Finite-step-integration: An original method for the forward kinematics analysis of parallel manipulators

Abstract

In today’s society, parallel manipulators (PMs) are widely used in industrial production, aerospace, and other fields, where their forward kinematic analyses often serve as the foundation for various tasks, such as design, calibration, and control. In the past few decades, this issue has seemingly been repeatedly addressed using various numerical methods, intelligent algorithms, and algebraic tools. While it is undeniable that solving the equations is easier with current technology, the problem of “how to formulate solvable equations” is often overlooked. This analysis issue typically involves establishing non-linear, multi-parameter, high-dimensional, and strong-coupled mathematical equations, which, from a geometric perspective, is also considered a process of solving a spatial polyhedron. When considering the temporal dimension of motion between two isomorphic polytopes, based on calculus theory, it has been found that this non-linear problem can be transformed into the superposition of multiple iteratively linear equations. Consequently, we propose an original method for the forward kinematic analysis of PMs, namely the finite-step-integration (FSI) method. In this study, the mathematical principles and modeling methods of the FSI method are elucidated, and the modeling and programming processes of the FSI method are demonstrated using general 6-UPS and 3-UPS/S manipulators as examples. Through the analysis of its unique algebraic structure, the methods for singularity determination and motion tracking characteristic analysis are investigated. This method addresses the long-standing challenges in the forward kinematic modeling of PMs, which is applicable for design, calibration, and control, while also offering novel insights for modeling and solving certain non-linear engineering problems.

Graphical abstract

Keywords

finite-step-integration / forward kinematic analysis / parallel manipulator / singularity determination / motion tracking

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Higher Education Press