what is the control circuit for an inductive sensor?

it is being used in a magnetic pick and place machine using four bar linkage. The inductive switch is being used in such a way that if the metal ball will be detected in the detection area. the four bar linkage will move towards the ball where the inductive switch is located, and carry and transfer it to the desired area.

I think the circuit you are looking for is the same as that used in a metal detector of the beat frequency oscillator (BFO) type.  So a search using phrases like BFO and "metal detector" might lead you to some useful circuits.  For a typical BFO metal detector, the area of the search coil is very large, but I think for your application you want to build a coil that is roughly the same size as the metal ball you want to locate.

I've always wanted to build a metal detector like this, and had a specific circuit in mind too.  I found it in a book of mine years ago.  A picture of this page is attached.

Source: Graf, Rudolf F. Encyclopedia of Electronic Circuits. Volume 1. Tab Books 1985. p. 408