A simplified model of planar snake robot locomotion Pål Liljebäck, Kristin Y. Pettersen, Øyvind Stavdahl, and Jan Tommy Gravdahl Abstract— This paper presents a model of the kinematics and dynamics of a planar, wheelless snake robot aimed at control design and stability analysis purposes. The proposed model is significantly less complex than existing models of planar snake robot locomotion. The paper presents an analysis of an existing complex snake robot model which reveals a set of essential properties that characterize the overall motion of a planar snake robot. The proposed model is developed to capture only these essential properties of snake locomotion, thereby significantly reducing the complexity compared to the original model used in the analysis. The paper presents simulation results that indicate that the qualitative behaviour of the proposed model and the original complex model are similar, and that a quantitative similarity is achieved with a proper choice of numerical values of the friction coefficients in the two models. I. I NTRODUCTION Inspired by biological snakes, snake robots carry the potential of meeting the growing need for robotic mobility in challenging environments. Snake robots consist of serially connected modules capable of bending in one or more planes. The many degrees of freedom of snake robots make them difficult to control, but provide traversability in irregular environments that surpasses the mobility of the more conven- tional wheeled, tracked and legged forms of robotic mobility. Research on snake robots has been conducted for several decades and several models have been proposed to facilitate a better understanding of snake locomotion. Gray [1] con- ducted empirical and analytical studies of snake locomotion already in the 1940s. Hirose [2] studied biological snakes and developed mathematical relationships characterizing their motion, such as the serpenoid curve. Several models of wheelless snake robots influenced by ground friction have been developed [3]–[11]. All these models are, however, rather complex and thereby challenging to investigate an- alytically. An interesting exception is the work by Nilsson [12], which proposes and analyses a simplified model of the forward velocity of a planar snake robot based on energy arguments. In the authors’ opinion, our understanding of snake locomotion so far is for the most part based on empirical studies of biological snakes and simulation-based synthesis of relationships between parameters of the snake. This is due to the complexity of existing models of snake locomotion. Affiliation of Pål Liljebäck is shared between the Department of Engineering Cybernetics at the Norwegian University of Science and Technology, NO-7491 Trondheim, Norway, and SINTEF ICT, Dept. of Applied Cybernetics, N-7465 Trondheim, Norway. E-mail: Pal.Liljeback@sintef.no K. Y. Pettersen, Øyvind Stavdahl, and Jan Tommy Gravdahl are with the Department of Engineering Cybernetics at the Norwegian University of Science and Technology, NO-7491 Trondheim, Norway. E-mail: {Kristin.Y.Pettersen, Oyvind.Stavdahl, Tommy.Gravdahl}@itk.ntnu.no Fig. 1. Kinematic parameters for the snake robot. This paper has two contributions. The first contribution is an analysis of an existing complex model of a planar snake robot that identifies a set of essential properties of snake locomotion. This analysis forms the basis of the second contribution, which is a simplified model of planar snake locomotion aimed at simplifying analytical investigations of the equations of motion. The proposed model is also devel- oped to facilitate synthesis of new control strategies for snake robots. The basic idea behind the modelling approach is to capture only the essential part of the snake robot dynamics, i.e. the features that determine the overall behaviour of the snake. The paper is organized as follows. Section II presents an existing complex model of a snake robot. Section III analyses the complex model in order to identify fundamental proper- ties of snake locomotion. Section IV presents the simplified model of a snake robot. Section V presents a controller for the robot. Section VI presents simulation results. Finally, Section VII presents concluding remarks. II. A COMPLEX MODEL OF A PLANAR SNAKE ROBOT This section summarizes an existing complex model of a planar snake robot previously presented in [11]. The model will be analysed in Section III in order to identify some essential properties of snake locomotion. This analysis will be used as a basis for the development of a simplified model of a planar snake robot in Section IV. A. Kinematics of the snake robot We consider a planar snake robot consisting of  links of length  interconnected by  −1 active joints. The kinematics of the robot is defined in terms of the symbols illustrated in Fig. 1. All  links have the same mass  and moment of inertia  . The total mass of the robot is therefore . The mass of each link is uniformly distributed so that the link CM (center of mass) is located at its center point. The snake robot moves in the horizontal plane and has a total of  +2 degrees of freedom. The CM (center of mass)