# Generalized Singularity Analysis of Snake-Like Robot
## 2 Notation
### Kinematics around a Joint Adjacent Two Links with Side-Slipping  
From (3) and its derivation process, the kinematics around a joint including side-slipping links are derived as well. In this paper, the passive wheel represents that the link is side-slipping or non side-slipping in order to distinguish clearly (Notation 1). Combinations of side-slipping of two adjacent link regarding around a joint are shown in Figure 3.  
The kinematics of Figure 3 are formulated as follows:  
1. Both adjacent links are non side-slipping (Figure 3a). \[\mathbf{v}_{i} =\mathbf{A}^{\prime}_{\mathbf{i}}\mathbf{v}_{i-\mathbf{1}},\] (4) \[\mathbf{A}^{\prime}_{\mathbf{i}} =\begin{bmatrix}-\frac{l_{i-1}}{d_{i}}\cos\phi_{i}&\frac{1}{d_{i}} \sin\phi_{i}&0\\ l_{i-1}\sin\phi_{i}&\cos\phi_{i}&0\\ 0&0&0\end{bmatrix}.\]
2. Only hind link is side-slipping (Figure 3b). \[\mathbf{v}_{i} =\mathbf{A}^{\prime\prime}_{\mathbf{hi}}\mathbf{v}_{i-\mathbf{1}}+\mathbf{\Delta}_{i} \dot{\phi}_{i},\] (5) \[\mathbf{A}^{\prime\prime}_{\mathbf{hi}} =\mathbf{A}^{\prime}_{\mathbf{hi}}\mathbf{I}^{2\times 3}\] \[\mathbf{A}^{\prime}_{\mathbf{hi}} =\begin{bmatrix}0&0\\ l_{i-1}\sin\phi_{i}&\cos\phi_{i}\\ l_{i-1}\cos\phi_{i}&-\sin\phi_{i}\end{bmatrix}+\begin{bmatrix}1&0\\ 0&0\\ d_{i}&0\end{bmatrix}.\]
3. Both adjacent links are side-slipping (Figure 3c). \[\mathbf{v}_{i} =\mathbf{A}^{\prime}_{\mathbf{si}}\mathbf{v}_{i-\mathbf{1}}+\mathbf{\Delta}_{i}\dot{ \phi}_{i},\] (6) \[\mathbf{A}^{\prime}_{\mathbf{si}} =\begin{bmatrix}0&0&0\\ l_{i-1}\sin\phi_{i}&\cos\phi_{i}&\sin\phi_{i}\\ l_{i-1}\cos\phi_{i}&-\sin\phi_{i}&\cos\phi_{i}\end{bmatrix}+\begin{bmatrix}1& 0&0\\ 0&0&0\\ d_{i}&0&0\end{bmatrix}.\]
4. Only front link is side-slipping (Figure 3d). \[\mathbf{v}_{i} =\mathbf{A}^{\prime\prime}_{\mathbf{fi}}\mathbf{v}_{i-\mathbf{1}},\] (7) \[\mathbf{A}^{\prime\prime}_{\mathbf{fi}} =\begin{bmatrix}-\frac{l_{i-1}}{d_{i}}\cos\phi_{i}&\frac{1}{d_{i}} \sin\phi_{i}&-\frac{1}{d_{1}}\cos\phi_{i}\\ l_{i-1}\sin\phi_{i}&\cos\phi_{i}&\sin\phi_{i}\\ 0&0&0\end{bmatrix}=\begin{bmatrix}\mathbf{A}^{\prime}_{\mathbf{fi}}\\ \mathbf{0}^{T}\end{bmatrix}.\]  
Figure 3: Combinations of side-slipping of \(i\)th link and \(i-1\)th link. (**a**): Both adjacent links are non side-slipping. (**b**): Only hind link is side-slipping. (**c**): Both adjacent links are side-slipping. (**d**): Only front link is side-slipping. A link with the passive wheel represents a non side-lipping link.  
Submatrices are defined as follows:  
\[\begin{aligned} \mathbf{A}_{i}&:=\begin{bmatrix}-\frac{l_{i-1}}{d_{i}} \cos\phi_{i}&\frac{1}{d_{i}}\sin\phi_{i}\\ l_{i-1}\sin\phi_{i}&\cos\phi_{i}\end{bmatrix},&\delta_{i}&:=\begin{bmatrix}0 \\ d_{i}\end{bmatrix},\\ \mathbf{A}_{hi}&:=\begin{bmatrix}l_{i-1}\sin\phi_{i}&\cos\phi_{i}\\ l_{i-1}\cos\phi_{i}&-\sin\phi_{i}\end{bmatrix},&\mathbf{a}_{hi}&:=\begin{bmatrix} \sin\phi_{i}\\ \cos\phi_{i}\end{bmatrix},\\ \mathbf{A}_{si}&:=\begin{bmatrix}\cos\phi_{i}&\sin\phi_{i}\\ -\sin\phi_{i}&\cos\phi_{i}\end{bmatrix},&\mathbf{a}_{si}&:=\begin{bmatrix}l_{i-1} \sin\phi_{i}\\ l_{i-1}\cos\phi_{i}\end{bmatrix},\\ \mathbf{A}_{fi}&:=\begin{bmatrix}\frac{1}{d_{i}}\sin\phi_{i}&-\frac{1}{d_{i}}\cos \phi_{i}\\ \cos\phi_{i}&\sin\phi_{i}\end{bmatrix},&\mathbf{a}_{fi}&:=\begin{bmatrix}-\frac{l_ {i-1}}{d_{i}}\cos\phi_{i}\\ l_{i-1}\sin\phi_{i}\end{bmatrix}.\end{aligned}\]