Lie Group Structure of SE(3)

• SE(3) is the Lie group of 3D rigid-body motions, combining rotations and translations through a semidirect product structure.
• Its Lie algebra, se(3), represents infinitesimal rotations and translations as twists using skew-symmetric matrices and the hat operator.
• Applications include geometric control, state estimation, trajectory planning, and equivariant deep learning using Jacobian-based optimization techniques.

The Special Euclidean group in dimension three, denoted SE(3), is the Lie group of orientation-preserving rigid-body motions in three-dimensional space. SE(3) forms the mathematical foundation for modern geometric approaches to robotics, state estimation, geometric control, and equivariant deep learning. The group structure encapsulates all 3D rotations and translations and underpins a suite of representation, inference, and optimization techniques ubiquitous in computational science and engineering.

1. Definition and Fundamental Structure

SE(3) is defined as the group of all rigid transformations in ℝ³: $$\mathrm{SE}(3) = \left\{\, g = \begin{bmatrix} R & p \ 0 & 1 \end{bmatrix} \,\middle|\, R \in \mathrm{SO}(3),\, p \in \mathbb{R}^3 \right\}$$ where $R$ is a rotation matrix and $p$ a translation vector. The group operation is matrix multiplication: $$g_1 g_2 = \begin{bmatrix} R_1 R_2 & R_1 p_2 + p_1 \ 0 & 1 \end{bmatrix}$$ The identity element is $$e = \begin{bmatrix} I_3 & 0 \ 0 & 1 \end{bmatrix}$$, and the inverse is $$g^{-1} = \begin{bmatrix} R^T & -R^T p \ 0 & 1 \end{bmatrix}$$ (Seo et al., 12 Mar 2025).

SE(3) is a 6-dimensional real matrix Lie group and forms a semidirect product $\mathbb{R}^3 \rtimes \mathrm{SO}(3)$, reflecting the fact that translations compose additively, while their composition is intertwined with the rotation action (Kumar et al., 2023, Gallo, 2022).

2. Lie Algebra se(3) and Algebraic Structure

The Lie algebra $\mathfrak{se}(3)$ consists of "twists," written in block form: $$\hat{\xi} = \begin{bmatrix} \hat{\omega} & v \ 0 & 0 \end{bmatrix}$$ where $\hat{\omega} \in \mathfrak{so}(3)$ is a 3×3 skew-symmetric matrix corresponding to angular velocity $\omega \in \mathbb{R}^3$, and $v \in \mathbb{R}^3$ corresponds to linear velocity (Seo et al., 12 Mar 2025, Hu, 4 Apr 2025).

The "hat" map $\xi \mapsto \hat{\xi}$ is an isomorphism between $\mathbb{R}^6$ and $\mathfrak{se}(3)$; the inverse map is called the "vee" operator.

The Lie bracket for two twists $\xi_1 = (\omega_1, v_1)$, $\xi_2 = (\omega_2, v_2)$ reads: $$[(\omega_1,v_1),\,(\omega_2,v_2)] = \left(\omega_1 \times \omega_2,\, \omega_1 \times v_2 - \omega_2 \times v_1\right)$$ This structure reflects the semidirect product and encodes the geometry of screw theory (Kumar et al., 2023, Luo et al., 2020).

A convenient basis is given by:

• $E_{\omega_i}$: rotation about axis $i$, with $i=1,2,3$
• $E_{v_j}$: translation along axis $j$, with $j=1,2,3$

with commutators: $$[E_{\omega_i}, E_{\omega_j}] = \varepsilon_{ij}^k E_{\omega_k} \quad [E_{\omega_i}, E_{v_j}] = \varepsilon_{ij}^k E_{v_k} \quad [E_{v_i}, E_{v_j}] = 0$$ where $\varepsilon_{ij}^k$ is the Levi-Civita symbol (Seo et al., 12 Mar 2025).

3. Exponential, Logarithm, and Jacobians

The exponential map $\exp: \mathfrak{se}(3) \to \mathrm{SE}(3)$ has a closed form: $$\exp \hat{\xi} = \begin{bmatrix} \exp \hat{\omega} & V(\omega) v \ 0 & 1 \end{bmatrix}$$ where $\exp \hat{\omega}$ is given by Rodrigues' formula and

$$V(\omega) = I + \frac{1 - \cos\theta}{\theta^2} \hat{\omega} + \frac{\theta - \sin\theta}{\theta^3} \hat{\omega}^2, \quad \theta = \|\omega\|$$

(Seo et al., 12 Mar 2025, Gallo, 2022).

The logarithm map $\log: \mathrm{SE}(3) \to \mathfrak{se}(3)$ is:

• Compute $\hat{\omega} = \log R$ via the axis-angle parameterization.
• Set $v = V(\omega)^{-1} p$.

The right and left Jacobians ($J_r$, $J_\ell$) of SE(3) linearize the exponential map and are essential for uncertainty propagation, state estimation, and geometric numerical integration. They have closed-form $6\times6$ block structure: $$J_r(\xi) = \begin{bmatrix} J_r^{SO(3)}(\omega) & 0 \ C_r(\omega, v) & J_r^{SO(3)}(\omega) \end{bmatrix}$$ with $J_r^{SO(3)}(\omega)$ the SO(3) right Jacobian and $C_r$ explicit from the series expansion or from closed form as detailed in (Sjøberg et al., 2020, Mueller, 2023). The inverse right Jacobian plays a prominent role in the propagation of the logarithm in filtering and estimation (Sjøberg et al., 2020, Mueller, 2023).

Derivative and Hessian expressions are critical for second-order optimization and variational integrators on SE(3), often invoked in geometric numerical integration and filtering schemes (Mueller, 2023).

4. Adjoint Representation and Group Actions

The adjoint representation $$\mathrm{Ad}_g : \mathfrak{se}(3) \to \mathfrak{se}(3)$$, encapsulating how twists transform under the group action, is given by: $$\mathrm{Ad}_g(\omega, v) = \left( R\omega,\; R v + p \times (R\omega) \right)$$ or, in $6 \times 6$ block matrix form,

$$\mathrm{Ad}_g = \begin{pmatrix} R & \widehat{p} R \ 0 & R \end{pmatrix}$$

where $g = (R, p)$ (Seo et al., 12 Mar 2025, Luo et al., 2020).

The adjoint is fundamentally used in geometric control, estimation, and in defining group-invariant error measures critical for Kalman-type filtering, log-linear invariant observer designs, and equivariant neural architectures (Chang, 2020, Barrau et al., 2022).

5. Applications: State Estimation, Control, and Deep Learning

State Estimation.

Filters such as the Invariant/Geometric Extended Kalman Filter (IEKF/GEKF) operate directly on SE(3) by modeling process and observation updates on the group and its algebra (Chang, 2020, Berger et al., 2015, Barrau et al., 2022). The Lie algebra structure enables invariant error definitions (left/right), log-linearization, and consistent propagation of uncertainty via the Jacobians (Sjøberg et al., 2020, Gallo, 2022).

Optimal Control.

Continuous-time LQR and potential shaping methods on SE(3) use canonical exponential coordinates, the derivatives of the exp/log maps, and left/right trivialized velocities to pose control designs and tracking intrinsically on the manifold, avoiding issues of coordinate singularities and double covering that arise with quaternions or Euler angles (Kumar et al., 2023, Wotte et al., 25 Jan 2024). These methods exploit the natural geometry for trajectory tracking, stabilization, and optimal control (Wotte et al., 25 Jan 2024).

Spline and Trajectory Generation.

Interpolations on SE(3) utilize product-of-exponentials (POE) splines, with explicit BCH truncation, to respect group geometry and prescribed kinematic constraints. Such approaches (e.g., POE splines) efficiently resolve high-order trajectory problems and ensure smoothness by operating in the algebra and reconstructing curves via the group exponential (Mueller, 14 Aug 2025).

Equivariant Deep Learning.

SE(3)-equivariant neural networks impose group-level symmetry by construction, with group actions realized via the adjoint and tensor representations. These networks leverage the equivariance of SE(3) in 3D vision, manipulation, and protein structure generation, yielding robust, sample-efficient architectures (Seo et al., 12 Mar 2025, Yim et al., 2023).

6. Metrics, Potentials, and Error Functions

Quantification of configuration errors on SE(3) is foundational for geometric control and state estimation. Two principal classes are:

• Group-based metric: Frobenius norm of the matrix difference (trace-based), $V_G(E) = \frac{1}{2}\|I-E\|_F^2$, with $E$ a configuration error on SE(3).
• Algebra-based metric: Quadratic form in the algebra, $V_A(E) = \frac{1}{2}\|\log(E)\|^2$, with more uniform feedback and absence of spurious minima for large motions (Seo et al., 24 Jan 2024).

Algebraic metrics are generally favored for large-angle tracking and almost-global convergence, while group-based norms are used for computational simplicity in many applications (Seo et al., 24 Jan 2024).

7. Spectral Theory, Canonical Decompositions, and Minimal Polynomials

The eigenstructure of SE(3) elements admits a principal-axis decomposition: every $T \in \mathrm{SE}(3)$ is similar to a block corresponding to a rotation and a translation along a unique axis (the Mozzi–Chasles theorem). The spectrum consists of $\lambda = e^{\pm i\phi}$ (rotation) and $\lambda = 1$ (translation), with the minimal polynomial reflecting the coupling of rotation and translation (D'Eleuterio et al., 2022). The adjoint representation shares this spectral decomposition, and all analytic functions of $T$ reduce to low-degree polynomials, providing efficient tools for uncertainty analysis and principal axis computations.

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These structural elements of SE(3)—group law, Lie algebra, exp/log maps, adjoints, and Jacobians—are foundational in contemporary geometric approaches to robotics, estimation, control, and machine learning (Seo et al., 12 Mar 2025, Hu, 4 Apr 2025, Wotte et al., 25 Jan 2024, Kumar et al., 2023, Barrau et al., 2022, D'Eleuterio et al., 2022). The consistent exploitation of these properties leads to algorithms and architectures that are coordinate-free, globally consistent, and robust to the physical symmetries inherent in 3D spaces.