Well, when you have a variable link length, all you have to do is assign the length of the link to a variable and then solve the rest of the arm. You will have your joint solutions in terms of a variable link length. You then substitute values in for you link lengths. End of solver.
Linear joints just need to be stored as a simple variable in order to propagate through a solver. Rotational joints involve trigonometry, which is less straightforward. You know how tool tip poses are represented? XYZ, rpy
They aren't represented in RcosTheta, RsineTheta, etc.
They are in Cartesian coordinates.
This is because Cartesian coordinates are easier to understand during presentation.
In other words, the convention makes translation joints trivial.
Link lengths are always assumed to be a constant in solvers, and the algebra used to solve angular joints does not require link lengths be constant or variable, they are always dependent on link lengths anyways. In other words, link lengths don't matter, because the math doesn't change.
It is assumed, given your question, that the maximum and minimum link length values are known. Given this assumption, you need to run your IK solver twice - once with the minimum link length, and once with the maximum. That will give you the range of possible angular joint values. That is what you are looking for.
When you have linear links that change values, you are creating a range of valid angular joint values. This is the crux of your question. You don't have 1 value, you have infinite values.
