In my understanding of standard robot control, "Yacobian" is the keyword. The kinematics of multi DOF robot arms consist of a lot of sin, cos, and tan. Hence, it is very hard to obtain the inverse solution. However, for a very small movement of the arm, the kinematics is described with linear simultaneous equations. So, you can solve them with derivatives of the kinematics. An actual robot control is a repetition of small movements, Hence, this is good enough to create a control program.
I agree with Figen Ozen. The inverse kinematics in open kinematics chain has multiple solutions which require a program to solve it and exclude solutions don't match with the system constraints