# Design and Implementation of a Lizard-Inspired Robot
## 2 Kinematic Analysis
### Modeling  
Figure 3 shows a schematic figure of the robot, and Table 1 defines the physical parameters. Our main idea was to set the robot's own coordinate, called the \"robot coordinate,\" on the robot. The \(y\) axis of the robot coordinate (say the \(y_{r}\) axis) is set on a line through both a middle point of \(P_{1}\) and \(P_{2}\) (say \(O_{r}\)) and a middle point of \(P_{3}\) and \(P_{4}\) (say \(O_{r}^{\prime}\)) as positive to the \(O_{r}\) direction. The \(x\) axis of the robot coordinate (say \(x_{r}\)) is set on a line perpendicular to the \(y_{r}\) axis and through \(O_{r}\) as positive to the \(P_{2}\) direction. Let \(\theta_{1}(t)\), \(\theta_{2}(t)\), \(\theta_{3}(t)\), and \(\theta_{4}(t)\) be relative angles between each link. \(\theta_{b}(t)\) is an angle from \(\mathbf{p_{1}4}\) to the \(x_{r}\) axis, where \(\mathbf{p_{n}m}\) denotes a vector from \(\mathbf{P_{n}}\) to \(\mathbf{P_{m}}\). \(\theta(t)\) is defined as the input angle. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameters** & **Notation** & **Value** \\ \hline Length of \(\mathbf{P_{1}}\) to \(\mathbf{P_{2}}\) and \(\mathbf{P_{3}}\) to \(\mathbf{P_{4}}\) & \(l\) [m] & 0.1 \\ Length of \(\mathbf{P_{1}}\) to \(\mathbf{P_{4}}\) and \(\mathbf{P_{2}}\) to \(\mathbf{P_{3}}\) & \(d\) [m] & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters. In Figure 3, the robot switches the support legs appropriately while walking. The support legs are link-\(\overline{P_{1}P_{4}}\) and link-\(\overline{P_{2}P_{3}}\). In this paper, the model with link-\(\overline{P_{1}P_{4}}\) as the support legs was called \"Model-1,\" and \"Model-2\" is that with link-\(\overline{P_{2}P_{3}}\). For each model, the coordinates of each pin joint were formulated as the kinematics of the robot. The kinematics of Model-1 were derived. In this model, the coordinates of both the pin joint \(P_{1}\) and \(P_{4}\) are known because both joints are fixed on the ground as the support legs. Hence, \(\mathbf{p_{41}}=\overline{P_{4}P_{1}}\) is also known. Thus, the coordinates of both \(P_{2}\) and \(P_{3}\) were formulated here. From Figure 3:  
\[\theta_{b}(t) =\cos^{-1}\biggl{(}\frac{l}{d}\cos\theta(t)\biggr{)},\] \[\theta_{1}(t) =\theta(t)+\theta_{b}(t).\]  
By the cosine theorem:  
\[s_{24}=l^{2}+d^{2}-2 ld\cos\theta_{1}(t),\]  
where \(s_{ij}\) is defined as a square of the distance between the pin joints; that is, \(s_{ij}=\big{\|}\mathbf{p_{ij}}\big{\|}^{2}\). By the bilateration problem:  
\[\mathbf{p_{42}}=\mathbf{Z_{412}}\mathbf{p_{41}},\]  
where \(\mathbf{Z_{ijk}}\) is the bilateration matrix, and:  
\[\mathbf{Z_{ijk}} =\frac{1}{2s_{ij}} \begin{bmatrix}s_{ij}+s_{ik}-s_{jk}&-4A_{ijk}\\ 4A_{ijk}&s_{ij}+s_{ik}-s_{jk}\end{bmatrix} ,\] \[A_{ijk} =\pm\frac{1}{4}\sqrt{(s_{ij}+s_{ik}+s_{jk})^{2}-2(s_{ij}^{2}+s_{ ik}^{2}+s_{jk}^{2})}.\]  
Figure 3: A schematic figure of the proposed lizard-inspired robot. \(A_{ijk}\) is defined as positive if \(P_{k}\) is to the left of vector \(p_{ij}\), and negative otherwise. Additionally:  
\[p_{\mathbf{43}}=Z_{\mathbf{423}}p_{\mathbf{42}}.\]  
Hence, the coordinates of the pin joint \(P_{\mathbf{2}}\) and \(P_{\mathbf{3}}\) were obtained as follows:  
\[ \begin{cases}P_{\mathbf{2}}&=P_{\mathbf{4}}+p_{\mathbf{42}},\\ P_{\mathbf{3}}&=P_{\mathbf{4}}+p_{\mathbf{43}}.\end{cases}  \tag{1}\]  
The kinematics of Model-2 were also derived. In this model, the coordinates of both the pin joint \(P_{\mathbf{2}}\) and \(P_{\mathbf{3}}\) are known because both joints are fixed on the ground as the support legs. Hence, \(p_{\mathbf{23}}=\overline{P_{\mathbf{2}}P_{\mathbf{3}}}\) is also known. Thus, the coordinates of both \(P_{\mathbf{1}}\) and \(P_{\mathbf{4}}\) were formulated here. By the bilateration matrix:  
\[p_{\mathbf{24}}=Z_{\mathbf{234}}p_{\mathbf{23}}. \tag{2}\]  
Additionally:  
\[p_{\mathbf{21}}=Z_{\mathbf{241}}p_{\mathbf{24}}. \tag{3}\]  
Hence, the coordinates of the pin joint \(P_{\mathbf{1}}\) and \(P_{\mathbf{4}}\) were obtained as follows:  
\[