  Symmetries, e.g. rotational and translational invariances for the class of
mechanical systems, allow to characterize solution trajectories of nonlinear
dynamical systems. Thus, the restriction to symmetry-induced dynamics, e.g. by
using the concept of motion primitives, may be considered as a quantization of
the system. Symmetry exploitation is well-established in both motion planning
and control. However, the linkage between the respective techniques to optimal
control is not yet fully explored. In this manuscript, we want to lay the
foundation for the usage of symmetries in Model Predictive Control (MPC). To
this end, we investigate a mobile robot example in detail where our
contribution is twofold: Firstly, we establish asymptotic stability of a
desired set point w.r.t. the MPC closed loop, which is also demonstrated
numerically by using motion primitives applied to the parallel parking
scenario. Secondly, if the optimization criterion is not consistent with the
symmetry action, we provide guidelines to rigorously derive stability
guarantees based on symmetry exploitation.
