## 2 Modeling of the Snake-Like Robot Switching Side-Slipping Passively
### Modeling of the Five-Link Snake-Like Robot with a Rotational Elastic Actuator  
First, a dynamic model of the snake-like robot without a constraint for the passive wheels without side-slipping is derived. A schematic figure of the model is shown in Figure 1, and its physical parameters are defined in Table 1. From Figure 1, the passive wheel is installed at the center of gravity of each link. The subscript \(s\) represents the element related to the spring; \(\theta_{i}(i=1,\cdots,5)\) and \(\theta_{js}(j=2,\cdots,5)\) represent the absolute angle of each link and spring, respectively; \(\omega_{i},v_{xi},v_{yi}\) and \(\omega_{js},v_{xjs},v_{yjs}(i=1,\ \cdots,\ 5,j=2,\ \cdots,\ 5)\) represent quasi-velocities at each center of gravity. First, a model of the constraint-free system is derived. From Figure 1, the generalized coordinate \(\mathbf{x_{p}}\) of this system is defined as  
\[\mathbf{x_{p}} = \begin{bmatrix}\mathbf{x_{p1}}&\mathbf{x_{p2s}}&\mathbf{x_{p2}}&\cdots&\mathbf{x_{ p5s}}&\mathbf{x_{p5}}\end{bmatrix} ^{T}, \tag{1}\] \[\mathbf{x_{pi}}: = \begin{bmatrix}\theta_{i}&x_{i}&y_{i}\end{bmatrix} ,\qquad i=1, \,\cdots,\,5,\] \[\mathbf{x_{pjs}}: = \begin{bmatrix}\theta_{js}&x_{js}&y_{js}\end{bmatrix} ,\qquad j= 2,\,\cdots,\,5.\]  
As deriving the model of the constraint free system is equivalent to formulating motion equations, both in the rotational direction and in the translation direction of (1) (according to [PERSON]'s equation of motion), the model is derived as follows,  
\[\mathbf{M_{p}}\ddot{\mathbf{x}_{p}}=\mathbf{h_{p}}, \tag{2}\]  
where \(\mathbf{M_{p}}\) and \(\mathbf{h_{p}}\) represent the generalized mass matrix and the generalized force matrix, respectively, which are defined as follows,  
\[\mathbf{M_{p}} =\text{diag}(\mathbf{M_{p1}},\mathbf{M_{p2s}},\mathbf{M_{p2}},\,\cdots,\,\mathbf{ M_{p5s}},\mathbf{M_{p5}}),\] \[\mathbf{h_{p}} = \begin{bmatrix}\mathbf{h_{p1}}&\mathbf{h_{p2s}}&\mathbf{h_{p2}}&\cdots&\mathbf{ h_{p5s}}&\mathbf{h_{p5}}\end{bmatrix} ^{T},\] \[\mathbf{M_{pi}}: =\text{diag}(j_{i},m_{i},m_{i}),\qquad i=1,\,\cdots,\,5,\] \[\mathbf{M_{pjs}}: =\text{diag}(j_{js},m_{js},m_{js}),\qquad j=2,\,\cdots,\,5,\] \[\mathbf{h_{pi}}: = \begin{bmatrix}-\tau_{i+1}-k_{i}(\theta_{i}-\theta_{is})-C_{i}( \theta_{i}-\dot{\theta}_{is})+C_{i+1s}(\dot{\theta}_{i+1s}-\dot{\theta}_{i})&0 &0\end{bmatrix} ,\] \[\mathbf{h_{pjs}}: = \begin{bmatrix}\tau_{j}+k_{j}(\theta_{j}-\theta_{js})-C_{j}( \dot{\theta}_{js}-\dot{\theta}_{j-1})+C_{j}(\dot{\theta}_{j}-\dot{\theta}_{js} )&0&0\end{bmatrix} .\]  
\(\tau_{i}(i=1,\,\cdots,\,6)\) denotes the input torques into each joint; note that \(\tau_{1}=\tau_{6}=0\) and \(C_{1}=C_{6s}=0\), from Figure 1, at the first and fifth links, have no joint at their front and rear, respectively. Furthermore, from Figure 1b, the quasi-velocity \(\mathbf{v}\) is defined as follows,  
\[\mathbf{v} = \begin{bmatrix}\mathbf{v_{1}}&\mathbf{v_{2s}}&\mathbf{v_{2}}&\cdots&\mathbf{v_{ 5s}}&\mathbf{v_{5}}\end{bmatrix} ^{T},\] \[\mathbf{v_{i}}: = \begin{bmatrix}\omega_{i}&v_{xi}&v_{yi}\end{bmatrix} ,\qquad i=1, \,\cdots,\,5,\] \[\mathbf{v_{js}}: = \begin{bmatrix}\omega_{js}&v_{xjs}&v_{yjs}\end{bmatrix} ,\qquad j =2,\,\cdots,\,5.\]  
From Figure 1, the velocity transformation matrix \(\mathbf{A}\) relates the generalized coordinate \(\mathbf{x_{p}}\) and the quasi-velocity \(\mathbf{v}\) as follows,  
\[\mathbf{A} =\text{diag}(\mathbf{A_{1}},\mathbf{A_{2s}},\mathbf{A_{2}},\,\cdots,\,\mathbf{A_ {5s}},\mathbf{A_{5}}) \tag{3}\] \[\mathbf{A_{i}}: = \begin{bmatrix}1&0&0\\ 0&\cos\theta_{i}&-\sin\theta_{i}\\ 0&\sin\theta_{i}&\cos\theta_{i}\end{bmatrix} ,\qquad i=1,\,\cdots,\,5,\] \[\mathbf{A_{jjs}}: = \begin{bmatrix}1&0&0\\ 0&\cos\theta_{js}&-\sin\theta_{js}\\ 0&\sin\theta_{js}&\cos\theta_{js}\end{bmatrix} ,\qquad j=2,\,\cdots,\,5,\]  
By Equation (3), Equation (2) is transformed as follows:  
\[\mathbf{M}\dot{\mathbf{v}}=\mathbf{h},\]where, \(\mathbf{M}=\mathbf{A}^{T}\mathbf{M}_{p}\mathbf{A}\), \(\mathbf{h}=\mathbf{A}^{T}(\mathbf{h}_{p}-\mathbf{M}_{p}\dot{\mathbf{A}}\mathbf{v})\).