From 908ba707062fdbf698101ab1d4a29f84697ddf39 Mon Sep 17 00:00:00 2001 From: Gerry Chen Date: Sat, 16 Oct 2021 03:34:13 -0400 Subject: [PATCH] documentation --- doc/math.lyx | 179 +++++++++++++++++++++++++++++++++++++++++++++++++++ doc/math.pdf | Bin 261727 -> 264527 bytes 2 files changed, 179 insertions(+) diff --git a/doc/math.lyx b/doc/math.lyx index 2533822a7..f14ebc66f 100644 --- a/doc/math.lyx +++ b/doc/math.lyx @@ -5084,6 +5084,185 @@ reference "ex:projection" \end_layout +\begin_layout Subsection +Derivative of Adjoint +\end_layout + +\begin_layout Standard +Consider +\begin_inset Formula $f:SE(3)\rightarrow\mathbb{R}^{6}$ +\end_inset + + is defined as +\begin_inset Formula $f(T)=Ad_{T}y$ +\end_inset + + for some constant +\begin_inset Formula $y=\begin{bmatrix}\omega_{y}\\ +v_{y} +\end{bmatrix}$ +\end_inset + +. + Defining +\begin_inset Formula $\xi=\begin{bmatrix}\omega\\ +v +\end{bmatrix}$ +\end_inset + + for the derivative notation (w.r.t. + pose +\begin_inset Formula $T$ +\end_inset + +): +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +f'(T)=\frac{\partial Ad_{T}y}{\partial\xi}=\begin{bmatrix}\frac{\partial f}{\omega} & \frac{\partial f}{v}\end{bmatrix} +\] + +\end_inset + +Then we can compute +\begin_inset Formula $f'(T)$ +\end_inset + + by considering the rotation and translation separately. + To reduce confusion with +\begin_inset Formula $\omega_{y},v_{y}$ +\end_inset + +, we will use +\begin_inset Formula $R,t$ +\end_inset + + to denote the rotation and translation of +\begin_inset Formula $T\exp\xi$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\frac{\partial}{\partial\omega}\begin{bmatrix}R & 0\\{} +[t]_{\times}R & R +\end{bmatrix}\begin{bmatrix}\omega_{y}\\ +v_{y} +\end{bmatrix}=\frac{\partial}{\partial\omega}\begin{bmatrix}R\omega_{y}\\{} +[t]_{\times}R\omega_{y}+Rv_{y} +\end{bmatrix}=\begin{bmatrix}-R[\omega_{y}]_{\times}\\ +-[t]_{\times}R[w_{y}]_{\times}-R[v_{y}]_{\times} +\end{bmatrix} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\frac{\partial}{\partial t}\begin{bmatrix}R & 0\\{} +[t]_{\times}R & R +\end{bmatrix}\begin{bmatrix}\omega\\ +v +\end{bmatrix}=\frac{\partial}{\partial t}\begin{bmatrix}R\omega_{y}\\{} +[t]_{\times}R\omega_{y}+Rv_{y} +\end{bmatrix}=\begin{bmatrix}0\\ +-[R\omega_{y}] +\end{bmatrix} +\] + +\end_inset + +Applying chain rule for the translation, +\begin_inset Formula $\frac{\partial}{\partial v}=\frac{\partial}{\partial t}\frac{\partial t}{\partial v}$ +\end_inset + +: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\frac{\partial}{\partial v}\begin{bmatrix}R & 0\\{} +[t]_{\times}R & R +\end{bmatrix}\begin{bmatrix}\omega_{y}\\ +v_{y} +\end{bmatrix}=\begin{bmatrix}0\\ +-[R\omega_{y}]_{\times} +\end{bmatrix}R=\begin{bmatrix}0\\ +-[R\omega_{y}]_{\times}R +\end{bmatrix}=\begin{bmatrix}0\\ +-R[\omega_{y}]_{\times} +\end{bmatrix} +\] + +\end_inset + +The simplification +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $[R\omega_{y}]_{\times}R=R[\omega_{y}]_{\times}$ +\end_inset + + can be derived by considering the result for when +\begin_inset Formula $\omega_{y}$ +\end_inset + + is each of the 3 standard basis vectors +\begin_inset Formula $\hat{e}_{i}$ +\end_inset + +: +\begin_inset Formula $-[R\hat{e}_{i}]_{\times}R$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Now putting together the rotation and translation: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +f'(T)=\frac{\partial Ad_{T}y}{\partial\xi}=\begin{bmatrix}\frac{\partial f}{\omega} & \frac{\partial f}{v}\end{bmatrix}=\begin{bmatrix}-R[\omega_{y}]_{\times} & 0\\ +-[t]_{\times}R[w_{y}]_{\times}-R[v_{y}]_{\times} & -R[\omega_{y}]_{\times} +\end{bmatrix} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +We can apply a similar procedure to compute the derivative of +\begin_inset Formula $Ad_{T}^{T}y$ +\end_inset + +. +\end_layout + \begin_layout Subsection Instantaneous Velocity \end_layout diff --git a/doc/math.pdf b/doc/math.pdf index 8dc7270f1139ce7e246a63925d29302f3afad2aa..06dec078d5fb590fe6dfe74dcdc13aace1a22328 100644 GIT binary patch delta 127232 zcmZshQ)8G7lx}0&YGXEbW81cECyn0Nwr$&Xlg74fJ2~H++nL|6Z`Rt++WtVEPC$th z0cA~fyalHQHZ^tS@x?KGw)L4}cOj3ap4Ak|l;DJ{)?gHJJskUBYx=|tB-Zt}ojDlK zZ>KIY4A8Y;%~CfXyN`36`A6VksjC$9I8h}Q0>U691qET}giw*hERG@m7?NE<8fJ|m z8WvY1p-tiBS2^@x!Li>IRfv{l&(x8eYgr%`;)x*uF|H_U@JelE;4=7VFU0%0raR+C zOjzQ`_7~`1hys81cjXlj<*}ZuB1u1_Jes4#_Hcsceo9DC2{$;wwb6o*@Zh5m#l8By zgWx4$=z)ZdP$S4jWxH6s zQ6JO;qC>602hx;-gbpBtI1sB1W~uHlB_T?|2IAFlCaa`F(T9d^aY!|h!V_l{V9dhg z4VIyLh6zw}>nM;2Lup19RZv|(i;2*@)B73(@zGb1P%<<4A$_V$67qwSqqoE~7B|4w zBVI$q)B?E!Bxc0aA^fW#O(D{&jDr{rdDoeMD##U5?>Q(&$+&T-FU}?>v_4{5xfSLG zr#MxV1E-l&`6Kov;=1|5sXf$RLJ20?gPIQ9-$iI|z1gyt!w97Fp 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