## 2 Materials and Methods
### Setup and Protocol  
The experimentation was performed in a single session. The study consists of three different conditions (Figure 4):  
* Fixed Assistance. Fixed assistance level provided by the robot by the use of a force field. * Local Therapist Assistance. Assistance provided by the therapist. * Remote Therapist Assistance. Assistance provided by the therapist remotely (approach to a tele-rehabilitation environment). Figure 3: Overview of the game. (**a**) A screenshot of the game. (**b**) Condition to successfully reach the target, where \(d\) is the distance of the player cursor to the target and \(r\) is the minimum distance to reach the target successfully. (**c**) Condition to fail the reach of the target. (**d**) Movement of the robot in correlation with the proposed game. In this study, in addition to using a force-field-based assistance mode, two different master-slave strategies were implemented for the therapist to assist the patient. This technique is similar to that used in other studies [15]  
In both modes, the patient uses a Rubidium device configured with a control algorithm based on force fields (Figure 2a). This force field will help the patient to perform a linear trajectory but will not help the patient to reach the target. It will be the therapist who will assist the patient to reach the targets. In one case, the therapist applies forces on the end effector of the Rubidium device the patient is using (Local Therapist Assistance, Figure 5). For this purpose, the system was provided with a force sensor placed on the end effector to measure the force applied by the therapist to assist the patient. Figure 4: Graphical representation of the different assistance modes evaluated. Figure 5: Image of the experimental setup taken during the local and the remote therapist assistance condition. In the other case, the therapist interacts remotely with the patient through another Rubidium device (Remote Therapist Assistance, Figure 5). The Rubidium device is also provided with a force sensor on the end effector to measure the assistance force that the therapist wants to apply to the patient. The device with which the therapist interacts replicates in real-time the current position of the patient. The forces applied by the therapist will not be able to move the position of the robot with which he is interacting. Figure 6 summarizes the study protocol carried out during the experimentation. Before starting the session, the patient is placed in front of the robotic device in a comfortable position. Firstly, we carry out a familiarization period where we explain how the game works and take the opportunity to make a general evaluation of the patient to establish some parameters such as the range of movement, the maximum time to reach the targets, or the magnitude of the force fields. This parameter will be used to configure the force fields and establish an adequate assistance level in the case of the fixed assistance mode. After the familiarization period, we start with the experimental session. The conditions of the study were performed in random order. In each of them, the patient will have to perform 32 trials, and in each trial, the patient will have to randomly reach one of the 8 roulette targets before the time is up and then return to the center. Before each trial, the patient must hold the player cursor (Figure 3) within the center of the roulette for 2 s. Then, one of the circles on the outer edge of the roulette lights up, indicating the next target to be reached. The time to complete the movement was adapted based on the initial evaluation of the patient, and once the target has been reached (or the time has expired), the patient has to go back to the roulette center. For this movement, there is no time limit. Each of the conditions lasts about 7-8 min. After each condition, the patient will have a 3-5 min rest. This rest period is also used to fill in the questionnaires. All the sessions were carried out with the same therapist so that the strategy for assisting patients is the same.