The author in https://arxiv.org/pdf/1812.01537.pdf gives an explicitly formula for what he defines as the right-jacobian which is used to approximate the exponential of the sum of a rotation vector with a small perturbation.
$Exp(\tau + \delta \tau) = Exp(\textbf{J}\ _r(\tau)\delta\tau)Exp(\tau)$
The formula is given in equation 143,
$\textbf{J}_r(\theta) = \textbf{I} - \frac{1-cos(\theta)} {\theta^2} [\theta]_\times + \frac{\theta-sin(\theta)}{\theta^3}[\theta]^2_\times$
He references a book:
G. Chirikjian, Stochastic Models, Information Theory, and Lie Groups, Volume 2: Analytic Methods and Modern Applications, ser. Applied and Numerical Harmonic Analysis. Birkhauser Boston, 2011. [Online]. ¨ Available:
The book doesn't derive the formula and instead references a PhD thesis which I can't get it.
Can someone explain how to derive the formula?
It can't just be computing the jacobian of the matrix exponential of $[\theta]_\times$ using Rodrgiuez formula since I think that would produce a different result.
In the case of a general Lie algebra $\mathfrak g$ we have, for $\mathbf v\in\mathfrak g$, $$\mathbf{J}_r(\mathbf v) = \int_0^1\exp(s\,[\mathbf v,\cdot])\,ds$$ see e.g. this answer.
For your Lie algebra $\mathfrak{so}(3)=\mathbb R^3$, where the Lie bracket is the $\times$-product, the linear transformation (or $3\times3$-matrix) $\exp([\mathbf v,\cdot])=\exp(\mathbf v\times\cdot)$ is the rotation given by the vector $\mathbf v\in\mathbb R^3$. If $\mathbf v=\theta\mathbf u$, where $\mathbf u$ is a unit vector and $\theta>0$, we have $$\exp(\theta\,\mathbf u\times\cdot) = \mathbf I + \sin\theta\,(\mathbf u\times\cdot)+ (1-\cos\theta)(\mathbf u\times(\mathbf u\times\cdot)).$$ Therefore $$\mathbf{J}_r(\theta\,\mathbf u)= \int_0^1\exp(s\,\theta\,\mathbf u\times\cdot)\,ds $$ $$ = \mathbf{I} + \frac{1-\cos\theta}{\theta}(\mathbf u\times\cdot) + (1-\frac{\sin\theta}{\theta})(\mathbf u\times(\mathbf u\times\cdot)).$$