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Errors dynamics captures the evolution of the state errors between two
distinct trajectories, that are governed by the same system rule but initiated
or perturbed differently. In particular, state observer error dynamics analysis
in matrix Lie group is fundamental in practice. In this paper, we focus on the
error dynamics analysis for an affine group system under external disturbances
or random noises. To this end, we first discuss the connections between the
notions of affine group systems and linear group systems. We provide two
equivalent characterizations of a linear group system. Such characterizations
are based on the homeomorphism of its transition flow and linearity of its Lie
algebra counterpart, respectively. Next, we investigate the evolution of a
linear group system and we assume it is diffused by a Brownian motion in
tangent spaces. We further show that the dynamics projected in the Lie algebra
is governed by a stochastic differential equation with a linear drift term. We
apply these findings in analyzing the error dynamics. Under differentiable
disturbance, we derive an ordinary differential equation characterizing the
evolution of the projected errors in the Lie algebra. In addition, the
counterpart with stochastic disturbances is derived for the projected errors in
terms of a stochastic differential equation. Explicit and accurate derivation
of error dynamics is provided for matrix group $SE_N(3)$, which plays a vital
role especially in robotic applications.
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