提供四类机器人运动学常见代数方程的解析求解器，提升逆运动学求解效率与可靠性。
Analytical Solvers for Common Algebraic Equations Arising in Kinematics Problems
- 针对四类典型三角方程，给出闭式解与数值算法，支持一般系数与奇点处理。
- 可高效枚举所有实数解，验证表明方法准确且稳定，适用于多种机器人结构。
- 代码开源并可由语言模型单次提示生成，适合快速开发与集成到机器人系统中。
本文针对机器人运动学中常见的四类代数方程——单变量三角方程、单角三角系统、双角三角系统及双线性双角系统——提出解析求解方法。包括闭式表达式、数值算法与鲁棒性分析。求解器支持任意系数，能有效处理奇点并高效枚举全部实数解。所实现的Python工具包可在GitHub公开获取，并可通过提示语言模型（如OpenAI Codex 5.1）在单一指令下自动生成完整求解代码。多个示例提示已提供，便于在编程代理中复现。大量验证与基准测试证明了该方法的有效性与可靠性。本工作为构建各类机器人构型的完整逆运动学求解器奠定了基础。
原文摘要 · Abstract (English)
This paper presents analytical solvers for four common types of algebraic equations encountered in robot kinematics: single trigonometric equations, single-angle trigonometric systems, two-angle trigonometric systems, and bilinear two-angle systems. These equations arise frequently in the kinematics problems, particularly in robot kinematics. We provide detailed solution methods, including closed-form expressions, numerical algorithms, and robustness considerations. The solvers are designed to handle general coefficients, manage singularities, and enumerate all real solutions efficiently. These solvers are implemented in Python packages and can be reproduced by prompting Language Lanuage Models. Sampe prompts are also provided in the public code space Github repo. These prompts can generate a working solver code with one single prompt in coding agent such as OpenAI's Codex 5.1. This work serves as a foundation for developing complete inverse kinematics solvers for various robot architectures. Extensive validation and benchmarking demonstrate the effectiveness and reliability of the proposed methods.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。