Several methods and theories have been used to create a smooth trajectory for machine assisted physiotherapy . We have introduced a novel theory to create such a smooth trajectory. The mathematical model introduced in this paper can be used for different robotics projects to create this type of trajectories. This model has been used in a European project to deliver a system capable of correcting movement in machine assisted stroke rehabilitation. This system can be used for errorless learning techniques and intensive rehabilitation treatment for patient recovering post-stroke.