Dynamics on Lie groups with applications to attitude estimation

The paper proves that for group-affine dynamics f on a matrix Lie group G, one can write f(e^X) = e^X J_g(X) D X + e^X Y1 + Y2 e^X (Theorem 1), which implies the left-trivialized error X obeys the linear ODE Ẋ = (D + adY2)X and, consequently, concentrated Gaussians TG(μ,Σ) are invariant with Σ̇ = (D + adY2)Σ + Σ(D + adY2)^T (Corollary 1) . The candidate solution follows the same backbone: (i) derive ṙ = f(r) − r f(I) for r = g μ^{-1} using the group-affine identity, (ii) substitute the Theorem 1 form of f and use the standard commutator–Jacobian identity to obtain ṙ = e^X J_g(X) (D + adY2)X, (iii) choose Ẋ = (D + adY2)X so that r(t) = e^{X(t)}, and (iv) push forward the initial TG distribution to get the covariance ODE stated in the corollary. These steps reproduce the paper’s equations (15)–(20) and Corollary 1 essentially verbatim, with only minor stylistic differences (the paper derives (17) then invokes Theorem 1 to reach (20); the model uses the same identities directly) .