direction (Complementarity Ratio CompR), in the same direction (Supplementarity Ratio SuppR) or that at least one of them moved slower than a threshold value of 0.05 m/s (Freezing Ratio FreezR). The evolution of these three variables over blocks, split per target size condition, is shown in Fig. 5. Using a 3 × 6 repeated measures ANOVA, we compared all ratios across target size conditions and practice blocks. As can be seen in the left panel of Fig. 5, CompR increased significantly over blocks ($F(2.16, 15.14) = 4.37, p = 0.029, \eta_p^2 = 0.39$). CompR tended to be lower for the smallest target size ($F(1.13, 7.91) = 5.18, p = 0.050, \eta_p^2 = 0.43$) while CompR's evolution over blocks was not significantly different across target sizes ($F(10, 70) = 0.72, p = 0.703, \eta_p^2 = 0.09$). The change in FreezR over the 6 blocks did not reach statistical significance ($F(2.10, 14.70) = 3.36, p = 0.061, \eta_p^2 = 0.32$), despite showing a decreasing trend over practice (Fig. 5, middle panel). A comparison between the target sizes revealed that the FreezR was not significantly different across the conditions ($F(1.12, 7.87) = 2.49, p = 0.153$,

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** The vertical positions (in cm) of the two beam extremities as a function of the standardized [0 to 1] ball position on the beam in block 6 of the joint action session. Each row represents one target size conditions (large, medium, small), while the columns represent the eight dyads (D1-D8). In each graph, the beam extremity of the high performer is displayed in red, while the beam extremity of the low performer is represented in blue. Generally, dyad members moved the beam extremities in opposite directions as a function of the position of the ball, leading to crossing trajectories. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Complementarity ratio (CompR), freezing ratio (FreezR) and supplementarity ratio (SuppR) as a function of practice blocks under each of the three target size conditions (small: green, medium: blue, and large: red). Error bars represent between participant standard deviations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

$\eta_p^2 = 0.26)$. Lastly, the SuppR (Fig. 5, right panel) did not change significantly over practice blocks ($F(1.21, 8.44) = 3.82, p = 0.417, \eta_p^2 = 0.35$) and was not significantly different across target sizes ($F(1.62, 11.37) = 0.89, p = 0.080, \eta_p^2 = 0.11$).

In summary, the results show that the dyads in this experiment moved complementary to each other for the majority of the trial duration. Moreover, the proportional amount of time that they did so increased over blocks, showing an increased tendency to display a mutually dependent form of interpersonal coordination. The comparisons across target sizes did not indicate any noteworthy differences in the between-agent mode of coordination.

### 3.3.3. Continuous relative phase

We assessed the continuous relative phase between the vertical positions of the two beam extremities. To provide an indication of the variability in the CRP over the course of a trial, Fig. 6 presents examples of the CRP timeseries of two representative dyads (D1 and D3) under the large and small target size condition of block 6. As can be appreciated from these graphs, the variability was generally higher for the smaller targets. In Fig. 7, we show the distributions of the CRP of all dyads in block 6 of the session. The relative phase in most dyads varied around 180 degrees. This confirms that most dyads, indeed, roughly moved the beam handles in opposite directions. However, the bandwidth of variability with which they did so varied strongly between the dyads. While some like D3 and D7 displayed a low CRP variability, others like D5 and D8 showed a much larger variability. To compare the variability across target size conditions (3 levels) and blocks of practice (6 levels), we performed a 3 × 6 repeated measures ANOVA on the CRP<sub>STD</sub>. As shown in the left panel of Fig. 8, the CRP<sub>STD</sub> decreased significantly over practice blocks ($F(2.73, 19.11) = 8.76, p < 0.001, \eta_p^2 = 0.56$). Also, the CRP<sub>STD</sub> was significantly different across conditions ($F(1.33, 9.30) = 39.90, p < 0.001, \eta_p^2 = 0.85$), such that the smaller target sizes evoked a larger variability in the interpersonal coordination. The decrease in the CRP<sub>STD</sub> was not significantly different across conditions ($F(10, 70) = 0.68, p = 0.744, \eta_p^2 = 0.09$).

The graphs in Fig. 7 not only show differences in the bandwidth of the CRP distributions. They also provide an indication of the symmetry of the interaction in each dyad’s performance. Any systematic deviation from 180 degrees (i.e., anti-phase) suggests that the two individuals did not move symmetrically, but that one took the lead, while the other followed. To compare such asymmetries across conditions (3 levels) and blocks (6 levels), we conducted a 3 × 6 repeated measures ANOVA on the CRP<sub>DEV</sub>, the mean absolute deviation from anti-phase. As can be seen in the right panel of Fig. 8, the CRP<sub>DEV</sub> did not significantly change over practice blocks ($F(172.32, 11.42) = 0.33, p = 0.684, \eta_p^2 = 0.05$). Moreover, this variable was not significantly different between target size conditions ($F(1.33, 9.35) = 3.50, p = 0.085, \eta_p^2 = 0.34$), although in the first blocks the deviation from symmetry in the large target sizes appeared larger than in the other two target size conditions. In addition to the absolute deviations from symmetry, we also considered the directionality of the deviation. We recall that a CRP > 180 deg. indicated a leading role for high performer HP, while a CRP < 180 deg. indicated a leading role for the low performer LP. Following this convention, Fig. 7 demonstrates that in for instance D1 and D6 the HP was leading, while in D4 and D8 the LP was leading. To investigate whether this directionality in the leader-follower dynamics could somehow be predicted from (that is, explained by) their individual differences in solo action, we conducted a final solo-to-joint transfer analysis.

### 3.3.4. Solo-to-joint transfer

We assessed whether the dyads’ mean CRP in the joint action session could be related to any (ball and beam level) differences in the dyad members’ solo action performances. In other words, we analyzed whether the leader-follower dynamics of dyads could be partially explained by their differences in the number of target hits, beam variabilities or any other dependent variable. In Fig. 9, we plotted the dyad’s mean CRP values as a function of their inter-individual differences in the (second) solo action session. We did not observe a consistent linear relationship between the inter-individual differences in the dependent variable and the directionality of the phase relationship in any of these six graphs ($0.00 < R^2 \cdot s < 0.42$). Nevertheless, for some variables the (positive/negative) sign of the
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Exemplary timeseries of the continuous relative phase (CRP, in deg) of two representative dyads in block 6 of the joint action session, for the largest (upper row, red) and smallest (lower row, green) target size condition. D1 (left) had a relatively high variability, while D3 (right) had a lower variability. The solid black lines indicate a relative phase of 180 degrees. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

inter-individual solo difference was congruent with the directionality (leader/follower) of the joint action CRP. This was, for instance, the case for the ball's number of target hits and the beam's range of motion. Table 5 shows, for each variable and per target size condition, the percentage of the datapoints located within congruent quadrants (white/white or gray/gray). Notably, in 75 % of the cases (18/24) the individual with a larger number of target hits (TH) took the lead in the interaction. This was also true for the number of half cycles (HC) of the ball, in 67 % of the cases (16/24). The individual with the highest range of beam motion (ROM) in the solo condition took the lead in 71 % of the joint action cases (17/24). At the same time, the individual with the higher ball accuracy ratio (AR) and relative beam variability (CV) only took the lead in 58 % of the cases (14/24), suggesting no relationship between the solo action accuracy or variability and the joint action leadership. For the shape ratio, we found a negative relationship: In 79 % (19/24) of the cases the individual with the *lower* shape ratio in solo action took the lead in the joint action.

**4. Discussion**

In this contribution we studied the performances of dyads in the joint action setting of the ball-and-beam task. In a two-step study, of which the current contribution formed the second step, we investigated the transfer of skill from a solo action to a joint action setting of this novel experimental paradigm. In the first step, reported in Hafkamp et al. (2024), individual participants performed two solo action sessions of the task. This allowed us to identify the pertinent performance characteristics on the level of the ball and the level of
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**Fig. 7.** Circular frequency distributions of the continuous relative phase (in deg) between the beam extremities of all dyads (D1-D8) in block 6 of the joint action session. The graphs in red represent the large target conditions, in blue the medium targets and in green the small targets sizes. For reasons of visibility, the r-axis varies in scale, with scale provided by a circle of proportional frequency 0.1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the beam. In the second step, those same participants were paired into dyads to perform a subsequent joint action session, allowing us to assess the solo-to-joint propagation of these characteristics. In discussing our results, we start by addressing the general performances of the dyads in the joint action session of this study. Thereafter, we discuss the transfer of skill from the solo to the joint action context, thereby moving progressively from the level of the ball, via the level of the beam, to the level of the interpersonal coordination. By touching upon each of these levels, our aim is to provide an account of interpersonal coordination in which pattern formation and affordance actualization are covered together, as two dimensions of one and the same joint action. In the remainder of the discussion we reflect on the repercussions of our account for future research into joint action.

In the ball-and-beam paradigm participants had to control a ball rolling on a long and hand-held beam. By continuously adjusting the beam's inclination angle, they were to roll the ball fast and accurately between two targets, with the goal to hit these targets as often as possible during a trial. In general, dyads were increasingly successful in accomplishing that goal. Over blocks of practice, they improved the number of target hits (TH) per trial, by increasing not only the average speed of the ball (as revealed by an increase in the number of half cycles, HC) but also its average accuracy (as demonstrated by an increase in the accuracy ratio, AR) (Fig. 2, upper row). This performance enhancement over practice was in alignment with earlier research into the solo and joint action versions of this paradigm (Hafkamp et al., 2023), as was the finding of a significant speed-accuracy trade-off in the motion of the ball (Hafkamp et al., 2024). When the target sizes were reduced, the speed of the ball decreased accordingly –in both a solo and a joint action context. This suggests that the relevant behavioral organization occurred at the level of the ball in relation to the targets. This frame of reference, in which the pertinent dynamics of a task are defined, is also known as a *task space* (Fernandez & Bootsma, 2008; Mottet et al., 2001; Saltzman & Kelso, 1987). When considering the ball motion variables of each dyad, we found a strong transfer of skill from the solo to the joint action context. In particular, we found that the TH and the HC of a dyad were quite well predicted by the solo action performances of the two dyad members (mean R²'s of 0.60 and 0.73, respectively, Fig. 2, lower row). The AR, in contrast, was less well explained by the corresponding individual's solo performances (mean R² = 0.43). Together, this suggests that each participant contributed a particular ability to bring the ball up to speed, while a dyad's capacity to be accurate emerged primarily from the interaction. This interpretation turned out to be confirmed by what we found at the underlying level of the beam.

Across dyads, the relative variability of the beam's angle of inclination (coefficient of variation, CV) decreased over blocks of practice. That is, dyads were increasingly consistent in tilting the beam up and down to roll the ball back-and-forth. The range of motion (ROM) and the timing (shape ratio, SR) of the average beam each cycle, however, did not change significantly over blocks (Fig. 3, upper row). The improvements in the ball speed and accuracy over blocks were thus not caused by large adaptations in the jointly produced beam movements, but by subtle improvements in the consistency and accuracy of the performance. The speed-

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**Fig. 8.** The standard deviation (STD, left) and the mean deviation from anti-phase (DEV, right) of the continuous relative phase (CRP) between the movements of the two beam extremities. Both variables are plotted as a function of the practice blocks, under each of the three target size conditions (small: green, medium: blue and large: red). Error bars represent between participant standard deviations. The CRP<sub>STD</sub> decreased significantly over blocks and was significantly higher for smaller target sizes. The CRP<sub>DEV</sub> did not change significantly over blocks and was not significantly different across target size conditions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
<figure><img src="image_35.png" /><figcaption>Differences in SAS performances</figcaption></figure>

**Fig. 9.** Relations between performance differences among dyad members as observed in the solo action sessions (SAS) and the mean continuous relative phase (CRP) observed during joint action performance. Each panel represents a different dependent variable, at the level of ball motion (upper row: TH: number of target hits, HC: number of ball half cycles, AR: accuracy ratio) and at the level of beam motion (lower row: ROM: range of motion, CV: coefficient of variability, SR: shape ratio). Colors refer to the different targets size conditions (small: green; medium: blue and large: red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

<caption>**Table 5**</caption>

The percentage of dyads (%) in which the directionality of the CRP (leader/follower) was congruent with the sign (positive/negative) of the inter-individual differences in the solo action session performances on each of the six dependent variables (TH: target hits, HC: half cycles, AR: accuracy ratio, ROM: range of motion, CV: coefficient of variation and SR: Shape ratio).

<table><thead><tr><th>Condition</th><th>TH</th><th>HC</th><th>AR</th><th>ROM</th><th>CV</th><th>SR</th></tr></thead><tbody><tr><td>L</td><td>75</td><td>75</td><td>38</td><td>75</td><td>63</td><td>75</td></tr><tr><td>M</td><td>75</td><td>75</td><td>63</td><td>75</td><td>63</td><td>87</td></tr><tr><td>S</td><td>75</td><td>50</td><td>75</td><td>63</td><td>50</td><td>75</td></tr><tr><td>Average</td><td>75</td><td>67</td><td>58</td><td>71</td><td>58</td><td>79</td></tr></tbody></table>

accuracy trade-off on the level of the ball was also reflected in the beam's motion. A smaller target size resulted in a lower ROM and SR, but a higher CV. This suggests that dyads used more compensatory movements to control the ball motion when the demands on accuracy increased. This was in line with what we found earlier in the solo action setting (Hafkamp et al., 2024). In fact, the dyadic performances could to quite a large extent be explained by the individual performances of the dyad members in solo action. The ROM and the SR, in particular, were strongly propagated from the solo to the joint action context (mean R²'s of 0.79 and 0.67, respectively, Fig. 3, lower row). Given that the range and the timing of beam's cycle of motion were crucial to the production of the ball speed, this finding explained why the ball's speed (or more specifically, its number of half cycles) was well predicted by the solo performances. The transfer of the beam's CV was notably smaller (mean R² = 0.52), especially in the small target size condition (R² = 0.27). This implied that each dyad's variability was relatively poorly predicted by the corresponding individual's solo action performances. How variable the two dyad members manipulated the beam was thus an emergent property of their interaction, rather than a combination of each of their solo contributions. It is no coincidence that the same was true for the ball's accuracy, for which we saw earlier that it was only moderately explained by the participant's solo action performances, with a mean R² < 0.50. But as the beam's variability decreased over blocks, the ball's accuracy increased with practice, suggesting that dyads somehow learned to coordinate their movements. To confirm whether this was indeed the case, we proceeded to the interpersonal level, at which a stable pattern of
coordination is required to support goal achievement.

To begin with, we found that dyads generally moved the beam extremities simultaneously and in opposite directions, with one dyad member moving up while the other moved down (Figs. 4 and 5). On average, this resulted in an anti-phase mode of coordination with a continuous relative phase (CRP) between the handle heights of somewhere around 180 degrees (Figs. 6 and 7). As such, the interpersonal coordination was compliant with earlier research into the ball-and-beam paradigm. In Hafkamp et al. (2023), we also found that a complementary (anti-phase) task solution, in contrast with a sequential solution, prevailed under dyads that had already practiced the task individually in a solo action setting. Furthermore, the coordination between the two beam handles reflected the speed-accuracy that we found on the level of the ball. A reduction in the size of the targets not only evoked a lower ball speed, but also a significant increase in the standard deviation of the CRP between the beam handles (CRP<sub>STD</sub>, Fig. 8, left panel). This aligned with the increased coefficient of variation we found on the level of the beam and suggested the emergence of compensatory movements. Over blocks of practice, however, the CRP<sub>STD</sub> decreased significantly, while the amount of trial time that dyad members moved complementary to each other increased. Thus, dyads managed to improve the stability of their interpersonal coordination. Notwithstanding these improvements, we note that a stable phase locking of the two beam handles for long durations –let alone for the entire duration of a trial– was particularly rare. Even though the dyad members were usually entrained in their frequency, the relative phase between their movements remained variable, also in the latter stages of practice (Fig. 6). As hypothesized, fully stable phase locking did not occur. Each dyad member retained its unique phase progression profile (at least partially), as evidenced by the high solo-to-joint transfer of the SR. Moreover, the CRP of some of the dyads considerably deviated from anti-phase (CRP<sub>DEV</sub> > 0). The mean absolute CRP<sub>DEV</sub> decreased nor increased over blocks of practice (Fig. 8, right panel), suggesting a systematicity in the leader follower-dynamics of the interactions. As a last step, we investigated whether the phase lag between the dyad members in the joint action was statistically related to any differences in their performances in the solo action.

For none of the ball/beam performance indicators was the difference in the dyad members' solo action performances proportionally related to the mean phase lag in the interaction of a dyad (0.00 < R² < 0.42). It was thus not possible to accurately predict who would lead and who would follow (and with what phase lag) on the basis of differences in the corresponding individual's solo performances. This result is in contrast with what has been found in dyadic pendulum swinging tasks (Amazeen et al., 1995; Schmidt et al., 1994; Schmidt & Survey, 1994). In this classic paradigm, the phase lag between dyad members was proportionally related to the difference in preferred swinging frequency in solo action, an effect known as detuning. Although such a difference was not found here, the results did point out that for some variables the (positive/negative) sign of the solo action difference between a dyad's members was congruent with the CRP directionality (leader/follower) in joint action (Fig. 9). This congruency was most prominent for those parameters that were (indirectly) related to the speed of the ball, such as TH (75 %), but also ROM (71 %) and SR (79 %). Those individuals that had scored more target hits in the solo action session, or that had moved with a larger range of motion and a stronger inclination for a middle-stop strategy, were more likely to lead the interaction in joint action. The congruency was smaller, however, for those variables that were (indirectly) related to the ball's accuracy, such as the AR (58 %) and the CV (58 %). Earlier, we already noted that the AR and the CV were only moderately propagated from solo to joint action, in contrast with the other variables (TH, HC, ROM and SR), which were strongly transferred from one setting to another. Now we show that this transfer also had an impact the leader-follower dynamics in their interaction. The variability and the accuracy of a dyadic performance, in contrast, arose from the interaction of the contributions. This suggests that the stability of the interpersonal coordination was indeed an emergent property of the interaction, while being constrained by the goal of the task.

The solo-to-joint transfer analysis, as presented for the first time in this study, provides a window into the relationship between pattern stability and affordance actualization in joint action. It shows that individual ball-and-beam players first and foremost learn to hit the targets, a skill that they then transpose to the joint action setting. Our data suggests that the formation of a stable pattern of interpersonal coordination is subordinate to this process and thus emergent from the interaction. In our view, this result calls for a conceptual extension of interpersonal synergies theory (IST). Riley et al. (2011, p. 1) defined an interpersonal synergy as a "higher-order control system formed by coupling movement system degrees of freedom of two or more actors". In this perspective, synergies feature dimensional compression of the social system as well as reciprocal compensations between the degrees of freedom, placing IST within a rich tradition of pattern formation theories. However, a synergy is more than just a stable pattern of coordination that subserves goal achievement. It is also a collective effectivity that allows for goal achievement *beyond the individual action capabilities*. If analyses of the motor behavior during joint action remain at the level of interpersonal coordination, this critical feature of a synergy remains obscured. What is required, instead, is a careful analysis of the possibilities for action that become available upon interaction with another agent. When encountering their environment together, dyads, triads or groups of individuals acquire an increased capability for action. The extent to which this ecological fact pulls them into the orbit of interpersonal coordination, so as to form a synergy and actualize that new affordance, is a topic to that requires further empirical study. The current work is only a first step in this direction.

An interesting issue that may shed some light on the formative process of an interpersonal synergy, is gaze behavior. Where do participants *look* when they learn to perform the ball-and-beam task together? If, indeed, ball control is primary and interpersonal coordination secondary, one would expect the two dyad members to attend to the rolling ball from the onset of the joint action. Gaze behavior in solo and in joint action would not be fundamentally different, reflecting a one-to-one transfer of skill from the one setting to the other. If, in contrast, the set-up of a stable pattern of interpersonal coordination would be prioritized above ball control, one would expect both dyad members to attend frequently to each other while moving the beam up and down. In that case, ball control would follow from the spatiotemporal organization between the dyad members. In a pioneering study into manual tool making, Van Dijk and Bongers (2014) showed that gaze behavior gets functionally organized over a learning process, so as to accommodate for the actualization of a newly discovered affordance. In the first stages of learning, gaze is exploratory, with short fixations on various aspects of
the environment. In later stages, gaze becomes more goal-directed, with fixations on a longer timescale. Extrapolating these findings to the current experimental setting, we speculate that dyad members who are first confronted with the task (group J/S in Hafkamp et al., 2023) may start by exploring the ball and the beam as well as each other, in an effort to establish a coordinative structure. The explorative character of their gaze may contribute to the set-up of a relatively ‘cautious’ sequential pattern of coordination (see Hafkamp et al., 2023). Dyads formed by individuals that had already figured out the task individually (as in the present study and in group S/J in Hafkamp et al., 2023), may already have organized their gaze behavior to accommodate for goal achievement. This could explain why they managed to adopt a more efficient simultaneous pattern of interpersonal coordination (Hafkamp et al., 2023). Eye tracking devices could serve as a tool to empirically test these hypotheses, so as to unravel the formative processes of an interpersonal synergy.

In conclusion, this study employed the ball-and-beam paradigm to provide an ecological account of interpersonal coordination. Through an analysis of skill transfer from solo to joint action, we empirically demonstrated that dyads engaged in a collaboration not only strive to maintain a stable coordination, but also and perhaps primarily actualize an affordance so as to achieve a common goal. We suggest that future research into joint action further examines the cardinal role of affordances in shaping interpersonal coordination, for the key to explaining social behavior lies in understanding the relationship of agents to their populated environment.

## CRediT authorship contribution statement

**Marijn S.J. Hafkamp:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Remy Casanova:** Writing – review & editing, Validation, Software, Methodology, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Reinoud J. Bootsma:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Project administration, Methodology, Funding acquisition, Conceptualization.

## Funding

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 956003.

## Declaration of competing interest

The authors have no relevant financial or non-financial interests to disclose.

## Acknowledgements

The authors gratefully acknowledge Harjo de Poel for his helpful insights, in particular with respect to the continuous relative phase analysis.

## Data availability

Data will be made available on request.

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