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![Examples of biological snakes employing a lifting body wave in addition to lateral undulation
a A sidewinding rattlesnake (Crotalus cerastes) asymmetrically lifts up only one side of its body⁵². b A corn snake (Pantherophis guttatus) slithering on a flat surface and symmetrically lifting regions of high lateral curvature on both sides of its body13, 14. c Schematic of the planar snake model. Note that the arc-length s goes from tail to head to retain consistency with13, 14. The local position x is related to the center of mass x¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{{{{{{{\bf{x}}}}}}}}}$$\end{document} through zero-mean integration function I[t] (Methods). d Three different stereotypes of body lifting. Top: Zero body lifting leads to classical undulatory planar gaits. Middle: Symmetric body lifting, the snake symmetrically lifts both sides of its body13, 21. Bottom: Asymmetric body lifting, the snake lifts one side of its body off the ground and maintains the other in contact with the ground. Asymmetric lifting has been well-documented in sidewinding snakes2, 21, 22, 26, 27. Net forces and torques acting on the snake over one undulation period are computed via Fnet=∫01∫01F(s,t)dsdt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\bf{F}}}}}}}}}_{{{{{{{{\rm{net}}}}}}}}}=\int\nolimits_{0}^{1}\int\nolimits_{0}^{1}{{{{{{{\bf{F}}}}}}}}(s,t)\,ds\,dt$$\end{document} and Tnet=∫01∫01(x−x¯)×F(s,t)dsdt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\bf{T}}}}}}}}}_{{{{{{{{\rm{net}}}}}}}}}=\int\nolimits_{0}^{1}\int\nolimits_{0}^{1}({{{{{{{\bf{x}}}}}}}}-\overline{{{{{{{{\bf{x}}}}}}}}})\times {{{{{{{\bf{F}}}}}}}}(s,t)\,ds\,dt$$\end{document}, respectively.](publication/355399925/figure/fig3/AS:1080858744238082@1634708214681/Examples-of-biological-snakes-employing-a-lifting-body-wave-in-addition-to-lateral.png)
Examples of biological snakes employing a lifting body wave in addition to lateral undulation a A sidewinding rattlesnake (Crotalus cerastes) asymmetrically lifts up only one side of its body⁵². b A corn snake (Pantherophis guttatus) slithering on a flat surface and symmetrically lifting regions of high lateral curvature on both sides of its body13, 14. c Schematic of the planar snake model. Note that the arc-length s goes from tail to head to retain consistency with13, 14. The local position x is related to the center of mass x¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{{{{{{{\bf{x}}}}}}}}}$$\end{document} through zero-mean integration function I[t] (Methods). d Three different stereotypes of body lifting. Top: Zero body lifting leads to classical undulatory planar gaits. Middle: Symmetric body lifting, the snake symmetrically lifts both sides of its body13, 21. Bottom: Asymmetric body lifting, the snake lifts one side of its body off the ground and maintains the other in contact with the ground. Asymmetric lifting has been well-documented in sidewinding snakes2, 21, 22, 26, 27. Net forces and torques acting on the snake over one undulation period are computed via Fnet=∫01∫01F(s,t)dsdt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\bf{F}}}}}}}}}_{{{{{{{{\rm{net}}}}}}}}}=\int\nolimits_{0}^{1}\int\nolimits_{0}^{1}{{{{{{{\bf{F}}}}}}}}(s,t)\,ds\,dt$$\end{document} and Tnet=∫01∫01(x−x¯)×F(s,t)dsdt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\bf{T}}}}}}}}}_{{{{{{{{\rm{net}}}}}}}}}=\int\nolimits_{0}^{1}\int\nolimits_{0}^{1}({{{{{{{\bf{x}}}}}}}}-\overline{{{{{{{{\bf{x}}}}}}}}})\times {{{{{{{\bf{F}}}}}}}}(s,t)\,ds\,dt$$\end{document}, respectively.
Source publication
Motivated by a possible convergence of terrestrial limbless locomotion strategies ultimately determined by interfacial effects, we show how both 3D gait alterations and locomotory adaptations to heterogeneous terrains can be understood through the lens of local friction modulation. Via an effective-friction modeling approach, compounded by 3D simul...
Citations
... snow, sand or granular media), their drag will be determined by friction in addition to viscosity and hence, their performance will depend on the texture of the material (Hosoi & Goldman, 2015;Maladen et al., 2011;Li, Zhang & Goldman, 2013). The result will depend on the mode of locomotion: legged organisms may use friction as leverage to pull themselves forward (Persson, 2007), while undulatory or crawling organisms will be impeded by rougher substrates (Zhang et al., 2021). Roughness is thus an important physical property that affects locomotion of ground-dwelling organisms (Clifton et al., 2023). ...
Understanding the factors that determine the occurrence and strength of ecological interactions under specific abiotic and biotic conditions is fundamental since many aspects of ecological community stability and ecosystem functioning depend on patterns of interactions among species. Current approaches to mapping food webs are mostly based on traits, expert knowledge, experiments, and/or statistical inference. However, they do not offer clear mechanisms explaining how trophic interactions are affected by the interplay between organism characteristics and aspects of the physical environment, such as temperature, light intensity or viscosity. Hence, they cannot yet predict accurately how local food webs will respond to anthropogenic pressures, notably to climate change and species invasions. Herein, we propose a framework that synthesises recent developments in food‐web theory, integrating body size and metabolism with the physical properties of ecosystems. We advocate for combination of the movement paradigm with a modular definition of the predation sequence, because movement is central to predator–prey interactions, and a generic, modular model is needed to describe all the possible variation in predator–prey interactions. Pending sufficient empirical and theoretical knowledge, our framework will help predict the food‐web impacts of well‐studied physical factors, such as temperature and oxygen availability, as well as less commonly considered variables such as wind, turbidity or electrical conductivity. An improved predictive capability will facilitate a better understanding of ecosystem responses to a changing world.
... Bio-hybrid soft robots, which integrate living cells or tissues with soft robotic structures, are also gaining attention for applications in tissue engineering and drug delivery [24]. Additionally, soft robot locomotion is a prominent area of research, with robots capable of crawling, slithering, or undulating like snakes, making them suitable for tasks such as search and rescue missions [25] [26]. In the medical field, soft robotic exoskeletons are being designed to provide support and assistance to individuals with mobility impairments or to enhance the physical abilities of healthy users, offering a new dimension to rehabilitation and physical augmentation [27] [28]. ...
Soft slender robots have attracted more and more research attentions in these years due to their continuity and compliance natures. However, mechanics modeling for soft robots interacting with environment is still an academic challenge because of the non-linearity of deformation and the non-smooth property of the contacts. In this work, starting from a piece-wise local strain field assumption, we propose a nonlinear dynamic model for soft robot via Cosserat rod theory using Newtonian mechanics which handles the frictional contact with environment and transfer them into the nonlinear complementary constraint (NCP) formulation. Moreover, we smooth both the contact and friction constraints in order to convert the inequality equations of NCP to the smooth equality equations. The proposed model allows us to compute the dynamic deformation and frictional contact force under common optimization framework in real time when the soft slender robot interacts with other rigid or soft bodies. In the end, the corresponding experiments are carried out which valid our proposed dynamic model.
... Its controllability and workspace are exceedingly high, and the compliance endows elephants with an unrivalled muscular multi-tool on the meter scale 7 . Nature holds many other examples of such large-scale structures which simultaneously offer large range of motions, precision, and compliance including octopus tentacles 8 , snake bodies 9 , and monkey tails 10 . ...
... For the experiments requiring trajectory planning (Fig. 7) and the human-robot interaction demonstrations (Figs. 8,9), an inverse static approach was used. The method computes the tendon actuation needed to achieve the desired trajectory at the gripper's end effector. ...
The development and use of architectured structures is opening up new possibilities for designing and fabricating soft robots. These materials utilize their unique topology and geometry to modify the physical and mechanical properties of the structure. We propose a novel architectured structure based on trimmed helicoids that allows for independent regulation of the bending and axial stiffness. This allows for control of the many mechanical properties of the structure which facilitates far greater tuneability of the resulting soft robot properties. Leveraging FEA and computational analysis we select a geometry that provides an optimal trade-off between controllability, sensitivity to errors in control, and compliance. By combining these modular trimmed helicoid structures in conjunction with control methods, we demonstrate a meter-scale soft manipulator with unprecedented capabilities in terms of control precision, workspace size, and compliant interactions with the environment. Combining these properties, we demonstrate how the robot arm can be used to perform complex tasks that leverage robot-human and robot-environment interactions such as human feeding and collaborative object manipulation.
... The former offers higher accuracy, while the latter is simpler to implement (Fig. 4(a1) and (b1)) [22,93]. Anisotropic characteristics of snakes on heterogeneous terrains were explored using the Coulomb friction model to capture the frictional effects [94]. ...
Purpose of review
In this review, we briefly summarize the numerical methods commonly used for the nonlinear dynamic analysis of soft robotic systems. The underlying mechanical principles as well as the geometrical treatment tailored for soft robots are introduced with particular emphasis on one-dimensional models. Additionally, the review encompasses three-dimensional frameworks, available simulation packages, and various types of interaction models, shedding light on the design, actuation, motion control, and internal and external forces of soft robots.
Recent findings
Reduced-order models can offer high efficiency in characterizing nonlinear deformations, allowing convenient tailoring based on specific structural and material configurations. For pursuing high simulation accuracy and detailed mechanics, the finite element method proves to be a valuable tool through numerous off-the-shelf platforms. Furthermore, machine learning has emerged as a promising tool to effectively address the challenges within the mechanics community.
Summary
A wide range of kinematic and dynamic numerical models is available for simulating the behaviors of soft robots, offering exceptional adaptability to different geometries and structures based on existing modeling theories and numerical solution algorithms. However, the trade-off between computational complexity and simulation accuracy remains a challenge in achieving fast, accurate, and robust control of soft robots in complex environments.
... Furthermore, a recent work by Zhang et al. [27] developed a framework for modeling and simulating complex musculoskeletal structures. The framework has been implemented in the study of the effect of friction modulation in snake slithering locomotion [28]. Additionally, there is an increasing interest in computer graphics, and physics-based simulation for utilizing muscle-based actuation to simulate the locomotion of 3D bipedal characters and imaginary creatures [29][30][31]. ...
This paper presents a study of energy efficiency and kinematic-based optimal design locomotion of a pneumatic artificial muscle (PAM)-driven snake-like robot. Although snake-like robots have several advantages over wheeled and track-wheeled mobile robots, their low energy-locomotion has limited their applications in long-range and outdoor fields. This work continues our previous efforts in designing and prototyping a muscle-driven snake-like robot to address their low energy efficiency limitation. An electro-pneumatic control hardware was developed to control the robot's locomotion and a control algorithm for generating the lateral undulation gait. The energy efficiency of a single muscle (i.e., PAM), a single 2-link module of the robot, and a 6-link snake robot were also studied. Moreover, the power consumption was derived for the snake locomotion to determine the cost of transportation as the index for measuring the performance of the robot. Finally, the performance of the robot was analyzed and compared to similar models. Our analysis showed that the power consumption efficiency for our robot is 0.21, which is comparable to the reported range of 0.016-0.32 from other robots. In addition, the cost of transportation for our robot was determined to be 0.19 compared to the range of 0.01-0.75 reported for the other mobile robots. Finally, the range of motion for the joints of the robot is ±30 • , which is comparable to the reported range of motion of other snake-like robots, i.e., 25 •-45 • .
... In addition, future studies should test how generalist snakes modify their 3-D body bending to adapt to terrain properties change. Does their preference of using lateral or vertical bending depend on their habitat terrain properties, such as the spatial density of lateral and vertical push points available (Jayne and Herrmann, 2011;Schiebel et al., 2020b;Sponberg and Full, 2008) or friction (Alben, 2013;Gray, 1946;Marvi and Hu, 2012;Zhang et al., 2021)? ...
Snakes can traverse almost all types of environments by bending their elongate bodies in 3-D to interact with the terrain. Similarly, a snake robot is a promising platform to perform critical tasks in various environments. Understanding how 3-D body bending effectively interacts with the terrain for propulsion and stability can not only inform how snakes traverse natural environments, but also allow snake robots to achieve similar performance. How snakes and snake robots move on flat surfaces has been understood well. However, such ideal terrain is rare in natural environments and little was understood about how to generate propulsion and maintain stability in 3-D terrain, except for some studies on arboreal snake locomotion and on robots using geometric planning. To bridge the knowledge gap, we integrated animal experiments and robotic studies in three representative environments: a large smooth step, an uneven arena of blocks of large height variation, and large bumps. We discovered that vertical body bending induces stability challenges but can generate large propulsion. When traversing a large smooth step, a snake robot is challenged by roll instability that increases with the amplitude of vertical bending. The instability can be reduced by body compliance that statistically improves body-terrain contact. Despite this, vertical body bending can potentially allow snakes to push against terrain for propulsion, as demonstrated by corn snakes traversing an uneven arena. A snake robot can generate large propulsion like this if contact is well maintained. Contact feedback control can help accommodate perturbations such as novel terrain geometry or excessive external forces by improving contact. Our findings provide insights into how snakes and snake robots can use vertical body bending for efficient and versatile traversal of the 3-D world stably.
... Indeed, material non-linearities, instabilities, continuum mechanics, distributed actuation, and conformability to the environment all render the control problem challenging [4,5]. In search of potential solutions, roboticists have turned to nature for inspiration [6], from the investigation of elephant trunks [7,8], mammalian tongues [9,10], and inchworms [11,12,13] to plant tendrils [14,15], starfish [16,17], snakes [18,19], and octopuses [20,21,22,23,24,25,26]. The latter, in particular, have received a great deal of attention owing to their unmatched ability to orchestrate multiple soft arms for swimming, crawling, and hunting, as well as manipulating complex objects, from coconut shells to lidded jars [27,28]. ...
... 2). Elastica has been demonstrated across a range of biophysical applications from soft [40] and biohybrid [39,41,42,43] robots to artificial muscles [44] and biolocomotion [39,18,35,25]. In Elastica, our CyberOctopus consists of a head and eight arms, of which only a subset (gradually increased throughout the paper) is actively engaged. ...
Inspired by the unique neurophysiology of the octopus, we propose a hierarchical framework that simplifies the coordination of multiple soft arms by decomposing control into high-level decision making, low-level motor activation, and local reflexive behaviors via sensory feedback. When evaluated in the illustrative problem of a model octopus foraging for food, this hierarchical decomposition results in significant improvements relative to end-to-end methods. Performance is achieved through a mixed-modes approach, whereby qualitatively different tasks are addressed via complementary control schemes. Here, model-free reinforcement learning is employed for high-level decision-making, while model-based energy shaping takes care of arm-level motor execution. To render the pairing computationally tenable, a novel neural-network energy shaping (NN-ES) controller is developed, achieving accurate motions with time-to-solutions 200 times faster than previous attempts. Our hierarchical framework is then successfully deployed in increasingly challenging foraging scenarios, including an arena littered with obstacles in 3D space, demonstrating the viability of our approach.
... The serpentine mode often features lifting at the curvature peaks, termed 'sinus-lifting' [3,4] and modelled in [22,23] using planar shapes together with a weight distribution function. [38] used planar shapes together with the weight distribution function approach of [22,23] to model sidewinding and other three-dimensional motions. Using a travelling-wave weight distribution function that is phase shifted from a travelling-wave body curvature they found a wide range of turning, slithering and sidewinding motions. ...
We optimize three-dimensional snake kinematics for locomotor efficiency. We assume a general space-curve representation of the snake backbone with small-to-moderate lifting off the ground and negligible body inertia. The cost of locomotion includes work against friction and internal viscous dissipation. When restricted to planar kinematics, our population-based optimization method finds the same types of optima as a previous Newton-based method. With lifting, a few types of optimal motions prevail. We have an s-shaped body with alternating lifting of the middle and ends at small-to-moderate transverse friction. With large transverse friction, curling and sliding motions are typical at small viscous dissipation, replaced by large-amplitude bending at large viscous dissipation. With small viscous dissipation, we find local optima that resemble sidewinding motions across friction coefficient space. They are always suboptimal to alternating lifting motions, with average input power 10–100% higher.
... And active extended objects undergoing shapechanging motions have appeared as a novel class of systems with their own repertoire of rich behavior,especially in regard to their locomotion [6][7][8][9][10]. Novel behaviors such as mechanical diffraction [11][12][13] and emergent locomotion [14][15][16] have been observed in shape-changing active matter systems. Furthermore, emergent collective behaviors such as collision-mediated gait synchronization in clusters of nematode worms [17] and in undulatory robots [18], and collective transport by worm blobs [19], have also been observed in this novel class of systems. ...
We report in experiment and simulation the spontaneous formation of dynamically bound pairs of shape changing smarticle robots undergoing locally repulsive collisions. Borrowing terminology from Conway's simulated Game of Life, these physical `gliders' robustly emerge from an ensemble of individually undulating three-link two-motor smarticles and can remain bound for hundreds of undulations and travel for multiple robot dimensions. Gliders occur in two distinct binding symmetries and form over a wide range of angular flapping extent. This parameter sets the maximal concavity which influences formation probability and translation characteristics. Analysis of dynamics in simulation reveals the mechanism of effective dynamical attraction -- a result of the emergent interplay of appropriately oriented and timed repulsive interactions.
... [46], we use a population-based optimization method. Three-dimensional (nonplanar) motions [20,47,48] will not be considered here for simplicity. ...
... We select a random set of Fourier coefficients that are widely distributed in the space given by (20) as follows. ...
... (21) We need 4n + 2 parameters to describe a given trajectory up to frequency n, so adding higher frequencies enlarges the search space dimension considerably. Also, there is no longer a simple rule like (20) to ensure that the body does not self-intersect. These factors lead to a large number of invalid and suboptimal trajectories during the optimization. ...
Many previous studies of sliding locomotion have assumed that body inertia is negligible. Here we optimize the kinematics of a three-link body for efficient locomotion and include among the kinematic parameters the temporal period of locomotion, or equivalently, the body inertia. The optimal inertia is non-negligible when the coefficient of friction for sliding transverse to the body axis is small. Inertia is also significant in a few cases with relatively large coefficients of friction for transverse and backward sliding, and here the optimal motions are less sensitive to the inertia parameter. The optimal motions seem to converge as the number of frequencies used is increased from one to four. For some of the optimal motions with significant inertia we find dramatic reductions in efficiency when the inertia parameter is decreased to zero. For the motions that are optimal with zero inertia, the efficiency decreases more gradually when we raise the inertia to moderate and large values.
