Soft robots achieve functionality through tight coupling among geometry, material composition, and actuation. As a result, effective design optimization requires these three aspects to be considered jointly rather than in isolation. This coupling is computationally challenging: nonlinear large-deformation mechanics increase simulation cost, while contact, collision handling, and non-smooth state transitions limit the applicability of standard gradient-based approaches. We introduce a smooth, low-dimensional design embedding for soft robots that unifies shape morphing, multi-material distribution, and actuation within a single structured parameter space. Shape variation is modeled through continuous deformation maps of a reference geometry, while material properties are encoded as spatial fields. Both are constructed from shared basis functions. This representation enables expressive co-design while drastically reducing the dimensionality of the search space. In our experiments, we show that design expressiveness increases with the number of basis functions, unlike comparable neural network encodings whose representational capacity does not scale predictably with parameter count. We further show that joint co-optimization of shape, material, and actuation using our unified embedding consistently outperforms sequential strategies. All experiments are performed independently of the underlying simulator, confirming compatibility with black-box simulation pipelines. Across multiple dynamic tasks, the proposed embedding surpasses neural network and voxel-based baseline parameterizations while using significantly fewer design parameters. Together, these findings demonstrate that structuring the design space itself enables efficient co-design of soft robots.

翻译：软体机器人通过几何结构、材料组成与驱动机制的紧密耦合实现功能。因此，有效的设计优化需要将这三个方面进行联合考量而非孤立处理。这种耦合在计算上面临挑战：非线性大变形力学增加了仿真成本，而接触、碰撞处理与非光滑状态转换限制了标准梯度优化方法的适用性。本文提出一种平滑的低维设计嵌入方法，将形状变形、多材料分布与驱动机制统一于单一结构化参数空间内。形状变化通过对参考几何的连续变形映射进行建模，材料属性则编码为空间场。二者均基于共享的基函数构建。该表征方法在实现高表达能力协同设计的同时，显著降低了搜索空间的维度。实验表明，设计表达能力随基函数数量增加而提升，这与参数量无法预测表征能力的神经网络编码方式形成对比。进一步实验证明，使用本统一嵌入方法对形状、材料与驱动进行联合协同优化，其性能始终优于顺序优化策略。所有实验均独立于底层仿真器进行，验证了与黑箱仿真流程的兼容性。在多个动态任务中，所提出的嵌入方法在使用更少设计参数的同时，超越了基于神经网络与体素化的基线参数化方法。综上，这些结果表明：通过对设计空间本身进行结构化，能够实现软体机器人的高效协同设计。