## 2 Methods and Results  
### The Chain Model  
We considered a planar musculoskeletal chain composed of a generic number \(n\) of rigid links connected by ideal revolute joints. Each link of the chain has a generic shape and size (Figure 1). Two adjacent links, link \(i-1\) and link \(i\), articulate at joint \(i\)\((i=2,\ 3,\dots,\ n)\). The proximal link of the chain (link 1) is also articulated with a fixed base at joint 1, and the distal link (link \(n\)) only articulates with link \(n-1\) at joint \(n\). The fixed base of the chain is referred to as \"link 0\". The center of rotation of joint \(i\) is denoted by J\({}_{i}\), and two points, J\({}_{0}\) (distinct from J\({}_{1}\)) and J\({}_{n+1}\) (distinct from J\({}_{n}\)), were arbitrarily selected within the base and link \(n\), respectively. Single-joint and multi-joint muscles attached to the chain and base were modeled as linear force actuators that can span 1 to \(n_{\text{S}}\) joints \((n_{\text{S}}\leq n)\). Figure 1: Mechanical diagram of the individual link \(i\) of the chain. The diagram includes the forces acting on the link, and the relevant points, unit vectors, and angular quantities associated with the link. The definition of these quantities is also reported in the ”Nomenclature”. The following parameters are associated with each link \(i\) (\(i=0,\,1,\,\ldots,n\)) of the chain (Figure 1): the distance \(l_{i}=|\text{J}_{i}|_{i+1}|\) between \(\text{J}_{i}\) and \(\text{J}_{i+1}\), the unit vector \(\hat{u}_{i}\) of \(\text{J}_{\text{J}}|_{i+1}\) (\(\text{J}_{i}|_{i+1}=l_{i}\hat{u}_{i}\)), the angle \(\theta_{i}\) between \(\hat{u}_{i}\) and the horizontal \(X\)-axis (the counterclockwise direction of rotation is considered positive); the distance \(l_{C_{i}}=|\text{J}_{i}\text{C}_{i}|\) from \(\text{J}_{i}\) of the center of mass \(\text{C}_{i}\) of the link, the unit vector \(\hat{u}_{\text{C}_{i}}\) of \(\text{J}_{i}\text{C}_{i}\) (\(\text{J}_{i}\text{C}_{i}=l_{C_{i}}\hat{u}_{\text{C}_{i}}\)), the angle \(\varepsilon_{\text{C}_{i}}\) between \(\hat{u}_{\text{C}_{i}}\) and \(\hat{u}_{i}\); the mass \(m_{i}\), the moments of inertia \(\text{J}_{i}\) and \(\text{J}_{i}\) and \(\text{C}_{i}\); the accelerations \(\overset{\rightarrow}{a}_{\text{J}_{i}}\) and \(\overset{\rightarrow}{a}_{\text{C}_{i}}\) of \(\text{J}_{i}\) and \(\text{C}_{i}\); the angular velocity \(\overset{\rightarrow}{\omega}_{i}=\hat{\theta}_{i}\hat{k}\) and acceleration \(\overset{\rightarrow}{\omega}_{i}=\hat{\theta}_{i}\hat{k}\) (\(\hat{k}\) is the unit vector normal to the plane of the chain); and the unit vectors \(\hat{w}_{i}=\ \hat{k}\times\hat{u}_{i}\) and \(\hat{w}_{\text{C}_{i}}=\ \hat{k}\times\hat{u}_{\text{C}_{i}}\). The external forces acting on each link \(i\) (\(i=0,\ 1,\ \ldots,n\)) are listed below (Figure 1):  
* The weight \(m_{i}\overset{\rightarrow}{S}\) of the link applied at \(\text{C}_{i}\). * A contact external resistance \(\overset{\rightarrow}{R}_{i}\) may act on a point \(\text{P}_{R_{i}}\) of the link at distance \(l_{R_{i}}\) from \(\text{J}_{i}\) (\(\varepsilon_{R_{i}}\) is the angle between \(\text{J}_{i}\text{P}_{R_{i}}\) and \(\hat{u}_{i}\), and \(\gamma_{R_{i}}\) the angle between \(\overset{\rightarrow}{R}_{i}\) and \(\hat{u}_{i}\)). The contact external force \(\overset{\rightarrow}{R}_{0}\) acting on the base is typically the ground reaction force. The case of multiple contact external resistances acting on a link is examined in the discussion section. * The sum of muscle forces \(\sum\limits_{m}\overset{\rightarrow}{F}_{ij}^{m}\) exerted on points \(\text{P}_{ij}^{m}\) of link \(i\) by the muscles joining link \(i\) to the other links \(j\) (\(j=i\pm 1\) for single-joint muscles, \(j=i\pm 2\) for two-joint muscles, etc.). The sum is extended over all muscles joining link \(i\) to link \(j\). The joint reaction force \(\overset{\rightarrow}{\Phi}_{i,i-1}\), exerted on link \(i\) by link \(i-1\), and \(\overset{\rightarrow}{\Phi}_{i,i+1}\), entered on link \(i\) by link \(i+1\).