Key Takeaways & Executive Findings
- •• Introduces a novel CGA-based method for forward kinematics of parallel mechanisms with 3-RE structure, eliminating complex algebraic elimination. • Transforms angular constraints into distance constraints, simplifying geometric modeling and solution. • Achieves a high-order univariate polynomial equation via tangent half-angle substitution, enabling efficient and coordinate-invariant solutions. • Validates the approach with a numerical case, demonstrating its effectiveness for complex limb structures.
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.
1. Introduction
Traditional parallel mechanisms (PMs) with fixed degrees of freedom have been extensively studied, such as the Stewart platform [1]. However, certain tasks do not require a 6-degree of freedom (DOF) fully PM. A robotic manipulator with fewer than 6-DOF can fulfill the requirements, offering advantages such as reduced structural complexity, design redundancy, and cost savings. Simultaneously, there is a growing expectation for robots and mechanisms in manufacturing to possess adaptable structures and mobility, enabling them to accommodate diverse operational environments and meet varying task requirements. This has led to the development of low-DOF PMs [2–4] and reconfigurable parallel mechanisms (RPMs) [5–7].
This paper focuses on a class of parallel mechanisms composed of special limbs. Fang et al. [8] synthesized two types of 5-DOF serial chains based on screw theory. The chains providing a constraint force were denoted as F-limbs and others providing a constraint couple were denoted as C-limbs, and several PMs with 3, 4, and 5-DOF were assembled by them [9]. Li et al. [10], using group theory obtained similar results. Ye et al. [11] extracted the common part of C-limbs and F-limbs and named them FC-limbs. Combining an FC-limb with a kinematotropic linkage, a hybrid kinematics chain with variable constraints was obtained and then used to synthesize a class of reconfigurable PMs. Similar approaches are also found in the works of Tian et al. [12] and Jia et al. [13], which share analogous principles. These studies provide a method for synthesizing mechanisms based on task requirements, enabling the design of limbs according to the desired DOF and motion modes of the moving platform (MP).
The forward kinematics of PMs has consistently been an extremely crucial but meaningful issue. However, for the aforementioned mechanisms, the relatively complex limb structures make their forward kinematics problems challenging. One must contend with issues such as the difficulty in intuitively identifying spatial relationships among geometric elements used to determine constraint equations, as well as the complexity involved in simplifying and solving multivariate high-order equation systems. Therefore, there are also fewer research outcomes related to this issue.
The essence of the forward kinematics problem in PMs lies in the solution of a system of nonlinear equations. The complexity of nonlinear systems of equations is closely tied to the modeling method and significantly influences the difficulty.
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Duanling Li, Yuankai Zhang, Ying Zhang, Zhonghai Zhang, Longjie Fan, Xiao Su, Shuaimin Gao (2025). CGA-Based Approach to Forward Kinematics of Parallel Mechanisms with the 3-RE Structure. Chinese Journal of Mechanical Engineering. https://doi.org/10.1186/s10033-025-01223-9
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Frequently Asked Questions
What is the main contribution of this paper?
The paper proposes a CGA-based method for solving forward kinematics of parallel mechanisms with 3-RE structure, which avoids complex algebraic elimination and provides a coordinate-invariant solution.
How does the method simplify the forward kinematics problem?
It transforms angular constraints into distance constraints and uses tangent half-angle substitution to derive a high-order univariate polynomial equation, simplifying the solution process.
What are the advantages of using Conformal Geometric Algebra?
CGA provides intuitive geometric entities and operations, enabling straightforward modeling of spatial relationships and reducing the complexity of solving nonlinear equations.
Is the method applicable to other parallel mechanisms?
While demonstrated on 3-RE structures, the approach can be extended to other PMs with similar geometric constraints, offering a general framework for forward kinematics.
What is the significance of the numerical case presented?
The numerical case validates the effectiveness of the proposed method, confirming its accuracy and applicability to practical parallel mechanisms.
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