Bio-Hybrid Feathered Robot Model

• The paper details a bio-hybrid robotic wing model that segments into 40% rigid and 60% flexible regions, enabling predictive FSI simulations under low-Re flows.
• The study applies tailored material properties and geometric segmentation inspired by avian biomechanics, validated by CFD and wind-tunnel results with <4% deviation.
• The research demonstrates that spatially graded stiffness optimizes aeroelastic behavior, providing design guidelines to balance load damping and structural integrity.

A bio-hybrid feathered robot model represents an engineering framework for morphing aerial structures that emulate avian wing morphology by segmenting into rigid and flexible components and prescribing spatially tailored material properties. This configuration is numerically modeled under low Reynolds number ($Re = 5 \times 10^{5}$) aerodynamic loading, capturing dynamic fluid–structure interaction (FSI) effects by explicitly coupling an incompressible Navier–Stokes solver to an elastodynamic solid mechanics solver. The model is grounded in geometric, material, and computational parameters derived from experimental biology and implemented in a partitioned FSI scheme, enabling predictive simulations for optimizing robotic wing performance under biologically relevant flow regimes (Boughou et al., 2023).

1. Geometric Segmentation and Computational Mesh

The geometry of the bio-hybrid feathered robot wing is defined by a NACA 6409 airfoil with a chord length $c$ (representative value: $0.1$ m) and unit span (2D approximation, one cell in the $z$ direction). The model explicitly divides the wing along the chordwise axis:

• Leading-edge (Bone/Muscle Analog): $0 \leq x/c \leq 0.40$ is treated as a rigid structure, representing a composite of avian bone and muscle.
• Trailing-edge (Feather Analog): $0.40 < x/c \leq 1.00$ is modeled as an elastic shell, capturing the morphodynamics of feather vane and barbs.

The computational domain employs a structured O-grid mesh surrounding the airfoil, with an outer radial extent of approximately $20c$ and a wake region extending $30c$ downstream. Solid mesh discretization encompasses the shell-like flexible region with surface faces and volumetric cells corresponding to the prescribed thickness, ensuring $y^{+}<1$ at fluid–solid interfaces for boundary layer resolution.

2. Biologically Inspired Material Modeling and Stiffness Tailoring

Material distributions use empirical data from avian feather and bone studies (Bachmann et al. 2012, Bonser 1995, Cameron 2003, Pabisch et al. 2010):

• Rigid Segment (“Bone/Muscle”):
  • Young’s modulus: $E_{\mathrm{rigid}} \approx 3$ GPa
  • Density: $\rho_{\mathrm{rigid}} \approx 1\,200$ kg/m³
  • Poisson’s ratio: $\nu_{\mathrm{rigid}} \approx 0.3$
  • Bending stiffness: $E_{\mathrm{rigid}} I_{\mathrm{rigid}}$ with constant thickness
• Flexible Segment (“Feather”):
  • Young’s modulus at base ($x/c = 0.40$): $E_{\mathrm{base}} \approx 7$ GPa (keratin, rachis root)
  • Modulus at tip: $E_{\mathrm{tip}} \rightarrow 0$ (feather vane)
  • Linear spatial variation:

  $$E_{\mathrm{feather}}(x) = E_{\mathrm{base}} \left[1 - \frac{x/c - 0.40}{0.60}\right]$$ - Density: $\rho_{\mathrm{feather}} \approx 1\,300$ kg/m³ - Second moment of area: $I(x)$ based on linearly tapering thickness $t(x)$ ($t_{\mathrm{base}}\approx 0.5$ mm → $t_{\mathrm{tip}}\approx 0.05$ mm) - Flexural stiffness distribution: $D(x) = E_{\mathrm{feather}}(x)I(x)$

This parameterization enables tailoring of mechanical compliance across the flexible region, allowing systematic investigation of tip deflection and aeroelastic stability as functions of material gradation.

3. Fluid–Structure Interaction Computational Framework

The FSI simulation is implemented by tightly coupling OpenFOAM’s incompressible finite-volume CFD solver (V9 or foam-extend) with an updated Lagrangian finite-volume solid mechanics module. The two-way, partitioned coupling utilizes the foam-extend FSI libraries (Cardiff et al.), with primary steps at each pseudo-time step:

1. Fluid Solve: PIMPLE algorithm (merged PISO/SIMPLE), turbulence closure via $k$–$\omega$ SST, no-slip BC on deforming surface.
2. Interface Load Transfer: Pressure and wall shear are interpolated to the solid interface.
3. Structural Solve: Elastodynamics for hyperelastic/orthotropic material, including mass, damping, and tangent stiffness contributions.
4. Mesh Update: Interface displacements inform dynamic mesh morphing using radial basis functions.
5. Under-Relaxation: Aitken relaxation scheme stabilizes interface motion; coefficient updated each iteration.

Boundary conditions comprise uniform inflow at the far-field, zero-gradient outflow, symmetry on upper/lower boundaries, fluid–solid velocity and traction continuity at the interface ($u_f=u_s$, $\sigma_f\cdot n=\sigma_s\cdot n$), fixed structural clamping at $x/c=0$, and a free trailing-edge.

4. Governing Equations and Constitutive Laws

The simulation relies on the following system of equations:

• Incompressible Navier–Stokes (Low Re):

$$\frac{\partial \mathbf{u}}{\partial t} + \nabla \cdot (\mathbf{u} \otimes \mathbf{u}) = -\frac{1}{\rho_f}\nabla p + \nu \nabla^2 \mathbf{u}$$

with $\nabla \cdot \mathbf{u} = 0$, $Re = U_{\infty}c/\nu = 5\times 10^5$.

• Solid Elastodynamics (Finite-Volume Discretization):

$$\mathbf{M} \ddot{\mathbf{u}} + \mathbf{C} \dot{\mathbf{u}} + \mathbf{K} \mathbf{u} = \mathbf{F}_{\mathrm{aero}}$$

where $\mathbf{F}_{\mathrm{aero}}$ results from mapping fluid pressures and shear.

• Fluid–Structure Coupling:

$$\sigma_f \cdot n_f + \sigma_s \cdot n_s = 0, \quad u_f = u_s \ \text{(on interface)}$$

• Aerodynamic Force Integration (2D):

$$F_{\mathrm{aero}}(x) = \int_0^{L_{\mathrm{out}}} \left[-p(x,s) n(x,s) + \tau(x,s)\cdot n(x,s)\right] ds$$

• Flexural Stiffness Variation along Feather:

$$E(x) = E_{\mathrm{base}} \left(1-\frac{x/c-0.40}{0.60}\right), \quad I(x) = \frac{b\,t(x)^3}{12}$$

where $t(x)$ tapers linearly from $t_{\mathrm{base}}$ to $t_{\mathrm{tip}}$.

Time-stepping is performed with $\Delta t_f \sim 10^{-4}$ s (CFL $<0.5$), using 3–5 coupling iterations per step to reduce interface residuals below $10^{-5}$.

5. Validation and Performance Assessment

Aerodynamic validation at $15^\circ$ angle of attack and $Re=5\times 10^5$ shows $C_{l,CFD}=1.10$ compared to experimental $C_{l,exp}\approx 1.15$ (UIUC wind-tunnel, Selig 1995) with less than $4\%$ deviation. Surface $C_p$ distributions agree within $\pm0.05$.

For a moderate modulus ($E=2.5$ GPa), tip deflection stabilizes at approximately $2.2\%$ of the $0.6c$ flexible length, with maximum bending stress below $60$ MPa (well below keratin yield of $\sim200$ MPa). Lowering $E$ to $689$ MPa increases $\Delta y_{tip}$ to $10\%L$ and amplifies stress fluctuation, indicating substantial aeroelastic sensitivity to stiffness tailoring.

Fluid–structure trends demonstrate that moderate compliance maintains stable FSI coupling (small separation bubble, minimal unsteadiness), whereas excessive flexibility induces large deflections and stall delay but increases risk of material fatigue.

6. Robotic Wing Design Recommendations

Design guidelines for robotic applications derived from simulation results include:

• Spatial tailoring of $E(x)$ to constrain $\Delta_{tip} \lesssim 3\%L$ for performance–integrity trade-off.
• Fine mesh discretization near the FSI interface and Aitken relaxation to maintain coupling stability.
• Emulation of feather-like modulus and thickness tapering to leverage nature-inspired dynamic load damping.
• Pre-validation of aerodynamic coefficients using XFoil or experimental benchmarks prior to full FSI simulation.

The prescribed FSI modeling approach, segmenting the wing into $40\%$ rigid and $60\%$ flexible regions with linearly graded biomimetic material properties, enables accurate prediction and optimization of bio-hybrid feathered robot wing behavior under low-Reynolds-number flow (Boughou et al., 2023).