how to understand this comparison of screw theory to other methods? there is a paragraph in the second page of chapter 2 in the book "A mathematical introduction to robotic manipulation" as follows:
"There are two main advantages to using screws, twists, and wrenches
for describing rigid body kinematics. The first is that they allow a global
description of rigid body motion which does not suffer from singularities
due to the use of local coordinates. Such singularities are inevitable when
one chooses to represent rotation via Euler angles, for example. ..."
So what I dont understand is why this method "does not suffer from singularities
due to the use of local coordinates", while other method like euler angles will meet the singularities. Could you please give some examples? 
Thank you.
 A: I think the best way to understand this is something like epicycloids.

The picture above is linked to Mathworld's Epicycloid Page
Consider a circle rolling around another circle. Mark a point on the rolling circle. You can think of this like a hand on a robot arm where the arm is a rod attached to another rod where each rod can be rotated around.
Say you rotate both rods around their fixed ends making the angles change at a constant rate. Doing this will make the hand follow a epicycloid path (like the one of the red paths above). 
Notice that the red paths have cusps (singular points)! However, the change in the angles is nice and smooth (even linear). 
This means if I look at the how the robot's hand moves in terms of local coordinates, the hand's path has singularities and is very difficult to deal with directly. But if I consider just how the angles change, I have "linear" movement, which is much easier to analyze. 
So does the hand move smoothly? Local coordinates say "No". The screws/wrenches/etc viewpoint says "Yes." 
I hope this helps! :)
