## 2 Materials and Methods
### Mathematical Model  
We start from the model proposed by [PERSON] and associates [14]. The arm is represented as a double pendulum as shown in Figure 1 (right) with \(\theta\) being the shoulder angle and \(\varphi\) being the elbow angle. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**ID** & **SC III OFF** & **SC III ON** & **BDI-II** & **MMSE** & **PDQ-39** & **LEDDs** \\ \hline PD1 & 42 & 27 & 32 & 28 & 77 & 400 \\ PD2 & 24 & 6 & 6 & 30 & 20 & 400 \\ PD3 & 35 & 20 & 22 & 30 & 34 & 700 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Demographic and clinical features of participants. BDI-II: Beck Depression Inventory-II; SC stands for MDS-UPDRS: Movement Disorder Society Unified Parkinson’s Disease Rating Scale; MMSE: Mini-Mental State Evaluation; PDQ-39: Parkinson’s Disease Questionnaire. Figure 1: (**Left**) Configuration of the Movit System Units on the participant’s body. (**Right**) Graphical representation of the double pendulum system. Indeed, the model relies on the Euler-Lagrange equations, which involve the Lagrangian function \(\mathcal{L}\) as the sum of the total kinetic energy \(\mathcal{T}\) and the total potential energy \(\mathcal{U}\) of the double pendulum. In the following formulas, (i) the subscripts 1 and 2 will refer to the upper link and lower link, respectively, whereas (ii) \(L\) represents the length of the segment, \(m\) its mass, \(p_{\rm cm}\) the position of the center of mass, and \(I\) the moment of inertia. Anthropometric quantities are taken, for each subject, according to the related tables [34]. Furthermore, normalized \(\tau_{\rm CCE}(\cdot)\) and \(\tau_{\rm CEe}(\cdot)\) depend on the force applied by the shoulder muscles and the elbow muscles, respectively (CE stands for contractile element). On the other hand, \(\tau_{s}(\theta,t)=a_{1}(t)m_{1}L_{1}\cos(\theta)\) depends on the force due to the linear acceleration of the shoulder, whereas \(\tau_{e}(\theta,\varphi,t)=a_{2}(t)m_{2}L_{2}\cos(\theta+\varphi)\) depends on the force due to the acceleration of the elbow (\(a\) stands for the acceleration of the center of mass, coming from the IMU sensor system). Finally, \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), and \(\beta_{4}\) represent the elastic constant, viscous constant, resting position, and CE of the shoulder, respectively, whereas \(\beta_{5}\), \(\beta_{6}\), \(\beta_{7}\), and \(\beta_{8}\) denote the same parameters for the elbow, respectively. In fact, it is known that biological tissues, like muscles, present a viscoelastic behavior [35] with several models having been proposed over the years [36; 37] to reproduce the muscle response. Indeed, a classical Kelvin-Voight model of viscoelastic materials is also adopted hereafter to model the viscoelastic properties of the muscles, as illustrated by Figure 2.  
On the other hand, a source of non-conservative force in the upper limb is the elbow constraint, which forbids a complete extension of the forearm, while a forward flexion is free (\(\varphi_{0}\) is the position of the elbow at rest). As a matter of fact, the resulting model reads (dependence of state variables on time \(t\) is omitted for the sake of simplicity):  
\[\left[ \begin{array}{c}\ddot{\theta}\\ \ddot{\varphi}\end{array} \right] = M^{-1}(\varphi)\left(S(\theta,\dot{\theta},\varphi,\phi)+\Gamma( \theta,\dot{\theta},\varphi,\dot{\varphi},t)\beta+\left[ \begin{array}{c}0\\ \exp\Bigl{(}-\frac{180}{\pi}(\varphi-\varphi_{0})\Bigr{)}\end{array} \right]\right. \tag{1}\] \[\left.+\left[ \begin{array}{c}\tau_{s}(\theta,t)\\ \tau_{e}(\theta,\varphi,t)\end{array} \right]\right)\]  
in which  
\[M^{-1}(\varphi) = \left[ \begin{array}{cc}M_{11}(\varphi)&M_{12}(\varphi)\\ M_{21}(\varphi)&M_{22}\end{array}