Closed Form Relations and Higher-Order Approximations of First and Second Derivatives of the Tangent Operator on SE(3)

2026-04-24

AI Summary

This work addresses the challenge of high-fidelity mechanical modeling of multibody systems, robotic mechanisms, and Cosserat continua on the special Euclidean group SE(3), where existing approaches rely on block-matrix representations and lack compact closed-form expressions for the tangent operator and its higher-order derivatives. The authors derive block-free closed-form formulas for the SE(3) tangent operator along with its first and second derivatives, and provide analytical expressions for the Jacobian and Hessian of the associated evaluation maps, as well as higher-order approximations. Built upon the exponential map, right-trivialized differentials, and local Taylor expansions, the proposed formulation ensures both numerical robustness and computational efficiency. Its accuracy and stability are demonstrated through applications to deformation fields and strain-rate computations in elastic Cosserat–Simo–Reissner rods.

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Abstract

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.

Problem

• SE(3)
• tangent operator
• higher-order derivatives
• closed-form expressions
• numerical robustness

Innovation

• SE(3)
• tangent operator
• closed-form derivatives
• higher-order approximations
• Cosserat rod