## 4 Results and Discussion
The use of control barrier function (CBF)-based algorithms represents a significant advancement over conventional approaches, offering efficient, real-time obstacle avoidance capabilities. These algorithms are particularly advantageous in dynamic environments, where traditional methods often struggle to maintain safety and performance. By integrating CBFs into control systems, the WMR achieves globally asymptotically stable tracking, ensuring reliable and safe trajectories even in the presence of external disturbances. This stability is critical for applications in unpredictable environments, such as agricultural fields, where obstacles and terrain conditions can change rapidly. Moreover, CBFs have demonstrated considerable potential in hybrid system verification, a critical aspect of ensuring that robotic systems adhere to predefined safety standards throughout their operation. Hybrid systems, which combine continuous and discrete dynamics, are inherently complex and challenging to verify. However, CBFs provide a systematic framework for enforcing safety constraints across both types of dynamics, ensuring that the system remains within safe operational bounds at all times. This capability is particularly valuable in safety-critical applications, where even minor deviations from safe behavior can lead to catastrophic outcomes. The ability of CBF-based algorithms to handle nonlinearities and disturbances makes them highly suitable for real-world applications. For instance, in the case of the WMR navigating double obstacles, the nonlinear recovery of velocity and the linear restoration of position demonstrate the system's robustness and adaptability. The rapid decline in control effort for velocity regulation, coupled with the gradual reduction for position stabilization, reflects the efficient allocation of energy to achieve optimal performance. Additionally, the variations in controller barrier gains highlight the system's ability to dynamically adjust its control strategies based on the specific requirements of the scenario. Table 1 highlights the significant performance improvements achieved by the control barrier function quadratic programming (CBF-QP) approach compared to the rapidly exploring random tree (RRT) method for wheeled mobile robots (WMRs). For WMR position, the CBF-QP approach demonstrates improvements of 11% in settling time, 34% in rise time, and 75% in steady-state error. Similarly, for WMR velocity, the improvements are even more pronounced, with 27% in settling time, 37% in rise time, and 99% in steady-state error. Furthermore, the control gain metrics show substantial enhancements, with 50% in settling time, 59% in rise time, and 93% in steady-state error when using CBF-QP over RRT. These results underscore the superior efficiency and precision of the CBF-QP approach in optimizing the performance of WMRs. Control barrier functions (CBFs) are widely recognized for their ability to ensure obstacle avoidance and stabilize dynamic systems by defining rigorous safety constraints in motion planning. CBFs operate within a quadratic programming framework, where they enforce safety by ensuring that the system remains within predefined safe limits. This approach is particularly effective in dynamic environments, where real-time adaptability and safety are paramount. By integrating CBFs into control algorithms, robotic systems can navigate complex scenarios while maintaining robust safety guarantees, even in the presence of disturbances or unexpected obstacles. Figure 10: Wheeled mobile collision avoidance at double obstacle states response. In contrast, RRTs are sampling-based motion planning algorithms that excel in exploring and identifying feasible paths in high-dimensional and complex environments. RRTs are designed to efficiently discover collision-free paths by randomly sampling the configuration space and incrementally building a tree of possible trajectories.