Generation of eight new classes of the 2UPS-UP mechanism with analytical forward kinematics solution

Abstract

The forward kinematic equation of a general 2UPS-UP mechanism is an 8-th degree polynomial that cannot be analytically solved. Two special cases that reduce the degree from 8 to 4 have been presented in the literature, while a systematical study on the generation of special classes of the mechanism with analytical forward kinematics solution is still lacking. This paper proposes an algebraic method to study this issue. The mechanism and its forward kinematic equations in a general case are introduced first. Then seven cases are drawn and discussed according to whether the coefficients in the trigonometric expressions are zero. By considering the singularity and inspecting the equation forms, several conditions and propositions are addressed to judge if the equations are analytical. Guided by the conditions and propositions, all possibilities of each case are studied to find the special classes with analytical forward kinematics solution. Totally 8 special classes are generated and the geometric conditions are demonstrated. An example is given to verify the results. The new discovery of this paper can deepen our understanding of the mechanism, thereby guiding the design of the mechanism.