机器人的广义非线性和Finsler几何
Generalized Nonlinear and Finsler Geometry for Robotics
摘要：
机器人研究发现了黎曼几何的许多重要应用。尽管如此，这一概念对许多机器人专家来说仍然具有挑战性，因为背景材料复杂而且非常陌生。除了{ \em黎曼}几何之外，在数学文献中还有许多自然的概括--例如芬斯勒几何和喷雾几何--但这些概括在很大程度上是不可访问的，因此在机器人技术中的应用仍然很少。本文提出了一种重新推导的喷雾和芬斯勒几何形状，我们发现我们最近的工作发展的一个强大的行为设计工具，我们称之为几何织物的关键。这些推导建立在高级微积分和变分法的基本工具上，使它们比标准演示更容易被机器人观众所接受。我们专注于实用和可计算的结果，避免使用张量符号来吸引更广泛的受众，强调几何路径一致性的想法周围的连接和曲率。我们希望，这些推导将有助于增加了解广义非线性，甚至经典黎曼，几何机器人社区内，并激发未来的研究到新的应用。
英文摘要：
Robotics research has found numerous important applications of Riemannian geometry. Despite that, the concept remain challenging to many roboticists because the background material is complex and strikingly foreign. Beyond {\em Riemannian} geometry, there are many natural generalizations in the mathematical literature -- areas such as Finsler geometry and spray geometry -- but those generalizations are largely inaccessible, and as a result there remain few applications within robotics. This paper presents a re-derivation of spray and Finsler geometries we found critical for the development of our recent work on a powerful behavioral design tool we call geometric fabrics. These derivations build from basic tools in advanced calculus and the calculus of variations making them more accessible to a robotics audience than standard presentations. We focus on the pragmatic and calculable results, avoiding the use of tensor notation to appeal to a broader audience, emphasizing geometric path consistency over ideas around connections and curvature. We hope that these derivations will contribute to an increased understanding of generalized nonlinear, and even classical Riemannian, geometry within the robotics community and inspire future research into new applications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
