# Hybrid Control

Assuming the grasp $G$ is force closure, the robot-object dynamics equation is given by

The velocity constraint for pure rolling is given by

Differentiating above equation, we obtain the acceleration constraint

which implies

Substituting the $\ddot{\qb}$ and $\tilde{P}\ddot{\xb}$ in the system dynamics equation by the pure rolling acceleration constraints above separately, we obtain

and