Path Following Control for an Eel-like Robot Lionel Lapierre LIRMM Montpellier, France Email: lapierre@lirmm.fr Bruno Jouvencel LIRMM Montpellier, France Email: jouvencel@lirmm.fr Abstract— We investigate the problem of controlling the motion of an eel like-robot. Recent work has shown promising results for this type of systems, opening new issues in the field of efficient propulsion and high manoeuvrability systems. The motion planning is decoupled in the two subproblems of thrust generation and heading control. In this paper we investigate a new type of autonomous gait generation, explicitly controlling the local system curvatures. The solution is then coupled with a path following controller, using the virtual target principle. The controller design is based on Lyapunov techniques. Simulations illustrate the performances of the proposed solution. I. I NTRODUCTION Since a long time, biologically inspired solutions have motivated a large number of robotic applications. In the domain of underwater robotics, the question of efficient and low consumption underwater propulsion, in conjunction with a high manoeuvrability is an active field of research. A whale tail is an ’unreachable’ reference in terms of thrust efficiency; the lamprey is able of incredible maneuvers. The marine biologists have studied this type of locomotion, from the lamprey to the tuna fish, and have identified the body shape evolution to produce a thrust along the body axis. From a robotics point of view, this body shape evolution is a propagating wave (also called ’gait’), adjusting the signal phase according to the joint situation located on each vertebrae. From these results, a large number of interesting robotic applications have been performed in the last decades. The first question to be solved, concerning the model derivation of such a flexible system, is generally solved considering a highly coupled, non linear, hyper redundant model. The motion control problem is decoupled into three sub-problems: • the gait generation, • the control of the joint actuation in order to follow the previous reference, • the adaptation of the gait parameters according to the system situation with respect to the main path the system has to reach and follow. The gait generation is a crucial question in this problem. Indeed, a bad chosen gait will agitate the robot, without guarantying that the maximum number of the system elements (vertebrae) are involved in the movement, loosing by the way the desired efficiency, in terms of thrust and manoeuvrability. The solutions of the literature propose some actuation gaits, directly controlling the joints as a trajectory tracker, or a kinematic joint control to follow the desired body shape ([1], Fig. 1. Problem Pose [3], [2]). The loop is closed on the gait parameters, adapting them in function of the system situation with respect to the main goal (path). An interresting optimal approach is described in [4]. Neural approach has been also tested, and promising results can be found in [5] and [6]. In this paper, we propose a method to drive the system according to a reference body curvature, which is not time dependant and that explicitly takes into account the system dynamics. The control is derived using Lyapunov theory and Backstepping techniques, in order to guarantee the system to asymptotically converge to the desired shape defined in function of the curvilinear abscissa of each joints on the reference to be followed. The reference parameters are then adjusted in order for the system to reach a desired path, at a desired forward velocity. The path following algorithm is using the ’virtual target’ guidance principle. The paper is organized as follow : chapter II describes the system modeling, chapter III introduces the control design method we are proposing and chapter IV indicates some sim- ulation results to illustrate the performances of our solution. II. SYSTEM MODELING We model the eel-like robot as a planar, serial chain of N links of length d i , mass m i and inertia I i , where i =1...N . Referring to the figure 1, we define two groups of kinematic variables called configuration variables ˙ q and shape variable ˙ s, denoted as : ˙ q =  u 1 v 1 r 1 ˙ φ 2 ˙ φ 3 ... ˙ φ N-1 ˙ φ N  T ˙ s =[u 2 v 2 u 3 v 3 ...u N v N ] T (1) where u i is the forward velocity of the link i, v i is the side-slip velocity, and r i the rotational velocity, expressed in the body axis {B i }. φ i is the relative angle between the link i - 1 and i. Note that the variables φ i can be called