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

= 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&times;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 &#9742; 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.  &#x2013; b_jonas 21:32, 20 January 2012 (UTC)
 * Yes, I did mention that. &#x2013; 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)