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The dominant scheme for the last stage is impedance control rather than pure position control, and the distinction matters for exactly the reason a robot has to touch things. A pure position controller tries to force a joint to a commanded angle regardless of what resists it, which is dangerous the instant the arm meets an object, a surface, or a person, because the controller will keep applying more torque to fight the very contact it should be responding to. An impedance controller instead makes the joint behave like a programmable spring and damper around the commanded target: $\tau = K_p (q_d - q) + K_d (\dot{q}_d - \dot{q}) + \tau_{ff}$. Here $q_d$ and q are the desired and measured joint positions, $K_p$ and $K_d$ set how stiff and…

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