SE₂(3) Lie Group Framework Overview

• SE₂(3) is a Lie group defined as a 5×5 matrix structure that unifies attitude, velocity, and position with exact group operations.
• Its exponential and logarithm maps along with closed-form Jacobians enable precise log-linear error propagation and robust covariance analysis.
• Applications include rigorous Kalman filtering, LQR, MPC, and safety-certified control for spacecraft, UAVs, and integrated navigation systems.

The SE₂(3) Lie group framework is a matrix-Lie-group formalism that unifies rotational, translational, and velocity states in a single geometric space, enabling globally consistent modeling, analysis, and control of rigid-body systems with full attitude, velocity, and position information. Originally motivated by the limitations of classical local linearizations in navigation and spacecraft control, SE₂(3) allows exact log-linear error propagation and robust filter/controller synthesis based on linear systems theory, while strictly preserving the underlying manifold structure.

1. Definition and Algebraic Structure

SE₂(3) is defined as the set of 5×5 real matrices encoding the triplet (attitude, velocity, position) as follows: $$X = \begin{bmatrix} R & v & p \ 0_{1\times3} & 1 & 0 \ 0_{1\times3} & 0 & 1 \end{bmatrix}$$ where $R \in SO(3)$ is a rotation matrix, $v \in \mathbb{R}^3$ is velocity, and $p \in \mathbb{R}^3$ is position. The group law is ordinary matrix multiplication, which couples rotations, velocities, and positions. The identity and inverse are standard: the inverse is

$$X^{-1} = \begin{bmatrix} R^T & -R^T v & -R^T p \ 0_{1\times3} & 1 & 0 \ 0_{1\times3} & 0 & 1 \end{bmatrix}$$

The Lie algebra $\mathfrak{se}_2(3)$ consists of matrices

$$\xi^{\wedge} = \begin{bmatrix} [\xi_R]_\times & \xi_v & \xi_p \ 0_{1\times3} & 0 & 0 \ 0_{1\times3} & 0 & 0 \end{bmatrix}$$

with the commutator serving as the Lie bracket.

2. Exponential, Logarithm, and Jacobians

The exponential map on SE₂(3) is given by

$$\exp(\xi^{\wedge}) = \begin{bmatrix} \exp([\xi_R]_\times) & J_r(\xi_R)\xi_v & J_r(\xi_R)\xi_p \ 0_{1\times3} & 1 & 0 \ 0_{1\times3} & 0 & 1 \end{bmatrix}$$

where $J_r$ is the right-Jacobian of SO(3). The logarithm is computed by extracting rotation, then solving for velocity and position components using the inverse Jacobian. These exponential/logarithm mappings enable exact conversion between group and algebra coordinates for nonlinear system analysis and uncertainty propagation.

Closed-form expressions for the left- and right-Jacobians, including block-triangular structure, have been established, permitting analytic evaluation of derivatives required for covariance propagation and linearization of error dynamics (Lin et al., 8 Nov 2025, Brossard et al., 2020).

3. Group-Affine Dynamics and Log-Linear Error Propagation

A key property enabling SE₂(3)–based control and filtering is the group-affine formulation of rigid-body kinematics: $\dot{X} = (M - C)X + X(N + C)$ where $M, N$ are elements of the Lie algebra, and $C$ is a constant coupling matrix. With proper encoding of gravity, thrust, and angular velocity, the true system dynamics (whether for spacecraft, UAV, or navigation) are mapped to this form (Condie et al., 5 Dec 2025, Lin et al., 8 Nov 2025).

Defining a left-invariant error as

$\eta = X_d^{-1}X, \quad \xi = \log(\eta)$

the error propagation in the Lie algebra coordinates becomes

$\dot{\xi} = A(t)\xi$

where $A(t)$ is determined by the reference trajectory and system parameters. Under suitable feed-forward compensation and construction, all trajectory-dependent terms (e.g., gravity mismatch, Coriolis in non-inertial frames) can be eliminated, yielding exact log-linear (autonomous) error dynamics irrespective of the system state (Condie et al., 5 Dec 2025, Cui et al., 22 Jan 2026, Cui et al., 22 Jan 2026). This allows all classical linear analysis tools (LQR, H₂/H∞, LMIs, convex safety certification) to be directly applied.

4. SE₂(3) Framework in Filtering and Smoothing

Embedding navigation and sensor-fusion states on SE₂(3) enables rigorous Kalman filtering and smoothing. The EKF operates by lifting IMU increments to SE₂(3), propagating covariance via the adjoint operator and Jacobians, and performing measurement updates in the group via the exponential map:

• Prediction: propagate state by IMU or process model as group-affine flow,
• Covariance: propagate by linearized adjoint or Jacobian,
• Correction: form innovation on the manifold and update by group-exponential lifting (Cui et al., 22 Jan 2026, Luo et al., 2021).

Autonomous error propagation is preserved only if velocity is represented in the inertial frame; otherwise, state-dependent Coriolis and curvature terms break log-linearity (Cui et al., 22 Jan 2026).

5. Control Synthesis on SE₂(3): LQR, MPC, and Backstepping

Control architectures leveraging SE₂(3) benefit from globally linear error models:

• Left-invariant LQR designs use the group error to linearize the dynamics and solve Riccati equations for time-varying or constant-gain feedback (Silveria et al., 19 Nov 2025).
• Model Predictive Control (MPC) optimizes constrained trajectories by propagating the error in SE₂(3)-algebra coordinates; both LQR and MPC designs exploit the block structure of the error ODE (Silveria et al., 19 Nov 2025).
• Log-linear backstepping yields exponentially stable, block-triangular error systems. The use of closed-form Jacobians permits systematic LMI or H∞ synthesis for robust gain selection (Lin et al., 8 Nov 2025).

Empirical results demonstrate significant improvements in trajectory tracking, robustness and computational efficiency for UAVs and spacecraft in simulation and hardware (Silveria et al., 19 Nov 2025).

6. Applications: Satellite Docking, Navigation, and Safety Certification

The SE₂(3) framework is foundational in:

• Satellite proximity operations and docking: globally valid linear error propagation supports convex reachable set computation and safety guarantees (Condie et al., 5 Dec 2025).
• Integrated inertial navigation (SINS/ODO, IMU/LiDAR fusion): group-affine EKF and smoothing on SE₂(3) handle Earth rotation, Coriolis and sensor bias effects exactly, yielding improved real-world and Monte-Carlo performance (Cui et al., 22 Jan 2026, Brossard et al., 2020, Luo et al., 2021).
• Robust controller synthesis and safety: all convex design and certification methods for linear systems apply directly in the SE₂(3)–error coordinates, eliminating the limitations of local-linearization ([TH]/[YA], HCW models) (Condie et al., 5 Dec 2025).

7. Extensions and Broader Context

The SE₂(3) Lie group framework generalizes to SEₖ(3) for higher-order kinematic chains and admits full group-algebra convolutional signal processing, supporting deep learning models using invariant filter banks (Kumar et al., 2023). Recent results in generative modeling leverage the algebraic decomposition of SE₂(3) to train universal score-matching networks and flow-matching policies for data residing on general Lie groups (Bertolini et al., 4 Feb 2025). This suggests broad utility in domains requiring equivariant learning, bridge-sampling, and structure-preserving generative manipulation.

The SE₂(3) group thus provides a rigorous and practical geometric foundation for modeling, estimation, and control of rigid-body systems where coupled rotation, translation, and velocity must be treated as unified state. The log-linear property, supported by group-affine dynamics and closed-form Jacobians, is central in unlocking global linearization and the full suite of modern control and estimation techniques.