In the first experiment, six human subjects performed the CST with the only instruction to ‘perform the task without failing to the best of your ability’. Failure occurred if the cursor escaped the boundaries of the screen (±10 cm from the center) within the trial duration of 6 s. Subjects received categorical feedback about the outcome at the end of each trial in a text appearing on the screen reading ‘Well done!’ for success, and ‘Failed!’ for failure. The degree of difficulty, set by λ, was increased stepwise across trials until the subject could no longer perform the task (see Methods for the specifics about the setting of λ values).

We first sought to examine the main characteristics of behavior in CST performance and how it compared between humans and monkeys. To quantify the overall behavior, four main metrics were employed as described and motivated below. To begin, we considered the overall success rate in the task among different individuals, before focusing on the kinematics of task performance. Figure 2A illustrates the success rates and how they dropped as the task difficulty increased. Both humans and monkeys showed a similar pattern of decrease in success rate which was well-captured with a sigmoidal function. Expectedly, individuals varied in their ability to achieve high difficulty levels as a measure of skillful performance, indicated by their ‘critical λ value’ λc, that is, the value of λ when the success rate dropped below 50%.

To investigate the performance in more detail, the kinematics of movement were examined, specifically the hand and cursor position during each trial. As indicated in Equation 1, the hand position p was the control input to the system which aimed to control the cursor position x as the variable of interest. Due to the unstable nature of the task, drifting of the cursor towards the edge of the screen demanded a response by the hand movement to avoid failure. As such, two simple metrics characterized control, one quantifying how the movement of hand and cursor correlated, and a second one to what degree the hand response lagged cursor displacements. Figure 2B shows the correlation between the cursor and hand movements as a function of task difficulty. The strength of the correlation increased as trials became more challenging in both monkeys and humans, asymptoting towards –1. According to Equation 1, this behavior was equivalent to reducing the sum p+x to mitigate the effect of large λ values on cursor velocity x˙ , and, hence, reduce the chance of failure.

The response lag from the cursor movement (observed feedback) to the hand movement (control response) is an important characteristic of a control system. As shown in Figure 2C, by increasing the task difficulty λ, the lag decreased for all subjects. Coupled with the increase in the strength of the correlation with increasing λ, these findings indicate that subjects generated faster and more precise corrective responses to cursor displacements in more difficult trials.

As the fourth metric, we calculated the ratio of root mean squared (RMS) of hand position to the RMS of cursor position for each trial, as a measure of response strength. This measure determined to what extent the control signal (hand movement) compared in magnitude to the cursor movement. A large RMS ratio meant that on average across a trial, the hand exhibited larger movements than necessary to correct for cursor deviations. Figure 2D illustrates the calculated RMS ratio as a function of task difficulty for humans and monkeys. Except for Monkey J, the RMS ratio showed a gradual decrease as the task difficulty increased for most individuals. The seemingly divergent behavior of Monkey J was likely due to subject-to-subject variability, as also observed in human performers (Appendix 1—figure 1). Such decrease could be justifiable due to larger cursor movements at higher difficulty levels, and perhaps more efficient corrective hand responses to cursor displacements. It is worth noting that for high λ values, small hand movements could cause large cursor displacements, which were detrimental to the task success. Therefore, pruning any task-irrelevant hand movements, consistent with promoting efficiency, seemed essential to succeed in more difficult trials.

Overall, the control metrics presented in Figure 2 give insight into how the CST was performed: as the task difficulty increased, subjects tended to respond to cursor displacements faster (that is, with lower lag), more precisely (seen in the stronger hand-cursor correlation), and more efficiently (with lower RMS ratio). Behavior was comparable between humans and monkeys, which suggests that there were similar control strategies used by both species. Next, we sought to detect those control strategies.

The CST, as described earlier, affords redundancy in the behavioral strategies that could result in task success. Although covert in aggregate level of performance (i.e. Figure 2), single trial observations of hand and cursor trajectories suggested that different underlying control objectives might be at play. Two types of behavioral patterns appeared recognizable in the data. In one case, the cursor seemed to be always balanced around the center of the screen, and any deviations from the center induced a response to bring the cursor back to the center. This was reflected in the oscillatory movements of the cursor around the center, shown in example trials in Figure 1C and D (first row). In other trials, the cursor either exhibited a slow drift from the center or remained relatively still anywhere within the boundaries of the screen, with only limited attempts to bring the cursor back to the center (for example, Figure 1C and D, second row). We hypothesized that these patterns of behavior arise from different control objectives, each focused on a different state variable in the state-space of the cursor movement. In the former case, the position of the cursor appeared to be the primary control variable. Under this strategy, subjects might pursue the objective of keeping the cursor near the center of the screen. We refer to this strategy as Position Control. In the latter case, the cursor velocity seemed to be of primary importance for control, with the objective to slow down cursor velocity regardless of its position in the workspace. We refer to this strategy as Velocity Control.

Can we distinguish between different control strategies by examining behavior? To test this idea, we took a computational approach by developing a generative model based on optimal feedback control (Todorov and Jordan, 2002) that could simulate the task under different conditions and with different objectives (Todorov and Jordan, 2002). The model involved a controller that generated optimal motor commands based on a given objective to perform the CST via a simple effector model. The model also contained a state estimation block that estimated the states of the system based on the given feedback (Todorov, 2005). In this case, cursor position and cursor velocity were used as feedback to the controller at each time step. Figure 3A illustrates a block diagram of this model.

The control gains used in the controller to generate the motor commands were optimally found by minimizing the sum of two cost functions: the cost of effort to reduce energy, as well as the cost of accuracy that prevented the states of the system from making large deviations (2):(2)J=∑t=1n(xtTQxt+utTUut)

where u and x represented the motor command and the state vector of the system, respectively. In this model, the state vector consisted of six states: the position, velocity and acceleration of the hand, as well as the position, velocity and acceleration of the cursor (see Methods). Variables t and n represent the time, and the total number of time steps in a trial, respectively. The matrix Q and the scalar U determined the weight of accuracy and effort in the cost function, respectively. Importantly, the matrix Q allowed for determining which states of the system were of primary importance in the control process. Therefore, the implementation of different control objectives in the controller was done through setting the Q matrix appropriately. As such, a Position Control strategy was implemented by setting the weight of cursor position in the Q matrix to a large value, emphasizing the primacy of cursor position as a control variable. Similarly, to implement the Velocity Control strategy, the weight of the cursor velocity in the Q matrix was set to a large value (see Methods). By simulating the task for each control strategy, we could generate synthetic behavior similar to that of humans and monkeys. Figure 3B and C illustrate a few example simulations of the task under different difficulty levels for the Position Control and Velocity Control, respectively. As exemplified, the simulated trials for Position Control show oscillatory movements of the cursor around the center, whereas the trials generated based on Velocity Control, exhibited slow drift of the cursor from the center with minimal attempt to correct for such drift. These characteristics were similar to the observed patterns of behavior in human and monkey data (Figure 1C and D).

To further identify the behavioral signatures associated with each control objective, beyond the apparent differences between single trials, we conducted a series of simulations in which the model performance was examined for a range of task difficulties. Novel predictions of the model for each control objective were assessed. For each objective, the task was simulated for different difficulty levels, ranging from λ=1.5 to λ=7, with increments of ∆λ=0.2. For each difficulty level, 500 trials were simulated (see Methods for details). In the first step, we performed the same set of analyses as reported in Figure 2 to evaluate how the model compared to human and monkey behavior at an aggregate level of CST performance. Figure 4A illustrates the overall performance of the model for both Position Control and Velocity Control. As shown, for each metric, the model exhibited comparable behavior to the experimental data with regard to the task difficulty: the success rate dropped in a sigmoidal fashion, the strength of the correlation between hand and cursor position increased, and the response lag between hand and cursor as well as the hand/cursor RMS ratio decreased. These results showed that, overall, both simulated control objectives produced similar behavioral characteristics as humans and monkeys. More interestingly, the model predicted that Position and Velocity Control performed comparably in success rate and hand-cursor correlation (Figure 4A, top two panels), but differed significantly in the response lag and the hand/cursor RMS ratio (Figure 4A, bottom two panels). Specifically, Position Control consistently showed larger values for lag and RMS ratio for most task difficulty levels.

The model indicated that differences in behavioral metrics existed for Position vs Velocity Control. This led to a new experiment for which we recruited two new groups of human subjects (n=6 per group). Each group performed the CST under the same procedure as described in Experiment 1, except that this time each group was explicitly instructed to use a specific control objective. One group was asked to perform the task with ‘keeping the cursor at the center of the screen at all times’. This instruction intended to induce Position Control. The second group was asked to ‘keep the cursor still anywhere within the boundaries of the screen’. This instruction aimed to induce Velocity Control (see Methods for details). In each group, the kinematic behavior of hand and cursor was collected, and the control metrics were calculated. The goal was to elicit differences in performance between the two groups and, if such differences were found, to determine whether they matched the behavior of the corresponding model.

The summary of performance for both human subject groups is shown in Figure 4B. The general trends of all four measures with respect to the task difficulty were consistent with the data generated by the model, as well as the human data from Experiment 1 (Figure 2). Importantly, the behavioral differences between the two control strategies in human data matched the predictions of the model relatively well (Figure 4A and B): the rate of success was similar, and with the exception of hand-cursor correlation, the group with Position Control instruction showed a significantly larger hand-cursor lag (unpaired t-test: t₁₀=3.79, p=0.004) and hand/cursor RMS ratio (unpaired t-test: t₁₀=5.27, p<10^–3) compared to the group with Velocity Control instructions (Figure 4C).

These results showed that the model not only captured the overall performance features observed in the data, it also successfully demonstrated the redundancy of control strategies in CST performance. Importantly, it could qualitatively distinguish between such strategies at an aggregate level of performance. To ask further, can we identify in a quantitative way the control objective employed by an individual? Can we do so even in a given trial, when no explicit information about their preferred objective was available? To this end, we examined performance at a single-trial level and introduced quantitative measures that evaluated the degree to which a particular control objective was used in that trial, as described in the next section.

To further investigate what control objective was preferred by an individual or in a given trial, we examined the predictions of the model about the cursor behavior in state space, and then tested these predictions using experimental data from Experiment 2. Two metrics were defined that captured the state-space behavior of the cursor in each trial. First, we examined the average cursor position and cursor velocity in each trial, represented in the state space of cursor movement. This provided a single data point for each trial in state space, indicating whether on average there was a drift in cursor position and its velocity away from zero (x=x˙=0). It was expected that for Position Control, all trials scattered around the origin of the state space, whereas for Velocity Control, they could deviate from the origin. We also examined whether there was a correlation between the cursor mean position and its mean velocity. This, in essence, was equivalent to the autocorrelation of cursor position, i.e., correlation between mean position and final position. Figure 5A illustrates the state-space representation of cursor movement based on model simulations for both Position Control (top) and Velocity Control (bottom), where each data point represents one simulated trial. As shown, the distribution of trials in this space differed markedly between the two control objectives (also see Appendix 1—figures 2 and 3). Position Control resulted in a distribution with little correlation between cursor position and its velocity, and closely scattered around the center. In contrast, Velocity Control revealed an elongated distribution with a relatively strong correlation between the cursor position and its velocity.

We further examined whether such distinction in behavior was solely due to a change in the control objective, or whether varying other parameters in the model simulations, such as motor noise, delay, or effort cost, could also generate similar distinctive patterns. These sensitivity analyses showed that different magnitudes of noise and delay only affected the success rates, but no other features, as to be expected. Effort cost also could not account for the observed differences in the above mentioned movement distributions (Appendix 1—figures 4–6). Therefore, this robustness to variations in the model parameters allowed us to probe performance based on the underlying control objective.

To validate the model predictions, the same analyses of cursor position and velocity were performed on the empirical data from Experiment 2. Figure 5B illustrates three example subjects from the Position Control and Velocity Control groups, and Figure 5C shows a summary of how the correlation values differed across control objectives for the model and the empirical data. As shown, overall, subjects in the Velocity Control group showed significantly larger correlations than individuals in the Position Control group (unpaired t-test on the Pearson correlation coefficient: t₁₀=4.06, p=0.002). Based on the within-group variability, this allowed us to determine how pronounced a subject executed their respective strategy compared to other subjects in the same group.

It should be noted that Figure 5B shows the data for an ensemble of trials ranging in difficulty levels from easy up to the critical λc values, corresponding to success rates ranging from 100% to 50%. Additional analyses probed whether these behavioral features changed as a function of the difficulty levels. When grouping the trials into easy and moderate difficulty trials, the cursor position and velocity relations did not change and the correlations for the two control objectives continued to reveal the same relative difference between the control objectives (see Appendix 1—figures 7 and 8).

Due to the task’s redundancy, the control objective may not be fixed for an individual throughout their performance and might vary from one trial to the next. It is therefore of great interest to determine, in a given trial, to what extent the behavior is the outcome of Position or Velocity Control. To this end, we examined the magnitude of cursor movement calculated as the root mean squared (RMS) of its position and velocity in each trial. This was directly related to the cost functions used in the model (Equation 2), which provided a more direct comparison regarding the primacy of Position versus Velocity Control of the cursor: Position Control aimed to minimize the RMS of cursor position, while Velocity Control aimed to minimize the RMS of cursor velocity. This distinction could be well represented in the state-space of the cursor movement.

Figure 6A illustrates the model prediction for the RMS of cursor position and cursor velocity plotted against each other for Position Control (top) and Velocity Control (bottom). For Position Control, the distribution of trials leans towards the vertical axis (restricting cursor position but allowing large cursor velocities), whereas for Velocity Control, it leans mainly towards the horizontal axis (a larger range of cursor positions but restricted velocities). This distinction could be quantified by the slope of a fitted regression line to the data, with relatively larger slopes indicating Position Control and smaller slopes signaling Velocity Control. Similar patterns of behavior could be observed in the human data from Experiment 2 as illustrated in Figure 6B and C, with the Position Control group showing significantly larger regression slope than the Velocity Control group (unpaired t-test, t₁₀=6.33, p<0.001). The regression slope could more clearly distinguish between individual trials than could the correlation coefficient metric shown in Figure 5, regarding their corresponding control objective: if a given trial in the RMS space of the cursor movement lay below/above a certain slope threshold, its performance could be considered the result of Velocity/Position Control. We could therefore use this behavioral feature to develop a classifier that inferred, with a certain level of confidence, the underlying control objective in the performance of an individual in any given trial.

When monkeys performed the CST, we lacked explicit knowledge about which strategy they might have employed. This resembles Experiment 1 when humans performed the CST with no specific instructions and their control objective was not explicitly available. To achieve the goal of inferring an individual’s control objective based on their performance, we used the control characteristics that our computational approach introduced to distinguish between different control objectives. To this end, the simulation results based on the cursor movement in its RMS space (Figure 6A) were used to train a simple classifier, a support vector machine (see Methods). This classifier then determined, based on the learned regression slopes from the RMS distributions (Figure 7A), whether a given trial was likely performed as Position Control, or Velocity Control.

We first evaluated the accuracy of the classifier based on simulated data, where the classifier was trained on synthetic data generated with Position and Velocity Control (2250 trials each), and then tested on a separate set of trials from either control objective (2500 trials). For any given trial, the classifier obtained a posterior probability indicating to what extent that trial was generated under Position Control. The probability of 95% and higher identified the given trial as Position Control, while the probability less than 5% labeled the trial as Velocity Control; anything in between was considered ‘Uncertain’ as to the underlying control objective. We also measured how often the classifier misclassified a Position Control trial as Velocity Control, and vice versa. The results showed that 6.2% of trials that were generated with Position Control were misclassified as Velocity Control, and 5.5% of trials that were generated with Velocity Control were misclassified as Position Control. This provided a reasonable accuracy for the classifier applied to the experimental data.

Next, we tested the performance of the classifier on the empirical data from Experiment 2, where the intended control objective used by each subject was known. We asked how well the classifier could recover the control objective used by each subject. Figure 7B shows the cursor RMS data from three example subjects in each instructed group (similar to Figure 6). For each trial (data point) the probability of Position Control was estimated by the classifier, and the trial was identified as Position or Velocity Control based on the estimated probability (>95% for Position, and <5% for Velocity Control). As shown in Figure 7B, for the Position Control group, most of the trials were rightfully classified as Position Control, and similarly for the Velocity Control group, the majority of trials were classified as Velocity Control. The average probability across all trials for each individual was also obtained as an overall measure of the control objective for that subject. This average measure is shown in Figure 7B for the example subjects and summarized in Figure 7C for all subjects in each group. This showed that the classifier correctly determined the control strategy of each individual without being trained on any experimental data.

The ultimate test of our approach would be to infer the control strategy used by individuals whose control objective was unknown, that is the monkeys and the humans who received no instructions about the control strategy in Experiment 1. After representing the performance of each subject in the RMS space, the classifier was used to determine what control objective was used in each trial. Figure 8 illustrates the classification results for human subjects of Experiment 1 as well as two monkeys (Monkey I and J from Quick et al., 2018). The model simulations are also provided as reference in Figure 8A. Figure 8B and C show the data from three example human subjects, as well as two monkeys, in which each trial is either labeled as Position Control (brown), Velocity Control (cyan), or Uncertain (grey). Two example trials, one from each inferred control strategy are also singled out from each subject’s performance in Figure 8B and C (bottom row) to show how the hand and cursor movement behaved under each control objective. Calculating the average probability of control objective for each individual, similar to Figure 7, we could infer which control objective was of primary importance for each subject (Figure 8D). For example, human subject NI-S2 more likely adopted Velocity Control, while human subject NI-S4 mainly performed the task with Position Control (Figure 8B). Similarly, Monkey I seemed to prefer Velocity Control, while Monkey J most likely adopted Position Control (Figure 8C).

Ultimately, our procedure enabled us to not only infer the underlying control objective at a single trial level, but also identify which objective was overall preferred by humans and monkeys when no explicit knowledge about their strategy was available. These results are encouraging as they constitute an important step towards bridging our findings between human and monkey research, and ultimately guide neurophysiological analyses to identify the neural underpinnings of control objectives in the primates’ brain.

18 healthy, right-handed university students (age: 18 to –25 years; 8 females) with no self-reported neuromuscular pathology volunteered to take part in the experiments. All participants were naïve to the purpose of the experiment and provided informed written consent prior to participation. The experimental paradigm and procedure were approved by the Northeastern University Institutional Review Board (IRB# 22-02-15).

The data from two adult male Rhesus monkeys (Macaca mulatta, wild type, supplied by Alpha Genesis, Ages: 7 and 8 years old) used in this study was taken from a previously published work (Quick et al., 2018). All animal procedures were approved by the University of Pittsburgh Institutional Animal Care and Use Committee, in accordance with the guidelines of the US Department of Agriculture, the International Association for the Assessment and Accreditation of Laboratory Animal Care, and the National Institutes of Health. For details of experimental rig and procedure see the Methods in Quick et al., 2018.

The CST involved balancing an unstable cursor displayed on the screen using the movement of the hand (Jex et al., 1966; Quick et al., 2014; Quick et al., 2018). The CST dynamics was governed by a first-order differential equation as shown in Equation 1. The difficulty of the task was manipulated by changing the parameter λ: by increasing λ the task became more unstable, hence more difficult to accomplish. To perform the task, subjects sat on a sturdy chair behind a small table, with their right hand free to move above the table (Figure 1). A reflective marker was attached to the subject’s back of the hand on the third metacarpal bone. The hand position was recorded using a 12-camera motion capture system at a sampling rate of 250 Hz (Qualisys, 5+, Goetheburg, SE). The mediolateral component of the hand position was used to solve the CST dynamics with the initial condition of xt=0=0 (Quick et al., 2018). The calculated cursor position was real-time projected as a small blue disk (diameter: 4 mm, approximately 0.8° in visual angle) on a large vertical screen in front of the subject at a 150 cm distance. The processing delay of the visual rendering was roughly 50 ms.

Each human subject participated in the experiment for three consecutive days. At the beginning of the first day, subjects were familiarized with the experimental setup and the objectives of the task. Familiarization consisted of five CST trials with moderate difficulty level. These trials were later excluded from the analyses. The main experiment consisted of three main phases that were repeated on each day. The first and second phases of the experiment involved 15 reaction time trials and 10 tracking trials, respectively (data for reaction time and tracking trials were not reported in this study). Phase three involved the CST trials, which were performed in three blocks. In Block 1, subjects performed 30 CST trials, where the difficulty level was determined in each trial using an up-down method: starting from λ=2.5 in the first trial, if subjects succeeded/failed on the current trial, λ was increased/decreased by Δλ=0.2 in the next trial. By the end of Block 1, subjects had gradually converged to λ values in which the success rate was approximately 50%. This value was considered as the critical instability value (Quick et al., 2018), denoted by λc , and was obtained by averaging the λ’s of the last five trials of Block 1.

In Block 2, a stepwise increase in λ was adopted: subjects started with a difficulty level of λ=70%λc (using λc from the previous block). They continued until they completed 10 successful trials, or 20 trials in total (whichever occurred first). The difficulty level was then increased by Δλ=0.2, and the procedure repeated. This incremental increase of λ continued until the subjects’ success rate for the ongoing λ dropped below 10% (i.e., less than 2 successful trials out of 20). This marked the end of the second block. In total, subjects performed approximately 120–200 trials in Block 2, depending on the individual’s performance.

In Block 3, subjects performed the CST under three selected difficulty levels of easy, medium, and hard, with 20 trials for each difficulty level. These levels corresponded to λ values that led to 75% success rate (easy), 50% success rate (medium) and 25% success rate (hard) obtained from each individual’s performance in Block 2. The exact values of λ75% , λ50% , and λ25% were calculated by fitting a psychometric curve to the success rate data from Block 2 as a function of λ (see Figure 2). The order of difficulty was pseudo-randomly selected for each subject. For this study, we only analyzed the CST data from Block 2 (stepwise increase in λ) as it matched the procedure used in the monkey experiment (Quick et al., 2018). Subjects repeated the same experimental procedure on Day 2 and 3.

Three groups of human subjects participated in the experiment, where each group received different instructions about the task goal. The first group was instructed to perform the CST ‘without failing to the best of their ability’ (no-instruction group); the second group was instructed to ‘keep the cursor at the center of the screen at all times’ (Position Control group); and the third group was instructed to ‘keep the cursor still anywhere within the bounds of the screen’ (velocity control group).

To evaluate the overall performance of humans and monkeys during the CST, four quantities were calculated: success rate, correlation between hand and cursor position, hand-cursor lag, and hand/cursor RMS ratio. For each individual, the quantities were calculated as the average across trials for each bin of λ values (bin size: 0.3, starting from λ=1.5).

The success rate was obtained as the percentage of successful trials within each λ bin. A psychometric curve (a Gaussian cumulative distribution function) was then fitted to the success rate data as a function of λ to estimate λc (critical stability, where success rate was 50%):(3)%Success=501-erfλ-λc2σ

where, ‘erf’ indicates the error function, and σ denotes the standard deviation of the Gaussian cumulative. The correlation and lag quantities (Figure 2, B and C) were obtained by first cross-correlating the hand and cursor position trajectories in each trial, and then finding the peak correlation and the corresponding lag (Figure 2, see also Quick et al., 2018). The hand/cursor RMS ratio (Figure 2, D) was defined as the ratio of the root mean squared (RMS) value of hand position over the RMS value of the cursor position in each trial.

Finally, to perform the classification analysis used in Figure 7 and Figure 8, a Support Vector Machine method was applied to learn the two-class control objective labels. In order to build and train a classifier, we used ‘fitcsvm.m’ function in MATLAB, where synthetic data (RMS of cursor position and cursor velocity) was used as training set. To classify experimental data using the trained classifier, the MATLAB function ‘predict.m’ was used. Finally, the posterior probabilities over each classification (i.e. the confidence on classification) was calculated using the ‘fitPosterior.m’ function in MATLAB.

An optimal control model was used to build control agents that performed the CST with different control strategies. The model involved an optimal feedback controller that moved the hand, a point mass of M=1 kg, through a simple muscle-like actuator (Todorov, 2005; Todorov and Jordan, 2002). The muscle model was approximated by a first-order low-pass filter that generated forces on the hand in the lateral direction as in Equations 4; 5:(4)τF˙=-F+u(5)p¨=1MF

where F is the actuator force acting on the hand, τ is the time constant of the low-pass filter, u is the control input to the muscle, and p¨ is the second derivative of the hand position. Combined with Equation 1, the model includes cursor position x, hand position P, and actuator force F as the states of the system. By taking the first derivative of Equation 1, the hand and cursor velocity are also included in the state space of the system:(6)x¨=λ(x˙+p˙)

Finally, by combining Equations 1; 6, the CST dynamics can be derived as follows:(7)x¨=λ2x+λ2p+λp˙

Note that by taking the higher derivative of CST dynamics in Equation 6, we practically made the cursor velocity x˙ available to the controller as a state of the system, which allowed us to explore different control strategies directly related to the cursor velocity. This was done with the caveat that the initial conditions of the resultant CST dynamics in Equation 7 should always satisfy Equation 1.

In this case, the behavior of the system could be represented by the state vector, x=[x,x˙,p,p˙,F] , using the state-space form of the system dynamics as shown below:(8)x˙=Ax+Bu

where A and B represent the dynamics of the system, and u is the control input:(9)A=[01000λ20λ2λ00001000001/M0000−1/τ]B=[00001/τ]T

In order to implement the feedback controller, the state-space equations were first discretized using the time steps of δ=10 ms. Further, three noise terms were included in the system dynamics representing the motor additive noise ξ, signal dependent noise ε and sensory additive noise ω, according to the models of biological systems (Harris and Wolpert, 1998; Todorov, 2005). The resultant equations of the system dynamics were presented as shown below:(10)xt+1=Adxt+Bd(1+εtC)ut+ξtyt=Hxt+ωt

where εt , ξt, and ωt are zero-mean Gaussian noise term, C is the signal-dependent noise scalar, yt represents the sensory feedback, and matrix H determines the available sensory feedback from the vector of states. For our simulations, all the states were available as feedback, therefore, we considered H=diag1,1,1,1,1 . The matrices Ad and Bd were modified according to Equation 9 for discrete-time representation of the system: Ad=I+δA and Bd=δB, where I was an identity matrix.

Given the overall dynamics of the system, the feedback controller aimed to calculate the optimal motor command ut based on the sensory feedback yt by minimizing the cost function J (Todorov, 2005):(11)ut=argmin(J)J=∑t=1n(xtTQxt+utTUut)

where n was the number of time samples throughout the movement, and Q and U determined the contribution of state accuracy and effort in the cost function, respectively. In all simulations, U=10. The matrix Q, however, was appropriately manipulated to implement different state-dependent control strategies (see below). Accordingly, the optimal control law was obtained in the form:(12)ut=−Ltx^t

where Lt was the optimal control gain that was solved recursively by minimizing the cost function J (see equation 4.2 of Todorov, 2005 for detailed calculation of Lt). Also, x^t represented the estimated states of the system based on the provided feedback yt , which were obtained using a state estimator as shown below:(13)x^t+1=(Ad−BdLt)x^t+Kt(yt−Hx^t)

Here, Kt was the filter gain matrix which was calculated in a recursive procedure along with the control gains (see equation 5.2 of Todorov, 2005).

The aim of the Position Control strategy was to maintain the cursor at the center of the screen throughout the trial. This was implemented by penalizing the deviation of the cursor position x from the center. In this case, the matrix Q was set to Q=diagq,0,0,0,0,0, where q≫0 was a constant. As such, the cost of deviation from the center for the cursor position was dominant represented in the value J of the cost function, making the regulation of cursor position at the center, the primary goal of control.

The Velocity Control strategy aimed to keep the cursor still at any point within the boundaries of the workspace. In this case, upon deviation of the cursor from the center, the main goal was to bring the cursor to a stop regardless of the location. This was implemented through penalizing the cursor velocity x˙ by setting the matrix Q=diag0,v,0,0,0,0 , where v≫0 was a constant.

Given a control strategy, the model generated 500 trials of CST for each level of task difficulty from λ=1.5 to λ=7, with increments of ∆λ=0.2. The parameters of the hand and the muscle model (Equations 4; 5) were fixed to M=1 kg and τ=0.06 s. A sensory delay of 50 ms was considered when simulating the task with the optimal feedback controller (Cluff et al., 2019; Todorov, 2005). To implement the delay, system augmentation was used by adding the states from the current time step with all the states from the 5 preceding time steps (Crevecoeur et al., 2019; Todorov, 2004). The signal-dependent noise terms were set to εt∼N(0,1) and C=1.5. The motor noise was ξt∼N(0,Σ) , where Σ=0.4BBT . For each trial, the simulation started from the initial condition of x=0, and ran for 8 s. Only the first 6 s of each simulation were considered in the analysis for consistency with the experimental paradigm. The success or failure in each simulated trial was decided post-hoc, by determining whether the cursor position x exceeded the limits of the workspace (±10 cm from the center) within the 6 s duration of the trial.