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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

https://search-ext.abb.com/library/Download.aspx?DocumentID=3HAC041346-001&LanguageCode=en&DocumentPartId=&Action=Launch&fbclid=IwAR2atlGRywE3gEGu7Or-LGl5_xKxuD6r_MAAIS5EQxEb7MJwwd2hJRmoi7Q enter image description here

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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.

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  • $\begingroup$ Sorry for the late response. Thanks for the answer I totally understood the things you mentioned. After my search I ended up that this manipulator has 8 possible solutions, like you mentioned. (This is the max number of possible solutions . In some cases solutions are rejected because of singularities) . Taking a look in Craig's book Introduction to Robotics I found out that the more nonzero offsets the manipulator has the more IK possible solutions exist. In some cases IK solutions can reach 16 . $\endgroup$ – Giannis D Jun 20 at 7:34

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