We present a comprehensive study on discrete morphological symmetries of dynamical systems, which are commonly observed in biological and artificial locomoting systems, such as legged, swimming, and flying animals/robots/virtual characters. These symmetries arise from the presence of one or more planes/axis of symmetry in the system's morphology, resulting in harmonious duplication and distribution of body parts. Significantly, we characterize how morphological symmetries extend to symmetries in the system's dynamics, optimal control policies, and in all proprioceptive and exteroceptive measurements related to the system's dynamics evolution. In the context of data-driven methods, symmetry represents an inductive bias that justifies the use of data augmentation or symmetric function approximators. To tackle this, we present a theoretical and practical framework for identifying the system's morphological symmetry group $\G$ and characterizing the symmetries in proprioceptive and exteroceptive data measurements. We then exploit these symmetries using data augmentation and $\G$-equivariant neural networks. Our experiments on both synthetic and real-world applications provide empirical evidence of the advantageous outcomes resulting from the exploitation of these symmetries, including improved sample efficiency, enhanced generalization, and reduction of trainable parameters.

翻译：本文对动力系统中常见的离散形态对称性进行了全面研究，这类对称性普遍存在于生物与人工运动系统中，例如腿式、游泳式及飞行式动物/机器人/虚拟角色。此类对称性源于系统形态结构中存在一个或多个对称平面/轴，导致身体部件的和谐复制与分布。重要的是，我们刻画了形态对称性如何延伸至系统动力学、最优控制策略以及所有与系统动力学演化相关的本体感受与外感受测量中的对称性。在数据驱动方法背景下，对称性作为一种归纳偏置，为数据增强或对称函数逼近器的应用提供了合理性。为解决该问题，我们提出了一套理论与实践的框架，用于识别系统的形态对称群$\G$，并刻画本体感受与外感受数据测量中的对称性。随后，我们通过数据增强与$\G$-等变神经网络利用这些对称性。在合成数据与真实场景中的实验提供了经验证据，证明利用这些对称性可带来样本效率提升、泛化能力增强及可训练参数数量减少等优势。