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What is the Degree of Freedom for a k-link planar serial chain for k = 2, 3, 6? Using Grüber Formula, I get k-link chain has k Degrees of Freedom. But the maximum DoF in a plane can be 3. I am not sure what I am missing.

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In robotics, the Degrees of Freedom (DoF) typically refers to the dimension of the Joint Space. In this case, DoF = k.

In the planar case, 3 is the maximum dimension of the Task Space, instead. When k > 3, we basically deal with a redundant planar manipulator.

Here's a quite illustrative series of slides on the topic of Kinematic Redundancy.

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