Results

Measured and optimal arm movements

As shown in Table 3, the measured movement times were close to the movement time specified during the practice session, although they were slightly longer in a few movement directions. Because the movement distance was approximately 20 cm, the measured fingertip paths were mostly straight (Fig. 4), but slight curvature was observed in some directions. Inter-participant variability in fingertip paths was greater for MD4, MD6, and MD8 than for the other directions. The measured arm postures also showed clear directional dependence (Fig. 5): in some directions, such as MD1 and MD5, the wrist rotated substantially, whereas in others, such as MD3 and MD7, wrist rotation was minimal. Although the initial posture differed across participants because of differences in link lengths, the temporal profiles of the joint angles were broadly similar.

Figure 6 shows the optimal fingertip paths generated by four selected computational models. Across all evaluated models, the AJ, \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 produced less curved trajectories and smaller inter-participant variability than the other models. In MD1, MD5, and MD6, differences among the models were small. In the remaining directions, however, differences in fingertip paths were observed among the computational models, with the MTRC model in particular generating markedly curved fingertip paths. The MTC model also produced somewhat greater curvature than the AJ 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, especially in MD3 and MD4. In MD3, the fingertip path predicted by MTC curved in the opposite direction from those predicted by the AJ and MTRC models. In addition, the AJ model produced curvature opposite to that of the other models in MD4 and MD8, whereas the MTRC model produced curvature opposite to that of the AJ model in MD7. Figure 7 shows the optimal arm postures predicted by each 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 yielded smaller wrist rotations than the other models in all movement directions, with the smallest rotations observed for \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. By contrast, the other models predicted larger wrist rotations during movement, and the MTRC model produced the largest wrist rotation overall.

Fig. 8: Model-dependent errors in trajectory and posture reproduction. Vertical bars are the standard deviation. Horizontal connectors indicate significant pairwise differences. (Muscle selection: S22. PCSA: PCSA1. *: p<p< 0.05)

Fig. 9: RMS errors for each movement direction. (Muscle selection: S22. PCSA: PCSA1)

_Fig. 10: Influence of PCSA on the optimal fingertip paths. (Computational model: \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, Muscle selection: S22)

Fig. 11: Contribution rate of the wrist joint. Vertical lines indicate the standard error of the mean and are shown only for the measured data; error bars for the computational models are omitted for clarity. (Muscle selection: S22, PCSA: PCSA1)

As shown in Fig. 8a, the RMS errors of the fingertip trajectory were smallest for the AJ 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, followed by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MSC}_3$$\end{document}MSC3 model. The direction-specific errors varied across models (Fig. 9a). \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 produced larger errors in MD3, MD4, and MD8. The AJ model produced smaller errors in MD3 and MD4, but a larger error in MD8. In the remaining directions, both models showed similarly small errors. When 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 were compared, \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 produced slightly larger errors in all directions, with the difference again being most pronounced in MD3 and MD4. As shown in Fig. 8b, the RMS errors of the arm posture were smallest for \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, followed by the AJ 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. Direction-specific analysis (Fig. 9b) further showed that 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}$$\text {MTC}_s$$\end{document}MTCs models produced large posture errors in MD4, with the error being larger for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {MTC}_s$$\end{document}MTCs model. In contrast, 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 showed small posture errors in all movement directions, 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 consistently produced smaller errors 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.

The effects of PCSA and muscle selection were also examined (see Supplementary Materials for the results). As shown in Fig. 10, the optimal fingertip trajectories generated 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 varied across the three PCSA sets. However, when the RMS error of the optimal fingertip trajectory was aggregated across all movement directions, no substantial differences were found among the three PCSA sets. The errors in the MTC model also showed little difference among the PCSA sets. For arm posture, the error obtained with PCSA2 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 was slightly larger than those obtained with the other PCSA sets. For the MTC model, the error obtained with PCSA3 was smaller than those obtained with the other PCSA sets. For muscle selection, 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 were also evaluated under the four muscle-selection conditions listed in Table 2. The fingertip-trajectory and arm-posture errors of \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 changed only slightly across the muscle-selection conditions. For the MTC model, the fingertip-trajectory error was somewhat larger for muscle selection S12 and slightly smaller for S21 than for the other conditions. For arm posture, the errors were slightly larger for S11 and S21. Therefore, the arm movements 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 were not substantially affected by the choice of PCSA dataset or by muscle selection with different numbers and types of muscles.

Wrist-joint contribution for different movement directions

The measured arm postures included directions with substantial wrist rotation and directions with almost no wrist rotation (Fig. 5). As shown in Fig. 11, the contribution rate of the wrist joint, \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, in the measured arm postures exhibited a bimodal profile, with peaks in MD2 and MD5. In directions where \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 was small (MD3, MD4, MD7, and MD8), the shoulder-rotation angle was larger than that in the other movement directions. All computational models also produced a bimodal pattern of \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, but the magnitude of \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 and the directions of its peaks differed across models. The MTRC model overestimated \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. 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}$$\text {MTC}_s$$\end{document}MTCs models reproduced the overall waveform shape reasonably well, but they overestimated \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 in all directions except MD6, leading to excessive wrist rotation. The \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 model produced larger errors than MTC, and its peak directions were slightly shifted. In the AJ model, the peak and trough directions were not well aligned with the measured data, and \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 was large in all directions except MD2 and MD6. 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 the bimodal pattern and the peaks in MD2 and MD5 more closely 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 a waveform shape similar to that of the measured data, but the errors in \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 were larger than 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, particularly in MD3 and MD4.

Muscle tensions selected by the computational models

Examples of the muscle tensions predicted during movement 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 are shown in Fig. 12. Because the temporal profiles of muscle tension were broadly similar across participants, the muscle tensions were averaged after normalizing time by the movement duration of each trial. To compare muscle recruitment across all movement directions, the activation rate of each muscle was computed under muscle selection S22 and PCSA1 (Table 4). Muscle recruitment in the MTC model differed from that in the other models for many muscles. 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 more muscles, including muscles that were rarely active 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.

To examine the strategy by which each model determined muscle tension, Figs. 13 and 14 show the relationships between peak muscle tension and PCSA and between peak muscle tension and moment arm, respectively. The moment-arm magnitude was defined as the mean absolute moment arm during each movement trial. 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 suggested different recruitment principles. In the MTC model, no clear relationship was found between PCSA and the largest peak tensions. A notable feature of this model, however, was that muscles with very small PCSAs, such as muscles 4, 10, 14, 17, 18, 19, 22, 23, 24, and 26, tended to generate larger tensions in the MTC model than 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, whereas the tensions of muscles with large PCSAs, such as muscles 5, 7, and 8, remained small. 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 largest peak tension decreased with decreasing PCSA for muscles with PCSAs below 7 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {cm}^2$$\end{document}cm2, indicating that peak tension was related to PCSA. In addition, muscles with small PCSAs at each joint (e.g., muscles 4, 14, and 23) were rarely activated, whereas muscles with large PCSAs tended to generate larger tensions than in the MTC model. A different tendency was observed with respect to moment arm. In the MTC model, the largest peak tension decreased as moment arm decreased, particularly when the moment arm was below 3 cm. 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, however, no clear relationship was found between peak muscle tension and moment arm.