## 2 Architecture and Deployment of the Mobile Robotic Network Developed for Emergency Situations
For the purpose of kinematic analysis, each leg-wheel locomotion unit is approximated as a single conventional wheel, allowing the system to be modeled as a four-wheeled skid-steering mobile robot, as shown in Figure 3. The velocities associated with the locomotion unit are described, where the angular velocity of the motor is developing all four \(\omega_{i}\) angular velocities through a planetary gear system. The robot is assumed to operate on a planar surface, where its linear velocity in the local frame is expressed as \(v=(v_{x}\), \(v_{y}\), 0)T, and its angular velocity is given by \(\omega=(0\), \(\omega_{2})\)T. The generalized state vector q = (X, Y, \(\theta\))T describes the robot's pose, including the position of the center of mass in the global frame and the orientation angle \(\theta\) of the local frame relative to the global one. Consequently, the vector of generalized velocities is \(\dot{\mathrm{q}}=(\dot{\mathrm{x}},\dot{\mathrm{y}},\dot{\mathrm{\theta}})\)T. The transformation between the velocity components in the local and global coordinate systems can then be derived in a straightforward manner. \[ \begin{bmatrix}\dot{\mathrm{X}}\\ \dot{\mathrm{Y}}\\ \dot{\mathrm{\theta}}\\ \end{bmatrix} = \begin{bmatrix}cos\theta&-sin\theta&0\\ sin\theta&cos\theta&0\\ 0&0&1\\ \end{bmatrix}\begin{bmatrix}v_{x}\\ v_{y}\\ v_{z}\\ \end{bmatrix}  \tag{1}\]  
Figure 3: Mobile robot’s ICR and wheel velocities. The angular velocities of the wheels, denoted by \(\omega_{i}\), where i = 1, 2, 3, 4 correspond to the front-left, rear-left, front-right, and rear-right wheels, respectively. According to the second modeling assumption, the wheels on each side of the robot are constrained to rotate at the same angular speed, leading to the following conditions:  
\[\omega_{L}=\omega_{1}=\omega_{2},\ \ \omega_{R}=\omega_{3}=\omega_{4} \tag{2}\]  
Based on these assumptions, the forward kinematic model of the robot on a planar surface can be expressed as follows [28]:  
\[ \begin{bmatrix}v_{x}\\ v_{y}\\ \omega_{z}\end{bmatrix} =f \begin{bmatrix}\omega_{L}r\\ \omega_{R}r\end{bmatrix}  \tag{3}\]  
where \(v=(v_{x},v_{y})\) is the robot's translational velocity with respect to its local frame, \(\omega_{x}\) is its angular velocity, and r is the radius of the wheel. To fully define the kinematic model of the mobile robot, it is necessary to consider additional velocity components relative to the inertial frame. Based on the geometric configuration, it can be shown that the velocity coordinates of the four wheel-ground contact points O\({}_{1}\), O\({}_{2}\), O\({}_{3}\), and O\({}_{4}\) satisfy the following relationships [28]:  
\[v_{L}\triangleq v_{1x}=v_{2x},v_{R}\triangleq v_{3x}=v_{4x},\ \ v_{F} \triangleq v_{2y}=v_{3y},\ \ v_{B}\triangleq v_{1y}=v_{4y} \tag{4}\]  
Here, \(v_{L}\) and \(v_{R}\) represent the longitudinal velocity components of the wheels on the left and right sides of the robot, respectively, while \(v_{F}\) and \(v_{B}\) denote the lateral velocity components of the front and rear wheels. **Case 1**: From Figure 3, it becomes evident that the lateral velocity \(v_{iy}\), (i = 1..4), equals zero only in the case of pure forward motion, when \(\omega_{z}\) = 0. In all other cases, \(v_{iy}\), does not equal zero, implying that lateral slip is essential for changing the robot's orientation [28]. The angular velocities of the left and right wheel groups denoted \(\omega_{L}\) and \(\omega_{R}\), respectively, can be treated as control inputs in the kinematic model. These can be used to compute both the longitudinal linear velocity \(v_{x}\) and the angular velocity \(\omega_{z}\) of the robot using the following expressions:  
\[v_{x}=r\frac{\omega_{L}+\omega_{R}}{2},\ \omega=r\frac{-\omega_{L}+\omega_{R}}{B} \tag{5}\]  
Here, \(r\) is the effective wheel radius, and B represents the wheel track width, as shown in Figure 2.  
**Case 2**: It is important to emphasize that the equations in (5) assume the absence of longitudinal wheel slip.