提出积分形式简化矩阵李群计算，让复杂公式更紧凑易懂。
Integral Forms in Matrix Lie Groups
- 用积分表达替代无穷级数，推导过程更简洁
- 发现积分形式间存在递归关联，揭示公式共性
- 适合机器人、视觉领域研究者快速推导运动模型
矩阵李群为机器人学、计算机视觉和图形学中的运动描述提供了语言。在使用这些工具时，常需将无穷级数表达转化为更紧凑的有限形式（如欧拉-罗德里格斯公式），但这一过程往往繁琐。本文识别出矩阵李群表达中一些有用的积分形式，为获得紧凑解析结果提供了更流畅的路径。此外，我们揭示了这些积分形式中的递归结构，表明许多表达式彼此关联。本方法的关键在于在推导早期即应用李代数的极小多项式，使表达始终紧凑。而传统级数法通常在最后才应用极小多项式，难以识别结果中的共性。我们证明该积分方法可复现文献中若干级数推导的结果。
原文摘要 · Abstract (English)
Matrix Lie groups provide a language for describing motion in such fields as robotics, computer vision, and graphics. When using these tools, we are often faced with turning infinite-series expressions into more compact finite series (e.g., the Euler-Rodrigues formula), which can sometimes be onerous. In this paper, we identify some useful integral forms in matrix Lie group expressions that offer a more streamlined pathway for computing compact analytic results. Moreover, we present some recursive structures in these integral forms that show many of these expressions are interrelated. Key to our approach is that we are able to apply the minimal polynomial for a Lie algebra quite early in the process to keep expressions compact throughout the derivations. With the series approach, the minimal polynomial is usually applied at the end, making it hard to recognize common analytic expressions in the result. We show that our integral method can reproduce several series-derived results from the literature.
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