A Unified Low-Dimensional Design Embedding for Joint Optimization of Shape, Material, and Actuation in Soft Robots

TL;DR

This work introduces a smooth, low-dimensional design embedding for soft robots that unifies shape morphing, multi-material distribution, and actuation within a single structured parameter space and demonstrates that structuring the design space itself enables efficient co-design of soft robots.

Abstract

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.

Figures (7)

• Figure 1: Overview of the Proposed Basis Function Design Embedding. (A) Material distribution obtained by element-wise $\arg\max$ evaluation of the score field $\phi$, assigning each element to its dominant material phase. (B) Basis function parameterization: a finite set of spatial basis functions can approximate a continuous field. (C) Optimized swimmer trajectories (with lateral drift penalty): the basis function encoder with joint optimization (middle) produces a straight forward-moving trajectory, outperforming sequential optimization (bottom) and the neural field encoder (top), which exhibit a lateral drift.
• Figure 2: Co-design Optimization Pipeline. The simulator evaluates a proposed design and computes its trajectory, CMA-ES updates parameter vector $\bm{c}$ to minimize the task loss (top). $\bm{c}$ is mapped by the encoder $\mathcal{E}$ to material fields, shape, and actuation signals via precomputed basis matrices $A$ and $B$ (bottom).
• Figure 3: Scaling of Material Expressiveness. Material distribution matching for 2D torus (top) and cross (bottom). With increasing basis resolution, we find a better match with the target pattern.
• Figure 4: Basis Functions Expand the Design Space. Novelty score distribution $\nu_i$ (nearest-neighbor Chamfer distance) for $N{=}2000$ randomly decoded shapes (top) using basis function (left) and neural field encoder (right).
• Figure 5: Optimization Dynamics: Sequential vs Co-Optimization. Loss over CMA-ES generations for swimmer (left) and jumper (right) under sequential and co-optimization.