Measurement of three-joint reaching movements

Nine healthy right-handed adults participated in the experiment (age: 22.1 ± 1.1 years; height: 1.71 ± 0.08 m; two females). All participants reported normal or corrected-to-normal vision and were not informed of the detailed hypothesis before the experiment. Written informed consent was obtained from each participant, and participants received book coupons after participation. The study protocol was approved by the Human Research Ethics Committee of the Division of Human and Artificial Intelligent Systems at University of Fukui (#H181201).

The experimental apparatus and basic recording procedure were the same as those described in Katayama (2025), whereas the present analysis used an eight-direction target arrangement designed to evaluate movement and muscle recruitment without strong directional bias. Briefly, as illustrated in Fig. 2, participants were seated and stabilized with a four-point harness, and the hand, forearm, and upper arm were partially suspended from the ceiling to reduce gravitational loading during horizontal reaching. Four infrared markers were attached to the shoulder, elbow, wrist, and fingertip, and their three-dimensional positions were recorded at 200 Hz using an OPTOTRAK3020 system. All reaching movements were performed in a horizontal plane containing the shoulder, elbow, wrist, and fingertip. Thus, the upper arm, forearm, and hand moved within the horizontal reaching plane throughout each movement. The measured arm configuration was displayed in real time as a stick figure on a screen positioned in front of the participant and parallel to the participant’s coronal plane. The target positions shown in Fig. 3 were predefined in the horizontal reaching coordinate system but were not physically placed in the reaching plane. Instead, the start point and the target circle for each movement were displayed on the screen. Participants performed the reaching movements while viewing the real-time stick figure and the displayed target. This arrangement prevented the target display from being obscured by the participant’s moving arm.

Before data recording, participants practiced the task to become familiar with the apparatus and to perform relaxed reaching movements, using the same procedure as Katayama (2025). In the present experiment, targets were arranged in eight directions at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {45}^\circ $$\end{document}45∘ intervals around the start position, as illustrated in Fig. 3. The movement distance was 20 cm so that all participants could perform all eight directions within the reachable workspace. Fifteen trials were recorded for each direction, and the order of directions was randomized. In each trial, participants briefly shook the hand near the start position after the first auditory cue, reached to the target after the second cue, and maintained the final posture until the final cue.

Data analysis. Marker trajectories were low-pass filtered in both the forward and reverse directions with a cutoff frequency of 10 Hz. For the fingertip position (x, y), the apparent planar curvature, C, was calculated as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C = |\dot{x} \ddot{y} - \ddot{x} \dot{y}| / (\dot{x}^2 + \dot{y}^2)^{3 \over 2}$$\end{document}C=|x˙y¨-x¨y˙|/(x˙2+y˙2)32. The calculated apparent curvature became extremely large near movement onset and offset, where tangential velocity was low, and its temporal profile clearly differed from that of tangential velocity. Movement onset was defined as the first point at which C exceeded 0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$,\textrm{mm}^{-1}$$\end{document}mm-1 when searching backward from the midpoint of the movement, following the method of Katayama (2025). Movement offset was defined similarly by searching forward from the midpoint. The term “apparent planar curvature” is used because the calculated value is sensitive to small positional fluctuations when tangential velocity is low and does not imply that the physical fingertip trajectory had such a large curvature near movement onset or offset. For each movement direction and participant, RMS errors between the measured and model-predicted movements were computed separately for the planar fingertip trajectory and the three joint-angle trajectories. Multiple comparisons among models were performed using the Tukey–Kramer method.

The characteristics of arm posture during reaching movements are particularly evident in wrist-joint rotation. In this study, these characteristics were evaluated using the ratio of the wrist-joint rotation angle to the total rotation angle of the three joints during movement. 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, was defined as follows:where _n = 1, 2,_n=1,2, and 3 correspond to the shoulder, elbow, and wrist joints, respectively, and N is the number of time samples in each trial. Thus, \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 reflects the relative contribution of the wrist throughout the movement, rather than only the final arm posture at the endpoint.

13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {C_w}= & \frac{\Delta \theta _3}{\Delta \theta _1 + \Delta \theta _2 + \Delta \theta _3}, \nonumber \\ \Delta \theta _n= & {\sum ^{N}_{t=1} |{\dot{\theta }_n(t)}| \Delta t}, \end{aligned}$$\end{document}Cw=Δθ3Δθ1+Δθ2+Δθ3,Δθn=∑t=1N|θ˙n(t)|Δt,

Fig. 4: Mean fingertip paths across all participants. Solid lines represent the reaching movements for MD1–MD4, and dashed lines represent those for MD5–MD8. Because target locations varied across participants, the origin was placed at the center of each participant’s movement range. The shaded area indicates the standard error of the mean in the direction perpendicular to the line connecting the start and end points

Fig. 5: Mean joint angles across all participants. Time was normalized to the movement time of each trial. Joint angles of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {0}^\circ $$\end{document}0∘ at the shoulder, elbow, and wrist indicate that the upper arm points to the right, the upper arm and forearm are aligned (i.e., the elbow is fully extended), and the forearm and hand are aligned, respectively. The shaded area indicates the standard error of the mean at each time point

Fig. 6: Fingertip trajectories generated by selected models. Solid traces show MD1–MD4, and dashed traces show MD5–MD8. Coordinates were centered on each participant’s movement range; shading represents the SEM perpendicular to the start-target axis. (Muscle selection: S22. PCSA: PCSA1)

Fig. 7: Optimal joint angles predicted by each computational model. Joint angles of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {0}^\circ $$\end{document}0∘ at the shoulder, elbow, and wrist indicate that the upper arm points to the right, the upper arm and forearm are aligned (i.e., the elbow is fully extended), and the forearm and hand are aligned, respectively. (Muscle selection: S22. PCSA: PCSA1)