Final answer:
To determine the passivity range for force gain g in a robotic system, analogies to mechanical systems like a damped mass-spring system and principles of dynamics and energy conservation are used to set up equations and analyze the system's energy properties.
Step-by-step explanation:
The question at hand is asking for the derivation of the range of force gain g that ensures that the closed-loop admittance y(s) of a robotic system is passive. Passivity, in the context of this robotic system, refers to an energy property where the system does not generate energy but can store it or dissipate it. To derive the range of g for passivity, one needs to apply the definition of passivity, which usually includes conditions such as the system being stable, the total energy remaining non-negative, and the energy dissipated by the system being greater than or equal to zero.
To approach this problem, consider the analogy to the mechanical system composed of a mass (m), a damper (b), and a spring (k). First, one must ensure the system is underdamped, which occurs when the square root of k/m is greater than b/2m. From this point, applying the passivity conditions will allow one to derive the range for the force feedback gain g for the given closed-loop system to be passive. By considering Newton's laws of motion and the principles of energy conservation, the process will involve setting up equations based on the dynamics of force and motion, analyzing the system's energy properties, and finding the constraints for g that uphold the system's passive behavior.