Automated design of soft-rigid hybrid robots for dynamic locomotion

Technical Summary: Automated Design of Soft–Rigid Hybrid Robots for Dynamic Locomotion

Problem Statement
Traditional rigid robots offer precision but lack the compliance needed for unstructured environments, while fully soft robots, though adaptable, struggle with scalability and load capacity due to their inability to efficiently generate or transmit large forces. Biological musculoskeletal systems address this by integrating hard and soft tissues. However, designing artificial soft–rigid hybrid robots remains challenging. Current design processes rely heavily on manual iteration and biological inspiration, which are difficult to translate into concrete, optimized configurations. Furthermore, computational design methods face limitations: explicit time integration often restricts material stiffness contrasts, and continuous design representations (common in topology optimization) struggle to depict discrete structural features like skeletal load paths or mechanism-like force transmission.

Methodology
The authors propose a computational framework for the automated, gradient-based design of soft–rigid hybrid robots. The method simultaneously optimizes three components:

1. Soft-body morphology: Represented as a continuous deformable body.
2. Skeletal structure: Represented as a discrete network of stiff truss elements (bones).
3. Actuation signals: Time-series control inputs for solenoid actuators.

Differentiable Simulation Framework
The core of the methodology is a differentiable simulator that couples two distinct physics solvers:

• Soft Body: Modeled using the Material Point Method (MPM), specifically the moving least squares variant (MLS-MPM). This handles large deformations and contact with the environment.
• Stiff Skeleton: Modeled using Extended Position-Based Dynamics (XPBD). This represents the truss network and solenoid mechanisms as a system of mass points connected by bar constraints. XPBD allows for explicit stiffness control via a compliance parameter without solving large global systems, making it suitable for differentiable optimization.

Optimization Strategy
The system uses an augmented Lagrangian formulation with the Adam optimizer to minimize a loss function. The primary objective is to maximize forward locomotion distance. The optimization includes:

• Penalties: To suppress undesirable behaviors (e.g., collapse, falling, lateral drifting) by penalizing deviations from rigid-body translation.
• Binarization Constraints: To drive design variables (soft density, skeletal density, and actuation pulse parameters) toward binary values (0 or 1), ensuring manufacturable designs.
• Skeletal Constraints: A specific constraint controls the number of bone elements to prevent over-design.
• Actuation Modeling: Solenoids are modeled with state-dependent stiffness (low stiffness during free sliding, high stiffness upon contact) and stroke-dependent force attenuation.

Key Contributions

1. Unified Optimization Framework: The paper presents a method to co-optimize soft body shape, discrete skeletal topology, and actuation timing within a single differentiable simulation loop.
2. Hybrid Physics Solver: By coupling MPM (for soft continua) with XPBD (for discrete trusses), the framework overcomes numerical limitations in simulating high stiffness contrasts and enables the explicit design of skeletal load paths.
3. Automated Discovery: The method generates functional soft–rigid morphologies without manual design, revealing how rigid elements should be distributed to balance flexibility and force transmission.

Results

• Optimization Process: The optimization converges to a design where the soft body maintains a dense structure with specific limb-like features, while the skeletal structure evolves from a dense ground structure into a sparse, truss-like skeleton. The actuation signal converges early, establishing a frequency and phase relationship that the morphology subsequently adapts to.
• Fabrication and Validation: The optimized design was fabricated using silicone rubber (soft body) and 3D-printed PLA (skeleton), actuated by solenoids.
  • Performance: The fabricated robot achieved a sustained walking speed of approximately 69.7 mm/s (mean of 5 trials) over a 5-second duration, traveling 294 mm.
  • Role of Skeleton: Control experiments without the skeletal structure showed significantly reduced performance (~19.9 mm/s) and failed to generate coordinated global deformation, confirming the skeleton's role in transmitting actuation forces to distant body parts.
• Dynamic Analysis: Frequency sweep analysis and experiments revealed that locomotion performance is highly frequency-dependent, with a peak around 10–12 Hz. The optimized actuation frequency (10 Hz) aligns with this peak, suggesting the design exploits resonance-like behaviors where the skeleton enables specific deformation modes (lifting the front limb while stabilizing the rear) that are not possible with a purely soft body.

Significance and Claims
The paper claims that its framework provides a scalable strategy for designing soft–rigid hybrid robots by automating the discovery of effective morphologies that balance flexibility and structural integrity. The authors emphasize that:

• The method successfully bridges the gap between continuous soft-body design and discrete structural design.
• The resulting robots operate at a relatively large scale (150 mm, 1.33 kg), demonstrating that incorporating rigid load-bearing elements can enable scalable soft robotics, consistent with biological strategies for supporting increased body size.
• The optimized structures exhibit both biologically reminiscent features (alternating gait) and engineering-driven characteristics (triangular truss reinforcements), suggesting that optimization-based design can adapt structures to engineering requirements while retaining functional behaviors, potentially offering advantages over direct biological imitation.

The authors note that the current approach is inherently local (gradient-based) and that future work could explore expanding the design space (e.g., optimizing node positions, moment-transmitting structures, and actuator placement) or integrating complementary strategies like evolutionary algorithms.