The elbow joint is a complex articulation composed of the humeroulnar and humeroradial joints (for flexion-extension movement) and the proximal radioulnar articulation (for pronation-supination movement). During the flexion-extension movement of the elbow joint, the rotation center changes and this articulation cannot be truly represented as a simple hinge joint. The main goal of this project is to design and assemble a medical rehabilitation exoskeleton for the elbow with one degree of freedom for flexion-extension, using the rotation center for proper patient elbow joint articulation. Compared with the current solutions, which align the exoskeleton axis with the elbow axis, this offers an ergonomic physical human-robot interface with a comfortable interaction. The exoskeleton is actuated with shape memory alloy wire-based actuators having minimum rigid parts, for guiding the actuators. Thanks to this unusual actuation system, the proposed exoskeleton is lightweight and has low noise in operation with a simple design 3D-printed structure. Using this exoskeleton, these advantages will improve the medical rehabilitation process of patients that suffered stroke and will influence how their lifestyle will change to recover from these diseases and improve their ability with activities of daily living, thanks to brain plasticity. The exoskeleton can also be used to evaluate the real status of a patient, with stroke and even spinal cord injury, thanks to an elbow movement analysis.