Differential Geometric Modelling and Robust Path Following Control of Snake Robots Using Sliding Mode Techniques Ehsan Rezapour, Kristin Y. Pettersen, P˚ al Liljeb¨ ack, and Jan T. Gravdahl Abstract— This paper considers straight line path following control of wheel-less planar snake robots using sliding mode techniques. We first derive the Poincar´ e representation of the equations of motion of the robot using the techniques of differential geometry. Furthermore, we use partial feedback linearization to linearize the directly actuated part of the system dynamics. Subsequently, we propose an analytical solution to the robust path following control problem in two steps. In the first step, we use sliding mode techniques to design a robust tracking controller for the joints of the robot to track a desired gait pattern. In the second step, we stabilize an appropriately defined sliding manifold for the underactuated configuration variables of the robot, thereby guaranteeing convergence of the robot to the desired straight path. The paper presents simulation results which validate the theoretical results. I. INTRODUCTION Wheels and legs have been the primary locomotion tools for biologically inspired robots on flat surfaces. However, challenging environments where the surfaces are irregular and unstructured, may significantly degrade the performance of such robots. Under these circumstances, snake robots are an interesting alternative to wheel and leg based robots due to their long and slender body. The many degrees-of-freedom (DOF) of snake robots provide adaptability properties and enable them to maintain mechanical stability even during failure of some of their actuators. Motion control of snake robots is, however, challenging due to the underactuation, which is characterized by fewer independent control inputs than DOF, complex gait patterns, and complicated force interactions with their environments. Mine detection and elimination, firefighting operations, and industrial operations in narrow environments are typical areas where the structural flexibility properties of snake robots have made them an interesting choice for applications. This paper considers path following control of snake robots. Path following involves making the outputs of the motion control system converge to and follow a desired planar path while guaranteeing forward motion along the path and boundedness of the system states. This problem is particularly relevant for snake robots since it can automate Ehsan Rezapour, Kristin Y. Pettersen, and Jan T. Gravdahl are with the Department of Engineering Cybernetics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway. emails: {ehsan.rezapour, kristin.y.pettersen, jan.tommy.gravdahl}@itk.ntnu.no. The affiliation of P˚ al Liljeb¨ ack is shared between the Department of Engineering Cybernetics and the Department of Applied Cybernetics, SINTEF ICT, NO-7465 Trondheim, Norway. email: {pal.liljebaeck@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. their applications in environments where human presence is unsafe or unwanted. However, underactuation, non-minimum phase zero dynamics, and complex motion patterns make this a challenging task where many research challenges still remain. In this paper, we show how sliding mode control techniques can be used to solve this problem. Our main motivation for using this technique for motion control of snake robots is the fact that these robots move on different surfaces with different friction properties. Accordingly, the necessity of developing control methods for snake robots which are robust w.r.t. changes in the environment of the robot is well-justified. Path following control of snake robots has been considered in several previous works. The majority of these works consider snake robots with passive wheels, which is inspired by the world’s first snake robot developed in 1972 [1], and which introduce sideslip constraints (i.e. nonholonomic ve- locity constraints) on the links of the robot. These constraints allow the control input to be specified directly in terms of the desired propulsion of the snake robot, which is employed in e.g. [2-5] for computed torque control of the position and heading of wheeled snake robots. Path following control of wheel-less (i.e. without velocity constraints) snake robots is only considered in a few previous works. In [6], path following control of swimming snake robots is achieved by moving the joints according to a predetermined gait pattern while introducing an angular offset in each joint to steer the robot to some desired path. Methods based on numerical optimal control are considered in [7] for determining optimal gaits during positional control of snake robots. In [8,9] cas- caded systems theory is employed to achieve path following control of a snake robot described by a simplified model. Sliding mode control of the joint angles of a snake robot is considered in [10]. This work does, however, not consider the underactuated DOF of the robot. The first contribution of this paper is to derive a partially feedback linearized Poincar´ e representation of the equations of motion of a snake robot without velocity constraints, which gives a detailed mathematical description of the sys- tem behaviour that can be used for analysis and model-based control design. To our best knowledge, the only previous work which derives the dynamic model of unconstrained (i.e. without velocity constraints) snake robots in a geo- metric mechanics framework is [11]. However, that work employs general affine differential geometry in contrast with the particular Poincar´ e representation in the present work. Furthermore, we add parametric modelling uncertainties due to changes in the friction coefficients to this model. We also