## 2 Methods and Results
### Moment Equations and Muscle Torques
\begin{subarray}{c}j_{2}=k\\ (j_{2}\leq n)\end{subarray} }^{k+n_{k}-1}\sum_{ \begin{subarray}{c}j_{1}=j_{2}-n _{\mathrm{s}}\\ (j_{1}\geq 0)\end{subarray} }^{k-1}\left(\sum_{m}\mathrm{J}_{k}\mathrm{P}_{j_{2} j_{1}}^{m}\times\overset{\rightarrow}{F}_{j_{2}j_{1}}^{m}\right).\hat{k} \tag{22}\]  
The second term, \(\tau_{k+1}\), has the same meaning but refers to joint \(k+1\). The final step is the determination of the closed-form expression of each single torques \(\tau_{k}\) (\(k=1,\ 2,\ \ldots,n-1\)), instead of the differences \(\tau_{k}-\tau_{k+1}\). The distal torque \(\tau_{n}\) has already been determined by Equation (13). Adding together the moment Equation (20) or (21) with \(k=k^{*},\ k^{*}+1,\ldots,n\) gives two equivalent expressions for \(\  
shape and size, and driven by single-joint and multi-joint muscle actuators that can span from one joint to all the joints of the chain. These closed-form analytical equations can be applied to multilink musculoskeletal chains with deep/superficial and segmental/global muscles. Equations (8)-(11) highlight that the joint reaction force \(\overset{\rightarrow}{\Phi}_{k,k+1}\) exerted by link \(k+1\) on link \(k\) through joint \(k+1\) is determined by the forces \(\overset{\rightarrow}{F}_{ij}^{m}\) of single-joint and multi-joint muscles spanning the joint (\(i\leq k\) and \(j\geq k+1\)), with an additional one of the following two equivalent sets of data: (1) the angular velocity and acceleration of each individual link of the chain (link \(1\) to link \(n\)), and the weight and the contact external resistance acting on each link distal to the joint (link \(k+1\) to link \(n\)); (2) the angular velocity and acceleration of each link proximal to the joint (link \(1\) to link \(k\)), and the weight and contact external resistance acting on these links and the base (link \(0\) to link \(k\)). The choice of data set depends on the unknown contact external forces \(\overset{\rightarrow}{R}_{i}\). For example, in many strengthening exercises executed lying down or sitting on a bench, the force acting on the base \(\overset{\rightarrow}{R}_{0}\) is the only unknown contact external force, and the joint reaction forces can be determined using the first set of data (Equation (9)). In contrast, in the dynamic wall-squat exercise [19], the force exerted by the wall on the back is unknown, and the load acting on the lower-limb joints can be determined using the second set of data, provided that the ground reaction force \(\overset{\rightarrow}{R}_{0}\) is determined by a force plate (Equation (10)). In both cases, the force of the muscles spanning the joint of interest should be properly estimated [20; 21; 22; 23; 24], which is beyond the scope of the study. The reaction force \(\overset{\rightarrow}{\Phi}_{k,k+1}\) acting on joint \(k+1\) is indirectly affected by the force \(\overset{\rightarrow}{F}_{ij}^{m}\) of muscles not spanning the joint (\(i<j\leq k\) or \(k+1\leq i<j\)). The torque developed by \(\overset{\rightarrow}{F}_{ij}^{m}\) yields inertial forces on links \(i\) and \(j\), which propagate along the chain, contributing to the load acting on each joint (Equation (9)). This is, for example, the case in the effect of the soleus muscle force on the tibofemoral joint during single-leg landing [25]. Nevertheless, the present study highlights that, in the static condition, a muscle force cannot affect the load acting on joints not spanned by the muscle. Notably, with the use of the global reference frame, the link angular velocities \(\dot{\theta}_{i}\) occur in the closed-form dynamic equations only through the centripetal terms proportional to \(\dot{\theta}_{i}^{2}\), avoiding the Coriolis terms proportional to the products of link angular velocities in the local joint frames. This considerably simplifies the closed-form dynamic equations (Appendix A). The limitations of this study are mainly related to the bidimensional nature of the adopted musculoskeletal chain model.