Movement selection by computational models

Muscle-level computational models

The models evaluated here were selected to clarify how muscle-level cost functions determine both arm movements and muscle recruitment. Following the two-stage optimization framework described by Katayama (2025), optimal solutions were obtained for each muscle-level computational model. Common implementation details are therefore summarized briefly, whereas the muscle-level cost functions and differences from the previous eight-muscle model are emphasized.

Prior isometric force-production studies indicate that muscle tensions inferred from EMG can be described by stress-based optimization criteria, particularly quadratic and cubic formulations (Crowninshield and Brand 1981; van Bolhuis and Gielen 1999; Gomi 2001). This evidence provides physiological motivation for evaluating the MSC framework as a rule for muscle recruitment in the present anatomically expanded arm models. The evaluation functions based on the MSC framework are as follows:Here, m is the number of muscles in each three-joint arm model; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i$$\end{document}Si and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_i$$\end{document}Ti denote the stress and tension of the i-th muscle at time t, respectively; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{\tau }$$\end{document}τ is the joint-torque vector; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document}T is the vector of muscle tensions; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{\theta }$$\end{document}θ is the vector of joint angles; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{A}$$\end{document}A is the moment-arm matrix that depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{\theta }$$\end{document}θ; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PCSA_i$$\end{document}PCSAi is the physiological cross-sectional area of the i-th muscle; and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_f$$\end{document}tf is the movement duration from the initial to the final position. Muscle stress is defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i=T_i/PCSA_i$$\end{document}Si=Ti/PCSAi. \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 denote the cases p=2_p=2 and p=3_p=3, respectively. In the first stage of the two-stage optimization, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document}T was determined at each time step by minimizing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E{MSC_p}$$\end{document}EMSCp subject to (i) the joint torque–muscle tension relationship \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{\tau }=\textbf{A}(\boldsymbol{\theta })\textbf{T}$$\end{document}τ=A(θ)T and (ii) non-negativity and upper bounds on muscle tension, with maximal voluntary contraction stress limited to approximately 60 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {N/cm^2}$$\end{document}N/cm2 (Ikai and Fukunaga 1968; Crowninshield 1978). At the beginning and end of the movement, the arm was stationary: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\boldsymbol{\theta }}(0)=\dot{\boldsymbol{\theta }}(t_f)=\boldsymbol{0}$$\end{document}θ˙(0)=θ˙(tf)=0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\boldsymbol{\theta }}(0)=\ddot{\boldsymbol{\theta }}(t_f)=\boldsymbol{0}$$\end{document}θ¨(0)=θ¨(tf)=0. Accordingly, the joint torques at both endpoints were set to zero. In the second-stage optimization, the muscle stresses calculated from the muscle tensions determined at all time steps in the first-stage optimization were used to calculate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C{MSC_p}$$\end{document}CMSCp. Following the RCGA framework described by Katayama (2025), this calculation was repeated while varying the fingertip trajectories, x and y, and the wrist-joint angle trajectory, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta 3$$\end{document}θ3, until \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C{MSC_p}$$\end{document}CMSCp was minimized. The optimal fingertip and joint-angle trajectories and the corresponding muscle-tension profiles were thereby obtained. No additional constraints were imposed in the second-stage optimization because the muscle tensions determined in the first stage already satisfied all prescribed constraints.

1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \text {First stage:}\nonumber \\ & {E_{MSC_p}} = \sum _{i=1}^{m}{S_i}^p, ~~~~~ S_i = \frac{T_i}{PCSA_i}, \end{aligned}$$\end{document}First stage:EMSCp=∑i=1mSip,Si=TiPCSAi,

2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & s.t.\boldsymbol{\tau } = \textbf{A}(\boldsymbol{\theta })\textbf{T},~0 \le T_i \le PCSA_i \times 60\; \mathrm {[N/cm^2]}, \nonumber \\ & \text {Second stage:}\nonumber \\ & {C_{MSC_p}} = \frac{1}{2}\int _0^{t_f} \sum _{i=1}^m \left| \frac{d S_{i}}{dt} \right| ^p dt. \end{aligned}$$\end{document}s.t.τ=A(θ)T,0≤Ti≤PCSAi×60[N/cm2],Second stage:CMSCp=12∫0f∑i=1mdSidtpdt.

The minimum muscle-tension-change (MTC) model was also evaluated as a muscle-level model because it determines the muscle-tension vector directly (e.g., Dornay et al. 1996). Following the two-stage approach used for the MSC framework, the MTC model computed optimal fingertip trajectories, arm postures, and recruitment patterns by minimizing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{MTC}$$\end{document}EMTC in the first stage and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{MTC}$$\end{document}CMTC in the second stage:The MSC models were primarily compared with the MTC model, but two additional computational models were evaluated for reference. First, the combined model (\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), which minimized \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E{MSC_2}$$\end{document}EMSC2 in the first-stage optimization and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{MTC}$$\end{document}CMTC in the second-stage optimization, was evaluated. Second, the minimum muscle-torque-change (MTRC) model was examined. In this model, the second-stage objective function, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{MTRC}$$\end{document}CMTRC, was minimized using the muscle tensions obtained by minimizing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{MSC_2}$$\end{document}EMSC2 in Eq. 1:Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{j,i}$$\end{document}Qj,i denotes the torque produced by the i-th muscle about the j-th joint, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _j$$\end{document}Ωj denotes the set of muscles acting about that joint. A biarticular muscle belongs to both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _1$$\end{document}Ω1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _2$$\end{document}Ω2, whereas each monoarticular muscle belongs to only one of the three sets. Therefore, the total number of muscle-torque terms is _m+α _m+α, where _α _α is the number of biarticular muscles.

3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \text {First stage:} \nonumber \\ & {E_{MTC}} = \sum _{i=1}^{m} {T_i}^2, \end{aligned}$$\end{document}First stage:EMTC=∑i=1mTi2,

4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & s.t.\boldsymbol{\tau } = \textbf{A}(\boldsymbol{\theta })\textbf{T},~~ 0 \le T_i \le PCSA_i \times 60\; \mathrm {[N/cm^2]}, \nonumber \\ & \text {Second stage:} \nonumber \\ & {C_{MTC}} = \frac{1}{2}\int _0^{t_f} \sum _{i=1}^m \left( \frac{d T_{i}}{dt} \right) ^2 dt. \end{aligned}$$\end{document}s.t.τ=A(θ)T,0≤Ti≤PCSAi×60[N/cm2],Second stage:CMTC=12∫0f∑i=1mdTidt2dt.

5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {Second stage:} \nonumber \\ C_{MTRC}= & \frac{1}{2} \int _0^{t_f} \sum _{j=1}^{3} \sum _{i\in \Omega _j} \left( \frac{dQ_{j,i}}{dt} \right) ^2 dt, \qquad \nonumber \\ Q_{j,i}= & a_{j,i}(\theta _j)T_i. \end{aligned}$$\end{document}Second stage:CMTRC=12∫0f∑j=13∑i∈ΩjdQj,idt2dt,Qj,i=aj,i(θj)Ti.

The minimum angular-jerk (AJ) model was used only as a benchmark because it has been shown to reproduce human arm movements well but does not determine muscle recruitment (e.g., Katayama 2025). Specifically, the AJ model generated the optimal arm movements by minimizing the objective function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{AJ}$$\end{document}CAJ, defined as follows:where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _j$$\end{document}θj denotes the joint angle of the j-th joint.

6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {C_{AJ}} = \frac{1}{2}\int _0^{t_f} \sum _{j=1}^3 \left( \frac{d^3 \theta _j}{dt^3} \right) ^2 dt, \end{aligned}$$\end{document}CAJ=12∫0f∑j=13d3θjdt32dt,

Model predictions were obtained using the RCGA framework described by Katayama (2025), but the present implementation was adapted to the expanded 19–26-muscle arm models. In addition to 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 muscle-level computational models \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, MTC, \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, and MTRC were implemented. Quadratic first-stage problems were solved by quadratic programming, whereas the cubic stress criterion was optimized using the C++ interface of the NLopt library (Johnson 2008). Each condition was computed three times from different initial populations; the resulting solutions showed no substantive differences.

Anatomically expanded arm models

A central difference from Katayama (2025) is the musculoskeletal representation. The previous model used eight muscles with constant moment arms, which was sufficient for evaluating movement-level reproduction but insufficient for discussing how individual anatomical muscles are recruited. In the present study, the model was expanded to 19–26 muscles, and moment arms were defined as joint-angle-dependent functions. This extension allows evaluation of whether each computational model recruits muscles in a manner consistent with anatomical PCSA and moment-arm properties.

The planar three-joint arm dynamics were represented bywhere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{\tau }$$\end{document}τ is the three-dimensional joint-torque vector for the shoulder, elbow, and wrist, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{\tau }= (\tau _1, \tau _2, \tau _3)^T$$\end{document}τ=(τ1,τ2,τ3)T. This joint-torque vector is generated by the muscle-tension vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{T}}$$\end{document}T through the moment-arm matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{A}}(\boldsymbol{\theta })$$\end{document}A(θ). The state vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\boldsymbol{\theta }$$\end{document}θ represents the shoulder, elbow, and wrist angles, and its first and second derivatives represent angular velocity and angular acceleration. The matrices and vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{M}(\boldsymbol{\theta })$$\end{document}M(θ), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{D}(\boldsymbol{\tau })$$\end{document}D(τ), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{H}(\boldsymbol{\theta },\boldsymbol{\dot{\theta }})$$\end{document}H(θ,θ˙) describe inertial, viscous, and velocity-dependent dynamic components, respectively. Unlike the previous eight-muscle formulation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{A}}(\boldsymbol{\theta })$$\end{document}A(θ) has three rows and m columns, where m is the number of muscles included in each expanded arm model.

7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \boldsymbol{\tau }= \textbf{A}(\boldsymbol{\theta })\textbf{T},\end{aligned}$$\end{document}τ=A(θ)T,

8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \textbf{M}(\boldsymbol{\theta }) \boldsymbol{\ddot{\theta }} + \textbf{H}(\boldsymbol{\theta },\boldsymbol{\dot{\theta }}) + \textbf{D}(\boldsymbol{\tau }) \boldsymbol{\dot{\theta }} = {\boldsymbol{\tau }}, \end{aligned}$$\end{document}M(θ)θ¨+H(θ,θ˙)+D(τ)θ˙=τ,

The muscle-tension vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document}T is defined asand the moment-arm matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{A}(\boldsymbol{\theta })$$\end{document}A(θ) is expressed as where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{j,i}$$\end{document}aj,i denotes the moment arm of the i-th muscle about the j-th joint and is a function of the corresponding joint angle. For S22 in Table 2, subscripts were assigned as follows: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s_1} = 1$$\end{document}s1=1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s_2} = 11$$\end{document}s2=11 for the shoulder; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e_1} = 12$$\end{document}e1=12 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e_2} = 17$$\end{document}e2=17 for the elbow; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d_1} = 18$$\end{document}d1=18 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d_2} = 20$$\end{document}d2=20 for biarticular muscles spanning the shoulder and elbow; and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${w_1} = 21$$\end{document}w1=21 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_2 = m = 26$$\end{document}w2=m=26 for the wrist. As listed in Table 2, S11, S12, S21, and S22 include 19, 21, 21, and 26 muscles, respectively.

9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{T} = \left( \begin{array}{cccccccccccc} T_{s_1}&\ldots&T_{{s_2}}&T_{{e_1}}&\ldots&T_{{e_2}}&T_{{d_1}}&\ldots&T_{{d_2}}&T_{{w_1}}&\ldots&T_{{w_2}} \end{array}\right) ^{T}, \end{aligned}$$\end{document}T=Ts1…Ts2Te1…Te2Td1…Td2Tw1…Tw2T,

10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{A}(\boldsymbol{\theta }) = \left( \begin{array}{cccccccccccc} a_{1,{s_1}}(\theta _1) & \ldots & a_{1,{s_2}}(\theta _1) & 0 & \ldots & 0 & a_{1,{d_1}}(\theta _1) & \ldots & a_{1,{d_2}}(\theta _1) & 0 & \ldots & 0 \\ 0 & \ldots & 0 & a_{2,{e_1}}(\theta _2) & \ldots & a_{2,{e_2}}(\theta _2) & a_{2,{d_1}}(\theta _2) & \ldots & a_{2,{d_2}}(\theta _2) & 0 & \ldots & 0 \\ 0 & \ldots & 0 & 0 & \ldots & 0 & 0 & \ldots & 0 & a_{3,{w_1}}(\theta _3) & \ldots & a_{3,{w_2}}(\theta _3) \end{array}\right) , \end{aligned}$$\end{document}A(θ)=a1,s1(θ1)…a1,s2(θ1)0…0a1,d1(θ1)…a1,d2(θ1)0…00…0a2,e1(θ2)…a2,e2(θ2)a2,d1(θ2)…a2,d2(θ2)0…00…00…00…0a3,w1(θ3)…a3,w2(θ3),

The joint-viscosity matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{D}$$\end{document}D was defined asIn the viscosity matrix, the diagonal elements (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{11}$$\end{document}D11, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{22}$$\end{document}D22, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{33}$$\end{document}D33) describe viscosity at the shoulder, elbow, and wrist, whereas the off-diagonal shoulder-elbow terms represent inter-joint viscous coupling. The shoulder and elbow viscosity coefficients were derived from empirical measurements of human planar reaching (Gomi and Kawato 1995). The wrist coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{33}$$\end{document}D33 was specified as one-half of the average shoulder-elbow baseline value. Although joint viscosity was fixed in Katayama (2025), it was modeled here as a function of joint torque, which changes during movement, based on the results reported by Gomi and Kawato (1995).Here, viscosity values are expressed in \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.

11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{D}(\boldsymbol{\tau })= & \left[ \begin{array}{ccc} D_{11}(\tau _1) & D_{12}(\tau _2) & 0 \\ D_{21}(\tau _2) & D_{22}(\tau _2) & 0 \\ 0 & 0 & D_{33}(\tau _3) \end{array} \right] . \end{aligned}$$\end{document}D(τ)=D11(τ1)D12(τ2)0D21(τ2)D22(τ2)000D33(τ3).

12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} D_{11}(\tau _1)&= 0.63 + 0.095 |\tau _1|, \\ D_{21}(\tau _2)&= D_{12}(\tau _2) = 0.175 + 0.0375 |\tau _2|, \\ D_{22}(\tau _2)&= 0.76 + 0.185 |\tau _2|, \\ D_{33}(\tau _3)&= 0.348 + 0.140 |\tau _3|. \end{aligned} \end{aligned}$$\end{document}D11(τ1)=0.63+0.095|τ1|,D21(τ2)=D12(τ2)=0.175+0.0375|τ2|,D22(τ2)=0.76+0.185|τ2|,D33(τ3)=0.348+0.140|τ3|.

For the muscle-level computational models, the key anatomical extension was the use of joint-angle-dependent moment arms. Based on measurements reported by Kuechle et al. (1997), Pigeon et al. (1996), and Amis et al. (1979), each moment arm was approximated as a polynomial function of joint angle (Fig. 1 and Table 5). This extension is particularly important for shoulder muscles, whose moment arms change substantially with joint angle. Because the MSC framework also requires PCSAs, model performance and recruitment were evaluated using the three PCSA sets listed in Table 2. In PCSA1, monoarticular shoulder-muscle PCSAs were determined from Meek et al. (1990), and the remaining PCSAs were determined from An et al. (1981). In PCSA2, monoarticular shoulder-muscle PCSAs were determined from Meek et al. (1990), and the remaining PCSAs were determined from Amis et al. (1979). In PCSA3, all PCSA values were determined from Holzbaur et al. (2005). The muscle abbreviations are as follows: (1) DC, deltoid (clavicular part); (2) PMC, pectoralis major (clavicular part); (3) PMA, pectoralis major (abdominal part); (4) Co, coracobrachialis; (5) Su, subscapularis; (6) DS, deltoid (scapular part); (7) DA, deltoid (acromial part); (8) LD, latissimus dorsi; (9) In, infraspinatus; (10) TMi, teres minor; (11) TMa, teres major; (12) Br, brachialis; (13) PT, pronator teres; (14) BR, brachioradialis; (15) MHTr, triceps brachii (medial head); (16) LtHTr, triceps brachii (lateral head); (17) A, anconeus; (18) LHBi, biceps brachii (long head); (19) SHBi, biceps brachii (short head); (20) LHTr, triceps brachii (long head); (21) FCU, flexor carpi ulnaris; (22) FCR, flexor carpi radialis; (23) PL, palmaris longus; (24) ECRB, extensor carpi radialis brevis; (25) ECU, extensor carpi ulnaris; and (26) ECRL, extensor carpi radialis longus. The muscle-set labels follow the convention of the earlier study, but their anatomical contents were redefined for the present multi-muscle model. S11 and S12 were based on Ito and Takano (2012), whereas S21 and S22 were based on Nakamura and Saito (1992); supplementary muscles were included only in S12 and S22. Participant-specific physical parameters used in the dynamics calculation of Eq. 8 were taken from the estimation procedure described in Katayama (2025) and are listed in Table 1.

Fig. 1: Moment-arm functions for the anatomical muscles in the three-joint arm models. Positive and negative moment arms indicate flexion- and extension-producing actions, respectively. Zero degrees correspond to a rightward upper arm, a fully extended elbow, and a hand aligned with the forearm. Muscle indices correspond to Table 2

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document}L1 (m) | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document}L2 (m) | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_3$$\end{document}L3 (m) | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{g1}$$\end{document}Lg1 (m) | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{g2}$$\end{document}Lg2 (m) | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{g3}$$\end{document}Lg3 (m)
0.284 ± 0.020 | 0.245 ± 0.014 | 0.188 ± 0.019 | 0.112 ± 0.008 | 0.0909 ± 0.0043 | 0.0667 ± 0.0071
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_1$$\end{document}M1 [kg] | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_2$$\end{document}M2 [kg] | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_3$$\end{document}M3 [kg] | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_1$$\end{document}I1 [kg \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {m}^2$$\end{document}m2] | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_2$$\end{document}I2 [kg \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {m}^2$$\end{document}m2] | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_3$$\end{document}I3 [kg \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {m}^2$$\end{document}m2]
1.654 ± 0.375 | 0.831 ± 0.130 | 0.455 ± 0.088 | 0.0383 ± 0.0124 | 0.0126 ± 0.0030 | 0.0039 ± 0.0014

Fig. 2: Planar-reaching apparatus used for the present experiment, based on the setup described by Katayama (2025). The display provided the start position, target circle, and online arm-configuration feedback, while ceiling suspension reduced gravitational loading of the arm

Fig. 3: Eight-direction target arrangement defined in the horizontal reaching coordinate system. Target positions were defined 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. Because the target coordinates varied among participants according to their body dimensions, numerical coordinate axes are not shown; the scale bar represents 10 cm. W/2 denotes half the shoulder width, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document}L1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document}L2, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_3$$\end{document}L3 denote the upper-arm, forearm, and hand lengths, respectively

Muscle | Muscle | Type | Muscle selection | PCSA [\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]
# | S11 | S12 | S21 | S22 | PCSA1 | PCSA2 | PCSA3
Monoarticular muscles of the shoulder joint
1 | DC | SF | ○◯ | ○◯ | ○◯ | ○◯ | 4.52 | ←← | 8.2
2 | PMC | SF | ○◯ | ○◯ | ○◯ | ○◯ | 5.16 | ←← | 2.6
3 | PMA | SF | ○◯ | ○◯ | ○◯ | ○◯ | 3.87 | ←← | 3.7
4 | Co | SF | ○◯ | ○◯ | ○◯ | ○◯ | 1.29 | ←← | 1.7
5 | Su | SF |  |  | ○◯ | ○◯ | 9.68 | ←← | 9.8
6 | DS | SE | ○◯ | ○◯ | ○◯ | ○◯ | 3.87 | ←← | 1.9
7 | DA | SE |  |  | ○◯ | ○◯ | 13.55 | ←← | 8.2
8 | LD | SE |  | △▵ |  | △▵ | 12.90 | ←← | 7.6
9 | In | SE |  |  | ○◯ | ○◯ | 5.81 | ←← | 8.6
10 | TMi | SE |  |  | ○◯ | ○◯ | 2.58 | ←← | 2.5
11 | TMa | SE |  |  |  | △▵ | 5.81 | ←← | 3.0
Monoarticular muscles of the elbow joint
12 | Br | EF | ○◯ | ○◯ | ○◯ | ○◯ | 7.0 | 9.49 | 7.1
13 | PT | EF | ○◯ | ○◯ |  | △▵ | 3.4 | 4.37 | 4.0
14 | BR | EF |  | △▵ |  | △▵ | 1.5 | 3.22 | 1.9
15 | MHTr | EE | ○◯ | ○◯ | ○◯ | ○◯ | 6.1 | 14.46 | 4.5
16 | LtHTr | EE | ○◯ | ○◯ | ○◯ | ○◯ | 6.0 | 15.42 | 4.5
17 | A | EE | ○◯ | ○◯ |  | △▵ | 2.5 | 2.00 | 2.5
Biarticular muscles of the shoulder and elbow joints
18 | LHBi | DF | ○◯ | ○◯ | ○◯ | ○◯ | 2.5 | 4.13 | 4.5
19 | SHBi | DF | ○◯ | ○◯ | ○◯ | ○◯ | 2.1 | 3.96 | 3.1
20 | LHTr | DE | ○◯ | ○◯ | ○◯ | ○◯ | 6.7 | 20.44 | 5.7
Monoarticular muscles of the wrist joint
21 | FCU | WF | ○◯ | ○◯ | ○◯ | ○◯ | 3.2 | 5.97 | 2.9
22 | FCR | WF | ○◯ | ○◯ | ○◯ | ○◯ | 2.0 | 2.94 | 1.6
23 | PL | WF | ○◯ | ○◯ | ○◯ | ○◯ | 0.9 | 0.70 | 0.6
24 | ECRB | WE | ○◯ | ○◯ | ○◯ | ○◯ | 2.9 | 4.72 | 2.2
25 | ECU | WE | ○◯ | ○◯ | ○◯ | ○◯ | 3.4 | 4.72 | 2.1
26 | ECRL | WE | ○◯ | ○◯ | ○◯ | ○◯ | 2.4 | 3.30 | 2.2

MD1 | MD2 | MD3 | MD4
0.538 ± 0.065 | 0.533 ± 0.060 | 0.637 ± 0.070 | 0.645 ± 0.074
MD5 | MD6 | MD7 | MD8
0.527 ± 0.071 | 0.569 ± 0.073 | 0.642 ± 0.073 | 0.663 ± 0.087