Analytical form solution of the direct kinematics of a 4-4 fully in-parallel actuated six degree-of-freedom mechanism

This paper presents the direct position analysis in analytical form of a six-degree-of-freedom 4-4 fully-parallel mechanism. For a given set of actuator displacements the mechanism becomes a structure and the analysis finds all the possible closures of the structure. The analysis is performed in two steps. First, the two closures of the tetrahedron-like subchain of the structure are found. Then, for each tetrahedron closure, two transcendental equations are determined that represent the closure of the remaining part of the 4-4 structure. The two equations can he reduced to algebraic equations and, after eliminating the unwanted unknowns, a final 8th order equation in only one unknown is obtained. Hence, the maximum number of possible real closures of the 4-4 structure is sixteen. Numerical examples are reported which illustrate and confirm the new theoretical result.