The F-16C had such a rigid stick. It didn't move at all.
Funny enough when the Thrustmaster Cougar was released a lot of people ranted the stick moved, while the original was static and you can indeed buy a mod which replaces the gimbals with a system that only senses the forces on the stick, but renders the stick itself immobile. Realistic for the F-16, but there are only a handful of planes using this method.
As posted above, the pilots didn't like it for the little feel.
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