# Analysis of Dynamic Behavior of ParReEx Robot Used in Upper Limb Rehabilitation
## 2 Constructive and Functional Description of the ParReEx-Wrist Robot  
The ParReEx-wrist robotic system is designed for the rehabilitation of the wrist, namely flexion and extension of the wrist and ulnar and radial deviation of the wrist [49]. The robot has two degrees of freedom (DoF), one for each type of motion (Figure 1), namely: the revolute joint \(\mathrm{q}_{1}\) to perform flexion-extension and the revolute joint \(\mathrm{q}_{2}\) to perform radial-ulnar deviation. Each joint separately actuates a kinematic chain as follows: \(\mathrm{q}_{1}\) actuates an RR chain consisting of the \(\mathrm{R}_{1}\) and \(\mathrm{R}_{3}\) revolute joints connected through link \(\mathrm{d}\) around the OZ axis; \(\mathrm{q}_{2}\) actuates link \(e\) around the OY axis, fixed between two bearings. Links \(\mathrm{d}\) and \(\mathrm{e}\) are connected via two revolute joints, namely \(\mathrm{R}_{2}\) and \(\mathrm{R}_{3}\), required to perform the circumduction motion (which is a combination of flexion-extension and ulnar-radial deviation motions). Notations used in Figure 1 represents: \(\mathrm{q}_{1}\)--flexion-extension motion; \(\mathrm{q}_{2}\)--radial-ulnar deviation motion; \(\mathrm{R}_{1}\), \(\mathrm{R}_{2}\), \(\mathrm{R}_{3}\), \(\mathrm{R}_{4}\)--revolute joints; \(\mathrm{e}\), \(\mathrm{d}\)--mechanism links.  
The ParReEx-wrist CAD model is presented in Figure 2. Motor (1) achieves the flexion-extension motion of the wrist through a set of bevel gears (3) and a custom-made circular rail and sledge guiding (4). Motor (2) performs radial-ulnar deviation using a timing belt and pulleys (5). The patient's hand is supported using the wrist support (6), which acts as active anchor, and the forearm support (7), which acts as passive anchor. Since the patient's  
Figure 1: The ParReEx-wrist kinematic scheme [47].  
wrist is placed above the fixed coordinate system (Figure 1), as the intersection between the flexion-extension and radial-ulnar deviation robot motion axis, an additional straight rail-and-sledge mechanism (8) is required, as well as a spherical joint (9) to perform the circumduction motion.  
Notations used in Figure 2 represents: 1--motor for flexion-extension motion; 2--motor for radial-ulnar deviation motion; 3--bevel gears; 4--sledge pulleys; 5--belt transmission; 6--wrist support; 7--forearm support; 8--straight rail and sledge mechanism; 9--spherical joint.  
The ParReEx-wrist experimental model is presented in Figure 3. Based on the virtual prototype of the ParReEx robot, the execution drawings for the component parts of the assembly were made. The housing of the robotic system and the component parts, such as those numbered 6, 7 and 8 in Figure 2, are manufactured by additive manufacturing. For this purpose, we used a 3D printer, which works on the fused deposit manufacturing principle. For the execution of the printed parts, we used 3D models, and, using Cura Ultimaker software, we made the G-code for 3D printing. The printer uses ABS (acrylonitrile butadiene styrene) wire, which is melted by the print head, and successive layers are deposited to achieve the shape of the parts. Thus, these parts are assembled to realize the prototype of the robotic system, as shown in Figure 3, where the robotic system is shown in the situation of its use by a patient.  
Figure 3: The ParReEx-wrist experimental model.  
Figure 2: The ParReEx-wrist CAD model.