Snake Robots pp 89-101 | Cite as

# Path Following Control and Analysis of Snake Robots Based on the Poincaré Map

## Abstract

In this chapter, we turn to the problem of controlling the heading and position of the snake robot, and in particular, we consider the problem of enabling the robot to track a straight path. Straight line path following capabilities are important for many future applications of snake robots since they enable a snake robot to follow a desired path given by waypoints interconnected by straight lines.

Control design for snake robots is challenging since these mechanisms are underactuated. The underactuated degrees of freedom, i.e. the heading and position of the robot, make it impossible to independently control all degrees of freedom of the robot. During path following control, there is additionally the challenge that the position and heading of the snake robot does not trace out a straight path during forward locomotion, but rather oscillates periodically about the straight line pointing in the forward direction of the robot. With these challenges in mind, it becomes clear that we need a mathematical tool which allows us to study the periodically oscillating behaviour of the system states. We find such a tool in the theory of *Poincaré maps*. The Poincaré map represents a widely used tool for analysing the existence and stability of periodic orbits of dynamical systems.

In this chapter, we first propose a path following controller for planar snake robots, and subsequently we analyse the stability of the locomotion along the path by use of a *Poincaré map*. In particular, we show that all state variables of the snake robot, except for the position along the path, trace out an *exponentially stable* periodic orbit during path following with the proposed controller. We also present simulation results that illustrate the performance of the controller. The path following controller considered in this chapter is extended later in this book, where we employ cascaded systems theory to investigate the convergence of the snake robot to the desired path based on a simplified model of the snake robot.

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