## 2 Methods and Results
### Moment Equations and Muscle Torques
\begin{split}\mathsf{J}_{k}\mathsf{J}_{k+i}\times\sum\limits_{i=k +1}^{n}& m_{i}\overset{\rightarrow}{a}_{\text{C}_{i}}\cdot \hat{k}\\ &=l_{k}\Bigg{(}\sum\limits_{j=k+1}^{n}m_{j}\Bigg{)}\sum\limits_{i=0 }^{k-1}l_{i}[\ddot{\theta}_{i}\cos(\theta_{k}-\theta_{i})+\dot{\theta}_{i}^{2} \sin(\theta_{k}-\theta_{i})]+\Bigg{(}\sum\limits_{j=k+1}^{n}m_{j}\Bigg{)}l_{k} ^{2}\ddot{\theta}_{k}\\ &+(1-\delta_{k,n-1})l_{k}\sum\limits_{i=k+1}^{n-1}\Bigg{(}\sum \limits_{j=i+1}^{n}m_{j}\Bigg{)}l_{i}[\ddot{\theta}_{i}\cos(\theta_{i}-\theta _{k})-\dot{\theta}_{i}^{2}\sin(\theta_{i}-\theta_{k})]\\ &+l_{k}\sum\limits_{i=k+1}^{n}m_{i}l_{\text{C}_{i}}[\ddot{\theta }_{i}\cos(\theta_{i}+\varepsilon_{\text{C}_{i}}-\theta_{k})-\dot{\theta}_{i}^{2 }\sin(\theta_{i}+\varepsilon_{\text{C}_{i}}-\theta_{k})]\qquad\qquad(k=1, \ldots,n-1)\end{split}  \tag{18}\]  
\[ \begin{split}\mathsf{J}_{k}\mathsf{J}_{k+i}\times\sum\limits_{i=0 }^{k}m_{i}\overset{\rightarrow}{a}_{\text{C}_{i}}\cdot\hat{k}\\ &=l_{k}\sum\limits_{i=0}^{k-1}\Bigg{(}\sum\limits_{j=i+1}^{k}m_{j }\Bigg{)}l_{i}[\ddot{\theta}_{i}\cos(\theta_{k}-\theta_{i})+\dot{\theta}_{i}^{2 }\sin(\theta_{k}-\theta_{i})]\\ &+l_{k}\sum\limits_{i=1}^{k}m_{i}l_{\text{C}_{i}}[\ddot{\theta }_{i}\cos(\theta_{k}-\theta_{i}-\varepsilon_{\text{C}_{i}})+\dot{\theta}_{i}^{2 }\sin(\theta_{k}-\theta_{i}-\varepsilon_{\text{C}_{i}})]\qquad\qquad(k=1, \ldots,n)\end{split}  \tag{19}\]  
Equations (14) and (15), considering Equations (16)\(-\)(19), result in the following two equivalent moment equations for the individual link \(k\) (\(k=1,\ 2,\ldots,n-1\))  
\[\begin{split}\sum\limits_{ \begin{subarray}{c}j_{2}=k\\ (j_{2}\leq n)\end{subarray} }^{k+n_{\text{S}}-1}\sum\limits_{ \begin{subarray}{c}j_ {1}=-n_{\text{S}}\\ (j_{1}\geq 0)\end{subarray} }^{k-1}\Bigg{(}\sum\limits_{ \begin{subarray}{c}j_ {2}=k+1\\ (j_{2}\leq n)\end{subarray} }^{k-n}\Bigg{(}\sum\limits_{ \begin{subarray}{c}j_ {2}=k+1\\ (j_{2}\leq n)\end{subarray} }^{m}\Bigg{)}\cdot\hat{k}-m_{k}g\;l_{\text{C}_{k}}\cos \big{(}\theta_{k}+\varepsilon_{\text{C}_{k}}\big{)}\\ \qquad\qquad\qquad+l_{k}\sum\limits_{i=1}^{k}m_{j}\Bigg{(}\sum\limits_{i=0 }^{k-1}l_{i}[\ddot{\theta}_{i}\cos(\theta_{k}-\theta_{i})+\dot{\theta}_{i}^{2 }\sin(\theta_{k}-\theta_{i})]+\Bigg{(}l_{k}+\sum\limits_{j=k+1}^{n}m_{j}\cdot l_{ k}^{2}\Bigg{)}\ddot{\theta}_{k}\\ \qquad\qquad\qquad+(1-\delta_{k,n-1})l_{k}\sum\limits_{i=k+1}^{n-1}\Bigg{(} \sum\limits_{j=i+1}^{n}m_{j}\Bigg{)}l_{i}[\ddot{\theta}_{i}\cos(\theta_{i}- \theta_{k})-\dot{\theta}_{i}^{2}\sin(\theta_{i}-\theta_{k})]\\ \qquad\qquad\qquad+l_{k}\sum\limits_{i=k+1}^{n}m_{i}l_{\text{C}_{i}}[\ddot{ \theta}_{i}\cos(\theta_{i}+\varepsilon_{\text{C}_{i}}-\theta_{k})-\dot{\theta}_{i}^ {2}\sin(\theta_{i}+\varepsilon_{\text{C}_{i}}-\theta_{k})]\\ \qquad\qquad\qquad+l_{k}\sum\limits_{i=k+1}^{n}\sum\limits_{j=i+1}^{k-1}l_{i} [\ddot{\theta}_{i}\cos(\theta_{i}+\varepsilon_{\text{C}_{i}}-\theta_{k})-\dot{ \theta}_{i}^{2}\sin(\theta_{i}+\varepsilon_{\text{C}_{i}}-\theta_{k})]\\ \qquad\qquad\qquad\qquad+l_{k}\sum\limits_{i=k+1}^{n}m_{i}l_{\text{C}_{i}}[ \ddot{\theta}_{i}\cos(\theta_{i}+\varepsilon_{\text{C}_{i}}-\theta_{k})-\dot{ \theta}_{i}^{2}\sin(\theta_{i}+\varepsilon_{\text{C}_{i}}-\theta_{k})]\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquadThe first term on the left-hand side of Equations (20) and (21) represents the moment \(\tau_{k}\) about \(J_{k}\) of the muscle forces exerted on the links distal to joint \(k\) by the single-joint and multi-joint muscles that span joint \(k\)  
\[\tau_{k}=\sum_{