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.

翻译：矩阵李群为机器人学、计算机视觉和图形学等领域中的运动描述提供了一种语言。在使用这些工具时，我们常常面临将无穷级数表达式转化为更紧凑的有限级数（例如欧拉-罗德里格斯公式）的问题，这一过程有时颇为繁琐。本文识别了矩阵李群表达式中一些有用的积分形式，为计算紧凑的解析结果提供了一条更为简化的途径。此外，我们展示了这些积分形式中的一些递归结构，表明许多此类表达式是相互关联的。我们方法的关键在于，能够在推导过程的早期应用李代数的最小多项式，从而在整个推导过程中保持表达式的紧凑性。在级数方法中，最小多项式通常在最后才被应用，这使得结果中难以识别出共同的解析表达式。我们证明了我们的积分方法能够复现文献中多个由级数推导出的结果。