# Head-Raising of Snake Robots Based on a Predefined Spiral Curve Method
## 1 Introduction
The key points to use this method include two aspects. First, how to find a proper curve to describe the shape of robots should be considered. Second, if the curve is obtained, there should be an approach to accomplish the fitting process. Examples of using shape control method can be found in [16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. In [16], the authors presented a geometric spline method for the shape control of planar manipulators with hyper-degrees of freedom. With the fitting of the robotic  
Figure 1: The concepts of 2D and 3D head-raising. Label 1 and 2 represent 2D head-raising, and label 3 and 4 show 3D head-raising. macro configuration and the referent spline curve, real-time control could be easily accomplished. In [17], the authors presented locomotion control using a phase oscillator network, which can produce a smooth transition of the body shape of a snake robot to avoid obstacles. In [18], the researchers introduced a framework to control the shape of robots, and the framework can be divided into four steps, namely, defining the shape control point, using the shape control point to define the Bezier curves, fitting the curves and the shape of the robot, and changing the parameters of the Bezier curves to control the movement of the snake robot according to the desired gait. In [19], the authors used the shape control point method to control the position and orientation of the head of a robot. In [20], the researchers presented a planar manipulator that moves according to a variable-structure polygon function produced by neural networks. In [21], the authors introduced and used a modal approach to generate a backbone curve and thus capture a hyper-redundant manipulator's macroscopic geometric feature. The backbone curve was represented by the linear combination of a set of shape functions with the appropriate coefficients. Then, inverse kinematics reduced to the solution of these coefficients to avoid the heavy computation required by the Jacobian pseudo-inverse method. In [22], the researchers analyzed the curvature of the sidewinding locomotion of a snake robot and developed algorithms to fit the robot's shape to a continuous curve. In [23], the authors proposed the annealed chain fitting algorithm and keyframe wave extraction algorithm; the former maps the shape of the snake robot to a continuous backbone curve and obtains a set of joint angles, whereas the latter obtains a sequence of backbone curves. A similar modeling strategy can also be found in [24; 25]. In [25], the researchers considered dynamics during the shape control of hyper-redundant manipulators. The goal of this paper is to propose a new approach to raise a snake robot's head in 3D environments. This mechanism allows the robot to obtain a 360\({}^{\circ}\) field of view and a large working space. In this paper, inspired by the shape control method, a head-raising motion control method based on a predefined spiral curve is proposed. The robot is represented by several line segments, and the corresponding local coordinate system is established. Every line segment corresponds to a module of the snake robot, and the connection point between adjacent line segments corresponds to a joint involving three degrees of freedom, namely, roll, pitch, and yaw angles, which can be obtained by matrix transformation of the local coordinate system. The predefined curve is introduced to constrain the macro shape of the snake robot, and a phase-shifting algorithm is developed to drive the snake robot along the curve. During the phase-shifting process, the angle trajectories for the head-raising motion in a given time can be obtained. With the angle trajectories, the head-raising motion can be achieved. The contributions of this work are summarized by following remarks:  
1. A new shape control curve, the predefined spiral curve, is proposed and it is utilized for 3D head-raising of a snake robot;
2. A shape-fitting algorithm is developed for adhering the line segments model of the snake robot to the predefined spiral curve;
3.