Soft-Rigid Co-Design Framework

• Soft–rigid co-design frameworks are comprehensive methodologies that jointly optimize compliant and rigid subcomponents using joint parameterizations and surrogate-driven techniques.
• They employ mixed-integer, constrained multi-objective formulations with tailored parameterizations and both analytical and neural surrogate models to map design variables to physical performance metrics.
• Practical implementations demonstrate up to 90% reduction in prototyping iterations while effectively bridging the sim-to-real gap and integrating control policy co-design.

Soft–rigid co-design frameworks systematically integrate the design optimization of compliant and rigid subcomponents in robotic systems, enabling performance, robustness, and manufacturability targets to be achieved efficiently. These frameworks replace traditional ad-hoc or sequential workflows with joint parameterizations, surrogate-driven optimization, and fast prototyping across a wide range of soft–rigid devices, including manipulators, grippers, and protective exoskeletons (Mansueto et al., 2024, Patterson et al., 3 Jun 2025, Wang et al., 6 Jan 2026, Yi et al., 26 May 2025, Stölzle et al., 20 Apr 2025, Patterson et al., 2024). They address the sim-to-real gap arising from manufacturing tolerances and accommodate geometric, material, and actuation variables, often via black-box or surrogate models, to learn optimal mappings from desired mechanical or task-specific metrics to physical realizations.

1. Mathematical Problem Formulations for Co-Design

Soft–rigid co-design is typically cast as a mixed-integer, constrained, multi-objective optimization. The design variable vector $x$ collects both continuous (geometry, material) and discrete (segment count, layer, mold partition) parameters. For a generalized description:

• Objective:

$$\min_{x\in\mathcal X} F(x) = [f_1(x),\dots,f_J(x)]^\top$$

where each $f_j$ represents a metric such as task-level performance, mechanical compliance, durability, manufacturability, or safety (Stölzle et al., 20 Apr 2025).

• Constraints:
  • Bound constraints: $x_\mathrm{min} \leq x \leq x_\mathrm{max}$
  • Integrality/feasibility: $x_i \in \mathbb{Z}$ as required
  • Fabrication constraints: minimal wall thickness, maximal span, segment resolution, etc.
• Physics-based mapping: The forward model $\phi(x)$ provides a black-box relationship from $x$ to key physical properties (e.g., principal stiffness components) (Mansueto et al., 2024).
• Typical loss/residual:

$$\mathcal{R}(x) = \sum_{i}\left(\frac{\phi_i(x)-\hat{k}_i}{\hat{k}_i}\right)^2$$

to ensure matching of target mechanical properties, appropriately weighted for scale.

• Multi-objective scalarization: Weighted sum or Pareto front extraction as in $J_w(x) = \sum w_j f_j(x)$ (Stölzle et al., 20 Apr 2025).

2. Geometric and Material Parameterization Strategies

Frameworks for soft–rigid co-design employ detailed parametric forms tailored to the structural role of each subcomponent:

• WaveJoint Grippers: Geometry is parameterized by $\mathbf{x}=(l_t, n_r, h_t, t_h, \alpha)$, capturing “wavelength” length, ridge count, amplitude, filament thickness offset, and twist angle. Rigid segments use fixed-dimension ABS parallelepipeds; compliant joints are generated by extruding and twisting a 2D cosine profile (Mansueto et al., 2024).
• Trimmed Helicoid Hybrids: Segments are specified by height $H$, diameter $D$, wall width $w$, thickness $t$, and helix turn count $N_h$, with derived parameters such as axial/bending stiffness based on closed-form beam theory (Patterson et al., 3 Jun 2025).
• Neural Physics Soft Grippers: Blockwise Young’s modulus vector $\mathbf{k}$ is assigned to segment groups, with tendons modeled using uniform-pressure routing and each block mapped to print settings via empirical calibration (Yi et al., 26 May 2025).
• Modular Manipulators: Stacked soft modules use foam stiffness, plate dimensions, and tendon routing as core variables; rigid actuators interlink soft modules, forming a serial or hybrid configuration (Patterson et al., 2024).

The parameterization ensures explicit control over spatial distribution of compliance and stiffness, enabling direct modulation of task-relevant physical behavior.

3. Surrogate Models, Optimization Algorithms, and Prototyping

Due to the expense of high-fidelity simulations and real-world prototyping, soft–rigid co-design frameworks heavily utilize surrogate models:

• Radial Basis Function (RBF) Surrogates: Black-box mappings from design parameters to simulated/measured mechanical properties are interpolated using RBFOpt, supporting fast global optimization and integrating both “cheap/noisy” (FEM) and “expensive/exact” (physical) data points. Acquisition functions balance minimization of predicted residual and exploration in under-sampled regions (Mansueto et al., 2024).
• Analytical Closed-form Models: Axial and bending stiffness for architectured segments are estimated analytically for rapid multi-dimensional parameter sweeps, validated by targeted FEM runs and experimental compression tests (Patterson et al., 3 Jun 2025).
• Neural Surrogates: Supervised neural networks approximate the forward physics of soft grippers, predicting force, displacement, and collision from blockwise stiffness and pose. These surrogates are 1000× faster than differentiable-FEM for end-to-end optimization (Yi et al., 26 May 2025).
• Prototyping Workflow: Design–manufacture cycle is streamlined. 3D printing in ABS or molding in LSR enables rapid iteration, with measured properties fed back into the optimization loop (RBFOpt/GPs). Mold CAD tools are parameter-driven and produce batch-ready, high-fidelity molds within minutes (Mansueto et al., 2024, Patterson et al., 3 Jun 2025).

Such surrogate-driven pipelines reduce prototyping iterations by 70–90% versus naive approaches and are crucial for managing the sim-to-real gap and fabrication uncertainties.

4. Integrated Control and Co-Design

Advanced frameworks integrate controller optimization with mechanical design, capturing cross-domain dependencies:

• Control Policy Variables: PD, LQR, or model-predictive control parameters are co-optimized with morphological design, typically including feedback gains, MPC horizons, or neural policy weights (Stölzle et al., 20 Apr 2025, Patterson et al., 2024).
• Shape and Impedance Controllers: Modular manipulators embed both configuration-space PD+shape controllers, which compensate for geometry- and contact-dependent stiffness jumps, and Cartesian impedance layers to render target mass-spring-damper properties at the end effector (Patterson et al., 2024).
• Coupling Strategies: Foam/rubber stiffness, tendon routing, and rigid-link inertia directly affect achievable feedback gains and stability margins; mechanical bandwith informs required actuation and sensor placement (Patterson et al., 2024, Yi et al., 26 May 2025).
• Learning in the Loop: For dynamic safety (humanoid robots), RL policies are trained in MuJoCo+RT-FEM simulations, learning to reorient during falls to maximize energy absorption by soft overlays or regions with custom stiffness (Wang et al., 6 Jan 2026).

5. Experimental Validation and Performance Outcomes

Soft–rigid co-design frameworks are supported by rigorous simulation and physical validation:

• Optimization-Driven Grippers: RBFOpt converges to sub-$10^{-3}$ relative-error residuals in simulated scenarios, with hardware convergence (real-world $\mathcal{R}_{\rm real}\approx0.01$) after a single prototype and $>70\%$ reduction in physical trial count compared to grid search or neural inverse models (Mansueto et al., 2024).
• Trimmed Helicoid Hybrids: Closed-form models for bending/axial stiffness show 1–20% error against FEM and experiment; design sweeps allow rapid trade-off analysis between compliance, mass, workspace, and manufacturability (Patterson et al., 3 Jun 2025).
• Neural Surrogate Grippers: Co-designed grippers achieve 92% in-domain and 82% out-of-domain success rates (simulation), outperforming rigid or soft-uniform baselines. Real hardware tests confirm high grasp reliability and successful adaptation to diverse objects (Yi et al., 26 May 2025).
• Humanoid Protection: Soft-rigid overlays reduce joint peak impact forces by 70–87%, enable survival of $>40$ high-drop events, and decrease ground/object pressure areas by 79–99% (Wang et al., 6 Jan 2026).
• Manipulators: Modular, cable-driven arms transition from highly compliant to stiff states on demand, supporting up to 300 g payloads per module and achieving step responses within 0.5 s (Patterson et al., 2024).

Key experimental results are summarized below:

System	Method/Framework	Physical Trials Saved	Task Metric Outperformance	Reference
WaveJoint Gripper	RBFOpt surrogate, FEM+bench	70–90% fewer prints	$\mathcal{R}_{\rm real} \approx 0.01$	(Mansueto et al., 2024)
Trimmed Helicoid Hybrid Arm	Analytical stiffness model, molded parts	Mold batch in <30min	1–20% abs error on $k_{ax},k_{bend}$	(Patterson et al., 3 Jun 2025)
Neural Physics Soft Gripper	End-to-end NN surrogate	%%%%22$H$23%%%% faster	+32% sim/hardware success vs. baseline	(Yi et al., 26 May 2025)
Humanoid SRM Protector	Rate-dependent viscoplastic + RL	N/A	70–87% impact reduction, $>40$ drops	(Wang et al., 6 Jan 2026)
Modular Hybrid Manipulator	PCC+shape+impedance control	N/A	300 g/module, sub-5° tracking error	(Patterson et al., 2024)

6. Generalization and Current Limitations

The core black-box/surrogate architecture facilitates transfer to diverse soft–rigid architectures, as only the forward model $\phi(\cdot)$ or its surrogate requires system-specific implementation. This makes the methodology applicable to wave joints, flexures, trimmed helicoid bodies, self-contact modules, and dynamic overlays, provided appropriate parameterizations and interface constraints are established (Mansueto et al., 2024, Patterson et al., 3 Jun 2025, Yi et al., 26 May 2025, Stölzle et al., 20 Apr 2025, Patterson et al., 2024, Wang et al., 6 Jan 2026).

Major limitations include the computational scaling of RBF/interpolant surrogates ($O(k^3)$ cost with dataset size), lack of full co-design for rigid-link geometry and actuation in some treatments, and reliance on analytical or surrogatized models that may not capture all high-dimensional, nonlinear interactions or failure modes (e.g., extreme mesh collapse in FEM, non-Euclidean deformation spaces). Real-to-sim validation is therefore a critical final loop in pipeline design (Mansueto et al., 2024, Stölzle et al., 20 Apr 2025).

7. Future Directions

Research trajectories seek to expand soft–rigid co-design frameworks in several directions:

• Full-Stack Automation: Embedding black-box loops inside high-level topology and morphology optimization, supporting both shape and topology co-design across multiple modules or limbs (Mansueto et al., 2024).
• Advanced Surrogate Learning: Leveraging diffusion models, Gaussian processes, and spectral submanifold discovery for richer, multidimensional response surfaces and improved uncertainty quantification (Stölzle et al., 20 Apr 2025).
• Complex Hybrid Systems: Application to multi-fingered hands, systems with dynamic self-modulation (e.g., tunable jamming), and architectures with embedded, distributed sensing and actuation.
• Control-Integrated Design: Refining co-design to explicitly capture controller feasibility, stability, and safety constraints, with joint optimization of policy and device architecture (Stölzle et al., 20 Apr 2025, Patterson et al., 2024).
• Robustness and Durability: Incorporation of fatigue life, real-world environmental impact, manufacturability constraints, and stakeholder input through multi-objective and probabilistic modeling (Stölzle et al., 20 Apr 2025).

A plausible implication is the convergence toward holistic frameworks unifying materials, geometry, control policy, and real-world iterated prototyping into single, auditable design pipelines suitable for large-scale deployment in human-centered robotics (Stölzle et al., 20 Apr 2025, Patterson et al., 3 Jun 2025).