🤖 AI Summary
This work addresses the complexity of derivative computation and algebraic expressions on matrix Lie groups in state estimation, which has long hindered algorithmic understanding and implementation. For the first time, it introduces tensor notation together with Einstein summation convention into the differential calculus of matrix Lie groups, integrating concepts from differential geometry to establish a concise and unified mathematical formalism. This framework substantially enhances the clarity and readability of derivative derivations and algebraic manipulations, thereby facilitating more intuitive comprehension and efficient implementation of gradient-based estimation algorithms.
📝 Abstract
Several recent papers have demonstrated the utility of using Lie groups within estimation problems, yielding improved accuracy and consistency. This paper introduces a new tool for describing operations with matrix Lie groups: tensors and the Einstein summation notation. While tensors and Einstein notation are well-known in other research fields, applying this mathematical notation to represent and compute matrix Lie derivatives is novel. More importantly, this new notation greatly clarifies the derivatives and operations necessary to work with matrix Lie Groups in (gradient-based) estimation frameworks. Therefore, the main contribution of this paper is not a new capability, but a more perspicuous mathematical notation for working with matrix Lie groups.