We have assumed for the plate that it is free to rotate \(\theta\) and \(\phi\) around two orthogonal cartesian coordinates \(x\) and \(y\) respectively which are fixed to a joint so this system can not translate.
The sphere displacement mechanism only admits to roll over the plate without slip or loss of contact so with such simplifications we can relate all the angular cinematic of the ball to a correspondent linear quantity. The actuators action is included in Lagrange equations as two non conservative torques in each direction which we discuss better in the next section and then we are able to merge both dynamical subsystems.