## 3 Modeling and Simulation
### Kinematic Modeling  
The forward kinematics of the robot is modeled using the DH convention [26]. The Denavit and Hartenberg (DH) parameters uses four independent parameters, namely joint offset \(b_{i}\), joint angle \(\theta_{i}\), link length \(a_{i}\), and twist angle \(\alpha_{i}\) for the \(i\)th link to represent the transformation between two consecutive frames say \(i\) and \((i+1)\) in a kinematic chain. The Denavit-Hartenberg (DH) parameters convention used in this paper with the Homogeneous Transformation Matrix (HTM) for a single link are presented in Appendix A. Figure 8 shows the kinematic diagram of the robot platform and the legs. The DH parameters of a single leg for the _Tarantula_ robot are listed in Table 2.  
Figure 8b highlights the second leg and the joint axis vectors attached at each of the joints. The joint axis direction of each joint is denoted by the \(z\)-axis and \(W_{2}\) is the frame attached at the point of contact of the wheel. Assuming all the legs as symmetric, the pose of the wheel \(W_{k}\) w.r.t. the frame \(F_{k}\) on the moving platform attached to the trunk near the proximal revolting joint was calculated using the successive multiplication of the homogeneous transformation matrix (HTM) as:  
\[\mathbf{T}_{F,k}^{W,k}=\mathbf{T}_{F,k}^{1}\mathbf{T}_{1}^{2}\mathbf{T}_{2}^{3 }\mathbf{T}_{3}^{4}\mathbf{T}_{4}^{W,k}. \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & **Joint Limits** & **Remarks** \\ \# & \(\mathbf{\alpha}\) & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{\theta}\) & **Initial** & **Final** & \\ \hline
1 & 90 & \(a_{1}\) & 0 & \(\theta_{1}\) (JV) & 0 deg.\({}^{*}\) & 180 deg. & Joint near trunk \\
2 & \(-90\) & 0 & \(b_{2}\) (JV) & 0 & 470 mm & 950 mm & To adjust height \\
3 & 90 & 0 & 0 & \(\theta_{3}\) (JV) & 0 deg. & 180 deg. & \(\theta_{3}=-\theta_{1}\pm\psi\) \\
4 & 0 & 0 & \(b_{4}\) & \(\theta_{4}\) (JV) & 0 deg. & 360 deg. & For steering \\ \hline \hline \end{tabular} \({}^{*}\) deg. is degrees; \(\psi\) is the inclination angle of the ground as shown Figure 1.  
\end{table} 
Table 2: Denavit and Hartenberg (DH) parameters of the single leg of _Tarantula_ robot. Figure 7: System diagram of _Tarantula_. The subscript \(k\) is for the four legs of the quadruped, i.e., \(k=1,\cdots,4\). Here, it is assumed that the geometry of the legs are identical, and hence the DH parameters remain the same for the four legs. After substituting the values of DH parameters and post multiplying the HTM, the position of the wheels in the frame attached to the proximal revolute joint is:  
\[\mathbf{T}_{F,k}^{W,k}= \begin{bmatrix}C\theta_{12,k}C\theta_{4,k}&-C\theta_{1 2,k}S\theta_{4,k}&S\theta_{13,k}&b_{4,k}S\theta_{13,k}+a_{1,k}C\theta_{1,k}+b_ {2}S\theta_{1,k}\\ S\theta_{12,k}C\theta_{4,k}&-S\theta_{12,k}S\theta_{4,k}&0&0\\ S\theta_{4,k}&C\theta_{4,k}&0&b_{4,k}C\theta_{13,k}-a_{1,k}S\theta_{1,k}+b_ {2}C\theta_{1,k}\\ 0&0&0&1\end{bmatrix} . \tag{2}\]  
The above expression written in the fixed frame attached to the trunk of the robot by prep-multiplying it with the HTM (coordinates and orientation of the frame shown in Figure 8) is:  
\[\mathbf{T}_{T}^{W,k}=\mathbf{T}_{T}^{F,k}\mathbf{T}_{F,k}^{W,k}. \tag{3}\]  
The first three elements in the fourth column of Equation (2) give the position of the wheel, i.e., \([x_{w,k},y_{w,k},z_{w,k}]^{T}\). The \(y\)-coordinates are explicitly shown in Figure 8, and it does not vary w.r.t., the frame attached with the trunk. The two-dimensional graphical representation of the workspace of a single leg is depicted in Figure 9a and, with four legs considering the constraint in Equation (5), is shown in Figure 9b.