Riding the interface: differentiable physics for wave-driven locomotion

A vibrating “vibrobot” can surf across water by shaping the waves it creates. That simple idea opens a quantitative program: build a model where surface waves, body motion, and actuation co-evolve, then make the whole pipeline differentiable so design and control can be optimized directly.

Our work takes the SurferBot concept (Rhee et al., 2022) and turns it into a computational laboratory for interfacial locomotion, grounded in theory (Benham et al., 2024) and implemented in Julia with differentiable solvers.

Why this problem matters

At an air–water interface, surface tension, gravity waves, and added-mass effects dominate. Small actuators can create asymmetric wave fields that carry momentum and generate thrust. Designing such systems by trial-and-error is slow because performance depends on many coupled choices: body shape, mass distribution, motor location, drive frequency and waveform, and environmental parameters.

A differentiable simulator lets us optimize these choices directly against objectives such as average speed, thrust-per-power, or trajectory tracking, with gradients from physics, not heuristics.

Modeling approach

We model the robot as a buoyant, possibly flexible body constrained to the interface and driven by a time-varying internal actuator. The surrounding fluid is represented by an interface-resolving, small-amplitude free-surface model compatible with capillarity and viscous damping at the scales of interest. The body–wave coupling yields a net drift when the radiated waves are directionally biased (as in (Benham et al., 2024)).

Key outputs include time-averaged speed, thrust, hydrodynamic power, and wave momentum flux. These are computed from the state and used as loss functions for design and control.

Fully differentiable simulation in Julia

The pipeline is end-to-end differentiable with respect to parameters (\theta) (geometry, actuator placement, frequency, waveform coefficients):

• State update uses linear and nonlinear solves (A(\theta)\,y=b(\theta)) with custom reverse-mode rules so that (\nabla_\theta \mathcal{L}) is obtained by two linear solves (forward and adjoint) per time step, keeping memory bounded and gradients stable.
• Wave kinematics and body dynamics are coded to avoid non-differentiable switches; contact with the interface is handled via smooth constraints consistent with small-slope theory.
• We expose Jacobian-vector and vector-Jacobian products to AD, enabling first-order methods and quasi-Newton updates over large parameter spaces.

This yields physics-based gradients for objectives like speed, thrust-per-watt, or robustness to perturbations.

Optimization over design and control

With gradients available, we explore:

• Design parameters: hull length/width, mass distribution, motor position, mounting stiffness.
• Control parameters: drive frequency, multi-harmonic waveform coefficients, duty cycles.
• Environment: surface tension, viscosity, depth, background currents.

We run multi-start gradient optimization to locate high-performance regions, then fit surrogates for rapid sweeps and Bayesian optimization for global search under constraints (e.g., power budget, manufacturable geometries).

Scientific questions we address

1. Thrust generation: how do specific wave modes and phase relationships create directional momentum flux at the interface?
2. Efficiency: what combinations of placement and drive reduce wasted radiation while maximizing net drift?
3. Robustness: which designs maintain performance across changes in fluid properties or small manufacturing errors?
4. Control: can we shape wave packets in time to navigate or station-keep against disturbances?

From theory to hardware

The simulator predicts the operating windows where wave radiation produces propulsion without saturating losses to viscous damping or generating counter-productive modes. Because gradients come from the governing equations, the same framework supports parameter inference from experimental trajectories and controller tuning on real devices.

References and related work

• SurferBot concept and experiments: (Rhee et al., 2022)
• Theory of wave-driven propulsion at interfaces: (Benham et al., 2024)