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.
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.
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.
Whereas the transport of matter by wheeled vehicles or legged robots can be guaranteed in engineered landscapes such as roads or rails, locomotion prediction in complex environments such as collapsed buildings or crop fields remains challenging. Inspired by the principles of information transmission, which allow signals to be reliably transmitted over “noisy” channels, we developed a “matter-transport” framework that demonstrates that noninertial 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 with advances in gaits (coding) and sensor-based feedback control (error detection and correction) can lead to agile locomotion in complex terradynamic regimes.
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