Modeling of underwater snake robots E. Kelasidi, K. Y. Pettersen, J. T. Gravdahl and P. Liljeb¨ ack Abstract— Increasing efficiency by improving the locomotion methods is a key issue for underwater robots. Hence, an accurate dynamic model is important for both controller design and efficient locomotion methods. This paper presents a model of the kinematics and dynamics of a planar, underwater snake robot aimed at control design. Fluid contact forces and torques are modeled using analytical fluid dynamics. The model is derived in a closed form and can be utilized in modern model-based control schemes. The proposed model is easily implemented and simulated, regardless of the number of robot links. Simulation results with a ten link robotic system are presented. I. INTRODUCTION For centuries, engineers and scientists have gained inspi- ration from the natural world while searching for ideal solu- tions to technical problems. More recently, this process has been termed as biomimetics. Snake robots have been studied due to their ability to move in challenging environments, where other types of robots usually fail [1]. Empirical and analytical studies of snake locomotion were reported by Gray [2]. Among the first attempts to develop a snake prototype, the work of Hirose [3] is essential. The high number of DOFs of snake robots makes them difficult to control, but gives them the ability to traverse irregular environments, surpassing the mobility of conventional wheeled, tracked or legged robots [1]. Mobile robots continue to challenge researchers with new applications in a variety of environments [1]. The most recent fields of interest include the integration of robotic technology into underwater exploration, monitoring, and surveillance. Comparing amphibious snake robots to the traditional snake robots, the former ones have the advantage of adaptability to aquatic environments. The research on amphibious snake robots (also referred to as lamprey robots or eel-like robots) is, however, much less extensive, and fewer prototypes have been developed [4], [5], [6]. Increasing efficiency by improving the locomotion meth- ods is a key issue for underwater robots. Increased agility and maneuverability are connected to a general decrease in E. Kelasidi, and J. T. Gravdahl are with the Department of Engi- neering Cybernetics at NTNU, NO-7491 Trondheim, Norway. E-mail: {Eleni.Kelasidi,Tommy.Gravdahl}@itk.ntnu.no K. Y. Pettersen is with the Centre for Autonomous Marine Operations and Systems, Dept. of Engineering Cybernetics at NTNU, NO-7491 Trondheim, Norway. E-mail: Kristin.Y.Pettersen@itk.ntnu.no Affiliation of P. Liljeb¨ ack is shared between the Dept. of Engineer- ing Cybernetics at NTNU, NO-7491 Trondheim, Norway, and SINTEF ICT, Dept. of Applied Cybernetics, N-7465 Trondheim, Norway. E-mail: Pal.Liljeback@sintef.no. This work was partly supported by the Research Council of Norway through project no. 205622 and its Centres of Excellence funding scheme, project no. 223254 the size of the robot, as well as more flexibility in its internal shape. In order to improve these properties, researchers begun studying aquatic biological systems and their methods of locomotion [7], [8], [9], [10]. There exist many underwater robots, and these can be classified into autonomous underwater vehicles (AUVs), remotely-operated underwater vehicles (ROVs), and bottom- crawling-legged underwater robots. More recently, there has been growing interest in the design, modeling and control of underwater robots that propel themselves and maneuver by mimicking the movement of a fish. A number of researchers have developed analytical models for the forces generated during the motion of these devices in the water [8]. The dynamics of snake robots moving on land have been derived by utilizing various modeling techniques [1]. The friction between the snake robot and the ground significantly affects its motion. In addition to many models of snake robots that consider sideslip constraints, there have been reported cases with anisotropic ground friction properties similar to biological snakes, providing the opportunity to model lateral undulation locomotion patterns. In [1], the authors provide an overview on modeling and analysis of snake robot loco- motion emphasizing the growing trend toward locomotion in unknown and challenging environments. When it comes to swimming snake robots, only a few modeling approaches have been presented for eel-like robots [11], [12], [13]. Generally, studies of hyper-redundant mechanisms (HRMs), also known as snake robots, have focused on land based studies. An emerging field of study considers such multi-link systems suited for aquatic propulsion as well, and several prototypes of multi-link swimmers have been developed [5], [6], [11]. Two fundamental works in the field, of Taylor [9] and Lighthill [14], provide analytical models of fluid forces acting on the body during undulatory swimming. However, their analytical methods require a number of major simpli- fying assumptions. McIsaac and Ostrowski [11] presented a dynamic model of anguilliform swimming for eel-like robots and Boyer et al. [12] present the dynamic modeling of a continuous three-dimensional swimming eel-like robot. However, the majority of swimming robots modeling omit fluid moments which are supposed to have a negligible effect on the overall motion of the system [11], [15]. It should be noted that fluid moments are directly related to the power consumption of the system (see e.g. [13]) and thus, they are neglected in these modeling approaches in order to simplify the hydrodynamic effects. It is also worth noting that, in [13], [12] and [16] fluid moments are modeled, but the drag force and moment are integrated numerically at each sample time