Analytical Solvers for Common Algebraic Equations Arising in Kinematics Problems

TL;DR

This work develops analytical solvers for four algebraic equation forms commonly encountered in robot kinematics: a single trig equation $a \cos \theta + b \sin \theta + c = 0$, a single-angle system $A\cos\theta\sin\theta = \mathbf c$, a two-angle system $A\cos\theta_1\sin\theta_1 + B\cos\theta_2\sin\theta_2 = \mathbf c$, and a bilinear two-angle system $K\mathbf m = \mathbf 0$ with $\mathbf m = (1, c_1, s_1, c_2, s_2, c_1 c_2, c_1 s_2, s_1 c_2, s_1 s_2)^T$. It develops robust, closed-form or elimination-based methods (including the Weierstrass substitution $t=\tan(\theta/2)$ and Sylvester resultant techniques) to enumerate all real solutions, handling degeneracies such as singular matrices and free-angle cases. Implemented in Python and accompanied by LLM-assisted prompts, the solvers are validated across extensive benchmark cases, demonstrating sub-millisecond performance and complete solution coverage. This work provides reusable building blocks for inverse kinematics solvers across robotic architectures and emphasizes reproducibility through open-source prompts and code.

Abstract

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.