path following control using kinematics and dynamics

Hello,

I have a general question regarding what is the best controller (PI, PD, etc.) to control the position of a mobile robot in order to follow a known path.

The mobile robot at hand is a mechanism (a 4 bar linkage with its two base gliding on two parallel but not straight rails) and I want the end effector to follow the center-line between the two rails.

I have a path following controller working perfectly with the kinematics using just a P controller.

The problem occurs when I introduce the Dynamics. The Dynamics controller I am using is a PD.

I am controlling the Dynamics based on the kinematics path following control.

I do not know if this kind of control has a name ???

Thank you for your answer.