# A Snake-Like Robot with Envelope Wheels and Obstacle-Aided Gaits
## 2 Methods
### Control System
#### 2.3.2 Gaits Design  
Obstacle-Aided Concertina Gait  
In our previous study [32], we proposed obstacle-aided rectilinear gait modeled on the base of a 2D snake-like robot, which enables a snake-like robot to advance in a narrow path by locking its tail or head by yawing joints and then pull its body forward by telescoping joints. However, the efficiency will decrease when the telescoping joints move slowly or the ground is very smooth. Therefore, the  
Figure 11: A complete snake-like robot. Figure 12: The control command system. Figure 10: (a) Assembled module; (b) assembled module without casing. obstacle-aided concertina gait as shown in Figure 13 is proposed to improve the efficiency. This gait is modeled on the base of the snake-like robot designed in this paper, which uses yawing and rolling joints. Figure 13 can be treated as a side view of a snake-like robot in a narrow path. Modules in this figure are defined first to fourth from left to right. In the initial state, the joints are set in zero states and the yawing joints are parallel to the ground. The whole robot acts like a line. Firstly, the yawing joint between first and second modules rotates to make these two tail modules lift as shown in the 1 st step, then this joint rotates to zero while the following joint rotates as shown in the 2 nd step. These two steps make the robot advance for a distance. Next, the rolling and yawing joints between first and second modules rotate little ranges, making the first module close to the wall so that this module can be locked in the position as the 3 rd step. Finally, the yawing joint between second and third modules puts down as the 4 th step. After that, the first module unlocks by rotating the rolling and yawing joints between first and second modules as the 5 th step. Then a cycle of obstacle-aided concertina gait is completed. Obstacle-Aided Gaits through Narrow Corner  
The above gait enables the snake-like robot to advance in a narrow path. However, it is difficult to continue when it encounters a corner in a narrow passage. Especially when the corner width is lower than length of one module as shown in Figure 14.  
This paper proposes a method through the corner based on the above design. The mechanical analysis of the robot is shown in Figure 15.  
Figure 14: Narrow corner. Figure 13: Obstacle-aided concertina gait model. When a module is going to cross the corner, the modules should be lift as shown in Figure 15. Assuming the modules are first to third from left, there are 4 joints (rolling and yawing joints of first and third modules) rotating during a cycle, their torques are \(T_{1 Roll}\), \(T_{1 Yaw}\), \(T_{2 Roll}\), and \(T_{2 Yaw}\), respectively. The equivalent forces are \(F_{1}^{\prime}\) and \(F_{2}^{\prime}\) perpendicular to the outer wall, which are offset by support, and \(F_{1}\) and \(F_{2}\) parallel to the outer wall. \(F_{1}^{\prime}\) and \(F_{2}^{\prime}\) are not drawn in Figure 15. \(F_{1}\) and \(F_{2}\) have the decomposition of \(F_{1x}\), \(F_{1y}\), \(F_{2x}\), and \(F_{2y}\) as the figure. To simplify the analysis, it is assumed that the friction coefficients between the robot and the wall and the ground are stable. Set the friction coefficient of the wall and ground are \(\mu_{w}\) and \(\mu_{g}\), the mass of the robot is \(m\). The following conditions must be met to make the robot forward  
\[F_{1x}+F_{2x}>\mu_{w}\Big{(}F_{1}^{\prime}+F_{2}^{\prime}\Big{)}+\mu_{g} \Big{(}mg+F_{2y}-F_{1y}\Big{)} \tag{7}\]  
Figure 16 shows the process of the obstacle-aided gait through narrow corner based on a snake-like robot with 5 modules. Phases a-d is the process of the first module passing through the corner, which can be called a cycle. Each subsequent module's passing through the corner is based on this process. The first cycle will be analyzed in detail below. Firstly, the snake robot acts like a straight line while the yawing joint axis is parallel to the ground, as shown by Phase (**a**).