This paper presents AquaMILR+, an untethered limbless robot designed for agile navigation in complex aquatic environments. The robot features a bilateral actuation mechanism that models musculoskeletal actuation in many anguilliform swimming organisms which propagates a moving wave from head to tail allowing open fluid undulatory swimming. This actuation mechanism employs mechanical intelligence through programmable body compliance, enhancing the robot's open-loop maneuverability when interacting with obstacles. AquaMILR+ also includes a compact depth control system inspired by the swim bladder and lung structures of eels and sea snakes. The mechanism, driven by a syringe and telescoping leadscrew, enables depth and pitch control - capabilities that are difficult for most anguilliform swimming robots to achieve. Additional structures, such as fins and a tail, further improve stability and propulsion efficiency. Our tests in both open water and laboratory models of 2D and 3D heterogeneous aquatic environments highlight AquaMILR+'s capabilities and suggest a promising system for complex underwater tasks such as search and rescue and deep-sea exploration.

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 multilegged robots by propagating the two waves of lateral body undulation and leg stepping, which we will refer to as the twowave 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 twowave template that we used to prescribe the multi-legged locomotion. Seeking effective alternative waves for obstacleclimbing, we designed a five-segment robot with static (nonactuated) 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 multilegged robots, paving the way towards all-terrain 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. Robust mechanisms for such elongate multi-legged robots to self-right remain unstudied. Here, we use a comparative biological and robophysical approach to investigate self-righting strategies. We first released 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's 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.

While undulatory swimming of elongate limbless robots has been extensively studied in open hydrodynamic environments, less research has been focused on limbless locomotion in complex, cluttered aquatic environments. Motivated by the concept of mechanical intelligence [1], where controls for obstacle navigation can be offloaded to passive body mechanics in terrestrial limbless locomotion, we hypothesize that principles of mechanical intelligence can be extended to cluttered hydrodynamic regimes. To test this, we developed an untethered limbless robot capable of undulatory swimming on water surfaces, utilizing a bilateral cable-driven mechanism inspired by organismal muscle actuation morphology to achieve programmable anisotropic body compliance. We demonstrated through robophysical experiments that, similar to terrestrial locomotion, an appropriate level of body compliance can facilitate emergent swim through complex hydrodynamic environments under pure open-loop control. Moreover, we found that swimming performance depends on undulation frequency, with effective locomotion achieved only within a specific frequency range. This contrasts with highly damped terrestrial regimes, where inertial effects can often be neglected. Further, to enhance performance and address the challenges posed by nondeterministic obstacle distributions, we incorporated computational intelligence by developing a real-time body compliance tuning controller based on cable tension feedback. This controller improves the robot's robustness and overall speed in heterogeneous hydrodynamic environments.

Centipedes exhibit great maneuverability in diverse environments due to their many legs and body-driven control. By leveraging similar morphologies and control strategies, 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 because of the difficulty in coordinating a high degree-of-freedom robot to follow predictable, planar trajectories. To resolve these challenges, we take inspiration from control schemes based on geometric mechanics(GM) in elongate systems’ locomotion through highly damped environments. We model the elongate, multi-legged system as a "terrestrial swimmer" in highly frictional environments and implement steering schemes derived from low-order templates. We identify an effective turning strategy by superimposing two traveling waves of lateral body undulation and further explore variations of the "turning wave" to enable a spectrum of arc-following steering primitives. We test our hypothesized modulation scheme on a robophysical model and validate steering trajectories against theoretically predicted displacements producing steering radii between 0 and 0.6 body length. We then apply our control framework to Ground Control Robotics’ elongate multi-legged robot, Major Tom, using these motion primitives to autonomously navigate around obstacles and corners on indoor and outdoor terrain. Our work creates a systematic framework for controlling these highly mobile devices in the plane using a low-order model based on sequences of body shape changes.

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 [1]. 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 [2], [3] 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 feedback control framework to enhance speed of multi-legged robots on challenging terrains remains underexplored due to diverse environmental interactions. Such complexity makes it difficult to identify the key parameters to control in these high-degree-of-freedom systems. Here, using laboratory and field experiments, we demonstrate that a vertical body undulation wave helps mitigate environmental disturbances that affect robot speed. These findings are supported by probabilistic models. Using such insights, 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 50% to 60% increase in speed and a 30% to 50% 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. Project website:https://juntaohe.github.io/TRO_2025/

Future planetary exploration missions will require reaching challenging regions such as craters and steep slopes. Such regions are ubiquitous and present science-rich targets potentially containing information regarding the planet’s internal structure. Steep slopes consisting of low-cohesion regolith are prone to flow downward under small disturbances, making it challenging for autonomous rovers to traverse. Moreover, the navigation trajectories of rovers are heavily limited by the terrain topology and future systems will need to maneuver on flowable surfaces without getting trapped, allowing them to further expand their reach and increase mission efficiency.In this work, we used a robophysical rover model and performed maneuvering experiments on a steep granular slope of poppy seeds to explore the rover’s turning capabilities. The rover is capable of lifting, sweeping, and spinning its wheels, allowing it to execute leg-like gait patterns. The high-dimensional actuation capabilities of the rover facilitate effective manipulation of the underlying granular surface. We used Bayesian Optimization (BO) to gain insight into successful turning gaits in high dimensional search space and found strategies such as differential wheel spinning and pivoting around a single sweeping wheel. We then used these insights to further fine-tune the turning gait, enabling the rover to turn nearly 90 degrees at just above 4 seconds with minimal downhill slip. Combining gait optimization and human-tuning approaches, we found that fast turning is empowered by creating anisotropic torques with the sweeping wheel.

Sidewinding, a locomotion strategy characterized by the coordination of lateral and vertical body undulations, is frequently observed in rattlesnakes and has been successfully implemented by limbless robotic systems for effective movement across diverse terrestrial terrains. However, the integration of compliant mechanisms into sidewinding limbless robots remains less explored, posing challenges for navigation in complex, rheologically diverse environments. Inspired by a notable control simplification via mechanical intelligence in lateral undulation [1], which offloads feedback control to passive body mechanics and interactions with the environment, we present an innovative design of a mechanically intelligent limbless robot for sidewinding. This robot features a decentralized bilateral cable actuation system that resembles organismal muscle actuation mechanisms. We develop a feedforward controller that incorporates programmable body compliance into the sidewinding gait template. Our experimental results highlight the emergence of mechanical intelligence when the robot is equipped with an appropriate level of body compliance. This allows the robot to 1) locomote more energetically efficiently, as evidenced by a reduced cost of transport, and 2) navigate through terrain heterogeneities, all achieved in an open-loop manner, without the need for environmental awareness.

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 [1]. 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.

In a geometric mechanics framework, the configuration space is decomposed into a shape space and a position space. The internal motion of the system is prescribed by a closed loop in the shape space, which causes net motion in the position space. If the shape space is a simply connected domain in an Euclidean space, then with an optimal choice of the body frame, the displacement in the position space is reasonably approximated by the surface integral of the height function, a functional relationship between the internal shape and position space variables. Our recent work has extended the scope of geometric methods from limbless undulatory system to those with legs; interestingly, the shape space for such systems has a torus structure. However, to the best of our knowledge, the optimal choice of the body frame on the torus shape space was not explored. In this paper, we develop a method to optimally choose the body frame on the torus which results in good approximation of displacement by the integral of the height function. We apply our methods to the centipede locomotion system and observe quantitative agreement of our prediction and experimental results.

Snake robots have the potential to locomote through tightly packed spaces, but turning effectively within unmodelled and unsensed environments remains challenging. Inspired by a behavior observed in the tiny nematode worm C. elegans, we propose a novel in-place turning gait for elongated limbless robots. To simplify the control of the robots' many internal degrees-of-freedom, we introduce a biologically-inspired template in which two co-planar traveling waves are superposed to produce an in-plane turning motion, the omega turn. The omega turn gait arises from modulating the wavelengths and amplitudes of the two traveling waves. We experimentally test the omega turn on a snake robot, and show that this turning gait outperforms previous turning gaits: it results in a larger angular displacement and a smaller area swept by the body over a gait cycle, allowing the robot to turn in highly confined spaces.

Mobile robots of all shapes and sizes move through the air, water, and over ground. However, few robots can move through the ground. Not only are the forces resisting movement much greater than in air or water, but the interaction forces are more complicated. Here we propose a soft robotic device that burrows through dry sand while requiring an order of magnitude less force than a similarly sized intruding body. The device leverages the principles of both tip-extension and granular fluidization. Like roots, the device extends from its tip; the principle of tip-extension eliminates skin drag on the sides of the body, because the body is stationary with respect to the medium. We implement this with an everting, pressure-driven thin film body. The second principle, granular fluidization, enables a granular medium to adopt a dynamic fluid-like state when pressurized fluid is passed through it, reducing the forces acting on an object moving through it. We realize granular fluidization with a flow of air through the core of the body that mixes with the medium at the tip. The proposed device could lead to applications such as search and rescue in mudslides or shallow subterranean exploration. Further, because it creates a physical conduit with its body, electrical lines, fluids, or even tools could be passed through this channel.

The varied and complex dynamics of deformable terrain are significant impediments toward real-world viability of locomotive robotics, particularly for legged machines. We explore vertical jumping on granular media (GM) as a model task for legged locomotion on uncharacterized deformable terrain. By integrating (Gaussian process) GP-based regression and evaluation to estimate ground forcing as a function of state, a one-dimensional jumper acquires the ability to learn forcing profiles exerted by its environment in tandem to achieving its control objective. The GP-based dynamical model initially assumes a baseline rigid, non-compliant surface. As part of an iterative procedure, the optimizer employing this model generates an optimal control to achieve a target jump height while respecting known hardware limitations of the robot model. Trajectory and forcing data recovered from evaluation on the true GM surface model simulation is applied to train the GP, and in turn, provide the optimizer a more richly informed dynamical model of the environment. After three iterations, predicted optimal control trajectories coincide with execution results, within 1.2% jumping height error, as the GP-based approximation converges to the true GM model.

Achieving stable bipedal robotic walking on deformable terrain is an open and challenging problem at the intersection of robotics and physics. Ground deformation introduces underactuation; uncertainty in terrain dynamics further complicates dynamical modeling and control methods. This work provides a stability criterion for flat-footed bipedal locomotion and allows model-based control methods to function on homogeneous deformable granular terrain (e.g. sand and dirt). By characterizing static reaction forces from granular materials, in conjunction with granular resistive force theory (RFT), we model and validate a static stability region for the center of mass (CoM) projection of a biped on a granular surface. We show that this stability region approximates the admissible Zero Moment Point (ZMP) region for walking, rendering common Linear Inverted Pendulum Model (LIPM) methods valid with our foot placement strategy. By interpreting the stability region as the maximum reaction moment of the terrain, we formulate walking as a hybrid dynamical system and utilize the partial hybrid zero dynamics (PHZD) based methodology to generate walking gaits. Finally, we experimentally validate both the ZMP and PHZD walking gaits on a planar bipedal robot, showing that the stability region criterion permits stable dynamic walking on homogeneous granular terrain.

This work demonstrates fast motion planning for robot locomotion that is optimized for terrain with complex dynamics, specifically, rapid penetration of granular media. Gait planning is critical for many legged locomotion control approaches, but they typically assume rigid ground contact. We aim to extend these planning methods to include terrain dynamics we see in the natural world, like sand and dirt, which can both deform and fluidize. Using an added-mass description of collective grain motion, we formulated a model of hydrostatic and hydrodynamic terrain effects that is both principled and representable with closed-form dynamics. As a result, we present a model and fast optimization formulation which solves accurate motion plans on granular media with tractable solving times (6.4-3.8 seconds). For validation, we optimized open-loop motor trajectories for a testbed jumping robot to jump to a target apex height from a bed a loosely packed poppy seeds, a model granular medium. While jumps optimized for rigid ground were anemic on granular media, terrain-aware trajectories hit within 6% of their target. This demonstrates the potential for robot locomotion which meets practical task demands, all while being aware of the terrain beneath it.

Biologically inspired robots perform many interesting and useful behaviors, but to effectively emulate their biological counterparts, robots often need to possess many degrees of freedom, complicating their mechanical design and making it difficult to apply standard control and motion planning strategies. To address this complexity, the robotics community has derived low-dimensional parameter-based controllers that naturally coordinate many degrees of freedom such as the serpenoid curves used to control snake robots. Controllers utilizing this parameterization for snake robots have been able to induce behaviors similar to that of the robots' biological counterparts. A similar concept, called a control template, is used in the study of animal movements. However, much of the prior work on control templates has been limited to in-plane motion. In this work, we extend the usage of control templates to three dimensions to both better model and understand biology, as well as to help us gain better intuition into how we can use pre-existing control paradigms to create new behaviors for biologically inspired robots.

Our research group has started a collaboration that analyzes data collected from biological snakes to provide insight on how to better program snake robots. Most data collected on biological snakes views the snakes from above and thus can only detect motion in the horizontal plane. However, both our robots and biological snakes are capable of generating motions both in the horizontal and vertical planes. Vertical waves naturally play a major role in limbless locomotion in that they simultaneously provide thrust motion and make-and-break contact between the mechanism and environment. Analysis on the data, collected from sidewinder rattle snakes, revealed that disparate modes of locomotion emerged from different contact patterns. We conclude that the same horizontal undulation can cause dramatically different motions for both the biological and robotic snakes depending upon the choice of contacts. With this knowledge, we introduce contact scheduling, a technique that plans positions of contacts along the body to design gaits for snake robots. Contact scheduling results in a novel turning gait, which can reorient a snake robot more than 90 degrees in one gait cycle.

The sandfish lizard (Scincus scincus) swims within granular media (sand) using axial body undulations to propel itself without the use of limbs. In previous work we predicted average swimming speed by developing a numerical simulation that incorporated experimentally measured biological kinematics into a multibody sandfish model. The model was coupled to an experimentally validated soft sphere discrete element method simulation of the granular medium. In this paper, we use the simulation to study the detailed mechanics of undulatory swimming in a "granular frictional fluid" and compare the predictions to our previously developed resistive force theory (RFT) which models sand-swimming using empirically determined granular drag laws. The simulation reveals that the forward speed of the center of mass (CoM) oscillates about its average speed in antiphase with head drag. The coupling between overall body motion and body deformation results in a non-trivial pattern in the magnitude of lateral displacement of the segments along the body. The actuator torque and segment power are maximal near the center of the body and decrease to zero toward the head and the tail. Approximately 30% of the net swimming power is dissipated in head drag. The power consumption is proportional to the frequency in the biologically relevant range, which confirms that frictional forces dominate during sand-swimming by the sandfish. Comparison of the segmental forces measured in simulation with the force on a laterally oscillating rod reveals that a granular hysteresis effect causes the overestimation of the body thrust forces in the RFT. Our models provide detailed testable predictions for biological locomotion in a granular environment.

Previously we modeled the undulatory subsurface locomotion of the sandfish lizard with a sand-swimming robot which displayed performance comparable to the organism. In this work we control the lift forces on the robot by varying its head shape and demonstrate that these granular forces predict the vertical motion of the robot. Inspired by the tapered head of the sandfish lizard, we drag a wedge shaped object horizontally and parallel to its lower face through a granular medium and show that by varying the angle of the upper leading surface of the wedge, α, the lift force can be varied from positive to negative. Testing the robot with these wedges as heads results in vertical motion in the same direction as the lift force in the drag experiments. As the robot moves forward, the force on its head normal to the body plane results in a net torque imbalance which pitches the robot causing it to rise or sink within the medium. Since repeatedly varying α for a wedge head to achieve a desired lift is impractical, we test robot heads that approximate a wedge head inclined at varying angles by changing the angle of the bottom and top surfaces of the wedge, and show that similar lift control is achieved. Our results provide principles for the construction of robots that will be able to follow arbitrary trajectories within complex substrates like sand, and also lend support to hypotheses that morphological adaptations of desert-dwelling organisms aid in their subsurface locomotion.