CGA-Based Approach to Forward Kinematics of Parallel Mechanisms with the 3-RE Structure

• DOI: 10.1186/s10033-025-01223-9
• Publication Date: 2025-05-13T10:00:26Z
• Authors: Duanling Li; Yuankai Zhang; Ying Zhang; Zhonghai Zhang; Longjie Fan; Xiao Su; Shuaimin Gao

Abstract To investigate the forward kinematics problem of parallel mechanisms with complex limbs and to expand the applicability of the powerful tool of Conformal Geometric Algebra (CGA), a CGA-based modeling and solution method for a class of parallel platforms with 3-RE structure after locking the actuated joints is proposed in this paper. Given that the angle between specific joint axes of limbs remains constant, a set of geometric constraints for the forward kinematics of parallel mechanisms (PM) are determined. After translating unit direction vectors of these joint axes to the common starting point, the geometric constraints of the angle between the vectors are transformed into the distances between the endpoints of the vectors, making them easier to handle. Under the framework of CGA, the positions of key points that determine the position and orientation of the moving platform can be intuitively determined by the intersection, division, and duality of basic geometric entities. By employing the tangent half-angle substitution, the forward kinematic analysis of the parallel mechanisms leads to a high-order univariate polynomial equation without the need for any complex algebraic elimination operations. After solving this equation and back substitution, the position and pose of the MP can be obtained indirectly. A numerical case is utilized to confirm the effectiveness of the proposed method.