## Appendix A Moment Equations for Chains with Two or Three Links  
The closed-form Equations (13), (23), and (24) for the joint torque \(\tau_{k}\) developed by single-joint and multi-joint muscles can be compared to the corresponding closed-form equations that have been reported for chains driven by ideal torque actuators and composed of two [4] or three [8] links modeled as slender rods. * _Moment equations for a two-link chain_ Equations (13) and (23) with \(n=2\) give the closed-form expressions of \(\tau_{1}\) and \(\tau_{2}\) for a two-link chain: \[\tau_{2}=m_{2}g\;l_{\text{C}_{2}}\cos\bigl{(}\theta_{2}+\varepsilon_{\text{C}_{ 2}}\bigr{)}-R_{2}l_{R_{2}}\sin\bigl{(}\gamma_{R_{2}}-\varepsilon_{R_{2}}\bigr{)} +m_{2}l_{\text{C}_{2}}l_{1}[\tilde{\theta}_{1}\cos\bigl{(}\theta_{2}+ \varepsilon_{\text{C}_{2}}-\theta_{1}\bigr{)}+\dot{\theta}_{1}^{2}\sin\bigl{(} \theta_{2}+\varepsilon_{\text{C}_{2}}-\theta_{1}\bigr{)}]+l_{\text{J}_{2}} \dot{\theta}_{2}\] \[\tau_{1}=\sum\limits_{k=1}^{2}\quad\bigl{[}m_{k}g\;l_{\text{C}_{k} }\cos\bigl{(}\theta_{k}+\varepsilon_{\text{C}_{k}}\bigr{)}-R_{k}l_{R_{k}}\sin \bigl{(}\gamma_{R_{k}}-\varepsilon_{R_{k}}\bigr{)}\bigr{]}+\bigl{[}m_{2}g\;l_{1} \cos\theta_{1}-R_{2}l_{1}\sin\bigl{(}\gamma_{R_{2}}+\theta_{2}-\theta_{1} \bigr{)}\bigr{]}\] \[\qquad\qquad\qquad\qquad+m_{2}l_{\text{C}_{2}}l_{1}[\tilde{\theta }_{1}\cos\bigl{(}\theta_{2}+\varepsilon_{\text{C}_{2}}-\theta_{1}\bigr{)}+ \dot{\theta}_{1}^{2}\sin\bigl{(}\theta_{2}+\varepsilon_{\text{C}_{2}}-\theta_ {1}\bigr{)}]+\bigl{(}l_{\text{J}_{1}}+m_{2}l_{1}^{2}\bigr{)}\tilde{\theta}_{1} +l_{\text{J}_{2}}\tilde{\theta}_{2}\] \[\qquad\qquad\qquad\qquad+l_{1}m_{2}l_{\text{C}_{2}}[\tilde{\theta }_{2}\cos\bigl{(}\theta_{2}+\varepsilon_{\text{C}_{2}}-\theta_{1}\bigr{)}- \dot{\theta}_{2}^{2}\sin\bigl{(}\theta_{2}+\varepsilon_{\text{C}_{2}}-\theta_ {1}\bigr{)}]\] These equations, for \(\varepsilon_{\text{C}_{k}}=\varepsilon_{R_{k}}=\gamma_{R_{k}}=0\), \(R_{k}=0\), and \(\theta_{k}=\sum\limits_{i=1}^{k}\alpha_{i}\), coincide with the closed-form equations derived by [PERSON] [4] (p. 380) in the joint space. The above equations for \(\tau_{2}\) and \(\tau_{1}\), derived in the global reference frame, include 3 and 7 inertial terms, respectively, whereas the corresponding equations in the joint space include 6 and 11 inertial terms [4]. * _Moment equations for a three-link chain_ For the three-link chain \((n=3)\), Equations (13) and (23) give \[\tau_{3}=m_{3}g\,l_{\mathrm{C}_{3}}\cos\bigl{(}\theta_{3}+\varepsilon_{\mathrm{C }_{3}}\bigr{)}-R_{3}  
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* _[PERSON] et al.