## 2 Methods and Results
### Moment Equations and Muscle Torques  
The dynamic moment equation of link \(n\) about \(J_{n}\) is given by  
\[ \begin{array}{ll}\sum\limits_{j=n-n_{s}}^{n-1}\left(\sum\limits_{m}J_{n}P_{nj }^{m}\times\overset{\rightarrow}{F}_{nj}^{m}\right)\cdot\dot{k}-m_{n}g\;l_{C _{n}}\cos(\theta_{n}+\varepsilon_{C_{n}})+R_{n}l_{R_{n}}\sin\left(\gamma_{R_{n} }-\varepsilon_{R_{n}}\right)=I_{\eta}\ddot{\theta}_{n}+J_{n}C_{n}\times m_{n} \overset{\rightarrow}{\mathcal{G}}_{J_{n}}\dot{k}\\ =I_{C_{n}}\ddot{\theta}_{n}+J_{n}C_{n}\times m_{n}\overset{\rightarrow}{ \mathcal{G}}_{C_{n}}\dot{k}\end{array} \]  
where \(I_{\eta}=I_{C_{n}}+m_{n}l_{C_{n}}^{2}\) (Huygens-Steiner theorem). This equation, considering Equation (5) for \(\overset{\rightarrow}{\mathcal{G}}_{C_{n}}\), becomes  
\[ \begin{array}{ll}\sum\limits_{j=n-n_{s}}^{n-1}\left(\sum\limits_{m}J_{n}P_{nj }^{m}\times\overset{\rightarrow}{F}_{nj}^{m}\right)\cdot\dot{k}-m_{n}g\;l_{C _{n}}\cos(\theta_{n}+\varepsilon_{C_{n}})+R_{n}l_{R_{n}}\sin\left(\gamma_{R_{n} }-\varepsilon_{R_{n}}\right)\\ =I_{\eta}\ddot{\theta}_{n}+m_{n}l_{C_{n}}\overset{n-1}{\underset{i=1}{\mathcal{G }}}_{i=1}^{n-1}l_{i}[\ddot{\theta}_{i}\cos(\theta_{n}+\varepsilon_{C_{n}}- \theta_{i})+\dot{\theta}_{i}^{2}\sin(\theta_{n}+\varepsilon_{C_{n}}-\theta_{i})] \end{array}  \tag{13}\]  
For \(k=1,\;2,\ldots,n-1\), the moment equation of link \(k\) about \(J_{k}\) includes the moment of the joint reaction force \(\overset{\rightarrow}{\Phi}_{k,\;k+1}\):  
\[ \begin{array}{ll}\sum\limits_{j=k-n_{s}}^{k-1}\left(\sum\limits_{m}J_{k}P_{kj }^{m}\quad\times\overset{\rightarrow}{F}_{kj}^{m}\right)\cdot\dot{k}+\sum \limits_{\begin{subarray}{c}k+n_{s}\\ j=k+1\end{subarray}}^{k+n_{s}}\left(\sum\limits_{m}J_{k}P_{kj}^{m}\overset{ \rightarrow}{F}_{kj}^{m}\right)\cdot\dot{k}-m_{k}g\;l_{C_{k}}\cos\left(\theta_{ k}+\varepsilon_{C_{k}}\right)+R_{k}l_{R_{k}}\sin\left(\gamma_{R_{k} }-\varepsilon_{R_{k}}\right)\\ \hskip 14.226378 pt(j\leq n)\\ +J_{k}J_{k+1}\times\overset{\rightarrow}{\Phi}_{k,\;k+1}\cdot\dot{k}=I_{C_{k}} \ddot{\theta}_{k}+J_{k}C_{k}\times m_{k}\overset{\rightarrow}{\mathcal{G}}_{C_{k}} \cdot\dot{k}\end{array} \]  
Substitution of \(\overset{\rightarrow}{\Phi}_{k,\;k+1}\) with the right-hand side of Equation (2) or (3) gives Equations (14) and (15), respectively:  
Theinerinertial factors \(\mathsf{J}_{k}C_{k}\times m_{k}\overset{\rightarrow}{a}_{\text{C}_{k}}\cdot\hat{k}\), \(\mathsf{J}_{k}\mathsf{J}_{k+1}\times\sum_{i=k+1}^{n}m_{i}\overset{ \rightarrow}{a}_{\text{C}_{i}}\cdot\hat{k}\), and \(\mathsf{J}_{k}\mathsf{J}_{k+1}\times\sum_{i=0}^{k}m_{i}\overset{ \rightarrow}{a}_{\text{C}_{i}}\cdot\hat{k}\) (where \(\mathsf{J}_{k}C_{k}=l_{\text{C}_{k}}\hat{a}_{\text{C}_{k}}\) and \(\mathsf{J}_{k}\mathsf{J}_{k+1}=l_{k}\hat{a}_{k}\)) in Equations (14) and (15) are readily deduced from Equations (5)-(7), respectively \((\theta_{0}=\dot{\theta}_{0}=\ddot{\theta}_{0}=0)\):  
\[\mathsf{J}_{k}C_{k}\times m_{k}\overset{\rightarrow}{a}_{\text{C}_{k}}\cdot \hat{k}=m_{k}l_{\text{C}_{k}}\sum_{i=0}^{k-1}l_{i}[\ddot{\theta}_{i}\cos\big{(} \theta_{k}+\varepsilon_{\text{C}_{k}}-\theta_{i}\big{)}+\dot{\theta}_{i}^{2} \sin\big{(}\theta_{k}+\varepsilon_{\text{C}_{k}}-\theta_{i}\big{)}]+m_{k}l_{ \text{C}_{k}}^{2}\ddot{\theta}_{k}\qquad\qquad(k=1,\ldots,n) \tag{17}\]  
\[