Integral Forms in Matrix Lie Groups

2025-03-04

AI Summary

Matrix Lie group operations in robotics, computer vision, and graphics traditionally rely on cumbersome infinite series expansions, hindering analytical tractability, interpretability, and computational efficiency. Method: This paper introduces a novel analytical paradigm based on closed-form integral representations. It systematically replaces power series with compact integral expressions; embeds minimal polynomials of the Lie algebra early in derivation to preserve expression conciseness; and exploits recursive structures and algebraic relationships among integral kernels. Contribution/Results: The approach successfully recovers classical results—including the Euler–Rodrigues formula—and significantly simplifies derivations of analytical solutions in rigid-body kinematics, dynamics, and related domains. It enhances symbolic computation in terms of interpretability, consistency, and efficiency. By unifying treatment across matrix Lie groups, the method provides a more concise, general, and theoretically grounded framework for analytical Lie group modeling.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Visual Reasoning & Symbolic Representations
• Intelligent Robots: Motion and Path Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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-Rodriques 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.

Problem

• Simplify infinite-series expressions in matrix Lie groups
• Identify integral forms for compact analytic computations
• Use minimal polynomial early to maintain expression compactness

Innovation

• Integral forms streamline matrix Lie group computations.
• Recursive structures reveal interrelations in integral expressions.
• Early minimal polynomial application maintains compact derivations.