cs.ROMar 6, 2026

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

Authors: Vittorio Candiello, Manuel Mekkattu, Mike Y. Michelis, Robert K. Katzschmann

Organizations: Soft Robotics Lab, D-MAVT, ETH Zurich, Switzerland. · ETH AI Center, ETH Zurich, Switzerland.

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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 & tables

Fig. 1 : Overview of the Proposed Basis Function Design Embedding. (A) Material distribution obtained by element-wise arg⁡max⁡\arg\max evaluation of the score field ϕ\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.
Fig. 2 : Co-design Optimization Pipeline. The simulator evaluates a proposed design and computes its trajectory, CMA-ES updates parameter vector c\bm{c} to minimize the task loss (top). c\bm{c} is mapped by the encoder E\mathcal{E} to material fields, shape, and actuation signals via precomputed basis matrices AA and BB (bottom).
Fig. 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.
EncoderConfig.Paramsd95d_{95}
Basis function2×2×2×32\times 2\times 2\times 32470
4×4×4×34\times 4\times 4\times 3192244
6×6×6×36\times 6\times 6\times 3648394
Neural field3→4→33\to 4\to 331203
3→6→12→6→33\to 6\to 12\to 6\to 3207227
3→8→16→16→8→33\to 8\to 16\to 16\to 8\to 3611238
TABLE I : Intrinsic dimensionality d95d_{95} , the number of principal components capturing 95% of design variance, for basis function and neural field encoders at three parameter counts.
Fig. 4 : Basis Functions Expand the Design Space. Novelty score distribution νi\nu_{i} (nearest-neighbor Chamfer distance) for N=2000N{=}2000 randomly decoded shapes (top) using basis function (left) and neural field encoder (right).
Fig. 5 : Optimization Dynamics: Sequential vs Co-Optimization. Loss over CMA-ES generations for swimmer (left) and jumper (right) under sequential and co-optimization.
Fig. 6 : Co-Optimization Yields Superior Jump Dynamics. Snapshots at rest, ground-contact compression, and mid-flight for co-optimized (front) and sequentially optimized (back) jumper designs.
Fig. 7 : Basis Functions Outperform Alternative Encodings. Task loss for the swimmer (left) using neural field vs. basis function encoding, and the jumper (right) using voxel-based vs. basis function encoding.
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