Discussion

Comparison of computational model performance

Most previous studies of reaching movements have examined point-to-point reaches over distances of approximately 30–50 cm (e.g., Hogan 1984; Uno et al. 1989; Nakano et al. 1999; Katayama 2025). In contrast, the present study focused on shorter reaches (20 cm), so that all participants, including shorter individuals, could perform movements in eight directions within the reachable workspace. Although differences among computational models might be expected to diminish for shorter reaches, the present results showed that clear differences remained, particularly in the reproduction of arm posture.

Comparison of the computational models further showed that small errors in fingertip trajectory and arm posture do not necessarily guarantee accurate reproduction of the direction-dependent contribution of the wrist joint. The AJ model produced small errors in fingertip trajectory and arm posture, but its direction-dependent wrist contribution did not match the characteristic pattern observed in the measurements. By contrast, although the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model showed relatively large fingertip-trajectory errors in MD3 and MD4, it yielded small posture errors across all movement directions and reproduced the measured wrist contribution more accurately than the other models. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 model showed qualitative tendencies similar to those of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model, but the trajectory and posture errors were slightly larger, and the mismatch in wrist contribution was also greater. These results indicate that the apparent reproducibility of human arm movements depends on the evaluation metric used—fingertip trajectory, arm posture, or wrist contribution—and therefore should be assessed from multiple perspectives.

The optimal fingertip trajectories and arm postures predicted by each computational model depend on the joint-viscosity parameters. Katayama (2025) evaluated the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 and TC models under five joint-viscosity conditions and reported that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 was largely insensitive to joint viscosity, whereas the TC model showed larger errors in fingertip trajectories and arm postures under low-viscosity conditions but reproduced them more accurately under high-viscosity conditions. In the present study, additional analyses were conducted to examine the viscosity dependence of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 and MTC models (see Supplementary Materials for the results). The influence of joint viscosity on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 was again small. For the MTC model, the mismatch with the measurements was reduced under higher-viscosity conditions, similar to the tendency observed for the TC model. Therefore, it is important to evaluate the models using joint-viscosity values that are as close as possible to the true values for the human arm. Although many previous studies have reported estimates of wrist-joint viscosity (e.g., Sinkjær and Hayashi 1989; Falzarano et al. 2021), the reported range is broad. This variability likely reflects differences in cocontraction of flexor and extensor muscles, the presence or absence of grasping, and the method used to apply perturbations, all of which can alter viscosity during movement. Reported wrist-joint viscosity values therefore range widely, from nearly zero to 0.6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {N,m,s/rad}$$\end{document}Nms/rad. In the present study, the baseline wrist-joint viscosity used in the expanded arm models was set near the middle of the reported range. In addition, to prevent participants from exerting excessive force during movement measurements, they completed sufficient practice beforehand, including several wrist-movement exercises, and performed a brief wrist movement immediately before each trial (corresponding to a so-called waggle). These procedures likely reduced cocontraction, suggesting that joint viscosity during movement was not excessively high.

Changing the joint-viscosity value also changes the muscle-tension profiles of agonist and antagonist muscles. In general, higher viscosity increases viscous force; compensating for this requires greater agonist tension and reduced antagonist tension. As shown in Fig. 12, antagonist tension at the elbow and wrist was already small under the viscosity used in the present study. Because this tendency was observed across all movement directions, the joint viscosity adopted here may even have been somewhat overestimated. If viscosity were increased further, antagonist tension would likely decrease even more, approaching a pattern dominated by agonist tension alone; however, such a pattern would not be consistent with the agonist–antagonist activation typically observed in human movement. From the standpoint of joint viscosity, the values used in the present study—or possibly smaller values—therefore appear plausible. This consideration also suggests that the TC and MTC models should be interpreted cautiously, because their ability to reproduce the measured three-joint point-to-point reaching movements may depend strongly on the assumed viscosity values.

_Fig. 12: Muscle tensions predicted by the MTC, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}3$$\end{document}MSC3 models. Time was normalized to the movement time of each trial. Muscle tensions were averaged across all participants. (Muscle selection: S22, PCSA: PCSA1, Movement direction: MD8)

Fig. 13: Relation between PCSA and peak predicted tension. Each point corresponds to one muscle in one movement trial. (Muscle selection: S22, PCSA: PCSA1)

Fig. 14: Relation between trial-averaged moment arm and peak predicted tension. The moment-arm magnitude was defined as the mean absolute moment arm during each movement trial, and peak tension was computed separately for each muscle. (Muscle selection: S22, PCSA: PCSA1)

Muscle # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
MTRC | 29 ± 8 | 68 ± 13 | 50 ± 6 | 0 ± 0 | 6 ± 12 | 47 ± 14 | 85 ± 10 | 44 ± 13
MTC | 33 ± 17 | 69 ± 16 | 69 ± 16 | 56 ± 25 | 0 ± 0 | 79 ± 14 | 4 ± 8 | 0 ± 0
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MTC}_s$$\end{document}MTCs | 43 ± 22 | 88 ± 0 | 72 ± 14 | 0 ± 0 | 17 ± 19 | 54 ± 17 | 85 ± 13 | 47 ± 19
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 | 46 ± 24 | 92 ± 8 | 65 ± 21 | 0 ± 0 | 12 ± 20 | 68 ± 16 | 88 ± 10 | 56 ± 23
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 | 65 ± 21 | 88 ± 16 | 68 ± 21 | 0 ± 0 | 67 ± 20 | 69 ± 15 | 93 ± 6 | 86 ± 7
9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17
78 ± 13 | 0 ± 0 | 0 ± 0 | 65 ± 18 | 3 ± 8 | 3 ± 8 | 56 ± 16 | 54 ± 14 | 0 ± 0
68 ± 19 | 50 ± 16 | 0 ± 0 | 28 ± 10 | 1 ± 4 | 49 ± 7 | 61 ± 18 | 61 ± 18 | 46 ± 8
79 ± 14 | 0 ± 0 | 0 ± 0 | 71 ± 18 | 1 ± 4 | 1 ± 4 | 57 ± 12 | 57 ± 12 | 0 ± 0
82 ± 16 | 3 ± 8 | 0 ± 0 | 75 ± 21 | 11 ± 14 | 11 ± 14 | 58 ± 8 | 58 ± 8 | 1 ± 4
85 ± 13 | 24 ± 26 | 14 ± 16 | 72 ± 21 | 43 ± 9 | 22 ± 11 | 65 ± 14 | 65 ± 14 | 22 ± 10
18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26
25 ± 17 | 10 ± 16 | 88 ± 6 | 50 ± 14 | 40 ± 8 | 29 ± 6 | 71 ± 8 | 74 ± 9 | 65 ± 10
33 ± 18 | 33 ± 18 | 90 ± 5 | 29 ± 13 | 28 ± 10 | 33 ± 20 | 61 ± 9 | 56 ± 6 | 58 ± 8
31 ± 20 | 12 ± 16 | 90 ± 5 | 47 ± 19 | 33 ± 8 | 28 ± 5 | 56 ± 9 | 54 ± 6 | 54 ± 6
28 ± 19 | 12 ± 20 | 90 ± 5 | 36 ± 19 | 17 ± 12 | 3 ± 8 | 46 ± 14 | 32 ± 10 | 18 ± 18
49 ± 21 | 32 ± 20 | 93 ± 6 | 43 ± 17 | 32 ± 9 | 11 ± 16 | 57 ± 6 | 57 ± 6 | 56 ± 6

Limitations of the first-stage optimization

Under the constraints imposed in the first-stage optimization, the arm is stationary at the start and end points. These boundary conditions require the joint torques at both endpoints to be zero; consequently, the first-stage optimization yields zero muscle tensions at the beginning and end of the movement (Fig. 12). Under these constraints, the predicted cocontraction of flexor and extensor muscles during movement tends to be limited. In the present experiment, participants practiced extensively so that they could perform the reaching movements smoothly while reducing excessive cocontraction. In addition, the movement speed was set slightly lower than that of typical everyday arm movements, a relatively large target area was used, and participants were instructed not to force the fingertip into the target at the end of the movement. Because fingertip overshoot was very small in the measured movements, these endpoint constraints appear to be reasonable approximations for the present task.

In human reaching movements, however, triphasic muscle activity is often observed: an initial agonist burst is followed by antagonist activity and then by a second agonist burst (e.g., Kambara et al. 2013). Because the optimization used in the present study imposes zero muscle tension at the endpoint, the muscle-tension profiles predicted by the MSC and MTC models do not exhibit such triphasic patterns. Nevertheless, the present feedforward-planning framework could in principle be extended to account for triphasic activity. Two possible approaches can be considered. The first is to combine the present feedforward-planning model with feedback control driven by movement errors, under the assumption that a mismatch between the internal dynamics model used for planning and the actual arm dynamics causes the realized trajectory to deviate from the optimal trajectory. The second is to introduce signal-dependent noise in muscle tension, as in Harris and Wolpert (1998), and to combine it with feedback control that reduces the resulting trajectory deviations.

Muscle recruitment

This section discusses the strategy by which each computational model determines muscle recruitment, represented here by the muscle-tension distribution. This evaluation is central to the present study and distinguishes it from Katayama (2025), because the present 19–26-muscle model with joint-angle-dependent moment arms allows anatomical muscle recruitment to be examined, rather than only trajectory and posture reproduction.

Compared with the MTC model, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 models tended to suppress activation of muscles with small PCSAs and to recruit muscles with larger PCSAs more strongly. This tendency is consistent with the fact that, in both models, the objective function directly affects muscle-tension distribution through muscle stress (muscle tension divided by PCSA), thereby favoring solutions that avoid imposing high stress on muscles with small PCSAs. From a mechanical standpoint, however, a muscle with a smaller moment arm produces less torque for the same muscle tension, so selecting muscles with larger moment arms is more mechanically efficient. Indeed, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model also tended to avoid recruiting muscles with small moment arms, but muscles 5, 7, and 8 were recruited despite their relatively small moment arms because they have large PCSAs in PCSA1. This result suggests that, in the MSC framework, PCSA strongly influences muscle recruitment in addition to moment arm.

When muscle recruitment under PCSA1 and PCSA3 was compared for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model, muscles with large PCSAs in PCSA3, such as muscle 1, tended to be recruited more frequently, whereas muscles with small PCSAs, such as muscles 2, 6, 7, and 8, were recruited less frequently. Thus, in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model, the set of recruited muscles and their recruitment levels varied depending on the PCSA values used in the optimization. Furthermore, S11, S12, and S21 in Table 2 include fewer muscles than S22; several muscles included in S22 are omitted or included only as supplementary muscles in these muscle selections. These muscles have short moment arms, small PCSAs, and/or possible roles in joint stabilization. However, as shown in Table 4, many of these muscles were in fact recruited. This suggests that muscles with short moment arms or small PCSAs should be considered rather than excluded a priori. Therefore, within the assumptions of the model, the muscle-recruitment strategy of the MSC framework is physiologically interpretable because it takes both PCSA and moment arm into account and may reduce excessive loading of muscles with smaller PCSAs. However, whether the muscle-recruitment patterns predicted by the MSC framework correspond to actual human muscle recruitment during reaching has not been directly evaluated. Direct evaluation would require comparing the predicted muscle-recruitment patterns with EMG recorded during the same reaching task, together with participant-specific estimates of PCSA and moment arms.

Compared with the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 model still tended not to recruit muscles with extremely small PCSAs, but it recruited more muscles, including some that were rarely recruited by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2. This tendency is consistent with the fact that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 model uses an objective function that includes the cube of muscle stress (_p=3_p=3), so the penalty for concentrating high stress in a specific muscle is larger than in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 (_p=2_p=2), making it easier to distribute muscle tension across multiple muscles (Crowninshield and Brand 1981). In fact, the number of muscles with activation rates above 10 % was 21 for both the MTC and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 models, whereas it increased to 25 for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 model, meaning that nearly all muscles except muscle 4 under muscle-selection condition S22 were recruited. Therefore, the tendency of muscle recruitment varies depending on the form of the objective function in the MSC models (_p=2_p=2 or _p=3_p=3) and on the PCSA and moment-arm values used in the optimization.

Overall, movement-level predictions of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model were relatively robust, whereas its muscle-recruitment predictions were more sensitive to PCSA and moment-arm values. This difference indicates that reproduction of fingertip trajectories and arm postures alone is insufficient for evaluating muscle-level computational models. The validity of such models should ideally be assessed not only in terms of errors in fingertip trajectory, arm posture, and posture-related characteristics such as the wrist-joint contribution rate, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_w$$\end{document}Cw, introduced in this study, but also in terms of muscle recruitment, for example using electromyographic measurements during movement. However, such evaluation is challenging because predicted recruitment patterns may vary depending on the PCSA and moment-arm values assigned to individual muscles in each participant.

In addition, the present models do not explicitly account for task-dependent agonist–antagonist cocontraction. Such cocontraction has been reported when stability against external perturbations is required (Burdet et al. 2001), in responses to joint perturbations (Lewis et al. 2010; Armstrong et al. 2025), under increased accuracy demands (Gribble et al. 2003), and during motor learning (Osu et al. 2002). Because cocontraction can alter muscle recruitment, the recruitment patterns predicted by the present models may differ from actual human muscle recruitment under these conditions. Accordingly, the present predictions should be interpreted primarily in relation to the unperturbed reaching task examined here, for which the experimental procedures were designed to reduce excessive cocontraction, as described in Sect. 5.2.

Relation to neuromuscular synergy theories

The present results are relevant to neuromuscular synergy theories. Muscle synergies are often extracted from electromyographic (EMG) signals using non-negative matrix factorization, and previous studies have suggested that complex upper-limb movements can be represented by combinations of a relatively small number of coordinated muscle-activation patterns (e.g., d’Avella et al. 2006; Bizzi and Cheung 2013). However, synergy extraction does not necessarily capture all task-relevant muscle activity, because residual EMG components may still influence task performance (Barradas et al. 2020).

The comparison between the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 models provides a useful computational perspective on this issue. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model reproduced fingertip trajectories, arm postures, and the wrist-joint contribution rate more accurately than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 and, as shown in Table 4, recruited fewer muscles. Conversely, under muscle-selection condition S22, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 model recruited almost all muscles, but its reproduction accuracy for these components of arm movement was slightly lower than that of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model. These results suggest that human reaching movements do not simply reflect maximally distributed muscle activity. Rather, coordinated muscle activity resembling muscle synergies may emerge from an optimization process that constrains muscle stress to change smoothly while taking into account muscle-specific differences in PCSA and moment arms. Thus, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model may provide a candidate computational mechanism for generating coordinated muscle-recruitment patterns without assuming predefined fixed muscle synergies.

Because EMG signals were not recorded in the present study, however, muscle synergies could not be extracted from actual muscle activity. Therefore, an additional muscle-synergy analysis was not performed. Even if synergies were extracted from the muscle tensions determined by each computational model, it would be difficult to draw direct conclusions about physiological muscle synergies. Future studies should record EMG signals during reaching movements under the same movement conditions as those used in the present study and examine whether the recruitment patterns predicted by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model correspond to extracted muscle synergies and how such patterns depend on PCSA and moment-arm properties.

Feedforward planning and optimal feedback control

The present model is also relevant to the distinction between feedforward planning and optimal feedback control (OFC). In feedforward planning, smoothness-based computational models, such as the MTC and MSC models, define optimal movement trajectories and can therefore predict internal variables such as joint torque, muscle tension, and muscle recruitment. However, because these models do not include an explicit feedback mechanism and movement duration must be specified as a boundary condition, they cannot fully account for online correction of movement errors or trial-to-trial variability.

By contrast, OFC emphasizes flexible online regulation based on sensory feedback and the minimum intervention principle, whereby deviations in task-irrelevant dimensions are tolerated, whereas deviations related to the behavioral goal are corrected (e.g., Todorov and Jordan 2002; Diedrichsen et al. 2010; Kambara et al. 2021). Although OFC has these advantages, standard feedback-dependent OFC frameworks may have difficulty accounting for goal-directed movements after deafferentation in monkeys, because such movements can be generated even when sensory feedback from the limb is severely impaired (Polit and Bizzi 1979). In addition, how high-dimensional muscle redundancy is resolved in a three-joint arm model with muscle-specific PCSAs, joint-angle-dependent moment arms, and up to 26 muscles remains an open question.

Thus, feedforward planning and OFC should be regarded not as mutually exclusive alternatives but as complementary frameworks. The MSC framework can determine muscle-tension distributions in a feedforward manner by taking into account physiological properties of the arm musculoskeletal system, such as PCSA and joint-angle-dependent moment arms. Because these distributions avoid excessive loading of muscles with smaller PCSAs and tend to assign larger tensions to muscles with larger PCSAs, they are physiologically interpretable. During the execution of movements generated by feedforward planning, as discussed in Sect. 5.2, feedback control may then correct movement errors online. In this sense, the MSC framework may complement OFC by specifying how anatomically constrained muscle redundancy is resolved in the feedforward component of arm movement generation.

Limitations and future directions

Several limitations of this study should be noted. First, all participants were right-handed healthy young adults; therefore, the generalizability of the present findings to other populations, such as left-handed individuals or older adults, remains unclear. Second, the experimental task was limited to reaching movements in the horizontal plane. Third, the solution obtained with the two-stage optimization framework is not guaranteed to coincide with the solution that would be obtained by directly minimizing the second-stage objective function alone, such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{MSC_2}$$\end{document}CMSC2. Fourth, the validity of the muscle recruitment patterns predicted by the MSC framework was not directly verified by EMG. Future studies should combine the present eight-direction reaching task with EMG measurements, estimate participant-specific anatomical parameters where possible, and examine whether muscle synergies or other EMG-derived coordination patterns correspond to the recruitment patterns predicted by the MSC framework. Overall, the present findings support the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_2$$\end{document}MSC2 model as a muscle-level optimization model that links task-space behavior, joint-space coordination, and physiologically interpretable muscle recruitment.