inserted figures
Frank Dellaert
parent
e9ea8b426a
commit
d4447cbe48
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@ -348,7 +348,7 @@ Aff(2),6
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\begin_layout Standard
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\begin_inset Note Comment
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status open
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status collapsed
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\begin_layout Plain Layout
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SL(3),8
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@ -390,36 +390,6 @@ SL(3),8
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Motivation: Rigid Motions in the Plane
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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placement h
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wide false
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sideways false
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status collapsed
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\begin_layout Plain Layout
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IMAGINE A FIGURE HERE
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\begin_inset Caption
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\begin_layout Plain Layout
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(a) A robot translating.
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(b) A robot rotating.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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We will start with a small example of a robot moving in a plane, parameterized
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by a
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@ -480,6 +450,7 @@ where
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\end_inset
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in counterclockwise direction.
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\end_layout
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\begin_layout Standard
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@ -490,7 +461,13 @@ sideways false
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status collapsed
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\begin_layout Plain Layout
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IMAGINE A FIGURE HERE
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\align center
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\begin_inset Graphics
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filename images/circular.pdf
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\end_inset
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\begin_inset Caption
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\begin_layout Plain Layout
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@ -520,7 +497,7 @@ The reason is that, if we move the robot a tiny bit according to the velocity
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\begin_inset Formula $(v_{x},\, v_{y},\,\omega)$
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\end_inset
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, we do have to first order
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, we have (to first order)
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\begin_inset Formula \[
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(x_{t+\delta},\, y_{t+\delta},\,\theta_{t+\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)\]
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@ -608,7 +585,13 @@ sideways false
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status open
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\begin_layout Plain Layout
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IMAGINE A FIGURE HERE
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\align center
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\begin_inset Graphics
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filename images/n-steps.pdf
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\end_inset
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\begin_inset Caption
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\begin_layout Plain Layout
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@ -718,18 +701,13 @@ T(t)=\lim_{n\rightarrow\infty}\left(I+\frac{t}{n}\xihat\right)^{n}\]
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For real numbers, this series is familiar and is actually a way to compute
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the exponential function:
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\begin_inset Formula \[
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e^{x}=\lim_{n\rightarrow\infty}\left(I+\frac{x}{n}\right)^{n}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\]
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e^{x}=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\]
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\end_inset
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The series can be similarly defined for square matrices,
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\begin_inset Formula \[
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e^{A}=\lim_{n\rightarrow\infty}\left(I+\frac{A}{n}\right)^{n}=\sum_{k=0}^{\infty}\frac{A^{k}}{k!}\]
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\end_inset
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Our final result is that we can write the motion of a robot along a circular
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trajectory, resulting from the 2D twist
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The series can be similarly defined for square matrices,and the final result
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is that we can write the motion of a robot along a circular trajectory,
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resulting from the 2D twist
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\begin_inset Formula $\xi=(v,\omega)$
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\end_inset
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@ -757,7 +735,7 @@ exponential map.
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\end_layout
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\begin_layout Standard
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The above has all elemtns of Lie group theory.
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The above has all elements of Lie group theory.
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We call the space of 2D rigid transformations, along with the composition
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operation, the
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\emph on
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@ -1212,7 +1190,7 @@ The Lie group
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\begin_inset Formula \[
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\hat{}:\theta\rightarrow\that=\skew{\theta}\]
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\hat{}:\omega\rightarrow\what=\skew{\omega}\]
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\end_inset
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@ -1256,13 +1234,46 @@ which maps the angle
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The exponential map can be computed in closed form as
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\begin_inset Formula \[
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R=e^{\skew{\theta}}=\left[\begin{array}{cc}
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R=e^{\skew{\omega t}}=e^{\skew{\theta}}=\left[\begin{array}{cc}
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\cos\theta & -\sin\theta\\
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\sin\theta & \cos\theta\end{array}\right]\]
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\end_inset
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This can be proven
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\begin_inset CommandInset citation
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LatexCommand cite
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key "Hall00book"
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\end_inset
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by realizing
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\begin_inset Formula $\skew 1$
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\end_inset
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is diagonizable with eigenvalues
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\begin_inset Formula $-i$
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\end_inset
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and
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\begin_inset Formula $i$
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\end_inset
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, and eigenvectors
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\begin_inset Formula $\left[\begin{array}{c}
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1\\
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i\end{array}\right]$
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\end_inset
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and
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\begin_inset Formula $\left[\begin{array}{c}
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i\\
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1\end{array}\right]$
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\end_inset
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.
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Readers familiar with projective geometry will recognize these as the circular
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points when promoted to homogeneous coordinates.
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\end_layout
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\begin_layout Subsection
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@ -1336,13 +1347,13 @@ We would now like to know what an incremental rotation parameterized by
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would do:
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\begin_inset Formula \[
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q(\text{\theta})=Re^{\skew{\theta}}p\]
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q(\text{\omega t})=Re^{\skew{\omega t}}p\]
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\end_inset
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hence the derivative is:
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\begin_inset Formula \[
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\deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\skew{\theta}}p\right)=R\deriv{}{\omega}\left(\skew{\theta}p\right)=RH_{p}\]
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\deriv{q(\omega t)}t=R\deriv{}t\left(e^{\skew{\omega t}}p\right)=R\deriv{}t\left(\skew{\omega t}p\right)=RH_{p}\]
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\end_inset
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@ -2872,7 +2883,7 @@ Applying a homography is then a tensor contraction
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.
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\end_layout
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\begin_layout Plain Layout
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\begin_layout Standard
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\begin_inset Note Note
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status collapsed
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Binary file not shown.
Binary file not shown.
182
doc/math.lyx
182
doc/math.lyx
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@ -54,7 +54,20 @@ Geometry Derivatives and Other Hairy Math
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Frank Dellaert
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Box Frameless
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position "t"
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hor_pos "c"
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has_inner_box 1
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inner_pos "t"
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use_parbox 0
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width "100col%"
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special "none"
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height "1in"
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height_special "totalheight"
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status collapsed
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\begin_layout Plain Layout
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\begin_inset Note Comment
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status open
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@ -67,7 +80,7 @@ Derivatives
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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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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@ -88,7 +101,7 @@ Derivatives
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Note Comment
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status open
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@ -101,7 +114,7 @@ Lie Groups
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset FormulaMacro
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\newcommand{\xhat}{\hat{x}}
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{\hat{x}}
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@ -122,7 +135,7 @@ Lie Groups
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset FormulaMacro
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\newcommand{\define}{\stackrel{\Delta}{=}}
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{\stackrel{\Delta}{=}}
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@ -143,7 +156,7 @@ Lie Groups
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Note Comment
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status open
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@ -156,7 +169,7 @@ SO(2)
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset FormulaMacro
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\newcommand{\Rtwo}{\mathfrak{\mathbb{R}^{2}}}
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{\mathfrak{\mathbb{R}^{2}}}
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@ -189,7 +202,7 @@ SO(2)
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Note Comment
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status open
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@ -202,7 +215,7 @@ SE(2)
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset FormulaMacro
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\newcommand{\SEtwo}{SE(2)}
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{SE(2)}
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@ -217,7 +230,7 @@ SE(2)
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Note Comment
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status open
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@ -230,7 +243,7 @@ SO(3)
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset FormulaMacro
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\newcommand{\Rthree}{\mathfrak{\mathbb{R}^{3}}}
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{\mathfrak{\mathbb{R}^{3}}}
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@ -263,7 +276,7 @@ SO(3)
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Note Comment
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status open
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@ -276,7 +289,7 @@ SE(3)
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset FormulaMacro
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\newcommand{\Rsix}{\mathfrak{\mathbb{R}^{6}}}
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{\mathfrak{\mathbb{R}^{6}}}
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@ -301,12 +314,121 @@ SE(3)
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Section
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Derivatives of Lie Group Mappings
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\end_layout
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\begin_layout Subsection
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New
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\end_layout
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\begin_layout Standard
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The following is relevant
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\begin_inset CommandInset citation
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LatexCommand cite
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after "page 45"
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key "Hall00book"
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\end_inset
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: suppose that
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\begin_inset Formula $\Phi:G\rightarrow H$
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\end_inset
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is a a mapping (Lie group homomorphism).
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Then there exists a unique linear map
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\begin_inset Formula $\phi:\gg\rightarrow\mathfrak{h}$
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\end_inset
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\begin_inset Formula \[
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\phi(\xhat)\define\lim_{t\rightarrow0}\frac{d}{dt}\Phi\left(e^{t\xhat}\right)\]
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\end_inset
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such that
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\end_layout
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\begin_layout Enumerate
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\begin_inset Formula $\Phi\left(e^{\xhat}\right)=e^{\phi\left(\xhat\right)}$
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\end_inset
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\end_layout
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\begin_layout Enumerate
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\begin_inset Formula $\phi\left(T\xhat T^{-1}\right)=\Phi(T)\phi(\xhat)\Phi(T^{-1})$
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\end_inset
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\end_layout
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\begin_layout Enumerate
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\begin_inset Formula $\phi\left([\xhat,\yhat]\right)=\left[\phi(\xhat),\phi(\yhat)\right]$
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\end_inset
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\end_layout
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\begin_layout Standard
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In other words, the map
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\begin_inset Formula $\phi$
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\end_inset
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is the derivative of
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\begin_inset Formula $\Phi$
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\end_inset
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at the identity.
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It suffices to compute
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\begin_inset Formula $\phi$
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\end_inset
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for a basis of
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\begin_inset Formula $\gg$
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\end_inset
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.
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Since
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\begin_inset Formula \[
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e^{-\xhat}=\left(e^{-\xhat}\right)^{-1}\]
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\end_inset
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clearly
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\begin_inset Formula $\phi(\xhat)=-\xhat$
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\end_inset
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for the inverse mapping.
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\end_layout
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\begin_layout Standard
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Let us define two mappings
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\begin_inset Formula \[
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\Phi_{1}(A)=AB\mbox{ and }\Phi_{2}(B)=AB\]
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\end_inset
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Then
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\begin_inset Formula \[
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\phi_{1}(\xhat)=\lim_{t\rightarrow0}\frac{d}{dt}\Phi_{1}\left(e^{t\xhat}B\right)=\]
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\end_inset
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\end_layout
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\begin_layout Subsection
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Old
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\end_layout
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\begin_layout Standard
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The derivatives for
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\emph on
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@ -962,11 +1084,11 @@ We would now like to know what an incremental rotation parameterized by
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would do:
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\begin_inset Formula \[
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q(\text{\theta})=Re^{\skew{\theta}}p\]
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q(\text{\omega t})=Re^{\skew{\omega t}}p\]
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\end_inset
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hence the derivative (following the exposition in Section
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The derivative is (following the exposition in Section
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "sec:Derivatives-of-Actions"
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@ -975,7 +1097,7 @@ reference "sec:Derivatives-of-Actions"
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):
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\begin_inset Formula \[
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\deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\skew{\theta}}p\right)=R\deriv{}{\omega}\left(\skew{\theta}p\right)=RH_{p}\]
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\deriv{q(\omega t)}t=R\deriv{}t\left(e^{\skew{\omega t}}p\right)=R\deriv{}t\left(\skew{\omega t}p\right)\]
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\end_inset
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@ -1004,22 +1126,34 @@ cross product
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\begin_inset Formula \[
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\skew{\theta}p=\left[\begin{array}{c}
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-y\\
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x\end{array}\right]\theta=H_{p}\theta\]
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x\end{array}\right]\theta=\omega R_{pi/2}pt\]
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\end_inset
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with
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\begin_inset Formula $H_{p}=R_{pi/2}p$
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Hence, the final derivative of an action in its first argument is
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\begin_inset Formula \[
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\deriv{q(\omega t)}{\omega t}=\omega RR_{pi/2}p=\omega R_{pi/2}Rp=\omega R_{pi/2}q\]
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\end_inset
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\end_layout
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\begin_layout Standard
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Really need to think of relationship
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\begin_inset Formula $\omega$
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\end_inset
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and
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\begin_inset Formula $t$
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\end_inset
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.
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Hence, the final derivative of an action in its first argument is
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\begin_inset Formula \[
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\deriv{q(\theta)}{\theta}=RH_{p}=RR_{pi/2}p=R_{pi/2}Rp=R_{pi/2}q\]
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We don't have a time
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\begin_inset Formula $t$
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\end_inset
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in our code.
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\end_layout
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\begin_layout Standard
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