Re: [Toon-members] SE3: relation of input-translation to output-translation

[Ethan Eade]

Hi Rafael,

The exponential map in the group SE(3) is different than the exponential 
map in the rotation group SO(3) followed by translation  in R^3 .  In 
algebraic terms, rigid transformations in SE(3) are a semi-direct 
product of rotations and translations, so the two interact.  You can 
imagine exp() as doing a series of tiny [translation+rotation] motions 
in series. 

The upshot of this algebraic property is that the rotation and 
translation components in the tangent space (your vector mu) interact 
under the exponential map, yielding an element of SE(3) which doesn't 
have the translation you were expecting.  This is mathematically correct 
behavior -- it corresponds to motion along the geodesic given by mu.

In summary, the exponential map in SE(3) is not, in the general case, 
equivalent to any independent combination of exp()ing a translation and 
then a rotation and composing the result.

Elements of the tangent space are sometimes called twists, and their 
exponentials in SE(3) are called screws, because they correspond to a 
rotation around an axis followed by a  translation along that same axis. 

For some more detail, see page 544 of this chapter:
http://www.cis.upenn.edu/~cis610/cis610lie1.ps

Regards,
Ethan Eade


Rafael Spring wrote:
> Hello gentlemen,
>
> I am using TooN for a while now and I really appreciate it. Scope, 
> usability and speed are really great and I couldn't imagine working 
> without it.
> However, today, I've had some weird behavior and I'm unsure if TooN is 
> supoosed to behave like that (Btw. I am using TooN beta3) or if my 
> understanding of SE3s is insufficient. The question boils down to what 
> happens to an input translation after exponentiating.
> Consider the following sourcecode:
>
> ----
> Vector<3> rotation = makeVector(0.4, 0.6, 0.15);    //arbitrary numbers..
> Vector<3> translation = makeVector(-1, 0, 0.5); //arbitrary numbers..
> Vector<6> mu;
> mu.slice(0,3) = translation;
> mu.slice(3,3) = rotation;
>     
> SE3<> se = SE3<>::exp(mu);
> Vector<3> t = se.get_translation();
>
> ----
> Now, I was expecting:
>
> t == (-1, 0, 0.5)
>
> following the assumption that a euclidean transformation by an SE3 
> equals to an orthogonal transformation (exp'ing the rotation part of 
> mu) + a translation.
> The true value, however, is:
>
> t == (-0.789750 , -0.19886, 0.734775)
>
> So I reversed my assumption: It could be a translation by t followed 
> by an orthogonal transform (this would correspond to the 
> translation-part of mu being the inverse camera translation as seen 
> from camera 1 and get_translation() being the inverse translation as 
> seen from camera 2). This assumption, however, was neither confirmed:
>
> ----
> // make a rotation-only SE3
> Vector<6> mu_ = mu;
> mu_.slice(0,3) = makeVector(0,0,0);
> SE3<> se_ = SE3<>::exp(mu_);
>
> // make a translation-only SE3   
> Vector<6> mu__ = mu;
> mu__.slice(3,3) = makeVector(0,0,0);
> SE3<> se__ = SE3<>::exp(mu__);
>     
> // 1st translate then rotate
> Vector3 t_ = (se_ * se__).get_translation();
>
> //  t_ == (-0.5293 , -0.412372 , 0.89431)
>
> ----
>
> Now to the part beyond my understanding: The second assumption holds 
> (roughly) if I halve the rotation amount in mu:
>
> ----
> Vector<6> mu_;
> mu_.slice(0,3) = makeVector(0,0,0);
> mu_.slice(3,3) = rotation / 2.0;
> SE3<> se_ = SE3<>::exp(mu_); //se_ is half the rotation
>     
> Vector<6> mu__ = mu;
> mu__.slice(3,3) = makeVector(0,0,0);
> SE3<> se__ = SE3<>::exp(mu__); //se__ is only translation
>     
> Vector3 t__ = (se_ * se__).get_translation();    // 1st translate then 
> rotate
>
> // t__ == (-0.8024 , -0.19517 , 0.75371)
> // t__ is roughly  == t
>
> ----
> Is this within the general logic of SE3 members (am I missing 
> something?) or might there be something wrong with the SE3 class?
>
> Thanks!
> Rafael
>
>
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