Derivatives of group actions, in progress
Frank Dellaert
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28f4493a2b
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03e8641a61
634
doc/math.lyx
634
doc/math.lyx
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@ -511,11 +511,103 @@ Hence, now we undo
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\end_layout
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\begin_layout Section
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Important Lie Groups
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Derivatives of Actions
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\end_layout
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\deriv}[2]{\frac{\partial#1}{\partial#2}}
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{\frac{\partial#1}{\partial#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\at}[2]{#1\biggr\rvert_{#2}}
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{#1\biggr\rvert_{#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\Jac}[3]{ \at{\deriv{#1}{#2}} {#3} }
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{\at{\deriv{#1}{#2}}{#3}}
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\end_inset
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\end_layout
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\begin_layout Standard
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When a Lie group
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\begin_inset Formula $G$
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\end_inset
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acts on a vector space
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\begin_inset Formula $V$
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\end_inset
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, we are interested in the derivatives of
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\begin_inset Formula \[
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f_{1}\left(g\right)=gv\mbox{ and }f_{2}(v)=gv\]
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\end_inset
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with
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\begin_inset Formula $f_{1}:G\rightarrow V$
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\end_inset
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and
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\begin_inset Formula $f_{2}:V\rightarrow V$
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\end_inset
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.
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The brilliance of Lie group theory is that we only need to know how the
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generators of the group act around the group's identity element
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\begin_inset Formula $g=id$
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\end_inset
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, and then we can use the Adjoint map to effectuate that action in the correct
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frame of reference.
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Specifically, if
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\begin_inset Formula \[
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H_{v}=\left[\begin{array}{ccc}
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\frac{\partial f_{1}}{\partial x_{1}} & \ldots & \frac{\partial f_{1}}{\partial x_{n}}\end{array}\right]\rvert_{g=id}\]
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\end_inset
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is the
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\begin_inset Formula $m\times n$
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\end_inset
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Jacobian of the group action on
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\begin_inset Formula $\mbox{v\in}V$
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\end_inset
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with respect to an incremental change
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\begin_inset Formula $x$
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\end_inset
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, we have
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\begin_inset Formula \[
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\Jac{f_{1}}xg=H_{v}\Ad g\]
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\end_inset
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The meaning of
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\begin_inset Formula $H$
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\end_inset
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will depend on the group
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\begin_inset Formula $G$
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\end_inset
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and will be very intuitive!
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\end_layout
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\begin_layout Section
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3D Rotations
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\end_layout
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\begin_layout Subsection
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3D Rotations
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Basics
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\end_layout
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\begin_layout Standard
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@ -654,6 +746,10 @@ and this is defines the canonical parameterization of
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.
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\end_layout
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\begin_layout Subsection
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The Adjoint Map
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\end_layout
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\begin_layout Standard
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We can prove the following identity for rotation matrices
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\begin_inset Formula $R$
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@ -787,6 +883,10 @@ q=e^{\Skew{R\omega}}p\]
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\end_layout
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\begin_layout Subsection
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Derivatives of Mappings
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\end_layout
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\begin_layout Standard
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Hence, we are now in a position to simply posit the derivative of
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\series bold
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@ -837,6 +937,181 @@ and between in its second argument,
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\end_layout
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\begin_layout Subsection
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Derivatives of Actions
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\end_layout
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\begin_layout Standard
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The beauty of Lie group theory comes in play when we talk about the derivative
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of a group action.
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In the case of
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\begin_inset Formula $\SOthree$
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\end_inset
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the vector space is
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\begin_inset Formula $\Rthree$
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\end_inset
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, and the group action corresponds to rotating a point
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\begin_inset Formula \[
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q=Rp\]
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\end_inset
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We would now like to know what an incremental rotation parameterized by
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\begin_inset Formula $\omega$
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\end_inset
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would do
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\begin_inset Formula \[
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\deriv q{\omega}=\deriv{}{\omega}\left(Rp\right)=\deriv{}{\omega}\left(e^{\Skew{\omega}}p\right)\]
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\end_inset
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Since
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\begin_inset Formula \[
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e^{A}=I+A+\frac{A^{2}}{2!}+\frac{A^{3}}{3!}+\ldots\]
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\end_inset
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and derivative is linear and we are only interested in first order we have
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\begin_inset Formula \[
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\deriv{e^{\Skew{\omega}}}{\omega}=\deriv{\Skew{\omega}}{\omega}=\omega_{x}G_{1}+\omega_{y}G_{2}+\omega_{z}G_{3}\]
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\end_inset
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Specifically, the generators for
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\begin_inset Formula $\SOthree$
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\end_inset
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are
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\begin_inset Formula \[
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G_{1}=\left(\begin{array}{ccc}
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0 & 0 & 0\\
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0 & 0 & -1\\
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0 & 1 & 0\end{array}\right)\mbox{}G_{2}=\left(\begin{array}{ccc}
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0 & 0 & 1\\
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0 & 0 & 0\\
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-1 & 0 & 0\end{array}\right)\mbox{ }G_{1}=\left(\begin{array}{ccc}
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0 & -1 & 0\\
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1 & 0 & 0\\
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0 & 0 & 0\end{array}\right)\]
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\end_inset
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corresponding to a rotation around
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\begin_inset Formula $X$
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\end_inset
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,
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\begin_inset Formula $Y$
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\end_inset
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, and
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\begin_inset Formula $Z$
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\end_inset
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, respectively.
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When given an incremental angular velocity
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\begin_inset Formula $\omega$
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\end_inset
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, we obtain the effect of the group action around the identity:
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\begin_inset Formula \[
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H\omega=\omega_{x}G_{1}+\omega_{y}G_{2}+\omega_{z}G_{3}=\left[\begin{array}{ccc}
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0 & -\omega_{z} & \omega_{y}\\
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\omega_{z} & 0 & -\omega_{x}\\
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-\omega_{y} & \omega_{x} & 0\end{array}\right]=\Skew{\omega}\]
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\end_inset
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Hence, at the origin, the effect of an incremental action
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\begin_inset Formula $\omega$
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\end_inset
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on a point
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\begin_inset Formula $p$
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\end_inset
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is a velocity
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\begin_inset Formula \[
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\Skew{\omega}p=\omega\times p\]
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\end_inset
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We can write this as a
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\begin_inset Formula $3\times3$
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\end_inset
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Jacobian
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\begin_inset Formula $H_{p}$
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\end_inset
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that is multipled with
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\begin_inset Formula $\omega$
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\end_inset
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,
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\begin_inset Formula \[
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\omega\times p=-p\times\omega=-\Skew p\omega=H_{p}\omega\]
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\end_inset
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Now, if we want to apply this in a frame
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\begin_inset Formula $R$
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\end_inset
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, we need to do something quite similar to the Adjoint map: (a) transform
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the point
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\begin_inset Formula $p$
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\end_inset
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to the origin using
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\begin_inset Formula $R$
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\end_inset
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, apply the action
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\begin_inset Formula $\Skew{\omega}$
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\end_inset
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, and transform back with
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\begin_inset Formula $R^{T}$
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\end_inset
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.
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In short
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\begin_inset Formula \[
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q=R^{T}e^{\Skew{\omega}}Rp=\exp\left(\Ad{R^{T}}\Skew{\omega}\right)p=\exp\left(\Skew{R^{T}\omega}\right)p\]
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\end_inset
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and hence the velocity becomes
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\begin_inset Formula \[
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\Skew{R^{T}\omega}\times p=-\Skew pR^{T}\omega=-R^{T}R\Skew pR^{T}\omega=-R^{T}\Skew{Rp}\omega=R^{T}H_{Rp}\omega\]
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\end_inset
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This is quite intuitive in hindsight: we transform
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\begin_inset Formula $p$
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\end_inset
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to
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\begin_inset Formula $Rp$
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\end_inset
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, calculate the velocity by
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\begin_inset Formula $H_{Rp}$
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\end_inset
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, and transform back by the rotation
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\begin_inset Formula $R^{T}$
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\end_inset
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.
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\end_layout
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\begin_layout Section
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3D Rigid Transformations
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\end_layout
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@ -961,6 +1236,10 @@ key "Murray94book"
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.
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\end_layout
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\begin_layout Subsection
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The Adjoint Map
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\end_layout
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\begin_layout Standard
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The adjoint is
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\begin_inset Formula \begin{eqnarray*}
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@ -1001,6 +1280,14 @@ v\end{array}\right]\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Derivatives of Mappings
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\end_layout
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\begin_layout Standard
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Hence, as with
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\begin_inset Formula $\SOthree$
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\end_inset
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@ -1065,6 +1352,207 @@ and between in its second argument,
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\end_layout
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\begin_layout Subsection
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The derivatives of Actions
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\end_layout
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\begin_layout Standard
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The action of
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\begin_inset Formula $\SEthree$
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\end_inset
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on 3D points is done by embedding the points in
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\begin_inset Formula $\mathbb{R}^{4}$
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\end_inset
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by using homogeneous coordinates
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\begin_inset Formula \[
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q=\left[\begin{array}{cc}
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R & t\\
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0 & 1\end{array}\right]\left[\begin{array}{c}
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p\\
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1\end{array}\right]\]
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\end_inset
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\end_layout
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\begin_layout Standard
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The generators for
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\begin_inset Formula $\SEthree$
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\end_inset
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are
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\begin_inset Formula \[
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G_{1}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & -1 & 0\\
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0 & 1 & 0 & 0\\
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0 & 0 & 0 & 0\end{array}\right)\mbox{}G_{2}=\left(\begin{array}{cccc}
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0 & 0 & 1 & 0\\
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0 & 0 & 0 & 0\\
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-1 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\end{array}\right)\mbox{ }G_{1}=\left(\begin{array}{cccc}
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0 & -1 & 0 & 0\\
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1 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\end{array}\right)\]
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\end_inset
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\begin_inset Formula \[
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G_{4}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\end{array}\right)\mbox{}G_{5}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\end{array}\right)\mbox{ }G_{6}=\left(\begin{array}{cccc}
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 0\\
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0 & 0 & 0 & 1\\
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0 & 0 & 0 & 0\end{array}\right)\]
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\end_inset
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and hence a twist around the origin applies to homogeneous coordinates
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\begin_inset Formula $\hat{p}\in\mathbb{R}^{4}$
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\end_inset
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as
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\begin_inset Formula \[
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\sum_{i}\xi_{i}G_{i}=\left(\begin{array}{cccc}
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0 & -\omega_{z} & \omega_{y} & v_{x}\\
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\omega_{z} & 0 & -\omega_{x} & v_{y}\\
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-\omega_{y} & \omega_{x} & 0 & v_{z}\\
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0 & 0 & 0 & 0\end{array}\right)=\left[\begin{array}{cc}
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\Skew{\omega} & v\\
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0 & 0\end{array}\right]=\xihat\]
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\end_inset
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Hence, at the origin, an incremental action
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\begin_inset Formula $\xi$
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\end_inset
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on a point
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\begin_inset Formula $p$
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\end_inset
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induces the velocity
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\begin_inset Formula \[
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\left[\begin{array}{cc}
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\Skew{\omega} & v\\
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0 & 0\end{array}\right]\left[\begin{array}{c}
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p\\
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1\end{array}\right]=\left[\begin{array}{c}
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\omega\times p+v\\
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0\end{array}\right]\]
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\end_inset
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We can write this as a velocity in
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\begin_inset Formula $\Rthree$
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\end_inset
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as the product of a
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\begin_inset Formula $3\times6$
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\end_inset
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matrix
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\begin_inset Formula $H(p)$
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\end_inset
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that acts upon the exponential coordinates
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\begin_inset Formula $\xi$
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\end_inset
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directly:
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\begin_inset Formula \[
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\omega\times p+v=-p\times\omega+v=\left[\begin{array}{cc}
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-\Skew p & I_{3}\end{array}\right]\left[\begin{array}{c}
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\omega\\
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v\end{array}\right]=H(p)\xi\]
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\end_inset
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Now, if we want to apply this in a frame
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\begin_inset Formula $T$
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\end_inset
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, we can (1) transform the point
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\begin_inset Formula $p$
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\end_inset
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to the origin using
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\begin_inset Formula $T^{-1}$
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\end_inset
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, apply the action
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\begin_inset Formula $\xihat$
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\end_inset
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, and transform back to
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\begin_inset Formula $T$
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\end_inset
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.
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In short
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\begin_inset Formula \[
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\left[\begin{array}{c}
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q\\
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1\end{array}\right]=Te^{\xihat}T^{-1}\left[\begin{array}{c}
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p\\
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1\end{array}\right]=\exp\left(\Ad T\xihat\right)\left[\begin{array}{c}
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p\\
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1\end{array}\right]\]
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\end_inset
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To get the velocity of the point in frame
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\begin_inset Formula $T$
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\end_inset
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, we have
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\begin_inset Formula \[
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H(p)\Ad T\xi=\left[\begin{array}{cc}
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-\Skew p & I_{3}\end{array}\right]\left[\begin{array}{cc}
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R & 0\\
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\Skew tR & R\end{array}\right]\xi=R\left[\begin{array}{cc}
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-\Skew{T^{-1}p} & I_{3}\end{array}\right]\xi=H(T^{-1}p)\xi\]
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\end_inset
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where I made use of
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\begin_inset Formula $\Skew{t-p}R=-RR^{T}\Skew{p-t}R=-R\Skew{R^{T}(p-t)}=-R\Skew{T^{-1}p}$
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\end_inset
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.
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This is intuitive in hindsight: we transform
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\begin_inset Formula $p$
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\end_inset
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back to the orgin by
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\begin_inset Formula $T^{-1}p$
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\end_inset
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, apply
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\begin_inset Formula $H(.)$
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\end_inset
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to get a velocity, and only need the rotation
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\begin_inset Formula $R$
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\end_inset
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to transform it back to the orginal frame (as velocities are not affected
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by translation).
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\end_layout
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\begin_layout Section
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2D Rotations
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\end_layout
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@ -1228,6 +1716,14 @@ R=e^{\skew{\theta}}=\left[\begin{array}{cc}
|
|||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
Derivatives of Mappings
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The adjoint map for
|
||||
\begin_inset Formula $\sotwo$
|
||||
\end_inset
|
||||
|
|
@ -1281,7 +1777,7 @@ and between in its second argument,
|
|||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
\begin_layout Section
|
||||
2D Rigid Transformations
|
||||
\end_layout
|
||||
|
||||
|
|
@ -1383,6 +1879,10 @@ T=\exp\xihat\]
|
|||
A closed form solution for the exponential map is in the works...
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
The Adjoint Map
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The adjoint is
|
||||
\begin_inset Formula \begin{eqnarray*}
|
||||
|
|
@ -1415,6 +1915,14 @@ v\\
|
|||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
Derivatives of Mappings
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
We can just define all derivatives in terms of the above adjoint map:
|
||||
\begin_inset Formula \begin{eqnarray*}
|
||||
\frac{\partial T^{^{-1}}}{\partial\xi} & = & -\Ad T\end{eqnarray*}
|
||||
|
|
@ -1736,126 +2244,6 @@ and with respect to
|
|||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The derivative of
|
||||
\begin_inset Formula $inverse=R^{T},-R^{T}t=R^{T}(I,-t)$
|
||||
\end_inset
|
||||
|
||||
, first derivative of rotation in rotation argument:
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The partials
|
||||
\begin_inset Formula \[
|
||||
\frac{\partial\omega^{\prime}}{\partial\omega}=\frac{\partial inv(R)}{\partial\omega}=-R\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula \[
|
||||
\frac{\partial t^{\prime}}{\partial\omega}=\frac{-\partial unrot(R,t)}{\partial\omega}=-skew(R^{T}t)\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula \[
|
||||
\frac{\partial\omega^{\prime}}{\partial t}=\mathbf{0}\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula \[
|
||||
\frac{\partial t^{\prime}}{\partial t}=\frac{-\partial unrot(R,t)}{\partial t}=-R^{T}\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\series bold
|
||||
old stuff:
|
||||
\series default
|
||||
|
||||
\begin_inset Formula \begin{eqnarray*}
|
||||
(I+\Omega')R^{T} & = & \left((I+\Omega)R\right)^{T}\\
|
||||
R^{T}+\Omega'R^{T} & = & R^{T}(I-\Omega)\\
|
||||
\Omega'R^{T} & = & -R^{T}\Omega\\
|
||||
\Omega' & = & -R^{T}\Omega R=-\Skew{R^{T}\omega}\\
|
||||
\omega' & = & -R^{T}\omega\end{eqnarray*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Now
|
||||
\series bold
|
||||
\emph on
|
||||
compose
|
||||
\series default
|
||||
\emph default
|
||||
, first w.r.t.
|
||||
a change in rotation in the first argument:
|
||||
\begin_inset Formula \begin{align*}
|
||||
AB & =\left(T_{A}R_{A}T_{B}\right)\left(R_{A}R_{B}\right)\\
|
||||
\left(T_{A}R_{A}T_{B}\left(I+T^{\prime}\right)\right)\left(R_{A}R_{B}\left(I+\Omega^{\prime}\right)\right) & =\left(T_{A}R_{A}\left(I+\Omega\right)T_{B}\right)\left(R_{A}\left(I+\Omega\right)R_{B}\right)\\
|
||||
\textrm{translation only:}\\
|
||||
T_{A}R_{A}T_{B}\left(I+T^{\prime}\right) & =T_{A}R_{A}\left(I+\Omega\right)T_{B}\\
|
||||
T_{B}\left(I+T^{\prime}\right) & =\left(I+\Omega\right)T_{B}\\
|
||||
T_{B}+T_{B}T^{\prime} & =T_{B}+\Omega T_{B}\\
|
||||
T^{\prime} & =T_{B}^{-1}skew(\omega)T_{B}\\
|
||||
T^{\prime} & =skew(T_{B}\omega)\,???\\
|
||||
\textrm{rotation only:}\\
|
||||
R_{A}R_{B}\left(I+\Omega^{\prime}\right) & =R_{A}\left(I+\Omega\right)R_{B}\\
|
||||
R_{B}\Omega^{\prime} & =\Omega R_{B}\\
|
||||
\Omega^{\prime} & =R_{B}^{T}\Omega R_{B}\\
|
||||
& =skew(R_{B}^{T}\omega)\\
|
||||
\omega^{\prime} & =R_{B}^{T}\omega\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
And w.r.t.
|
||||
a rotation in the second argument:
|
||||
\begin_inset Formula \begin{align*}
|
||||
\left(T_{A}R_{A}T_{B}\left(I+T^{\prime}\right)\right)\left(R_{A}R_{B}\left(I+\Omega^{\prime}\right)\right) & =\left(T_{A}R_{A}T_{B}\right)\left(R_{A}R_{B}\left(I+\Omega\right)\right)\\
|
||||
\left(R_{A}R_{B}\left(I+\Omega^{\prime}\right)\right) & =\left(R_{A}R_{B}\left(I+\Omega\right)\right)\\
|
||||
\omega^{\prime} & =\omega\\
|
||||
t^{\prime} & =0\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
w.r.t.
|
||||
a translation in the second argument:
|
||||
\begin_inset Formula \begin{align*}
|
||||
\left(T_{A}R_{A}T_{B}\left(I+T^{\prime}\right)\right)\left(R_{A}R_{B}\left(I+\Omega^{\prime}\right)\right) & =\left(T_{A}R_{A}T_{B}\left(I+T\right)\right)\left(R_{A}R_{B}\right)\\
|
||||
\omega^{\prime} & =0\\
|
||||
t^{\prime} & =t\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Finally,
|
||||
\series bold
|
||||
\emph on
|
||||
between
|
||||
\series default
|
||||
\emph default
|
||||
in the first argument:
|
||||
\begin_inset Formula \begin{align*}
|
||||
\frac{\partial A^{-1}B}{\partial A} & =\frac{\partial c\left(A^{-1},B\right)}{\partial A^{-1}}\frac{\partial inv(A)}{A}\\
|
||||
\frac{\partial A^{-1}B}{B} & =\frac{\partial c\left(A^{-1},B\right)}{\partial B}\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
|
|
|
|||
Loading…
Reference in New Issue