推导SE(3)切向算子的高阶导数闭式解，提升机器人与连续体仿真精度。
Closed Form Relations and Higher-Order Approximations of First and Second Derivatives of the Tangent Operator on SE(3)
- 不依赖分块结构，直接推导切向算子及其一、二阶导数的闭式表达
- 提出高阶近似公式，在弹性杆形变场与应变率计算中验证有效
- 适合需高精度微分的刚体运动建模与优化场景
SE(3)群用于建模多体系统、机器人及柯塞拉特连续体。其数值模拟与优化需用到指数映射及其右平凡化微分（常称切向算子）及高阶导数的闭式表达。本文在避免使用3×3分块结构的前提下，给出了切向算子、其一阶与二阶导数，以及评估映射\mathbf{dexp}_{\mathbf{X}}\mathbf{Z}和\mathbf{dexp}_{\mathbf{X}}^{T}\mathbf{Z}的雅可比与海森矩阵的闭式表达，并推导了所有量的高阶近似形式。相比传统方法，新表达更紧凑且结合局部近似时具有更强数值鲁棒性。通过弹性柯塞拉特-Simo-Reissner杆的形变场与应变率计算，验证了所提方法的有效性。
原文摘要 · Abstract (English)
The Lie group SE(3) of isometric orientation preserving transformation is used for modeling multibody systems, robots, and Cosserat continua. The use of these models in numerical simulation and optimization schemes necessitates the exponential map, its right-trivialized differential (often referred to as tangent operator), as well as higher derivatives in closed form. The $6\times 6$ matrix representation of the differential, $\mathbf{dexp}_{\mathbf{X}}:se\left( 3\right) \rightarrow se\left( 3\right) $ , and its first derivative were reported using a $3\times 3$ block partitioning. In this paper, the differential, its first and second derivative, as well as the Jacobian and Hessian of the evaluation maps, $\mathbf{dexp}_{\mathbf{X}}\mathbf{Z}$ and $\mathbf{dexp}_{\mathbf{X}}^{T}% \mathbf{Z}$, are reported avoiding the block partitioning. For all of them, higher-order approximations are derived. Besides the compactness, the advantage of the presented closed form relations is their numerical robustness when combined with the local approximation. The formulations are demonstrated for computation of the deformation field and the strain rates of an elastic Cosserat-Simo-Reissner rod.
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