The desert-dwelling sandfish (Scincus scincus) moves within dry sand, a material that displays solid and fluidlike behavior. High-speed x-ray imaging shows that below the surface, the lizard no longer uses limbs for propulsion but generates thrust to overcome drag by propagating an undulatory traveling wave down the body. Although viscous hydrodynamics can predict swimming speed in fluids such as water, an equivalent theory for granular drag is not available. To predict sandfish swimming speed, we developed an empirical model by measuring granular drag force on a small cylinder oriented at different angles relative to the displacement direction and summing these forces over the animal movement profile. The agreement between model and experiment implies that the noninertial swimming occurs in a frictional fluid.

Discovery of fundamental principles which govern and limit effective locomotion (self-propulsion) is of intellectual interest and practical importance. Human technology has created robotic moving systems that excel in movement on and within environments of societal interest: paved roads, open air and water. However, such devices cannot yet robustly and efficiently navigate (as animals do) the enormous diversity of natural environments which might be of future interest for autonomous robots; examples include vertical surfaces like trees and cliffs, heterogeneous ground like desert rubble and brush, turbulent flows found near seashores, and deformable/flowable substrates like sand, mud and soil. In this review we argue for the creation of a physics of moving systems-a 'locomotion robophysics'-which we define as the pursuit of principles of self-generated motion. Robophysics can provide an important intellectual complement to the discipline of robotics, largely the domain of researchers from engineering and computer science. The essential idea is that we must complement the study of complex robots in complex situations with systematic study of simplified robotic devices in controlled laboratory settings and in simplified theoretical models. We must thus use the methods of physics to examine both locomotor successes and failures using parameter space exploration, systematic control, and techniques from dynamical systems. Using examples from our and others' research, we will discuss how such robophysical studies have begun to aid engineers in the creation of devices that have begun to achieve life-like locomotor abilities on and within complex environments, have inspired interesting physics questions in low dimensional dynamical systems, geometric mechanics and soft matter physics, and have been useful to develop models for biological locomotion in complex terrain. The rapidly decreasing cost of constructing robot models with easy access to significant computational power bodes well for scientists and engineers to engage in a discipline which can readily integrate experiment, theory and computation.

We integrate biological experiment, empirical theory, numerical simulation and a physical model to reveal principles of undulatory locomotion in granular media. High-speed X-ray imaging of the sandfish lizard, Scincus scincus, in 3 mm glass particles shows that it swims within the medium without using its limbs by propagating a single-period travelling sinusoidal wave down its body, resulting in a wave efficiency, h, the ratio of its average forward speed to the wave speed, of approximately 0.5. A resistive force theory (RFT) that balances granular thrust and drag forces along the body predicts h close to the observed value. We test this prediction against two other more detailed modelling approaches: a numerical model of the sandfish coupled to a discrete particle simulation of the granular medium, and an undulatory robot that swims within granular media. Using these models and analytical solutions of the RFT, we vary the ratio of undulation amplitude to wavelength (A/l) and demonstrate an optimal condition for sand-swimming, which for a given A results from the competition between h and l. The RFT, in agreement with the simulated and physical models, predicts that for a single-period sinusoidal wave, maximal speed occurs for A/l % 0.2, the same kinematics used by the sandfish.

A previous study of a sand-swimming lizard, the sandfish, revealed that it swims within granular media at speeds up to 0.4 body-lengths/cycle using body undulations (approximately a single period sinusoidal traveling wave) without limb use. Inspired by the organism, we develop a numerical model of a robot swimming in a simulated granular medium to guide the design of a physical device. Both in simulation and experiment the robot swims limblessly subsurface at speeds up to 0.3 body-lengths/cycle and, like the animal, increases its speed by increasing its oscillation frequency. The performance of the robot measured in terms of its wave efficiency η, the ratio of its forward speed to wave speed, is 0.34 ± 0.02, within 8% of the simulation prediction. Both in simulation and experiment, η increases with increasing particle-particle friction but decreases with increasing body-particle friction. On a flat, rigid surface the robot fails to move forward, as expected, due to the frictional isotropy between the interacting surfaces. However, the surface and subsurface performance of the robot on low friction particles are comparable. Our work provides a validated simulation tool and the design of a robot that can move on or within yielding terrestrial substrates.

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