So how are we going to escape this singularity? How do we create a square or circle printing area?
The singularity cannot be escaped, you can select a printing area in the blue area of the first graph, e.g. you could select X from -100 to 100 and Y from 100 to 200. Or if you want to print a square object, X from -80 to 80 and Y from 100 to about 250. A very long object could be from X from -200 to 200 and Y from 100 to 150. A circle printing area could be a circle around the center X0Y200.
I would avoid including the two special areas which I analyzed separately, because moving there means a massive angle change, which will result in high required time and probably results in printing problems. Therefore I would limit angle 1 to below 100 and angle 2 below 150.
(all those values are for your specific configuration of actuator distance and arm lengths)
Is it logical to use the closed loop stepper?
My personal opinion, the views are somewhat controversial:
a closed loop doesn't help, it only detects lost steps better. But if you don't have a very clever recover solution, you'll have a lost print whether you use closed loop or not. Steppers without closed loops are very reliable, if you don't overload them (like near a singularity with high load, or too high microsteps with too high processing load so that the stepper doesn't receive all steps). You can check steppers by homing before every layer. This was discussed in this forum in a thread, but it requires additional print time. A discussion with different opinions was here: https://forum.duet3d.com/topic/15139/layer-shift-detection Although main focus is stepper vs servo, there is some information about closed loop and homing at every layer shift also.