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Intrusions into granular media are common in natural and engineered settings (e.g. during animal locomotion and planetary landings). While intrusion of complex shapes in dry non-cohesive granular materials is well studied, less is known about intrusion in cohesive powders. Granular resistive force theory (RFT) -- a reduced-order frictional fluid model -- quantitatively predicts intrusion forces in dry, non-cohesive granular media by assuming a linear superposition of angularly dependent elemental stresses acting on arbitrarily shaped intruders. Here we extend RFT's applicability to cohesive dry powders, enabling quantitative modeling of forces on complex shapes during intrusion. To do so, we first conduct intrusion experiments into dry cornstarch powder to create stress functions. These stresses are similar to non-cohesive media; however, we observe relatively higher resistance to horizontal intrusions in cohesive powder compared to non-cohesive media. We use the model to identify geometries that enhance resistance to intrusion in such materials, aiming to minimize sinkage. Our calculations, supported by experimental verification, suggest that a flat surface generates the largest stress across various intrusion angles while a curved surface exhibits the largest resistance for vertical intrusion. Our model can thus facilitate optimizing design and movement strategies for robotic platforms (e.g. extraterrestrial landers) operating in such environments., Comment: 23 pages, 10 figures, submitted to the Journal of Terramechanics

Achieving robust legged locomotion on complex terrains poses challenges due to the high uncertainty in robot-environment interactions. Recent advances in bipedal and quadrupedal robots demonstrate good mobility on rugged terrains but rely heavily on sensors for stability due to low static stability from a high center of mass and a narrow base of support. We hypothesize that a multi-legged robotic system can leverage morphological redundancy from additional legs to minimize sensing requirements when traversing challenging terrains. Studies suggest that a multi-legged system with sufficient legs can reliably navigate noisy landscapes without sensing and control, albeit at a low speed of up to 0.1 body lengths per cycle (BLC). However, the control framework to enhance speed on challenging terrains remains underexplored due to the complex environmental interactions, making it difficult to identify the key parameters to control in these high-degree-of-freedom systems. Here, we present a bio-inspired vertical body undulation wave as a novel approach to mitigate environmental disturbances affecting robot speed, supported by experiments and probabilistic models. Finally, we introduce a control framework which monitors foot-ground contact patterns on rugose landscapes using binary foot-ground contact sensors to estimate terrain rugosity. The controller adjusts the vertical body wave based on the deviation of the limb's averaged actual-to-ideal foot-ground contact ratio, achieving a significant enhancement of up to 0.235 BLC on rugose laboratory terrain. We observed a $\sim$ 50\% increase in speed and a $\sim$ 40\% reduction in speed variance compared to the open-loop controller. Additionally, the controller operates in complex terrains outside the lab, including pine straw, robot-sized rocks, mud, and leaves., Comment: Submitted to IEEE Transactions on Robotics (T-RO)

Centipede-like robots offer an effective and robust solution to navigation over complex terrain with minimal sensing. However, when climbing over obstacles, such multi-legged robots often elevate their center-of-mass into unstable configurations, where even moderate terrain uncertainty can cause tipping over. Robust mechanisms for such elongate multi-legged robots to self-right remain unstudied. Here, we developed a comparative biological and robophysical approach to investigate self-righting strategies. We first released \textit{S. polymorpha} upside down from a 10 cm height and recorded their self-righting behaviors using top and side view high-speed cameras. Using kinematic analysis, we hypothesize that these behaviors can be prescribed by two traveling waves superimposed in the body lateral and vertical planes, respectively. We tested our hypothesis on an elongate robot with static (non-actuated) limbs, and we successfully reconstructed these self-righting behaviors. We further evaluated how wave parameters affect self-righting effectiveness. We identified two key wave parameters: the spatial frequency, which characterizes the sequence of body-rolling, and the wave amplitude, which characterizes body curvature. By empirically obtaining a behavior diagram of spatial frequency and amplitude, we identify effective and versatile self-righting strategies for general elongate multi-legged robots, which greatly enhances these robots' mobility and robustness in practical applications such as agricultural terrain inspection and search-and-rescue.

Centipedes exhibit great maneuverability in diverse environments due to their many legs and body-driven control. By leveraging similar morphologies, their robotic counterparts also demonstrate effective terrestrial locomotion. However, the success of these multi-legged robots is largely limited to forward locomotion; steering is substantially less studied, in part due to the challenges in coordinating their many body joints. Furthermore, steering behavior is complex and can include different combinations of desired rotational/translational displacement. In this paper, we explore steering strategies in multi-legged robots based on tools derived from geometric mechanics (GM). We characterize the steering motion in the plane by the rotation angle, the steering radius, and the heading direction angle. We identify an effective turning strategy by superimposing two traveling waves in the lateral body undulation and further explore variations of the "turning wave" to enable a broad spectrum of steering behaviors. By combining an amplitude modulation and a phase modulation, we develop a control strategy for steering behaviors that enables steering with a range of rotation angles (from 0{\deg} to 20{\deg}) and steering radius (from 0.28 to 0.38 body length) while keeping the heading direction angle close to 0. Lastly, we test our control framework on an elongate multi-legged robot model to verify the effectiveness of our proposed strategy. Our work demonstrates the generality of the two-wave template for effective steering of multi-legged elongate robots.

Characterized by their elongate bodies and relatively simple legs, multi-legged robots have the potential to locomote through complex terrains for applications such as search-and-rescue and terrain inspection. Prior work has developed effective and reliable locomotion strategies for multi-legged robots by propagating the two waves of lateral body undulation and leg stepping, which we will refer to as the two-wave template. However, these robots have limited capability to climb over obstacles with sizes comparable to their heights. We hypothesize that such limitations stem from the two-wave template that we used to prescribe the multi-legged locomotion. Seeking effective alternative waves for obstacle-climbing, we designed a five-segment robot with static (non-actuated) legs, where each cable-driven joint has a rotational degree-of-freedom (DoF) in the sagittal plane (vertical wave) and a linear DoF (peristaltic wave). We tested robot locomotion performance on a flat terrain and a rugose terrain. While the benefit of peristalsis on flat-ground locomotion is marginal, the inclusion of a peristaltic wave substantially improves the locomotion performance in rugose terrains: it not only enables obstacle-climbing capabilities with obstacles having a similar height as the robot, but it also significantly improves the traversing capabilities of the robot in such terrains. Our results demonstrate an alternative actuation mechanism for multi-legged robots, paving the way towards all-terrain multi-legged robots.

This paper presents AquaMILR+, an untethered limbless robot designed for agile navigation in complex aquatic environments. The robot features a bilateral actuation mechanism that models musculoskeletal actuation in many anguilliform swimming organisms which propagates a moving wave from head to tail allowing open fluid undulatory swimming. This actuation mechanism employs mechanical intelligence, enhancing the robot's maneuverability when interacting with obstacles. AquaMILR+ also includes a compact depth control system inspired by the swim bladder and lung structures of eels and sea snakes. The mechanism, driven by a syringe and telescoping leadscrew, enables depth and pitch control-capabilities that are difficult for most anguilliform swimming robots to achieve. Additional structures, such as fins and a tail, further improve stability and propulsion efficiency. Our tests in both open water and indoor 2D and 3D heterogeneous aquatic environments highlight AquaMILR+'s capabilities and suggest a promising system for complex underwater tasks such as search and rescue and deep-sea exploration.

Navigating rugged landscapes poses significant challenges for legged locomotion. Multi-legged robots (those with 6 and greater) offer a promising solution for such terrains, largely due to their inherent high static stability, resulting from a low center of mass and wide base of support. Such systems require minimal effort to maintain balance. Recent studies have shown that a linear controller, which modulates the vertical body undulation of a multi-legged robot in response to shifts in terrain roughness, can ensure reliable mobility on challenging terrains. However, the potential of a learning-based control framework that adjusts multiple parameters to address terrain heterogeneity remains underexplored. We posit that the development of an experimentally validated physics-based simulator for this robot can rapidly advance capabilities by allowing wide parameter space exploration. Here we develop a MuJoCo-based simulator tailored to this robotic platform and use the simulation to develop a reinforcement learning-based control framework that dynamically adjusts horizontal and vertical body undulation, and limb stepping in real-time. Our approach improves robot performance in simulation, laboratory experiments, and outdoor tests. Notably, our real-world experiments reveal that the learning-based controller achieves a 30\% to 50\% increase in speed compared to a linear controller, which only modulates vertical body waves. We hypothesize that the superior performance of the learning-based controller arises from its ability to adjust multiple parameters simultaneously, including limb stepping, horizontal body wave, and vertical body wave., Comment: Submitted to ICRA 2025

Organisms that locomote by propagating waves of body bending can maintain performance across heterogeneous environments by modifying their gait frequency $\omega$ or wavenumber $k$. We identify a unifying relationship between these parameters for overdamped undulatory swimmers (including nematodes, spermatozoa, and mm-scale fish) moving in diverse environmental rheologies, in the form of an active `dispersion relation' $\omega\propto k^{\pm2}$. A model treating the organisms as actively driven viscoelastic beams reproduces the experimentally observed scaling. The relative strength of rate-dependent dissipation in the body and the environment determines whether $k^2$ or $k^{-2}$ scaling is observed. The existence of these scaling regimes reflects the $k$ and $\omega$ dependence of the various underlying force terms and how their relative importance changes with the external environment and the neuronally commanded gait.

Humanity has long sought inspiration from nature to innovate materials and devices. As science advances, nature-inspired materials are becoming part of our lives. Animate materials, characterized by their activity, adaptability, and autonomy, emulate properties of living systems. While only biological materials fully embody these principles, artificial versions are advancing rapidly, promising transformative impacts across various sectors. This roadmap presents authoritative perspectives on animate materials across different disciplines and scales, highlighting their interdisciplinary nature and potential applications in diverse fields including nanotechnology, robotics and the built environment. It underscores the need for concerted efforts to address shared challenges such as complexity management, scalability, evolvability, interdisciplinary collaboration, and ethical and environmental considerations. The framework defined by classifying materials based on their level of animacy can guide this emerging field encouraging cooperation and responsible development. By unravelling the mysteries of living matter and leveraging its principles, we can design materials and systems that will transform our world in a more sustainable manner.

While undulatory swimming of elongate limbless robots has been extensively studied in open hydrodynamic environments, less research has been focused on limbless locomotion in complex, cluttered aquatic environments. Motivated by the concept of mechanical intelligence, where controls for obstacle navigation can be offloaded to passive body mechanics in terrestrial limbless locomotion, we hypothesize that principles of mechanical intelligence can be extended to cluttered hydrodynamic regimes. To test this, we developed an untethered limbless robot capable of undulatory swimming on water surfaces, utilizing a bilateral cable-driven mechanism inspired by organismal muscle actuation morphology to achieve programmable anisotropic body compliance. We demonstrated through robophysical experiments that, similar to terrestrial locomotion, an appropriate level of body compliance can facilitate emergent swim through complex hydrodynamic environments under pure open-loop control. Moreover, we found that swimming performance depends on undulation frequency, with effective locomotion achieved only within a specific frequency range. This contrasts with highly damped terrestrial regimes, where inertial effects can often be neglected. Further, to enhance performance and address the challenges posed by nondeterministic obstacle distributions, we incorporated computational intelligence by developing a real-time body compliance tuning controller based on cable tension feedback. This controller improves the robot's robustness and overall speed in heterogeneous hydrodynamic environments.

BoLD is a new dispute resolution protocol that is designed to replace the originally deployed Arbitrum dispute resolution protocol. Unlike that protocol, BoLD is resistant to delay attacks. It achieves this resistance without a significant increase in onchain computation costs and with reduced staking costs.

We study a slender filament beating in a viscous fluid with novel curvature-dependent bending stiffness. Our numerical and experimental investigations reveal that such differential stiffness can sustain planar bending waves far along flexible filaments, in stark contrast to the uniform-stiffness case which requires more sophisticated control. In particular, we establish basal actuation as a viable, parsimonious mechanism for generating high-amplitude planar bending waves. Moreover, the resulting beat patterns closely resemble the power-and-recovery strokes of propulsive biological filaments such as cilia, suggesting extensive applications in robotic and engineered systems., Comment: 7 pages, 4 figures

Future planetary exploration missions will require reaching challenging regions such as craters and steep slopes. Such regions are ubiquitous and present science-rich targets potentially containing information regarding the planet's internal structure. Steep slopes consisting of low-cohesion regolith are prone to flow downward under small disturbances, making it very challenging for autonomous rovers to traverse. Moreover, the navigation trajectories of rovers are heavily limited by the terrain topology and future systems will need to maneuver on flowable surfaces without getting trapped, allowing them to further expand their reach and increase mission efficiency. In this work, we used a laboratory-scale rover robot and performed maneuvering experiments on a steep granular slope of poppy seeds to explore the rover's turning capabilities. The rover is capable of lifting, sweeping, and spinning its wheels, allowing it to execute leg-like gait patterns. The high-dimensional actuation capabilities of the rover facilitate effective manipulation of the underlying granular surface. We used Bayesian Optimization (BO) to gain insight into successful turning gaits in high dimensional search space and found strategies such as differential wheel spinning and pivoting around a single sweeping wheel. We then used these insights to further fine-tune the turning gait, enabling the rover to turn 90 degrees at just above 4 seconds with minimal slip. Combining gait optimization and human-tuning approaches, we found that fast turning is empowered by creating anisotropic torques with the sweeping wheel., Comment: 6 pages, 6 figures, conference paper submission for ICRA2024

Sidewinding, a locomotion strategy characterized by the coordination of lateral and vertical body undulations, is frequently observed in rattlesnakes and has been successfully reconstructed by limbless robotic systems for effective movement across diverse terrestrial terrains. However, the integration of compliant mechanisms into sidewinding limbless robots remains less explored, posing challenges for navigation in complex, rheologically diverse environments. Inspired by a notable control simplification via mechanical intelligence in lateral undulation, which offloads feedback control to passive body mechanics and interactions with the environment, we present an innovative design of a mechanically intelligent limbless robot for sidewinding. This robot features a decentralized bilateral cable actuation system that resembles organismal muscle actuation mechanisms. We develop a feedforward controller that incorporates programmable body compliance into the sidewinding gait template. Our experimental results highlight the emergence of mechanical intelligence when the robot is equipped with an appropriate level of body compliance. This allows the robot to 1) locomote more energetically efficiently, as evidenced by a reduced cost of transport, and 2) navigate through terrain heterogeneities, all achieved in an open-loop manner, without the need for environmental awareness.

Much of the Earth and many surfaces of extraterrestrial bodies are composed of in-cohesive particle matter. Locomoting on granular terrain is challenging for common robotic devices, either wheeled or legged. In this work, we discover a robust alternative locomotion mechanism on granular media -- generating movement via self-vibration. To demonstrate the effectiveness of this locomotion mechanism, we develop a cube-shaped robot with an embedded vibratory motor and conduct systematic experiments on diverse granular terrains of various particle properties. We investigate how locomotion changes as a function of vibration frequency/intensity on granular terrains. Compared to hard surfaces, we find such a vibratory locomotion mechanism enables the robot to move faster, and more stable on granular surfaces, facilitated by the interaction between the body and surrounding granules. The simplicity in structural design and controls of this robotic system indicates that vibratory locomotion can be a valuable alternative way to produce robust locomotion on granular terrains. We further demonstrate that such cube-shape robots can be used as modular units for morphologically structured vibratory robots with capabilities of maneuverable forward and turning motions, showing potential practical scenarios for robotic systems.

While the transport of matter by wheeled vehicles or legged robots can be guaranteed in engineered landscapes like roads or rails, locomotion prediction in complex environments like collapsed buildings or crop fields remains challenging. Inspired by principles of information transmission which allow signals to be reliably transmitted over noisy channels, we develop a ``matter transport" framework demonstrating that non-inertial locomotion can be provably generated over ``noisy" rugose landscapes (heterogeneities on the scale of locomotor dimensions). Experiments confirm that sufficient spatial redundancy in the form of serially-connected legged robots leads to reliable transport on such terrain without requiring sensing and control. Further analogies from communication theory coupled to advances in gaits (coding) and sensor-based feedback control (error detection/correction) can lead to agile locomotion in complex terradynamic regimes.

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)]., Comment: 11 pages, 5 figures, 8 supplementary figures

Limbless locomotors, from microscopic worms to macroscopic snakes, traverse complex, heterogeneous natural environments typically using undulatory body wave propagation. Theoretical and robophysical models typically emphasize body kinematics and active neural/electronic control. However, we contend that because such approaches often neglect the role of passive, mechanically controlled processes (those involving "mechanical intelligence"), they fail to reproduce the performance of even the simplest organisms. To uncover principles of how mechanical intelligence aids limbless locomotion in heterogeneous terradynamic regimes, here we conduct a comparative study of locomotion in a model of heterogeneous terrain (lattices of rigid posts). We used a model biological system, the highly studied nematode worm Caenorhabditis elegans, and a robophysical device whose bilateral actuator morphology models that of limbless organisms across scales. The robot's kinematics quantitatively reproduced the performance of the nematodes with purely open-loop control; mechanical intelligence simplified control of obstacle navigation and exploitation by reducing the need for active sensing and feedback. An active behavior observed in C. elegans, undulatory wave reversal upon head collisions, robustified locomotion via exploitation of the systems' mechanical intelligence. Our study provides insights into how neurally simple limbless organisms like nematodes can leverage mechanical intelligence via appropriately tuned bilateral actuation to locomote in complex environments. These principles likely apply to neurally more sophisticated organisms and also provide a design and control paradigm for limbless robots for applications like search and rescue and planetary exploration., Comment: Published in Science Robotics

Limbless robots have the potential to maneuver through cluttered environments that conventional robots cannot traverse. As illustrated in their biological counterparts such as snakes and nematodes, limbless locomotors can benefit from interactions with obstacles, yet such obstacle-aided locomotion (OAL) requires properly coordinated high-level self-deformation patterns (gait templates) as well as low-level body adaptation to environments. Most prior work on OAL utilized stereotyped traveling-wave gait templates and relied on local body deformations (e.g., passive body mechanics or decentralized controller parameter adaptation based on force feedback) for obstacle navigation, while gait template design for OAL remains less studied. In this paper, we explore novel gait templates for OAL based on tools derived from geometric mechanics (GM), which thus far has been limited to homogeneous environments. Here, we expand the scope of GM to obstacle-rich environments. Specifically, we establish a model that maps the presence of an obstacle to directional constraints in optimization. In doing so, we identify novel gait templates suitable for sparsely and densely distributed obstacle-rich environments respectively. Open-loop robophysical experiments verify the effectiveness of our identified OAL gaits in obstacle-rich environments. We posit that when such OAL gait templates are augmented with appropriate sensing and feedback controls, limbless locomotors will gain robust function in obstacle rich environments.

Contact planning is crucial in locomoting systems.Specifically, appropriate contact planning can enable versatile behaviors (e.g., sidewinding in limbless locomotors) and facilitate speed-dependent gait transitions (e.g., walk-trot-gallop in quadrupedal locomotors). The challenges of contact planning include determining not only the sequence by which contact is made and broken between the locomotor and the environments, but also the sequence of internal shape changes (e.g., body bending and limb shoulder joint oscillation). Most state-of-art contact planning algorithms focused on conventional robots (e.g.biped and quadruped) and conventional tasks (e.g. forward locomotion), and there is a lack of study on general contact planning in multi-legged robots. In this paper, we show that using geometric mechanics framework, we can obtain the global optimal contact sequence given the internal shape changes sequence. Therefore, we simplify the contact planning problem to a graph optimization problem to identify the internal shape changes. Taking advantages of the spatio-temporal symmetry in locomotion, we map the graph optimization problem to special cases of spin models, which allows us to obtain the global optima in polynomial time. We apply our approach to develop new forward and sidewinding behaviors in a hexapod and a 12-legged centipede. We verify our predictions using numerical and robophysical models, and obtain novel and effective locomotion behaviors., Comment: SI video: https://doi.org/10.5281/zenodo.7608693

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 matter". This field can provide insight into important principles in living systems as well as aid engineering of responsive, robust and functional collectives., Comment: opinion piece

Study of the locomotion of a centipede (L. forficatus) at the air-water interface reveals that it does not predominantly use its 14 leg pairs to locomote; unlike most swimmers which propagate head-to-tail body bending waves, this species propels via tail-to-head waves. Its low mass and body-fluid contact yield locomotion dynamics in which fluid wave drag forces dominate inertia. Microorganism-inspired wave drag resistive force theory captures swimming performance of the macroscale animal, motivating a control hypothesis for the animals self-propulsion body and limb kinematics., Comment: Drag measurements presented in this paper are partially incorrect. New drag measurements that are accurate and correct will be included in an updated manuscript. Once the manuscript is complete, it will be submitted as a replacement

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 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 controls. We discover in laboratory experiments that terrestrial locomotion of a meter scale multi-segmented/legged robophysical model resembles undulatory fluid swimming. Experiments varying waves of limb 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 akin to microscopic-scale swimming. Theoretical analysis demonstrates that the high-dimensional multi-segmented/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 S. polymorpha) moving at relatively high speeds (~0.5 body lengths/sec). Our results could facilitate control of multilegged robots in complex terradynamic scenarios.

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 arises from the local interactions of individual particles are not well understood. In this study, we examine the emergent properties of amorphous entangled systems in three different examples: an in-silico "smarticle" collection, its robophysical chain, and living entangled aggregate of worm blobs (L. variegatus). In simulations, we examine how material properties change for a collective composed of dynamic three-link robots. We compare three methods of controlling entanglement in a collective: externally oscillations, shape-changes, and internal oscillations. We find that large-amplitude changes of the particle's shape using the shape-change procedure produced the highest average number of entanglements, with respect to the aspect ratio (l/w), improving the tensile strength of the collective. We demonstrate application of these simulations in two experimental systems: robotic chains and entangled worm blobs. In the robophysical models, we find emergent auxeticity behavior upon straining the confined collective. And finally, we show 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. Taken together, 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., Comment: 15 pages, 19 figures, W.S. and H.T. contributed equally to this work

We report in experiment and simulation the spontaneous formation of dynamically bound pairs of shape changing smarticle robots undergoing locally repulsive collisions. Borrowing terminology from Conway's simulated Game of Life, these physical `gliders' robustly emerge from an ensemble of individually undulating three-link two-motor smarticles and can remain bound for hundreds of undulations and travel for multiple robot dimensions. Gliders occur in two distinct binding symmetries and form over a wide range of angular flapping extent. This parameter sets the maximal concavity which influences formation probability and translation characteristics. Analysis of dynamics in simulation reveals the mechanism of effective dynamical attraction -- a result of the emergent interplay of appropriately oriented and timed repulsive interactions.

Many types of organisms utilize group aggregation as a method for survival. The freshwater oligochaete, California blackworms Lumbriculus variegatus form tightly entangled structures, or worm "blobs", that have adapted to survive in extremely low levels of dissolved oxygen (DO). Individual blackworms adapt to hypoxic environments through respiration via their mucous body wall and posterior ciliated hindgut, which they wave above them. However, the change in collective behavior at different levels of DO is not known. Using a closed-loop respirometer with flow, we discover that the relative tail reaching activity flux in low DO is $\sim$75x higher than in the high DO condition. Additionally, when flow rate is increased to suspend the worm blobs upward, we find that the average exposed surface area of a blob in low DO is $\sim$1.4x higher than in high DO. Furthermore, we observe emergent properties that arise when a worm blob is exposed to extreme DO levels. Here we show that internal stress is generated when worm blobs are exposed to high DO levels, allowing them to be physically lifted off from the bottom of a conical container using a serrated endpiece. Our results demonstrate how both collective behavior and the emergent generation of internal stress in worm blobs change to accommodate differing levels of oxygen, which could be used to model and simulate swarm robots, self-assembly structures, or soft material entanglements.

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 global-scale physics are often desired to predict how external bodies interact with soft media, as occurs in diverse situations from impact and penetration problems to locomotion over natural terrains. This work proposes a systematic program to develop three-dimensional reduced-order models for soft materials from a fundamental basis using continuum symmetries and rheological principles. In particular, we derive a reduced-order technique for modeling intrusion in granular media which we term three-dimensional Resistive Force Theory (3D-RFT), which is capable of accurately and quickly predicting the resistive stress distribution on arbitrary-shaped intruding bodies. 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 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., Comment: 12 pages, 7 figures, 1 SI document (12 pages, 8 figures, and 4 tables)

Systems consisting of spheres rolling on elastic membranes have been used to introduce a core conceptual idea of General Relativity (GR): how curvature guides the movement of matter. However, such schemes cannot accurately represent relativistic dynamics in the laboratory because of the dominance of dissipation and external gravitational fields. Here we demonstrate that an ``active" object (a wheeled robot), which moves in a straight line on level ground and can alter its speed depending on the curvature of the deformable terrain it moves on, can exactly capture dynamics in curved relativistic spacetimes. Via the systematic study of the robot's dynamics in the radial and orbital directions, we develop a mapping of the emergent trajectories of a wheeled vehicle on a spandex membrane to the motion in a curved spacetime. Our mapping demonstrates how the driven robot's dynamics mix space and time in a metric, and shows how active particles do not necessarily follow geodesics in the real space but instead follow geodesics in a fiducial spacetime. The mapping further reveals how parameters such as the membrane elasticity and instantaneous speed allow the programming of a desired spacetime, such as the Schwarzschild metric near a non-rotating blackhole. Our mapping and framework facilitate creation of a robophysical analog to a general relativistic system in the laboratory at low cost that can provide insights into active matter in deformable environments and robot exploration in complex landscapes.

Reorientation (turning in plane) plays a critical role for all robots in any field application, especially those that in confined spaces. While important, reorientation remains a relatively unstudied problem for robots, including limbless mechanisms, often called snake robots. Instead of looking at snakes, we take inspiration from observations of the turning behavior of tiny nematode worms C. elegans. Our previous work presented an in-place and in-plane turning gait for limbless robots, called an omega turn, and prescribed it using a novel two-wave template. In this work, we advance omega turn-inspired controllers in three aspects: 1) we use geometric methods to vary joint angle amplitudes and forward wave spatial frequency in our turning equation to establish a wide and precise amplitude modulation and frequency modulation on omega turn; 2) we use this new relationship to enable robots with fewer internal degrees of freedom (i.e., fewer joints in the body) to achieve desirable performance, and 3) we apply compliant control methods to this relationship to handle unmodelled effects in the environment. We experimentally validate our approach on a limbless robot that the omega turn can produce effective and robust turning motion in various types of environments, such as granular media and rock pile., Comment: Accepted to ICRA 2022

Although typically possessing four limbs and short bodies, lizards have evolved a diversity of body plans, from short-bodied and fully-limbed to elongate and nearly limbless. Such diversity in body morphology is hypothesized as adaptations to locomotion cluttered terrestrial environments, but the mode of propulsion -- e.g., the use of body and/or limbs to interact with the substrate -- and potential body/limb coordination remain unstudied. Here, we use biological experiments, a geometric theory of locomotion, and robophysical experiments to comparatively and systematically investigate such dynamics in a diverse sample of lizard morphologies. Locomotor field studies in short-limb, elongated lizards (Brachymeles) and laboratory studies of full-limbed lizards (Uma scoparia and Sceloporus olivaceus) and a limbless laterally undulating organism (Chionactis occipitalis) reveal that the body wave dynamics can be described by a combination of traveling and standing waves; the ratio of the amplitudes of these components is inversely related to limb length. We use geometric theory to analyze and explain the wave dynamics and body-leg coordination observations; the theory predicts that leg thrust modulates the body weight distribution and self-propulsion generation mechanism, which in turn facilitates the choice of body waves. We test our hypothesis in biological experiments by inducing the use of traveling wave in stereotyped lizards by modulating the ground penetration resistance, as well as in controlled non-biological experiments involving an undulating limbed robophysical model. Our models could be valuable in understanding functional constraints on the evolutionary process of elongation and limb reduction in lizards, as well as advancing robot designs.

Locomotion by shape changes (spermatozoon swimming, snake slithering, bird flapping) or gas expulsion (rocket firing) is assumed to require environmental interaction, due to conservation of momentum. As first noted in (Wisdom, 2003) and later in (Gu\'eron, 2009) and (Avron et al, 2006), in curved space or spacetime the non-commutativity of translations permits translation without momentum exchange, just as falling cats and lizards can self-deform to reorient in flat space without environmental interaction. Translation in curved space can occur not only in gravitationally induced curved spacetime (where translation is predicted to be on the order of $10^{-23}$ m per gait cycle) but also in the curved surfaces encountered by locomotors in real-world environments. Here 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 (impulse) via shape changes that can generate gauge potentials that manifest as translations. Our system produces shape changes comparable to the environment's inverse curvatures and generates from zero momentum forward movement of $10^{-1}$ cm per gait cycle even while resisted by weak gravitational and frictional forces. Dissipation via friction eventually arrests the robot but also imbues it with momentum which can be released upon a cessation of shape changes. This work demonstrates how the interaction between environmental curvature, active driving and geometric phases yields rich, exotic phenomena.

Serially connected robots are promising candidates for performing tasks in confined spaces such as search-and-rescue in large-scale disasters. Such robots are typically limbless, and we hypothesize that the addition of limbs could improve mobility. However, a challenge in designing and controlling such devices lies in the coordination of high-dimensional redundant modules in a way that improves mobility. Here we develop a general framework to control serially connected multi-legged robots. Specifically, we combine two approaches to build a general shape control scheme which can provide baseline patterns of self-deformation ("gaits") for effective locomotion in diverse robot morphologies. First, we take inspiration from a dimensionality reduction and a biological gait classification scheme to generate cyclic patterns of body deformation and foot lifting/lowering, which facilitate generation of arbitrary substrate contact patterns. Second, we use geometric mechanics methods to facilitates identification of optimal phasing of these undulations to maximize speed and/or stability. Our scheme allows the development of effective gaits in multi-legged robots locomoting on flat frictional terrain with diverse number of limbs (4, 6, 16, and even 0 limbs) and body actuation capabilities (including sidewinding gaits on limbless devices). By properly coordinating the body undulation and the leg placement, our framework combines the advantages of both limbless robots (modularity) and legged robots (mobility). We expect that our framework can provide general control schemes for the rapid deployment of general multi-legged robots, paving the ways toward machines that can traverse complex environments under real-life conditions.

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., Comment: 14 pages, 8 figures, and 4 movies

Self-organization is frequently observed in active collectives, from ant rafts to molecular motor assemblies. General principles describing self-organization away from equilibrium have been challenging to identify. We offer a unifying framework that models the behavior of complex systems as largely random, while capturing their configuration-dependent response to external forcing. This allows derivation of a Boltzmann-like principle for understanding and manipulating driven self-organization. We validate our predictions experimentally in shape-changing robotic active matter, and outline a methodology for controlling collective behavior. Our findings highlight how emergent order depends sensitively on the matching between external patterns of forcing and internal dynamical response properties, pointing towards future approaches for design and control of active particle mixtures and metamaterials.

Snake robots composed of alternating single-axis pitch and yaw joints have many internal degrees of freedom, which make them capable of versatile three-dimensional locomotion. In motion planning process, snake robot motions are often designed kinematically by a chronological sequence of continuous backbone curves that capture desired macroscopic shapes of the robot. However, as the geometric arrangement of single-axis rotary joints creates constraints on the rotations in the robot, it is challenging for the robot to reconstruct an arbitrary 3D curve. When the robot configuration does not accurately achieve the desired shapes defined by these backbone curves, the robot can have unexpected contacts with the environment, such that the robot does not achieve the desired motion. In this work, we propose a method for snake robots to reconstruct desired backbone curves by posing an optimization problem that exploits the robot's geometric structure. We verified that our method enables fast and accurate curve-configuration conversions through its applications to commonly used 3D gaits. We also demonstrated via robot experiments that 1) our method results in smooth locomotion on the robot; 2) our method allows the robot to approach the numerically predicted locomotive performance of a sequence of continuous backbone curve., Comment: Accepted for publication in IEEE RA-L

The impact of single passive intruders into granular particles has been studied in detail. However, the impact force produced by multiple intruders separated at a distance from one another, and hence the effect of their presence in close proximity to one another, is largely unexplored. Here, we use numerical simulations and laboratory experiments to study the force response of two parallel rods intruding vertically into granular media while varying the gap spacing between them. We also explored the effect of variations in friction, intruder size, and particle size on the force response. The net work ($W$) of the two rods over the depth of intrusion was measured, and the instantaneous velocities of particles over the duration of intrusion were calculated by simulations. We found that the work done by the intruders changes with distance between them. We observed a peak in $W$ at a gap spacing of $\sim$3 particle diameters, which was up to 25% greater than $W$ at large separation (\textgreater 11 particle diameters), beyond which the net work plateaued. This peak was likely due to less particle flow between intruders as we found a larger number of strong forces---identified as force chains---in the particle domain at gaps surrounding the peak force. Although higher friction caused greater force generation during intrusion, the gap spacing between the intruders at which the peak work was generated remained unchanged. Larger intruder sizes resulted in greater net work with the peak in $W$ occurring at slightly larger intruder separations. Taken together, our results show that peak work done by two parallel intruders remained within a narrow range, remaining robust to most other tested parameters., Comment: 12 pages, 8 figures, 6 supplemental figures, and appendix

Active matter physics and swarm robotics have provided powerful tools for the study and control of ensembles driven by internal sources. At the macroscale, controlling swarms typically utilizes significant memory, processing power, and coordination unavailable at the microscale, e.g., for colloidal robots, which could be useful for fighting disease, fabricating intelligent textiles, and designing nanocomputers. To develop principles that that can leverage physics of interactions and thus can be utilized across scales, we take a two-pronged approach: a theoretical abstraction of self-organizing particle systems and an experimental robot system of active cohesive granular matter that intentionally lacks digital electronic computation and communication, using minimal (or no) sensing and control, to test theoretical predictions. We consider the problems of aggregation, dispersion, and collective transport. As predicted by the theory, as a parameter representing interparticle attraction increases, the robots transition from a dispersed phase to an aggregated one, forming a dense, compact collective. When aggregated, the collective can transport non-robot "impurities" in their environment, thus performing an emergent task driven by the physics underlying the transition. These results point to a fruitful interplay between algorithm design and active matter robophysics that can result in new nonequilibrium physics and principles for programming collectives without the need for complex algorithms or capabilities.

Granular intrusions, such as dynamic impact or wheel locomotion, are complex multiphase phenomena where the grains exhibit solid-like and fluid-like characteristics together with an ejected gas-like phase. Despite decades of modeling efforts, a unified description of the physics in such intrusions is as yet unknown. Here we show that a continuum model based on the simple notions of frictional flow and tension-free separation describes complex granular intrusions near free surfaces. This model captures dynamics in a variety of experiments including wheel locomotion, plate intrusions, and running legged robots. The model reveals that three effects (a static contribution and two dynamic ones) primarily give rise to intrusion forces in such scenarios. Identification of these effects enables the development of a further reduced-order technique (Dynamic Resistive Force Theory) for rapid modeling of granular locomotion of arbitrarily shaped intruders. The continuum-motivated strategy we propose for identifying physical mechanisms and corresponding reduced-order relations has potential use for a variety of other materials., Comment: 41 pages including supplementary document, 10 figures, and 8 video

In many systems motion occurs on deformed and deformable surfaces, setting up the possibility for dynamical interactions solely mediated by the coupling of the entities with their environment. Here we study the "two-body" dynamics of robot locomotion on a highly deformable spandex membrane in two scenarios: one in which a robot orbits a large central depression and the other where the two robots affect each other's motion solely through mutual environmental deformations. Inspired by the resemblance of the orbits of the single robot with those of general relativistic orbits around black holes, we recast the vehicle plus membrane dynamics in physical space into the geodesic motion of a "test particle" in a fiducial curved space-time and demonstrate how this framework facilitates understanding the observed dynamics. The two-robot problem also exhibits a resemblance with Einstein's general relativistic view of gravity, which in the words of Wheeler: "spacetime tells matter how to move; matter tells spacetime how to curve." We generalize this case the mapping to include a reciprocal coupling that translates into robotic curvature-based control schemes which modify interaction (promoting avoidance or aggregation) without long-range sensing. Our work provides a starting point for developing a mechanical analog gravity system as well as develops a framework that can provide insights into active matter in deformable environments and robot exploration in complex landscapes.