Probabilistic approach to feedback control enhances multi-legged locomotion on rugged landscapes

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.

Addition of a peristaltic wave improves multi-legged locomotion performance on complex terrains

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.

Steering Elongate Multi-legged Robots By Modulating Body Undulation Waves

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.

Effective self-righting strategies for elongate multi-legged robots

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.

Learning to enhance multi-legged robot on rugged landscapes

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.

Multi-legged matter transport: a framework for locomotion on noisy landscapes

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.

Mechanical Intelligence Simplifies Control in Terrestrial Limbless Locomotion

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 (i.e., those involving mechanical intelligence), they fail to reproduce the performance of even the simplest organisms. To discover 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 use a model biological system, the highly studied nematode worm C. elegans, and a novel robophysical device whose bilateral actuator morphology models that of limbless organisms across scales. The robot's kinematics quantitatively reproduce the performance of the nematodes with purely open-loop control; mechanical intelligence simplifies 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, robustifies 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 new design and control paradigm for limbless robots for applications like search and rescue and planetary exploration.

Gait design for limbless obstacle aided locomotion using geometric mechanics

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.

Geometry of contact: contact planning for multi-legged robots via spin models duality

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.

Generalized Omega Turn Gait Enables Agile Limbless Robot Turning in Complex Environments

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.