Limbless organisms like snakes can navigate nearly all terrain. In particular, desert-dwelling sidewinder rattlesnakes (C. cerastes) operate effectively on inclined granular media (like sand dunes) that induce failure in field-tested limbless robots through slipping and pitching. Our laboratory experiments reveal that as granular incline angle increases, sidewinder rattlesnakes increase the length of their body in contact with the sand. Implementing this strategy in a physical robot model of the snake enables the device to ascend sandy slopes close to the angle of maximum slope stability. Plate drag experiments demonstrate that granular yield stresses decrease with increasing incline angle. Together these three approaches demonstrate how sidewinding 1 arXiv:1410.2945v1 [physics.bio-ph] 11 Oct 2014 with contact-length control mitigates failure on granular media.The majority of terrestrial mobile robots are restricted to laboratory environments, in part because such robots are designed to roll on hard flat surfaces. It is difficult to systematically improve such terrestrial robots because we lack understanding of the physics of interaction with complex natural substrates like sand, dirt and tree bark. We are thus limited in our ability to computationally explore designs for potential all-terrain vehicles; in contrast, many of the recent developments in aerial and aquatic vehicles have been enabled by sophisticated computationaldynamics tools that allow such systems to be designed in silico (1).Compared with human-made devices, organisms such as snakes, lizards, and insects move effectively in nearly all natural environments. In recent years, scientists and engineers have sought to systematically discover biological principles of movement and implement these in robots (2). This "bioinspired robotics" approach (3) has proved fruitful to design laboratory robots with new capabilities (new gaits, morphologies, control schemes) including rapid running (2, 4), slithering (5), flying (6), and swimming in sand (7). Fewer studies have transferred biological principles into robust field-ready devices (4, 8) capable of operating in, and interacting with, natural terrain.Limbless locomotors like snakes are excellent systems to study to advance real-world allterrain mobility. Snakes are masters of most terrains: they can move rapidly on land (9, 10) and through water (11), burrow and swim through sand and soil (12), slither through tiny spaces (13), climb complex surfaces (14), and even glide through the air (15). Relative to legged locomotion, limbless locomotion is less studied, and thus broad principles which govern multi-environment movement are lacking. Recently developed limbless robotic platforms (5), based generally on the snake body plan, are appealing for multi-functional robotics study because they are also capable of a variety of modes of locomotion. These robots can traverse confined spaces, climb trees and pipes, and potentially dive through loose material. However, 2 the gaits that carry these robots across fir...

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

Many organisms move using traveling waves of body undulation, and most work has focused on single-plane undulations in fluids. Less attention has been paid to multiplane undulations, which are particularly important in terrestrial environments where vertical undulations can regulate substrate contact. A seemingly complex mode of snake locomotion, sidewinding, can be described by the superposition of two waves: horizontal and vertical body waves with a phase difference of ±90°. We demonstrate that the high maneuverability displayed by sidewinder rattlesnakes (Crotalus cerastes) emerges from the animal's ability to independently modulate these waves. Sidewinder rattlesnakes used two distinct turning methods, which we term differential turning (26°change in orientation per wave cycle) and reversal turning (89°). Observations of the snakes suggested that during differential turning the animals imposed an amplitude modulation in the horizontal wave whereas in reversal turning they shifted the phase of the vertical wave by 180°. We tested these mechanisms using a multimodule snake robot as a physical model, successfully generating differential and reversal turning with performance comparable to that of the organisms. Further manipulations of the two-wave system revealed a third turning mode, frequency turning, not observed in biological snakes, which produced large (127°) in-place turns. The two-wave system thus functions as a template (a targeted motor pattern) that enables complex behaviors in a high-degree-offreedom system to emerge from relatively simple modulations to a basic pattern. Our study reveals the utility of templates in understanding the control of biological movement as well as in developing control schemes for limbless robots.sidewinder | biomechanics | robotics | template | control P ropagating waves of flexion along the axis of a long, slender body (henceforth "axial waves") to produce propulsion is common in biological locomotion in aquatic and terrestrial environments. The majority of biological studies of axial wave propulsion at different scales have occurred in aquatic environments (1, 2). Understanding the efficacy of given wave patternswhich are often assumed to act in a single plane (e.g., mediolateral axial bending)-can be gained through full solution of the equations of hydrodynamics (3) or approximations (4). Terrestrial environments such as sand, mud, and cluttered heterogeneous substrates encountered by limbless axial undulators such as snakes can display similar (if not greater) complexity, yet far less attention has been paid to such locomotion (5, 6).Snake axial propulsion in terrestrial environments differs from fluid locomotion in two key ways. First, most substrates are not yet described at the level of fluids (7), making it a challenge to understand how substrate-body interactions affect locomotor performance, and therefore requiring robotic physical models. Second, the body may be both laterally and/or dorsoventrally flexed (5) to allow different elements of the body to contact ...

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.

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Many animals generate propulsive forces by coordinating legs, which contact and push against the surroundings, with bending of the body, which can only indirectly influence these forces. Such body–leg coordination is not commonly employed in quadrupedal robotic systems. To elucidate the role of back bending during quadrupedal locomotion, we study a model system: the salamander, a sprawled-posture quadruped that uses lateral bending of the elongate back in conjunction with stepping of the limbs during locomotion. We develop a geometric approach that yields a low-dimensional representation of the body and limb contributions to the locomotor performance quantified by stride displacement. For systems where the damping forces dominate inertial forces, our approach offers insight into appropriate coordination patterns, and improves the computational efficiency of optimization techniques. In particular, we demonstrate effect of the lateral undulation coordinated with leg movement in the forward, rotational, and lateral directions of the robot motion. We validate the theoretical results using numerical simulations, and then successfully test these approaches using robophysical experiments on granular media, a model deformable, frictional substrate. Although our focus lies primarily on robotics, we also demonstrate that our tools can accurately predict optimal body bending of a living salamander Salamandra salamandra.

Snake robots are highly articulated mechanisms that can perform a variety of motions that conventional robots cannot. Despite many demonstrated successes of snake robots, these mechanisms have not been able to achieve the agility displayed by their biological counterparts. We suggest that studying how biological snakes coordinate whole-body motion to achieve agile behaviors can help improve the performance of snake robots. The foundation of this work is based on the hypothesis that, for snake locomotion that is approximately kinematic, replaying parameterized shape trajectory data collected from biological snakes can generate equivalent motions in snake robots. To test this hypothesis, we collected shape trajectory data from sidewinder rattlesnakes executing a variety of different behaviors. We then analyze the shape trajectory data in a concise and meaningful way by using a new algorithm, called conditioned basis array factorization, which projects high-dimensional data arrays onto a low-dimensional representation. The low-dimensional representation of the recorded snake motion is able to reproduce the essential features of the recorded biological snake motion on a snake robot, leading to improved agility and maneuverability, confirming our hypothesis. This parameterized representation allows us to search the low-dimensional parameter space to generate behaviors that further improve the performance of snake robots.

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.