Design Embedding | Project Page

Abstract

Soft robots achieve functionality through tight coupling among geometry, material composition, and actuation, so effective design optimization requires these three aspects to be considered jointly. This coupling is computationally challenging: nonlinear large-deformation mechanics increase simulation cost, while contact, collision handling, and non-smooth state transitions limit 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, and material properties as spatial fields; both constructed from shared basis functions, enabling expressive co-design while drastically reducing the dimensionality of the search space. In our experiments, design expressiveness increases with the number of basis functions, unlike comparable neural network encodings whose representational capacity does not scale predictably with parameter count, and joint co-optimization of shape, material, and actuation consistently outperforms sequential strategies. The proposed embedding, which operates independently of the underlying simulator and is compatible with black-box pipelines, surpasses neural network and voxel-based baselines across multiple dynamic tasks while using significantly fewer design parameters. These findings demonstrate that structuring the design space itself enables efficient co-design of soft robots.

Overview

Geometry, material distribution, and actuation are each infinite-dimensional in principle. Direct optimization over such spaces is intractable, and naive element-wise encodings scale poorly with mesh resolution. Our design embedding tackles this by representing all three through a shared set of smooth basis functions, evaluated once on the mesh and reused across the whole pipeline.

Unified basis-function embedding. A single low-dimensional parameter vector jointly encodes multi-material distribution and shape morphing via shared Gaussian RBF bases.
Controlled expressiveness. Representational capacity scales predictably with the number of basis functions, unlike neural-field encodings whose capacity saturates early.
Black-box compatible. The embedding requires no differentiable simulator; we pair it with CMA-ES to optimize shape, material, and actuation jointly under contact-rich, non-smooth dynamics.

Overview of the basis function design embedding — **Overview of the Proposed Basis Function Design Embedding.** (A) Material distribution obtained by element-wise argmax evaluation of the score field, assigning each element to its dominant material phase. (B) Basis function parameterization: a finite set of spatial basis functions approximates a continuous field. (C) Shape, material, and actuation are all decoded from one low-dimensional design embedding.

Methods at a Glance

Shared Basis Function Spaces

Multi-material score fields and vector-valued deformation fields are both expanded in Gaussian RBF bases, pre-evaluated once as basis matrices B and A on the mesh, so decoding a design reduces to a matrix–vector product.

Shape via Morphing & Thresholding

Smooth deformation maps morph a reference geometry while preserving mesh topology; an implicit occupancy field built from the material scores enables stronger topology change through element-wise thresholding.

Black-Box CMA-ES Optimization

The reduced design vector is optimized with CMA-ES, a gradient-free evolution strategy well suited to noisy, non-convex, contact-rich objectives evaluated through an opaque forward simulator.

Co-design optimization pipeline — **Co-design Optimization Pipeline.** The simulator evaluates a proposed design and computes its trajectory; CMA-ES updates the parameter vector c to minimize the task loss (top). c is mapped by the encoder to material fields, shape, and actuation signals via precomputed basis matrices A and B (bottom).

Expressiveness of the Design Embedding

We first study representational power independently of task dynamics: how does capacity scale with the number of basis functions, both for material distribution and for shape variation?

Material Distribution Matching

Material score coefficients are optimized to reproduce a prescribed pattern (a torus and a cross) on a fixed 50×50 grid with two materials, at three basis resolutions. At N_Φ=16 the encoder cannot reproduce sharp boundaries; at 36 the topology is recovered; at 64 the match is visually precise — finer basis grids translate directly into higher geometric fidelity.

	N_Φ = 16	N_Φ = 36	N_Φ = 64	Target
Torus
Cross

Intrinsic Dimensionality & Novelty

We sample 2000 random parameter vectors, decode them into shapes by morphing a reference sphere, and measure pairwise Chamfer distances. For the basis function encoder, the intrinsic dimensionality d₉₅ grows from 70 to 395 as the number of basis functions increases; a parameter-matched neural field baseline saturates quickly, rising only from 178 to 228 despite a 20× increase in parameters. A novelty score (nearest-neighbor Chamfer distance) shows the same pattern: steadily increasing for basis functions, nearly flat for neural fields.

Novelty score distribution for the basis function encoder — **Basis Functions Expand the Design Space.** Novelty score distribution (nearest-neighbor Chamfer distance) for 2000 randomly decoded shapes, using the basis function encoder (left) and a parameter-matched neural field encoder (right).

Novelty score distribution for the neural field encoder — **Basis Functions Expand the Design Space.** Novelty score distribution (nearest-neighbor Chamfer distance) for 2000 randomly decoded shapes, using the basis function encoder (left) and a parameter-matched neural field encoder (right).

Encoder	Configuration	Params	d₉₅
Basis function	2×2×2×3	24	70
	4×4×4×3	192	245
	6×6×6×3	648	395
Neural field	3→4→3	31	178
	3→6→12→6→3	207	216
	3→8→16→16→8→3	611	228

Co-Optimization of Design and Actuation

We test whether jointly optimizing shape, material, and actuation outperforms a sequential strategy (fix morphology, then tune actuation) on two dynamic tasks — swimming and jumping — under the same 10,000-simulation budget.

Swimming

An ellipsoid reference body with two actuated muscle groups is optimized for forward displacement with lateral-drift, rotation, and muscle-fraction penalties. The co-optimized swimmer reaches a minimal loss of −67.32 versus −61.89 for the sequential strategy, producing a stable, nearly straight trajectory, while the sequential design exhibits axial drift visible in the rotating muscle group.

Jumping

A cubic reference body with a Gaussian-pulse muscle phase is optimized for jump height and rotation under gravity and ground contact, with topology change through element removal only. The co-optimized jumper reaches a minimal loss of −9.14 versus −7.72 for the sequential strategy, with clearer design asymmetry that enhances rotational movement.

Swimmer loss over generations: sequential vs co-optimization — **Optimization Dynamics: Sequential vs. Co-Optimization.** Loss over CMA-ES generations for the swimmer (left) and jumper (right). In the sequential runs, actuation is released at generation 150, producing a visible jump that still falls short of the jointly optimized result.

Jumper loss over generations: sequential vs co-optimization — **Optimization Dynamics: Sequential vs. Co-Optimization.** Loss over CMA-ES generations for the swimmer (left) and jumper (right). In the sequential runs, actuation is released at generation 150, producing a visible jump that still falls short of the jointly optimized result.

Jumper at rest, t=0ms — **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.

Jumper at ground contact, t=50ms — **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.

Comparison to Baseline Encodings

The neural field baseline for swimming (149 parameters, matched to our 147) makes almost no progress during the first half of optimization and converges to a design with visible lateral drift, while the basis function encoder improves within the first few generations — evidence of a more navigable loss landscape from a linear, spatially local parameterization. For jumping, a per-voxel baseline (346 parameters, roughly 6× more than our 57) yields a fragmented, scattered muscle distribution and a worse final loss despite the larger parameter count, while the basis function jumper stays compact and coherent.

Swimmer task loss: neural field vs basis function encoding — **Basis Functions Outperform Alternative Encodings.** Task loss for the swimmer (left) using neural field vs. basis function encoding, and the jumper (right) using per-voxel vs. basis function encoding.

Jumper task loss: per-voxel vs basis function encoding — **Basis Functions Outperform Alternative Encodings.** Task loss for the swimmer (left) using neural field vs. basis function encoding, and the jumper (right) using per-voxel vs. basis function encoding.

Fragmented per-voxel jumper design — **The Per-Voxel Baseline Fragments.** Despite its larger parameter count, the per-voxel jumper converges to this scattered, disconnected muscle distribution, unlike the compact and coherent basis-function design.

Rollout Comparisons

Animated side-by-side rollouts for every comparison above — co-optimization versus sequential optimization, and the basis-function encoding versus its neural-field and per-voxel baselines.

Co-optimized versus sequential swimmer animation

Swimmer: co-optimized vs. sequential.

Basis-function versus neural-field swimmer animation

Swimmer: basis function vs. neural field.

Co-optimized versus sequential jumper animation

Jumper: co-optimized vs. sequential.

Basis-function versus per-voxel jumper animation

Jumper: basis function vs. per voxel.

Known Limitations

Multi-objective losses require manual weighting of task-specific terms.
Actuation is open-loop; state-dependent feedback control is not addressed.
Fabrication constraints are not explicitly enforced, only handled implicitly by the simulator.
Basis function placement is currently fixed on a uniform grid rather than adaptively refined.
Experimental fabrication and real-world validation of co-designed robots remain future work.

Citation

@article{candiello2026designembedding,
  author  = {Candiello, Vittorio and Mekkattu, Manuel and Michelis, Mike Y. and Katzschmann, Robert K.},
  title   = {A Unified Low-Dimensional Design Embedding for Joint Optimization of Shape, Material, and Actuation in Soft Robots},
  journal = {arXiv preprint arXiv:2603.06497},
  year    = {2026},
  url     = {https://arxiv.org/pdf/2603.06497}
}