# How big is the set of kinematics singularities?

Suppose we have an $n$-DOF robot manipulator and let $q \in \mathbf{R}^n$ denotes a robot joint configuration. Then a singular configuration $q'$ is a configuration at which the Jacobian $J(q')$ does not have maximum rank. Let $S$ be the set of all singular configurations of a given robot

Is there any (general) result regarding characterization of the set $S$? Any work discussing or answering questions such as "Is $S$ a manifold?", "Does $S$ contain only isolated points?", etc.?

So far I have found quite a few work talking about classification of singular configurations. I think they still do not really answer my questions. Can anyone point me to some related stuff?

• I don't know the answer but this is a very interesting question. You might start with Peiper's thesis from 1968 and follow the chain of Bernie Roth's students from there. Also check out Waldron, Duffy, Stanisic, Lenarcic, Hollerbach, Trevelyan. I'm sure there are many I am missing who contributed to this field. Mar 28, 2017 at 1:59
• Try this paper! dtic.mil/get-tr-doc/pdf?AD=ADA210116 Mar 28, 2017 at 2:23
• @SteveO, interesting! Thank you for your suggestion. Mar 28, 2017 at 14:29
• Partial answer: There is no isolated singularities. I found this written as a result of Theorem 1 from this paper. Mar 30, 2017 at 8:39

## 1 Answer

According to Pai and Leu, 1989, for a generic manipulator (which includes all 6R robots with a spherical wrist), the set of singular points of rank r are smooth manifolds in joint space of codimension (j - r)(k - r), where j is the dimension of joint space and k is the dimension of task space.

• I think the paper actually says that the method to determine genericity of a robot (Theorem 2) is applicable to common classes of robots including those 6R robots with a spherical wrist; not that all 6R robots with a spherical wrist are generic. Apr 3, 2017 at 6:54
• In the Conclusion they state that all 6-joint manipulators with a so-called "spherical wrist" are of this class. Apr 3, 2017 at 9:44
• Yeah, but I still think that they refer to the class in which the method for checking genericity is applicable rather than the class of generic robots. Apr 3, 2017 at 10:07
• I think you are correct about their derivation. Still, I cannot think of any 6R robot with a spherical wrist that would be an exception to this, except for trivial robots whose arms do not span 3-space. Apr 3, 2017 at 13:24