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Contact planning is crucial to the locomotion performance of robots: to properly self-propel forward, it is not only important to determine the sequence of internal shape changes (e.g., body bending and limb shoulder joint oscillation) but also the sequence by which contact is made and broken between the mechanism and its environment. Prior work observed that properly coupling contact patterns and shape changes allows for computationally tractable gait design and efficient gait performance. The state of the art, however, made assumptions, albeit motivated by biological observation, as to how contact and shape changes can be coupled. In this paper, we extend the geometric mechanics (GM) framework to design contact patterns. Specifically, we introduce the concept of “contact space” to the GM framework. By establishing the connection between velocities in shape and position spaces, we can estimate the benefits of each contact pattern change and therefore optimize the sequence of contact patterns. In doing so, we can also analyze how a contact pattern sequence will respond to perturbations. We apply our framework to sidewinding robots and enable (1) effective locomotion direction control and (2) robust locomotion performance as the spatial resolution decreases. We also apply our framework to a hexapod robot with two back-bending joints and show that we can simplify existing hexapod gaits by properly reducing the number of contact state switches (during a gait cycle) without significant loss of locomotion speed. We test our designed gaits with robophysical experiments, and we obtain good agreement between theory and experiments. 

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. 

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 10 − 1  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. 

Abstract—Contact planning is crucial to the locomotion per-formance of limbless robots. Typically, the pattern by which contact is made and broken between the mechanism and its environment determines the motion of the robot. The design of these patterns, often called contact patterns, is a difficult problem. In previous work, the prescription of contact patterns was derived from observations of biological systems or determined empirically from black-box optimization algorithms. However, such contact pattern prescription is only applicable to specific mechanisms, and is challenging to generalize. For example, the stable and effective contact pattern prescribed for a 12-link limbless robot can be neither stable nor effective for a 6-link limbless robot. In this paper, using a geometric motion planning scheme, we develop a framework to design, optimize, and analyze contact patterns to generate effective motion in the desired directions. Inspired by prior work in geometric mechanics, we separate the configuration space into a shape space (the internal joint angles), a contact state space, and a position space; then we optimize the function that couples the contact state space and the shape space. Our framework provides physical insights into the contact pattern design and reveals principles of empirically derived contact pattern prescriptions. Applying this framework, we can not only control the direction of motion of a 12-link limbless robot by modulating the contact patterns, but also design effective sidewinding gaits for robots with fewer motors (e.g., a 6-link robot). We test our designed gaits by robophysical experiments and obtain excellent agreement. We expect our scheme can be broadly applicable to robots which make/break contact. 

At the macroscale, controlling robotic swarms typically uses substantial 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 can leverage physical interactions and thus be used 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. As predicted by theory, as interparticle attraction increases, the collective transitions from dispersed to a compact phase. When aggregated, the collective can transport non-robot “impurities,” thus performing an emergent task driven by the physics underlying the transition. These results reveal a fruitful interplay between algorithm design and active matter robophysics that can result in principles for programming collectives without the need for complex algorithms or capabilities. 

Robot locomotion is typically generated by coordinated integration of single-purpose components, like actuators, sensors, body segments, and limbs. We posit that certain future robots could self-propel using systems in which a delineation of components and their interactions is not so clear, becoming robust and flexible entities composed of functional components that are redundant and generic and can interact stochastically. Control of such a collective becomes a challenge because synthesis techniques typically assume known input-output relationships. To discover principles by which such future robots can be built and controlled, we study a model robophysical system: planar ensembles of periodically deforming smart, active particles—smarticles. When enclosed, these individually immotile robots could collectively diffuse via stochastic mechanical interactions. We show experimentally and theoretically that directed drift of such a supersmarticle could be achieved via inactivation of individual smarticles and used this phenomenon to generate endogenous phototaxis. By numerically modeling the relationship between smarticle activity and transport, we elucidated the role of smarticle deactivation on supersmarticle dynamics from little data—a single experimental trial. From this mapping, we demonstrate that the supersmarticle could be exogenously steered anywhere in the plane, expanding supersmarticle capabilities while simultaneously enabling decentralized closed-loop control. We suggest that the smarticle model system may aid discovery of principles by which a class of future “stochastic” robots can rely on collective internal mechanical interactions to perform tasks.