# Generalized Singularity Analysis of Snake-Like Robot
## 6 Conclusions  
This paper has elucidated the generalized singularity analysis of the snake-like robot affected by the designable parameters such as the link length and/or the position of the passive wheel. This paper has addressed the model installing the passive wheel on arbitrary position, while the previous studies had addressed the model installing the wheel on the center of each link. Figure 14: **Case 2:** The singular configuration of 4-link snake-like robot including side-slipping when setting \(\phi_{2}=1.2\) rad, \(\phi_{3}=-1.2\) rad as slipping link, \(d_{i}=0.7\) m and \(l_{i}=0.5\) m. \(\phi_{4}=1.98\) rad on this situation. **Brack:** The axial of the passive wheel. The posture is not arc shape, however, these intersect at a point. Figure 15: **Case 3:** The singular configuration of 4-link snake-like robot including side-slipping when setting \(\phi_{2}=\pi/3\) rad, \(\phi_{3}=-1.2\) rad as slipping link, \(d_{i}=0.2\) m and \(l_{i}=0.8\) m. \(\phi_{4}=-1.50\) rad on this situation. **Brack:** The axial of the passive wheel. The posture is not arc shape, however, these intersect at a point. By applying the method using redundancy of the snake-like robot, we have reduced high complexity due to different set-up corresponding to the non-unique designable parameters such as link length and/or the position of the passive wheel. First, the kinematics around a joint of two adjacent links installing the passive wheel on arbitrary position have been derived corresponding to four patterns of side-slipping. By composing the joint kinematics redundantly, the kinematics of whole snake-like robot is formulated. Second, the Jacobian matrices of both the one including side-slipping and the one with non side-slipping have been formulated by using the kinematics. Finally, the rank deficient conditions of the Jacobian matrices--i.e., the singular configuration of the snake-like robot have been analyzed. The analysis creates two contributions. The first contribution is that the singular configuration depends on the designable parameters such as link length and/or the position of the passive wheel as well as the state vector consisted of the relative angles. Theorem 1 and 3 have revealed the relation between the singular configuration of the snake-like robot and the designable parameters, while the previous studies have reported that the singular configuration depends on only the state vector. The second contribution is that the singular configuration of the snake-like robot is able to be characterized by the axials of the passive wheel of all non side-slipping link. Theorem 2 and 4 have proven that the singular configuration of the snake-like robot is identified as the following two conditions regardless of whether the side-slipings actually exist. Axials of passive wheels of all non side-slipping links intersect at a common point. Axials of passive wheels of all non side-slipping links are parallel. These analyses lead shows discussions on the system design in the previous studies was insufficient. Conversely, this paper has shown importance, necessity and possibility of the discussion with respect to the system design including the locomotion control system as well as the designable parameters in the area of the snake-like robot. The discussion of the system design has a lot of possibilities to be a new index of mechanical design. For example in the area of the snake-like robot, we will be able to design a robot which is unlikely to result in the singular configuration as well as a novel locomotion control system. For another instance, the discussion will lead to invent a novel snake-like robot which can control the position of passive wheels actively. Also for another robotic system, the discussion helps robot's mechanical design corresponding to its intended use. Especially for unstable system, the discussion contributes to maximize its stability region. [PERSON], [PERSON] and [PERSON] conceptualized the paper. [PERSON] conceived and proofed the theorems and the lemmas. [PERSON] performed the numerical simulations.