# Sensorimotor Control Using Adaptive Neuro-Fuzzy Inference for Human-Like Arm Movement
## 2 Musculoskeletal Modeling
### Muscle Model  
This study adopts a muscle model developed by [PERSON] [30] and subsequently refined by [31]. This well-known muscle model calculates the muscle tensile force with the muscle activation level. Muscle length and velocity are the variables employed for precise calculation of the muscle tensile force, and each muscle force \(F_{im}\) is found by:  
\[F_{im}=\alpha_{i}f_{ia}(l_{i},\
u_{i})F_{imax}+f_{ip}(l_{i})F_{imax} \tag{6}\]  
where \(f_{ia}\) and \(f_{ip}\) denote active and passive scaling functions, \(F_{imax}\) is the maximal isometric muscle force, \(\
u_{i}\) denotes the muscle length velocity, and \(\alpha_{i}\in[0,1]\) indicates the muscle activation level. The contraction at constant length and zero muscle activation (\(\alpha_{i}=0\)), \begin{table}
\begin{tabular}{c c c} \hline \hline
**\# Muscle** & **Value (m)** \\ \hline \(l_{1}\) & \(m_{10}=0.055,\) & \(m_{11}=0.08\) \\ \(l_{2}\) & \(m_{20}=0.055,\) & \(m_{21}=0.08\) \\ \(l_{3}\) & \(m_{31}=0.03,\) & \(m_{32}=0.12\) \\ \(l_{4}\) & \(m_{41}=0.03,\) & \(m_{42}=0.12\) \\ \(l_{5}\) & \(m_{50}=0.04,\) & \(m_{52}=0.045\) \\ \(l_{6}\) & \(m_{60}=0.04,\) & \(m_{62}=0.045\) \\ \(l_{7}\) & \(m_{72}=0.035,\) & \(m_{73}=0.01\) \\ \(l_{8}\) & \(m_{82}=0.05,\) & \(m_{83}=0.01\) \\ \(l_{9}\) & \(m_{91}=0.04,\) & \(m_{93}=0.01\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The muscle model parameter values based on [29]. where the velocity of muscle shortening is zero, called the passive muscle force, is calculated from the force-length relationship. The maximum exerted muscle force is generated with \(\alpha_{i}=1\) when the force of contraction remains constant, and the velocity of contraction becomes progressively less with regards to the force-velocity relationship. The muscle model is written in a compact form as follows:  
\[F_{m}=\mathbf{F}_{a}\alpha+F_{p} \tag{7}\]  
where \(\mathbf{F}_{a}=\text{diag}(F_{a1},\cdots,F_{a9})\in\mathbb{R}^{9\times 9}\) is the active muscle force term, \(F_{p}=[F_{p1},\cdots,F_{p9}]^{\mathsf{T}}\in\mathbb{R}^{9}\) is the passive muscle force term, and \(\alpha=[\alpha_{1},\cdots,\alpha_{9}]^{\mathsf{T}}\in\mathbb{R}^{9}\) is the muscle activation vector. The relationship between muscle tensile force \(F_{m}=[F_{1m},F_{2m},\cdots,F_{9m}]^{\mathsf{T}}\in\mathbb{R}^{9}\) and joint torques \(\tau=[\tau_{1},\tau_{2},\tau_{3}]^{\mathsf{T}}\in\mathbb{R}^{3}\) is defined as:  
\[\tau=\mathbf{W}_{m}F_{m} \tag{8}\]  
where \(\mathbf{W}_{m}=\mathbf{J}_{m}^{\mathsf{T}}\in\mathbb{R}^{3\times 9}\) is transpose of the muscle Jacobian matrix. The number of muscles is larger than the degrees-of-freedom (DoFs) of the link arm, resulting in the infinite admission solutions. The solution can be expressed as follows:  
\[F_{m}=(\mathbf{W}_{m})^{\mathsf{T}}\tau+(\mathbf{I}-(\mathbf{W}_{m})^{\mathsf{ T}}(\mathbf{W}_{m}))\vartheta \tag{9}\]  
where \(\mathbf{W}_{m}^{\mathsf{T}}=\mathbf{W}_{m}^{\mathsf{T}}(\mathbf{W}_{m}\mathbf{ W}_{m}^{\mathsf{T}})^{-1}\in\mathbb{R}^{9\times 3}\) is called the pseudo-inverse matrix of \(\mathbf{W}_{m}\), and \(\mathbf{I}\in\mathbb{R}^{9\times 9}\) is the identity matrix. The first term of Equation (9) defines a minimum two-norm solution vector, and, therefore, the muscle force might be negative. The second term of Equation (9) is an arbitrary vector from the null space of \(\mathbf{W}_{m}\), depending on \(\vartheta\in\mathbb{R}^{9}\). Equation (9) is rewritten as  
\[F_{m}=(\mathbf{W}_{m})^{\mathsf{T}}\tau+\mathbf{N}h \tag{10}\]  
where \(\mathbf{N}h\) equals an arbitrary muscle force vector that balances the redundant muscles among all the muscles without affecting the end-point pose (internal muscle forces). Thus, the second term represents the internal muscle force generated by the redundant muscles. Note that this term is employed to keep all muscle tensions positive. \(\mathbf{N}\in\mathbb{R}^{9\times 6}\) represents the null space or kernel of \(\mathbf{W}_{m}\), and \(h\in\mathbb{R}^{6}\) needs to be determined to ensure that all muscle forces are positive.