I've read that 6 dof manipulators give eight closed form solutions, according to a paper. What is the number of iK solutions for the ABB IRB 140? (last three axes intersect but there is a offset between joint 1-2. Does it affect the number of solutions?) Thanks
Actually, in certain configurations you can have an infinite number of IK solutions. For example if joints 4, 5, and 6 are straight and inline as in your image. You can rotate joint 4 by $X$ and 6 by $-X$ to get the same end-effector pose.
This is a great video on arm singularities which also shows this phenomenon nicely: https://www.youtube.com/watch?v=zlGCurgsqg8
You don't typically have these joints inline like this, but in general you can have 2 solutions at the wrist. In one solution, joints 4, 5, and 6 will have values: $X$, $Y$, and $Z$. And in the other solution, they will have values: $180-X$, $-Y$, and $180-Z$. (I think, please verify). I am not sure if there is a standard term for this but I have always called it a "wrist-flip".
You can have another flip with joints 2, 3, and 4 ("elbow-flip"). And another with joints 1, 2 and 3 ("shoulder-flip"). These flips can be combined, creating 2*2*2 = 8 possible permutations of IK solutions.
The presence of an offset joint will not affect the number of IK solutions. You will still be able to do a shoulder-flip. Although I think this means the joint angles won't simply be 180 degrees off from each other. And maybe other joints will need to compensate too.
Note that you will not always get the same number of solutions. Joint limits, self collisions, and construction details can reduce the number of valid solutions. Here is where the offset between joints 1 and 2 comes into play. For example, near the limits of the workspace on one side, you won't be able to get that close if joint 1 is rotated 180 degrees. Eliminating the possibility of a shoulder-flip.