Symmetry and synchrony of bimanual movements are not predicated on interlimb control coupling

Abstract

Previous studies suggest that bimanual coordination recruits neural mechanisms that explicitly couple control of the arms, resulting in symmetric kinematics. However, the higher symmetry for actions that require congruous joint motions compared with noncongruous joint motions calls into question the concept of control coupling as a general policy. An alternative view proposes that codependence might emerge from an optimal feedback controller that minimizes control effort and costs in task performance. Support for this view comes from studies comparing conditions in which both hands move a shared or independent virtual objects. Because these studies have mainly focused on congruous bimanual movements, it remains unclear if kinematic symmetry emerges from such control policies. We now examine movements with congruous or noncongruous joint motions (inertially symmetric or asymmetric, respectively) under shared or independent cursors conditions. We reasoned that if a control policy minimizes kinematic differences between limbs, spatiotemporal symmetry should remain relatively unaffected by inertial asymmetries. As shared tasks reportedly elicit greater interlimb codependence, these conditions should elicit higher bilateral covariance regardless of inertial asymmetries. Our results indicate a robust spatiotemporal symmetry only under inertially symmetric conditions, regardless of cursor condition. We simulated bimanual reaching using an optimal feedback controller with and without explicit costs of kinematic asymmetry, finding that only the latter mirrored our empirical data. Our findings support the hypothesis that bimanual control policies do not include kinematic asymmetry as a cost when it is not demanded by task constraints suggesting that kinematic symmetry depends critically on mechanical movement conditions.

NEW & NOTEWORTHY Previously, the control coupling hypothesis and task-dependent control hypothesis have been shown to be robust in the bimanually symmetrical movement, but whether the same policy remains robust in the bimanually asymmetrical movement remains unclear. Here, with evidence from empirical and simulation data, we show that a spatiotemporal symmetry between the arms is not predicated on control coupling, but instead it is predicated on the symmetry of mechanical conditions (e.g. limb inertia) between the arms.

INTRODUCTION

Coordination of bilateral arm movements is an essential feature of human motor control that allows the stable and efficient performance of many activities of daily living. These activities can range from simple tasks like carrying a box with two hands to more complex tasks like tying shoelaces. Previous research on bimanual coordination has demonstrated that the movements of two arms tend to produce similar spatiotemporal characteristics even when the task does not explicitly require symmetry. This phenomenon has often been referred as “interlimb coupling” (1). There has been extensive research into the factors that determine whether movements of the two arms can be performed simultaneously with or without “coupling,” but despite several proposed hypotheses, the basis of these effects remains unclear.

An early hypothesis proposed by Kelso and colleagues (2, 3) was that the central nervous system (CNS) preferentially organizes groups of muscles as a bilateral synergistic unit, or “coordinative structure” to control movements of the two arms together. In their early studies, Kelso et al. (2, 3) asked participants to make discrete bilateral reaches that were mirror-symmetric with respect to the body midline but to targets that differed in size and distance, thus modulating movement difficulty according to Fitts law (4). Movements of the two arms were initiated and ended roughly at the same times, regardless of differences in the index of difficulty between each arm’s target. These initial observations led the authors to suggest that bilaterally homologous muscle groups may be activated through a single, shared motor command that couples control of the two arms (2, 3). In later studies (5), findings of high temporal synchronicity between arms were extended to nonmirror symmetric movements, suggesting that coupling may not be limited to bilaterally homologous muscle groups alone (5). The tendency that two hands become similar has been reported in not only the temporal domain but also the spatial domain of movements (6–12). For example, studies in which different movement amplitudes and directions were assigned to either arm reported that the two arms deviated from their intended trajectories and became more similar; a spatial assimilation effect (for discrete movement: 13–17, for continuous movement: 6–9, 12). Notably, such spatial assimilation appears to become greater as the interlimb differences in task requirements increases and, importantly, this effect is not limited to movements that recruit bilaterally homologous muscle pairs (1, 15, 18–20).

Such spatial assimilation and temporal synchrony findings in bilateral arm movements have been interpreted from the perspective of control “coupling” hypotheses, which suggest that bilateral arm movements might be coordinated through motor commands that couple activation of muscle groups across the arms (2, 3). From a control theory perspective, this implies that spatiotemporal asymmetry should be considered as a cost that the controller seeks to minimize. However, while Kelso and colleagues attributed the spatiotemporal similarities between the two arms to control coupling, the extent of spatiotemporal similarity between the arms during bimanual movement has also been suggested to depend on the biomechanical symmetry of the bilateral movement. Critically, bimanual movements have been shown to have the greatest similarity when the required joint displacements are mirror symmetric (21–23). Also, a series of studies have shown variant temporal synchrony depending on symmetry in peripheral conditions (such as reaching distances and weight) when a similar modified Fitt’s task was used as demonstrated by Kelso and colleagues (14, 18, 24) and when other discrete bimanual tasks were used (10, 25–27). This raises the question of whether motor commands are coupled or whether observations of high spatiotemporal similarity simply arise from peripheral conditions such as inertial symmetry. In other words, if spatiotemporal asymmetry reflects a control cost, then strong kinematic similarities should arise despite asymmetric inertial dynamics in the periphery.

It should be stressed that some studies have interpreted the spatiotemporal correlation between the hands as an epiphenomenon resulting from optimal control that seeks to minimize the costs of task performance errors and control effort (28–32). Diedrichsen (28) examined bimanual reaching movements under two task conditions: a “two-cursor” condition in which each hand was represented by an independent cursor directed toward a separate target, and a “shared cursor” condition in which both hands cooperatively controlled a single cursor toward a common target. Although required movements were symmetric, a velocity-dependent force was applied in random trials to one of the hands in one direction. During the perturbation, the other hand compensated by moving to the opposite direction of the perturbation in the single-cursor condition but not in the two-cursor condition, reflecting interlimb compensatory processes. Diedrichsen et al. (33) interpreted these findings within the framework of an optimal control theory (OCT), in which the two arms are controlled according to a “task-dependent bimanual control policy.” This policy minimizes control costs, such as task-related errors, movement variability, and motor effort. The spatial covariance between the arms resulting from the compensatory action can then be interpreted as secondary to the optimal control process. This view contrasts with the coupling hypothesis, which posits that the underlying bimanual control policy seeks to minimize spatiotemporal asymmetry as a primary cost. An important note, however, is that the concept of bimanual control policies at the task level (e.g., two or shared cursor tasks) has primarily been based on studies that involve mirror-symmetric bimanual movements, which inherently have biomechanically symmetrical movement requirements and the activation of bilaterally homologous muscle pairs. It should be stressed that in all of the previous studies cited earlier, the task did not require or encourage symmetry between the hands, and thus symmetry was interpreted as an emergent result of the control strategy.

We now ask whether bimanual symmetry during reaching tasks reflects a feature or cost of the controller, or whether it simply emerges from the symmetry of peripheral mechanical conditions in a more natural form of movement that is not perturbed and not forced into repetition. We designed a bimanual planar reaching task with targets that either elicited symmetrical or asymmetrical inertial requirements, under conditions where target direction changes frequently. To examine the extent of recruitment of bilateral coordination mechanisms due to visual feedback, participants performed these tasks under two feedback conditions: a shared end point condition (shared cursor; SC) and an independent end point condition (two cursor; TC) (28). From the perspective of control coupling, we hypothesized that bimanual control should reduce kinematic asymmetry costs, regardless of potential asymmetries in biomechanics. Therefore, this hypothesis predicts that kinematic variables between the two arms should remain symmetrical and synchronous, irrespective of the imposed inertial asymmetry. However, our empirical findings failed to support this hypothesis. Instead, they provided strong support for an alternative hypothesis that bimanual control policies do not include kinematic asymmetry as a control cost. To better interpret our empirical findings, we then developed two optimal feedback control simulations: one with explicit costs for kinematic asymmetry and one without such costs. These simulations were conducted under conditions in which the two arms had either symmetric or asymmetric inertial loadings. Both controllers minimized a cost function that included terms for control effort and limb state deviations from a target state, but one controller also included explicit costs for kinematic asymmetry. We simplified the dynamics of end point motion as much as possible to exclusively examine performance differences attributable to the control policy. Our simulation results were consistent with our empirical results for only the controller that did not include kinematic asymmetry as a control cost. We conclude that while bilateral symmetry may be observed in some cases of bimanual movements, the kinematic asymmetry is generally not included as a control cost if it is not explicitly required by the task. Thus, control” coupling” appears inappropriate to describe bimanual control that we perform on a regular basis.

METHODS

Participants

Ten right-handed participants between 18 and 35 yr of age provided written informed consent to participate in this experiment. Participants who scored 90% or above on the Edinburgh Handedness Inventory (34) were considered to be right-handed. We recruited only right-handed subjects because left handers do not represent a behaviorally (34) or a neurologically (35) homogeneous population, and thus one cannot assume that bimanual coordination should be analogous between groups of right and left handers (36–40). Participants with any orthopedic injuries on upper extremity or with neurological conditions that would affect the performance of task were excluded from the study. All of the experimental procedures were approved by the Institutional Review Board of the Pennsylvania State University. The data were collected and analyzed at the Pennsylvania State University.

Experimental Setup

The experimental paradigm was implemented using the Kinereach system (Sainburg) (see Fig. 1A). In this setup, participants sat in a chair in front of a table, with both arms supported on air-sleds that floated above the table. Wrist movements were constrained with braces and straps attached to the air-sleds. With pressurized air turned on, the arms moved freely on the table with negligible friction. A 52 in. TV was positioned above the table, facing downward, and a mirror was placed between the TV and the table, allowing participants to view images projected from the screen above while performing the task. The mirror’s location prevented participants from seeing their hands during the experiment, so all visual feedback related to task performance was provided via the screen. The position and orientation of the forearm segment and the upper arm segment were recorded using a six-degree-of-freedom magnetic sensor attached to each segment of both arms (as shown in Fig. 1A) (trakStar; Ascension Technology). We digitized seven bony landmarks of the arm to estimate finger, wrist, elbow, and shoulder joint positions. We used the estimated joint positions to calculate the shoulder and elbow angles.

Figure 1.

The schematics of Kinereach system and the experimental paradigm. A: the schematics of general setup of the Kinereach system. B: target setup for the two-cursor (TC) condition. In the two-cursor condition, two cursors were shown in the locations of each hand in near real-time during the trials. C: target setup for the shared cursor (SC) condition. In the shared cursor condition, participants were instructed to move each hand to a respective starting position (shown as dotted circles). Once the two hands were in their respective starting positions, a shared cursor appeared in the middle location of two hands. The position of shared cursor was determined by the averaged position from two hands (see the equations in the figure). Targets were located 15 cm away from the starting position in three directions (30° vs. 90° vs. 150°) relative to the positive x-axis. In each trial, a pair of targets (or a single target in the shared cursor condition) in the same direction was presented.

Experimental Task

Starting positions, cursor(s) representing the location of fingertip, and targets (see Fig. 1B) were displayed on the screen. Participants performed bimanual reaching movements toward targets in three directions (150°, 90°, and 30°) relative to the right horizontal axis (Fig. 1B). This target setup induced relatively noncongruous joint motions between the arms in the 150° and 30° directions and congruous joint motions in the 90° direction, influencing the symmetry of inertia requirements between the arms during the reaching movement (41). Before starting each trial, participants moved the cursor(s) to the starting position(s), which had a diameter of 1.5 cm and 2 cm, respectively. Once both cursors (one for each hand) were remained in the starting positions for 1 s, the target(s) with a diameter of 3 cm appeared 15 cm away from the starting position(s) in either one of three target directions. At this cue, participants were instructed to make bimanual reaching movement to the center of target(s) as accurately as possible, with a minimum peak velocity of 1.5 m/s encouraged. Once both cursor(s) left their respective starting positions, movement completion was determined when the velocity of both hands fell below 0.02 m/s. According to these criteria, all movements of all participants were completed within a 1-s window. The cursors remained visible in near real-time, with only a 17-ms delay, throughout the movement. After completing the movement, the final position(s) of the cursor(s) was revealed, and scores were assigned based on their proximity to the center of the target. There were two cursor conditions: the two-cursor (TC) and shared cursor (SC) conditions, which displayed the cursor in two different ways during the trial. In the two-cursor condition, each hand’s location was displayed as an individual cursor, and the two hands reached toward their respective targets (see Fig. 1B). In the shared cursor condition, once participants moved each cursor to the respective starting position, individual cursors for each hand were extinguished before movement, and a single cursor appeared in the center between the starting positions (Fig. 1C). The x and y coordinates of the “shared” cursor [x_C, y_C] were defined by the average x and y coordinates of the right and left hands:

$$X_c=\left(X_R+X_L\right)/2$$ (1a)

$$Y_c=\left(Y_R+Y_L\right)/2$$ (1b)

All participants completed a total of 120 trials of bimanual reaching movements, with the cursor condition alternating every 15 trials. Within each block of 15 trials, the target direction was determined pseudorandomly (with 5 trials of each target direction). We chose to switch the target direction between trials to reflect more natural conditions without forcing repeated stereotyped movements which is the condition that previous studies have implemented (28, 31). The initial task condition was counterbalanced across participants.

Data Processing and Analysis

All data were processed and analyzed using custom developed software, using the IgorPro programming environment (v6.37; WaveMetrics). The kinematic data were low-pass filtered at 12 Hz using a third-order dual-pass Butterworth filter. Subsequently, differentiation was performed on the filtered data to generate velocity profiles for each trial. Movement onset was determined by the first tangential velocity minimum that fell below 8% of the peak tangential velocity when searching backward from peak velocity. Movement end was defined as the first minimum under 8% of peak velocity, searching forward in time from peak velocity.

For our measures of temporal parameters, we calculated the movement duration by subtracting the movement onset time from the movement end time. In addition, we determined the time of peak velocity by finding the time difference between the movement onset and the time when the hand reached its peak tangential velocity. For our measures of spatial parameters, we computed the reaching distance by calculating the magnitude of displacement between the hand’s positions at movement onset and movement end. To assess interlimb differences in these measures, we computed the magnitude of the difference between the right arm values and the left arm values within each trial. For task performance, we computed the final position error by measuring the distance between the final cursor position at the movement’s end and the center of the assigned target. Because we had two cursors in the two-cursor condition and a single cursor in the shared cursor condition, we averaged the final position errors of the two hands in the two-cursor condition and compared with that of shared cursor condition. All dependent measures were calculated at the trial level and averaged across target directions and cursor conditions for each participant for the analysis.

For the analysis of spatial codependence between the hands, we performed a simple linear regression analysis using the position data of each hand at movement end. We decomposed the hand position into components parallel to the target direction and perpendicular to the target direction. The position in the parallel axis indicates the excursion of the hand from start position until the movement end along the target direction. The position in the perpendicular axis indicates the perpendicular deviation of the hand from the straight line between the start position and the target. We used R-values and correlation slopes to evaluate the coordination strength and codependence between the hands. In the parallel axis correlation, the slope of correlation line would indicate degrees of symmetry in reaching distances between the hands. The slope deviating from 1 indicates more asymmetry. In the perpendicular axis correlation, the sign of correlation between the hand positions would reveal whether the hands moved sideways in the same (positive correlation, nonmirror-symmetric) or in the opposite (negative correlation, mirror-symmetric) directions relative to the target direction. Using these measures, we can compare across task conditions (cursor conditions and target directions) the extent to which motions of the two hands are spatially codependent. Based on the task-dependent control hypothesis, we would predict that under the shared cursor condition, parallel axis correlation slope deviations (from 1) will be greater under asymmetric inertial resistance conditions (i.e., the 30° and 150° target directions). Specifically, the hand with higher inertial resistance should contribute less to the movement of the shared cursor compared with the hand with lower inertial resistance to reduce the motor effort, such that the slope will lean toward the hand with lower inertial resistance. In the perpendicular axis correlation, we would predict negative correlation regardless of the inertial symmetry of two hands as two hands would coordinate to reduce the directional errors of the shared cursor based on the idea of task-dependent bimanual control policy.

For regression, we used an orthogonal linear regression model, also referred to as Deming regression, to fit the measures between the two hands. Unlike the ordinary least-squares regression model, which only considers errors in the y-axis variable, the orthogonal regression model minimizes the sum of squared differences perpendicular to the linear model. This approach accounts for errors in both the y-axis and the x-axis variables. We chose to use this model because it aligns better with our analytical objectives, specifically quantifying the effects of experimental conditions, such as inertial symmetry and cursor conditions, on the spatial covariation between the arms. We conducted these analyses at the final position of movement. This analysis was previously used to calculate covariation between the two hands (31).

Kinetic Data

We calculated the mean squared net joint torque for both the elbow and shoulder. This calculation was performed using an inverse dynamics model that included four degrees of freedom in each arm (shoulder horizontal flexion/extension, elbow flexion/extension, shoulder point linear displacements in x and y axes) (42). Joint torques were computed for the shoulder and elbow using Eqs. 2a and 2b under the assumption that the upper extremity consisted of two rigid segments connected by frictionless joints at the shoulder and elbow, with the shoulder center of rotation free to accelerate in the horizontal plane. The inertia and mass of the forearm support were 0.0247 kg/m² and 0.58 kg, respectively. We derived limb segment inertia, center of mass, and mass values from regression equations using each subjects’ body mass and measured limb segment lengths (42, 43).

$$T_{e}N=\left(I_e+m_e{r}_e^2\right){\overline{\theta }}_e$$ (2a)

$$T_{s}N=\left(I_s+m_s{r}_s^2+m_e{I}_s^2+m_e{l}_s{r}_e\cos \hspace{0.17em}\left({\theta }_e\right)\right){\overline{\theta }}_s$$ (2b)

where m is mass of segment, r is center of mass of segment, l is length of segment, I is inertia of segment, θ_s is shoulder angle, θ_e is the elbow angle, T_eN is elbow net torque, and T_sN is shoulder net torque. The subscripts are defined as follows: s is upper arm segment and e is lower arm segment (including support and air sled device). To calculate the squared net joint torque for the shoulder and elbow joints, we first calculated the net joint torque profiles for both elbow and shoulder from movement onset to the movement end. Then, we squared the torque profiles to quantify the magnitude of the net torques. We combined the squared net joint torques of the elbow and the shoulder to calculate total magnitude of net joint torque for each arm. Then, we calculated the mean of this value separately for right and left arm across all trials for each target direction and cursor condition.

Statistical Analysis

We conducted a two-way repeated-measures ANOVA with cursor condition (TC vs. SC) and target direction (150° vs. 90° vs. 30°) as within-participant factors to analyze dependent variables including: interlimb difference of movement onset, movement end, peak velocity time, peak velocity, movement duration, and reaching distance. Two-way repeated-measures ANOVA was also used to compare final position errors across cursor conditions and target directions. To compare mean squared net torque, a three-way repeated-measures ANOVA with hand (right vs. left) and cursor condition (TC vs. SC) and target direction (150° vs. 90° vs. 30°) as within-participant factors was used. For post hoc analysis, we performed the Tukey honestly significant difference test for post hoc analyses, which accounted for potential Bonferroni errors in each family of analyses. To ensure normality of the data, we conducted the Shapiro–Wilk test. In case where the data were not normally distributed, we applied the Johnson transformation method (44). All statistical analyses were carried out using JMP software (SAS Institute, Cary, NC) with a family-wise error rate set at α = 0.05.

Optimal Control Simulations

We developed a simplified model of the bimanual reaching task under conditions of peripheral mechanical asymmetry. We used the model to obtain predictions of reach kinematics generated by an optimal feedback controller operating both with and without explicit costs of mechanical asymmetry. The purpose of the model was to examine the impact of imposing mechanical asymmetry on the kinematics of reaching with two “hands” operating under optimal feedback control. We simplified the model as much as possible to reduce complexities such as nonlinear musculoskeletal mechanics, while also retaining second-order dynamics that dominate limb motion under many experimental conditions (45–47). By doing so, the simulations highlight differences in kinematic performance that may arise solely due to differences in control policy. To that end, we modeled the two hands as independent point masses moving along a straight line from starting position to target. Motion of each hand is governed by the following dynamic equation (Eqs. 3a and 3b):

$$\left[\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& -\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$M$}\right.\end{array}\right]\left[\begin{array}{c}{q}_1\\ q_2\end{array}\right]+\left[\begin{array}{c}0\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$M$}\right.\end{array}\right]u$$ (3a)

$$y=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{q}_1\\ q_2\end{array}\right]$$ (3b)

where the time-varying state variables q₁ and q₂ correspond to “hand” position and velocity, respectively; the dot accent represents the first time derivative; M is the mass of the simulated hand, and B is an environmental viscosity that opposes u, the motive force produced by the optimal feedback controller. The output y of the system is simply the position q₁ of the simulated (point mass) hand.

We represented the composite system comprising the two hands by stacking the system equations for two independent point masses (Eqs. 4a and 4b):

$$\left[\begin{array}{c}\begin{array}{c}{\dot{q}}_1^L\\ {\dot{q}}_2^L\end{array}\\ \begin{array}{c}{\dot{q}}_1^R\\ {\dot{q}}_2^R\end{array}\end{array}\right]=\left[\begin{array}{cc}\begin{array}{cc}0& 1\\ 0& -\raisebox{1ex}{$B^L$}\!\left/ \!\raisebox{-1ex}{$M^L$}\right.\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 1\\ 0& -\raisebox{1ex}{$B^R$}\!\left/ \!\raisebox{-1ex}{$M^R$}\right.\end{array}\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{q}_1^L\\ q_2^L\end{array}\\ \begin{array}{c}{q}_1^R\\ q_2^R\end{array}\end{array}\right]+\left[\begin{array}{c}\begin{array}{cc}0& 0\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$M^L$}\right.& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$M^R$}\right.\end{array}\end{array}\right]\left[\begin{array}{c}{u}^L\\ u^R\end{array}\right]$$ (4a)

$$\left[\begin{array}{c}{y}^L\\ y^R\end{array}\right]=\left[\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 1& 0\end{array}\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{q}_1^L\\ q_2^L\end{array}\\ \begin{array}{c}{q}_1^R\\ q_2^R\end{array}\end{array}\right]$$ (4b)

Here, the superscripts L and R correspond to the left and right hands, respectively. Please note that even though the system is represented by a single set of vector-matrix equations, the dynamic equations of the two simulated hands remain uncoupled.

The task in all simulations was to move each hand from an initial position y₀ to a desired target. The initial position was located 15 cm away from the target. By defining the target location as the origin of the system dynamics for each hand, we simulate and solve the goal-directed movement task as a simple initial condition problem.

To implement an optimal feedback controller, we derived a set of feedback control gains K that would realize a linear quadratic regulator (MATLAB command: LQR). These gains minimize the following cost function:

$$J=\underset{0}{\overset{\infty }{{\displaystyle \int }}}\left(q^{T}Qq+u^{T}Ru\right)dt$$ (5)

where Q (positive semidefinite) and R (positive definite) are symmetric weighting matrices that define a tradeoff between kinematic performance [how fast the vector q(t) goes to 0] and control effort. J is the cost function to be minimized over an infinite time horizon. In this equation, the operator T corresponds to the vector transpose. As described elsewhere (48), the LQR minimizing gain matrix K is obtained by solving an algebraic Riccati equation, which is a function of the system dynamics (Eq. 4a) and the matrices Q and R.

We ran two sets of simulations under conditions of symmetric and asymmetric dynamic loads for the simulated left and right hands. For each set of simulations, we operationalized the experimental manipulation of moving the two hands along the 30˚ and 150˚ directions, thereby imposing peripheral asymmetry in limb mechanics, by increasing the mass M and viscosity B of one of the simulated hands by 50%. For the simulation where the optimal feedback controller operates without explicit costs of kinematic asymmetry, we defined the weighting matrices Q and R as in Eq. 6:

$$Q=W_{TargetCapture}*\left[\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}1& 0\\ 0& 0\end{array}\end{array}\right];R=W_{Effort}*\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$ (6)

For the cost function of Eq. 5 with the kinematic errors measured in units of centimeters and control effort measured in units of Newtons force, choosing nominal model parameters M = 1 kg, B = 1 Nm/s and the ratio W_{TargetCapture}:W_Effort = 1.5:1 yields simulated 15 cm hand movements with durations similar to those observed experimentally (cf., 49), thus realizing an apparent tradeoff between the minimization of kinematic errors and control effort. For simulations where the optimal feedback controller operates with explicit costs of kinematic asymmetry, we altered the weighting matrix Q to also include additional terms as in Eq. 7:

$$Q=W_{TargetCapture}*\left[\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& 0\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}1& 0\\ 0& 0\end{array}\end{array}\right]+W_{DisplacementAsymmetry}*\left[\begin{array}{cc}\begin{array}{cc}1& 0\\ 0& 0\end{array}& \begin{array}{cc}-1& 0\\ 0& 0\end{array}\\ \begin{array}{cc}-1& 0\\ 0& 0\end{array}& \begin{array}{cc}1& 0\\ 0& 0\end{array}\end{array}\right]+W_{VelocityAsymmetry}*\left[\begin{array}{cc}\begin{array}{cc}0& 0\\ 0& 1\end{array}& \begin{array}{cc}0& 0\\ 0& -1\end{array}\\ \begin{array}{cc}0& 0\\ 0& -1\end{array}& \begin{array}{cc}0& 0\\ 0& 1\end{array}\end{array}\right]$$ (7)

The set of weight matrices in Eq. 7 adds the cost of asymmetry to the costs in Eq. 6 by also including terms (W_{DisplacementAsymmetry} and W_{VelocityAsymmetry}), which weight the squared difference between the positions [i.e., ${\left(q_1^L-q_1^R\right)}^2$] and velocities [${\left(q_2^L-q_2^R\right)}^2$] of the two hands, respectively. By doing so, the simulations generate predictions of bimanual kinematic performance, based on first principles, allowing comparison of the kinematic consequences of controllers implementing control policies that do or do not consider explicit kinematic costs of mechanical asymmetry analogous to those examined experimentally.

RESULTS

Inertial Differences across Target Directions

We designed our task to vary the inertial requirements during reaching movements to different target directions, considering inertial anisotropy (41). Figure 2 shows representative hand trajectories and hand velocity profiles for movements into each target direction under both the two-cursor and shared cursor conditions. The insets in Fig. 2 show the mean-squared net torques of both arms. Net torque is the inertial torque that describes the inertial resistance to acceleration at the joint and accounts for the inertial distribution of the limb distal to the joint in question (see methods for details). For the 150° target, the dominant arm reaching to a contralateral target exhibited higher mean-squared net torque compared with the accompanying nondominant arm reaching to an ipsilateral target (Fig. 2, A and D, insets). For the 30° target, the nondominant arm reaching to a contralateral target showed higher mean-squared net torque than the accompanying dominant arm reaching to an ipsilateral target (Fig. 2, C and F, insets). In contrast, for the 90° target, mean-squared net torque was comparable for both arms (Fig. 2, B and E, insets). ANOVA confirmed this pattern by showing a significant effect of target direction [F(2,18) = 28.3160, P < 0.0001)], a two-way interaction between hand and target direction [F(2,18) = 33.0145, P < 0.0001)], and a three-way interaction [F(2,18) = 7.3533, P = 0.0046)] between hand, cursor condition, and target direction. In post hoc analysis of the three-way interaction effect, we found a significant difference of the mean-squared torque between the two arms only in the 30° and 150° targets (all P < 0.0038) but not in the 90° target (both P > 0.9677) regardless of cursor condition. Hence, the data indicate that our targets induced statistically significant asymmetrical inertial loading of the two arms in 30° and 150° directions but symmetrical inertial loading in 90° direction under both cursor conditions.

Figure 2.

A–C: representative hand trajectories and hand velocity profiles in the two-cursor (TC) condition. D–F: representative hand trajectories and hand velocity profiles in the shared cursor (SC) condition. The insets show the mean-squared net joint torque for both arms along with standard error bars. For the hand velocity profiles, movement onset, peak velocity time, and movement end are indicated with dotted lines, with different colors used for each arm (blue for the left arm, and red for the right arm).

Temporal Measures

Figure 2 also shows tangential hand velocity profiles for both arms, overlaid on each other, corresponding to each representative hand trajectory. In the 90° reaching trajectory (two cursor: Fig. 2B, shared cursor: Fig. 2E), movement onset, peak velocity timing, and movement end were highly synchronous. However, in the 150° and 30° reaching movements, movement onset, peak velocity timing, and movement end exhibited more asynchrony compared with 90° reaching. This asynchrony was mainly driven by the hand under the higher inertial resistance starting the movement earlier and ending later than the opposite hand under lower inertial resistance (Supplemental Fig. S1). These observations from representative trials were supported by our group data (Fig. 3). Figure 3, A–C, displays the interlimb difference of each temporal variable for both cursor conditions. In terms of movement onset, peak velocity time, and movement end, the interlimb difference was substantially larger in the 150° and 30° targets than in the 90° target (Fig. 3, A–C). Our ANOVA confirmed this pattern, revealing a significant effect of target direction on interlimb difference of the movement onset [F(2,18) = 9.1291, P = 0.0018)], peak velocity timing [F(2,18) = 54.9454, P < 0.0001], and movement end [F(2,18) = 30.6172, P < 0.0001]. There was no significant effect of cursor condition nor interaction (cursor condition × target direction) on the interlimb difference values of the movement onset, peak velocity time, and movement end (all P > 0.1548).

Figure 3.

The absolute interlimb differences in the temporal variables for both cursor conditions. A–E: the magnitude of interlimb difference in movement onset, peak velocity time, movement end, movement duration, and peak velocity. In all plots, a higher value indicates a greater asynchrony or asymmetry. Each dot represents an averaged data point from one participant. The error bars represent one standard error. SC, shared-cursor condition; TC, two-cursor condition.

As expected by the pattern of movement onset and movement end, movement duration was also asymmetric for the 30° and 150° targets, but not for the 90° target. Figure 3D shows the interlimb differences in the movement duration overlaid on each other for both cursor conditions. The mean interlimb difference in movement duration was highest in the 30° target, followed by the 150° target as the next highest, with the lowest interlimb difference for 90° target (Fig. 3D). Interestingly, the interlimb difference in movement duration appeared larger when the left arm moved under higher inertial resistance (30° target) compared with when the right arm moved under higher inertial resistance (150° target). This pattern was consistent across both cursor conditions. ANOVA found a significant effect of target direction [F(2,18) = 43.4450, P < 0.0001], but effects of cursor condition and the two-way interaction (cursor condition × target direction) were not significant (both P > 0.5731). In the post hoc analysis of movement duration across target directions, we found that the 90° movements showed the lowest interlimb differences [30° vs. 90°, P < 0.0001; 150° vs. 90°, P < 0.0001], while the 30° movements showed the greater interlimb differences [30° vs. 150°, P = 0.0042].

Figure 3E shows the interlimb difference in peak velocity for all target directions and both cursor conditions. The mean interlimb difference of peak velocity was higher in the 150° and 30° targets than in the 90° target. Although the mean interlimb difference of peak velocity appeared highest in the 30° target when the left arm was under higher inertial resistance, this difference was not statistically significant (Fig. 3E) and these patterns were consistent across both cursor conditions. In our ANOVA, we found a significant effect of target direction [F(2,18) = 11.8537, P = 0.0005], but effects of cursor condition and the two-way interaction (cursor condition × target direction) were not significant (both P > 0.2195).

To summarize our results on temporal measures, the data showed that the movement of two arms was most synchronous in the 90° target direction when the inertial resistance was symmetric and the degree of synchronization varied with asymmetry of the inertial resistance to movement. Also, asynchrony was generally higher when the nondominant left arm was under higher inertial resistance (30° target). These patterns of temporal asynchrony were consistent across cursor conditions.

Spatial Measures

Figure 4A shows the final position error for all three target directions in both cursor conditions overlaid on each other. As described in the methods, in the two-cursor condition, the final position errors of both hands were averaged, whereas in the shared cursor condition, the final position error was calculated from the shared cursor. The analysis of final position error of each hand in the two-cursor condition was performed separately in the supplementary section (Supplemental Fig. S2B). In both cursor conditions, mean final position error was higher in the 150° and 30° target directions than in the 90° target direction (Fig. 4A). In the shared cursor condition, the final position errors were generally lower than in the two-cursor condition. This was confirmed by ANOVA with a significant effect of target direction [F(2,18) = 10.0429, P = 0.0012] and cursor condition [F(1,9) = 9.1349, P = 0.0144]. There was no significant interaction effect of target direction and cursor condition [F(2,18) = 0.0140, P = 0.9861]. Figure 4B shows the absolute interlimb difference for reaching distance in both cursor conditions overlaid on each other. The mean interlimb difference of reaching distance was lowest in the 90° target. The mean interlimb difference of reaching distance was higher in the 30° than in the 90° target, and highest in the 150° target (Fig. 4B). The interlimb difference seemed to be higher in the shared cursor than in the two-cursor condition, especially in 30° and 150° targets. In our ANOVA, we found a significant effect of target direction [F(2,18) = 24.0470, P < 0.0001] and cursor condition [F(1,9) = 6.7826, P = 0.0285]. There was no significant interaction effect of target direction and cursor condition [F(2,18) = 1.7648, P = 0.1996]. In the post hoc analysis of the target direction effect, we found that the interlimb difference of reaching distance was significantly smaller in the 90° target direction than in the other two directions [150° vs. 90°, P = 0.0010; 30° vs. 90°, P < 0.0001; 30° vs. 150°, P = 0.0612]. In the post hoc analysis of the target direction × cursor condition interaction, the difference between the cursor conditions in 150° target direction was slightly above the α threshold [shared cursor-150° vs. two cursor-150°, P = 0.0574], while the difference between the cursor conditions in other two directions was not significantly different [shared cursor-90° vs. two cursor-90°, P = 0.9612; shared cursor-30° vs. two cursor-30°, P = 0.7363]. In the supplementary section, we performed additional analysis on signed interlimb difference of reaching distance, and we found that greater interlimb difference of reaching distance shown in 30° and 150° target directions was mainly due to the arm under higher inertial resistance reaching farther than the opposite arm (Supplemental Fig. S2D). In summary, our data indicate that two hands show lower final position errors in the 90° target direction when the inertial resistance was symmetric than in the other two targets. Participants performed the task more accurately in the shared cursor condition than in the two-cursor condition. Higher interlimb differences in reaching distance were found in the 30° and 150° targets compared with the 90° target, and interestingly, the interlimb difference in reaching distance was generally greater in the shared cursor condition than in the two-cursor condition, especially in the 150° target direction.

Figure 4.

Final position error and the interlimb differences in reaching distance. A: mean final position error in three target directions for both cursor conditions. B: the mean interlimb difference values for both cursor conditions. Each dot represents an averaged data point from one participant. In all graphs, the mean for line plots was calculated across participants, and the error bars represent one standard error. SC, shared-cursor condition; TC, two-cursor condition.

Interlimb Spatial Covariation

To investigate potential spatial covariations between the hands, we decomposed the final positions of the right and left hands into two axes: an axis parallel to the target direction and an axis perpendicular to the target direction. Figure 5 shows the group-level covariation between the final positions of the right and left hands in the parallel axis (Fig. 5A) and the perpendicular axis (Fig. 5B) for each of the three target directions in both the two-cursor condition (left) and shared cursor condition (right). In the parallel axis, if the slope deviates from 1, it indicates that the two hands are more asymmetric. In the perpendicular axis, the sign of covariation indicates whether the two arms deviated in the same direction (positive covariation, non-mirror symmetric) or in the opposite direction (negative covariation, mirror symmetric) relative to the target axis. We initially predicted that, under the shared cursor condition, a correlation line in the parallel axis should slant toward the arm that experienced lower inertial resistance. Furthermore, in the perpendicular axis, we predicted two hands should deviate in the opposite direction relative to the target axis, thereby indicating spatial compensation between the two hands to stabilize the cursor position perpendicular to the target axis. The correlation analysis was performed at the group level for each target direction and cursor condition.

Figure 5.

Group-level intermanual covariation of reaching distance separated by the parallel axis (A) and the perpendicular axis (B). Each dot represents a single trial, and data for each target direction were coded in a different color. Each ellipse was drawn at 95% confidence, and the correlation lines were fitted by the orthogonal regression model. The correlation coefficients (r value) and slopes of the regression fit are indicated at the bottom right of each graph.

All 12 group-level correlations (2 measures × 2 cursor conditions × 3 target directions) were significant (P < 0.05). The correlation slope of the parallel axis measures for the 90° target was closer to the slope of 1 compared with the other two targets (Fig. 5A). In the perpendicular axis measure, the sign of the interlimb correlation was varied by the target direction. The correlation was negative in the 90° target direction, while the correlation was positive in the 30° and 150° target directions (Fig. 5B). This result indicates that spatial codependence in the parallel and perpendicular axis that, respectively, stabilized the position of the single cursor either along the target axis or perpendicular to this axis, depends on the target directions. Interestingly, this was true for both cursor conditions, such that the reaching distances between the hands covaried more symmetrically in the 90° target, and the negative correlation of perpendicular axis measures was only shown in the 90° target.

Optimal Control Simulations

In the optimal control simulations, we examined the impact of imposing mechanical asymmetry on the kinematics of reaching with two “hands” operating under optimal feedback control policies that either did or did not include costs for kinematic asymmetry. The nominal mass (M) and viscosity (B) values of two point masses were matched in the symmetric condition (at 1 kg and 1 Ns/m, respectively). M and B values were increased by a factor of 1.5 for one of the point masses in the asymmetric condition. For both controllers (with and without asymmetry costs), the ratio of weight values W_{TargetCapture}:W_Effort was 1.5:1 when target capture errors were quantified in units of centimeters and effort was quantified in units of Newton force. For the controller with asymmetry costs, W_{DisplacementAsymmetry} and W_{VelocityAsymmetry}, both were set also to 1.5. Figure 6 shows the results of our optimal control simulation under conditions of symmetric and asymmetric dynamic loads for the simulated left and right hands. The displacement and velocity of two point masses are overlaid in each graph for comparison. In our simulation of a controller with costs for kinematic asymmetry performed under the symmetric dynamic loads condition, we found symmetry in the kinematics of the two simulated point masses (Fig. 6, A and C). This kinematic symmetry was maintained under asymmetric dynamic loads condition (Fig. 6, B and D). In our simulation of a controller without costs for kinematic asymmetry, the kinematic profiles of the two point masses overlapped each other under symmetric dynamic loads conditions (Fig. 6, E and G). However, kinematic profiles were largely asymmetric under asymmetric dynamic loads conditions (Fig. 6, F and H). The point mass having the larger dynamic load moved slower, and its peak velocity time and movement end time were more lagged than the point mass subject to less dynamic loading. Critically, when comparing our simulation results to the empirical results, the simulation of the controller without costs for kinematic asymmetry corresponded better to our empirical findings.

Figure 6.

Optimal feedback simulation results. A–D: results from a controller with costs for kinematic asymmetry. E–H: results from a controller without costs for kinematic asymmetry. In each controller simulation, the left column represents the symmetric dynamic loads condition, and the right column represents the asymmetric dynamic loads condition, in which the mass and viscosity of one of the two-point masses were increased by 50%. The first row shows the trajectory of position (y) over time in each simulation and condition, and the second row shows the change of velocity over time. Vertically dotted lines indicate the peak velocity time and movement end time. Blue and red colored lines represent the two different point masses that were simulated. Purple (blue + red) colored line is shown when two lines were completely overlapping each other.

DISCUSSION

This experiment was designed to determine whether reaching tasks that do not explicitly require kinematic symmetry show symmetrical kinematic patterns due to a feature or cost of the controller, or whether symmetrical kinematic patterns emerge secondary to the symmetry of peripheral mechanical conditions. Two previous hypotheses have been proposed to explain bimanual coordination. The coupling hypothesis proposes that bimanual coordination is governed by a neural process that considers kinematic dissimilarity as a control cost to be minimized during performance. The task-dependent control policy hypothesis (28, 33) posits that bimanual coordination is the product of a control policy that minimizes control effort as well as costs in task performance and accuracy. This is particularly evident from the high codependence observed between arms when the task requires cooperative transport of a shared virtual object, during which extrinsic perturbations are applied to one arm and are compensated by corrections made by the other arm (28). However, considering that these studies have primarily focused on movement directions involving mirror symmetric bimanual joint motions between the arms, it remains unclear whether the observed codependence between the arms would generalize to movements involving nonmirror symmetric bimanual joint motions between arms.

To test the robustness of these hypotheses when bimanual movements are biomechanically incongruent across the arms, we designed a planar bimanual reaching task in which participants were instructed to reach toward targets in three different directions (150° vs. 90° vs. 30°) and under two different cursor feedback conditions (two cursor vs. shared cursor). Although the required distances and directions in extrinsic space were equal for all targets and both arms, the target setup imposed either symmetric or asymmetric effective inertial loading during the reaching movement. This asymmetry was due to direction-dependent variations in effective limb inertia (41). In the case of 90° reaching, both arms moved in the same forward direction, resulting in congruent joint motions with symmetric limb inertia. However, in 30° and 150° reaching, one arm moved to the ipsilateral side while the other moved to the contralateral side, leading to noncongruent joint motions with asymmetric limb inertia. This relative inertial symmetry (90° target) or asymmetry (150° and 30° targets) between the arms for different target directions was confirmed through calculation of squared net joint torques, which is a measure of inertial resistance to joint acceleration. Importantly, a coupled control system seeking to minimize the costs of kinematic differences between limbs predicts high bilateral spatiotemporal symmetry, regardless of inertial incongruities between limbs (i.e., target direction) and visual feedback conditions (i.e., cursor conditions). On the other hand, a task-dependent control system seeking to minimize control effort and kinematic performance errors predicts that bilateral covariance should be higher in the shared cursor condition, regardless of the inertial incongruity between limbs imposed by different target directions.

Our results revealed a strong spatiotemporal similarity between the arms only when the movements required bilaterally symmetric limb inertia (90° target). By contrast, reach kinematics became grossly asymmetric in directions that induced asymmetric inertial resistances (30° and 150° targets). There were no effects of cursor condition on temporal kinematic measures, although there were lower final position errors as well as an increase in the interlimb difference in reaching distance in the shared cursor condition. Critically, the spatiotemporal asymmetry between limbs during movements with bilaterally asymmetric inertial requirements (30° and 150° targets) does not align with the predictions of a coupling control hypothesis.

To assess the degree of spatial covariation of the two arms, we then conducted orthogonal regressions either along the target axis (distance covariation) or perpendicular to the target axis (directional covariation), at the movement end. The slope of the directional covariation relationship between the final positions of the two arms was consistently positive in the asymmetric inertial resistance conditions (30° and 150° targets) under the shared cursor condition. This demonstrated that there were minimal active spatial compensations between the two hands. This therefore does not support the task-dependent control policy hypothesis, in which strong negative covariation representing codependent spatial control is predicted under all conditions of inertial asymmetry in the shared cursor condition. It should be noted however that previous studies of task-dependent bimanual control have focused on experiments in which extrinsic perturbations were applied to one arm (28, 50, 51), whereas no perturbations were applied in the current study.

Finally, we tested the coupling hypothesis using optimal feedback control simulations involving distinct controllers that implement control policies that either include or do not include explicit costs for kinematic asymmetry. The simulation results showed that the controller without asymmetry costs was most consistent with our empirical results: asymmetries in kinematics emerge when peripheral dynamics are asymmetric. Given that coupling control suggests bimanual coordination is governed by a control policy that considers kinematic asymmetry as control costs, our empirical and simulation findings did not support the coupling hypothesis. Instead, our findings support the idea that kinematic similarity can emerge as a result of peripheral mechanical symmetry. Furthermore, the visual feedback-driven codependence in bimanual movement does not generalize to unperturbed nonmirror symmetric movements.

Bimanual Control Policies Do Not Consider Kinematic Asymmetry as a Control Cost

Early research on bimanual coordination has shown that when the two arms reach toward targets that require temporally or spatially noncongruous movements, a significant disruption to performance accuracy occurs in both arms. This phenomenon has been referred to as “interference” because the task accuracy of the hands during the bimanual movement is often reduced when compared with unilateral performance (1, 52–54). During incongruous bimanual movements, the two hands appear to deviate from their intended trajectories and become more similar to each other, producing more similar temporal and spatial task outcomes. This phenomenon has been shown in both spatial and temporal domains, and it seems to occur universally across either discrete or continuous bimanual movements when the task requires different temporal or spatial profiles for each hand (1, 6, 12, 17, 19, 20, 23, 25, 55–57) (for review, see Ref. 58). Kelso and colleagues have attributed this phenomenon to bilateral “coupling,” which is a concept suggesting groups of muscles recruited across arms during bimanual movements might be activated as a functional unit by a single, shared motor command (2, 5, 59). This has been proposed to explain why incongruous bilateral movements are often reported to have high spatial and temporal symmetry. As a result, this theory implies that there is a control cost for kinematic asymmetry imposed during bimanual movements and thus that symmetric kinematics should be expected regardless of either biomechanical asymmetries between limbs or visual feedback manipulations.

In previous attempts to test this hypothesis, researchers compared kinematic symmetry during bilateral reaching movements under congruous and incongruous target distances (14, 18, 24, 26). These studies consistently reported greater spatial and temporal asymmetries in the incongruous condition compared with the congruous condition. In our current study, we extended this investigation by keeping the distances and direction of the two targets (for two hands) consistent but varying the symmetry of the effective limb inertia for the two arms. Our results showed that kinematic symmetry only occurs in the symmetric (90° target), but not in the asymmetric, inertial target condition shown by highly synchronized temporal movement parameters (i.e., movement initiation, peak velocity time, and movement end) and spatially symmetric movements (i.e., reaching distance). Furthermore, in the simulation of two separate controllers that include or do not include explicit costs for kinematic asymmetry, we found the strongest consistency between our empirical results in the simulation of the controller that did not include costs for asymmetry. Collectively, these findings do not support predictions made by the coupling hypothesis and further indicate that the control policy mediating bimanual control does not consider kinematic asymmetry as a significant control cost.

Further support for this conclusion comes from a recent study in which participants were asked to make bilateral reaches to move a shared cursor toward a midline target. In this study, the position of the shared cursor was controlled by an equal (1:1) gain contribution of the two hands. However, in some blocks of trials, the contribution of each arm to the cursor’s lateral displacement was covertly made asymmetric, such that the gain of one hand in the perpendicular direction increased while the perpendicular gain of the other hand was reduced. Importantly, the gain of each hand to cursor motion parallel to the target axis remained the same 1:1 throughout the session. The results showed highest spatial covariation under the equal gain condition, but under asymmetric gain conditions, the extent of spatial covariation between hands was graded to the magnitude and direction of the gain asymmetry (31). Thus, extent of bimanual spatial symmetry was modified depending on visual feedback conditions, despite the fact that there was no change in the required mirror-symmetric movement. This suggests that the bimanual controller does not obligatorily minimize kinematic asymmetry during bimanual movements. Consistent with this notion, our current findings suggest that spatiotemporal symmetry during bimanual movement is best characterized as a consequence of the biomechanical similarity in task requirements between the arms, as opposed to a coupling of central commands, or by minimizing a cost for asymmetry. Taken together, our current and recent findings provide clear evidence that kinematic symmetry changes depending on the central factors (i.e., transformations of visual feedback) or the peripheral factors (i.e., inertial requirements of the movement) involved in the task. These results would not be predicted if the bimanual control policy minimizes kinematic asymmetry as the coupling control hypothesis implies. We, therefore, conclude that the CNS does not control two arms under a bimanual control policy that includes a cost for kinematic asymmetries.

Task Redundancy Is Not Exploited in a Codependent Manner during Bilateral Reaches with Biomechanically Asymmetric Requirements across the Arms

When two arms cooperatively control a shared cursor, there is redundancy in the space that allows one hand to effectively and flexibly compensate for errors made by the other (and vice versa) during task performance. For example, there are infinite combinations of different directions and distances that the two hands could adopt relative to each other to yield an average position that places the shared cursor in the designated target, thereby maintaining task success. Previously, researchers have demonstrated that exploiting redundant space in this way during shared cursor tasks can lead to different patterns of bimanual coordination when the required bimanual movements are mirror-symmetric (28, 29, 33, 50, 51). For example, some researchers have used a task where either hand could be unexpectedly perturbed from its intended movement trajectory by an external force applied by a two-dimensional planar robotic handle that was grasped by the participant during task performance. Under the shared cursor condition, but not under the independent cursor condition, the unperturbed hand compensated for the displacement imposed by the force exerted on the perturbed (contralateral) hand by moving in the opposite direction (28, 29, 33). Other studies have shown that under shared cursor, but not independent cursor tasks, rapid feedback mechanisms that appear compensatory in nature can be elicited in the opposite arm in response to unilateral force perturbation (50, 51). This difference in bimanual coordination between shared cursor and independent cursor conditions has been interpreted within the framework of optimal control theory, which proposes a hypothesis that the controller establishes a task-dependent control policy that minimizes control effort and costs related to task errors during bilateral movements (33). Therefore, in cases where two hands can cooperatively solve the task (i.e., shared cursor condition), a bimanual control policy that allows for reciprocal spatial compensation between the hands to achieve shared goals is established. However, in cases where two hands should independently solve the task (i.e., two-cursor condition), a bimanual control policy that is more independent of each other’s goal is established.

The task-dependent control policy hypothesis (28, 33) would likely predict a negative covariation between left- and right-hand final positions (perpendicular to the target axis) for all three target directions under the shared cursor condition but not necessarily under the two-cursor condition. This negative covariation would indicate a spatial compensation between hands in which undesirable deviations away from the intended trajectory should be compensated by opposite direction motion of the other hand. However, our results did not show such compensation. Instead, our results showed that negative covariations between the hands only occurred in the 90° target direction, but not in the directions with asymmetrical inertial conditions (30° and 150° target directions). These results suggest that the task-dependent control policy hypothesis (28, 33) may not apply to movements that are not perturbed by an external force, or for which inertial conditions are asymmetric between the arms. It is possible that natural variability during unperturbed movements simply does not produce the amplitude of task error that might be expected to elicit interlimb compensation and resulting covariation such as that reported in studies with explicit force perturbations (28, 29). Given the frequent changes in target direction (every trial) and in cursor condition (every block of 15 trials) imposed in the current study, it may have been difficult for participants to settle on an “optimal” strategy given the high level of uncertainty regarding upcoming task conditions. This has been observed and conceptualized elsewhere as a cost associated with “switching” the control strategy between frequently changing task conditions (60, 61). As a result, participants may have weighted a switching cost more highly with frequently changing task conditions and chosen to persist with some default control scheme rather than optimize their behavior based on motor effort and performance errors for task conditions on a given trial. It remains to be determined whether such a model would predict the pattern of results shown in this study.

Conclusions

In conclusion, this study examined two prominent hypotheses regarding bimanual coordination: the coupling hypothesis and the task-dependent control policy hypothesis. The coupling hypothesis proposes that the bimanual control policy considers kinematic asymmetry as a control cost. The task-dependent control policy hypothesis suggests that the bimanual control policy is established with cost functions that minimize control effort and task error costs based on the specific task requirements. We examined bimanual coordination during planar reaching movements toward three target directions where the target setup imposed relatively symmetric or asymmetric limb inertias between arms. The task was performed under two-cursor conditions intended to provide different strategies to solve the task: a two-cursor condition, in which the two hands move each cursor independently toward their respective targets; and a shared cursor condition, in which the two hands cooperatively move a shared cursor to a common target. The coupling hypothesis predicts kinematic symmetry between the arms regardless of symmetric or asymmetric limb inertial conditions induced by the target directions. The task-dependent control policy hypothesis predicts high spatial covariation between the arms regardless of the target directions under the shared cursor condition. Contrary to these predictions, our empirical and simulation data demonstrated that spatiotemporal symmetry and spatial covariation between two arms are highly affected by the inertial symmetry of the bimanual movement. Also, there was no difference between cursor conditions in the spatial covariation patterns across target directions. Our findings provide strong support for the hypothesis that bimanual control policies do not include kinematic asymmetry as a control cost. When the symmetry was not an explicit requirement of task performance, kinematic symmetry seems to be an artifact of the mechanical conditions of the task. Furthermore, the effect of visual feedback during bilateral movement, previously shown to increase spatial covariance between the arms, does not generalize to the unperturbed bimanual movements with asymmetrical mechanical conditions in the periphery, nor to conditions where target direction changes frequently.

DATA AVAILABILITY

The experimental data and the supplementary data used for the analysis in this study are openly available on an Open Science Framework (OSF) repository at https://doi.org/10.17605/OSF.IO/GC65F.

SUPPLEMENTAL DATA

Supplemental Figs. S1 and S2: https://doi.org/10.17605/OSF.IO/GC65F.

GRANTS

This work was supported by NIH Grant R01HD059783 and Dorothy Foehr Huck and J. Lloyd Huck Distinguished Chair Endowment awarded to Robert L. Sainburg. R. A. Scheidt’s contributions were supported by NIH grants R15HD093086 and R21NS121624.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

A.P. and R.L.S. conceived and designed research; A.P. performed experiments; J.Y. and R.L.S. analyzed data; J.Y., N.M.K., R.A.S., and R.L.S. interpreted results of experiments; J.Y. prepared figures; J.Y. and R.A.S. drafted manuscript; J.Y., N.M.K., A.P., R.A.S., and R.L.S. edited and revised manuscript; J.Y., N.M.K., A.P., R.A.S., and R.L.S. approved final version of manuscript.