## 4 Results and Discussion  
In agricultural environments, collision avoidance for wheeled mobile robots (WMRs) relies on situational awareness and visual cues to navigate through complex and dynamic conditions. As illustrated in Figure 7, the WMRs successfully avoid collisions by dynamically maneuvering around obstacles, demonstrating the effectiveness of the proposed control algorithms. The graphical representation of the agricultural area provides a clear visualization of the system's operation. Black circles denote potential barriers or obstacles, while red lines represent the trajectories generated by the control algorithms guiding the WMRs. To enhance clarity, both the vertical and horizontal axes are labeled in meters, providing a precise spatial context for the WMRs' movements. Initially, the WMR follows a straight path toward its destination. However, upon detecting an obstacle, it dynamically adjusts its trajectory, opting for an efficient shortcut to avoid collisions while maintaining its course. The obstacles are strategically positioned across the first and second quadrants, while the WMRs operate primarily in the third quadrant, as indicated by the negative values on the axes. This spatial arrangement  
Figure 7: Wheeled mobile robotic collision avoidance at single obstacle. highlights the system's ability to navigate complex environments, even when obstacles are distributed across multiple regions. The ability of the WMRs to dynamically adapt their paths in real time underscores the robustness of the proposed control algorithms. By leveraging situational awareness and visual cues, the system ensures efficient and collision-free navigation, making it highly suitable for agricultural applications where dynamic obstacles and confined spaces are common. Wheeled mobile robots (WMRs) are designed to navigate agricultural environments without halting upon encountering obstacles or impacts. Instead, they dynamically adjust their trajectories to continue moving toward their intended positions. This capability requires WMRs to maintain appropriate stability and possess the ability to traverse obstacles effectively, as depicted in Figure 8. The position and velocity of the WMRs exhibit nonlinear characteristics during navigation, particularly when avoiding obstacles or recovering from disturbances, until they ultimately reach their destination. When a WMR experiences an impact, it may undergo temporary instability. However, the system's states are quickly stabilized through the instantaneous activation of control barrier mechanisms. The control algorithms expend a specific amount of energy to restore the system to its initial conditions. A positive control effort is applied to drive the states back to their desired values, while a negative control effort dissipates excess energy, ensuring the system remains stable and efficient. Initially, the control barriers are reduced, occasionally reverting to their initial conditions, before gradually increasing until equilibrium is achieved. Over time, both the control barriers and the control effort decline, but at different rates: the control barriers decrease  
Figure 8: Wheeled mobile collision avoidance at single obstacle states response. moderately, while the control effort diminishes rapidly. This differential reduction ensures that the system maintains stability and safety while minimizing energy consumption. Control barrier functions (CBFs) are essential mathematical tools in control theory, designed to enforce safety requirements during system operation. Analogous to how Lyapunov functions ensure stability, CBFs provide a systematic framework to guarantee that a system operates within predefined safe bounds. By integrating CBFs into control algorithms, robotic systems achieve enhanced safety and autonomy, particularly in dynamic and unpredictable environments such as agricultural fields.