Abstract
We examine the problem of snake-like locomotion by studying a system consisting of a planar inextensible elastic rod with adjustable spontaneous curvature, which provides an internal actuation mechanism that mimics muscular action in a snake. Using a Cosserat model, we derive the equations of motion in two special cases: one in which the rod can only move along a prescribed curve, and one in which the rod is constrained to slide longitudinally without slipping laterally, but the path is not fixed a priori (free-path case). The second setting is inspired by undulatory locomotion of snakes on flat surfaces. The presence of constraints leads in both cases to non-standard boundary conditions that allow us to close and solve the equations of motion. The kinematics and dynamics of the system can be recovered from a one-dimensional equation, without any restrictive assumption on the followed trajectory or the actuation. We derive explicit formulae highlighting the role of spontaneous curvature in providing the driving force (and the steering, in the free-path case) needed for locomotion. We also provide analytical solutions for a special class of serpentine motions, which enable us to discuss the connection between observed trajectories, internal actuation and forces exchanged with the environment.
Highlights
Snake locomotion has fascinated natural scientists for a long time
We have studied the motion of an active rod, arising from the interaction between external constraints and internal actuation by spontaneous curvature
Using Cosserat theory, we have formulated and solved both direct and inverse locomotion problems for two cases: one in which the system is forced to move along a prescribed path, and the other in which the path is not fixed a priori and the system slides along its tangential direction while subjected to lateral forces preventing lateral slipping
Summary
Snake locomotion has fascinated natural scientists for a long time. More recently, it has become a topic of great interest as one of the key examples of soft bioinspired. Inspired by the literature on snake-like locomotion recalled above, in this paper we study a model system similar to the one used in [27] in the context of undulatory swimming, and consisting of a planar inextensible elastic rod that is able to control its spontaneous curvature This is the curvature the rod would exhibit in the absence of external forces, which can be non-zero in the presence of internal actuation (see the sketch in figure 1b). The novelty of our approach consists in solving the equations of motion in the case of a system of finite length, with no a priori assumptions either on the followed path, which can be non-periodic, or on the reactive forces imposing no lateral slipping These both emerge as part of the solution of the problem, once a history of spontaneous curvatures is assigned. Possible connections of our results with observations made in the context of biological snake locomotion are briefly summarized in §5, while the existence and uniqueness of the solution of the equations of motion for the free-path case is proved in appendix A
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