Topology optimization is a powerful tool for designing structures in many fields, but has been limited to static or passively moving objects made of hard materials. Designing soft and actively moving objects, such as soft robots equipped with actuators, poses a challenge because the optimal structure depends on how the object will move and simulating dynamics problems is difficult. We propose "4D topology optimization," which extends density-based topology optimization to include time, as a method for optimizing the structure and movement of self-actuating soft bodies. The method represents the layout of both material and time-varying actuation using multi-index density variables distributed over the design domain, thus allowing simultaneous optimization of structure and movement using gradient-based methods. Forward and backward simulations of soft bodies are done using the material point method, a Lagrangian--Eulerian hybrid approach, implemented on a recent automatic differentiation framework. We present several numerical examples of designing self-actuating soft bodies targeted for locomotion, posture control, and rotation tasks. The results demonstrate that our method can successfully design complex shaped and biomimetic moving soft bodies due to its high degree of design freedom.

翻译：拓扑优化是众多领域中设计结构的强大工具，但此前仅限于由硬质材料制成的静态或被动运动物体。设计带有执行器的软体机器人等主动运动软体物体极具挑战性，因为最优结构取决于物体运动方式，且动力学问题模拟十分困难。我们提出“4D拓扑优化”——将基于密度的拓扑优化扩展至时间维度，作为优化自驱动软体结构与运动的方法。该方法利用分布在设计域上的多指标密度变量表示材料布局与时变驱动，从而可通过基于梯度的优化方法同步优化结构与运动。通过采用拉格朗日-欧拉混合方法——物质点法，在近期开发的自动微分框架上实现软体物体的正向与逆向模拟。本文展示了针对运动、姿态控制及旋转任务的自驱动软体设计数值实例。结果表明，该方法凭借高度设计自由度，可成功设计出复杂形状及仿生运动软体物体。