A method for the forward kinematic analysis of a class of 2 rotations and 1 translation parallel mechanisms based on conformal geometric algebra

Abstract

For a four-limb 2R1T parallel mechanism consisting of three coplanar limbs and one spatial limb, determining its exact position often involves complex elimination and cumbersome calculations. This paper introduces a conformal geometric algebra framework, integrating rigid-body motion operators with intersection operations of geometric elements, and proposes a differentiated forward kinematics solution strategy based on geometric feature classification. Based on the geometric perpendicular relationship between the line connecting the moving-platform articulation points of the passive coplanar limb and the spatial limb, and the common plane of the three coplanar limbs, this class of mechanisms is divided into two subclasses, namely mechanisms without perpendicular constraints and mechanisms with perpendicular constraints. For mechanisms without perpendicular constraints, a direct analytical solution method that does not require elimination has been developed. For mechanisms with perpendicular constraints, a low-order matrix can be constructed via resultant elimination to solve the problem, which also significantly reduces the scale of the elimination process and the difficulty of the solution. A comparative analysis under identical conditions shows that the proposed method, which combines rigid-body motion operators, is more efficient than the geometric intersection operations method alone. Taking the two spatial configurations 2UPU-2SPR as examples, the eight sets of real-number solutions obtained from the numerical examples are in close agreement with the 3D simulation results, thereby validating the correctness and reliability of the method. This work also provides a new approach that can be applied to the efficient solution of forward kinematics for other complex parallel mechanisms.