I have struggling with a problem of evaluating the gradient of a cost function on the Lie group of rotations: SO(3). The cost is the following:
\begin{equation} V_M(R)=\frac{1}{2}⟨\log(R),M\log(R)⟩,\;\;\;R\in\mathrm{SO(3)}\end{equation} for some matrix $M=M^{\mathsf T}>0$.
The inner product is given by $⟨A,B⟩:=\operatorname{tr}(A^{\mathsf T} B)$. The logarithm function on SO(3) is defined to be: \begin{equation} \log(R)=\frac{\theta}{2\sin(\theta)}(R-R^{\mathsf T})\in\mathfrak{so}(3) \end{equation} where $\theta$ represents the angle of rotation ($R:=\exp(\theta[u]_\times)=I+\sin(\theta)[u]_\times+(1-\cos(\theta))[u]_\times^2$ )
I have tried to use the fact that for some $X\in T_R\mathrm{SO(3)}$, we have \begin{equation} dV[X]=⟨\nabla V,X⟩ \end{equation} but I was not able even to determine the directional derivative $dV[X]$...
Thank you very much for your help!