两个旋转矩阵相乘的李代数扰动求导

神气爱哥

已于 2025-02-12 20:12:02 修改

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文章标签：矩阵算法人工智能

于 2023-12-07 11:38:41 首次发布

本文链接：https://blog.youkuaiyun.com/qingcaichongchong/article/details/134850848

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有一些非常有意思的求导方式，特此记录下来

$\frac{\partial R_1R_2}{\partial R_1} =\underset{\phi\rightarrow0}{lim}{}\frac{log(R_1exp(\phi)R_2\ominus (R_1R_2))}{\phi} \displaystyle \\=lim\frac{log((R_1R_2)^TR_1exp(\phi)R_2)}{\phi} \\=lim\frac{log(R_2^Texp(\phi)R_2)}{\phi}\\ =lim\frac{log(exp(R_2^T\phi))}{\phi} \\=lim\frac{R_2^T\phi}{\phi}\\=R_2^T$

$\frac{\partial{R_1R_2}}{\partial{R_1}}=lim\frac{log(R_1exp(\phi)R_2)-log(R_1R_2)}{\phi}\\=lim\frac{log(R_1R_2exp(R_2^T\phi))-log(R_1R_2)}{\phi}\\=lim\frac{log(R_1R_2)+J_r^{-1}(R_1R_2)log(exp(R_2^T\phi))-log(R_1R_2)}{\phi}\\=lim\frac{J_r^{-1}(R_1R_2)R_2^T\phi}{\phi}\\=J_r^{-1}(R_1R_2)R_2^T$

所以看起来 $J_r^{-1}$ 是个小量，可以去掉的

$\frac{\partial{R_1R_2}}{\partial{R_2}}=lim\frac{log(R_1R_2exp(\phi)\ominus (R_1R_2))}{\phi}\\=lim\frac{log((R_1R_2)^TR_1R_2exp(\phi))}{\phi}\\=lim\frac{log(exp(\phi))}{\phi}\\=lim\frac{\phi}{\phi}\\=I$

$\frac{\partial{R_1R_2}}{\partial{R_2}}=lim\frac{log(R_1R_2exp(\phi))-log(R_1R_2)}{\phi}\\=lim\frac{log(R_1R_2)-J_r^{-1}(R_1R_2)log(exp(\phi))-log(R_1R_2)}{\phi}\\=lim\frac{J_r^{-1}(R_1R_2)\phi}{\phi}\\=J_r^{-1}(R_1R_2)$