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Assuming the camera is mounted on the robot arm somewhere, hand-eye calibration is unavoidable. It will give you the pose of the camera w.r.t. robot base, bTc. Next, you get the pose of the teat w.r.t. camera, tTc. And you know the current position of the end effector (robot end point) w.r.t. robot base, bTe. From these, you can calculate the pose of teat w.r.t. end effector as eTt = inv(bTe) x bTc x cTt. Then you move your end effector by that amount, which should bring it to the teat.

Thanks @samarth, for providing solution