Kinematic Analysis of the vertebra of an eel like robot

The kinematic analysis of a spherical wrist with parallel architecture is the object of this article. This study is part of a larger French project, which aims to design and to build an eel like robot to imitate the eel swimming. To implement direct and inverse kinematics on the control law of the prototype, we need to evaluate the workspace without any collisions between the different bodies. The tilt and torsion parameters are used to represent the workspace. INTRODUCTION Parallel kinematic architectures are commonly claimed to offer several advantages over their serial counterparts, like high structural rigidity, high dynamic capacities and high accuracy [1]. Thus, they are interesting for applications where these properties are needed, such as flight simulators [2] and highspeed machines. Recently, new applications have used such mechanisms to build humanoid robots [3] or snake robots [4]. Over millions of years, fish have evolved swimming capacity far superior in many ways to what has been by nautical science and technology. They use their streamlined bodies to exploit fluid-mechanical principles. This way, they can achieve extraordinary propulsion efficiencies, acceleration and maneuverability not feasible by the best naval architects [5]. In [6], we have introduced a new architecture of spherical wrist able to reproduce the vertebra of an eel. The purpose of this article is to show the kinematic equations and to study its workspace taking into account mechanical constraints to avoid internal collisions. The Euler angles are classically used to compute the workspace of spherical wrists but they do not permit to visualize the symmetrical properties. Recently in [7], the Tilt-and-Torsion was introduced to represent the workspace of the agile eye. In this paper, we will present the kinematic equations of the spherical wrist and an algorithm to compute the workspace and the joint space. PRELIMINARIES Biomimetic robotics The aim of our project is to imitate the eel swimming and its biological systems and to conceive new technologies drawn from the lesson of their study [8]. Many researches have been made in the underwater field in America and in Japan [9]. In this context, two modes of locomotion mainly attract the attention of researchers, (i) the carangid swimming (family Carangidae as the one of a jack, a horse mackerel or a pompano [10]) based on oscillations of the body and (ii) the anguilliform swimming (of snake type, eel, lamprey, etc.) based on undulations of the body. An anguilliform swimmer propels itself forward by propagating waves of curvature backward along its body [5]. Figure 1: Change in body shape in swimming and a subdivision of its body