Formulation for actuatorsʹ number enumeration for main planar structures in robotics

Mechanisms and Machines Theory (MMT) contributed greatly to planar and spatial mechanism synthesis with different degrees of freedom. Some of the current industrial robots with planar chains have a main structure created by the kinematic graphs of MMT. This paper deals with a formulation allowing computing the number of possibilities for actuatorsʹ attachment for planar pin-jointed main driving mechanisms in robotics. It is based on the sub-chains symmetries of structures taking into account the position of the actuator of the robot. Pairs of groups of mutually symmetrical mechanisms are detected. Thus, eliminating the symmetrical ones confines the number of possible configurations. Different cases of symmetries have been studied. Expressions for calculation of the number of actuator position of a mechanism are presented. They enable the reduction of the number of structures by avoiding those that are isomorphic. Following this, examples for applications for various planar pin-jointed kinematic structures are presented, enabling the field of research to be restricted to the possible solutions.