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.

Mechanistic framework for reduced-order models in soft materials: Application to three-dimensional granular intrusion.

Soft materials often display complex behaviors that transition through apparent solid- and fluid-like regimes. While a growing number of microscale simulation methods exist for these materials, reduced-order models that encapsulate the macroscale physics are often desired to predict how external bodies interact with soft media. Such an approach could provide direct insights in diverse situations from impact and penetration problems to locomotion over natural terrains. This work proposes a systematic program to develop three-dimensional (3D) reduced-order models for soft materials from a fundamental basis using continuum symmetries and rheological principles. In particular, we derive a reduced-order, 3D resistive force theory (3D-RFT), which is capable of accurately and quickly predicting the resistive stress distribution on arbitrary-shaped bodies intruding through granular media. Aided by a continuum description of the granular medium, a comprehensive set of spatial symmetry constraints, and a limited amount of reference data, we develop a self-consistent and accurate 3D-RFT. We verify the model capabilities in a wide range of cases and show that it can be quickly recalibrated to different media and intruder surface types. The premises leading to 3D-RFT anticipate application to other soft materials with strongly hyperlocalized intrusion behavior.

Self-propulsion via slipping: Frictional swimming in multilegged locomotors.

Locomotion is typically studied either in continuous media where bodies and legs experience forces generated by the flowing medium or on solid substrates dominated by friction. In the former, centralized whole-body coordination is believed to facilitate appropriate slipping through the medium for propulsion. In the latter, slip is often assumed minimal and thus avoided via decentralized control schemes. We find in laboratory experiments that terrestrial locomotion of a meter-scale multisegmented/legged robophysical model resembles undulatory fluid swimming. Experiments varying waves of leg stepping and body bending reveal how these parameters result in effective terrestrial locomotion despite seemingly ineffective isotropic frictional contacts. Dissipation dominates over inertial effects in this macroscopic-scaled regime, resulting in essentially geometric locomotion on land akin to microscopic-scale swimming in fluids. Theoretical analysis demonstrates that the high-dimensional multisegmented/legged dynamics can be simplified to a centralized low-dimensional model, which reveals an effective resistive force theory with an acquired viscous drag anisotropy. We extend our low-dimensional, geometric analysis to illustrate how body undulation can aid performance in non-flat obstacle-rich terrains and also use the scheme to quantitatively model how body undulation affects performance of biological centipede locomotion (the desert centipede Scolopendra polymorpha) moving at relatively high speeds (∼0.5 body lengths/sec). Our results could facilitate control of multilegged robots in complex terradynamic scenarios.

Coordinating tiny limbs and long bodies: Geometric mechanics of lizard terrestrial swimming.

Although typically possessing four limbs and short bodies, lizards have evolved diverse morphologies, including elongate trunks with tiny limbs. Such forms are hypothesized to aid locomotion in cluttered/fossorial environments but propulsion mechanisms (e.g., the use of body and/or limbs to interact with substrates) and potential body/limb coordination remain unstudied. Here, we use biological experiments, a geometric theory of locomotion, and robophysical models to investigate body-limb coordination in diverse lizards. Locomotor field studies in short-limbed, elongate lizards (Brachymeles and Lerista) and laboratory studies of fully limbed lizards (Uma scoparia and Sceloporus olivaceus) and a snake (Chionactis occipitalis) reveal that body-wave dynamics can be described by a combination of standing and traveling waves; the ratio of the amplitudes of these components is inversely related to the degree of limb reduction and body elongation. The geometric theory (which replaces laborious calculation with diagrams) helps explain our observations, predicting that the advantage of traveling-wave body undulations (compared with a standing wave) emerges when the dominant thrust-generation mechanism arises from the body rather than the limbs and reveals that such soil-dwelling lizards propel via "terrestrial swimming" like sand-swimming lizards and snakes. We test our hypothesis by inducing the use of traveling waves in stereotyped lizards via modulating the ground-penetration resistance. Study of a limbed/undulatory robophysical model demonstrates that a traveling wave is beneficial when propulsion is generated by body-environment interaction. Our models could be valuable in understanding functional constraints on the evolutionary processes of elongation and limb reduction as well as advancing robot designs.

Field-mediated locomotor dynamics on highly deformable surfaces.

Studies of active matter-systems consisting of individuals or ensembles of internally driven and damped locomotors-are of interest to physicists studying nonequilibrium dynamics, biologists interested in individuals and swarm locomotion, and engineers designing robot controllers. While principles governing active systems on hard ground or within fluids are well studied, another class of systems exists at deformable interfaces. Such environments can display mixes of fluid-like and elastic features, leading to locomotor dynamics that are strongly influenced by the geometry of the surface, which, in itself, can be a dynamical entity. To gain insight into principles by which locomotors are influenced via a deformation field alone (and can influence other locomotors), we study robot locomotion on an elastic membrane, which we propose as a model of active systems on highly deformable interfaces. As our active agent, we use a differential driven wheeled robotic vehicle which drives straight on flat homogeneous surfaces, but reorients in response to environmental curvature. We monitor the curvature field-mediated dynamics of a single vehicle interacting with a fixed deformation as well as multiple vehicles interacting with each other via local deformations. Single vehicles display precessing orbits in centrally deformed environments, while multiple vehicles influence each other by local deformation fields. The active nature of the system facilitates a differential geometry-inspired mathematical mapping from the vehicle dynamics to those of test particles in a fictitious "spacetime," allowing further understanding of the dynamics and how to control agent interactions to facilitate or avoid multivehicle membrane-induced cohesion.

Robotic swimming in curved space via geometric phase.

Locomotion by shape changes or gas expulsion is assumed to require environmental interaction, due to conservation of momentum. However, as first noted in [J. Wisdom, Science 299, 1865-1869 (2003)] and later in [E. Guéron, Sci. Am. 301, 38-45 (2009)] and [J. Avron, O. Kenneth, New J. Phys, 8, 68 (2006)], the noncommutativity of translations permits translation without momentum exchange in either gravitationally curved spacetime or the curved surfaces encountered by locomotors in real-world environments. To realize this idea which remained unvalidated in experiments for almost 20 y, we show that a precision robophysical apparatus consisting of motors driven on curved tracks (and thereby confined to a spherical surface without a solid substrate) can self-propel without environmental momentum exchange. It produces shape changes comparable to the environment's inverse curvatures and generates movement of [Formula: see text] cm per gait. While this simple geometric effect predominates over short time, eventually the dissipative (frictional) and conservative forces, ubiquitous in real systems, couple to it to generate an emergent dynamics in which the swimming motion produces a force that is counter-balanced against residual gravitational forces. In this way, the robot both swims forward without momentum and becomes fixed in place with a finite momentum that can be released by ceasing the swimming motion. We envision that our work will be of use in a broad variety of contexts, such as active matter in curved space and robots navigating real-world environments with curved surfaces.

Robots as models of evolving systems.

Experimental robobiological physics can bring insights into biological evolution. We present a development of hybrid analog/digital autonomous robots with mutable diploid dominant/recessive 6-byte genomes. The robots are capable of death, rebirth, and breeding. We map the quasi-steady-state surviving local density of the robots onto a multidimensional abstract "survival landscape." We show that robot death in complex, self-adaptive stress landscapes proceeds by a general lowering of the robotic genetic diversity, and that stochastically changing landscapes are the most difficult to survive.

Colloidal robotics.

Robots have components that work together to accomplish a task. Colloids are particles, usually less than 100 µm, that are small enough that they do not settle out of solution. Colloidal robots are particles capable of functions such as sensing, computation, communication, locomotion and energy management that are all controlled by the particle itself. Their design and synthesis is an emerging area of interdisciplinary research drawing from materials science, colloid science, self-assembly, robophysics and control theory. Many colloidal robot systems approach synthetic versions of biological cells in autonomy and may find ultimate utility in bringing these specialized functions to previously inaccessible locations. This Perspective examines the emerging literature and highlights certain design principles and strategies towards the realization of colloidal robots.

Probing Hydrodynamic Fluctuation-Induced Forces with an Oscillating Robot.

We study the dynamics of an oscillating, free-floating robot that generates radially expanding gravity-capillary waves at a fluid surface. In open water, the device does not self-propel; near a rigid boundary, it can be attracted or repelled. Visualization of the wave field dynamics reveals that when near a boundary, a complex interference of generated and reflected waves induces a wave amplitude fluctuation asymmetry. Attraction increases as wave frequency increases or robot-boundary separation decreases. Theory on confined gravity-capillary wave radiation dynamics developed by Hocking in the 1980s captures the observed parameter dependence due to these "Hocking fields." The flexibility of the robophysical system allows detailed characterization and analysis of locally generated nonequilibrium fluctuation-induced forces [M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999)RMPHAT0034-686110.1103/RevModPhys.71.1233].

Collective dynamics in entangled worm and robot blobs.

Living systems at all scales aggregate in large numbers for a variety of functions including mating, predation, and survival. The majority of such systems consist of unconnected individuals that collectively flock, school, or swarm. However, some aggregations involve physically entangled individuals, which can confer emergent mechanofunctional material properties to the collective. Here, we study in laboratory experiments and rationalize in theoretical and robophysical models the dynamics of physically entangled and motile self-assemblies of 1-cm-long California blackworms (Lumbriculus variegatus, Annelida: Clitellata: Lumbriculidae). Thousands of individual worms form braids with their long, slender, and flexible bodies to make a three-dimensional, soft, and shape-shifting "blob." The blob behaves as a living material capable of mitigating damage and assault from environmental stresses through dynamic shape transformations, including minimizing surface area for survival against desiccation and enabling transport (negative thermotaxis) from hazardous environments (like heat). We specifically focus on the locomotion of the blob to understand how an amorphous entangled ball of worms can break symmetry to move across a substrate. We hypothesize that the collective blob displays rudimentary differentiation of function across itself, which when combined with entanglement dynamics facilitates directed persistent blob locomotion. To test this, we develop a robophysical model of the worm blobs, which displays emergent locomotion in the collective without sophisticated control or programming of any individual robot. The emergent dynamics of the living functional blob and robophysical model can inform the design of additional classes of adaptive mechanofunctional living materials and emergent robotics.

Functional consequences of convergently evolved microscopic skin features on snake locomotion.

The small structures that decorate biological surfaces can significantly affect behavior, yet the diversity of animal-environment interactions essential for survival makes ascribing functions to structures challenging. Microscopic skin textures may be particularly important for snakes and other limbless locomotors, where substrate interactions are mediated solely through body contact. While previous studies have characterized ventral surface features of some snake species, the functional consequences of these textures are not fully understood. Here, we perform a comparative study, combining atomic force microscopy measurements with mathematical modeling to generate predictions that link microscopic textures to locomotor performance. We discover an evolutionary convergence in the ventral skin structures of a few sidewinding specialist vipers that inhabit sandy deserts-an isotropic texture that is distinct from the head-to-tail-oriented, micrometer-sized spikes observed on a phylogenetically broad sampling of nonsidewinding vipers and other snakes from diverse habitats and wide geographic range. A mathematical model that relates structural directionality to frictional anisotropy reveals that isotropy enhances movement during sidewinding, whereas anisotropy improves movement during slithering via lateral undulation of the body. Our results highlight how an integrated approach can provide quantitative predictions for structure-function relationships and insights into behavioral and evolutionary adaptations in biological systems.

Mechanism and function of root circumnutation.

Early root growth is critical for plant establishment and survival. We have identified a molecular pathway required for helical root tip movement known as circumnutation. Here, we report a multiscale investigation of the regulation and function of this phenomenon. We identify key cell signaling events comprising interaction of the ethylene, cytokinin, and auxin hormone signaling pathways. We identify the gene Oryza sativa histidine kinase-1 (HK1) as well as the auxin influx carrier gene OsAUX1 as essential regulators of this process in rice. Robophysical modeling and growth challenge experiments indicate circumnutation is critical for seedling establishment in rocky soil, consistent with the long-standing hypothesis that root circumnutation facilitates growth past obstacles. Thus, the integration of robotics, physics, and biology has elucidated the functional importance of root circumnutation and uncovered the molecular mechanisms underlying its regulation.

Active and passive mechanics for rugose terrain traversal in centipedes.

Centipedes coordinate body and limb flexion to generate propulsion. On flat, solid surfaces, the limb-stepping patterns can be characterized according to the direction in which limb-aggregates propagate, opposite to (retrograde) or with the direction of motion (direct). It is unknown how limb and body dynamics are modified in terrain with terradynamic complexity more representative of these animal's natural heterogeneous environments. Here, we investigated how centipedes that use retrograde and direct limb-stepping patterns, Scolopendra polymorpha and Scolopocryptops sexspinosus, respectively, coordinate their body and limbs to navigate laboratory environments which present footstep challenges and terrain rugosity. We recorded the kinematics and measured the locomotive performance of these animals traversing two rugose terrains with randomly distributed step heights and compared the kinematics with those on a flat frictional surface. Scolopendra polymorpha exhibited similar body and limb dynamics across all terrains and a decrease in speed with increased terrain rugosity. Unexpectedly, when placed in a rugose terrain, S. sexspinosus changed the direction of the limb-stepping pattern from direct to retrograde. Further, for both species, traversal of these rugose terrains was facilitated by hypothesized passive mechanics: upon horizontal collision of a limb with a block, the limb bent and later continued the stepping pattern. Although centipedes have many degrees of freedom, our results suggest these animals negotiate limb-substrate interactions and navigate complex terrains leveraging the innate flexibility of their limbs to simplify control.

Physics of smart active matter: integrating active matter and control to gain insights into living systems.

We offer our opinion on the benefits of integration of insights from active matter physics with principles of regulatory interactions and control to develop a field we term "smart active matter". This field can provide insight into important principles in living systems as well as aid engineering of responsive, robust and functional collectives.

Amorphous entangled active matter.

The design of amorphous entangled systems, specifically from soft and active materials, has the potential to open exciting new classes of active, shape-shifting, and task-capable 'smart' materials. However, the global emergent mechanics that arise from the local interactions of individual particles are not well understood. In this study, we examine the emergent properties of amorphous entangled systems in an in silico collection of u-shaped particles ("smarticles") and in living entangled aggregate of worm blobs (L. variegatus). In simulations, we examine how material properties change for a collective composed of smarticles as they undergo different forcing protocols. We compare three methods of controlling entanglement in the collective: external oscillations of the ensemble, sudden shape-changes of all individuals, and sustained internal oscillations of all individuals. We find that large-amplitude changes of the particle's shape using the shape-change procedure produce the largest average number of entanglements, with respect to the aspect ratio (l/w), thus improving the tensile strength of the collective. We demonstrate applications of these simulations by showing how the individual worm activity in a blob can be controlled through the ambient dissolved oxygen in water, leading to complex emergent properties of the living entangled collective, such as solid-like entanglement and tumbling. Our work reveals principles by which future shape-modulating, potentially soft robotic systems may dynamically alter their material properties, advancing our understanding of living entangled materials, while inspiring new classes of synthetic emergent super-materials.

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.

Emergent Field-Driven Robot Swarm States.

We present an ecology-inspired form of active matter consisting of a robot swarm. Each robot moves over a planar dynamic resource environment represented by a large light-emitting diode array in search of maximum light intensity; the robots deplete (dim) locally by their presence the local light intensity and seek maximum light intensity. Their movement is directed along the steepest local light intensity gradient; we call this emergent symmetry breaking motion "field drive." We show there emerge dynamic and spatial transitions similar to gas, crystalline, liquid, glass, and jammed states as a function of robot density, resource consumption rates, and resource recovery rates. Paradoxically the nongas states emerge from smooth, flat resource landscapes, not rough ones, and each state can directly move to a glassy state if the resource recovery rate is slow enough, at any robot density.

Efficacy of simple continuum models for diverse granular intrusions.

Granular intrusion is commonly observed in natural and human-made settings. Unlike typical solids and fluids, granular media can simultaneously display fluid-like and solid-like characteristics in a variety of intrusion scenarios. This multi-phase behavior increases the difficulty of accurately modeling these and other yielding (or flowable) materials. Micro-scale modeling methods, such as DEM (Discrete Element Method), capture this behavior by modeling the media at the grain scale, but there is often interest in the macro-scale characterizations of such systems. We examine the efficacy of a macro-scale continuum approach in modeling and understanding the physics of various macroscopic phenomena in a variety of granular intrusion cases using two basic frictional yielding constitutive models. We compare predicted granular force response and material flow to experimental data in four quasi-2D intrusion cases: (1) depth-dependent force response in horizontal submerged-intruder motion; (2) separation-dependent drag variation in parallel-plate vertical-intrusion; (3) initial-density-dependent drag fluctuations in free surface plowing, and (4) flow zone development during vertical plate intrusions in under-compacted granular media. Our continuum modeling approach captures the flow process and drag forces while providing key meso- and macro-scopic insights. The modeling results are then compared to experimental data. Our study highlights how continuum modeling approaches provide an alternative for efficient modeling as well as a conceptual understanding of various granular intrusion phenomena.

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.