## 2 Materials and Methods
### Cable Routing Modeling  
The cable-driven exoskeleton proposed in this paper has a structure inspired by previous works [31]. It is composed of guiding rings which route the cable from the palm to the finger tip through anchor points. These rings must be adequately spaced to optimize force transmission and must be connected by flexible elements (i.e., compliant links) to allow finger flexion while maintaining the desired spacing in the resting condition.  
Figure 1a shows a geometric model developed for the thumb and long fingers which adapts the approach proposed in [31] to include the thickness of the guiding rings as parameter. Black dots indicate the anchor points. The model is part of an optimization process that takes as input anatomical data (Figure 1b) and outputs the position of the anchor points (\(k_{i}a_{i}\)) and the thickness of the guiding rings (\(s_{i}\)) to maximize the assistive torque provided to the user. The geometric model includes the intra-finger couplings found in [32]. In the model, friction is neglected.  
The tensile force (\(T\)) applied to the cable can be modeled at each anchor point as the sum of two components: one parallel \(T_{p}\) and one normal \(T_{n}\) to the phalanx axis. The total torque (Equation (1)) was calculated as the sum of the \(i\)-th contribution at each joint.  
\[\tau_{tot}=\sum_{i=1}^{q}\tau_{i} \tag{1}\]  
where \(q\) is the number of joints, i.e., 2 for the thumb and 3 for the long fingers. Each \(i\)-th contribution in Equation (1) is defined as:  
\[\tau_{i}=\begin{cases}(k_{i}a_{i}-s_{i})T\sin(\gamma_{i})-\Big{(}\frac{D_{i}}{ 2}+h_{i}\Big{)}T\cos(\alpha_{i+1})+k_{i}a_{i}T\sin(\alpha_{i+1})+\Big{(}\frac{ D_{i}}{2}+h_{i}\Big{)}T\cos(\gamma_{i}),\\ i\in[1,q-1]\\ (k_{i}a_{i}-s_{i})T\sin(\gamma_{i})+\Big{(}\frac{D_{i}}{2}+h_{i}\Big{)}T\cos( \gamma_{i}),\\ i=q\end{cases} \tag{2}\]  
where  
* \(a_{0}\) is the distance from joint 1 to the anchor point
* \(a_{i}\) is the length of the phalanx
* \(k_{i}a_{i}\) is the position of the ring as defined in Figure 1
* \(D_{i}\) is the diameter of the \(i\)-th phalanx
* \(h_{i}\) is the height of the anchor point
* \(\alpha_{i}\) and \(\gamma_{i}\) are defined as:  
\[\alpha_{i}=\arcsin\Bigg{(}\frac{(k_{i}a_{i}-s_{i})\sin(\pi-\theta _{i})}{\sqrt{(k_{i}a_{i}-s_{i})^{2}+(a_{i-1}-k_{i-1}a_{i-1})^{2}-2k_{i}a_{i}( a_{i-1}-k_{i-1}a_{i-1})\cos(\pi-\theta_{i})}}\Bigg{)} \tag{3}\] \[\gamma_{i}=\arcsin\Bigg{(}\frac{(a_{i-1}-k_{i-1}a_{i-1})\sin(\pi -\theta_{i})}{\sqrt{(k_{i}a_{i}-s_{i})^{2}+(a_{i-1}-k_{i-1}a_{i-1})^{2}-2k_{i} a_{i}(a_{i-1}-k_{i}a_{i})\cos(\pi-\theta_{i})}}\Bigg{)} \tag{4}\]  
The more distal ring (\(i=q\)) is a simple anchor point on the finger tip, so \(h_{q}=0\) and \(s_{q}=0\) where \(i=q\), \(h_{q}=0\) and \(s_{q}=0\) (i.e., the more distal ring is a simple anchor point on the finger tip).  
The optimization process aims at maximizing \(\tau_{tot}\) in the range of \([0,90]^{\circ}\) for \(\theta_{1}\). The solution is estimated in two steps: first, the optimal position of the anchor points is identified by constraining the ring thickness to 1 mm. The value of \(k_{i}\) is varied in the range [0.15, 0.85] with a step of 0.1. Subsequently, we constrained the position of the anchor points to the identified optimal positions \(k_{i}^{opt}a_{i}\) and varied the thickness of the guiding ring in the range of \([3,0.5a_{i}]\) mm with a step of 3 mm to identify the optimal thickness \(s_{i}^{opt}\). We defined this range and step based on anatomical limits and manufacturing constraints.