Supervised robotic platforms, able to perform a non-invasive therapy or minimal invasive surgery, represent one of the main achievements in recent years. Robotic-assisted medical procedures with medical doctor, patient and medical assistants interacting with a robotic platform can be seen as a paradigmatic example of the coexistence between system autonomy and human action in medicine. However, this can involve unpredicted and dangerous contacts between robotic structures and humans, contacts that have to be managed with appropriate safety strategies, often embedding specific sensitive components into the robot itself. In this paper, a smart sensorized polymeric skin based on textile multi-touch piezoresistive sensors, able to sense and safely manage pressure exerted during a collision with the surrounding environment (e.g., humans), has been designed, fabricated, integrated on a robotic manipulator and tested. The proposed system shows promising results in managing the pressure exerted during the collision, with a close correlation with the analytical analysis (difference lower than 5.6 kPa - error of 9%).