Error Dynamics in Affine Group Systems

Error dynamics captures the evolution of state errors between two distinct trajectories that are governed by the same dynamical system rule but initiated and/or perturbed differently. It is essential to analyze the error behavior of a state observer on a matrix Lie group for precise estimation. This article concentrates on the examination of error dynamics in affine group systems when exposed to external disturbances or random noises. To this end, we first characterize linear group systems equivalently by homomorphism of its transition flow and linearity of its Lie algebra. Next, we investigate linear group systems that are spread by a Brownian motion in tangent spaces. We show that the Itô stochastic differential equation in the Lie algebra space has a linear drift component and an additional “pinning” term. We apply these findings in analyzing error dynamics, a specific linear group system, with smooth disturbances and random noises. The error dynamics is expressed in the group and the Lie algebra. An explicit and accurate derivation of error dynamics is provided for a concrete example of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SE_{2}(3)$</tex-math></inline-formula>, which plays a vital role in inertial-based navigation and other robotics applications.