Clarification on proposition 2

[stefangachter]

Thanks a lot for providing the code on github and the paper on arxiv. I am studying the paper carefully and got stuck in section B on conditional infinitesimal symmetries of GVIO. If I understand it correctly, proposition 2 addresses the dot product of the pseudo-range rate measurements. It derives the infinitesimal symmetry of the pseudo-range rate measurements. What about the norm of the pseudo-range measurements? Is the infinitesimal symmetry of pseudo-range given by the one of the rate?

[ChangwuLiu]

Hi, thanks for your comments. Proposition 2 addresses the pseudo range and pseudo range rate measurements simultaneously. Verifications of the infinitesimal symmetries of the IMU process model and the visual update model are the same as the case in VIO (infinitesimal invariance is a subset of total invariance in VIO), and thus they're omitted. The remaining work in symmetry analysis is to discover whether the pseudo range and the pseudo range rate equations are infinitesimally invariant under the action of some group H. H must be a subgroup of S (the semi-direct product of yaw and 3-DoF translations). I first tackle the translations, this is intuitive: you can first take a translational direction $t_g$ from $null(N_g)$，then consider separately disturbing the position $p$ in the pseudo-range and pseudo-range rate measurement in Eq (4) along a curve $p\mapsto p+\lambda t_g$ ($\lambda$ being the curve parameterization). Taking the derivative along this curve, you can find both the pseudo-range and pseudo-range rate equation remain unchanged to first order (definition of infinitesimal invariance). A little more tricky one is the yaw, but the rationales are the same. Consider both the pseudo-range and pseudo-range rate equations. Rotating the IMU position $p$ and velocity $v$ along the gravity direction in the pseudo-range and pseudo-range rate measurement in Eq (4) is equivalent to disturbing the state $(p,v)$ on the state manifold along a curve $(p,v)\mapsto(\exp(\lambda g^\times)p,\exp(\lambda g^\times)v)$. (Sorry for the slight typo in the arxiv draft, will be corrected.) Taking the derivative with respect to the curve parameterization $\lambda$ and letting the result be zero yields the conditions for yaw to be unobservable, as shown in the proposition. In all, two easy steps: (a) disturbing the $(p,v)$ along a curve on the state manifold parameterized by $\lambda$ (b) taking the derivative to verify the infinitesimal symmetry for such measurement model. The two steps are separately conducted for pseudo-range and pseudo-range rate equations.

[stefangachter]

Sorry for my late reply and thanks a lot for the detailed explanation which I have yet to digest. I have to consult once more the paper and compare it with your explanation.