## 2 Materials and Methods
### Robotic Therapy  
Robotic therapy was administered using the commercial device Armeo(r) Power (Hocoma, Volketswil, Switzerland). This robot is an exoskeleton composed of an orthosis for the paretic upper limb with six degrees of freedom: three for the shoulder, one for the elbow flexion, one for the forearm supination and one for the hand closure. Each joint is powered by a motor and equipped with 2D sensors. The device can support the patient's upper limb weight, providing a feeling of fluctuation. The selected support is related to the maximum force (in percentage of upper limb weight) that the robot should exert to assist the movements of patients. A support > 100% could also compensate for adjunctive forces directed downwards. The device also includes an interface used for the exergame, designed to obtain task-oriented gestures in a specifically developed video game environment with exergames specifically developed for rehabilitative purposes [20; 21]. The computer-robot interface is designed to provide some exergames requiring specific upper limb movements consisting of single-joint movements around a single axis or combined movements of single joint around two or three axes, exercises for the opening and closing hand (grasping exercises) and multi-joint exercises along one, two or three spatial axes (coordination exercises). The training characteristics (difficulty level, duration, visual detail) were set conforming to the residual ability of the patient [21].  
At T1 and T2, the data related to joint ROMs and the total working volume obtained with passive mobilization of the paretic upper limb performed by a therapist up to the articular limit in the three planes of the space were collected, together with the weight support selected for performing the proposed exergames and analyzed with Armeo(r) Control Software (Figure 1). According to previous studies [11; 12; 13; 21; 22], the assessment procedure was performed with the measure of the patient's passive range of motion with the robotic device in a kinematic transparent mode, and the therapist moving the affected limb linked to the exoskeleton for exploring the entire reachable space to interact with the video game during therapy without experiencing resistance to the movement or elicit pain.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Baseline Parameters (T1)** & **Mean \(\pm\) SD** & **Correlation with WV** & **Correlation with WS** \\ \hline Working volume (WV) & 0.112 \(\pm\) 0.049 m\({}^{3}\) & - & **R = \(-\)0.387, \(p\) = 0.014** \\ Weight support (WS) & 97.6 \(\pm\) 22.2\% & **R = \(-\)0.387, \(p\) = 0.014** & \\ MRC score & 19 \(\pm\) 8 & R = 0.234, \(p\) = 0.145 & **R = \(-\)0.442, \(p\) = 0.004** \\ FMA-UE score & 28 \(\pm\) 15 & R = 0.046, \(p\) = 0.779 & R = \(-\)0.182, \(p\) = 0.26 \\ VAS score & 2 \(\pm\) 2 & R = 0.041, \(p\) = 0.803 & **R = 0.314, \(p\) = 0.048** \\ MAS total score & 4 \(\pm\) 2 & **R = \(-\)0.395, \(p\) = 0.012** & **R = 0.441, \(p\) = 0.004** \\ NIH-SS score & 4 \(\pm\) 2 & R = \(-\)0.181, \(p\) = 0.264 & **R = 0.607, \(p\) < 0.001** \\ BI score & 82 \(\pm\) 12 & R = \(-\)0.066, \(p\) = 0.687 & R = \(-\)0.274, \(p\) = 0.087 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mean \(\pm\) standard deviation (SD) of working volume and weight support (expressed in percentage of the upper limb weight) with respect to the clinical scale scores and their correlations (in terms of Spearman’s correlation coefficients R and the relevant \(p\)-values, in bold if statistically significant).