I am reading about IMU error propagation and to pre integrate the IMU one uses the formula.
$$Exp(\phi + \delta) \approx Exp(\phi)Exp(J_r(\phi)\delta)$$
I want to understand where it comes from since it is much different from the standard exponential property(particularly I don't understand why the jacobian comes into play).
We'll use the Frobenius norm for matrices. Let's say that a series of matrices $$A_0+A_1+A_2+...$$converges absolutely if the series $$\vert \vert A_0\vert\vert+\vert \vert A_1 \vert\vert+\vert \vert A_2 \vert \vert +...$$ converges. It's easy to show that if series of matrices converges absolutely, it converges. Thus for any square real or complex matrix $M$, the series $$I+M+\frac{1}{2!}M^2+...$$ converges absolutely and thus converges. The matrix to which this series converges is defined to be $\exp(M).$ Let $A$ and $B$ be $n \times n$ matrices that commute with each other. For any real number $r$ let $$f_{\nu}(r)=1+r+...+\frac{1}{\nu!}r^{\nu}$$ For any square matrix $M$ let $$g_{\nu}(M)=I+M+...+\frac{1}{\nu!}M^{\nu}$$ Let $$(A,B)_{\mu}^{\lambda}=\text{binomial expansion of} (A+B)^{\lambda} \text {with first and last}\mu \text{terms omitted.}$$Then $$g_{n}(A)g_{n}(B)=g_{n}(A+B)+\frac{1}{(n+1)!}(A,B)_1^{n+1}+...+\frac{1}{(2n)!}(A,B)_n^{2n}$$ Thus $$\vert \vert g_n(A)g_n(B)-g_n(A+B) \vert \vert\le f_{2n}( \vert \vert A \vert \vert +\vert \vert B \vert \vert)-f_n(\vert \vert A \vert \vert +\vert \vert B \vert \vert)$$. Take the limit as $n \to \infty$ and it follows that $ \text{exp}(A)\text{exp}(B)=\text{exp}(A+B).$