Friction modulation in limbless, three-dimensional gaits and heterogeneous terrains

Abstract

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 simulations, the emergence and disappearance of a range of locomotory behaviors observed in nature is systematically explained in relation to inhabited environments. Our approach also simplifies the treatment of terrain heterogeneity, whereby even solid obstacles may be seen as high friction regions, which we confirm against experiments of snakes 'diffracting' while traversing rows of posts, similar to optical waves. We further this optic analogy by illustrating snake refraction, reflection and lens focusing. We use these insights to engineer surface friction patterns and demonstrate passive snake navigation in complex topographies. Overall, our study outlines a unified view that connects active and passive 3D mechanics with heterogeneous interfacial effects to explain a broad set of biological observations, and potentially inspire engineering design.

Fig. 1. 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 body. b A corn snake (Pantherophis guttatus) slithering on a flat surface and symmetrically lifting regions of high lateral curvature on both sides of its body^, . c Schematic of the planar snake model. Note that the arc-length s goes from tail to head to retain consistency with^, . The local position x is related to the center of mass $\overline{\mathbf{x}}$ 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 body^, . 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 snakes^{, , , ,} . Net forces and torques acting on the snake over one undulation period are computed via ${\mathbf{F}}_{\mathrm{net}}={\int }_0^1{\int }_0^1\mathbf{F}\left(s,t\right)dsdt$ and ${\mathbf{T}}_{\mathrm{net}}={\int }_0^1{\int }_0^1\left(\mathbf{x}-\overline{\mathbf{x}}\right)\times \mathbf{F}\left(s,t\right)dsdt$, respectively.

Fig. 2. Locomotory behaviors available to a snake as function of body lifting A and phase offset Φ.

a Quantities used to analyze locomotion behavior. Steering rate $\dot{\theta }$ is the time-averaged angular velocity of the snake’s center of mass. Pose angle γ is the angle between the snake’s orientation and velocity direction (Methods). b Classifications of qualitatively different locomotion behaviors given $\dot{\theta }$ and γ (Supplementary Movie 1), based on experimentally observed pose angles of sidewinding snakes. Black lines are the snake’s center of mass (COM) trajectories. c Field map of steering rate $\dot{\theta }$ and pose angle γ for varying lifting wave amplitude A and phase offset Φ (for μ_t/μ_f = 2). d Phase space of locomotion behaviors available to a snake for the friction ratio μ_t/μ_f = 2. White spaces are transition regimes between different behaviors. Separatrices are zero contours for steering rate (dash line) and pose angle (dash-dot line). All simulations employ ϵ = 7 and k = 1, as in.

Fig. 3. Emergence of locomotory behaviors in context with the environment – influence of friction ratio.

a Phase space maps for different friction ratios μ_t/μ_f (full exploration in SI). Numbered labels indicate the location in terms of friction ratio, lifting amplitude, and phase offset of typical locomotion behaviors: (1) a snake sidewinding in a sandy desert or a tidal mudflat, (2) an undulating snake with no body lift based on measurements from, and (3) a wheeled robotic snake where the wheels can be viewed as inducing strong friction anisotropy. Note that for μ_t/μ_f = 1, straight sidewinding does not occur for A = 0 as the snake is unable to produce directional motion when there is no body lift. b Effective speed, steering rate, and pose angle of four different lifting strategies over a range of friction ratios. Different body lifting strategies lead to large differences in all three quantities at low friction anisotropy ratios, while there is a general convergence of behaviors to traveling forward in a straight trajectory as friction anisotropy increases. For isotropic friction, the no lifting case of A = 0 corresponding to planar slithering has ∣v_eff∣ = 0, while asymmetric lifting with A = 1 and Φ = 0.25, corresponding to sidewinding^, , exhibits high ∣v_eff∣.

Fig. 4. Consistency with 3D simulations and control via ground friction design – heterogeneity and optical analogy.

a Schematic of a 3D elastic snake model with internal muscular activation and out-of-plane body lift. b Field maps of steering rate and pose angle for μ_t/μ_f = 2. c Trajectories for (1) Φ = 0.5 and A ∈ [0, 1], (2) A = 1 and Φ ∈ [0.1, 0.6]. We note that A and Φ here are amplitude and phase offset of the lifting torque wave used to model the snake internal muscular activation (full details in SI). As such they are the dynamic counterparts of the kinematic parameters A and Φ in the theoretical model. While in both cases the same range and organization of locomotory behaviors is observed, the quantitative values of the two parameter sets match only approximately. d Complex trajectories possible by controlling lift amplitude and phase offset. e Diffraction pattern and probability density function (pdf) of diffracted angle α_p for simulations of snakes slithering through regularly spaced patches of high friction (friction model — FM, friction is modulated by scaling the local friction coefficients of the patches by a large factor p), and comparison with experimental observations of biological snakes traveling through rigid posts (Exp.) and with collision model (CM) simulations. f A snake moving on a flat surface (μ_t/μ_f = 10) patterned with frictionless strips of width w. g Snakes encountering a frictionless strip are either reflected or refracted depending on incidence angle α. h Demonstrations of passive trajectory control through friction surface patterning (additional details for all cases in SI). i Snakes interacting with heterogeneous ground features of diameter d_c for both increasing and decreasing friction modulation p. j Snakes passively meander through an heterogeneous frictional contour map. All snakes here utilize a lateral muscular activation function (torque wave) that produces planar gaits of waveform κ.