# How to update a new pose in iterative pose optimization problem?

There are multiple ways to update new pose in iterative pose optimization problem. The easiest one we often find in papers in robotics is as

SO3 + translation

Update on right $$\begin{split} \textbf {R}' &= \textbf{R} \textbf{e}^{[\boldsymbol{\omega}] _\times}, \textbf {t}' = \textbf{t} + \delta\textbf{t} \end{split}$$

Update on left $$\begin{split} \textbf {R}' &= \textbf{e}^{[\boldsymbol{\omega}] _\times}\textbf{R} , \textbf {t}' = \textbf{t} + \delta\textbf{t} \end{split}$$

where $\delta\textbf{t}\in R^3, \boldsymbol{\omega}\in R^3$ are estimated update on the pose with $\textbf{R}\in SO(3),\textbf{t}\in R^2$.

But there are other ways as well, such as

SE3 Matrix update

Right side update $\textbf {T}'=\delta\textbf{T}\textbf{T}$

$$\begin{split} \textbf {T}' &=\delta\textbf{T} \textbf{T}= \begin{bmatrix} \textbf{e}^{[\boldsymbol{\omega}] _\times}&\delta\textbf{t} \\ \textbf{0}&1 \end{bmatrix} \begin{bmatrix} \textbf{R}&\textbf{t} \\ \textbf{0}&1 \end{bmatrix} =\begin{bmatrix} \textbf{e}^{[\boldsymbol{\omega}] _\times}\textbf{R}&\textbf{} \delta\textbf{t}+\textbf{e}^{[\boldsymbol{\omega}] _\times}\textbf{t} \\ \textbf{0}&1 \end{bmatrix} \end{split} \label{eq:disp}$$

$$\begin{split} \textbf {T}' &= \textbf{e}^{[\boldsymbol{\xi}]_\times} \textbf{T}= \begin{bmatrix} \textbf{e}^{[\boldsymbol{\omega}] _\times}&\textbf{V} \delta\textbf{t} \\ \textbf{0}&1 \end{bmatrix} \begin{bmatrix} \textbf{R}&\textbf{t} \\ \textbf{0}&1 \end{bmatrix} =\begin{bmatrix} \textbf{e}^{[\boldsymbol{\omega}] _\times}\textbf{R}&\textbf{V} \delta\textbf{t}+\textbf{e}^{[\boldsymbol{\omega}] _\times}\textbf{t} \\ \textbf{0}&1 \end{bmatrix} \end{split}$$

Left side update $\textbf {T}'=\textbf{T}\delta\textbf{T}$

$$\begin{split} \textbf {T}' &= \textbf{T}\delta \textbf{T}=\begin{bmatrix} \textbf{R}&\textbf{t} \\ \textbf{0}&1 \end{bmatrix} \begin{bmatrix} \textbf{e}^{[\boldsymbol{\omega}] _\times}&\delta\textbf{t} \\ \textbf{0}&1 \end{bmatrix} =\begin{bmatrix} \textbf{R}\textbf{e}^{[\boldsymbol{\omega}] _\times}&\textbf{R}\delta\textbf{t}+\textbf{t} \\ \textbf{0}&1 \end{bmatrix} \end{split}$$

$$\begin{split} \textbf {T}' &= \textbf{T}\textbf{e}^{[\boldsymbol{\xi}] _\times}=\begin{bmatrix} \textbf{R}&\textbf{t} \\ \textbf{0}&1 \end{bmatrix} \begin{bmatrix} \textbf{e}^{[\boldsymbol{\omega}] _\times}&\textbf{V} \delta\textbf{t} \\ \textbf{0}&1 \end{bmatrix} =\begin{bmatrix} \textbf{R}\textbf{e}^{[\boldsymbol{\omega}] _\times}&\textbf{R}\textbf{V} \delta\textbf{t}+\textbf{t} \\ \textbf{0}&1 \end{bmatrix} \end{split}$$

se3 6x1 pose update($\boldsymbol{\mathfrak{I}}^{-1}$ could be left or right jacobian)

$$\boldsymbol{\xi}'=\boldsymbol{\xi} + \boldsymbol{\mathfrak{I}}^{-1}\boldsymbol{\xi}$$

My question is that which one is most stable or prefered. If you know any reference toward this problem, please let me know.

Thanks!