Geometric phase predicts locomotion performance in undulating living systems across scales.

Self-propelling organisms locomote via generation of patterns of self-deformation. Despite the diversity of body plans, internal actuation schemes and environments in limbless vertebrates and invertebrates, such organisms often use similar traveling waves of axial body bending for movement. Delineating how self-deformation parameters lead to locomotor performance (e.g. speed, energy, turning capabilities) remains challenging. We show that a geometric framework, replacing laborious calculation with a diagrammatic scheme, is well-suited to discovery and comparison of effective patterns of wave dynamics in diverse living systems. We focus on a regime of undulatory locomotion, that of highly damped environments, which is applicable not only to small organisms in viscous fluids, but also larger animals in frictional fluids (sand) and on frictional ground. We find that the traveling wave dynamics used by mm-scale nematode worms and cm-scale desert dwelling snakes and lizards can be described by time series of weights associated with two principal modes. The approximately circular closed path trajectories of mode weights in a self-deformation space enclose near-maximal surface integral (geometric phase) for organisms spanning two decades in body length. We hypothesize that such trajectories are targets of control (which we refer to as "serpenoid templates"). Further, the geometric approach reveals how seemingly complex behaviors such as turning in worms and sidewinding snakes can be described as modulations of templates. Thus, the use of differential geometry in the locomotion of living systems generates a common description of locomotion across taxa and provides hypotheses for neuromechanical control schemes at lower levels of organization.

Mechanical diffraction reveals the role of passive dynamics in a slithering snake.

Limbless animals like snakes inhabit most terrestrial environments, generating thrust to overcome drag on the elongate body via contacts with heterogeneities. The complex body postures of some snakes and the unknown physics of most terrestrial materials frustrates understanding of strategies for effective locomotion. As a result, little is known about how limbless animals contend with unplanned obstacle contacts. We studied a desert snake, Chionactis occipitalis, which uses a stereotyped head-to-tail traveling wave to move quickly on homogeneous sand. In laboratory experiments, we challenged snakes to move across a uniform substrate and through a regular array of force-sensitive posts. The snakes were reoriented by the array in a manner reminiscent of the matter-wave diffraction of subatomic particles. Force patterns indicated the animals did not change their self-deformation pattern to avoid or grab the posts. A model using open-loop control incorporating previously described snake muscle activation patterns and body-buckling dynamics reproduced the observed patterns, suggesting a similar control strategy may be used by the animals. Our results reveal how passive dynamics can benefit limbless locomotors by allowing robust transit in heterogeneous environments with minimal sensing.

Surprising simplicities and syntheses in limbless self-propulsion in sand.

Animals moving on and in fluids and solids move their bodies in diverse ways to generate propulsion and lift forces. In fluids, animals can wiggle, stroke, paddle or slap, whereas on hard frictional terrain, animals largely engage their appendages with the substrate to avoid slip. Granular substrates, such as desert sand, can display complex responses to animal interactions. This complexity has led to locomotor strategies that make use of fluid-like or solid-like features of this substrate, or combinations of the two. Here, we use examples from our work to demonstrate the diverse array of methods used and insights gained in the study of both surface and subsurface limbless locomotion in these habitats. Counterintuitively, these seemingly complex granular environments offer certain experimental, theoretical, robotic and computational advantages for studying terrestrial movement, with the potential for providing broad insights into morphology and locomotor control in fluids and solids, including neuromechanical control templates and morphological and behavioral evolution. In particular, granular media provide an excellent testbed for a locomotion framework called geometric mechanics, which was introduced by particle physicists and control engineers in the last century, and which allows quantitative analysis of alternative locomotor patterns and morphology to test for control templates, optimality and evolutionary alternatives. Thus, we posit that insights gained from movement in granular environments can be translated into principles that have broader applications across taxa, habitats and movement patterns, including those at microscopic scales.

An insect-scale robot reveals the effects of different body dynamics regimes during open-loop running in feature-laden terrain.

The transition from the lab to natural environments is an archetypal challenge in robotics. While larger robots can manage complex limb-ground interactions using sensing and control, such strategies are difficult to implement on small platforms where space and power are limited. The Harvard Ambulatory Microrobot (HAMR) is an insect-scale quadruped capable of effective open-loop running on featureless, hard substrates. Inspired by the predominantly feedforward strategy of rapidly-running cockroaches on uneven terrain (Sponberg, 2007), we used HAMR to explore open-loop running on two 3D printed heterogeneous terrains generated using fractional Brownian motion. The 'pocked' terrain had foot-scale features throughout while the 'jagged' terrain features increased in height in the direction of travel. We measured the performance of trot and pronk gaits while varying limb amplitude and stride frequency. The frequencies tested encompassed different dynamics regimes: body resonance (10-25 Hz) and kinematic running (30-40 Hz), with dynamics typical of biological running and walking, respectively, and limb-transmission resonance (45-60 Hz). On the featureless and pocked terrains, low mechanical cost-of-transport (mCoT) kinematic running combinations performed best without systematic differences between trot and pronk; indicating that if terrain features are not too tall, a robot can transition from homo-to heterogeneous environments in open-loop. Pronk bypassed taller features than trot on the jagged terrain, and higher mCoT, lower frequency running was more often effective. While increasing input power to the robot improved performance in general, lower frequency pronking on jagged terrain allowed the robot to bypass taller features compared with the same input power at higher frequencies. This was correlated with the increased variation in center-of-mass orientation occurring at frequencies near body resonance. This study established that appropriate choice of robot dynamics, as mediated by gait, frequency, and limb amplitude, can expand the terrains accessible to microrobots without the addition of sensing or closed-loop control.

Comparative study of snake lateral undulation kinematics in model heterogeneous terrain.

Terrestrial organisms that use traveling waves to locomote must leverage heterogeneities to overcome drag on the elongate body. While previous studies illuminated how habitat generalist snakes self-deform to use rigid obstacles in the surroundings, control strategies for multi-component terrain are largely unknown. We compared the sand-specialist Chionactis occipitalis to a habitat generalist, Pantherophis guttatus, navigating a model terrestrial terrain-rigid post arrays on a low-friction substrate. We found the waveshapes used by the generalist were more variable than the specialist. Principal component analysis revealed that while the specialized sand-swimming waveform was always present on C. occipitalis, the generalist did not have a similarly pervasive low-dimensional waveshape. We expected the generalist to thus outperform the specialist in the arrays, but body slip of both species was comparable on level ground and in all trials the snakes successfully traversed the arena. When we further challenged the snakes to ascend an inclined lattice, the sand-specialist had difficulty maintaining contact with the obstacles and was unable to progress up the steepest inclines in the largest lattice spacings. Our results suggest that species adapted to different habitats use different control modalities-the specialist is primarily controlling its kinematics to achieve a target shape while, consistent with previous research, the generalist is using force control and self-deforms in response to terrain contacts. While both strategies allowed progress on the uninclined low-friction terrain with posts, the more variable waveshapes of the generalist may be necessary when faced with more challenging locomotor tasks like climbing inclines.

Mitigating memory effects during undulatory locomotion on hysteretic materials.

While terrestrial locomotors often contend with permanently deformable substrates like sand, soil, and mud, principles of motion on such materials are lacking. We study the desert-specialist shovel-nosed snake traversing a model sand and find body inertia is negligible despite rapid transit and speed dependent granular reaction forces. New surface resistive force theory (RFT) calculation reveals how wave shape in these snakes minimizes material memory effects and optimizes escape performance given physiological power limitations. RFT explains the morphology and waveform-dependent performance of a diversity of non-sand-specialist snakes but overestimates the capability of those snakes which suffer high lateral slipping of the body. Robophysical experiments recapitulate aspects of these failure-prone snakes and elucidate how re-encountering previously deformed material hinders performance. This study reveals how memory effects stymied the locomotion of a diversity of snakes in our previous studies (Marvi et al., 2014) and indicates avenues to improve all-terrain robots.

Dynamics of scattering in undulatory active collisions.

Natural and artificial self-propelled systems must manage environmental interactions during movement. In complex environments, these interactions include active collisions, in which propulsive forces create persistent contacts with heterogeneities. Due to the driven and dissipative nature of these systems, such collisions are fundamentally different from those typically studied in classical physics. Here we experimentally and numerically study the effects of active collisions on a laterally undulating sensory-deprived robophysical model, whose dynamics are relevant to self-propelled systems across length scales and environments. Interactions with a single rigid post scatter the robot, and this deflection is dominated by head-post contact. These results motivate a model which reduces the snake to a circular particle with two key features: The collision dynamics are set by internal driving subject to the geometric constraints of the post, and the particle has an effective length equal to the wavelength of the snake. Interactions with a single row of evenly spaced posts (with interpost spacing d) produce distributions reminiscent of far-field diffraction patterns: As d decreases, distinct secondary peaks emerge as large deflections become more likely. Surprisingly, we find that the presence of multiple posts does not change the nature of individual collisions; instead, multimodal scattering patterns arise from multiple posts altering the likelihood of individual collisions to occur. As d decreases, collisions near the leading edges of the posts become more probable, and we find that these interactions are associated with larger deflections. Our results, which highlight the surprising dynamics that can occur during active collisions of self-propelled systems, can inform control principles for locomotors in complex terrain and facilitate design of task-capable active matter.

Tail use improves performance on soft substrates in models of early vertebrate land locomotors.

In the evolutionary transition from an aquatic to a terrestrial environment, early tetrapods faced the challenges of terrestrial locomotion on flowable substrates, such as sand and mud of variable stiffness and incline. The morphology and range of motion of appendages can be revealed in fossils; however, biological and robophysical studies of modern taxa have shown that movement on such substrates can be sensitive to small changes in appendage use. Using a biological model (the mudskipper), a physical robot model, granular drag measurements, and theoretical tools from geometric mechanics, we demonstrate how tail use can improve robustness to variable limb use and substrate conditions. We hypothesize that properly coordinated tail movements could have provided a substantial benefit for the earliest vertebrates to move on land.