Wikipedia:Reference desk/Archives/Mathematics/2012 January 20

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January 20

Derivative of log map on SE(3)

I have an application where I'm doing optimization on the manifold SE(3). Right now, I'm computing the Jacobian matrix by finite difference, but would like to do it symbolically. I can convert the residual to the form

$\mathbf {r} =\log \left(\exp(\delta )A\right)$

where exp and log implicitly convert 6-vectors containing axis-angle rotation and translation into 4×4 transformation matrices and back respectively. That is,

$\exp([\omega t]^{\top })=\exp {\begin{pmatrix}[\omega ]_{\times }&t\\0&0\end{pmatrix}}={\begin{pmatrix}R&T\\0&1\end{pmatrix}}$.

Also, A is also a rigid-body transform, of course. I'm thinking that with a closed from for $\mathbf {r} (\delta ,A)$, I can then numerically differentiate it with respect to the 6-vector $\delta$ to find the Jacobian.

This equation looks very related to the Baker–Campbell–Hausdorff formula which solves

$e^{Z}=e^{X}e^{Y}$

for Z, but which in general doesn't have a closed form. I think something similar to what I'm looking for is in section 10 of this paper; it seems to give a closed form for at least part of the infinite series, but I have a lot of trouble following that notation.

Any suggestions? If there is a closed form for the Baker–Campbell–Hausdorff formula for SE(3), it would be worth adding it to that page.

Thanks. —Ben FrantzDale (talk) 15:03, 20 January 2012 (UTC)

I do know a way of finding a closed-form solution for logarithms on the double cover of $S\mathbb {E} (3)$, which may help you. Are you familiar with Spin groups? For definite metric signatures, they double-cover the corresponding rotation groups The Spin group Spin$(4)$ can be described by a pair of quaternions, whose logarithms are straightforward to find. By describing translations as infinitesimal rotations (using dual numbers, for instance), it is possible to construct a representation of the double cover of $S\mathbb {E} (3)$ in this way. If you're willing to learn about dual quaternions, you may find them useful for your purposes. Typing "dual quaternion logarithm" into google certainly provides several links.--Leon (talk) 13:54, 21 January 2012 (UTC)

Thanks. I've encountered dual quaternions before. I'll investigate. —Ben FrantzDale (talk) 20:55, 21 January 2012 (UTC)

Are these equations chaotic

Can $F_{E}$ exhibit chaotic behavior given $a$, $b$, and $c$?

$F_{A}\left(a,x\right)=\left(x-2\cdot a\cdot {\mbox{floor}}\left({\frac {x}{2\cdot a}}+{\frac {1}{2}}\right)\right)\cdot \left(-1\right)^{{\mbox{floor}}\left({\frac {x}{2\cdot a}}+{\frac {1}{2}}\right)}$

$F_{B}\left(a,x\right)=a\cdot \left(-\left({\mbox{abs}}\left({\frac {x}{a}}\right)\%2-1\right)^{2}+1\right)\cdot \left(-1\right)^{{\mbox{floor}}\left({\frac {{\frac {x}{a}}-1}{2}}+{\frac {1}{2}}\right)}$

$F_{C}\left(a,x\right)=F_{B}\left(a,{\frac {2x}{a}}-a\right)+1+a$

$F_{D}\left(a,b,x\right)=F_{A}\left(a,{\frac {x^{2}}{F_{\mbox{C}}\left(b,x\right)}}\right)$

$F_{E}\left(a,b,c,x\right)=c\cdot \left(F_{D}\left(a,b,{\frac {{\mbox{floor}}\left(c\cdot x+1\right)}{c}}\right)-F_{D}\left(a,b,{\frac {{\mbox{floor}}\left(c\cdot x\right)}{c}}\right)\right)\cdot \left(x-{\frac {{\mbox{floor}}\left(c\cdot x+1\right)}{c}}\right)+F_{D}\left(a,b,{\frac {{\mbox{floor}}\left(c\cdot x+1\right)}{c}}\right)$

--Melab±1 ☎ 20:43, 20 January 2012 (UTC)

Last time you asked something like this, did I mention that you should read what Knuth says in chapter 3.1 about random generators? He concludes the morale that you can't generate good random numbers by just choosing a random method haphazardly. – b_jonas 21:32, 20 January 2012 (UTC)

Yes, I did mention that. – b_jonas 21:36, 20 January 2012 (UTC)

How did you come by such complicated formulae? Chaos doesn't require anything complicated to produce it. Dmcq (talk) 14:05, 21 January 2012 (UTC)

Those functions are not continuous. As I understand it the definition of dynamical chaos ("sensitive dependence on initial conditions") requires continuity in order to make sense. Looie496 (talk) 18:28, 21 January 2012 (UTC)