## 3 Modeling and Simulation
### Kinematics of Wheel  
The kinematics of the platform depend upon the arrangement of the wheels and its type selected. They are discussed in detail in [38]. The various architecture of mobile robots rely on the choice of wheel arrangement like differential drive robots, omnidirectional wheels, traction wheels, etc. The four motorized and steered standard wheels were selected, which resulted in greater maneuverability. The formulation is presented for the steering with the varying dimensions of the base plane \(\Pi_{B}\) (Figure 8b) that is dependent on the leg kinematics. Figure 10a shows the position of four wheels \((W_{1},W_{2},W_{3},W_{4})\) on the base plane. Note that the location of the wheels is subjected to vary as per the change in joint angles \(\theta_{1k}\) of the legs. It changes the dimension of the rectangle defined by the points of contact of the wheels in the base plane \(\Pi_{B}\). The position vector of the wheels in the frame attached with the base is denoted by \(\mathbf{1}_{w,k}\). The magnitude and the angle subtended by the position vector are given by:  
\[l_{w,k}=\sqrt{x_{k}^{2}+y_{k}^{2}}\ \ \ \text{and}\ \ \ \gamma_{k}=\tanh^{-1}(y_{k},x_{k}), \tag{6}\]  
where \(\tanh^{-1}\) is the inverse hyperbolic tangent function. The values of \(x_{k}\) were found from Equation (2) and \(y_{k}\) was obtained from the geometry. These point of contacts were experimentally identified in Section 4.3. In this section, the generalized kinematic modeling of the four wheels is presented. The sate vector of the robot's base frame is defined as:  
\[\mathbf{v}_{B}= \begin{bmatrix}\dot{x}_{B}&\dot{y}_{B}&\dot{a}_{B}\end{bmatrix}  ^{T}, \tag{7}\]  
where \(\dot{x}_{B},\dot{y}_{B}\) are the velocity vectors along the \(x\)- and \(y\)-directions and \(\dot{a}_{B}\) is the angular velocity vector about the \(z\)-axis as shown in Figure 10b. Now, taking the component of the velocity vectors at the origin of the base frame, the rolling wheel constraint of the \(k\)th wheel was written as:  
\[[\sin(\gamma_{k}+\beta_{k})\ \ \ -\cos(\gamma_{k}+\beta_{k})\ \ \ l_{r,k}\cos\beta_{k}] \mathbf{v}_{B}-r_{w}\dot{\phi}=0, \tag{8}\]  
Figure 9: Graphical representation of the workspace of Tarantula. where \(\dot{\varphi}\equiv[\varphi_{1}\ \varphi_{2}\ \varphi_{3}\ \varphi_{4}]^{T}\) is the vector of rate of rotation for the wheels. In addition, the no-sliding constraint for the \(k\)th wheel is written as:  
\[[\cos(\gamma_{k}+\beta_{k})\quad\sin(\gamma_{k}+\beta_{k})\quad l_{w,k}\sin \beta_{k}]\mathbf{v}_{B}=0, \tag{9}\]  
where \(l_{w,k}\) and \(\gamma_{k}\) were found as per Equation (6). Equations (8) and (9) were written for all the four wheels as:  
\[\mathbf{C}_{R}\mathbf{v}_{B}-\mathbf{W}\dot{\boldsymbol{\phi}} =\mathbf{0}, \tag{10}\] \[\mathbf{C}_{S}\mathbf{v}_{B} =\mathbf{0}. \tag{11}\]  
The matrix \(\mathbf{C}_{R}\) and \(\mathbf{C}_{S}\) are defined as  
\[\mathbf{C}_{R}\equiv \begin{bmatrix}\sin(\gamma_{1}+\beta_{1})&\cos(\gamma_{1} +\beta_{1})&l_{w,1}\cos\beta_{1}\\ \sin(\gamma_{2}+\beta_{2})&\cos(\gamma_{2}+\beta_{2})&l_{w,2}\cos\beta_{2}\\ \sin(\gamma_{3}+\beta_{3})&\cos(\gamma_{3}+\beta_{3})&l_{w,3}\cos\beta_{3}\\ \sin(\gamma_{4}+\beta_{4})&\cos(\gamma_{4}+\beta_{4})&l_{w,4}\cos\beta_{4}\\ \end{bmatrix} \ \ \text{and}\ \ \mathbf{C}_{S}\equiv \begin{bmatrix}\cos(\gamma_{1}+\beta_{1})&\sin( \gamma_{1}+\beta_{1})&l_{r,1}\sin\beta_{1}\\ \cos(\gamma_{2}+\beta_{2})&\sin(\gamma_{2}+\beta_{2})&l_{r,2}\sin\beta_{2}\\ \cos(\gamma_{3}+\beta_{3})&\sin(\gamma_{3}+\beta_{3})&l_{r,3}\sin\beta_{3}\\ \cos(\gamma_{4}+\beta_{4})&\sin(\gamma_{4}+\beta_{4})&l_{r,4}\sin\beta_{4}\\ \end{bmatrix} , \tag{12}\]  
where the values of \(\gamma_{k}\) for \(k=1,\cdots,4\) are obtained from Equation (6). The degree of maneuverability \(M\) was found using the sum of the mobility \(m\), i.e., the dimensionality of the null space of the matrix \(\mathbf{C}_{R}\) and steerability \(s\), i.e., the rank of the matrix \(\mathbf{C}_{S}\). In short, the maneuverability is found as 3 (\(M=m+s\equiv 0+3=3\)).