Log-linear Backstepping control on $SE_2(3)$

TL;DR

This paper introduces a novel log-linear backstepping control law on SE(2,3) that maintains geometric consistency and guarantees exponential stability, advancing control design for UAVs and spacecraft.

Contribution

It develops an exact, geometrically consistent backstepping control framework on SE(2,3) using Lie group theory, with explicit error dynamics and stability guarantees.

Findings

Exact logarithmic error dynamics derived for SE(2,3)

Closed-form Jacobian inverses provided for precise control

Ensures exponential stability via LMI and $H_$ gain design

Abstract

Most of the rigid-body systems which evolve on nonlinear Lie groups where Euclidean control designs lose geometric meaning. In this paper, we introduce a log-linear backstepping control law on SE2(3) that preserves full rotational-translational coupling. Leveraging a class of mixed-invariant system, which is a group-affine dynamic model, we derive exact logarithmic error dynamics that are linear in the Lie algebra. The closed-form expressions for the left- and right-Jacobian inverses of SE2(3) are expressed in the paper, which provides us the exact error dynamics without local approximations. A log-linear backstepping control design ensures exponential stability for our error dynamics; since our error dynamics is a block-triangular structure, this allows us to use Linear Matrix Inequality (LMI) formulation or $H_\infty$ gain performance design. This work establishes the exact backstepping framework for a class of mixed-invariant system, providing a geometrically consistent foundation for future Unmanned Aerial Vehicle (UAV) and spacecraft control design.