This paper demonstrates a novel geometric modeling and computational method of the family of spatial parallel mechanisms (PMs) with 3-R(P)S structure for direct kinematic analysis based on the point pair relationship. The point pair relationship, which is derived from the framework of conformal geometric algebra (CGA), consists of the relationship between the point and the point pair and two point pairs. The first research is on the distance relationship between the point and the point pair. Second, the derivation of the distance relationship between two point pairs is based on the aforementioned result, which shows the mathematical homogeneity. Third, two formulations for a point of the point pairs that satisfy the distance relationship between two point pairs are reduced. Fourth, the point pair relationship is applied to solve the direct kinematic analysis of the spatial parallel mechanism with 3-R(P)S structure. Finally, four numerical examples are provided to verify the validity of the proposed algorithm. Overall, the proposed method can be generalized for the direct kinematics of a series of spatial parallel mechanisms with 3-R(P)S structure.

This paper presents a novel geometric modeling method for direct displacement analysis of 6-4 Stewart platforms based on conformal geometric algebra (CGA). First, a geometric constraint relationship of four lines and a plane intersecting at a point is published. Second, a new coordinate-invariant geometric constraint equation of 6-4 Stewart platforms is deduced by CGA operation. Third, five polynomial equations are established by CGA theory. Fourth, based on the above six equations, a 5 × 5 Sylvester’s matrix is formulated by using Sylvester’s Dialytic elimination method and Gröbner bases method under the graded reverse lexicographical order. Finally, the coordinates of four points on the moving platform are revealed. Besides, a numerical example is used to prove the validity of the proposed method. The novelty of this study is that a whole geometric modeling method by geometric constraint relationship of four lines and a plane intersecting at a point is put forward under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. At the same time, Sylvester’s matrix constructed by this method is the smallest one in the known literature for forward displacement analysis of 6-4 Stewart platforms.