Contents I. Introduction
In this paper, we will present the preliminary results of a multidisciplinary research project supported by the French CNRS. The goal of the project is to design and control a three-dimensional (3-D) eel-like robot. As many authors working in the biomimetic robotics community have noted, eel-like robots present an interesting perspective for improving the efficiency and maneuverability of underwater vehicles [1]–[5]. The prototype we are designing will be a hyper-redundant robot made by connecting many parallel platforms. Moreover, in order to guarantee efficient propulsion, it will be covered with a continuous deformable organ, which will mimic the eel's skin. This paper essentially deals with the macroscopic modeling of the future prototype. By “macroscopic”, we mean a “high-level” model, which can be used as the basis for a preliminary design of the system and its control strategy. In particular, the macroscopic model does not take into account the detailed technology of the prototype but rather an ideal dynamic behavior that is useful to fix the guidelines of the project. The new results reported in this paper are multiple. First, and contrary to most previous research on the same topic, the investigated robot is capable of 3-D swimming. Second, it is based on a continuous model adapted to the macroscopic modeling of the future hyper-redundant prototype and to the continuous character of its skin. Some authors, using the concept of backbone curves [6]–[8], have previously studied continuous modeling of hyper-redundant manipulators. In order to apply this kind of idea to the dynamics of the 3-D framework, we adopted here the geometrically exact theory of beams in finite deformation, originally by Simo [9]–[11]. The idea consists of considering the eel robot as a beam defined by a continuous assembly of rigid cross sections and controlled through distributed laws of internal strains or torque. With this choice, just as in works dealing with articulated locomotion systems [12]–[14], the head dynamics of the continuous eel are derived on a fiber bundle. However, in our case, while the fiber is still the group SE of the head displacements, the shape space is no longer a finite dimensional manifold but rather a functional space of curves in a Lie algebra. In fact, the shape space will be parameterized by the field of the infinitesimal transformations of the cross sections along the eel's backbone. Moreover, in accordance with the works of Simo, the eel's body dynamics will be written on the space of position orientation of the beam cross sections with respect to the earth frame, i.e., a functional space of curves in a Lie group. However, contrary to the numerical approach proposed by Simo to integrate the dynamics of passive beams, the dynamics problem considered here is not solved with the standard numerical tools of nonlinear structural dynamics but using the “Newton–Euler philosophy” of rigid robotics [15]–[17]. Finally, the proposed approach turns out to be a generalization of the Newton–Euler-based algorithm of Luh and Walker [15] applied to the case of a continuous robot with a mobile base (here imitating the eel's head). The algorithm gives the motion of the eel and the control torque evolution as outputs in terms of the deformation-time law of its body as inputs. As is well known from rigid robotics, the recursive nature of the Newton–Euler approach allows us to obtain efficient and fast algorithms, which are very simple to implement. Moreover, it gives us a straightforward link to the modeling of the future poly-articulated prototype. Finally, as far as the interaction of the fluid with the eel is concerned, for control requirements, we need to model the contact in a simple manner regarding the robustness of our future closed-loop controllers. Two simple analytical models suited to our purposes exist. Both are based on the fluid mechanics' theory of the slender body [18]. The biomechanics community suggests the first one, and the second is offered by the oceanic engineering community. The first is a result of the “Large amplitude elongated body theory of fish locomotion” by Lighthill [19]. This model is based on the basic assumption of the existence of some slices of fluid transversally transported with the cross sections of the eel. Then, from kinetic conservation laws, the undulation of the eel's body generates the propulsion by reaction. Nevertheless, this model has been restricted until now to planar swimming. Hence, in order to investigate 3-D swimming, we use the second model, which is, today, devoted to the dynamics of underwater flexible cables [20]. In this second model, as in Lighthill's planar solution, fluid forces are introduced through a local analytical model written for each transverse slice of the cable. Moreover, this model takes into account not only some inertial terms (like the Lighthill model does), but also some drag (transversal) and viscous (tangential) forces.
