The following content has been provided by the University of Erlangen-Nürnberg.

Okay … so welcome to the Tuesday's session, 45 minutes,

and we will conclude today the lecture on hand eye calibration.

I'm not explaining anymore the big picture,

because the big picture I was able to explain yesterday.

Today I'm going to explain the maths that I wasn't able to explain yesterday,

formulas. It's way simpler than the slides shows. It's a slide that actually I

haven't prepared that was done by one of my PhD students but that's no

excuse you know. Things look way more complicated than they actually are. So

what type of picture do we have to keep in mind? Basically we have oops

Hmm basically we have the following

We have here our transformation X. What is X the transformation X the transformation X tells us the

transform from the hand

coordinate system to the image coordinate system yeah, and then we have here a transformation B

that denotes the

Transformation of the robot arm of the ultrasound probe of the endoscope, and then we have down there

the transformation that we call a

And this is the transformation in between the two

coordinate systems of the of the images

yeah, that's the

structure and this diagram commutes and

If it commutes you might know it from algebra that just tells you it makes no different

Difference whether you go from here to here through this path or through that path these two passes are the same and that basically means that

XA is equal to

BX

Yeah, and while I'm moving along the path I'm driving my car and usually I mean a male driver

Usually moves a car in a rigid manner from A to B you move it and it's looking the same as before

So no deformations so just rotations and translations

This isn't that nice

That's nice. Yeah, so so we have here a rotation and

translation rotation translation rotation

translation rotation and translation and if I want to know the final rotation of

Course, I have to combine the rotations and how do I characterize rotations?

I could characterize them by a three by three rotation matrix. Yeah, and

If you combine them, you just have to multiply the rotation matrices. I

Should not make jokes like this before the evaluation. I'm very sorry. Okay

Good that means we also can write there not the the rotations like that and that's what written what's written here

Yeah, it says

rotation

X and after that rotation a is the same as rotation B and after that rotation X

And the same is true for the translations

We have a translation from here to here in this coordinate frame

and then we have a translation TB from there to there and a translation from there to there and

the relationship between these translations and

The rotations is written in this equation can sit down and and draw the arrows and and look at them how they move

And I did this this morning and it seems to be right

Okay, so these are the two equations and now I can

Rewrite ra in terms of our X and our B. How can I?

Bring the our X to the other side

I just multiply from the right with a transpose because the rotation matrix is a