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.

No abstract

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.

Kinematic motion planning using geometric mechanics tends to prescribe a trajectory in a parameterization of a shape space and determine its displacement in a position space. Often this trajectory is called a gait. Previous works assumed that the shape space is Euclidean when often it is not, either because the robotic joints can spin around forever (i.e., has an S 1 configuration space component, or its parameterization has an S 1 dimension). Consider a shape space that is a torus; gaits that "wrap" around the full range of a shape variable and return to its starting configuration are valid gaits in the shape space yet appear as line segments in the parameterization. Since such a gait does not form a closed loop in the parameterization, existing geometric mechanics methods cannot properly consider them. By explicitly analyzing the topology of the underlying shape space, we derive geometric tools to consider systems with toroidal and cylindrical shape spaces.

Many quadrupedal animals use lateral degrees of freedom in their backs to assist locomotion. This paper seeks to use a robotic model to demonstrate that back bending assists not only forward motion, but also lateral and turning motions. We present a simple planner that uses geometric mechanics to prescribe gaits that coordinate both leg movements and back bending motion. Using these geometric tools, we show that back-bending can improve stride displacement in the forward, rotational, and lateral directions. Interestingly, we observe that the animal's (salamander, Salamandra salamandra) back-bending is close to our calculated back-bending to improve forward displacement. In addition to locomotion performance improvement, back bending can also expand the target position space a robot can reach within one gait cycle. Our results are verified by conducting experiments with a robot moving on granular materials.

Sidewinding is a form of locomotion executed by certain snakes and has been reconstructed in limbless robots; the gait is beneficial because it is effective in diverse terrestrial environments. Sidewinding gaits are generated by coordination of horizontal and vertical traveling waves of body undulation: the horizontal wave largely sets the direction of sidewinding with respect to the body frame while the vertical traveling wave largely determines the contact pattern between the body and the environment. When the locomotor’s center of mass leaves the supporting polygon formed by the contact pattern, undesirable locomotor behaviors (such as unwanted turning or unstable oscillation of the body) can occur. In this article, we develop an approach to generate desired translation and turning by modulating the vertical wave. These modulations alter the distribution of body–environment contact patches and can stabilize configurations that were previously statically unstable. The approach first identifies the spatial frequency of the vertical wave that statically stabilizes the locomotor for a given horizontal wave. Then, using geometric mechanics tools, we design the coordination between body waves that produces the desired translation or rotation. We demonstrate the effectiveness of our technique in numerical simulations and on experiments with a 16-joint limbless robot locomoting on flat hard ground. Our scheme broadens the range of movements and behaviors accessible to sidewinding locomotors at low speeds, which can lead to limbless systems capable of traversing diverse terrain stably and/or rapidly.

Colloidal particles with asymmetric surface chemistry (Janus particles) have unique bifunctional properties. The size of these particles is an important determinant for their applications in diverse fields from drug delivery to chemical catalysis. The size of Janus particles, with a core surface coated with carboxylate and a partially encapsulating silica shell, depends upon several factors, including the core size and the concentration of carboxylate coating. The role of the carboxylate coating on the Janus particle size is well-understood; however, the role of the core size is not well defined. The role of the carboxylated polystyrene (cPS) core size on the cPS-silica Janus particle morphology (its size and shape) was examined by testing two different silica sizes and five different cPS core sizes. Results from electron microscopy (EM) and dynamic light scattering (DLS) analysis indicate that the composite cPS-silica particle acquires two distinct shapes: (i) when the size of the cPS core is much smaller than the non-cPS silica (b-SiO2) sphere, partially encapsulated Janus particles are formed, and (ii) when the cPS core is larger than or equal to the b-SiO2 sphere, a raspberry-like structure rather than a Janus particle is formed. The cPS-silica Janus particles of ∼100-500 nm size were obtained when the size of the cPS core was much smaller than the non-cPS silica (b-SiO2) sphere. These scalable nanoscale Janus particles will have wide application in a multifunctional delivery platform and catalysis.

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.