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.

对称性，例如机械系统类别的旋转和平移不变性，可以表征非线性动力系统的解轨迹。因此，例如通过使用运动原语的概念对对称性引起的动力学的限制可以被认为是系统的量化。在运动计划和控制中都充分利用了对称性。但是，尚未充分探讨各个技术与最佳控制之间的联系。在本手稿中，我们希望为模型预测控制（MPC）中对称性的使用奠定基础。为此，我们详细研究了一个移动机器人示例，其中，我们的贡献是双重的：首先，我们通过MPC闭环建立所需设定点的渐近稳定性，这也通过使用应用于平行停车场景的运动图元进行了数值演示。其次，如果优化准则与对称作用不一致，则我们提供了基于对称利用严格导出稳定性保证的准则。