# Generalized Singularity Analysis of Snake-Like Robot
## 2 Notation
### Snake-Like Robot with Side-Slipping  
The singular configuration of the snake-like robot including side-slipping is analyzed next. It also analyzed as same as the one with non side-slipping as well. An analysis result shows that it satisfies the two conditions mentioned at beginning of this section. With respect to the snake-like robot including side-slipping, following 2 theories and 2 lemmas are obtained. **Theorem 3**. : _Except for \(\mathbf{e_{1}^{T}(B_{i}-I)=0^{T}}\), (9) is rank deficient if and only if \(\forall\phi_{i}\),_  
\[\left(\frac{d_{i}}{l_{i}}\cos{}^{t}\phi_{i}+\sum\limits_{m=-\phi_{i+1}}^{i-1} \frac{L_{m}}{l_{i}}\cos{}^{z}\phi_{m}+1\right)\frac{1}{d_{x}}\sin{}^{x}\phi_{ i}-\frac{1}{l_{i}}\sin{}^{t}\phi_{i}\left(\frac{l_{i}}{d_{x}}\cos{}^{x}\phi_{ i}+\sum\limits_{m=+\phi_{i+1}}^{i-1}\frac{L_{m}}{d_{x}}\cos{}^{x}\phi_{i}+1 \right)=0. \tag{15}\]  
Figure 4: Schematic figure of the axials of the passive wheels in \(i-1\)th and \(i+1\)th link, when \(i\)th link is focused on. Proof. : From (9), since columns after 3 rd column are clearly rank sufficient and \(\mathbf{J}_{s}\in\mathbb{R}^{\eta-1\times 2}\), rank of (9) drops and the snake-like robot is singular, when rank\(\mathbf{J}_{s}\)\(<2\). (10) is analyzed as well as (8). Thus, conditions of which rank\(\mathbf{\bar{J}}_{si}\) drops are formulated. Where,  
\[\mathbf{\bar{J}}_{si}= \begin{bmatrix}\mathbf{e}_{\mathbf{1}}^{T}(\mathbf{I}-\mathbf{B}_{i}^{-1})\\ \mathbf{e}_{\mathbf{1}}^{T}(\mathbf{B}_{i+1}^{-1}+\mathbf{I})\end{bmatrix} .\]  
\(\mathbf{B}_{i}\) is expanded with supposing \(\iota-\zeta>2\) as the most complex case. \[\mathbf{B}_{i}= \begin{bmatrix}-\frac{l_{\zeta}}{d_{i}}\cos{{}^{i}\phi_{\zeta}}-\sum _{m=\zeta+1}^{i-1}\frac{L_{m}}{d_{i}}\cos{{}^{i}\phi_{m}}&\frac{1}{d_{i}}\sin {{}^{i}\phi_{\zeta}}\\ l_{\zeta}\sin{{}^{i}\phi_{\zeta}}+\sum_{m=\zeta+1}^{i-1}L_{m}\sin{{}^{i}\phi_{ m}}&\cos{{}^{i}\phi_{\zeta}}\end{bmatrix} .\]  
Thus,  
\[\mathbf{B}_{i}^{-1}= \begin{bmatrix}\frac{\cos{{}^{i}\phi_{\zeta}}}{| \mathbf{B}_{i}|}&-\frac{\sin{{}^{i}\phi_{\zeta}}}{d_{i}|\mathbf{B}_{i}|}\\ *&*\end{bmatrix} ,\] \[\therefore\Big{|}\mathbf{B}_{i}\Big{|}=-\frac{l_{\zeta}}{d_{i}}-\sum_{m= \zeta+1}^{i-1}\frac{L_{m}}{d_{i}}\cos{{}^{\zeta}\phi_{m}}. \tag{16}\]  
\(\mathbf{\bar{J}}_{si}\) is transformed as:  
\[ \begin{bmatrix}-\frac{\cos{{}^{i}\phi_{\zeta}}}{|\mathbf{B}_{i}|}+1& \frac{\sin{{}^{i}\phi_{\zeta}}}{d_{i}|\mathbf{B}_{i}|}\\ -\frac{l_{\zeta}}{d_{\kappa}}\cos{{}^{\kappa}\phi_{l}}-\sum_{m=i+1}^{\kappa-1} \frac{L_{m}}{d_{\kappa}}\cos{{}^{\kappa}\phi_{m}}-1&\frac{1}{d_{\kappa}}\sin {{}^{\kappa}\phi_{l}}\end{bmatrix} \] \[\to \begin{bmatrix}\frac{d_{\iota}}{l_{\zeta}}\left(\cos{{}^{i} \phi_{\zeta}}-|\mathbf{B}_{i}|\right)&\frac{d_{\iota}\sin{{}^{i}\phi_{\zeta}}}{l_{ \zeta}d_{i}}\\ -\frac{l_{\zeta}}{d_{\kappa}}\cos{{}^{\kappa}\phi_{l}}-\sum_{m=i+1}^{\kappa-1} \frac{L_{m}}{d_{\kappa}}\cos{{}^{\kappa}\phi_{m}}-1&\frac{1}{d_{\kappa}}\sin {{}^{\kappa}\phi_{l}}\end{bmatrix} \] \[= \begin{bmatrix}\frac{d_{\iota}}{l_{\zeta}}\cos{{}^{i}\phi_{\zeta} }+\sum_{m=\zeta+1}^{i-1}\frac{L_{m}}{l_{\zeta}}\cos{{}^{\zeta}\phi_{m}}+1&- \frac{1}{l_{\iota}}\sin{{}^{i}\phi_{\zeta}}\\ -\frac{l_{\iota}}{d_{\kappa}}\cos{{}^{\kappa}\phi_{l}}-\sum_{m=i+1}^{\kappa-1} \frac{L_{m}}{d_{\kappa}}\cos{{}^{\kappa}\phi_{m}}-1&\frac{1}{d_{\kappa}}\sin{{ }^{\kappa}\phi_{l}}\end{bmatrix} .\]  
Therefore, except for \(\mathbf{e}_{\mathbf{1}}^{T}(\mathbf{B}_{i}-\mathbf{I})=\mathbf{0}^{T}\), (10) is rank deficient if and only if \(\forall\phi_{i}\), (15). Equation (15) also indicates as well as (11) that the singular configuration of the snake-like robot depends on the designable parameters as well as the state vector. Lemma 2.: \(\mathbf{B}_{i}\) _is singular if and only if the joint of next non side-lipping link is placed on a axial of the passive wheel of current non side-slipping link (see Figure 5)._  
Proof.