## 2 Materials & Methods
### Software and Control
The movements of the first two joints are functions of the desired horizontal abduction and abduction, respectively, thus depending on J3 and J4:  
\[\theta_{j1}=f\left(\theta_{j\lambda},\ j_{1 coupling}\right)\]  
\[\theta_{j2}=f\left(\theta_{j\lambda},\ j_{2 coupling}\right)\]  
where \(\theta_{ji}\) represents the angular position of joint Ji, for \(i=1,\ \ldots,\ 4\), while \(j_{1 coupling}\) defines the coupling level between J1 and J3 and its value is 9, whereas \(j_{2 coupling}\) defines the relation between J2 and J4 and its value is 4. The corresponding functions can be defined as shown below:  
\[\theta_{j1}=\frac{\theta_{j3}}{j_{1 coupling}}\]  
\[\theta_{j2}=\left\{ \begin{array}{cl}0&if\ \theta_{j4}<threshold\\ \frac{\theta_{j4}}{j_{2 coupling}}&if\ \theta_{j4}>threshold\end{array} \right.\]  
The coupling between J2 and J4 is applied exclusively after the elevation in the coronal plane raises over the threshold value of 30\({}^{\circ}\), thus making J4 work alone in the early phases of abduction to better emulate the physiological movement. In Transparency mode the patient is active and can freely move the limb (and thus the exoskeleton) without feeling resistance thanks to a friction compensation at the joints. This function enables the patient to perform exercises by interacting with daily life objects (Figure 7). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Exercise** & **Min Angle [deg]** & **Max Angle [deg]** \\ \hline sHA/HA & 0 & 100 \\ sF/E & 30 & 90 \\ sA/A & 15 & 110 \\ sIER & \(-\)55 & 55 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: List of the ROM for the physiological exercises. During transparency, it is possible to activate the additional features Assistance and Support. Assistance helps the patient in performing upwards vertical motions by applying an extra push which depends on J4 current angle. Support provides a viscous resistance in downwards vertical motions, in order to support the weight of the arm and avoid rapid descending movements. These modalities intervene on joint 4 and their intensity can be selected between three values: low, medium and high. Empirical values for these levels were chosen over the average weight of a human arm. Transparency mode is also activated to record an arbitrary trajectory under the guidance of the therapist. In this operational mode, the therapist freely moves the patient's limb following the desired trajectory, which is recorded by the device in terms of motor positions and can be subsequently replicated (Repeat Trajectory mode). Transparency is here intended as compensation of friction torques at each single joint, by means of a suitably applied motor torque as a function of the dynamic variables. In Float device, transparency is achieved by means of a model-based feed-forward compensation of friction at each of the involved joints, which is described below. Despite being sub-optimal, this technique proved very well suited for the application. Inertial effects are intrinsically minimized in the application, as the movements performed during therapy are designed to be slow and continuous in order to minimize accelerations and avoid patient discomfort. The equation of motion referred to the output (slow) shaft of the single joint gearmotor can be written as follows:  
\[\tau_{ext}+\tau_{f}+\tau_{mot}=\mathrm{J}\dot{\omega}\]  
where:  
1. \(\tau_{ext}\) is the external torque acting on the joint output shaft, due to contact interaction with the rest of the system via downstream connecting link, and to inertial forces due to accelerated motion of the joint base. 2. \(\tau_{f}\) is the total friction torque in motor and gearbox, output shaft referred. 3. \(\tau_{mot}\) is the electromagnetic torque exerted on motor rotor by motor stator, multiplied by gearbox gear ratio. 4. J is the total moment of inertia of motor rotor plus gearbox, output shaft referred. 5. \(\dot{\omega}\) is the angular velocity of the output shaft. Figure 7: Transparency mode taking advantage of the polyarticulated arm to perform activities of daily living.