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I thought I understood the Jacobian and singularities pretty well, but then I was asked: UR robots (and others) have 2 joints that are always in parallel. Doesn't this put the robot at a singularity because the Jacobian is rank-deficient?
On the other hand, I've printed the condition number of UR robots and it's not so large. Typically under 30. So what am I missing; how is it that the condition number is not large all the time?
Parallel axes marked in red:
Wrist 1 is also always parallel on the UR robots.
As far as I know, if you're only considering two joints in isolation, the manipulator loses rank when they become coaxial.
That can't really happen with the first four joints, since the link lengths preclude exact coincidence and the wrist cluster would hit the base/shoulder before the wrist 1 axis gets close to coincident with the shoulder lift axis.
This article illustrates the singularities of this style of arm, including both Mecademic and UR examples:
https://www.mecademic.com/academic_articles/singularities-6-axis-robot-arm/
The video linked from there is great:
If the last axis ends up parallel with the others then you get the wrist singularity. I hit that frequently when I was experimenting with joystick teleop with a UR.
The native UR controller limits the robot motion with that keep-out cylinder coaxial with the base, which I believe is to slow and stop the robot as it approaches the shoulder singularity.
