4D topology optimization: Integrated optimization of the structure and movement of self-actuating soft bodies

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.