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A new method for the geometric modeling of lower pairs and its application to the kinematic spaces of spatial robots

A new notation is developed to model and analyze spatial robots. This method is formulated based on the homogeneous cylindrical coordinates and Bryant angles transformation matrices and thus termed “C-B notation.” At first, shape matrix for a binary link is derived. Then, the characteristic matrices for the five most commonly used kinematic lower pairs—revolute (R), prismatic (P), cylindrical (C), helical (H), and spherical (S)—are formulated. The “exact” joint positions, that is, the actual location of the physical joint center in space, can be defined using this method. The governing kinematic equations for the analysis of spatial robots are derived as well. Finally, the “kinematic spaces” for the open-chain robot are defined as “the group of space of work (displacement) space, velocity space (the velocity envelope in the displacement differential Vx-Vy-Vz coordinates), and acceleration space (the acceleration envelope in the velocity differential Ax-Ay-Az coordinates).” Numerical examples for parallel projections of kinematic spaces on both sagittal and frontal planes are illustrated by the industrial Unimate 2000 spherical (SP/RRR) robot, Bendix AA/CNC (RRP/RRR) robot, and 6R Cincinnati Milacron T3 robot.