Log-linear Backstepping control on $SE_2(3)$
Li-Yu Lin, Benjamin Perseghetti, James Goppert
TL;DR
The paper develops a geometry-aware control framework for rigid-body dynamics on the Lie group $SE_2(3)$, preserving full rotational-translational coupling. It introduces a log-linear backstepping approach that yields exact log-error dynamics linear in the Lie algebra, leveraging explicit left/right Jacobian inverses. The resulting closed-loop design provides exponential stability and is compatible with LMIs or $H_\infty$-type performance criteria, with applicability to UAV and spacecraft control. Overall, it establishes a geometrically consistent backstepping methodology for a class of mixed-invariant systems on $SE_2(3)$, laying a foundation for future geometry-aware control 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.