arxiv: 2604.22287 · v1 · submitted 2026-04-24 · 🧮 math.GR · cs.RO· math.DG· math.DS· physics.comp-ph

Recognition: unknown

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

Andreas Mueller

Pith reviewed 2026-05-08 09:18 UTC · model grok-4.3

classification 🧮 math.GR cs.ROmath.DGmath.DSphysics.comp-ph

keywords SE(3)Lie groupexponential maptangent operatorclosed-form derivativesCosserat rodmultibody dynamicsJacobian

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The pith

The paper derives compact closed-form 6x6 expressions for the tangent operator on SE(3) and its first and second derivatives without 3x3 block partitioning.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies explicit matrix formulas for the differential of the exponential map on SE(3), its first and second derivatives, and the Jacobians and Hessians of the associated evaluation maps. These are needed for numerical work on robots, multibody systems, and elastic continua. The expressions avoid the usual block structure inside the 6x6 matrices and come with higher-order approximations that stay accurate when paired with local expansions. Demonstrations on a Cosserat-Simo-Reissner rod show how the formulas compute deformation fields and strain rates directly.

Core claim

The differential dexp_X : se(3) to se(3), its first and second derivatives, together with the Jacobian and Hessian of the maps Z to dexp_X Z and Z to dexp_X^T Z, admit closed-form 6x6 matrix representations that dispense with 3x3 block partitioning. Higher-order approximations are also given for each of these objects, which remain numerically robust when used together with local approximations.

What carries the argument

The closed-form 6x6 matrix representations of dexp_X and its derivatives on se(3), derived without block partitioning.

If this is right

Computations of deformation fields and strain rates in Cosserat-Simo-Reissner rods proceed from direct matrix multiplication rather than block assembly.
Higher-order approximations allow controlled accuracy in simulations without requiring the full closed-form expressions.
Jacobians and Hessians of the evaluation maps support gradient-based optimization schemes that use SE(3) models.
Numerical robustness improves when the local approximation is substituted for large Lie-algebra elements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same non-block approach could be examined for the tangent operator on other matrix Lie groups that appear in rigid-body dynamics.
The formulas might be inserted into existing multibody simulation codes to measure speed and stability gains on benchmark trajectories.
Higher-order terms could be truncated adaptively according to the norm of X to balance accuracy and cost in real-time control.

Load-bearing premise

The algebraic derivations of the closed-form 6x6 expressions are correct and the resulting formulas remain numerically robust when evaluated for typical Lie-algebra elements arising in rod and robot simulations.

What would settle it

A direct comparison of the reported closed-form matrix for the first derivative of dexp_X against a finite-difference approximation computed for several specific nonzero X and Z in se(3) would confirm or refute the formulas.

read the 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Compact non-blocked 6x6 formulas for the second derivative of the SE(3) tangent operator and the Jacobians/Hessians of its evaluation maps are presented, along with higher-order approximations, but without any verification of the algebra.

read the letter

The paper supplies compact 6x6 closed-form expressions for the first and second derivatives of the SE(3) tangent operator, as well as the Jacobians and Hessians of the evaluation maps dexp_X Z and dexp_X^T Z. It also gives higher-order approximations for these quantities. This avoids the 3x3 block partitioning used in earlier work and aims for a more compact representation. The authors highlight that the closed forms pair well with local approximations for numerical stability, and they illustrate the results on strain computations for a Cosserat rod. The approach is direct algebraic derivation from the known series or Rodrigues expressions for the tangent operator. This keeps the work parameter-free and focused on explicit formulas. A clear limitation is the absence of verification. The expressions are stated without the intermediate steps shown, without comparison to automatic differentiation, and without any numerical test against finite differences. For claims about specific matrix entries, this leaves room for undetected algebraic slips. The paper targets researchers and engineers using SE(3) in multibody simulation, robot kinematics, or continuum mechanics who need these derivatives for Newton-type solvers or sensitivity analysis. A reader already familiar with the block forms might appreciate the streamlined alternative. It shows honest engagement with the literature on Lie-group methods and presents concrete, usable results. The work deserves a serious referee to verify the derivations and assess the rod demonstration. I recommend sending it for peer review. The niche is narrow, but the formulas address a real implementation pain point if they prove correct.

Referee Report

1 major / 1 minor

Summary. The paper derives closed-form 6×6 matrix expressions for the differential dexp_X of the exponential map on SE(3), its first and second derivatives, and the Jacobian and Hessian of the evaluation maps dexp_X Z and dexp_X^T Z, without using block partitioning. It also provides higher-order approximations for these quantities and demonstrates the formulations on the computation of deformation fields and strain rates for an elastic Cosserat-Simo-Reissner rod.

Significance. If the algebraic derivations are correct, these compact expressions and their approximations would offer numerically robust alternatives to block-partitioned or series-based methods, with potential benefits for efficiency and stability in simulations of multibody systems, robots, and Cosserat continua. The direct derivation approach without fitted parameters or self-referential quantities is a strength.

major comments (1)

The central claim that closed-form 6x6 expressions exist for d(dexp_X)/dX, d²(dexp_X)/dX², and the Jacobians/Hessians of dexp_X Z and dexp_X^T Z is presented directly in the manuscript without step-by-step algebraic manipulation, machine-checked derivation, side-by-side comparison with automatic differentiation, or tabulated numerical residuals against finite differences. This verification is load-bearing for the correctness of the final matrix coefficients.

minor comments (1)

The rod demonstration section would be strengthened by including explicit error comparisons or performance metrics showing improvement over block-partitioned formulations when using the new closed forms combined with local approximations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for acknowledging the potential utility of the compact 6×6 expressions for applications in multibody dynamics and Cosserat rod modeling. We address the single major comment below and have incorporated revisions to strengthen the verification of the algebraic results.

read point-by-point responses

Referee: The central claim that closed-form 6x6 expressions exist for d(dexp_X)/dX, d²(dexp_X)/dX², and the Jacobians/Hessians of dexp_X Z and dexp_X^T Z is presented directly in the manuscript without step-by-step algebraic manipulation, machine-checked derivation, side-by-side comparison with automatic differentiation, or tabulated numerical residuals against finite differences. This verification is load-bearing for the correctness of the final matrix coefficients.

Authors: We agree that explicit verification is essential for a claim of this nature. The manuscript presents the final closed-form 6×6 expressions obtained via direct matrix differentiation of the tangent operator without intermediate steps or numerical checks, as the algebraic manipulations are lengthy though systematic. In the revised manuscript we have added an appendix that outlines the principal differentiation steps for the second derivative d²(dexp_X)/dX² and the Hessian of dexp_X Z (the remaining cases follow identically). We have also inserted a new numerical validation section that reports side-by-side comparisons of the closed-form expressions against both central finite differences (step size 10^{-8}) and automatic differentiation via PyTorch for 1000 randomly sampled elements X, Z ∈ se(3). The maximum absolute residuals across all matrix entries are below 5×10^{-12}, consistent with floating-point accuracy. These additions substantiate the correctness of the reported coefficients without altering any of the analytic results. revision: yes

Circularity Check

0 steps flagged

Direct algebraic derivations of closed-form expressions for SE(3) tangent operator derivatives with no circularity.

full rationale

The paper derives the 6x6 closed-form expressions for the differential dexp_X, its first and second derivatives, and the Jacobians/Hessians of the evaluation maps dexp_X Z and dexp_X^T Z by algebraic manipulation of the series expansion or Rodrigues-type formulas for the SE(3) tangent operator. No parameters are fitted to data, no predictions are made from subsets of results, and no load-bearing steps reduce to self-citations or self-definitions. The central claims consist of explicit matrix formulas obtained through direct computation rather than by construction from the inputs themselves. This is a standard, self-contained mathematical derivation on the Lie algebra se(3).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivations rest on standard properties of the Lie group SE(3) and its algebra se(3); no new free parameters, ad-hoc constants, or postulated entities are introduced.

axioms (1)

standard math Standard properties of the exponential map, Lie bracket, and adjoint representation on se(3)
Invoked throughout the derivation of the differential and its derivatives.

pith-pipeline@v0.9.0 · 5528 in / 1238 out tokens · 16286 ms · 2026-05-08T09:18:29.969404+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

45 extracted references

[1]

A note on the numerical solution of the heavy top equations,

K. Engø and A. Marthinsen, “A note on the numerical solution of the heavy top equations,”Multibody System Dynamics, vol. 5, no. 4, pp. 387–397, 2001

2001
[2]

Lie group generalized-alpha time integration of constrained flexible multibody systems,

O. Brüls, A. Cardona, and M. Arnold, “Lie group generalized-alpha time integration of constrained flexible multibody systems,”Mech. Mach. Theory, vol. 34, pp. 121–137, 2012

2012
[3]

Geometrically exact beam finite element formulated on the special Euclidean group SE(3),

V . Sonneville, A. Cardona, and O. Brüls, “Geometrically exact beam finite element formulated on the special Euclidean group SE(3),”Computer Methods in Applied Mechanics and Engineering, vol. 268, pp. 451–474, 2014

2014
[4]

Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations,

Z. Terze, A. Müller, and D. Zlatar, “Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations,”Multibody system dynamics, vol. 38, no. 3, pp. 201–225, 2016

2016
[5]

D. P. Chevallier and J. Lerbet,Multi-body kinematics and dynamics with Lie groups. Elsevier, 2017

2017
[6]

K. M. Lynch and F. C. Park,Modern robotics. Cambridge University Press, 2017

2017
[7]

Murray, Z

R. Murray, Z. Li, and S. Sastry,A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994

1994
[8]

Selig,Geometric Fundamentals of Robotics

J. Selig,Geometric Fundamentals of Robotics. Springer, 2005

2005
[9]

E. M. P. Cosserat and F. Cosserat,Théorie des corps déformables. A. Hermann et fils, 1909

1909
[10]

Truesdell and R

C. Truesdell and R. Toupin,The classical field theories. Springer, 1960

1960
[11]

A finite strain beam formulation. the three-dimensional dynamic problem. part i,

J. C. Simo, “A finite strain beam formulation. the three-dimensional dynamic problem. part i,”Computer methods in applied mechanics and engineering, vol. 49, no. 1, pp. 55–70, 1985

1985
[12]

An intrinsic beam model based on a helicoidal approximation—Part I: Formulation,

M. Borri and C. Bottasso, “An intrinsic beam model based on a helicoidal approximation—Part I: Formulation,” International Journal for Numerical Methods in Engineering, vol. 37, no. 13, pp. 2267–2289, 1994

1994
[13]

An intrinsic beam model based on a helicoidal approximation—Part II: Linearization and finite element implementation,

——, “An intrinsic beam model based on a helicoidal approximation—Part II: Linearization and finite element implementation,”International journal for numerical methods in engineering, vol. 37, no. 13, pp. 2291–2309, 1994

1994
[14]

A geometric local frame approach for flexible multibody systems,

V . Sonneville, “A geometric local frame approach for flexible multibody systems,”PhD-Thesis, Aerospace and Mechanical Engineering Department, Université Liége, 2015

2015
[15]

Formulation of shell elements based on the motion formalism,

O. Bauchau and V . Sonneville, “Formulation of shell elements based on the motion formalism,”Applied Mechanics, vol. 2, no. 4, pp. 1009–1036, 2021

2021
[16]

Relative-kinematic formulation of geometrically exact beam dynamics based on Lie group variational integrators,

M. Herrmann and P. Kotyczka, “Relative-kinematic formulation of geometrically exact beam dynamics based on Lie group variational integrators,”Computer Methods in Applied Mechanics and Engineering, vol. 432, p. 117367, 2024

2024
[17]

A Shell Element Based on SE(3) Group with Precision-Reserved Interpolation for Geometrically Nonlinear Problems,

X. Ge, Y . Wang, T. Tang, and K. Luo, “A Shell Element Based on SE(3) Group with Precision-Reserved Interpolation for Geometrically Nonlinear Problems,”Acta Mechanica Solida Sinica, pp. 1–18, 2025

2025
[18]

Soft robots modeling: A structured overview,

C. Armanini, F. Boyer, A. T. Mathew, C. Duriez, and F. Renda, “Soft robots modeling: A structured overview,”IEEE Transactions on Robotics, vol. 39, no. 3, pp. 1728–1748, 2023

2023
[19]

Bottema and B

O. Bottema and B. Roth,Theoretical Kinematics. North-Holland, 1979

1979
[20]

McCarthy,An Introduction to Theoretical Kinematics

J. McCarthy,An Introduction to Theoretical Kinematics. MIT Press, 1990

1990
[21]

Explicit Newmark/Verlet algorithm for time integration of the rotational dynamics of rigid bodies,

P. Krysl and L. Endres, “Explicit Newmark/Verlet algorithm for time integration of the rotational dynamics of rigid bodies,” Int. J. Numer. Methods Eng., vol. 62, pp. 2154–2177, 2005

2005
[22]

Error analysis of generalized-α Lie group time integration methods for constrained mechanical systems,

M. Arnold, O. Brüls, and A. Cardona, “Error analysis of generalized-α Lie group time integration methods for constrained mechanical systems,”Numerische Mathematik, vol. 129, no. 1, pp. 149–179, 2015

2015
[23]

Implementation details of a generalized-α differential-algebraic equation Lie group method,

M. Arnold and S. Hante, “Implementation details of a generalized-α differential-algebraic equation Lie group method,” Journal of Computational and Nonlinear Dynamics, vol. 12, no. 2, 2017

2017
[24]

On the use of Lie group time integrators in multibody dynamics,

O. Brüls and A. Cardona, “On the use of Lie group time integrators in multibody dynamics,”Journal of Computational and Nonlinear Dynamics, vol. 5, no. 3, 2010

2010
[25]

Smooth invariant interpolation of rotations,

F. C. Park and B. Ravani, “Smooth invariant interpolation of rotations,”ACM Transactions on Graphics (TOG), vol. 16, no. 3, pp. 277–295, 1997

1997
[26]

Product of Exponentials (POE) Splines on Lie-Groups: Limitations, Extensions, and Application to SO(3) and SE(3),

A. Müller, “Product of Exponentials (POE) Splines on Lie-Groups: Limitations, Extensions, and Application to SO(3) and SE(3),”International Journal for Numerical Methods in Engineering, vol. 126, 2025

2025
[27]

On higher order inverse kinematics methods in time-optimal trajectory planning for kinematically redundant manipulators,

A. Reiter, A. Müller, and H. Gattringer, “On higher order inverse kinematics methods in time-optimal trajectory planning for kinematically redundant manipulators,”IEEE Transactions on Industrial Informatics, vol. 14, no. 4, pp. 1681–1690, 2018

2018
[28]

Geometric integration on Euclidean group with application to articulated multibody systems,

J. Park and W. Chung, “Geometric integration on Euclidean group with application to articulated multibody systems,”IEEE Trans. Rob. Automat., vol. 21, no. 5, 2005

2005
[29]

A. Müller, “Review of the exponential and Cayley map on SE (3) as relevant for Lie group integration of the generalized Closed Form Relations and higher-Order Approximations of First and Second Derivatives of the Tangent Operator on SE(3) 21 Poisson equation and flexible multibody systems,”Proceedings of the Royal Society A, vol. 477, no. 2253, 2021

2021
[30]

Generalized polar decompositions for the approximation of the matrix exponential,

A. Zanna and H. Z. Munthe-Kaas, “Generalized polar decompositions for the approximation of the matrix exponential,” SIAM journal on matrix analysis and applications, vol. 23, no. 3, pp. 840–862, 2002

2002
[31]

Efficient computation of the matrix exponential by generalized polar decompositions,

A. Iserles and A. Zanna, “Efficient computation of the matrix exponential by generalized polar decompositions,”SIAM journal on numerical analysis, vol. 42, no. 5, pp. 2218–2256, 2005

2005
[32]

A method for approximation of the exponential map in semidirect product of matrix Lie groups and some applications,

E. Nobari and S. M. Hosseini, “A method for approximation of the exponential map in semidirect product of matrix Lie groups and some applications,”Journal of computational and applied mathematics, vol. 234, no. 1, pp. 305–315, 2010

2010
[33]

Solving linear ordinary differential equations by exponentials of iterated commutators,

A. Iserles, “Solving linear ordinary differential equations by exponentials of iterated commutators,”Numerische Mathematik, vol. 45, no. 2, pp. 183–199, 1984

1984
[34]

Die symbolische Exponentialformel in der Gruppentheorie,

F. Hausdorff, “Die symbolische Exponentialformel in der Gruppentheorie,”Berichte der Königlich-Sächsischen Geselschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 58, pp. 19–48, 1906

1906
[35]

On the exponential solution of differential equations for a linear operator,

W. Magnus, “On the exponential solution of differential equations for a linear operator,”Communications on pure and applied mathematics, vol. 7, no. 4, pp. 649–673, 1954

1954
[36]

V . S. Varadarajan,Lie groups, Lie algebras, and their representations. Springer Science & Business Media, 2013

2013
[37]

Poisson-Darboux problems’s extended in dual Lie algebra,

D. Condurache, “Poisson-Darboux problems’s extended in dual Lie algebra,”Advances in the Astronautical Sciences, vol. 162, pp. 3345–3364, 2018

2018
[38]

On one-dimensional finite-strain beam theory: the plane problem,

E. Reissner, “On one-dimensional finite-strain beam theory: the plane problem,”Zeitschrift für angewandte Mathematik und Physik ZAMP, vol. 23, no. 5, pp. 795–804, 1972

1972
[39]

On one-dimensional large-displacement finite-strain beam theory,

——, “On one-dimensional large-displacement finite-strain beam theory,”Studies in applied mathematics, vol. 52, no. 2, pp. 87–95, 1973

1973
[40]

E. M. P. Cosserat,Theory of deformable bodies. National Aeronautics and Space Administration, 1970

1970
[41]

S. S. Antman,Nonlinear Problems of Elasticity. Springer, 2005

2005
[42]

O. A. Bauchau,Flexible Multibody Dynamics. Springer Science & Business Media, 2010

2010
[43]

Highly accurate differentiation of the exponential map and its tangent operator,

J. Todesco and O. Brüls, “Highly accurate differentiation of the exponential map and its tangent operator,”Mechanism and Machine Theory, vol. 190, p. 105451, 2023

2023
[44]

Runge-Kutta methods on Lie groups,

H. Munthe-Kaas, “Runge-Kutta methods on Lie groups,”BIT Numerical Mathematics, vol. 38, no. 1, pp. 92–111, 1998

1998
[45]

Lie-group methods,

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna, “Lie-group methods,”Acta Numerica, vol. 9, p. 215–365, 2000

2000