# A Snake-Like Robot with Envelope Wheels and Obstacle-Aided Gaits
## 2 Methods
### Modular Design of Robot
#### 2.2.1 Joint Mechanism  
A module of the snake-like robot used in this paper has three degrees of freedom, each of which is driven by a motor. The yaw motor's swing axes is orthogonal against the roll motor and the telescopic motor. Movement can be realized by changing the rotation of the roll motor. When the robot performs the plane movement, the roll motor is at the zero position, with the yaw motor's axis parallel to the ground. The telescopic motor drives the screw to rotate, thereby completing the telescopic function. The motors configuration diagram is shown in Figure 6.  
Figure 5: (a) Whole snake; (b) single joint. The casing of one module is designed to be cylindrical with the envelope passive wheels installed in the center of the module. A front view of a casing with envelope wheels is shown in Figure 7a. The size of these passive wheels is chosen properly to meet Equations (1) and (2). In addition, a passive support rotating arm is mounted on the swing axes of the yaw motor and a color of small bearings are used to support the yaw motor bracket to strengthen the joint. A complete module is shown in Figure 7b. For every module, the materials are mainly selected as polylactic acid (PLA) or UV Curable Resin. Only some components which require higher strength or smoothness are made of steel. This design can decrease the weight effectively. Some sizes and materials of per module are given in Table 1. Several joints can be articulated to form a complete snake-like robot as shown in Figure 8. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Labels** & **Part Name** & **Properties** \\ \hline
1 & Supporting mount & PLA \& SLA \\
2 & Guide bars & Chromed steel \\
3 & Envelope wheels & SLA \\
4 & Support bearings & Stainless steel \\
5 & Length of a module (mm) & 290 \\
6 & Length of telescopic joint (mm) & 30 \\
7 & Diameter with wheels (mm) & 140 \\
8 & Diameter without wheels (mm) & 110 \\
9 & Single module quality (g) & 950 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Components of the snake-like robot. Figure 6: Motors configuration diagram. Figure 7: (a) A casing with envelope wheels; (b) a complete module. In order to make the snake-like robot able to move steadily, some properties such as range, torque of joints must be considered. For this design, the joint with the highest torque requirement is the yawing joint, which is used to achieve motions such as swinging, lifting, and so on. To make the robot able to achieve 3D movements, a yawing joint is required to lift two modules in this design. The minimum lifting torque can be calculated as:  
\[\mathrm{T}=\sum_{i=0}^{n-1}(i+0.5)mgL \tag{4}\]  
where \(n\) is number of lifted modules, and takes 2, \(L\) is one module's length, \(m\) is one module's weight, \(g\) takes 9.8 N/kg. Thus T euqals 5.4 N \(\cdot\) m after calculated. As this calculation, the maximum output torque of yawing motor is selected as 6 N \(\cdot\) m and the range is \(\pm 90^{\circ}\), which can meet the requirements in this prototype. Conversely, the rolling joint requires a bigger range of motion to make each module well rotate along the axis, but doesn't need a large torque. In the prototype, the maximum output torque of this motor is selected as 1.3 N \(\cdot\) m and the range is \(\pm 140^{\circ}\). Because the telescopic joint is driven by screw, which highly improves the force of telescopic motor, this motor needs to use a 360\({}^{\circ}\) continuous rotating motor without a highly requirement of torque, which is actual 0.9 N \(\cdot\) m. The parameters of motors are shown in Table 2, where the velocity of each motor is the maximum speed of the motor selected. Assuming that the parameters of serpentine curve are same with that used in previous studies, per yawing joint will rotate on average approximately 193\({}^{\circ}\) during one cycle in total. In order to ensure every yawing joint keeps synchronic, it is necessary to keep the same control cycle of each module.