Closed Form Relations and Higher-Order Approximations of First and Second Derivatives of the Tangent Operator on SE(3)
TLDR
This paper provides closed-form expressions and higher-order approximations for the tangent operator and its derivatives on SE(3) without block partitioning.
Key contributions
- Provides closed-form expressions for the tangent operator's first and second derivatives on SE(3).
- Derives Jacobian and Hessian of evaluation maps `dexp_X Z` and `dexp_X^T Z`.
- Avoids 3x3 block partitioning, leading to more compact and robust numerical methods.
- Includes higher-order approximations for all derived closed-form relations.
Why it matters
Accurate and robust derivatives of the tangent operator on SE(3) are crucial for numerical simulations and optimization in robotics and multibody systems. This work provides compact, numerically robust closed-form solutions, improving the efficiency and stability of these applications.
Original 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.
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