## 2 Methods and Results
### Force Equations and Joint Reaction Forces  
The dynamic force equations of individual links \(i\) (\(i=0,\ 1,\ldots,n\)) are given by  
\[\sum\limits_{ \begin{subarray}{c}j=i-n_{\text{s}}\\ (j\geq 0)\end{subarray} }^{i-1}\left(\sum\limits_{m}\overset{\rightarrow}{F}_{ ij}^{m}\right)+\sum\limits_{ \begin{subarray}{c}j=i+1\\ (j\leq n)\end{subarray} }^{i+n_{\text{s}}}\left(\sum\limits_{m}\overset{ \rightarrow}{F}_{ij}^{m}\right)+m_{i}\overset{\rightarrow}{g}+\overset{ \rightarrow}{R}_{i}+\overset{\rightarrow}{\Phi}_{i,i-1}+\overset{ \rightarrow}{\Phi}_{i,i+1}=m_{i}\overset{\rightarrow}{a}_{\text{C}_{i}} \tag{1}\]  
where \(\overset{\rightarrow}{\Phi}_{-1,0}=\overset{\rightarrow}{\Phi}_{n,n+1}=0\) and the index \(j\) may only range between \(0\) and \(n\). Adding together the last \(n-k\) equations (\(i=k+1,\ k+2,\ldots,\ n\)) or the first \(k+1\) ones(\(i=0,\ 1,\ldots,\ k\)), and taking into account that \(\overset{\rightarrow}{F}_{ij}^{m}=-\overset{\rightarrow}{F}_{ji}^{m}\) and \(\overset{\rightarrow}{\Phi}_{i,i+1}=-\overset{\rightarrow}{\Phi}_{i+1,i}\), one can express the joint reaction forces \(\overset{\rightarrow}{\Phi}_{k,k+1}\) (\(k=0,\ 1,\ldots,n-1\)) in relation to the other parameters:  
\[\overset{\rightarrow}{\Phi}_{k,k+1}=\sum\limits_{i=k+1}^{n}\left(m_{i} \overset{\rightarrow}{g}+\overset{\rightarrow}{R}_{i}\right)-\sum\limits_{i=k+1 }^{n} \begin{array}{c}m_{i}\overset{\rightarrow}{a}_{\text{C}_{i}}+\\ (j_{2}\leq n)\end{array} \sum\limits_{ \begin{subarray}{c}j_{1}=i_{2}-n_{\text{s}} \\ (j_{1}\geq 0)\end{subarray} }^{k}\left(\sum\limits_{m}\overset{\rightarrow}{F}_{ j_{2}i}^{m}\right) \tag{2}\]  
\[\overset{\rightarrow}{\Phi}_{k,k+1}=-\sum\limits_{i=0}^{k}\left(m_{i} \overset{\rightarrow}{g}+\overset{\rightarrow}{R}_{i}\right)+\sum\limits_{i=0}^{k}m_{i} \overset{\rightarrow}{a}_{\text{C}_{i}}+\sum\limits_{ \begin{subarray}{c}j_{2}=k+1 \\ (j_{2}\leq n)\end{subarray} }^{k+n_{\text{s}}}\sum\limits_{ \begin{subarray}{c}j_{1}=j_{ 2}-n_{\text{s}}\\ (j_{1}\geq 0)\end{subarray} }^{k}\left(\sum\limits_{m}\overset{\rightarrow}{F}_{ j_{2}i_{1}}^{m}\right) \tag{3}\]These two equations are equivalent, as it can be directly proved considering the force equation applied to the whole system of links, including link 0:  
\[\sum\limits_{i=0}^{n}\biggl{(}m_{i}\overrightarrow{g}+\overrightarrow{R}_{i} \biggr{)}=\sum\limits_{i=0}^{n}m_{i}\overrightarrow{a}_{C_{i}} \tag{4}\]  
The accelerations \(\overrightarrow{a}_{C_{i}}\)\((i=1,\ 2,\ldots,n)\) can be expressed as a function of the angular velocity and acceleration of the individual links by applying recursively the equation the relates the accelerations of two arbitrary points of a rigid body:  
\[ \begin{array}{c}\overrightarrow{a}_{C_{i}}=\sum\limits_{j=0}^{i-1}\biggl{(} \overrightarrow{a}_{J_{j+1}}-\overrightarrow{a}_{J_{j}}\biggr{)}+\biggl{(} \overrightarrow{a}_{C_{i}}-\overrightarrow{a}_{J_{i}}\biggr{)}=\sum\limits_{ j=0}^{i-1}\left(\tilde{\theta}_{j}\hat{k}\times l_{j}\hat{u}_{j}-\tilde{\theta}_{j}^{2}l_{j} \hat{u}_{j}\right)+\left(\tilde{\theta}_{i}\hat{k}\times l_{C_{i}}\hat{u}_{C_ {i}}-\tilde{\theta}_{i}^{2}l_{C_{i}}\hat{u}_{C_{i}}\right)\\ =\sum\limits_{j=0}^{i-1}\left(\tilde{\theta}_{j}l_{j}\hat{w}_{j}-\tilde{\theta }_{j}^{2}l_{j}\hat{u}_{j}\right)+\left(\tilde{\theta}_{i}l_{C_{i}}\hat{w}_{C_ {i}}-\tilde{\theta}_{i}^{2}l_{C_{i}}\hat{u}_{C_{i}}\right)\end{array}  \tag{5}\]  
Equation (5) enables the closed-form determination of the inertial-term sum \(\sum\limits_{i=k+1}^{n}m_{i}\overrightarrow{a}_{C_{i}}\)\((k=0,\ 1,\ \ldots,n-1)\) in Equation (2)  
\[