An epicycloid or hypocycloid mechanism is capable of drawing an exact epicycloid or hypocycloid curve. Similar mechanism designs can be found abundantly in industrial machines or educational equipment. Currently, the major type of epicycloid or hypocycloid configurations is planetary gear trains, which contain a binary link that has one fixed and one moving pivot, and a singular link adjacent to the moving pivot. The main feature of the configurations is that any point on the singular link may describe an epicycloid or hypocycloid curve when the binary link is rotated. Presently, the major types of configurations of epicycloid (hypocycloid) mechanisms have one degree of freedom. However, at present, as far as the authors are concerned, there appears to be no approach in designing epicycloid (hypocycloid) mechanisms with two degrees of freedom. Thus, the main aim of this paper is to develop a new design method in designing new configurations of epicycloid (hypocycloid) mechanisms. This paper analyses the characteristics of the topological structures of existing planetary gear train type epicycloid (hypocycloid) mechanisms with one degree of freedom. The equation of motion and kinematical model of the mechanism was derived and appropriate design constraints and criteria were implemented. Subsequently, using the design constraints and criteria, this work designs a new and simple epicycloid (hypocycloid) mechanism that is a three-links robot and has two degrees of freedom. We can easily control the angular velocities of the binary and singular links to satisfy the criterion to draw an epicycloid (hypocycloid) curve. Additionally, an epicycloid (hypocycloid) path of a point on the three links robot is simulated by computer drawing to prove the feasibility of proposed theory. Finally, a prototype of three links robot for drawing an epicycloid (hypocycloid) path is done well. We know the methods of design and manufacture of the proposed epicycloid or hypocycloid mechanism in linkage is easily done.

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