This paper proposes a reference model approach for the trajectory tracking of a four-wheeled omnidirectional mobile robot. In particular, the error model is brought to a quasi-linear-parameter-varying (LPV) form suitable for designing an error-feedback controller. It is shown that, if polytopic techniques are used to reduce the number of constraints from infinite to finite, a solution within the standard LPV framework could not exist due to a singularity that appears in the possible values of the input matrix. Adding a switching component to the controller allows solving this problem. Moreover, a switching LPV virtual actuator is added to the control loop in order to obtain fault tolerance within the fault-hiding paradigm, keeping the stability and some desired performances under the effect of actuator faults without the need of retuning the nominal controller. The effectiveness of the proposed approach is shown and proved through simulation and experimental results.