A Robust Numerical Method for Solving Trigonometric Equations in Robotic Kinematics

2025-08-31

AI Summary

This paper addresses numerical instability in solving trigonometric equation systems arising in robotic inverse kinematics. We propose a highly robust hybrid analytical-numerical method. Its core innovation lies in integrating the Weierstrass substitution with geometric-constraint-driven singular value decomposition (SVD) to uniformly handle both nonsingular and singular configurations; further, polynomial substitution, eigenvalue decomposition, and numerical algebraic modeling enable stable reconstruction of full analytical solutions. Evaluated on over one thousand standard and boundary-case benchmarks, the method achieves machine precision (error < 1×10⁻¹⁵) with 100% solution success rate—significantly outperforming conventional numerical iterative and symbolic elimination approaches. The algorithm is released as a lightweight, open-source Python solver, supporting real-time inverse kinematics computation and singularity analysis.

Technology Category

• Intelligent Robots: State Estimation
• Natural Language Processing: Safety and Robustness
• Computer Vision: Motion & Tracking

Application Category

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• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper presents a robust numerical method for solving systems of trigonometric equations commonly encountered in robotic kinematics. Our approach employs polynomial substitution techniques combined with eigenvalue decomposition to handle singular matrices and edge cases effectively. The method demonstrates superior numerical stability compared to traditional approaches and has been implemented as an open-source Python package. For non-singular matrices, we employ Weierstrass substitution to transform the system into a quartic polynomial, ensuring all analytical solutions are found. For singular matrices, we develop specialized geometric constraint methods using SVD analysis. The solver demonstrates machine precision accuracy ($< 10^{-15}$ error) with 100% success rate on extensive test cases, making it particularly valuable for robotics applications such as inverse kinematics problems.

Problem

• Solving trigonometric equations in robotic kinematics
• Handling singular matrices with specialized methods
• Ensuring numerical stability and machine precision accuracy

Innovation

• Polynomial substitution with eigenvalue decomposition
• Weierstrass substitution for quartic transformation
• SVD-based geometric constraint methods