## 2 Kinematics Constraints and Robot Design Methodology
### Robot Design and Kinematics Constraints  
The mobile robot design comprises a chassis, a box for the delivery package, and six wheels individually controlled by each actuator, as shown in Figure 1. The front wheel \(w_{1}\) and the middle wheel \(w_{2}\) are attached to a passive rotation joint that denotes by bogie and Link 1, which has a triangle shape formed by \(l_{1}\), \(l_{2}\), and \(l_{4}\) parameters, where \(l_{1}\) is the distance between the center of the front wheel and the center of the bogie. \(l_{2}\) is the distance between the center of the middle wheel and the center of the bogie. \(l_{4}\) is the distance between the front and middle wheel. The rear wheel \(w_{3}\) is connected to the chassis forming an imaginary link between the center of the wheel and the bogie's center called Link 2 and \(l_{3}\) parameter. The center of mass of delivery package is assumed fixed above Link 1. It remains stable because the delivery box rotates and has a horizontal position with respect to the ground during the robot's motion. \(l_{5}\) is the distance between the bogie and the center of mass. \(l_{6}\) is the distance between the chassis and the LiDAR. The sensors installed are encoders in each bogie to obtain the rotation measurement of the passive joint. The LiDAR sensor on each side of the robot to measure the contact angle between the wheel-ground and detect when the wheel loses contact with the ground. IMU in the chassis to obtain the rotation and the robot's velocity. The dimensions and parameters of the robot are detailed in Table 1 and Figure 2 shows the schematic diagram of the 2D model. The kinematic constraints of the wheeled robot are analyzed based on the parameters and the links of the robot. Figure 2 shows the nine parameters of the robot, including the radius of the wheels, the center of mass, and the position of the LiDAR. Equation (1) details the restrictions based on the geometric relationships to prevent collision between the links and the wheels, as well as the LiDAR position according to its specifications defined as follows:  
1. Parameters \(l_{1}\), \(l_{2}\), and \(l_{4}\) form Link 1. The constraints are based on keeping the link as a triangle shape to improve the robot's stability when climbing and descending stairs. 2. The radius of the wheels \(w_{1}\), \(w_{2}\), and \(w_{3}\) must be smaller than the parameters \(l_{1}\), \(l_{2}\), and \(l_{3}\), respectively, to avoid collisions between the wheels and parameters. 3. The sum of \(r_{1}\) and \(r_{2}\) must be greater than \(l_{4}\) to avoid collisions between the front and middle wheel, \(w_{1}\) and \(w_{2}\), respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values(m)** \\ \hline \hline \multirow{2}{*}{
\begin{tabular}{c} chassis (L \(\times\) W \(\times\) H) \\ delivery box (L \(\times\) W \(\times\) H) \\ \(r_{1}\) \\ \(r_{2}\) \\ \(r_{3}\) \\ \(l_{1}\) \\ \(l_{2}\) \\ \(l_{3}\) \\ \(l_{4}\) \\ \(l_{5}\) \\ \(l_{6}\) \\ \end{tabular} & 0.22 \(\times\) 0.12 \(\times\) 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the wheeled robot. Units in m, length (L), width (W), and height (H). Figure 1: CAD model design of a wheeled robot. 4. The difference between \(l_{3}\) and \(r_{3}\) must be greater than the sum of \(l_{2}\) and \(r_{2}\) to avoid collisions between the middle and rear wheels, \(w_{2}\) and \(w_{3}\) respectively, when the robot climbs or descends stairs. 5. The distance between the LiDAR position to the ground must be greater than 120 mm to ensure the accuracy of the contact angle results. According to the datasheet of LDS-01 from (120-499) mm, the accuracy of the distance is \(\pm\)15 mm. \[ \begin{split}& l_{2}+l_{4}>l_{1};l_{1}+l_{4}>l_{2};l_{1}+l_{2}>l_{4}; \\ & r_{i}>l_{i};\\ & r_{1}+r_{2}<l_{4};\\ & l_{3}-r_{3}\geq l_{2}+r_{2};\\ & l_{6}+2r_{2}\geq 0.12;\end{split}