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Convergence with Link/Truss or Cable elements is very difficult if you have perfect geometry and impossible with imperfect geometry.
With perfect geometry, it is often required to have several steps, where step 1 does not apply any loads such as displacements, but holds the value at 0. Step 1 has Initial substeps set to 100 to allow the contact to be established while the INISTATE command adds a small amount of strain to the perfect geometry.
One way to help stabilize the model is to hold the fingertip from moving in the Z direction by using a Remote Displacement. Promote it to a Remote Point for later use. Solve the model and obtain the Reaction Force of that Remote Displacement. If it is large, then the tension in the wires is unbalanced. You can set a Remote Force on that Remote Point to be equal to the reaction force found earlier and have that on in step 1 and 2. You can Deactivate the Remote Displacement in Step 2 and the forces in the mechanism will be balanced after the constraint is removed.
Do you see how much work is required to use wires? None of that is required if you use the Spring method.