## 1 Introduction
developed a snake-like robot for coal mine rescue and proposed an algorithm to avoid obstacles using the particle swarm optimization weight coefficient of extreme learning machine (PSOELM) method [27]. [PERSON] et al. proposed a real-time obstacle avoidance algorithm based on multiobjective optimization [28]. [PERSON] et al. proposed an algorithm which runs through a curved pipe that connects straight pipes by spiral rotation [29; 30]. [PERSON] et al. analyzed the three actions of sidewinding locomotion, sinus-lifting motion, and lateral undulation of a snake-like robot [31]. [PERSON] et al. used an extended Kalman filter (EKF) to move a pipe that cannot be passed by a wheeled robot [32]. [PERSON] et al. proposed a control design method using side-thrust links, in which the amplitude varies with time [33; 34]. [PERSON] et al. analyzed three-dimensional motion, based on the Gibbs-Appell method [35; 36]. These studies have supposed either a uniform or infinite friction coefficient. However, the friction coefficient is generally nonuniform in an actual environment and is finite. Therefore, side-slipping normally occurs, even if the robot features passive wheels. To address such an issue, control systems based on a dynamic model with side-slipping have also proposed. [PERSON] et al. realized a localization on pipes by sounds during locomotion [37]. [PERSON] et al. implemented locomotion control over irregular ground, such as a flanged pipe or rubble terrain, by utilizing a dynamic model without considering side-slipping [38; 39]. Additionally, [PERSON] et al. designed a control system to fit configurations of a snake-like robot to targets, by connecting simple shapes, such as straight lines, arc shapes, and spirals [40]. In the work by the authors of [40], ladder climbing locomotion was implemented by the proposed method [40]. For another example, [PERSON] et al. proposed a shape-control system for a snake-like robot which preserves the head coordinate [18]. In the work by the authors of [18], the side-slipping was actively controlled by lifting up part of the robot's body. However, side-slipping occurs unpredictably and is uncontrollable. Thus, it can be regarded as passive for the control system. The purpose of this study is to design a head position control system which is capable of adapting to passive side-slipping. There are two key points to realize this: First, a dynamic model capable of representing passive side-slipping must be formulated. Second, the control system must be able to adapt to the side-slipping. A solution for the first key point is to switch a dynamic model of the snake-like robot, depending on occurrence of side-slipping. In this paper, the Projection Method [41; 42; 43] is utilized for this solution. The linear complementarity problem (LCP) is a general method for switching the dynamic model [44]. [PERSON] et al. [45] and [PERSON] [46], as well as our previous study, formulated a dynamic model capable of switching model depending on environmental load forces or frictional forces. However, [PERSON] noted that there is an issue in real-time processing, due to its heavy calculation load. The Projection Method provides a means to provide a further solution to this issue. The Projection Method is a system modeling method that can be used to derive a dynamic model focusing on constraints of its system, and is capable of formulating constraint forces in parallel with the dynamic model of the system. In addition, the constraint forces can be added to change the dynamics (i.e., is able to switch the dynamic model). Static frictional forces generated in the axial direction of the passive wheel can be regarded as a constraint force in the Projection Method. Therefore, by switching the added constraint forces, depending on the side-slipping, the dynamic model is switched. Based on this idea, the dynamic model of a snake-like robot without constraints, with respect to the side-slipping of the passive wheel, is derived first.